text stringlengths 4 2.78M |
|---|
---
abstract: 'We propose a data-driven learned sky model, which we use for outdoor lighting estimation from a single image. As no large-scale dataset of images and their corresponding ground truth illumination is readily available, we use complementary datasets to train our approach, combining the vast diversity of illumination conditions of SUN360 with the radiometrically calibrated and physically accurate Laval HDR sky database. Our key contribution is to provide a holistic view of both lighting modeling and estimation, solving both problems end-to-end. From a test image, our method can directly estimate an HDR environment map of the lighting without relying on analytical lighting models. We demonstrate the versatility and expressivity of our learned sky model and show that it can be used to recover plausible illumination, leading to visually pleasant virtual object insertions. To further evaluate our method, we capture a dataset of HDR 360$^\circ$ panoramas and show through extensive validation that we significantly outperform previous state-of-the-art.'
author:
- |
Yannick Hold-Geoffroy$^*$\
Adobe Research\
[holdgeof@adobe.com]{}
- |
Akshaya Athawale[^1]\
Indian Institute of Tech. Dhanbad\
[akshaya.15je001564@am.ism.ac.in]{}
- |
Jean-François Lalonde\
Université Laval\
[jflalonde@gel.ulaval.ca]{}
bibliography:
- 'refs.bib'
title: Deep Sky Modeling for Single Image Outdoor Lighting Estimation
---
Acknowledgements {#acknowledgements .unnumbered}
================
The authors wish to thank Sébastien Poitras for his help in collecting the HDR panorama dataset. This work was partially supported by the REPARTI Strategic Network and the NSERC Discovery Grant RGPIN-2014-05314. We gratefully acknowledge the support of Nvidia with the donation of the GPUs used for this research, as well as Adobe for generous gift funding.
[^1]: Parts of this work were completed while Y. Hold-Geoffroy and A. Athawale were at U. Laval.
|
---
abstract: 'Methods for inferring average causal effects have traditionally relied on two key assumptions: (i) the intervention received by one unit cannot causally influence the outcome of another; and (ii) units can be organized into non-overlapping groups such that outcomes of units in separate groups are independent. In this paper, we develop new statistical methods for causal inference based on a single realization of a network of connected units for which neither assumption (i) nor (ii) holds. The proposed approach allows both for arbitrary forms of interference, whereby the outcome of a unit may depend on interventions received by other units with whom a network path through connected units exists; and long range dependence, whereby outcomes for any two units likewise connected by a path in the network may be dependent. Under network versions of consistency and no unobserved confounding, inference is made tractable by an assumption that the network’s outcome, treatment and covariate vectors are a single realization of a certain chain graph model. This assumption allows inferences about various network causal effects via the *auto-g-computation algorithm*, a network generalization of Robins’ well-known g-computation algorithm previously described for causal inference under assumptions (i) and (ii).'
bibliography:
- 'references.bib'
---
*Original Article*
[Auto-G-Computation of Causal Effects on a Network ]{}
[Eric J. Tchetgen Tchetgen]{}$^1$[, Isabel Fulcher]{}$^2$ [and Ilya Shpitser]{}$^3$
$^1$[Wharton Statistics Department, University of Pennsylvania]{}\
$^2$[Department of Biostatistics, Harvard University]{}\
$^3$[Department of Computer Science, Johns Hopkins University]{}
**Key words:** Network, Interference, Direct Effect, Indirect Effect, Spillover effect.
1. INTRODUCTION {#introduction .unnumbered}
===============
Statistical methods for inferring average causal effects in a population of units have traditionally assumed (i) that the outcome of one unit cannot be influenced by an intervention received by another, also known as the no-interference assumption [@cox1958planning; @rubin1974estimating]; and (ii) that units can be organized into non-overlapping groups, blocks or clusters such that outcomes of units in separate groups are independent and the number of groups grows with sample size. Only fairly recently has causal inference literature formally considered settings where assumption (i) does not necessarily hold [@sobel2006randomized; @rosenbaum2007interference; @hudgens2008toward; @hong2006evaluating; @graham2008identifying; @manski2013identification; @tchetgen2012causal].
Early work on relaxing assumption (i) considered blocks of non-overlapping units, where assumptions (i) and (ii) held across blocks, but not necessarily within blocks. This setting is known as *partial interference* [@sobel2006randomized; @hong2006evaluating; @hudgens2008toward; @tchetgen2012causal; @liu2014large; @lundin2014estimation; @ferracci2014evidence].
More recent literature has sought to further relax the assumption of partial interference by allowing the pattern of interference to be somewhat arbitrary [@verbitsky2012causal; @aronow2017estimating; @liu2016inverse; @sofrygin2017semi], while still restricting a unit’s set of interfering units to be a small set defined by spacial proximity or network ties, as well as severely limiting the degree of outcome dependence in order to facilitate inference. A separate strand of work has primarily focused on detection of specific forms of spillover effects in the context of an experimental design in which the intervention assignment process is known to the analyst [@aronow2012general; @bowers2013reasoning; @athey2018exact]. In much of this work, outcome dependence across units can be left fairly arbitrary, therefore relaxing (ii), without compromising validity of randomization tests for spillover effects. Similar methods for non-experimental data, such as observational studies, are not currently available.
Another area of research which has recently received increased interest in the interference literature concerns the task of effect decomposition of the spillover effect of an intervention on an outcome known to spread over a given network into so-called contagion and infectiousness components [@vanderweele2012components]. The first quantifies the extent to which an intervention received by one person may prevent another person’s outcome from occurring because the intervention prevents the first from experiencing the outcome and thus somehow from transmitting it to another [@vanderweele2012components; @ogburn2014causal; @shpitser2017modeling]. The second quantifies the extent to which even if a person experiences the outcome, the intervention may impair his or her ability to transmit the outcome to another. A prominent example of such queries corresponds to vaccine studies for an infectious disease [@vanderweele2012components; @ogburn2014causal; @shpitser2017modeling]. In this latter strand of work, it is typically assumed that interference and outcome dependence occur only within non-overlapping groups, and that the number of independent groups is large.
We refer the reader to [@tchetgen2012causal], [@vanderweele2014interference], and [@halloran2016dependent] for extensive overviews of the fast growing literature on interference and spillover effects.
An important gap remains in the current literature: no general approach exists which can be used to facilitate the evaluation of spillover effects on a single network in settings where treatment outcome relationships are confounded, unit interference may be due not only to immediate network ties but also from indirect connections (friend of a friend, and so on) in a network, and non-trivial dependence between outcomes may exist for units connected via long range indirect relationships in a network.
The current paper aims to fill this important gap in the literature. Specifically, in this paper, the outcome experienced by a given unit could in principle be influenced by an intervention received by a unit with whom no direct network tie exists, provided there is a path of connected units linking the two. Furthermore, the approach developed in this paper respects a fundamental feature of outcomes measured on a network, by allowing for an association of outcomes for any two units connected by a path on the network. Although network causal effects are shown to in principle be nonparametrically identified by a network version of the g-formula [@robins1986new] under standard assumptions of consistency and no unmeasured confounding adapted to the network setting, statistical inference is however intractable given the single realization of data observed on the network and lack of partial interference assumption. Nonetheless, progress is made by an assumption that network data admit a representation as a graphical model corresponding to *chain graphs* [@lauritzen2002chain]. This graphical representation of network data generalizes that introduced in [@shpitser2017modeling] for the purpose of interrogating causal effects under partial interference and it is particularly fruitful in the setting of a single network as it implies, under fairly mild positivity conditions, that the outcomes observed on the network may be viewed as a single realization of a certain conditional Markov random field (MRF); and that the set of confounders likewise constitute a single realization of an MRF. By leveraging the local Markov property associated with the resulting chain graph which we encode in non-lattice versions of Besag’s auto-models [@besag1974spatial], we develop a certain Gibbs sampling algorithm which we call the *auto-g-computation algorithm* as a general approach to evaluate network effects such as direct and spillover effects. Furthermore, we describe corresponding statistical techniques to draw inference which appropriately account for interference and complex outcome dependence across the network. Auto-g-computation may be viewed as a network generalization of Robins’ well-known g-computation algorithm previously described for causal inference under no-interference and i.i.d data [@robins1986new]. We also note that while MRFs have a longstanding history as models for network data starting with [@besag1974spatial] (see also [@kolaczyk2014statistical] for a textbook treatment and summary of this literature), a general chain graph representation of network data appears not to have previously been used in the context of interference and this paper appears to be the first instance of their use in conjunction with g-computation in a formal counterfactual framework for inferring causal effects from observational network data.
[@ogburn2017causal] have recently proposed in parallel to this work, an alternative approach for evaluating causal effects on a single realization of a network, which is based on traditional causal directed acyclic graphs (DAG) and their algebraic representation as causal structural equation models. As discussed in [@lauritzen2002chain], such alternative representation as a DAG will generally be incompatible with our chain graph representation and therefore the respective contribution of these two manuscripts present little to no overlap. Specifically, similar to our setting, [@ogburn2017causal] allow for a single realization of the network which is fully observed; however, they assume (i) an underlying nonparametric structural equation model with independent error terms [@pearl2000causality] compatible with a certain DAG generated the network data. This assumption implies a large number of cross-world counterfactual independences which are largely unnecessary for identification but inherent to their model [@richardson2013single]. Furthermore, (ii) their approach precludes any dependence between outcomes not directly connected on the network nor does it allow for interference between units which are not network ties. Finally, (iii) inferences are primarily based on an assumption that outcome errors for the network are conditionally independent given baseline characteristics. Our proposed approach do not require any of assumptions (i)-(iii).
The remainder of this paper is organized as followed. In Section 2 we present notation used throughout. In Section 3 we review notions of direct and spillover effects which arise in the presence of interference. In this same section, we review sufficient conditions for identification of network causal effects by a network version of the g-formula, assuming the knowledge of the observed data distribution, or (alternatively) infinitely many realizations from this distribution. We then argue that the network g-formula cannot be empirically identified nonparametrically in more realistic settings where a *single* realization of the network is observed. To remedy this difficulty, we leverage information encoding network ties (which we assume is both available and accurate) to obtain a chain graph representation of observed variables for units of the network. This chain graph is then shown to induce conditional independences which allow versions of coding and pseudo maximum likelihood estimators due to [@besag1974spatial] to be used to make inferences about the parameters of the joint distribution of the observed data sample. These estimators are described in Section 4, for parametric auto-models of [@besag1974spatial]. The resulting parametrization is then used to make inferences about network causal effects via a specialized Gibbs sampling algorithm we have called the auto-g-computation algorithm, also described in Section 4. In Section 5, we describe results from a simulation study evaluating the performance of the proposed approach. Finally, in Section 6, we offer some concluding remarks and directions for future research.
2. NOTATION AND DEFINITIONS {#notation-and-definitions .unnumbered}
===========================
2.1 Preliminaries {#preliminaries .unnumbered}
-----------------
Suppose one has observed data on a population of $N$ interconnected units. Specifically, for each $i\in\{1,\ldots N\}$ one has observed $(A_{i},Y_{i})$, where $A_{i}$ denotes the binary treatment or intervention received by unit $i$, and $Y_{i}$ is the corresponding outcome. Let $\mathbf{A}\equiv(A_{1},\ldots,A_{N})$ denote the vector of treatments all individuals received, which takes values in the set $\{0,1\}^{N},$ and $\mathbf{A}_{-j}\equiv(A_{1},\ldots A_{N})\backslash A_{j}\equiv(A_{1},\ldots,A_{j-1},A_{j+1},\ldots A_{N})$ denote the $N-1$ subvector of $\mathbf{A}$ with the $jth$ entry deleted. In general, for any vector $\mathbf{X=}\left( X_{i},...,X_{N}\right) ,$ $\mathbf{X}_{-j}=(X_{1},...,X_{N})\backslash
X_{j}=(X_{1},...,X_{j-1},X_{j+1},...,X_{N}).$ Likewise if $X_{i}=(X_{1,i},...,X_{p,i})$ is a vector with $p$ components, $X_{\backslash
s,i}=(X_{1,i},...,X_{s-1,i},X_{s+1,i},...,X_{p,i}).$ Following [@sobel2006randomized] and [@hudgens2008toward], we refer to $\mathbf{A}$ as an intervention, treatment or allocation program, to distinguish it from the individual treatment $A_{i}.\ $Furthermore, for $n=1,2,\ldots,$ we define $\mathcal{A(}n)$ as the set of vectors of possible treatment allocations of length $n$; for instance $\mathcal{A(}2)\equiv\left\{ (0,0),(0,1),(1,0),(1,1)\right\} .$ Therefore, $\mathbf{A}$ takes one of $2^{N}$ possible values in $\mathcal{A(}N)$, while $\mathbf{A}_{-j}$ takes values in $\mathcal{A(}N-1)$ for all $j$.
As standard in causal inference, we assume the existence of counterfactual (potential outcome) data $\mathbf{Y}(\mathbf{\cdot})=\{Y_{i}(\mathbf{a}):\mathbf{a}\in\mathcal{A(}N)\mathcal{\}}$ where $\mathbf{Y}(\mathbf{a})=\{Y_{1}\left( \mathbf{a}\right) ,\ldots,Y_{N}(\mathbf{a})\}$, $Y_{i}\left( \mathbf{a}\right) $ is unit $i^{\prime}s$ response under treatment allocation $\mathbf{a}$; and that the observed outcome $Y_{i}$ for unit $i$ is equal to his counterfactual outcome $Y_{i}\left( \mathbf{A}\right) $ under the realized treatment allocation $\mathbf{A;}$ more formally, we assume the network version of the consistency assumption in causal inference: $$\mathbf{Y}\left( \mathbf{A}\right) =\mathbf{Y}\text{ a.e.} \label{NetCons}$$ Notation for the random variable $Y_{i}(\mathbf{a})$ makes explicit the possibility of the potential outcome for unit $i$ depending on treatment values of other units, that is the possibility of interference. The standard no-interference assumption [@cox1958planning; @rubin1974estimating] made in the causal inference literature, namely that for all $j$ if $\mathbf{a}$ and $\mathbf{a}^{\prime}$ are such that $a_{j}=a_{j}^{\prime}$ then $Y_{j}\left(
\mathbf{a}\right) =Y_{j}\left( \mathbf{a}^{\prime}\right) $ a.e., implies that the counterfactual outcomes for individual $j$ can be written in a simplified form as $\left\{ Y_{j}\left( a\right) :a\in\{0,1\}\right\} $. The partial interference assumption [@sobel2006randomized; @hudgens2008toward; @tchetgen2012causal], which weakens the no-interference assumption, assumes that the $N$ units can be partitioned into $K$ blocks of units, such that interference may occur within a block but not between blocks. Under partial interference, $Y_{i}\left( \mathbf{a}\right) =Y_{i}\left(
\mathbf{a}^{\prime}\right) $ a.s. only if $a_{j}=a_{j}^{\prime}$ for all $j$ in the same block as unit $i.$ The assumption of partial interference is particularly appropriate when the observed blocks are well separated by space or time such as in certain group randomized studies in the social sciences, or community-randomized vaccine trials. [@aronow2017estimating] relaxed the requirement of non-overlapping blocks, and allowed for more complex patterns of interference across the network. Obtaining identification required a priori knowledge of the interference set, that is for each unit $i$, the knowledge of the set of units $\left\{
j:Y_{i}\left( \mathbf{a}\right) \neq Y_{i}\left( \mathbf{a}^{\prime}\right)
\text{ a.s. if }a_{k}=a_{k}^{\prime}\text{ and }a_{j}\not =a_{j}^{\prime
}\text{ for all }k\neq j\right\} $. In addition, the number of units interfering with any given unit had to be negligible relative to the size of the network. See [@liu2016inverse] for closely related assumptions.
In contrast to existing approaches, our approach allows *full* rather than partial interference in settings where treatments are also not necessarily randomly assigned. The assumptions that we make can be separated into two parts: network versions of standard causal inference assumptions, given below, and independence restrictions placed on the observed data distribution which can be described by a graphical model, described in more detail later.
We assume that for each $\mathbf{a}\in\mathcal{A(}N)$ the vector of potential outcomes $\mathbf{Y}(\mathbf{a})$ is a single realization of a random field. In addition to treatment and outcome data, we suppose that one has also observed a realization of a (multivariate) random field $\mathbf{L}=\left(
L_{1},\ldots,L_{N}\right) ,$ where $L_{i}$ denotes pre-treatment covariates for unit $i$. For identification purposes, we take advantage of a network version of the conditional ignorability assumption about treatment allocation which is analogous to the standard assumption often made in causal inference settings; specifically, we assume that: $$\mathbf{A}\perp\!\!\!\perp\mathbf{Y}(\mathbf{a})|\mathbf{L}\text{ for all
}\mathbf{a}\in\mathcal{A(}N),\text{ } \label{NetIgn}$$ This assumption basically states that all relevant information used in generating the treatment allocation whether by a researcher in an experiment or by “nature” in an observational setting, is contained in $\mathbf{L.}$ Network ignorability can be enforced in an experimental design where treatment allocation is under the researcher’s control. On the other hand, the assumption cannot be ensured to hold in an observational study since treatment allocation is no longer under experimental control, in which case credibility of the assumption depends crucially on subject matter grounds. Equation $\left( \ref{NetIgn}\right) $ simplifies to the standard assumption of no unmeasured confounding in the case of no interference and i.i.d. unit data, in which case $A_{i}\perp\!\!\!\perp Y_{i}\left( a\right) |L_{i}$ for all $a\in\left\{ 0,1\right\} $. $\ $Finally, we make the following positivity assumption at the network treatment allocation level:$$f\left( \mathbf{a}|\mathbf{L}\right) >\sigma>0\text{ a.e. for all
}\mathbf{a}\in\mathcal{A(}N). \label{positivity}$$
2.2 Network causal effects {#network-causal-effects .unnumbered}
--------------------------
We will consider a variety of network causal effects that are expressed in terms of unit potential outcome expectations $\psi_{i}\left( \mathbf{a}\right) =E\left( Y_{i}\left( \mathbf{a}\right) \right) ,$ $i=1,...,N.$ Let $\psi_{i}\left( \mathbf{a}_{-i},a_{i}\right) =E\left( Y_{i}\left(
\mathbf{a}_{-i},a_{i}\right) \right) $ The following definitions are motivated by analogous definitions for fixed counterfactuals given in [@hudgens2008toward]. The first definition gives the average direct causal effect for unit $i$ upon changing the unit’s treatment status from inactive ($a=0)$ to active ($a=1)$ while setting the treatment received by other units to $\mathbf{a}_{-i}:$ $$DE_{i}\left( \mathbf{a}_{-i}\right) \equiv\psi_{i}\left( \mathbf{a}_{-i},a_{i}=1\right) -\psi_{i}\left( \mathbf{a}_{-i},a_{i}=0\right) ;$$ The second definition gives the average spillover (or “indirect") causal effect experienced by unit $i$ upon setting the unit’s treatment inactive, while changing the treatment of other units from inactive to $\mathbf{a}_{-i}:$$$IE_{i}\left( \mathbf{a}_{-i}\right) \equiv\psi_{i}\left( \mathbf{a}_{-i},a_{i}=0\right) -\psi_{i}\left( \mathbf{a}_{-i}=\mathbf{0},a_{i}=0\right) ;$$ Similar to [@hudgens2008toward] these effects can be averaged over a hypothetical allocation regime $\pi_{i}\left( \mathbf{a}_{-i};\alpha\right)
$ indexed by $\alpha$ to obtain allocatio $DE_{i}\left( \alpha\right) =\sum_{\mathbf{a}_{-i} \in\mathcal{A(}N)}\pi_{i}\left( \mathbf{a}_{-i};\alpha\right) DE_{i}\left(
\mathbf{a}_{-i}\right) $ $IE_{i}\left( \alpha\right)
=\sum_{\mathbf{a}_{-i}\in\mathcal{A(}N)}\pi_{i}\left( \mathbf{a}_{-i};\alpha\right) IE_{i}\left( \mathbf{a}_{-i}\right) ,$ respectively. One may further average over the units in the network to obtain allocation-specific network average direct and spillover effects $DE\left( \alpha\right) =$ $N^{-1}\sum_{i}DE_{i}\left( \alpha\right) $ and $IE\left( \alpha\right) =$ $N^{-1}\sum_{i}IE_{i}\left( \alpha\right) $, respectively. These quantities can further be used to obtain other related network effects such as average total and overall effects at the unit or network level analogous to [@hudgens2008toward] and [@tchetgen2012causal].
Identification of these effects follow from identification of $\psi_{i}\left(
\mathbf{a}\right) $ for each $i=1,...,N.$ In fact, under assumptions $\left(
\ref{NetCons}\right) $-$\left( \ref{positivity}\right) ,$ it is straightforward to show that $\psi_{i}\left( \mathbf{a}\right) $ is given by a network version of Robins’ g-formula: $\psi_{i}\left( \mathbf{a}\right)
=\beta_{i}\left( \mathbf{a}\right) $ where $\beta_{i}\left( \mathbf{a}\right) \equiv\sum_{\mathbf{l}}E\left( Y_{i}|\mathbf{A=a,L=l}\right)
f\left( \mathbf{l}\right) \mathbf{,}$ $f\left( \mathbf{l}\right) $ is the density of $\mathbf{l,}$ and $\sum$ may be interpreted as integral when appropriate.
Although $\psi_{i}\left( \mathbf{a}\right) $ can be expressed as the functional $\beta_{i}\left( \mathbf{a}\right) $ of the observed data law, $\beta_{i}\left( \mathbf{a}\right) $ cannot be identified nonparametrically from *a single realization* $(\mathbf{Y,A,L)}$ drawn from this law without imposing additional assumptions. In the absence of interference, it is standard to rely on the additional assumption that $(Y_{i},A_{i},L_{i}\mathbf{)}$, $i=1,...N~$are i.i.d., in which case the above g-formula reduces to the standard g-formula $\beta_{i}\left( \mathbf{a}\right)
=\beta\left( a_{i}\right) =\sum_{l}E\left( Y_{i}|A_{i}=a_{i},L_{i}=l\right) f(l\mathbf{)\,\ }$ which is nonparametrically identified [@robins1986new]. Since we consider a sample of interconnected units in a network, the i.i.d. assumption is unrealistic. Below, we consider assumptions on the observed data law that are much weaker, but still allow inferences about network effects to be made.
We first introduce a convenient representation of $E\left( Y_{i}|\mathbf{A=a,L=l}\right) $, and describe a corresponding Gibbs sampling algorithm which could in principle be used to compute the network g-formula under the unrealistic assumption that the observed data law is known. First, note that $\beta_{i}\left( \mathbf{a}\right) =\sum_{\mathbf{y,l}}y_{i}f\left( \mathbf{y|A=a,L=l}\right) f\left( \mathbf{l}\right) .$
Suppose that one has available the conditional densities (also referred to as Gibbs factors) $f\left( Y_{i}\mathbf{|Y}_{-i}=\mathbf{y}_{-i},\mathbf{a,l}\right) $ and $f\left( L_{i}\mathbf{|L}_{-i}\mathbf{=l}_{-i}\right) $, $i=1,...,N,\,$ and that it is straightforward to sample from these densities. Then, evaluation of the above formula for $\beta_{i}\left( \mathbf{a}\right)
$ can be achieved with the following Gibbs sampling algorithm.
: $$\begin{aligned}
\text{for }m & =0,\text{let }\left( \mathbf{L}^{(0)},\mathbf{Y}^{(0)}\right) \text{ denote initial values ;}\\
\text{for }m & =0,...,M\\
& \text{let }i=(m\mod N)+1;\\
& \text{draw }L_{i}^{(m+1)}\text{ from }f\left( L_{i}\mathbf{|L}_{-i}^{(m)}\right) \text{ and }Y_{i}^{(m+1)}\text{ from }f\left( Y_{i}\mathbf{|Y}_{-i}^{(m)},\mathbf{a,L}^{(m)}\right) ;\\
& \text{let }\mathbf{L}_{-i}^{(m+1)}\left. =\right. \mathbf{L}_{-i}^{(m)}\text{ and }\mathbf{Y}_{-i}^{(m+1)}\left. =\right. \mathbf{Y}_{-i}^{(m)}.\\
&\end{aligned}$$ The sequence $\left( \mathbf{L}^{(0)},\mathbf{Y}^{(0)}\right) ,\left(
\mathbf{L}^{(1)},\mathbf{Y}^{(1)}\right) ,\ldots,\left( \mathbf{L}^{(m)},\mathbf{Y}^{(m)}\right) $ forms a Markov chain, which under appropriate regularity conditions converges to the stationary distribution $f\left( \mathbf{Y|a,L}\right) \times f\left( \mathbf{L}\right) $ [@liu2008monte]. Specifically, we assume $M$ is an integer larger than the number of transitions necessary for the appropriate Markov chain to reach equilibrium from the starting state. Thus, for sufficiently large $m^{\ast}$ and $K,$$$\beta_{i}\left( \mathbf{a}\right) \approx K^{-1}\sum_{k=0}^{K}Y_{i}^{(m^{\ast}+k)}.$$ Thus, if Gibbs factors $f\left( Y_{i}\mathbf{|Y}_{-i}=\mathbf{y}_{-i},\mathbf{a,l}\right) $ and $f\left( L_{i}\mathbf{|L}_{-i}\mathbf{=l}_{-i}\right) $ are available for every $i$, all networks causal effects can be computed. This approach to evaluating the g-formula is the network analogue of Monte Carlo sampling approaches to evaluating functionals arising from the g-computation algorithm in the sequentially ignorable model, see for instance [@westreich2012parametric]. Unfortunately these factors are not identified from a single realization of the observed data law, without additional assumptions. In the following section we describe additional assumptions which will imply identification.
3. A GRAPHICAL STATISTICAL MODEL FOR NETWORK DATA {#a-graphical-statistical-model-for-network-data .unnumbered}
=================================================
To motivate our approach, we introduce a representation for network data proposed by [@shpitser2017modeling] and based on chain graphs. A chain graph (CG) [@lauritzen1996graphical] is a mixed graph containing undirected ($-$) and directed ($\to$) edges with the property that it is impossible to add orientations to undirected edges in such a way as to create a directed cycle. A chain graph without undirected edges is called a directed acyclic graph (DAG).
A statistical model associated with a CG $\mathcal{G}$ with a vertex set $\mathbf{O}$ is a set of densities that obey the following two level factorization: $$\begin{aligned}
p(\mathbf{O}) = \prod_{\mathbf{B} \in\mathcal{B}(\mathcal{G})} p(\mathbf{B}
\mid\text{pa}_{\mathcal{G}}(\mathbf{B})),\label{eqn:fact1}$$ where $\mathcal{B}(\mathcal{G})$ is the partition of vertices in $\mathcal{G}$ into *blocks*, or sets of connected components via undirected edges, and $\text{pa}_{\mathcal{G}}(\mathbf{B})$ is the set $\{ W : W \to B
\in\mathbf{B} \text{ exists in }\mathcal{G} \}$. This outer factorization resembles the Markov factorization of DAG models. Furthermore, each factor $p(\mathbf{B} \mid\text{pa}_{\mathcal{G}}(\mathbf{B}))$ obeys the following inner factorization, which is a clique factorization for a conditional Markov random field: $$\begin{aligned}
p(\mathbf{B} \mid\text{pa}_{\mathcal{G}}(\mathbf{B})) = \frac{1}{Z(\text{pa}_{\mathcal{G}}(\mathbf{B}))} \prod_{\mathbf{C} \in\mathcal{C} (\mathcal{G}^{a}_{\mathbf{B} \cup\text{pa}_{\mathcal{G}}(\mathbf{B}))});
\mathbf{C} \not \subseteq \text{pa}_{\mathcal{G}}(\mathbf{B})} \phi
_{\mathbf{C}}(\mathbf{C}),\label{eqn:fact2}$$ where $Z(\text{pa}_{\mathcal{G}}(\mathbf{B}))$ is a normalizing function which ensures a valid conditional density, $\mathcal{C}(\mathcal{G})$ is a set of maximal pairwise connected components (cliques) in an undirected graph $\mathcal{G}$, $\phi_{\mathcal{C}}(\mathbf{C})$ is a mapping from values of $\mathbf{C}$ to real numbers, and $\mathcal{G}^{a}_{\mathbf{B} \cup
\text{pa}_{\mathcal{G}}(\mathbf{B}))}$ is an undirected graph with vertices $\mathbf{B} \cup\text{pa}_{\mathcal{G}}(\mathbf{B})$ and an edge between any pair in $\text{pa}_{\mathcal{G}}(\mathbf{B})$ and any pair in $\mathbf{B}
\cup\text{pa}_{\mathcal{G}}(\mathbf{B})$ adjacent in $\mathcal{G}$.
A density $p(\mathbf{O})$ that obeys the two level factorization given by (\[eqn:fact1\]) and (\[eqn:fact2\]) with respect to a CG $\mathcal{G}$ is said to be Markov relative to $\mathcal{G}$. This factorization implies a number of Markov properties relating conditional independences in $p(\mathbf{O})$ and missing edges in $\mathcal{G}$. Conversely, these Markov properties imply the factorization under an appropriate version of the Hammersley-Clifford theorem, which does not hold for all densities, but does hold for wide classes of densities, which includes positive densities [@hammersley1971markov]. Special cases of these Markov properties are described further below. Details can be found in [@lauritzen1996graphical].
3.1 A chain graph representation of network data {#a-chain-graph-representation-of-network-data .unnumbered}
------------------------------------------------
Observed data distributions entailed by causal models of a DAG do not necessarily yield a good representation of network data. This is because DAGs impose an ordering on variables that is natural in temporally ordered longitudinal studies but not necessarily in network settings. As we now show the Markov property associated with CGs accommodates both dependences associated with causal or temporal orderings of variables, but also symmetric dependences induced by the network.
Let $\mathcal{E}$ denote the set of neighboring pairs of units in the network; that is $(i,j)\in\mathcal{E}$ only if units $i$ and $j$ are directly connected on the network. We represent data $\mathbf{O}$ drawn from a joint distribution associated with a network with neighboring pairs $\mathcal{E}$ as a CG $\mathcal{G}_{\mathcal{E}}$ in which each variable corresponds to a vertex, and directed and undirected edges of $\mathcal{G}_{\mathcal{E}}$ are defined as follows. For each pair of units $(i,j)\in\mathcal{E}$, variables $L_{i}$ and $L_{j}$ are connected by an undirected edge in $\mathcal{G}_{\mathcal{E}}$. We use an undirected edge to represent the fact that $L_{i}$ and $L_{j}$ are associated, but this association is not in general due to unobserved common causes, nor as the variables are contemporaneous can they be ordered temporally or causally [@shpitser2017modeling]. Vertices for $A_{i}$ and $A_{j},$ and $Y_{i}$ and $Y_{j}\,$ are likewise connected by an undirected edge in $\mathcal{G}_{\mathcal{E}}$ if and only if $(i,j)\in$ $\mathcal{E}$. Furthermore, for each $(i,j)\in$ $\mathcal{E}$, a directed edge connects $L_{i}$ to both $A_{i}$ and $A_{j}$ encoding the fact that covariates of a given unit may be direct causes of the unit’s treatment but also of the neighbor treatments, i.e. $L_{i}\rightarrow$ $\left\{ A_{i},A_{j}\right\} ;$ edges $L_{i}\rightarrow$ $\left\{ Y_{i},Y_{j}\right\} $ and $A_{i}\rightarrow$ $\left\{ Y_{i},Y_{j}\right\} $ should be added to the chain graph for a similar reason. As an illustration, the CG in Figure 1 corresponds to a three-unit network where $\mathcal{E=}\left\{ \left( 1,2\right) ,\left( 2,3\right) \right\} $.
We will assume the observed data distribution on $\mathbf{O}$ associated with our network causal model is Markov relative to the CG constructed from unit connections in a network via the above two level factorization [@lauritzen1996graphical]. This implies the observed data distribution obeys certain conditional independence restrictions that one might intuitively expect to hold in a network, and which serve as the basis of the proposed approach. Let $\mathcal{N}_{i}$ denote the set of neighbors of unit $i,$ i.e. $\mathcal{N}_{i}=\left\{ j:\left( i,j\right) \in\mathcal{E}\right\} $, and let $\mathcal{O}_{i}=\left\{ \mathbf{O}_{j},j\in\mathcal{N}_{i}\right\} $ denote data observed on all neighbors of unit $i.$ Given a CG $\mathcal{G}_{\mathcal{E}}$ with associated neighboring pairs $\mathcal{E}$, the following conditional independences follow by the global Markov property associated with CGs [@lauritzen1996graphical]:$$\begin{aligned}
Y_{i} & \perp\!\!\!\perp\{Y_{k},A_{k},L_{k}\}|(A_{i},L_{i},\mathcal{O}_{i})\ \text{for all }i\text{ and }k, \text{ }k\neq
i;\text{ }\label{Markov}\\
\text{ }L_{i} & \perp\!\!\!\perp\mathbf{L}_{-i}\setminus\mathcal{O}_{i}|\mathbf{L}_{-i}\cap\mathcal{O}_{i}\text{ for all }i\text{ and }k\notin\mathcal{N}_{i},\text{ }k\neq i. \label{Markovii}$$ In words, equation (\[Markov\]) states that the outcome of a given unit can be screened-off (i.e. made independent) from the variables of all non-neighboring units by conditioning on the unit’s treatment and covariates as well as on all data observed on its neighboring units, where the neighborhood structure is determined by $\mathcal{G}_{\mathcal{E}}$. That is $(A_{i},L_{i},\mathcal{O}_{i})$ is the *Markov blanket* of $Y_{i}$ in CG $\mathcal{G}_{\mathcal{E}}$. This assumption, coupled with a sparse network structure leads to extensive dimension reduction of the model specification for $\mathbf{Y|A,L}$. In particular, the conditional density of $Y_{i}|\left\{ \mathbf{O\backslash}Y_{i}\right\} $ only depends on $\left(
A_{i},L_{i}\right) $ and on neighbors’ data $\mathcal{O}_{i}.$ Similarly, $\mathbf{L}_{-i}\cap\mathcal{O}_{i}$ is the Markov blanket of $L_{i}$ in CG $\mathcal{G}_{\mathcal{E}}$.
3.2 Conditional auto-models {#conditional-auto-models .unnumbered}
---------------------------
Suppose that instead of $\left( \ref{positivity}\right) $, the following stronger positivity condition holds: $$\mathbb{P}\left( \mathbf{O=o}\right) >0,\text{ for all possible values
}\mathbf{o.} \label{positivie}$$
Since $\left( \ref{Markov}\right) $ holds for the conditional law of $\mathbf{Y}$ given $\mathbf{A},\mathbf{L}$, it lies in the conditional MRF (CMRF) model associated with the induced undirected graph $\mathcal{G}_{\mathcal{E}}^{a}$. In addition, since $\left( \ref{positivie}\right) $ holds, the conditional MRF version of the Hammersley-Clifford (H-C) theorem and $\left( \ref{Markov}\right) $ imply the following version of the clique factorization in (\[eqn:fact2\]), $$f\left( \mathbf{y|a,l}\right) =\left( \frac{1}{\kappa\left( \mathbf{a,l} \right) }\right) \exp\left\{ U\left( \mathbf{y;a,l}\right) \right\} ,$$ where $\kappa\left( \mathbf{a,l}\right) =\sum_{\mathbf{y}}\exp\left\{
U\left( \mathbf{y;a,l}\right) \right\} ,$ and $U\left( \mathbf{y;a,l}\right) $ is a *conditional energy function* which can be decomposed into a sum of terms called conditional clique potentials, with a term for every maximal clique in the graph $\mathcal{G}_{\mathcal{E}}^{a}$ [@besag1974spatial].Conditional clique potentials offer a natural way to specify a CMRF using only terms that depend on a small set of variables. Specifically, $$\begin{aligned}
& f\left( Y_{i}=y_{i}|\mathbf{Y}_{-i}=\mathbf{y}_{-i},\mathbf{a,l}\right)
\nonumber\\
& =f\left( Y_{i}=y_{i}|\mathbf{Y}_{-i}=\mathbf{y}_{-i},\left\{
a_{j}\mathbf{,}l_{j}:j\in\mathcal{N}_{i}\right\} \text{ }\right) \nonumber\\
& =\frac{\exp\left\{ \sum_{c\in\mathcal{C}_{i}}U_{c}\left( \mathbf{y;a,l} \right) \right\} }{\sum_{\mathbf{y}^{\prime}\mathbf{:y}_{-i}^{\prime
}=\mathbf{y}_{-i}}\exp\left\{ \sum_{c\in\mathcal{C}_{i}}U_{c}\left(
\mathbf{y}^{\prime}\mathbf{;a,l}\right) \right\} }, \label{gibbs factor}$$ where $\mathcal{C}_{i}$ are all maximal cliques of $\mathcal{G}_{\mathcal{E}}^{a}$ that involve $Y_{i}$.
Gibbs densities specified as in $\left( \ref{gibbs factor}\right) $ is a rich class of densities, and are often regularized in practice by setting to zero conditional clique potentials for cliques of size greater than a pre-specified cut-off. This type of regularization corresponds to setting higher order interactions terms to zero in log-linear models. For instance, closely following [@besag1974spatial], one may introduce conditions (a) only cliques $c\in\mathcal{C}$ of size one or two have non-zero potential functions $U_{c},$ and (b) the conditional probabilities in $\left( \ref{gibbs factor}\right) $ have an exponential family form. Under these additional conditions, given $\mathbf{a,l,}$ the energy function takes the form $$U\left( \mathbf{y;a,l}\right) =\sum_{i\in\mathcal{G}_{\mathcal{E}}}y_{i}G_{i}\left( y_{i}\mathbf{;a,l}\right) +\sum_{\left\{ i,j\right\}
\in\mathcal{E}}y_{i}y_{j}\theta_{ij}\left( \mathbf{a,l}\right) ,$$ for some functions $G_{i}\left( \cdot\mathbf{;a,l}\right) $ and coefficients $\theta_{ij}\left( \mathbf{a,l}\right) .$ Note that in order to be consistent with local Markov conditions $\left( \ref{Markov})\text{ and
(}\ref{Markovii}\right) ,G_{i}\left( \cdot\mathbf{;a,l}\right) $ can only depend on $\left\{ \left( a_{s},l_{s}\right) :s\in\mathcal{N}_{j}\right\}
,$ while because of symmetry $\theta_{ij}\left( \mathbf{a,l}\right)
$ **** can depend at most on $\left\{ \left( a_{s},l_{s}\right)
:s\in\mathcal{N}_{j}\cap\mathcal{N}_{i}\right\} $. Following [@besag1974spatial], we call the resulting class of models *conditional auto-models*.
Conditions $\left( \ref{Markovii}\right) $ and $\left( \ref{positivie}\right) $ imply that $\mathbf{L}$ is an MRF; standard Hammersley-Clifford theorem further implies that the joint density of $\mathbf{L}$ can be written as $$f\left( \mathbf{l}\right) =\left( \frac{1}{\nu}\right) \exp\left\{
W\left( \mathbf{l}\right) \right\}$$ where $\nu=\sum_{\mathbf{l}^{\prime}}\exp\left\{ W\left( \mathbf{l}^{\prime
}\right) \right\} $, and $W\left( \mathbf{l}\right) $ is an energy function which can be decomposed as a sum over cliques in the induced undirected graph $(\mathcal{G}_{\mathcal{E}})_{\mathbf{L}}$. Analogous to the conditional auto-model described above, we restrict attention to densities of $\mathbf{L}$ of the form: $$\begin{aligned}
W\left( \mathbf{L}\right) =\sum_{i\in\mathcal{G}_{\mathcal{E}}}\left\{
\sum_{k=1}^{p}L_{k,i}H_{k,i}\left( L_{k,i}\right) +\sum_{k\neq s}\rho_{k,s,i}L_{k,i}L_{s,i}\right\} +\sum_{\left\{ i,j\right\}
\in\mathcal{E}}\sum_{k=1}^{p}\sum_{s=1}^{p}\omega_{k,s,i,j}L_{k,i}L_{s,j},
\label{eqn:l-model}$$ for some functions $H_{k,i}\left( L_{k,i}\right) $ and coefficients $\rho_{k,s,i},\omega_{k,s,i,j}.$ Note that $\rho_{k,s,i}$ encodes the association between covariate $L_{k,i}$ and covariate $L_{s,i}$ observed on unit $i,$ while $\omega_{k,s,i,j}$ captures the association between $L_{k,i}$ observed on unit $i$ and $L_{s,j}$ observed on unit $j.$
3.3 Parametric specifications of auto-models {#parametric-specifications-of-auto-models .unnumbered}
--------------------------------------------
A prominent auto-regression model for binary outcomes is the so-called auto-logistic regression first proposed by [@besag1974spatial]. Note that as $\left(
\mathbf{a,l}\right) $ is likely to be high dimensional, identification and inference about $G_{i}$ and $\theta_{ij}$ requires one to further restrict heterogeneity by specifying simple low dimensional parametric models for these functions of the form$:$ $$\begin{aligned}
G_{i}\left( y_{i}\mathbf{;a,l}\right) & =\widetilde{G}_{i}\left(
\mathbf{a,l}\right) =\mathrm{log}\frac{\Pr\left( Y_{i}=1|\mathbf{a,l,Y} _{-i}=0\right) }{\Pr\left( Y_{i}=0|\mathbf{a,l,Y}_{-i}=0\right) }\\
& =\beta_{0}+\beta_{1}a_{i}+\beta_{2}^{\prime}l_{i}+\beta_{3}\sum
_{j\in\mathcal{N}_{i}}w_{ij}^{a}a_{j}+\beta_{4}^{\prime}\sum_{j\in
\mathcal{N}_{i}}w_{ij}^{l}l_{j};\\
\theta_{ij} & =w_{ij}^{y}\theta,\end{aligned}$$ where $w_{ij}^{a}$, $w_{ij}^{l}$, $w_{ij}^{y}$ are user specified weights which may depend on network features associated with units $i$ and $j$, with $\sum_{j}w_{ij}^{a}=\sum_{j}w_{ij}^{l}=\sum_{j}w_{ij}^{y}=1;$ e.g. $w_{ij}^{a}=1/\mathrm{card}\left( \mathcal{N}_{i}\right) $ standardizes the regression coefficient by the size of a unit’s neighborhood. We assume model parameters $\tau=\left( \beta_{0},\beta_{1},\beta_{2}^{\prime},\beta
_{3},\beta_{4}^{\prime},\theta\right) $ are shared across units in a network. In addition, network features can be incorporated into the auto-models as model parameters, which may be desirable in settings where network features are confounders for the relationship between exposure and outcome. For example, one could further adjust for a unit’s degree (i.e. number of ties).
For a continuous outcome, an auto-Gaussian model may be specified as followed: $$\begin{aligned}
G_{i}\left( y_{i}\mathbf{;a,l}\right) & =-\left( \frac{1}{2\sigma_{y} ^{2}}\right) (y_{i}-2\mu_{y,i}\left( \mathbf{a,l}\right) );\\
\mu_{y,i}\left( \mathbf{a,l}\right) & =\beta_{0}+\beta_{1}a_{i}+\beta
_{2}^{\prime}l_{i}+\beta_{3}\sum_{j\in\mathcal{N}_{i}}w_{ij}^{a}a_{j}+\beta_{4}^{\prime}\sum_{j\in\mathcal{N}_{i}}w_{ij}^{l}l_{j};\\
\theta_{ij} & =w_{ij}^{y}\theta,\end{aligned}$$ where $\mu_{y,i}\left( \mathbf{a,l}\right) =E\left( Y_{i}|\mathbf{a,l,Y}_{-i}=0\right) $, and $\sigma_{y}^{2}=\mathrm{var}\left( Y_{i}|\mathbf{a,l,Y}_{-i}=0\right) $. Similarly, model parameters $\tau
_{Y}=\left( \beta_{0},\beta_{1},\beta_{2}^{\prime},\beta_{3},\beta
_{4}^{\prime},\sigma_{y}^{2},\theta\right) $ are shared across units in the network. Other auto-models within the exponential family can likewise be conditionally specified, e.g. the auto-Poisson model.
Auto-model density of $\mathbf{L}$ is specified similarly. For example, fix parameters in (\[eqn:l-model\]) $$\begin{aligned}
\rho_{k,s,i} & =\rho_{k,s},\\
\omega_{k,s,i,j} & =\widetilde{\omega}_{k,s}v_{i,j},\end{aligned}$$ where $v_{i,j}$ is a user-specified weight which satisfies $\sum_{j}v_{i,j}=1$. For $L_{k}$ binary, one might take$$H_{k,i}\left( L_{k,i};\tau_{k}\right) =\tau_{k}=\mathrm{log}\frac
{\Pr\left( L_{k,i}=1|L_{\backslash k,i}=0\mathbf{,L}_{-i}=0\right) }{\Pr\left( L_{k,i}=0|L_{\backslash k,i}=0\mathbf{,L}_{-i}=0\right) },$$ corresponding to a logistic auto-model for $L_{k,i}|L_{\backslash
k,i}=0\mathbf{,L}_{-i}=0,$ while for continuous $L_{k}$$$H_{k,i}\left( L_{k,i};\tau_{k}=\left( \sigma_{k}^{2},\mu_{k}\right)
\right) =-\left( \frac{1}{2\sigma_{k}^{2}}\right) (L_{k,i}-2\mu_{k}),$$ corresponding to a Gaussian auto-model for $L_{k,i}|L_{\backslash
k,i}=0\mathbf{,L}_{-i}=0.$ As before, model parameters $\tau_{L}=(\tau
_{1}^{\prime},...,\tau_{p}^{\prime})$ are shared across units in the network.
3.4 Coding estimators of auto-models {#coding-estimators-of-auto-models .unnumbered}
------------------------------------
Suppose that one has specified auto-models for $\mathbf{Y}$ and $\mathbf{L}$ as in the previous section with unknown parameters $\tau_{Y}$ and $\tau_{L}$ respectively. To estimate these parameters, one could in principle attempt to maximize the corresponding joint likelihood function. However, such task is well-known to be computationally daunting as it requires a normalization step which involves evaluating a high dimensional sum or integral which, outside relatively simple auto-Gaussian models is generally not available in closed form. For example, to evaluate the conditional likelihood of $\mathbf{Y|A,L}$ for binary $Y$ requires evaluating a sum of $2^{N}$ terms in order to compute $\kappa\left( \mathbf{A,L}\right) \mathbf{.}$ Fortunately, less computationally intensive strategies for estimating auto-models exist including pseudo-likelihood estimation and so called-coding estimators [@besag1974spatial], which may be adopted here. We first consider *coding-type estimators*, mainly because unlike pseudo-likelihood estimation, standard asymptotic theory applies. To describe these estimators in more detail requires additional definitions.
We define a *stable set* or *independent set* on $\mathcal{G}_{\mathcal{E}}$ as the set of nodes, $\mathcal{S}\left(\mathcal{G}_{\mathcal{E}}\right)$, such that $$(i,j) \notin \mathcal{E} \ \forall (i,j) \in \mathcal{S}\left(\mathcal{G}_{\mathcal{E}}\right)$$ That is, a stable set is a set of nodes with the property that no two nodes in the set have an edge connecting them in the network. The size of a stable set is the number of units it contains. A maximal stable set is a stable set such that no unit in $\mathcal{G}_{\mathcal{E}}$ can be added without violating the independence condition. A maximum stable set $\mathcal{S}_{\max}\left( \mathcal{G}_{\mathcal{E}}\right) $ is a maximal stable set of largest possible size for $\mathcal{G}_{\mathcal{E}}$. This size is called the stable number or independence number of $\mathcal{G}_{\mathcal{E}}$, which we denote $n_{1,N}=n_{1}\left( \mathcal{G}_{\mathcal{E}}\right) $. A maximum stable set is not necessarily unique in a given graph, and finding one such set and enumerating them all is challenging but a well-studied problem of computer science. In fact, finding a maximum stable set is a well-known NP-complete problem. Nevertheless, both exact and approximate algorithms exist that are computationally more efficient than an exhaustive search. Exact algorithms which identify all maximum stable sets were described in [@robson1986algorithms; @makino2004new; @fomin2009measure]. Unfortunately, exact algorithms for finding maximum stable sets quickly become computationally prohibitive with moderate to large networks. In fact, the maximum stable set problem is known not to have an efficient approximation algorithm unless P=NP [@zuckerman2006linear]. A practical approach we take in this paper is to simply use an enumeration algorithm that lists a collection of maximal stable sets [@myrvold2013fast], and pick the largest of the maximal sets found. Let $\Xi_{1}=\left\{ \mathcal{S}_{\max}\left(
\mathcal{G}_{\mathcal{E}}\right) :\mathrm{card}\left( \mathcal{S}_{\max
}\left( \mathcal{G}_{\mathcal{E}}\right) \right) =\mathrm{n}_{1}\left(
\mathcal{G}_{\mathcal{E}}\right) \right\}$ denote the collection of all maximum (or largest identified maximal) stable sets for $\mathcal{G}_{\mathcal{E}}$.
The Markov property associated with $\mathcal{G}_{\mathcal{E}}$ implies that outcomes of units within such sets are mutually conditionally independent given their Markov blankets. This implies the (partial) conditional likelihood function which only involves units in the stable set factorizes, suggesting that tools from maximum likelihood estimation may apply. In the Appendix, we establish that this is in fact the case, in the sense that under certain regularity conditions, coding maximum likelihood estimators of $\tau$ based on maximum (or largest identified maximal) stable sets are consistent and asymptotically normal (CAN). Consider the coding likelihood functions for $\tau_{Y}$ and $\tau_{L}$ based on a stable set $\mathcal{S}_{\max}\left( \mathcal{G}_{\mathcal{E}}\right) \in$ $\Xi_{1}$: $$\begin{aligned}
\mathcal{CL}_{Y}\left( \tau_{Y}\right) & ={\displaystyle\prod\limits_{i\in\mathcal{S}_{\max}\left( \mathcal{G} _{\mathcal{E}}\right) }}
\mathcal{L}_{Y,\mathcal{S}_{\max}\left( \mathcal{G}_{\mathcal{E}}\right)
,i}\left( \tau_{Y}\right) ={\displaystyle\prod\limits_{i\in\mathcal{S}_{\max}\left( \mathcal{G} _{\mathcal{E}}\right) }}
f\left( Y_{i}|\mathcal{O}_{i},A_{i},L_{i};\tau_{Y}\right) ;\label{Res LIk}\\
\mathcal{CL}_{L}\left( \tau_{L}\right) & ={\displaystyle\prod\limits_{i\in\mathcal{S}_{\max}\left( \mathcal{G} _{\mathcal{E}}\right) }}
\mathcal{L}_{L,\mathcal{S}_{\max}\left( \mathcal{G}_{\mathcal{E}}\right)
,i}\left( \tau_{L}\right) ={\displaystyle\prod\limits_{i\in\mathcal{S}_{\max}\left( \mathcal{G} _{\mathcal{E}}\right) }}
f\left( L_{i}|\left\{ L_{j}:j\in\mathcal{N}_{i}\right\} ;\tau_{L}\right) .
\label{likL}$$ The estimators $\widehat{\tau}_{Y}=\arg\max_{\tau_{Y}}\log\mathcal{CL}_{Y}\left( \tau_{Y}\right) $ and $\widehat{\tau}_{L}=\arg\max_{\tau_{L}}\log\mathcal{CL}_{L}\left( \tau_{L}\right) $ are analogous to Besag’s coding maximum likelihood estimators. Consider a network asymptotic theory according to which $\{\mathcal{G}_{\mathcal{E}_{N}}: N\}$ is a sequence of chain graphs as $N\rightarrow\infty,$ with vertices $(\mathbf{A}_{\mathcal{E}}\mathbf{,L}_{\mathcal{E}}\mathbf{,Y}_{\mathcal{E}}\mathbf{)}$ that follow correctly specified auto-models with unknown parameters $\left( \tau_{Y},\tau_{L}\right) $, and with edges defined according to a sequence of networks $\mathcal{E}_{N}$, $N = 1, 2, \ldots$ of increasing size. We establish the following result in the Appendix
*Result 1: Suppose that*$~n_{1,N}\rightarrow\infty$ *as* $N\rightarrow\infty$ *then under conditions 1-6 given in the Appendix,* $$\begin{aligned}
& \widehat{\tau}_{L}\underset{N\longrightarrow\infty}{\longrightarrow}\tau\text{ \textit{in probability;}}\widehat{\tau}_{Y}\underset{N\longrightarrow\infty}{\longrightarrow}\tau\text{ \textit{in
probability.} }\\
& \sqrt{n_{1,N}}\Gamma_{n_{1,N}}^{1/2}\left( \widehat{\tau}_{L}-\tau
_{L}\right) \underset{N\longrightarrow\infty}{\longrightarrow}N\left(
0,I\right) ;\\
& \sqrt{n_{1,N}}\Omega_{n_{1,N}}^{1/2}\left( \widehat{\tau}_{Y}-\tau
_{Y}\right) \underset{N\longrightarrow\infty}{\longrightarrow}N\left(
0,I\right) ;\\
\Gamma_{n_{1,N}} & =\frac{1}{n_{1,N}}\sum_{i\in\mathcal{S}_{\max}\left(
\mathcal{G}_{\mathcal{E}_{N}}\right) }\left\{ \frac{\partial\log
\mathcal{CL}_{L,\mathcal{S}_{\max}\left( \mathcal{G}_{\mathcal{E}_{N} }\right) ,i}\left( \tau_{L}\right) }{\partial\tau_{L}}\right\} ^{\otimes
2},\\
\Omega_{n_{1,N}} & =\frac{1}{n_{1,N}}\sum_{i\in\mathcal{S}_{\max}\left(
\mathcal{G}_{\mathcal{E}_{N}}\right) }\left\{ \frac{\partial\log
\mathcal{CL}_{Y,\mathcal{S}_{\max}\left( \mathcal{G}_{\mathcal{E}_{N} }\right) ,i}\left( \tau_{Y}\right) }{\partial\tau_{Y}}\right\} ^{\otimes
2}.\end{aligned}$$ **
Note that by the information equality, $\Gamma_{n_{1,N}}$ and $\Omega
_{n_{1,N}}$ can be replaced by the standardized (by $n_{1,N}$) negative second derivative matrix of corresponding coding log likelihood functions. Note also that condition $n_{1,N}\rightarrow\infty$ as $N\rightarrow\infty$ essentially rules out the presence of an ever-growing hub on the network as it expands with $N$, thus ensuring that there is no small set of units in which majority of connections are concentrated asymptotically. Suppose that each unit on a network of size $N$ is connected to no more than $C_{\max}<N,$ then according to Brooks’ Theorem, the stable number $n_{1,N}$ satisfies the inequalities [@brooks1941colouring]: $$\frac{N}{C_{\max}+1}\leq n_{1,N}\leq N.$$ This implies that in a network of bounded degree, $n_{1,N}=O\left( N\right)
$ is guaranteed to be of the same order as the size of the network$;$ however $n_{1,N}$ may grow at substantially slower rates $(n_{1,N}=o(N))$ if $C_{\max
}$ is unbounded.
3.5 Pseudo-likelihood estimation {#pseudo-likelihood-estimation .unnumbered}
--------------------------------
Note that because $L_{i}$ is likely multivariate, further computational simplification can be achieved by replacing $f\left( L_{i}|\left\{
L_{j}:j\in\mathcal{N}_{i}\right\} ;\tau_{L}\right) $ with the pseudo-likelihood (PL) function $${\displaystyle\prod\limits_{s=1}^{p}}
f\left( L_{s,i}|L_{\backslash s,i},\left\{ L_{j}:j\in\mathcal{N}_{i}\right\} ;\tau_{L}\right)$$ in equation $\left( \ref{likL}\right) .$ This substitution is computationally more efficient as it obviates the need to evaluate a multivariate integral in order to normalize the joint law of $L_{i}$. Let $\widetilde{\tau}$ denote the estimator which maximizes the log of the resulting modified coding likelihood function $\mathcal{L}_{L,\mathcal{S} _{\max}\left( \mathcal{G}_{\mathcal{E}}\right) ,i}^{\ast}\left( \tau
_{L}\right) .$ It is straightforward using the proof of Result 1 to establish that its covariance may be approximated by the sandwich formula $\Phi
_{L,n_{1,N}}^{-1}\Gamma_{L,n_{1,N}}\Phi_{L,n_{1,N}}^{-1}$ [@guyon1995random], where $$\begin{aligned}
\Gamma_{L,n_{1,N}} & =\frac{1}{n_{1,N}}\sum_{i\in\mathcal{S}_{\max}\left(
\mathcal{G}_{\mathcal{E}}\right) }\left\{ \frac{\partial\log\mathcal{L} _{L,\mathcal{S}_{\max}\left( \mathcal{G}_{\mathcal{E}}\right) ,i}^{\ast
}\left( \tau_{L}\right) }{\partial\tau_{L}}\right\} ^{\otimes2},\\
\Phi_{L,n_{1,N}} & =\frac{1}{n_{1,N}}\sum_{i\in\mathcal{S}_{\max}\left(
\mathcal{G}_{\mathcal{E}}\right) }\left\{ \frac{\partial^{2}\log
\mathcal{L}_{L,\mathcal{S}_{\max}\left( \mathcal{G}_{\mathcal{E}}\right)
,i}^{\ast}\left( \tau_{L}\right) }{\partial\tau_{L}\partial\tau_{L}^{T}}\right\} .\end{aligned}$$
As later illustrated in extensive simulation studies, coding estimators can be inefficient, since the partial conditional likelihood function associated with coding estimators disregards contributions of units $i\not \in \mathcal{S}_{\max}\left( \mathcal{G}_{\mathcal{E}}\right) .$ Substantial information may be recovered by combining multiple coding estimators each obtained from a separate approximate maximum stable set, however accounting for dependence between the different estimators can be challenging.
Pseudo-likelihood (PL) estimation offers a simple alternative approach which is potentially more efficient than either approach described above. PL estimators maximize the log-PLs $$\begin{aligned}
\log\left\{ \mathcal{PL}_{Y}\left( \tau_{Y}\right) \right\} & ={\displaystyle\sum\limits_{i\in\mathcal{G}_{\mathcal{E}}}}
\log f\left( Y_{i}|\mathcal{O}_{i},A_{i},L_{i};\tau_{Y}\right) ;\\
\log\left\{ \mathcal{PL}_{L}\left( \tau_{L}\right) \right\} & ={\displaystyle\sum\limits_{i\in\mathcal{G}_{\mathcal{E}}}}
\log f\left( L_{k,i}| \left\{ L_{s,i}:s \in\{1,..,p\} \setminus k \right\} ,
\left\{ L_{s,j}: s \in\{1,..,p\}, j\in\mathcal{N}_{i}\right\} ;\tau
_{L}\right) .\end{aligned}$$ Denote corresponding estimators $\check{\tau}_{Y}$ and $\check{\tau}_{L}$, $\ $which are shown to be consistent in the Appendix. There however is generally no guarantee that their asymptotic distribution follows a Gaussian distribution due to complex dependence between units on the network prohibiting application of the central limit theorem. As a consequence, for inference, we recommend using the parametric bootstrap, whereby algorithm Gibbs sampler I of Section 2.2 may be used to generate multiple bootstrap samples from the observed data likelihood evaluated at $\left( \check{\tau
}_{Y},\check{\tau}_{L}\right) ,$ which in turn can be used to obtain a bootstrap distribution for $\left( \check{\tau}_{Y},\check{\tau}_{L}\right)
$ and corresponding inferences such as bootstrap quantile confidence intervals.
4. AUTO-G-COMPUTATION {#auto-g-computation .unnumbered}
=====================
We now return to the main goal of the paper, which is to obtain valid inferences about $\beta_{i}\left( \mathbf{a}\right) .$ The auto-G-computation algorithm entails evaluating $$\widehat{\beta}_{i}\left( \mathbf{a}\right) \approx K^{-1}\sum_{k=0}^{K}\widehat{Y}_{i}^{(m^{\ast}+k)},$$ where $\widehat{Y}_{i}^{(m)}$ are generated by Gibbs Sampler I algorithm under posited auto-models with estimated parameters $\left( \hat{\tau}_{Y},\hat{\tau}_{L}\right) .$ An analogous estimator $\breve{\beta}_{i}\left(
\mathbf{a}\right) $ can be obtained using $\left( \check{\tau}_{Y},\check{\tau}_{L}\right) $ instead of $\left( \hat{\tau}_{Y},\hat{\tau}_{L}\right) .$ In either case, the parametric bootstrap may be used in conjunction with Gibbs Sampler I in order to generate the corresponding bootstrap distribution of estimators of $\beta_{i}\left( \mathbf{a}\right) $ conditional on either $\widehat{\beta}_{i}\left( \mathbf{a}\right) $ or $\breve{\beta}_{i}\left( \mathbf{a}\right) $. Alternatively, a less computationally intensive approach first generates i.i.d. samples $\tau
_{Y}^{(j)}$ and $\tau_{L}^{(j)}$ $,$ $j=1,...J$ from $N\left( \hat{\tau}_{Y},\widehat{\Gamma}_{n_{1,N}}\right) $ and $N\left( \hat{\tau}_{L},\widehat{\Omega}_{n_{1,N}}\right) $ respectively, conditional on the observed data, where $\widehat{\Gamma}_{n_{1,N}}=\Gamma_{n_{1,N}}\left(
\widehat{\tau}_{L}\right) $ and $\widehat{\Omega}_{n_{1,N}}=\Omega_{n_{1,N}}\left( \widehat{\tau}_{Y}\right) $ estimate $\Gamma_{n_{1,N}}$ and $\Omega_{n_{1,N}}.$. Next, one computes corresponding estimators $\widehat{\beta}_{i}^{(j)}\left( \mathbf{a}\right) $ based on simulated data generated using Gibbs Sampler I algorithm under $\tau_{Y}^{(j)}$ and $\tau
_{L}^{(j)},$ $j=1,...,J.$ The empirical distribution of $\left\{
\widehat{\beta}_{i}^{(j)}\left( \mathbf{a}\right) :j\right\} $ may be used to obtain standard errors for $\widehat{\beta}_{i}\left( \mathbf{a}\right)
$, and corresponding Wald type or quantile-based confidence intervals for direct and spillover causal effects.
5. SIMULATION STUDY {#simulation-study .unnumbered}
===================
We performed an extensive simulation study to evaluate the performance of the proposed methods on networks of varying density and size. Specifically, we investigated the properties of the coding-type and pseudo-likelihood estimators of unknown parameters $\tau_{Y}$ and $\tau_{L}$ indexing the joint observed data likelihood. Additionally, we evaluated the performance of proposed estimators of the network counterfactual mean $\beta(\alpha
)=N^{-1}\sum_{i=1}^{N}\sum_{\mathbf{a}_{-i}\in\mathcal{A(}N)}\pi_{i}\left(
\mathbf{a}_{-i};\alpha\right) E\left( Y_{i}\left( \mathbf{a}\right)
\right) ,$ as well as for the direct effect $DE(\alpha),$ and the spillover effect $IE(\alpha)$, where $\alpha$ is a specified treatment allocation law described below.
We simulated three networks of size 800 with varying densities: low (each node has either 2, 3, or 4 neighbors), medium (each node has either 5, 6, or 7 neighbors), and high (each node has either 8, 9, or 10 neighbors). For reference, a depiction of the low density network of size 800 is given in Figure 2. Additionally, we simulated low density networks of size 200, 400, and 1,000. The network graphs were all simulated in Wolfram Mathematica 10 using the RandomGraph function. For each network, we obtained an (approximate) maximum stable set. The stable sets for the 800 node networks were of size $n_{1,low}=375$, $n_{1,med}=275$, $n_{1,high}=224$.
For units $i=1,...,N$, we generated using Gibbs Sampler I$\ $a vector of binary confounders $\{L_{1i},L_{2i},L_{3i}\}$, a binary treatment assignment $A_{i}$, and a binary outcome $Y_{i}$ from the following auto-models consistent with the chain graph induced by the simulated network: $$\begin{aligned}
Pr(L_{1,i}=1\mid\mathbf{L}_{\setminus1,i},\{\mathbf{L}_{1,j}:j\in
\mathcal{N}_{i}\}) & =\mathrm{expit}\bigg(\tau_{1}+\rho_{12}L_{2,i}+\rho_{13}L_{3,i}+\nu_{11}\sum_{j\in\mathcal{N}_{i}}L_{1,j}+\nu_{12}\sum
_{j\in\mathcal{N}_{i}}L_{2,j}+\nu_{13}\sum_{j\in\mathcal{N}_{i}}L_{3,j}\bigg)\\
Pr(L_{2,i}=1\mid\mathbf{L}_{\setminus2,i},\{\mathbf{L}_{j}:j\in\mathcal{N}_{i}\}) & =\mathrm{expit}\bigg(\tau_{2}+\rho_{12}L_{1,i}+\rho_{23}L_{3,i}+\nu_{21}\sum_{j\in\mathcal{N}_{i}}L_{1,j}+\nu_{22}\sum_{j\in
\mathcal{N}_{i}}L_{2,j}+\nu_{23}\sum_{j\in\mathcal{N}_{i}}L_{3,j}\bigg)\\
Pr(L_{3,i}=1\mid\mid\mathbf{L}_{\setminus3,i},\{\mathbf{L}_{j}:j\in
\mathcal{N}_{i}\}) & =\mathrm{expit}\bigg(\tau_{3}+\rho_{13}L_{1,i}+\rho_{23}L_{2,i}+\nu_{31}\sum_{j\in\mathcal{N}_{i}}L_{1,j}+\nu_{32}\sum
_{j\in\mathcal{N}_{i}}L_{2,j}+\nu_{33}\sum_{j\in\mathcal{N}_{i}}L_{3,j}\bigg)\end{aligned}$$ $$\begin{aligned}
Pr(A_{i}=1\mid L_{i},\{A_{j},\mathbf{L}_{j}:j\in\mathcal{N}_{i}\}) &
=\mathrm{expit}\bigg(\gamma_{0}+\gamma_{1}L_{1,i}+\gamma_{2}\sum
_{j\in\mathcal{N}_{i}}L_{1,j}+\gamma_{3}L_{2,i}\\
& \hspace{2cm}+\gamma_{4}\sum_{j\in\mathcal{N}_{i}}L_{2,j}+\gamma_{5}L_{3,i}+\gamma_{6}\sum_{j\in\mathcal{N}_{i}}L_{3,j}+\gamma_{7}\sum
_{j\in\mathcal{N}_{i}}A_{j}\bigg)\end{aligned}$$ $$\begin{aligned}
Pr(Y_{i}=1\mid A_{i},L_{i},\mathcal{O}_{i}) & =\mathrm{expit}\bigg(\beta
_{0}+\beta_{1}A_{i}+\beta_{2}\sum_{j\in\mathcal{N}_{i}}A_{j}+\beta_{3}L_{1,i}+\beta_{4}\sum_{j\in\mathcal{N}_{i}}L_{1,j}\\
& \hspace{2cm}+\beta_{5}L_{2,i}+\beta_{6}\sum_{j\in\mathcal{N}_{i}}L_{2,j}+\beta_{7}L_{3,i}+\beta_{8}\sum_{j\in\mathcal{N}_{i}}L_{3,j}+\beta
_{9}\sum_{j\in\mathcal{N}_{i}}Y_{j}\bigg)\end{aligned}$$
where $\mathrm{expit}\left( x\right) =(1+\exp\left( -x\right)
)^{-1},\tau_{L}=\{\tau_{1},\tau_{2},\tau_{3},\rho_{12},\rho_{13},\rho_{23},\nu_{11},\nu_{12},\nu_{13},\nu_{22},\nu_{21},\nu_{23},\nu_{33},\nu_{31},\nu_{32}\}$, $\tau_{A}=\{\gamma_{0},...,\gamma_{7}\}$, and $\tau_{Y}=\{\beta_{0},...,\beta_{9}\}$. We evaluated network average direct and spillover effects via the Gibbs Sampler I algorithm under true parameter values $\tau_{Y}$ and $\tau_{L}$ and a treatment allocation, $\alpha$ given by a binomial distribution with event probability equal to $0.7$. All parameter values are summarized in Table 1. We generated $S=1,000$ simulations of the chain graph for each of the 4 simulated network structures. For each simulation $s$, data were generated by running the Gibbs sampler I algorithm $4,000$ times with the first $1,000$ iterations as burn-in. Additionally, we thinned the chain by retaining every third realization to reduce autocorrelation.
\[c\][cc]{} **Parameter** & **Truth**\
$\tau_{L}$ & (-1.0,0.50,-0.50,0.1,0.2,0.1,0.1,0,0,0.1,0,0,0.1,0,0)\
$\tau_{A}$ & (-1.00,0.50,0.10,0.20,0.05,0.25,-0.08,0.30\
$\tau_{Y}$ & (-0.30,-0.60,-0.20,-0.20,-0.05,-0.10,-0.01,0.40,0.01,0.20)\
For each realization of the chain graph, $\mathcal{G}_{\mathcal{E} _{N},s}$, we estimated $\tau_{Y}$ via coding-type maximum likelihood estimation and $\tau_{L}$ via the modified coding estimator. Both sets of parameters were also estimated via maximum pseudo-likelihood estimation. For each estimator we computed corresponding causal effect estimators, their standard errors and $95\%$ Wald confidence intervals as outlined in previous Sections. The estimation of the auto-model parameters was computed in R using functions `optim()` and `glm()`(R Core Team, 2013). The network average causal effects were estimated using Gibbs Sampler I using the `agcEffect` function in the `autognet` R package by plugging in estimates for $(\tau_{L},\tau_{Y})$ using $K=50$ iterations and a burn-in of $m^{\ast}=10$ iterations. For variance estimation of the coding-type estimator, 200 bootstrap replications were used.
Simulation results for the various density networks of size 800 are summarized in Tables 2 and 3 for the following parameters: the network average counterfactual $\beta(\alpha),$ the network average direct effect, and the network average spillover effect. Both coding and pseudo-likelihood estimators had small bias in estimating $\beta(\alpha)$ regardless of network density (absolute bias $<0.01$). Coverage of the coding estimator ranged between $93.1\%$ and $94.5\%$. Biases were also small for both spillover and direct effects: the bias slightly increased with network density, but still stayed below an absolute bias of $0.01$. Coverage of coding-based confidence intervals for direct effects ranged from $92.5\%$ to $95.6\%$, while the coverage for spillover effects decreased slightly with network density from $93.7\%$ to $92.2\%$. It is important to note that as the network structure changes with network size and density, the corresponding estimated parameters likewise vary and therefore it is not necessarily straightforward to compare performance of the methodology across network structure. Table 3 gives the MC variance for the pseudo-likelihood estimator which confirms greater efficiency compared to the coding estimator given the significantly larger effective sample size used by pseudo-likelihood. Appendix Tables 1-3 report bias and coverage for the network causal effect parameters for low density networks of size 200, 400, and 1,000. Additionally, Appendix Figures 1 and 2 report bias and coverage for all 25 auto-model parameters in the low-density network of size 800. As predicted by theory, coding-type and pseudo-likelihood estimators exhibit small bias. Additionally, coding-type estimators had approximately correct coverage, while pseudo-likelihood estimators had coverage substantially lower than the nominal level for a number of auto-model parameters. These results confirm the anticipated failure of pseudo-likelihood estimators to be asymptotically Gaussian. Most notably, the coverage for the outcome auto-model coefficient capturing dependence on neighbors’ outcomes $\beta_{9}$ was $81\%$, while coverage of the coding-type Wald CI for this coefficient was $94\%$. Although not shown here, the coverage results for the auto-model parameters are consistent across all simulations.
We also assessed the performance of auto-g-computation in small, dense networks and in the presence of missing network edges. For the first, we generated one network of size 100 ($n_{1,100} = 25$) and an additional network of size 200 ($n_{1,200} = 57$). For the network of size 100, coding estimation of auto-model parameters in 437 of the 1,000 simulated samples had convergence issues due to the small size of the maximal independent set. Excluding results with convergence issues, the causal estimates were biased and did not have correct coverage (see Appendix Figure 3a). The performance for the network of size 200 was much improved across these endpoints, though oftentimes the confidence intervals were too wide to be informative. In both cases, the pseudo-likelihood estimator exhibited less bias than the coding estimator. In the previously described dense network of size 800, we randomly removed 564 (14%) of edges. The estimated parameters from the auto-models were unbiased and had correct coverage (see Appendix Figure 4). However, the causal estimates for both the coding and pseudo-likelihood estimators exhibited bias, and the coding estimator had coverage slightly below the nominal level with the estimated spillover effect shifted towards null (see Appendix Figure 5).
\[c\][cccccc]{} & & & & &\
& Truth & Bias & MC Variance & Robust Variance & 95% CI Coverage\
& & & & &\
$\beta(\alpha)$ & 0.211 & $0.001$ & 0.001 & 0.001 & 0.945\
Spillover & -0.166 & $0.002$ & 0.004 & 0.004 & 0.937\
Direct & -0.179 & $0.002$ & 0.002 & 0.002 & 0.943\
& & & & &\
& & & & &\
$\beta(\alpha)$ & 0.209 & $0.003$ & 0.001 & 0.002 & 0.931\
Spillover & -0.170 & $0.007$ & 0.013 & 0.015 & 0.925\
Direct & -0.178 & $<0.001$ & 0.003 & 0.003 & 0.925\
& & & & &\
& & & & &\
$\beta(\alpha)$ & 0.208 & 0.004 & 0.005 & 0.004 & 0.937\
Spillover & -0.171 & 0.001 & 0.032 & 0.027 & 0.922\
Direct & -0.177 & -0.001 & 0.004 & 0.004 & 0.956\
\[c\][cccc]{} & & &\
& Truth & Absolute Bias & MC Variance\
& & &\
$\beta(\alpha)$ & 0.211 & $0.001$ & $<0.001$\
Spillover & -0.166 & $0.002$ & 0.002\
Direct & -0.179 & $0.002$ & 0.001\
& & &\
& & &\
$\beta(\alpha)$ & 0.209 & $0.003$ & 0.001\
Spillover & -0.170 & $0.007$ & 0.005\
Direct & -0.178 & $<0.001$ & 0.001\
& & &\
& & &\
$\beta(\alpha)$ & 0.208 & $0.004$ & 0.001\
Spillover & -0.171 & $0.001$ & 0.006\
Direct & -0.177 & $-0.001$ & 0.001\
6. DATA APPLICATION {#data-application .unnumbered}
===================
We consider an application of the *auto-g-computation* algorithm to the Networks, Norms, and HIV/STI Risk Among Youth (NNAHRAY) study to assess the effect of past incarceration on infection with HIV, STI, or Hepatitis C accounting for the network structure [@khan2009incarceration]. The NNAHRAY study was conducted in a New York neighborhood with epidemic HIV and widespread drug use from 2002-2005 [@friedman2008relative]. Through in-person interviews, information was collected regarding the respondents’ demographic characteristics, incarceration history, sexual partnerships and histories, and past drug use. At the time of the interviews, respondents were also tested for HIV, gonorrhea, chlamydia, Herpes Simple Virus (HSV) 2, Hepatitis C virus (HCV), and syphilis. The study population we consider includes all interviewed persons with recorded results from their HIV, STI, and HCV tests ($n=8$ persons missing) for a total sample size of $N=457$ persons. We assume that HIV/STI/HCV status is missing completely at random. We defined a network tie (i.e. edge) as a sexual and/or injection drug use partnership in the past three months if at least one of the partners reported the relationship. The network structure is given in Figure 3. The number of partners (i.e. neighbors) for each respondent varied from none to 10 resulting in a maximal independent set of $n_1 = 274$.
We estimated the network-level spillover and direct effect of past incarceration on infection with HIV, STI, or Hepatitis C (HCV) under a Bernoulli allocation strategy with treatment probability equal to 0.50. Past incarceration was defined as any amount of jail time in the respondents’ history. We accounted for confounding by Latino/a ethnicity, age, education, and past illicit drug use. The same models and estimation procedure detailed in the simulation section were utilized; note that $\nu_{ij}$ where $i \neq j$ were assumed to be 0. For comparison, the auto-model parameters were estimated using the coding-type and pseudolikelihood estimators. Network average spillover and direct effects were restricted to persons with at least one network tie.
Table 4 gives the outcome auto-model parameter point estimates for the coding and pseudolikelihood estimators with 95% confidence intervals for the coding estimators excluding the covariate terms. Due to scaling by number of network ties, the outcome and exposure influence of network ties can be interpreted as the effect of average covariate value among network ties. Individuals who experienced prior incarceration had 2.12 \[95% CI: 1.07-4.21\] times the odds of infection with HIV/STI/HCV compared to those without prior incarceration. However, the incarceration status of network ties was not significantly associated with a person’s risk of HIV/STI/HCV (OR= 1.21 \[95% CI: 0.52-2.84\]) conditional on the neighbors’ outcomes. Individual’s with a greater proportion of their ties infected with HIV, STI, and/or HCV were much more likely to be infected with HIV, STI, and/or HCV (OR=3.07 \[95% CI: 1.33-7.09\]). The pseudolikelihood point estimates were similar to the coding results. The full results for auto-model parameters from both the covariate and outcome model are given in Appendix Figure 6. The network average direct effect is 0.14 \[95% CI: 0.02-0.28\] when the proportion of persons with prior history of incarceration is 0.50. There was no significant evidence of a spillover effect of incarceration on HIV/STI/HCV risk over the network, as increasing the proportion of persons with a history of incarceration from 0 to 0.50 resulted in a negligible increase in average HIV/STI/HCV risk of a person with no prior incarceration \[$\widehat{DE}$=0.04; 95% CI: -0.06-0.14\].
In the Appendix, we have included two alternate outcome auto-model specifications that incorporate the number of sexual and injection drug use partners for each person in the network. In an infection disease setting, the number of partners should in principle be accounted for in the analysis as it is likely a confounder for the effect of incarceration (both individual and neighbors’ status) on infection status [@khan2018dissolution]. As shown in the Appendix, adjusting for the number of network ties (e.g sexual and injection drug partners) did not change our conclusions. Following a reviewer’s recommendation, we performed a simulation study based on the NNAHRAY network under under the sharp null and verified that we have valid inference in this setting. Results are provided in the Appendix Table 7.
\[c\][lcc]{} & &\
& Estimates & 95% CI\
& &\
Past incarceration status (individual) & 2.12 & \[1.07, 4.21\]\
Past incarceration status (neighbors) & 1.21 & \[0.52, 2.84\]\
HIV/STI/HCV status (neighbors) & 3.07 & \[1.33, 7.09\]\
& &\
& &\
Past incarceration status (individual) & 2.36 & –\
Past incarceration status (neighbors) & 0.97 & –\
HIV/STI/HCV status (neighbors) & 2.62 & –\
7. CONCLUSION {#conclusion .unnumbered}
=============
We have described a new approach for evaluating causal effects on a network of connected units. Our methodology relies on the crucial assumption that accurate information on network ties between observed units is available to the analyst, which may not always be the case in practice. In fact, as demonstrated in our simulation study, bias may ensue if information about the network is incomplete, and therefore omits to account for all existing ties. In future work, we plan to further develop our methods to appropriately account for uncertainty about the underlying network structure.
Another limitation of the proposed approach is that it relies heavily on parametric assumptions and as a result may be open to bias due to model mis-specification. Although this limitation also applies to standard g-computation for i.i.d settings which nevertheless has gained prominence in epidemiology [@taubman2009intervening; @robins2004effects; @daniel2011gformula], our parametric auto-models which are inherently non-i.i.d may be substantially more complex, as they must appropriately account both for outcome and covariate dependence, as well as for interference. Developing appropriate goodness-of-fit tests for auto-models is clearly a priority for future research. In addition, to further alleviate concerns about modeling bias, we plan in future work to extend semiparametric models such as structural nested models to the network context. Such developments may offer a real opportunity for more robust inference about network causal effects.
SUPPLEMENTARY MATERIALS {#supplementary-materials .unnumbered}
=======================
Acknowledgments:
: We are grateful to Dr. Samuel R. Friedman at National Development and Research Institutes, Inc. for access to the Networks, Norms, and HIV/STI Risk Among Youth study data and contributions to the data application section.
Appendix:
: Theorems, detailed proofs, and additional simulation results (.zip file)
Code:
: Code for estimation and inference of network causal effects. To download, please visit: https://isabelfulcher.github.io/autoGnetworks/ (R)
[*Appendix*]{}
Throughout, we assume that we observe a vector of Random Fields $\left(
\mathbf{Y}_{N},\mathbf{A}_{N},\mathbf{L}_{N}\right) ,$ such that $\mathbf{Y}_{N}$ is a conditional Markov Random Field (MRF) given $(\mathbf{A}_{N},\mathbf{L}_{N})$ on the sequence of Chain Graphs (CG) $\mathcal{G}_{\mathcal{E}}$ associated with an increasing sequence of networks $\mathcal{E}_{N},$ with distribution uniquely specified by the parametric model for its Gibbs factors $f\left( Y_{i}|\partial_{i};\tau_{Y}\right) $ where $\partial_{i}=\left( \mathcal{O}_{i},A_{i},L_{i}\right) $ for all $i=1,...,N.$
Let $\mathcal{N}_{i}^{(k)}$ denote the $kth$ order neighborhood of unit $i,$ defined as followed: $\mathcal{N}_{i}^{(1)}=$ $\mathcal{N}_{i},$ $\mathcal{N}_{i}^{(2)}={\displaystyle\bigcup\limits_{j\in\mathcal{N}_{i}^{(1)}}}
\mathcal{N}_{j}^{(1)}\backslash\left( \mathcal{N}_{i}^{(1)}\cup\{i\}\right)
,...,\mathcal{N}_{i}^{(k)}={\displaystyle\bigcup\limits_{j\in\mathcal{N}_{i}^{(k-1)}}}
\mathcal{N}_{j}^{(1)}\backslash\left(
{\displaystyle\bigcup\limits_{s\leq k-1}}
\mathcal{N}_{i}^{(s)}\cup\{i\}\right) .$ A *k-stable set* $\mathcal{S}^{(k)}\left( \mathcal{G}_{\mathcal{E}},k\right) $ of $\mathcal{G}_{\mathcal{E}}$ is a set of units $\left( i,j\right) $ in $\mathcal{G}_{\mathcal{E}}$ such that ${\displaystyle\bigcup\limits_{s\leq k}}
\mathcal{N}_{i}^{(s)}$ and ${\displaystyle\bigcup\limits_{s\leq k}}
\mathcal{N}_{j}^{(s)}$ are not neighbors$.$ The size of a k-stable set is the number of units it contains. A maximal k-stable set is a k-stable set such that no unit in $\mathcal{G}_{\mathcal{E}}$ can be added without violating the independence condition. A maximum k-stable set $\mathcal{S}_{k,\max}\left(
\mathcal{G}_{\mathcal{E}}\right) $ is a maximal k-stable set of largest possible size for $\mathcal{G}_{\mathcal{E}}$. This size is called the k-stable number of $\mathcal{G}_{\mathcal{E}}$, which we denote $n_{k,N}=n_{k}\left( \mathcal{G}_{\mathcal{E}}\right) $. Let $\Xi_{k}=\left\{
\mathcal{S}_{k,\max}\left( \mathcal{G}_{\mathcal{E}}\right) :\mathrm{card}\left( \mathcal{S}_{k,\max}\left( \mathcal{G}_{\mathcal{E}}\right) \right)
=\mathrm{s}_{k}\left( \mathcal{G}_{\mathcal{E}}\right) \right\} $ denote the collection of all (approximate) maximum k-stable sets for $\mathcal{G}_{\mathcal{E}}$.
The unknown parameter $\tau_{Y}$ is in the interior of a compact $\Theta\subset\mathbb{R}^{p}.$Let $\partial\partial_{i}=\left\{
\mathcal{O}_{j},A_{j},L_{j}:j\in\mathcal{N}_{i}^{(2)}\right\} \cup\left\{
A_{j},L_{j}:j\in\mathcal{N}_{i}^{(3)}\right\} \cup\left\{ L_{j}:j\in\mathcal{N}_{i}^{4)}\right\} .$ Let $$\mathcal{M}_{Y,i}\left( \tau_{Y},t;\partial_{i}\right) =-E\left\{ \log
\frac{f\left( Y_{i}|\mathcal{O}_{i},A_{i},L_{i};t\right) }{f\left(
Y_{i}|\mathcal{O}_{i},A_{i},L_{i};\tau_{Y}\right) }|\partial_{i}\right\}
\geq0;$$ We make the following assumptions:
Suppose that
1. $\mathcal{S}\left( \mathcal{G}_{\mathcal{E}}\right) $ is a 1-stable set.
2. $\mathcal{S}\left( \mathcal{G}_{\mathcal{E}}\right) $ can be partitioned into $K$ 4-stable subsets $\left\{ \mathcal{S}^{k}\left(
\mathcal{G}_{\mathcal{E}}\right) :k=1,...,K\right\} $ such that $\mathcal{S}\left( \mathcal{G}_{\mathcal{E}}\right) = {\displaystyle\bigcup\limits_{k=1}^{K}}
\mathcal{S}^{k}\left( \mathcal{G}_{\mathcal{E}}\right) .$ Let $n^{(k)}$ denote the number of units in $\mathcal{S}^{k}\left( \mathcal{G} _{\mathcal{E}}\right) $ and $n=\sum_{k}n^{(k)}$ denote the total number of units in $\mathcal{S}\left( \mathcal{G}_{\mathcal{E}}\right) .$ We further assume that there is a $k_{0}$ such that:
1. $\lim\inf_{\mathcal{E}}$ $n^{(k_{0})}/n>0$ and $\lim\inf_{\mathcal{E}}$ $n/N>0.$
2. Let $\mathcal{D}_{i}$ denote the support $\partial_{i}=\left(
\mathcal{O}_{i},A_{i},L_{i}\right) .$ The joint support $\mathcal{D=} {\displaystyle\prod\limits_{j\in\mathcal{N}_{i}}}
\mathcal{D}_{j}$ is a fixed space of neighborhood configurations when $i\in\mathcal{S}^{k_{0}}\left( \mathcal{G}_{\mathcal{E}}\right) .$
3. $\exists c>0$ such that for all $i\in\mathcal{S}^{k_{0}}\left(
\mathcal{G}_{\mathcal{E}}\right) ,$ we have for all values y$_{i},$ $\partial_{i},\partial\partial_{i}$ and $t\in\Theta,$ $$\begin{aligned}
f\left( y_{i}|\partial_{i};t_{y}\right) & >c\\
f\left( \partial_{i}|\partial\partial_{i},t_{y}\right) & >c
\end{aligned}$$ w.here $f\left( y_{i}|\partial_{i};t_{y}\right) $ and $f\left( \partial
_{i}|\partial\partial_{i},t_{y}\right) $ are density functions where $f\left( y_{i}|\partial_{i};t_{y}\right) $ is uniformly (in $i,y_{i} ,\partial_{i})$ continuous in $t_{y.}$
4. There exists $\mathcal{M}_{y}\left( \tau_{Y},t;z\right) \geq0$ with $\left( t,z\right) \in\Theta\times\mathcal{D}$, $\mu-$integrable for all $t$ such that:
1. $\mathcal{M}_{Y,i}\left( \tau_{Y},t;z\right) \geq\mathcal{M} _{Y}\left( \tau_{Y},t;z\right) $ if $i\in\mathcal{S}^{k_{0}}\left(
\mathcal{G}_{\mathcal{E}}\right) .$
2. $t\rightarrow N\left( \tau_{Y},t\right) =\int_{\mathcal{D}} \mathcal{M}_{Y}\left( \tau_{Y},t;z\right) \mu\left( dz\right) $ is continuous and has a unique minimum at $t=\tau_{Y}.$
5. For all $i\in\mathcal{S}\left( \mathcal{G}_{\mathcal{E}}\right) ,$ $f\left( y_{i}|\partial_{i};t_{y}\right) $ admits three continuous derivatives at $t_{y}$ in a neighborhood of $\tau_{Y}$, and $1/f\left(
y_{i}|\partial_{i};t_{y}\right) $ and the $vth$ derivatives $f^{(v)}\left(
y_{i}|\partial_{i};t_{y}\right) $ $v=1,2,3$ are uniformly bounded in $i,y_{i},\partial_{i}$ and $t_{y}$ in a neighborhood of $\tau_{Y}.$
6. There exist a positive-definite symmetric non-random $p\times p$ matrix $I\left( \tau_{Y}\right) $ such that $$\lim\inf_{\mathcal{E}}\Omega_{n_{1,N}}\geq I\left( \tau_{Y}\right)$$
Under assumptions 1-4, the maximum coding likelihood estimator on $\mathcal{S}\left( \mathcal{G}_{\mathcal{E}}\right) $ is consistent as $N\rightarrow\infty,$ that is $$\hat{\tau}_{Y}\rightarrow\tau_{Y}\text{ in probability }$$ where $$\hat{\tau}_{Y}=\arg\max_{t}\sum_{i\in\mathcal{S}\left( \mathcal{G} _{\mathcal{E}}\right) }\log f\left( Y_{i}|\mathcal{O}_{i},A_{i} ,L_{i};t\right)$$
Define $$U_{\mathcal{E}}^{k_{0}}\left( t\right) =-\frac{1}{n^{(k_{0})}}\sum
_{i\in\mathcal{S}^{k_{0}}\left( \mathcal{G}_{\mathcal{E}}\right) }\log
f\left( Y_{i}|\partial_{i};t\right)$$ and write $$Z_{\mathcal{E}}=-\frac{1}{n^{(k_{0})}}\sum_{i\in\mathcal{S}^{k_{0}}\left(
\mathcal{G}_{\mathcal{E}}\right) }\left\{ \log\frac{f\left( Y_{i} |\partial_{i};t\right) }{f\left( Y_{i}|\partial_{i};\tau_{Y}\right)
}+\mathcal{M}_{Y,i}\left( \tau_{Y},t;\partial_{i}\right) \right\} +\frac
{1}{n^{(k_{0})}}\sum_{i\in\mathcal{S}^{k_{0}}\left( \mathcal{G}_{\mathcal{E} }\right) }\mathcal{M}_{Y,i}\left( \tau_{Y},t;\partial_{i}\right)$$ As the term in curly braces is the sum of centered random variables with bounded variance and conditionally independent given $\left\{ \partial
_{i}:i\in\mathcal{S}^{k_{0}}\left( \mathcal{G}_{\mathcal{E}}\right)
\right\} ,$ $$\lim_{N\rightarrow\infty}\frac{1}{n^{(k_{0})}}\sum_{i\in\mathcal{S}^{k_{0} }\left( \mathcal{G}_{\mathcal{E}}\right) }\left\{ \log\frac{f\left(
Y_{i}|\partial_{i};t\right) }{f\left( Y_{i}|\partial_{i};\tau_{Y}\right)
}+\mathcal{M}_{Y,i}\left( \tau_{Y},t;\partial_{i}\right) \right\} =0\text{
}a.s.$$ Then we deduce the following sequence of inequalities almost surely: $$\begin{aligned}
\lim\inf_{\mathcal{E}}Z_{\mathcal{E}} & =\lim\inf_{\mathcal{E}}\left\{
\frac{1}{n^{(k_{0})}}\sum_{i\in\mathcal{S}^{k_{0}}\left( \mathcal{G} _{\mathcal{E}}\right) }\mathcal{M}_{Y,i}\left( \tau_{Y},t;\partial
_{i}\right) \right\} \\
& \geq\lim\inf_{\mathcal{E}}\int_{\mathcal{D}}\mathcal{M}_{Y}\left( \tau
_{Y},t;z\right) F_{n}\left( \mathcal{S}^{k_{0}}\left( \mathcal{G} _{\mathcal{E}}\right) ,dz\right)
\end{aligned}$$ where $$F_{n}\left( \mathcal{S}^{k_{0}}\left( \mathcal{G}_{\mathcal{E}}\right)
,dz\right) =\frac{1}{n^{(k_{0})}}\sum_{i\in\mathcal{S}^{k_{0}}\left(
\mathcal{G}_{\mathcal{E}}\right) }1\left( \partial i\in dz\right)$$ By assumptions 1-3 and Lemma 5.2.2 of Guyon (1995), then there is a positive constant $c^{\ast}$ such that $$\lim\inf_{\mathcal{E}}F_{n}\left( \mathcal{S}^{k_{0}}\left( \mathcal{G} _{\mathcal{E}}\right) ,dz\right) \geq c^{\ast}\lambda\left( dz\right)$$ therefore $$\begin{aligned}
\lim\inf_{\mathcal{E}}Z_{\mathcal{E}} & \geq c^{\ast}\int_{\mathcal{D} }\mathcal{M}_{Y}\left( \tau_{Y},t;z\right) \lambda\left( dz\right) \\
& \equiv c^{\ast}N\left( \tau_{Y},t\right)
\end{aligned}$$ Then note that $$\begin{aligned}
U_{\mathcal{E}}\left( t\right) & =-\frac{1}{n}\sum_{i\in\mathcal{S}\left(
\mathcal{G}_{\mathcal{E}}\right) }\log f\left( Y_{i}|\partial_{i};t\right)
\\
& =\sum_{k=1}^{K}\frac{n^{(k)}}{n}U_{\mathcal{E}}^{k}\left( t\right)
\end{aligned}$$ where $$U_{\mathcal{E}}^{k}\left( t\right) =-\frac{1}{n^{(k)}}\sum_{i\in
\mathcal{S}^{k}\left( \mathcal{G}_{\mathcal{E}}\right) }\log f\left(
Y_{i}|\partial_{i};t\right)$$ Consistency of the coding maximum likelihood estimator then follows by assumption 1-4 and Corollary 3.4.1 of Guyon (1995). Consistency of the pseudo maximum likelihood estimator follows from $$\begin{aligned}
U_{\mathcal{E}}^{P}\left( t\right) & \equiv-\frac{1}{N}\sum_{i}\log
f\left( Y_{i}|\partial_{i};t\right) \\
& =\frac{n}{N}U_{\mathcal{E}}\left( t\right) +\frac{\left( N-n\right)
}{N}\overline{U}_{\mathcal{E}}\left( t\right)
\end{aligned}$$ $U_{\mathcal{E}}^{P}\left( t\right) \equiv\left( N-n\right) ^{-1} \sum_{i\not \in \mathcal{S}\left( \mathcal{G}_{\mathcal{E}}\right) }\log
f\left( Y_{i}|\partial_{i};t\right) $ and a further application of Corollary 3.4.1 of Guyon (1995).
The next result establishes asymptotic normality of $\hat{\tau}_{Y}.$
Under Assumptions 1-6, as $N\rightarrow\infty$*,* $$\begin{aligned}
& \sqrt{n}\Omega_{n}^{1/2}\left( \widehat{\tau}_{Y}-\tau_{Y}\right)
\underset{N\longrightarrow\infty}{\longrightarrow}N\left( 0,I\right) ;\\
\Omega_{n} & =\frac{1}{n}\sum_{i\in\mathcal{S}\left( \mathcal{G} _{\mathcal{E}}\right) }\left\{ \frac{\partial\log\mathcal{CL}_{Y,\mathcal{S} \left( \mathcal{G}_{\mathcal{E}}\right) ,i}\left( \tau_{Y}\right)
}{\partial\tau_{Y}}\right\} ^{\otimes2}.
\end{aligned}$$
$\hat{\tau}_{Y}$ solves $V_{n}\left( \hat{\tau}_{Y}\right) =\frac
{\partial\log\mathcal{CL}_{Y}\left( \tau_{Y}\right) }{\partial\tau_{Y} }|_{\hat{\tau}_{Y}}=0,$ therefore $$\begin{aligned}
0 & =\sqrt{n}V_{n}\left( \tau_{Y}\right) +\dot{V}_{n}\left( \tau_{Y} ,\hat{\tau}_{Y}\right) \sqrt{n}\left( \hat{\tau}_{Y}-\tau_{Y}\right) \\
V_{n}\left( \tau_{Y}\right) & =\frac{1}{\sqrt{n}}\sum_{i\in\mathcal{S} \left( \mathcal{G}_{\mathcal{E}}\right) }\mathcal{D}_{i}=\frac{1}{\sqrt{n} }\sum_{i\in\mathcal{S}\left( \mathcal{G}_{\mathcal{E}}\right) }\left\{
\frac{\partial\log\mathcal{CL}_{Y,\mathcal{S}\left( \mathcal{G}_{\mathcal{E} }\right) ,i}\left( \tau_{Y}\right) }{\partial\tau_{Y}}\right\} \\
\dot{V}_{n}\left( \tau_{Y},\hat{\tau}_{Y}\right) & =\int_{0}^{1} \frac{\partial^{2}\log\mathcal{CL}_{Y}\left( \tau_{Y}\right) }{\partial
\tau_{Y}\partial\tau_{Y}^{T}}|_{t\left( \hat{\tau}_{Y}-\tau_{Y}\right)
+\tau_{Y}}dt
\end{aligned}$$ As the variables $\mathcal{D}_{i}$ are centered, bounded and independent conditionally on $\left\{ \partial_{i}:i\in\mathcal{S}^{k_{0}}\left(
\mathcal{G}_{\mathcal{E}}\right) \right\} ,$ one may apply a central limit theorem for non-iid bounded variable (Breiman, 1992). Under assumptions 5 and 6, it follows that $\dot{V}_{n}\left( \tau_{Y},\hat{\tau}_{Y}\right)
+\Omega_{n}\left( \tau_{Y}\right) \rightarrow^{P}0_{p\times p},$ proving the result.
Proofs of consistency of coding and pseudo maximum likelihood estimators of $\tau_{L}$ , as well as asymptotic normality of coding estimator of $\tau_{L}$ follow along the same lines as above, upon substituting $\partial_{i}=\left\{
L_{j}:j\in\mathcal{N}_{i}\right\} ,$ and replacing Assumption 2 with the assumption that $\mathcal{S}\left( \mathcal{G}_{\mathcal{E}}\right) $ can be partitioned into $K$ 2-stable subsets $\left\{ \mathcal{S}^{k}\left(
\mathcal{G}_{\mathcal{E}}\right) :k=1,...,K\right\} $ such that $\mathcal{S}\left( \mathcal{G}_{\mathcal{E}}\right) ={\displaystyle\bigcup\limits_{k=1}^{K}}
\mathcal{S}^{k}\left( \mathcal{G}_{\mathcal{E}}\right) ,$ and that there is a $k_{0}$ such that Assumptions 2.a. and 2.b. are satisfied.
Additional simulation results: varying sample size {#additional-simulation-results-varying-sample-size .unnumbered}
==================================================
{width="\linewidth"}
{width="\linewidth"}
\[-1.8ex\] Truth Absolute Bias MC Variance Robust Variance 95% CI Coverage
-------------------- -------- --------------- ------------- ----------------- -----------------
$E(Y(\mathbf{a}))$ 0.211 $0.005$ $0.005$ $0.0004$ 0.935
Spillover -0.166 $0.002$ 0.021 0.015 0.922
Direct -0.179 $0.006$ $0.008$ $0.008$ 0.944
$E(Y(\mathbf{a}))$ 0.211 $0.005$ $0.002$ - -
Spillover -0.166 $0.002$ 0.010 - -
Direct -0.179 $0.006$ $0.004$ - -
: Simulation results of coding and pseudo-likelihood based estimators of network causal effects for low density network of size 200[]{data-label=""}
\[-1.8ex\] Truth Absolute Bias MC Variance Robust Variance 95% CI Coverage
-------------------- -------- --------------- ------------- ----------------- -----------------
$E(Y(\mathbf{a}))$ 0.211 $0.003$ $0.002$ 0.002 0.936
Spillover -0.166 $0.006$ $0.009$ 0.008 0.932
Direct -0.179 $-0.002$ 0.004 0.004 0.961
$E(Y(\mathbf{a}))$ 0.211 $0.003$ 0.001 - -
Spillover -0.166 $0.006$ 0.004 - -
Direct -0.179 $-0.002$ 0.002 - -
: Simulation results of coding and pseudo-likelihood based estimators of network causal effects for low density network of size 400[]{data-label=""}
\[-1.8ex\] Truth Bias MC Variance Robust Variance 95% CI Coverage
-------------------- -------- --------- ------------- ----------------- -----------------
$E(Y(\mathbf{a}))$ 0.211 $0.001$ $0.001$ $0.001$ 0.941
Spillover -0.167 $0.001$ 0.004 0.005 0.935
Direct -0.179 $0.001$ $0.002$ $0.002$ 0.943
$E(Y(\mathbf{a}))$ 0.211 $0.001$ $<0.001$ - -
Spillover -0.148 $0.001$ 0.002 - -
Direct -0.176 $0.001$ $0.001$ - -
: Simulation results of coding and pseudo-likelihood based estimators of network causal effects for low density network of size 1,000[]{data-label=""}
Additional simulation results: small, dense networks {#additional-simulation-results-small-dense-networks .unnumbered}
====================================================
\
Additional simulation results: missing edges {#additional-simulation-results-missing-edges .unnumbered}
============================================
\
Data application: auto-model parameters {#data-application-auto-model-parameters .unnumbered}
=======================================
\
Data application: alternate model specifications {#data-application-alternate-model-specifications .unnumbered}
================================================
We propose the two alternate specifications of the outcome auto-model that incorporate neighbor terms, $W_i$, excluding covariate terms ($L_i$, $\sum_j L_j$). Note that we accounted for covariates in estimation. The auto-model parameter estimates for each option are given in Table 4. The network causal effect estimates are given in Table 5.
$$\begin{aligned}
\textrm{logit}( Pr[Y_i = 1 | W_i, A_i, A_j, Y_i, Y_j ]) & = \beta_0 + \beta_1 A_i + \beta_2 \sum_j A_j/W_i + \beta_3 \sum_j Y_j/W_i + \beta_4 W_i \tag{A} \\
\textrm{logit}( Pr[Y_i = 1 | W_i, A_i, A_j, Y_i, Y_j ]) & = \beta_0 + \beta_1 A_i + \beta_2 \sum_j A_j + \beta_3 \sum_j Y_j + \beta_4 W_i \tag{B} \end{aligned}$$
\[c\][lcc]{} & &\
& Estimates & 95% CI\
& &\
Past incarceration status (individual) & 2.05 & \[1.03, 4.11\]\
Past incarceration status (proportion neighbors) & 1.15 & \[0.49, 2.69\]\
HIV/STI/HCV status (proportion neighbors) & 2.78 & \[1.19, 6.50\]\
Number of neighbors & 1.28 & \[0.84, 1.93\]\
& &\
& &\
Past incarceration status (individual) & 2.09 & \[1.04, 4.21\]\
Past incarceration status (sum neighbors) & 1.30 & \[0.69, 2.44\]\
HIV/STI/HCV status (sum neighbors) & 2.33 & \[1.16, 4.70\]\
Number of neighbors & 0.62 & \[0.30, 1.31\]\
\[c\][lcc]{} & &\
& Estimates & 95% CI\
& &\
Spillover & 0.028 & \[-0.053, 0.122\]\
Direct & 0.131 & \[0.001, 0.245\]\
& &\
& &\
Spillover & 0.121 & \[-0.063, 0.243\]\
Direct & 0.150 & \[0.016, 0.279\]\
Data application: simulation under the sharp null {#data-application-simulation-under-the-sharp-null .unnumbered}
=================================================
We conducted a simulation study to evaluate the operating characteristics of our proposed estimator under the sharp null in the NNAHRAY network. The network structure for the simulation study was based on the NNAHRAY observed network structure excluding singletons (i.e. persons with no ties), $N=412$ ($n_1 = 229$). The true parameter values for $\boldsymbol{\alpha}$ and $\boldsymbol{\beta}$ are given in Table 6. We generated 500 realizations of ($\mathbf{Y},\mathbf{A},\mathbf{L}$) by sampling from the Gibbs factors at their true parameter values.
--------------------- -- ------------------------- --------------------- ------------------------ ---------------------
$\boldsymbol{\tau}$ $\theta_0$ -1.5 $\beta_0$ -1.1
$\boldsymbol{\rho}$ $\boldsymbol{\theta_1}$ (-0.3,-0.6,1.1,1.1) $\beta_1$ 0
$\boldsymbol{\nu}$ $\theta_2$ 0 $\boldsymbol{\beta}_2$ (-0.3,0.0,0.8,-0.3)
$\boldsymbol{\theta_3}$ (-0.3,0.2,0.9,-0.9) $\beta_3$ 0
$\beta_4$ 0
$\boldsymbol{\beta}_5$ (-0.5,0.0,0.5,0.7)
--------------------- -- ------------------------- --------------------- ------------------------ ---------------------
: NNAHRAY simulation parameter values
We then calculated the “true" value of the direct and spillover effect using the same method described in the Simulation section 5. For all effects, we considered treatment assignment to be a Bernoulli random variable with probability $\gamma = 0.5$. The spillover effects compared network incarceration allocations of 50% to 0% and the direct effect was estimated at an incarceration allocation of 50%. Under the sharp null, the true network direct and spillover effects were 0. The results are as expected and given in Table 7.
\[-1.8ex\] Truth Mean estimate MC Variance 95% CI Coverage
------------ ------- --------------- ------------- ----------------- --
Spillover 0 $-0.0003$ 0.005 0.952
Direct 0 $-0.0005$ 0.002 0.955
: Simulation results of coding estimators of network causal effects in NNAHRAY network under sharp null[]{data-label=""}
|
---
abstract: 'In this paper, we consider optimal low-rank regularized inverse matrix approximations and their applications to inverse problems. We give an explicit solution to a generalized rank-constrained regularized inverse approximation problem, where the key novelties are that we allow for updates to existing approximations and we can incorporate additional probability distribution information. Since computing optimal regularized inverse matrices under rank constraints can be challenging, especially for problems where matrices are large and sparse or are only accessable via function call, we propose an efficient rank-update approach that decomposes the problem into a sequence of smaller rank problems. Using examples from image deblurring, we demonstrate that more accurate solutions to inverse problems can be achieved by using rank-updates to existing regularized inverse approximations. Furthermore, we show the potential benefits of using optimal regularized inverse matrix updates for solving perturbed tomographic reconstruction problems.'
author:
- 'Julianne Chung [^1]'
- 'Matthias Chung [^2]'
bibliography:
- 'orim.bib'
title: |
[[Optimal regularized inverse matrices\
for\
inverse problems]{}]{}
---
[*Keywords*]{}: ill-posed inverse problems, low-rank matrix approximation, regularization, Bayes risk
[*AMS*]{}: 65F22, 15A09, 15A29
Introduction {#sec:introduction}
============
Optimal low-rank inverse approximations play a critical role in many scientific applications such as matrix completion, machine learning, and data analysis [@Ye2005; @Drineas2007; @Markovsky2012]. Recent theoretical and computational developments on *regularized* low-rank inverse matrices have enabled new applications, such as for solving inverse problems [@Chung2015]. In this paper, we develop theoretical results for a general case for finding *optimal regularized inverse matrices* (ORIMs), and we propose novel uses of these matrices for solving linear ill-posed inverse problems of the form, $$\label{eqn:linearsystem}
{{\bf b}}= {{\bf A}}{{\boldsymbol{\xi}}}+{{\boldsymbol{\delta}}},$$ where ${{\boldsymbol{\xi}}}\in {\mathbb{R}}^n$ is the desired solution, ${{\bf A}}\in {\mathbb{R}}^{m \times n}$ models the forward process, ${{\boldsymbol{\delta}}}\in {\mathbb{R}}^m$ is additive noise, and ${{\bf b}}\in {\mathbb{R}}^m$ is the observed data. We assume that ${{\bf A}}$ is very large and sparse, or that ${{\bf A}}$ cannot be formed explicitly, but matrix vector multiplications with ${{\bf A}}$ are feasible (e.g., ${{\bf A}}$ can be an object or function handle). Furthermore, we are interested in ill-posed inverse problems, whereby small errors in the data may result in large errors in the solution [@Hadamard1923; @Hansen2010; @Vogel1987], and regularization is needed to stabilize the solution.
Next, we provide a brief introduction to regularization and ORIMs, followed by a summary of the main contributions of this work. Various forms of regularization have been proposed in the literature, including variational methods [@RuOsFa92; @tikhonov1977solutions] and iterative regularization, where early termination of an iterative methods provides a regularized solution [@HaHa93; @Hank95a]. Optimal regularized inverse matrices have been proposed for solving inverse problems and have been studied in both the Bayes and empirical Bayes framework [@Chung2011; @Chung2013a; @Chung2015]. Let ${{\bf P}}\in {\mathbb{R}}^{n \times m}$ be an initial approximation matrix (e.g., ${{\bf P}}={{\bf0}}_{n \times m}$ in previous works). Then treating ${{\boldsymbol{\xi}}}$ and ${{\boldsymbol{\delta}}}$ as random variables, the goal is to find a matrix $\widehat{{\bf Z}}\in {\mathbb{R}}^{n \times m}$ that gives a small reconstruction error. That is, $\rho(({{\bf P}}+\widehat{{\bf Z}}) {{\bf b}}- {{\boldsymbol{\xi}}})$ should be small for some given error measure $\rho:{\mathbb{R}}^n \to {\mathbb{R}}^+_0$. In this paper, we consider $\rho$ to be the squared Euclidean norm, and we seek an *optimal* matrix $\widehat{{\bf Z}}$ that minimizes the expected value of the errors with respect to the joint distribution of ${{\boldsymbol{\xi}}}$ and ${{\boldsymbol{\delta}}}$. Hence, the problem of finding an ORIM $\widehat{{\bf Z}}$ can be formulated as $$\label{eqn:Bayesmin}
\widehat{{\bf Z}}= \operatorname*{arg\,min}_{{{\bf Z}}} \ {\mathbb{E}}\, {\left\|(({{\bf P}}+ {{\bf Z}}) {{\bf A}}- {{\bf I}}_n) {{\boldsymbol{\xi}}}+ {{\bf Z}}{{\boldsymbol{\delta}}}\right\|_{2}}^2\,.$$ This problem is often referred to as a *Bayes risk minimization problem* [@Carlin2000; @Vapnik1998]. Especially for large scale problems, it may be advisable to include further constraints on ${{\bf Z}}$ such as sparsity, symmetry, block or cyclic structure, or low-rank structure. Here, we will focus on matrices ${{\bf Z}}$ of low-rank. Once computed, ORIM $\widehat{{\bf Z}}$ has mainly been used to efficiently solve linear inverse problems in an online phase as data ${{\bf b}}$ becomes available and requires therefore only a matrix-vector multiplication $({{\bf P}}+\widehat{{\bf Z}}) {{\bf b}}$.
#### Overview of our contributions
First, we derive a closed-form solution for problem under rank constraints with uniqueness conditions. The two key novelties are that we include matrix ${{\bf P}}$, thereby allowing for updates to existing regularized inverse matrices, and we incorporate additional information regarding the distribution of ${{\boldsymbol{\xi}}}$. More specifically, we allow non-zero mean for the distribution of ${{\boldsymbol{\xi}}}$ and show that our results reduce to previous results in [@Chung2015] that assume zero mean and ${{\bf P}}={{\bf0}}_{n \times m}$. These extension are not trivial and require a different approach than [@Chung2015] for the proof. Second, we describe an efficient rank-update approach for computing a global minimizer of under rank constraints, that is related to but different than the approach described in [@chung2014efficient] where training data was used as a substitute for knowledge of the forward model. We demonstrate the efficiency and accuracy of the rank-update approach, compared to standard SVD-based methods, for solving a sequence of ill-posed problems.
Third, we propose novel uses of ORIM updates in the context of solving inverse problems. An example from image deblurring demonstrates that updates to existing regularized inverse matrix approximations such as the Tikhonov reconstruction matrix can lead to more accurate solutions. Also, we use an example from tomographic image reconstruction to show that ORIM updates can be used to efficiently and accurately solve perturbed inverse problems. This contribution has significant implications for further research development, ranging from use within nonlinear optimization schemes to preconditioner updates.
The key benefits of using ORIMs for solution updates and for solving inverse problems are that (1) we approximate the regularized inverse directly, so reconstruction or application requires only a matrix-vector multiplication rather than a linear solve; (2) our matrix inherently incorporates regularization; (3) ORIMs and ORIM updates can be computed for any general rectangular matrix ${{\bf A}}$, even if ${{\bf A}}$ is only available via a function call, making it ideal for large-scale problems.
The paper is organized as follows. In Section \[sec:background\], we provide preliminaries to establish notation and summarize important results from the literature. Then, in Section \[sec:proof\_general\_case\], we derive a closed form solution to problem under rank constraints and provide uniqueness conditions (see Theorem \[thm:mainresult\] for the main result). For large-scale problems, computing an ORIM according to Theorem \[thm:mainresult\] may be computationally prohibitive, so in Section \[sub:computational\_methods\_for\_obtaining\_bfz\], we describe a rank-update approach for efficient computation. Finally, in Section \[sec:numerics\] we provide numerical examples from image processing that demonstrate the benefits of ORIM updates. Conclusions and discussions are provided in Section \[sec:conclusions\].
Background {#sec:background}
==========
In this section, we begin with preliminaries to establish notation.
Given a matrix ${{\bf A}}\in{\mathbb{R}}^{m\times n}$ with rank $k \leq \min(m,n)$, let ${{\bf A}}= {{\bf U}}_{{\bf A}}{{\boldsymbol{\Sigma}}}_{{\bf A}}{{\bf V}}_{{\bf A}}{^{\top}}$ denote the singular value decomposition (SVD) of ${{\bf A}}$, where ${{\bf U}}_{{\bf A}}= [{{\bf u}}_1,\ldots,{{\bf u}}_m] \in {\mathbb{R}}^{m \times m}$ and ${{\bf V}}_{{\bf A}}= [{{\bf v}}_1,\ldots,{{\bf v}}_n] \in {\mathbb{R}}^{n \times n}$ are orthogonal matrices that contain the left and right singular vectors of ${{\bf A}}$, respectively. Diagonal matrix ${{\boldsymbol{\Sigma}}}_{{\bf A}}= {{\rm diag\!}\left( \sigma_1({{\bf A}}),\ldots,\sigma_k({{\bf A}}), 0,\ldots,0 \right)}\in {\mathbb{R}}^{m \times n}$ contains the singular values $\sigma_1({{\bf A}})\geq \cdots \geq \sigma_k({{\bf A}}) > 0$ and zeros on its main diagonal. The truncated SVD approximation of rank $r\leq k$ of ${{\bf A}}$ is denoted by ${{\bf A}}_r = {{\bf U}}_{{{\bf A}},r} {{\boldsymbol{\Sigma}}}_{{{\bf A}}, r} {{\bf V}}_{{{\bf A}},r}{^{\top}}\in {\mathbb{R}}^{m\times n}$ where ${{\bf U}}_{{{\bf A}},r}$ and ${{\bf V}}_{{{\bf A}},r}$ contain the first $r$ vectors of ${{\bf U}}_{{{\bf A}}}$ and ${{\bf V}}_{{{\bf A}}}$ respectively, and ${{\boldsymbol{\Sigma}}}_{{{\bf A}},r}$ is the principal $r \times r$ submatrix of ${{\boldsymbol{\Sigma}}}_{{\bf A}}$. The TSVD approximation is unique if and only if $\sigma_r({{\bf A}})>\sigma_{r+1}({{\bf A}})$. Furthermore, the Moore-Penrose pseudoinverse of ${{\bf A}}$ is given by ${{\bf A}}^\dagger = {{\bf V}}_{{{\bf A}},k} {{\boldsymbol{\Sigma}}}_{{{\bf A}},k}^{-1} {{\bf U}}_{{{\bf A}},k}{^{\top}}$. Next we show that the problem of finding an *optimal regularized inverse matrix* (ORIM) (i.e., a solution to ) is equivalent to solving a matrix approximation problem. That is, assuming ${{\boldsymbol{\xi}}}$ and ${{\boldsymbol{\delta}}}$ are random variables, the goal is to find a matrix ${{\bf Z}}$ such that we minimize the expected value of the squared 2-norm error, i.e., $\min_{{\bf Z}}f({{\bf Z}})$, where $$f({{\bf Z}}) = {\mathbb{E}}\, {\left\|({{\bf P}}+{{\bf Z}}){{\bf b}}-{{\boldsymbol{\xi}}}\right\|_{2}}^2 = {\mathbb{E}}\, {\left\|({{\bf P}}+{{\bf Z}})({{\bf A}}{{\boldsymbol{\xi}}}+{{\boldsymbol{\delta}}}) -{{\boldsymbol{\xi}}}\right\|_{2}}^2$$ is often referred to as the *Bayes risk*.
Lets further assume that ${{\boldsymbol{\xi}}}$ and ${{\boldsymbol{\delta}}}$ are independent random variables with ${\mathbb{E}}[{{\boldsymbol{\xi}}}] = {{\boldsymbol{\mu}}}_{{\boldsymbol{\xi}}}$, the covariance matrix ${{\rm Cov}\!\left[ {{\boldsymbol{\xi}}}\right]} = {{\boldsymbol{\Gamma}}}_{{\boldsymbol{\xi}}}$ is symmetric positive definite, ${\mathbb{E}}[{{\boldsymbol{\delta}}}] = {{\bf0}}_{m\times 1}$, and ${{\rm Cov}\!\left[ {{\boldsymbol{\delta}}}\right]} = \eta^2{{\bf I}}_m$. First, due to the independence of ${{\boldsymbol{\xi}}}$ and ${{\boldsymbol{\delta}}}$ and since ${\mathbb{E}}[{{\boldsymbol{\delta}}}] = {{\bf0}}_{m\times 1}$, we can rewrite the Bayes risk as $$f({{\bf Z}}) = {\mathbb{E}}\, \left[{\left\|(({{\bf P}}+{{\bf Z}}){{\bf A}}-{{\bf I}}_n){{\boldsymbol{\xi}}}\right\|_{2}}^2\right] + {\mathbb{E}}\, \left[{\left\|({{\bf P}}+{{\bf Z}}){{\boldsymbol{\delta}}}\right\|_{2}}^2\right].$$ Then using the property of the quadratic form [@Seber2012], ${\mathbb{E}}\left[{{\boldsymbol{\epsilon}}}{^{\top}}{{\boldsymbol{\Lambda}}}{{\boldsymbol{\epsilon}}}\right] = {{\rm tr\!}\left( {{\boldsymbol{\Lambda}}}{{\boldsymbol{\Sigma}}}_{{{\boldsymbol{\epsilon}}}} \right)} + {{\boldsymbol{\mu}}}_{{{\boldsymbol{\epsilon}}}}{^{\top}}{{\boldsymbol{\Lambda}}}{{\boldsymbol{\mu}}}_{{{\boldsymbol{\epsilon}}}}$, where $\rm tr(\cdot)$ denotes the trace, ${{\boldsymbol{\Lambda}}}$ is symmetric, ${\mathbb{E}}[{{\boldsymbol{\epsilon}}}] = {{\boldsymbol{\mu}}}_{{\boldsymbol{\epsilon}}}$ and ${{\rm Cov}\!\left[ {{\boldsymbol{\epsilon}}}\right]} = {{\boldsymbol{\Sigma}}}_{{\boldsymbol{\epsilon}}}$, $$\begin{aligned}
f({{\bf Z}}) =& \, {{\boldsymbol{\mu}}}_{{\boldsymbol{\xi}}}{^{\top}}(({{\bf P}}+{{\bf Z}}){{\bf A}}-{{\bf I}}_n){^{\top}}(({{\bf P}}+{{\bf Z}}){{\bf A}}-{{\bf I}}_n){{\boldsymbol{\mu}}}_{{\boldsymbol{\xi}}}\\
&+ {{\rm tr\!}\left( (({{\bf P}}+{{\bf Z}}){{\bf A}}-{{\bf I}}_n){^{\top}}(({{\bf P}}+{{\bf Z}}){{\bf A}}-{{\bf I}}_n){{\bf M}}_{{\boldsymbol{\xi}}}{{\bf M}}_{{\boldsymbol{\xi}}}{^{\top}}\right)} + \eta^2\, {{\rm tr\!}\left( ({{\bf P}}+{{\bf Z}}){^{\top}}({{\bf P}}+{{\bf Z}}) \right)}\end{aligned}$$ with ${{\bf M}}_{{\boldsymbol{\xi}}}{{\bf M}}_{{\boldsymbol{\xi}}}{^{\top}}= {{\boldsymbol{\Gamma}}}_{{\boldsymbol{\xi}}}$ being any symmetric factorization, e.g., Cholesky factorization. Using the cyclic property of the trace leads to $$f({{\bf Z}}) = {\left\|(({{\bf P}}+{{\bf Z}}){{\bf A}}-{{\bf I}}_n){{\boldsymbol{\mu}}}_{{\boldsymbol{\xi}}}\right\|_{2}}^2 + {\left\|(({{\bf P}}+{{\bf Z}}){{\bf A}}-{{\bf I}}_n){{\bf M}}_{{\boldsymbol{\xi}}}\right\|_{{\rm F}}}^2 + \eta^2{\left\|({{\bf P}}+{{\bf Z}})\right\|_{{\rm F}}}^2,$$ where ${\left\|\,\cdot\,\right\|_{{\rm F}}}$ denotes the Frobenius norm. Next we rewrite $f({{\bf Z}})$ in terms of only one Frobenius norm. Let ${{\bf M}}= \begin{bmatrix} {{\bf M}}_{{\boldsymbol{\xi}}}& {{\boldsymbol{\mu}}}_{{\boldsymbol{\xi}}}\end{bmatrix} \in {\mathbb{R}}^{n \times (n+1)}$, then using the identities of the Frobenius and the vector 2-norm, as well as applying Kronecker product properties, we get $$\label{eq:reformulatedFcn}
f({{\bf Z}}) = {\left\| {{\bf Z}}\begin{bmatrix}
{{\bf A}}{{\bf M}}& \eta {{\bf I}}_m \end{bmatrix} - \begin{bmatrix}
{{\bf M}}- {{\bf P}}{{\bf A}}{{\bf M}}& -\eta {{\bf P}}\end{bmatrix}\right\|_{{\rm F}}}^2.$$
Thus, minimizing the Bayes risk in problem is equivalent to minimizing . Notice that so far we have not imposed any constraints on ${{\bf Z}}$. Although various constraints can be imposed on ${{\bf Z}}$, here we consider ${{\bf Z}}$ to be of low-rank, i.e., ${{\rm rank}\left( {{\bf Z}}\right)}\leq r$ for some $r \leq {{\rm rank}\left( {{\bf A}}\right)}$. Hence the low-rank matrix approximation problem of interest in this paper is $$\label{eqn:lrproblem}
\min_{{{\rm rank}\left( {{\bf Z}}\right)} \leq r} \,\,f({{\bf Z}}) = {\left\| {{\bf Z}}\begin{bmatrix}
{{\bf A}}{{\bf M}}& \eta {{\bf I}}_m \end{bmatrix} - \begin{bmatrix}
{{\bf M}}- {{\bf P}}{{\bf A}}{{\bf M}}& -\eta {{\bf P}}\end{bmatrix}\right\|_{{\rm F}}}^2.$$ We will provide a closed form solution for in Section \[sec:proof\_general\_case\], but it is important to remark that special cases of this problem have been previously studied in the literature. For example, a solution for the case where ${{\bf P}}={{\bf0}}_{n \times m}$ and ${{\boldsymbol{\mu}}}_{{\boldsymbol{\xi}}}= {{\bf0}}_{n\times 1}$ was provided in [@Chung2015] that uses the generalized SVD of $\left\{{{\bf A}}, {{\bf M}}_{{\boldsymbol{\xi}}}^{-1}\right\}$. If, in addition, we assume ${{\bf M}}_{{\boldsymbol{\xi}}}= {{\bf I}}_n,$ then an optimal regularized inverse matrix of at most rank $r$ reduces to a truncated-Tikhonov matrix [@Chung2015], $$\label{eqn:TTik}
\widehat {{\bf Z}}= {{\bf V}}_{{{\bf A}},r} {{\boldsymbol{\Psi}}}_{{{\bf A}},r} {{\bf U}}_{{{\bf A}},r}{^{\top}},$$ where ${{\boldsymbol{\Psi}}}_{{{\bf A}},r} = {{\rm diag\!}\left( \frac{\sigma_1({{\bf A}})}{\sigma_1^2({{\bf A}})+ \eta^2}, \ldots, \frac{\sigma_r({{\bf A}})}{\sigma_r^2 ({{\bf A}})+ \eta^2} \right)}$. Moreover, this $\widehat {{\bf Z}}$ is the [*unique*]{} global minimizer for $$\label{eq:objFcn}
\min_{{{\rm rank}\left( {{\bf Z}}\right)} \leq r} \ {\left\|{{\bf Z}}{{\bf A}}- {{\bf I}}_n\right\|_{{\rm F}}}^2 + \eta^2 {\left\|{{\bf Z}}\right\|_{{\rm F}}}^2,$$ if and only if $\sigma_r({{\bf A}}) > \sigma_{r+1}({{\bf A}})$.
Low-rank optimization problem {#sec:proof_general_case}
=============================
The goal of this section is to derive the unique global minimizer for problem , under suitable conditions. We actually consider a more general problem, as stated in Theorem \[thm:mainresult\], where ${{\bf M}}\in {\mathbb{R}}^{n \times p}$ with ${{\rm rank}\left( {{\bf M}}\right)} = n \leq p.$ Our proof uses a special case of Theorem 2.1 from Friedland & Torokhti [@Friedland2007] that is provided here for completeness.
\[thm:Friedland\] Let matrices ${{\bf B}}\in {\mathbb{R}}^{m\times n}$ and ${{\bf C}}\in {\mathbb{R}}^{q \times n}$ with $k = {{\rm rank}\left( {{\bf C}}\right)}$ be given. Then $$\widehat {{\bf Z}}= \left({{\bf B}}{{\bf V}}_{{{\bf C}},k}{{\bf V}}_{{{\bf C}},k}{^{\top}}\right)_r {{\bf C}}^\dagger$$ is a solution to the minimization problem $$\min_{{{\rm rank}\left( {{\bf Z}}\right)} \leq r} {\left\|{{\bf Z}}{{\bf C}}- {{\bf B}}\right\|_{{{\rm F}}}}^2,$$ having a minimal ${\left\|{{\bf Z}}\right\|_{{{\rm F}}}}$. This solution is unique if and only if either $$r \geq {{\rm rank}\left( {{\bf B}}{{\bf V}}_{{{\bf C}},k}{{\bf V}}_{{{\bf C}},k}{^{\top}}\right)}$$ or $$1 \leq r < {{\rm rank}\left( {{\bf B}}{{\bf V}}_{{{\bf C}},k}{{\bf V}}_{{{\bf C}},k}{^{\top}}\right)} \quad \mbox{and} \quad \sigma_r({{\bf B}}{{\bf V}}_{{{\bf C}},k}{{\bf V}}_{{{\bf C}},k}{^{\top}}) > \sigma_{r+1}({{\bf B}}{{\bf V}}_{{{\bf C}},k}{{\bf V}}_{{{\bf C}},k}{^{\top}}).$$
See [@Friedland2007].
To get to our main result we first provide the following Lemma.
\[lem:svdExtended\] Let ${{\bf B}}= [{{\bf A}}\ \ \eta \, {{\bf I}}_m]$ with ${{\bf A}}\in {\mathbb{R}}^{m\times n}$ and parameter $\eta \geq 0$, nonzero if ${{\rm rank}\left( {{\bf A}}\right)} < \max\{m,n\}$. Let further ${{\bf D}}_{{\bf A}}\in{\mathbb{R}}^{m \times m}$ with ${{\bf D}}_{{\bf A}}= {{\rm diag\!}\left( \sqrt{\sigma_1^2({{\bf A}})+\eta^2},\ldots, \sqrt{\sigma_n^2({{\bf A}})+\eta^2}, \eta, \ldots, \eta \right)}$ for $m\geq n$ and ${{\bf D}}_{{\bf A}}= {{\rm diag\!}\left( \sqrt{\sigma_1^2({{\bf A}})+\eta^2},\ldots, \sqrt{\sigma_m^2({{\bf A}})+\eta^2} \right)}$ for $m < n$. Then the SVD of ${{\bf B}}$ is given by ${{\bf B}}= {{\bf U}}_{{\bf B}}{{\boldsymbol{\Sigma}}}_{{\bf B}}{{\bf V}}_{{\bf B}}{^{\top}}$, where $${{\bf U}}_{{\bf B}}= {{\bf U}}_{{\bf A}}, \quad {{\boldsymbol{\Sigma}}}_{{\bf B}}= \left[ {{\bf D}}_{{\bf A}}\ \ {{\bf0}}_{m \times n} \right] \quad \mbox{and} \quad {{\bf V}}_{{\bf B}}= \begin{bmatrix}
{{\bf V}}_{{\bf A}}{{\boldsymbol{\Sigma}}}_{{\bf A}}{^{\top}}{{\bf D}}_{{\bf A}}^{-1} & {{\bf V}}_{12} \\
\eta \, {{\bf U}}_{{\bf A}}{{\bf D}}_{{\bf A}}^{-1} & {{\bf V}}_{22}
\end{bmatrix},$$ with arbitrary ${{\bf V}}_{12}$ and ${{\bf V}}_{22}$ satisfying ${{\bf V}}_{12}{^{\top}}{{\bf V}}_{12}+{{\bf V}}_{22}{^{\top}}{{\bf V}}_{22} = {{\bf I}}_n$ and ${{\bf A}}{{\bf V}}_{12}+\eta{{\bf V}}_{22} = {{\bf0}}_{m\times n}.$
Let the SVD of ${{\bf A}}={{\bf U}}_{{\bf A}}{{\boldsymbol{\Sigma}}}_{{\bf A}}{{\bf V}}_{{\bf A}}{^{\top}}$ be given. First, notice that the singular values $\sigma_j({{\bf B}}) = \sqrt{\lambda_j({{\bf B}}{{\bf B}}{^{\top}})}$, where $\lambda_j({{\bf B}}{{\bf B}}{^{\top}})$ defines the $j$-th eigenvalue of the matrix ${{\bf B}}{{\bf B}}{^{\top}}$ with $\lambda_1({{\bf B}}{{\bf B}}{^{\top}})\geq \cdots \geq \lambda_n({{\bf B}}{{\bf B}}{^{\top}})$. Since the eigenvalue decomposition of ${{\bf B}}{{\bf B}}{^{\top}}$ is given by $$\label{eq:eigendecompB}
{{\bf B}}{{\bf B}}{^{\top}}= {{\bf U}}_{{\bf A}}({{\boldsymbol{\Sigma}}}_{{\bf A}}{{\boldsymbol{\Sigma}}}_{{\bf A}}{^{\top}}+\eta^2 {{\bf I}}_m) {{\bf U}}_{{\bf A}}{^{\top}}$$ we have $${{\boldsymbol{\Sigma}}}_{{\bf B}}= \left[ {{\bf D}}_{{\bf A}}\ \ {{\bf0}}_{m \times n} \right]$$ with ${{\bf D}}_{{\bf A}}$, where $${{\bf D}}_{{\bf A}}= {{\rm diag\!}\left( \sqrt{\sigma_1^2({{\bf A}})+\eta^2},\ldots, \sqrt{\sigma_n^2({{\bf A}})+\eta^2}, \eta, \ldots, \eta \right)} \quad \mbox{if } m\geq n,$$ and $${{\bf D}}_{{\bf A}}= {{\rm diag\!}\left( \sqrt{\sigma_1^2({{\bf A}})+\eta^2},\ldots, \sqrt{\sigma_m^2({{\bf A}})+\eta^2} \right)} \quad \mbox{if } m < n.$$ Notice that, ${{\bf D}}_{{\bf A}}$ is invertible if $\eta>0$ or ${{\rm rank}\left( {{\bf A}}\right)} = \max\{m,n\}$. By equation the left singular vectors of ${{\bf B}}$ correspond to the left singular vectors of ${{\bf A}}$, i.e., ${{\bf U}}_{{\bf B}}= {{\bf U}}_{{\bf A}}$. As for the right singular vectors let $${{\bf V}}_{{\bf B}}= \begin{bmatrix} {{\bf V}}_{11} & {{\bf V}}_{12} \\ {{\bf V}}_{21} & {{\bf V}}_{22} \\ \end{bmatrix}$$ with ${{\bf V}}_{11} \in {\mathbb{R}}^{n \times m}, {{\bf V}}_{21} \in {\mathbb{R}}^{m \times m}, {{\bf V}}_{12} \in {\mathbb{R}}^{n \times n}$, and ${{\bf V}}_{22} \in {\mathbb{R}}^{m \times n}$. Then $${{\bf B}}= [{{\bf A}}\ \ \eta \, {{\bf I}}_m] = {{\bf U}}_{{\bf A}}\left[ {{\bf D}}_{{\bf A}}\ \ {{\bf0}}_{m \times n} \right] \begin{bmatrix} {{\bf V}}_{11}{^{\top}}& {{\bf V}}_{21}{^{\top}}\\ {{\bf V}}_{12}{^{\top}}& {{\bf V}}_{22}{^{\top}}\\ \end{bmatrix} = [{{\bf U}}_{{\bf A}}{{\bf D}}{{\bf D}}_{{\bf A}}{{\bf V}}_{11}{^{\top}}\ \ \ {{\bf U}}_{{\bf A}}{{\bf D}}_{{\bf A}}{{\bf V}}_{21}{^{\top}}m]$$ and $ {{\bf V}}_{11}= {{\bf V}}_{{\bf A}}{{\boldsymbol{\Sigma}}}_{{\bf A}}{^{\top}}{{\bf D}}_{{\bf A}}^{-1}$ and ${{\bf V}}_{21} = \eta \, {{\bf U}}_{{\bf A}}{{\bf D}}_{{\bf A}}^{-1}$. The matrices ${{\bf V}}_{12}$ and ${{\bf V}}_{22}$ are any matrices satisfying ${{\bf V}}_{12}{^{\top}}{{\bf V}}_{12}+{{\bf V}}_{22}{^{\top}}{{\bf V}}_{22} = {{\bf I}}_n$ and ${{\bf V}}_{11}{^{\top}}{{\bf V}}_{12}+{{\bf V}}_{21}{^{\top}}{{\bf V}}_{22} = {{\bf0}}_{m\times n}$ or equivalently ${{\bf A}}{{\bf V}}_{12}+\eta{{\bf V}}_{22} = {{\bf0}}_{m\times n}$.
Next, we provide a main result of our paper.
\[thm:mainresult\] Given matrices ${{\bf A}}\in {\mathbb{R}}^{m \times n}$, ${{\bf M}}\in {\mathbb{R}}^{n \times p}$, and ${{\bf P}}\in {\mathbb{R}}^{n \times m}$, with ${{\rm rank}\left( {{\bf A}}\right)} = k \leq n \leq m$, $ {{\rm rank}\left( {{\bf M}}\right)} = n \leq p$, let index $r \leq k$ and parameter $\eta \geq 0$, nonzero if $r < m$. Define ${{\bf F}}= ({{\bf I}}_n - {{\bf P}}{{\bf A}}){{\bf M}}{{\bf M}}{^{\top}}{{\bf A}}{^{\top}}- \eta^2 {{\bf P}}$. If ${{\rm rank}\left( {{\bf F}}\right)} \geq r$, then a global minimizer $\widehat {{\bf Z}}\in {\mathbb{R}}^{n \times m}$ of the problem $$\label{eqn:thmproblem}
\min_{{{\rm rank}\left( {{\bf Z}}\right)} \leq r} \,\,f({{\bf Z}}) = {\left\| {{\bf Z}}\begin{bmatrix}
{{\bf A}}{{\bf M}}& \eta {{\bf I}}_m \end{bmatrix} - \begin{bmatrix}
{{\bf M}}- {{\bf P}}{{\bf A}}{{\bf M}}& -\eta {{\bf P}}\end{bmatrix}\right\|_{{\rm F}}}^2$$ is given by $$\label{eq:zhat}
\widehat {{\bf Z}}= {{\bf U}}_{{{\bf H}},r}{{\bf U}}_{{{\bf H}},r}{^{\top}}{{\bf F}}({{\bf A}}{{\bf M}}{{\bf M}}{^{\top}}{{\bf A}}{^{\top}}+\eta^2 {{\bf I}})^{-1},$$ where symmetric matrix ${{\bf H}}= {{\bf F}}({{\bf A}}{{\bf M}}{{\bf M}}{^{\top}}{{\bf A}}{^{\top}}+\eta^2 {{\bf I}})^{-1} {{\bf F}}{^{\top}}$ has eigenvalue decomposition ${{\bf H}}= {{\bf U}}_{{\bf H}}{{\boldsymbol{\Lambda}}}_{{\bf H}}{{\bf U}}_{{\bf H}}{^{\top}}$ with eigenvalues ordered so that $\lambda_j \geq \lambda_i$ for $j < i \leq n$, and ${{\bf U}}_{{{\bf H}},r}$ contains the first $r$ columns of ${{\bf U}}_{{{\bf H}}}$. Moreover, $\widehat {{\bf Z}}$ is the unique global minimizer of if and only if $\lambda_r > \lambda_{r+1}$.
We will use Theorem \[thm:Friedland\] where ${{\bf B}}= \left[ \left( {{\bf I}}_n - {{\bf P}}{{\bf A}}\right) {{\bf M}}\, \ \ \, -\eta {{\bf P}}\right] $ and ${{\bf C}}= \left[ {{\bf A}}{{\bf M}}\, \ \ \, \eta {{\bf I}}_m \right]$. Let $${{\bf U}}{^{\top}}{{\bf A}}{{\bf G}}= {{\boldsymbol{\Sigma}}}\quad \mbox{and} \quad {{\bf V}}{^{\top}}{{\bf M}}{^{\top}}{{\bf G}}= {{\bf S}}$$ with $${{\boldsymbol{\Sigma}}}= \begin{bmatrix} {{\rm diag\!}\left( \sigma_1,\ldots,\sigma_n \right)} \\ {{\bf0}}_{(m-n)\times n}\end{bmatrix}
\quad \mbox{and} \quad
{{\bf S}}= \begin{bmatrix} {{\rm diag\!}\left( s_1,\ldots,s_n \right)} \\ {{\bf0}}_{(p-n)\times n}
\end{bmatrix}$$ denote the generalized SVD of $\left\{ {{\bf A}}, {{\bf M}}{^{\top}}\right\}$ and let ${{\bf L}}$ be defined by ${{\bf L}}= {{\boldsymbol{\Sigma}}}{{\bf G}}^{-1}{{\bf G}}^{-\top}{{\bf S}}{^{\top}}$ with its SVD given by $ {{\bf L}}= {{\bf U}}_{{\bf L}}{{\boldsymbol{\Sigma}}}_{{\bf L}}{{\bf V}}_{{\bf L}}{^{\top}}$. Then ${{\bf A}}{{\bf M}}= {{\bf U}}_{{{\bf A}}{{\bf M}}} {{\boldsymbol{\Sigma}}}_{{\bf L}}{{\bf V}}_{{{\bf A}}{{\bf M}}}{^{\top}}$, where ${{\bf U}}_{{{\bf A}}{{\bf M}}} = {{\bf U}}{{\bf U}}_{{\bf L}}$ and ${{\bf V}}_{{{\bf A}}{{\bf M}}} = {{\bf V}}{{\bf V}}_{{\bf L}}$. Using Lemma \[lem:svdExtended\], the SVD of ${{\bf C}}$ is given by $${{\bf U}}_{{\bf C}}= {{\bf U}}_{{{\bf A}}{{\bf M}}}, \quad {{\boldsymbol{\Sigma}}}_{{\bf C}}=
\begin{bmatrix}
{{\bf D}}_{{{\bf A}}{{\bf M}}} & {{\bf0}}_{m \times p}
\end{bmatrix}
\quad \mbox{and} \quad
{{\bf V}}_{{\bf C}}=
\begin{bmatrix}
{{\bf V}}_{{{\bf A}}{{\bf M}}}{{\boldsymbol{\Sigma}}}_{{\bf L}}{^{\top}}{{\bf D}}_{{{\bf A}}{{\bf M}}}^{-1} & {{\bf V}}_{12} \\[1ex]
\eta \, {{\bf U}}_{{{\bf A}}{{\bf M}}}{{\bf D}}_{{{\bf A}}{{\bf M}}}^{-1} & {{\bf V}}_{22}
\end{bmatrix},$$ with $$\begin{aligned}
{{\bf D}}_{{{\bf A}}{{\bf M}}} &= {{\rm diag\!}\left( \sqrt{\sigma_1^2({{\bf A}}{{\bf M}})+\eta^2},\ldots, \sqrt{\sigma_n^2({{\bf A}}{{\bf M}})+\eta^2}, \eta, \ldots, \eta \right)}, \quad\mbox{for } m\geq p, \\
{{\bf D}}_{{{\bf A}}{{\bf M}}} &= {{\rm diag\!}\left( \sqrt{\sigma_1^2({{\bf A}}{{\bf M}})+\eta^2},\ldots, \sqrt{\sigma_m^2({{\bf A}}{{\bf M}})+\eta^2} \right)},\quad
\mbox{for } m < p,\end{aligned}$$ and appropriately defined ${{\bf V}}_{12}$ and ${{\bf V}}_{22}$. Notice that ${{\bf D}}_{{{\bf A}}{{\bf M}}}$ is invertible and ${{\rm rank}\left( {{\bf C}}\right)} = m$, if either $\eta >0$ or ${{\rm rank}\left( {{\bf A}}{{\bf M}}\right)} = m$. Also acknowledge that ${{\bf D}}_{{{\bf A}}{{\bf M}}}^2 ={{\boldsymbol{\Sigma}}}_{{\bf L}}{{\boldsymbol{\Sigma}}}_{{\bf L}}{^{\top}}+\eta^2 {{\bf I}}_m$. Thus, the pseudoinverse of ${{\bf C}}$ is given by $${{\bf C}}^\dagger =
\begin{bmatrix}
{{\bf V}}_{{{\bf A}}{{\bf M}}} & {{\bf0}}_{p\times m} \\
{{\bf0}}_{m \times p} & {{\bf U}}_{{{\bf A}}{{\bf M}}}
\end{bmatrix}
\begin{bmatrix}
{{\boldsymbol{\Sigma}}}_{{\bf L}}{^{\top}}\\
\eta \, {{\bf I}}_m
\end{bmatrix}
{{\bf D}}_{{{\bf A}}{{\bf M}}}^{-2}{{\bf U}}_{{{\bf A}}{{\bf M}}}{^{\top}}$$ and $${{\bf V}}_{{{\bf C}},m}{{\bf V}}_{{{\bf C}},m}{^{\top}}=
\begin{bmatrix}
{{\bf V}}_{{{\bf A}}{{\bf M}}}{{\boldsymbol{\Sigma}}}_{{\bf L}}{^{\top}}{{\bf D}}_{{{\bf A}}{{\bf M}}}^{-2}{{\boldsymbol{\Sigma}}}_{{\bf L}}{{\bf V}}_{{{\bf A}}{{\bf M}}}{^{\top}}&
\eta \, {{\bf V}}_{{{\bf A}}{{\bf M}}}{{\boldsymbol{\Sigma}}}_{{\bf L}}{^{\top}}{{\bf D}}_{{{\bf A}}{{\bf M}}}^{-2}
{{\bf U}}_{{{\bf A}}{{\bf M}}}{^{\top}}\\[1ex]
\eta \, {{\bf U}}_{{{\bf A}}{{\bf M}}}{{\bf D}}_{{{\bf A}}{{\bf M}}}^{-2}{{\boldsymbol{\Sigma}}}_{{\bf L}}{{\bf V}}_{{{\bf A}}{{\bf M}}}{^{\top}}&
\eta^2 \, {{\bf U}}_{{{\bf A}}{{\bf M}}}{{\bf D}}_{{{\bf A}}{{\bf M}}}^{-2} {{\bf U}}_{{{\bf A}}{{\bf M}}}{^{\top}}\end{bmatrix}.$$ Let ${{\bf F}}= ({{\bf I}}_n - {{\bf P}}{{\bf A}}){{\bf M}}{{\bf V}}_{{{\bf A}}{{\bf M}}}{{\boldsymbol{\Sigma}}}_{{\bf L}}{^{\top}}{{\bf U}}_{{{\bf A}}{{\bf M}}}{^{\top}}-\eta^2 \, {{\bf P}}$, then $$\begin{aligned}
\label{eqn:defineK}
{{\bf K}}&= {{\bf B}}{{\bf V}}_{{{\bf C}},m}{{\bf V}}_{{{\bf C}},m}{^{\top}}= {{\bf F}}{{\bf U}}_{{{\bf A}}{{\bf M}}}{{\bf D}}_{{{\bf A}}{{\bf M}}}^{-2}
\begin{bmatrix} {{\boldsymbol{\Sigma}}}_{{\bf L}}{{\bf V}}_{{{\bf A}}{{\bf M}}}{^{\top}}& \eta \, {{\bf U}}_{{{\bf A}}{{\bf M}}}{^{\top}}\end{bmatrix}.\end{aligned}$$
Notice that ${{\rm rank}\left( {{\bf K}}\right)} \geq r$, since ${{\rm rank}\left( {{\bf F}}\right)} \geq r$ by assumption. Then, let symmetric matrix ${{\bf H}}= {{\bf K}}{{\bf K}}{^{\top}}= {{\bf F}}{{\bf U}}_{{{\bf A}}{{\bf M}}} {{\bf D}}_{{{\bf A}}{{\bf M}}}^{-2} {{\bf U}}_{{{\bf A}}{{\bf M}}}{^{\top}}{{\bf F}}{^{\top}}$ have eigenvalue decomposition ${{\bf H}}= {{\bf U}}_{{\bf H}}{{\boldsymbol{\Lambda}}}_{{\bf H}}{{\bf U}}_{{\bf H}}{^{\top}}$ with eigenvalues ordered so that $\lambda_j \geq \lambda_i,$ for $j < i \leq n$. Next we proceed to get an SVD of ${{\bf K}}$, $${{\bf K}}= {{\bf U}}_{{\bf H}}\left[ {{\boldsymbol{\Lambda}}}_{{\bf H}}^{1/2}\, |\, {{\bf0}}_{n \times(m+p-n)} \right] {{\bf V}}_{{\bf K}}{^{\top}}$$ with $${{\bf V}}_{{\bf K}}= \begin{bmatrix}
{{\bf V}}_{11} & {{\bf V}}_{12} & {{\bf V}}_{13}\\
{{\bf V}}_{21} & {{\bf V}}_{22} & {{\bf V}}_{23}
\end{bmatrix},$$ where ${{\bf V}}_{11} \in {\mathbb{R}}^{p \times r}, {{\bf V}}_{21} \in {\mathbb{R}}^{m \times r}, {{\bf V}}_{12} \in {\mathbb{R}}^{p \times(n-r)},$ and remaining matrices are defined accordingly. Then equating the SVD of ${{\bf K}}$ with and using a similar argument as in Lemma \[lem:svdExtended\], we get $${{\bf U}}_{{\bf H}}{^{\top}}{{\bf F}}{{\bf U}}_{{{\bf A}}{{\bf M}}} {{\bf D}}_{{{\bf A}}{{\bf M}}}^{-2}{{\boldsymbol{\Sigma}}}_{{\bf L}}{{\bf V}}_{{{\bf A}}{{\bf M}}}{^{\top}}= {{\boldsymbol{\Lambda}}}_{{\bf H}}^{1/2} \begin{bmatrix} {{\bf V}}_{11}{^{\top}}\\ {{\bf V}}_{12}{^{\top}}\end{bmatrix}$$ and $$\eta {{\bf U}}_{{\bf H}}{^{\top}}{{\bf F}}{{\bf U}}_{{{\bf A}}{{\bf M}}} {{\bf D}}_{{{\bf A}}{{\bf M}}}^{-2} {{\bf U}}_{{{\bf A}}{{\bf M}}}{^{\top}}= {{\boldsymbol{\Lambda}}}_{{\bf H}}^{1/2} \begin{bmatrix} {{\bf V}}_{21}{^{\top}}\\ {{\bf V}}_{22}{^{\top}}\end{bmatrix}.$$ Since ${{\boldsymbol{\Lambda}}}_{{{\bf H}},r}$ (the principal $r \times r$ submatrix of ${{\boldsymbol{\Lambda}}}_{{\bf H}}$) is invertible, the transpose of the first $r$ columns of ${{\bf V}}_{{\bf K}}$ have the form, $$\begin{aligned}
{{\bf V}}_{{{\bf K}},r}{^{\top}}&= \left[ {{\bf V}}_{11}{^{\top}}\,| \, {{\bf V}}_{21}{^{\top}}\right] \\
&= {{\boldsymbol{\Lambda}}}_{{{\bf H}},r}^{-1/2} \left[ {{\bf I}}_r \,| \, {{\bf0}}_{r\times (n-r)} \right] {{\bf U}}_{{\bf H}}{^{\top}}{{\bf F}}{{\bf U}}_{{{\bf A}}{{\bf M}}} {{\bf D}}_{{{\bf A}}{{\bf M}}}^{-2} \left[{{\boldsymbol{\Sigma}}}_{{\bf L}}{{\bf V}}_{{{\bf A}}{{\bf M}}}{^{\top}}\, | \, \eta \, {{\bf U}}_{{{\bf A}}{{\bf M}}}{^{\top}}\right]\\
&= {{\boldsymbol{\Lambda}}}_{{{\bf H}},r}^{-1/2} {{\bf U}}_{{{\bf H}},r}{^{\top}}{{\bf F}}{{\bf U}}_{{{\bf A}}{{\bf M}}} {{\bf D}}_{{{\bf A}}{{\bf M}}}^{-2} \left[{{\boldsymbol{\Sigma}}}_{{\bf L}}{{\bf V}}_{{{\bf A}}{{\bf M}}}{^{\top}}\, | \, \eta \, {{\bf U}}_{{{\bf A}}{{\bf M}}}{^{\top}}\right]\end{aligned}$$ and the best rank $r$ approximation of ${{\bf K}}$ is given by $$\begin{aligned}
{{\bf K}}_r &= {{\bf U}}_{{{\bf H}},r} {{\boldsymbol{\Lambda}}}_{{{\bf H}},r}^{1/2}{{\bf V}}_{{{\bf K}},r}{^{\top}}\\
&={{\bf U}}_{{{\bf H}},r}{{\bf U}}_{{{\bf H}},r}{^{\top}}{{\bf F}}{{\bf U}}_{{{\bf A}}{{\bf M}}} {{\bf D}}_{{{\bf A}}{{\bf M}}}^{-2} \left[{{\boldsymbol{\Sigma}}}_{{\bf L}}\, | \, \eta \, {{\bf I}}_m \right]
\begin{bmatrix}
{{\bf V}}_{{{\bf A}}{{\bf M}}}{^{\top}}& {{\bf0}}_{p \times m} \\ {{\bf0}}_{ m\times p} &{{\bf U}}_{{{\bf A}}{{\bf M}}}{^{\top}}\end{bmatrix}.\end{aligned}$$ Finally, using Theorem \[thm:Friedland\] we find that all global minimizers of $f$ with rank at most $r$ can be written as $$\begin{aligned}
\widehat{{\bf Z}}&= {{\bf K}}_r {{\bf C}}^\dagger\\
&={{\bf U}}_{{{\bf H}},r}{{\bf U}}_{{{\bf H}},r}{^{\top}}{{\bf F}}{{\bf U}}_{{{\bf A}}{{\bf M}}}{{\bf D}}_{{{\bf A}}{{\bf M}}}^{-2} \left({{\boldsymbol{\Sigma}}}_{{\bf L}}{{\boldsymbol{\Sigma}}}_{{\bf L}}{^{\top}}+\eta^2{{\bf I}}_m\right) {{\bf D}}_{{{\bf A}}{{\bf M}}}^{-2}{{\bf U}}_{{{\bf A}}{{\bf M}}}{^{\top}}\\
&={{\bf U}}_{{{\bf H}},r}{{\bf U}}_{{{\bf H}},r}{^{\top}}{{\bf F}}({{\bf A}}{{\bf M}}{{\bf M}}{^{\top}}{{\bf A}}{^{\top}}+\eta^2 {{\bf I}})^{-1},\end{aligned}$$ where $\widehat {{\bf Z}}$ is a [*unique*]{} global minimizer of if and only if $\lambda_r > \lambda_{r+1}$ since this condition makes the choice of ${{\bf U}}_{{{\bf H}},r}$ unique.
Efficient methods to compute ORIM $\widehat{{\bf Z}}$ {#sub:computational_methods_for_obtaining_bfz}
=====================================================
The computational cost to compute a global minimizer $\widehat {{\bf Z}}$ according to Theorem \[thm:mainresult\] requires the computation of a GSVD of $\left\{{{\bf A}},{{\bf M}}{^{\top}}\right\}$, an SVD of ${{\bf L}}$, and a partial eigenvalue decomposition of ${{\bf H}}$. For large-scale problems this may be computational prohibitive, so we seek an alternative approach to efficiently compute ORIM $\widehat {{\bf Z}}$. In the following we decompose the optimization problem into smaller subproblems and use efficient methods to solve the subproblems. The optimality of our update approach is verified by the following corollary of Theorem \[thm:mainresult\].
\[coro:thm\] Assume all conditions of Theorem \[thm:mainresult\] are fulfilled. Let $\widehat {{\bf Z}}_r$ be a global minimizer of of maximal rank $r$ and let $\widehat {{\bf Z}}_{r+\ell}$ be a global minimizer of of maximal rank $r+\ell$. Then $\tilde{{\bf Z}}_{\ell} = \widehat {{\bf Z}}_{r+\ell}-\widehat {{\bf Z}}_r$ is of maximal rank $\ell$ and the global minimizer of $$\label{eq:updateFormula}
\tilde{{\bf Z}}_{\ell} = \operatorname*{arg\,min}_{{{\rm rank}\left( {{\bf Z}}\right)}\leq \ell}{\left\|\left(\widehat{{\bf Z}}_r+{{\bf Z}}\right)
\begin{bmatrix} {{\bf A}}{{\bf M}}& \eta\,{{\bf I}}_m \end{bmatrix} -
\begin{bmatrix} {{\bf M}}-{{\bf P}}{{\bf A}}{{\bf M}}& -\eta\,{{\bf P}}\end{bmatrix}\right\|_{\rm F}}^2 .$$ Furthermore, $\tilde{{\bf Z}}_{\ell}$ is the unique global minimizer if and only if $\lambda_r > \lambda_{r+1}$ and $\lambda_{r+\ell} > \lambda_{r+\ell+1}$.
The significance of the corollary is as follows. Assume we are given a rank $r$ approximation $\widehat{{\bf Z}}_r$ and we are interested in updating our approximation to a rank $r+\ell$ approximation $\widehat {{\bf Z}}_{r+\ell}$. To calculate the optimal rank $r+\ell$ approximation $\widehat {{\bf Z}}_{r+\ell}$, we just need to solve a rank $\ell$ optimization problem of the form and then update the solution, $\widehat{{\bf Z}}_{r+\ell} = \widehat {{\bf Z}}_r + \tilde{{\bf Z}}_{\ell}$. Thus, computing a rank $r$ ORIM matrix $\widehat {{\bf Z}}_r$ can be achieved by solving a sequence of smaller rank problems and updating the solutions. Algorithm \[alg:rank1update\] describes such an rank-1 update approach.
${{\bf A}},{{\bf M}}, {{\bf P}}, \eta$ set $\widehat{{\bf Z}}_0 = {{\bf0}}_{n\times m}$, $r = 0$ $\displaystyle
\tilde{{\bf Z}}_r = \operatorname*{arg\,min}_{{{\rm rank}\left( {{\bf Z}}\right)}\leq 1}{\left\|\left(\widehat{{\bf Z}}_r+{{\bf Z}}\right)
\begin{bmatrix} {{\bf A}}{{\bf M}}& \eta\,{{\bf I}}_m \end{bmatrix} -
\begin{bmatrix} {{\bf M}}-{{\bf P}}{{\bf A}}{{\bf M}}& -\eta\,{{\bf P}}\end{bmatrix}\right\|_{\rm F}}^2$ \[alg:lineOpt\] $\widehat{{\bf Z}}_{r+1} = \widehat{{\bf Z}}_{r} + \tilde{{\bf Z}}_r$ $r = r+1$ optimal $\widehat{{\bf Z}}_r$
The main question in Algorithm \[alg:rank1update\] is how to efficiently solve the optimization problem in line \[alg:lineOpt\]. First, we reformulate the rank-1 constraint by letting ${{\bf Z}}= {{\bf x}}{{\bf y}}{^{\top}},$ where ${{\bf x}}\in{\mathbb{R}}^n$ and ${{\bf y}}\in{\mathbb{R}}^m$ and defining ${{\bf X}}_r = [{{\bf x}}_1,\ldots,{{\bf x}}_r] \in {\mathbb{R}}^{n \times r}$ and ${{\bf Y}}_r = [{{\bf y}}_1,\ldots,{{\bf y}}_r] \in {\mathbb{R}}^{m \times r}$. Then $\widehat{{\bf Z}}_r = {{\bf X}}_r{{\bf Y}}_r{^{\top}}$, and the optimization problem in line \[alg:lineOpt\] of Algorithm \[alg:rank1update\] reads $$\label{eq:optxy}
\resizebox{.92 \textwidth}{!}
{$\displaystyle
({{\bf x}}_{r+1},{{\bf y}}_{r+1}) = \operatorname*{arg\,min}_{({{\bf x}},{{\bf y}})}
{\left\|\left({{\bf X}}_r{{\bf Y}}_r{^{\top}}+{{\bf x}}{{\bf y}}{^{\top}}\right)
\begin{bmatrix} {{\bf A}}{{\bf M}}& \eta\,{{\bf I}}_m \end{bmatrix} -
\begin{bmatrix} {{\bf M}}-{{\bf P}}{{\bf A}}{{\bf M}}& -\eta\,{{\bf P}}\end{bmatrix}\right\|_{\rm F}}^2.
$}$$ Although standard optimization methods could be used, care must be taken since this quartic problem is of dimension $n+m$ and ill-posed since the decomposition ${{\bf Z}}= {{\bf x}}{{\bf y}}{^{\top}}$ is not unique. Notice that for fixed ${{\bf y}}$, optimization problem is quadratic and convex in ${{\bf x}}$ and vise versa. Thus, we propose to use an alternating direction optimization approach. Assume ${{\bf x}}\neq {{\bf0}}_{n\times 1}$, ${{\bf y}}\neq {{\bf0}}_{m \times 1}$, and $\eta>0$, then the partial optimization problems resulting from are ensured to have unique minimizers $$\label{eq:minx}
\widehat{{\bf x}}= \frac{{{\bf M}}{{\bf M}}{^{\top}}{{\bf A}}{^{\top}}{{\bf y}}-({{\bf P}}+{{\bf X}}_r{{\bf Y}}_r{^{\top}})\left({{\bf A}}{{\bf M}}{{\bf M}}{^{\top}}{{\bf A}}{^{\top}}+ \eta^2{{\bf I}}_m\right){{\bf y}}}
{{{\bf y}}{^{\top}}\left( {{\bf A}}{{\bf M}}{{\bf M}}{^{\top}}{{\bf A}}{^{\top}}+\eta^2{{\bf I}}_m\right){{\bf y}}} \quad \mbox{ for fixed } {{\bf y}},$$ and $$\widehat{{\bf y}}= \frac{\left( {{\bf A}}{{\bf M}}{{\bf M}}{^{\top}}{{\bf A}}{^{\top}}+\eta^2{{\bf I}}_m \right)^{-1}{{\bf A}}{{\bf M}}{{\bf M}}{^{\top}}{{\bf x}}- ({{\bf P}}+ {{\bf X}}_r{{\bf Y}}_r{^{\top}}){^{\top}}{{\bf x}}}{{{\bf x}}{^{\top}}{{\bf x}}} \quad \mbox{ for fixed } {{\bf x}}.$$ Notice that computing $\widehat{{\bf x}}$ in only requires matrix-vector products, while computing $\widehat {{\bf y}}$ requires a linear solve. Since decomposition ${{\bf Z}}= {{\bf x}}{{\bf y}}{^{\top}}$ is not unique, we propose to select the computationally convenient decomposition where ${\left\|{{\bf x}}\right\|_{2}} = 1$ and ${{\bf x}}\perp {{\bf X}}_r$. This results in a simplified formula for $\widehat {{\bf y}}$, i.e., $$\label{eq:miny}
\widehat{{\bf y}}= \left( {{\bf A}}{{\bf M}}{{\bf M}}{^{\top}}{{\bf A}}{^{\top}}+\eta^2{{\bf I}}_m \right)^{-1}{{\bf A}}{{\bf M}}{{\bf M}}{^{\top}}{{\bf x}}- {{\bf P}}{^{\top}}{{\bf x}}.$$ Noticing that is just the normal equations solution to the following least squares problem, $$\label{eq:minyls}
\min_{{\bf y}}{\left\|\begin{bmatrix} {{\bf M}}{^{\top}}{{\bf A}}{^{\top}}\\ \eta\, {{\bf I}}_m \end{bmatrix} {{\bf y}}- \begin{bmatrix}
{{\bf M}}{^{\top}}{{\bf x}}- {{\bf M}}{^{\top}}{{\bf A}}{^{\top}}{{\bf P}}{^{\top}}{{\bf x}}\\ -\eta {{\bf P}}{^{\top}}{{\bf x}}\end{bmatrix}\right\|_{2}}\,,$$ we propose to use a computationally efficient least squares solver such as LSQR [@PaSa82a; @PaSa82b], where various methods can be used to exploit the fact that the coefficient matrix remains constant [@Chen2005; @Benzi2002]. In addition, quasi Newton methods may improve efficiency by taking advantage of a good initial guess and a good approximation on the inverse Hessian [@Nocedal1999], but such comparisons are beyond the scope of this paper.
The alternating direction approach to compute a rank-1 update is provided in Algorithm \[alg:alternating\].
${{\bf A}}, {{\bf M}},\eta,{{\bf Z}}, {{\bf P}}, r$ set $\widehat{{\bf y}}= {{\bf1}}_{m \times 1}$ get $\widehat {{\bf x}}$ by normalize $\widehat{{\bf x}}= \widehat {{\bf x}}/ {\left\|\widehat {{\bf x}}\right\|_{2}}$ orthogonalize by $\widehat{{\bf x}}= \widehat{{\bf x}}- {{\bf X}}_{r}{{\bf X}}_{r}{^{\top}}\widehat{{\bf x}}$ get $\widehat{{\bf y}}$ by solving ${{\bf x}}_{r+1} = \widehat{{\bf x}}$ and ${{\bf y}}_{r+1} =\widehat{{\bf y}}$ optimal ${{\bf x}}_{r+1}$ and ${{\bf y}}_{r+1}$
In summary, our proposed method to compute low-rank ORIM $\widehat{{\bf Z}}$ combines Algorithms \[alg:rank1update\] and \[alg:alternating\]. An efficient [Matlab]{} implementation can be found at the following website:
`https://github.com/juliannechung/ORIM.git`
Before providing illustrations and examples of our method, we make a few remarks regarding numerical implementation.
1. *Storage*. Algorithmically $\widehat{{\bf Z}}_r$ need never be constructed, as we only require matrices ${{\bf X}}_r$ and ${{\bf Y}}_r$. This decomposition is storage preserving as long as $r\leq \frac{mn}{m+n}$ and is ideal for problems where ${{\bf Z}}$ is too large to compute or ${{\bf A}}$ can only be accessed via function call.
2. *Stopping criteria*. For Algorithm \[alg:rank1update\], the specific rank $r$ for $\widehat{{\bf Z}}_{r}$ may be user-defined, but oftentimes such information is not available a priori. However, the rank-1 update approach allows us to track the improvement in the function value from rank $r$ to rank $r+1$. Then an approximation of rank $r$ is deemed sufficient when $f({{\bf Z}}_{r-1}) - f({{\bf Z}}_r) < {\rm tol}\cdot f({{\bf Z}}_r)$, where our default tolerance is ${\rm tol} = 10^{-6}$. Standard stopping criteria [@Gill1981] can be used for Algorithm \[alg:alternating\]. In particular, we track improvement in the function values $f({{\bf X}}_r{{\bf Y}}_r{^{\top}})$, track changes in the arguments $\widehat{{\bf x}}$ and $\widehat{{\bf y}}$, and set a maximum iteration. Our default tolerance is $10^{-6}$.
3. \[item:fro\] *Efficient function evaluations.* Rather than computing the function value $f({{\bf X}}_r{{\bf Y}}_r{^{\top}})$ from scratch at each iteration (e.g., for determining stopping criteria), efficient updates can be done by observing that $$\begin{aligned}
f({{\bf X}}_{r+1}{{\bf Y}}_{r+1}{^{\top}}) =& f({{\bf X}}_r{{\bf Y}}_r{^{\top}}) \\
&+ {{\bf y}}{^{\top}}\left( {{\bf A}}{{\bf M}}{{\bf M}}{^{\top}}{{\bf A}}{^{\top}}+ \eta^2{{\bf I}}_m\right)\left({{\bf y}}+2{{\bf P}}{^{\top}}{{\bf x}}\right) - 2 {{\bf y}}{^{\top}}{{\bf A}}{{\bf M}}{{\bf M}}{^{\top}}{{\bf x}}\,,\end{aligned}$$ where $f({{\bf0}}_{n \times m}) = {\left\|({{\bf I}}_n-{{\bf P}}{{\bf A}}){{\bf M}}\right\|_{{\rm F}}}^2 + \eta^2\,{\left\|{{\bf P}}\right\|_{{\rm F}}}^2$. Since function evaluations are only relevant for the stopping criteria, they can be discarded, if desired, or approximated using trace estimators [@avron2011randomized].
4. *Initialization.* Equation requires an initial guess for ${{\bf y}}$. One uninformed choice may be ${{\bf y}}= {{\bf1}}_{m\times 1}$, and another option is to select ${{\bf y}}$ orthogonal to ${{\bf Y}}_r$, i.e., ${{\bf y}}= ({{\bf I}}_m-{{\bf Y}}_r{{\bf Y}}_r{^{\top}}){{\bf r}}$ with ${{\bf r}}\in {\mathbb{R}}^m$ chosen at random.
5. *Symmetry.* If ${{\bf A}}$ and ${{\bf P}}$ are symmetric, our rank-1 update approach could be used to compute a symmetric ORIM $\widehat{{\bf Z}}_r = {{\bf X}}_r{{\bf X}}_r{^{\top}}$, but the alternating direction approach should be replaced by an appropriate method for minimizing a quartic in ${{\bf x}}$.
6. *Covariance matrix.* Since ${{\bf M}}$ in our rank update approach only occurs in the product ${{\bf M}}{{\bf M}}{^{\top}}$ and since ${{\bf M}}{{\bf M}}{^{\top}}= {{\bf M}}_{{\boldsymbol{\xi}}}{{\bf M}}_{{\boldsymbol{\xi}}}{^{\top}}+ {{\boldsymbol{\mu}}}_{{\boldsymbol{\xi}}}{{\boldsymbol{\mu}}}_{{\boldsymbol{\xi}}}{^{\top}}= {{\boldsymbol{\Gamma}}}_{{\boldsymbol{\xi}}}+ {{\boldsymbol{\mu}}}_{{\boldsymbol{\xi}}}{{\boldsymbol{\mu}}}_{{\boldsymbol{\xi}}}{^{\top}}$, our algorithm can work directly with the covariance matrix. Thus, a symmetric factorization does not need to be computed, which is important for various classes of covariance kernels [@saibaba2012application].
Numerical Results {#sec:numerics}
=================
In this section, we provide three experiments that not only highlight the benefits of ORIM updates but also demonstrate new approaches for solving inverse problems that use ORIM updates. In Experiment 1, we use an inverse heat equation to investigate the efficiency and accuracy of our update approach. Then in Experiment 2, we use an image deblurring example to show that more accurate solutions to inverse problems can be achieved by using ORIM rank-updates to existing regularized inverse matrices. Lastly, in Experiment 3, we show that ORIM updates can be used in scenarios where perturbed inverse problems need to be solved efficiently and accurately.
Experiment 1: Efficiency of ORIM rank update approach {#sub:experiment_1_efficiency_of_orim_vs_svd}
-----------------------------------------------------
The goal of this example is to highlight our new result in Theorem \[thm:mainresult\] and to verify the accuracy and efficiency of the update approach described in Section \[sub:computational\_methods\_for\_obtaining\_bfz\]. We consider a discretized (ill-posed) inverse heat equation derived from a Volterra integral equation of the first kind on $[0,1]$ with kernel $a(s,t) = k(s-t)$, where $k(t) = \frac{t^{-3/2}}{2 \sqrt{\pi}\kappa}{\textnormal{e}}^{-\frac{1}{4\kappa^2 t}}$. Coefficient matrix ${{\bf A}}$ is $1,\!000 \times 1,\!000$ and is significantly ill-posed for $\kappa \in [1,2]$. We generate ${{\bf A}}$ using the *Regularization Tools* package [@Hansen1994].
As a first study, we compare ORIM $\widehat{{\bf Z}}$ with other commonly used regularized inverse matrices. Notice that $\widehat{{\bf Z}}$ is fully determined by ${{\bf A}},\eta,{{\bf M}}$, and ${{\bf P}}$.
For this illustration, we select ${{\bf P}}$ and ${{\bf M}}$ to be realizations of random matrices whose entries are i.i.d. standard normal ${\mathcal{N}}(0,1)$, and we select $\kappa = 1$ and $\eta = 0.02$. Then we compute ORIM $\widehat {{\bf Z}}$ as in Equation for various ranks $r$ and plot the function values $f(\widehat {{\bf Z}})$ in Figure \[fig:Example1\]. For comparison, we also provide function values for other commonly used rank-$r$ reconstruction matrices, including the TSVD matrix, ${{\bf A}}_{r}^\dagger,$ the truncated Tikhonov matrix (TTik), and the matrix provided from Theorem 1 of [@Chung2015], here referred to as ORIM$_0$. Notice that TTik and ORIM$_0$ matrices are just special cases of ORIM where ${{\bf M}}= [\,{{\bf I}}_n \ \ {{\bf0}}_{n \times 1}\,]$ and ${{\bf P}}= {{\bf0}}_{n \times m}$ for TTik and ${{\bf M}}= [\,{{\bf M}}_{{\boldsymbol{\xi}}}\ \ {{\bf0}}_{n \times 1}\,]$ and ${{\bf P}}= {{\bf0}}_{n \times m}$ for ORIM$_0$. Figure \[fig:Example1\] shows that, as expected, the function values for ORIM are smallest for all computed ranks.
![Comparison of the function values $f({{\bf Z}})$ where ${{\bf Z}}$ corresponds to different reconstruction matrices. The dotted line refers to TSVD, the dashed line to truncated-Tikhonov, the dash-dotted line to ORIM$_0$ (i.e., ORIM where ${{\bf M}}_{{\boldsymbol{\xi}}}= {{\bf I}}_n$ and ${{\boldsymbol{\mu}}}_{{\boldsymbol{\xi}}}={{\bf0}}_{n\times 1}$), and the solid line to ORIM $\widehat {{\bf Z}}$. Results correspond to a discretized Volterra integral equation.[]{data-label="fig:Example1"}](Example1Figure.pdf){width="80.00000%"}
We also verified our proposed rank-update approach by comparing function values computed with the rank update approach to those from Theorem \[thm:mainresult\]. We observed that the relative absolute errors remained below $2.9485\cdot10^{-3}$ for all computed ranks $r$, making the plot of the function values for the update approach indistinguishable from the solid line in Figure \[fig:Example1\]. Thus, we omit it for clarity of presentation.\
Next, we illustrate the efficiency of our rank update approach for solving a sequence of ill-posed inverse problems. Such scenarios commonly occur in nonlinear optimization problems such as variable projection methods where nonlinear parameters are moderately changing during the optimization process [@Nocedal1999; @GoPe73]. Consider again the inverse heat equation, and assume that we are given a sequence of matrices ${{\bf A}}(\kappa_j) \in {\mathbb{R}}^{n\times n}$, where the matrices depend nonlinearly on parameter $\kappa_j$, and we are interested in solving a sequence of problems, ${{\bf b}}(\kappa_j)= {{\bf A}}(\kappa_j){{\boldsymbol{\xi}}}+ {{\boldsymbol{\delta}}}_j$ for various $\kappa_j$.
For each problem in the sequence, one could compute a Tikhonov solution ${{\boldsymbol{\xi}}}_{\rm Tik}(\kappa_j) = {{\bf V}}_{{{\bf A}}(\kappa_j)}{{\boldsymbol{\Psi}}}_{{{\bf A}}(\kappa_j)}{{\bf U}}_{{{\bf A}}(\kappa_j)}{^{\top}}{{\bf b}}(\kappa_j)$, where $${{\boldsymbol{\Psi}}}_{{{\bf A}}(\kappa_j)} = {{\rm diag\!}\left( \frac{\sigma_1({{\bf A}}(\kappa_j))}{\sigma_1^2({{\bf A}}(\kappa_j))+ \eta^2}, \ldots, \frac{\sigma_n({{\bf A}}(\kappa_j))}{\sigma_n^2 ({{\bf A}}(\kappa_j))+ \eta^2} \right)},$$ but this approach requires an SVD of ${{\bf A}}(\kappa_j)$ for each $\kappa_j$. We consider an alternate approach, where the SVD is computed once for a fixed $\kappa_j$ and then ORIM updates are used to obtain improved regularized inverse matrices for other $\kappa_j$’s. This approach relies on the fact that small perturbations in ${{\bf A}}(\kappa_j)$ lead to small rank updates in its inverse [@Stewart2001].
Again for the inverse heat equation we use $n = 1,\!000$ and $\eta = 0.02$ and choose ${{\bf M}}= {{\bf I}}_n$ and ${{\boldsymbol{\mu}}}= {{\bf0}}_{n \times 1}$. We select equidistant values for $\kappa_j\in [1,2]$, $j = 1,\ldots, 100$, and let ${{\bf P}}^{(1)} = {{\bf V}}_{{{\bf A}}(\kappa_1)}{{\boldsymbol{\Psi}}}_{{{\bf A}}(\kappa_1)}{{\bf U}}_{{{\bf A}}(\kappa_1)}{^{\top}}$ be the Tikhonov reconstruction matrix corresponding to $\kappa_1$. Then for all other problems in the sequence, we compute reconstructions as $${{\boldsymbol{\xi}}}_{\rm ORIM}(\kappa_{j+1}) = {{\bf P}}^{(j+1)}{{\bf b}}(\kappa_{j+1})$$ where ${{\bf P}}^{(j+1)} = {{\bf P}}^{(j)} + {{\bf X}}^{(j+1)}\left( {{\bf Y}}^{(j+1)}\right){^{\top}}$, where ${{\bf X}}^{(j+1)}$ and ${{\bf Y}}^{(j+1)}$ are the low rank ORIM updates corresponding to ${{\bf A}}(\kappa_{j+1})$. We use a tolerance ${\rm tol} = 10^{-3}$. In Figure \[fig:Ex2timing\], we report computational timings for the ORIM rank update approach, compared to the SVD, and in Figure \[fig:Ex2error\] we provide corresponding relative reconstruction errors, computed as ${\rm rel} = {\left\|{{\boldsymbol{\xi}}}_\star - {{\boldsymbol{\xi}}}_{\rm true}\right\|_{2}} / {\left\|{{\boldsymbol{\xi}}}_{\rm true}\right\|_{2}}$, where ${{\boldsymbol{\xi}}}_\star$ is and approximation of ${{\boldsymbol{\xi}}}$ (here, ${{\boldsymbol{\xi}}}_{\rm ORIM}(\kappa_{j})$ and ${{\boldsymbol{\xi}}}_{\rm Tik}(\kappa_{j})$). We observe that the ORIM update approach requires approximately half the required CPU time compared to the SVD, and the ORIM update approach can produce relative reconstruction errors that are comparable to and even slightly better than Tikhonov. However, we also note potential disadvantages of our approach. In particular, the SVD can be more efficient for small $n$, although ORIM updates are significantly faster for larger problems (results not shown). Also, using different noise levels $\eta$ in each problem or taking larger changes in $\kappa_j$ may result in higher CPU times and/or higher reconstruction errors for the update approach. We assume that the noise levels and problems are not changing significantly.
![CPU times for computing a regularized inverse matrix using ORIM updates (solid line) and for computing the SVD to get a Tikhonov solution (dotted line) for a sequence of inverse problems varying in $\kappa$. We repeated the experiment 50 times and report the median as well as the 25-75th percentiles.[]{data-label="fig:Ex2timing"}](PaperExample2Time.png){width="\textwidth"}
![Relative reconstruction errors for reconstructions obtained using ORIM updates (solid line) and using Tikhonov regularization (dotted line). We report the median as well as the 25-75th percentiles for each $\kappa$ after repeating the experiment 50 times.[]{data-label="fig:Ex2error"}](PaperExample2Err.png){width="\textwidth"}
Experiment 2: ORIM Updates to Tikhonov {#sub:updating_inverse_matrices}
--------------------------------------
Here we consider a classic image deblurring problem, where the model is given in where ${{\boldsymbol{\xi}}}$ represents the desired image, ${{\bf A}}$ models the blurring process, and ${{\bf b}}$ is the blurred, observed image. The true image was taken to be the $15$-th slice of the 3D MRI image dataset that is provided in [MATLAB]{}, which is $256 \times 256$ pixels. We assume spatially invariant blur, where the point spread function (PSF) is a $11 \times 11$ box-car blur. We assume reflexive boundary conditions for the image. Since the PSF is doubly symmetric, blur matrix ${{\bf A}}$ is highly structured and its singular value decomposition is given by ${{\bf A}}={{\bf U}}_{{\bf A}}{{\boldsymbol{\Sigma}}}_{{\bf A}}{{\bf V}}_{{\bf A}}{^{\top}}$, where here ${{\bf V}}_{{\bf A}}{^{\top}}$ and ${{\bf U}}_{{\bf A}}$ represent the 2D discrete cosine transform (DCT) matrix and inverse 2D DCT matrix respectively [@Hansen2006]. Here we use the RestoreTools package [@Nagy2004]. Noise ${{\boldsymbol{\delta}}}$ was generated from a normal distribution, with zero mean, and scaled such that the noise level was ${\left\|{{\boldsymbol{\delta}}}\right\|_{2}}^2/ {\left\|{{\bf A}}{{\boldsymbol{\xi}}}\right\|_{2}}^2= 0.01$. The true and observed images, along with the PSF, are provided in Figure \[fig:mriproblem\].
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Image deblurring example. The true (desired) MRI image is given (a). The observed, blurred image is provided in (b), and the PSF is provided in (c).[]{data-label="fig:mriproblem"}](Truemri "fig:"){width="25.00000%"} ![Image deblurring example. The true (desired) MRI image is given (a). The observed, blurred image is provided in (b), and the PSF is provided in (c).[]{data-label="fig:mriproblem"}](Blurredmri "fig:"){width="25.00000%"} ![Image deblurring example. The true (desired) MRI image is given (a). The observed, blurred image is provided in (b), and the PSF is provided in (c).[]{data-label="fig:mriproblem"}](PSF "fig:"){width="25.00000%"}
\(a) True image \(b) Observed, blurred image \(c) Point spread function
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
As an initial regularized inverse approximation, we use a Tikhonov reconstruction matrix, ${{\bf P}}= {{\bf V}}_{{\bf A}}({{\boldsymbol{\Sigma}}}_{{\bf A}}{^{\top}}{{\boldsymbol{\Sigma}}}_{{\bf A}}+ \eta^2{{\bf I}})^{-1} {{\boldsymbol{\Sigma}}}_{{\bf A}}^{-1} {{\bf U}}_{{\bf A}}{^{\top}}$, where regularization parameter $\eta$ was selected to provide minimal reconstruction error. That is, we used $\eta = 2.831\cdot10^{-2}$, which corresponded to the minimum of error function, ${\left\|{{\bf P}}{{\bf b}}-{{\boldsymbol{\xi}}}\right\|_{2}}$. Although this approach uses the true image (which is not known in practice), our goal here is to demonstrate the improvement that can be obtained using the rank-update approach. In practice, a standard regularization parameter selection method such as the generalized cross-validation could be used, which for this problem gave $\eta = 2.713\cdot10^{-2}$. The Tikhonov reconstruction, ${{\bf P}}{{\bf b}}$, is provided in Figure \[fig:recon\](a) along with the computed relative reconstruction error.
Next we consider various ORIM updates to ${{\bf P}}$ and evaluate corresponding reconstructions. For the mean vector ${{\boldsymbol{\mu}}}_{{\boldsymbol{\xi}}}$, we use the image shown in Figure \[fig:recon\](b), which was obtained by averaging images slices 8–22 of the MRI stack (omitting slice 15, the image of interest). For efficient computations and simplicity, we assume ${{\boldsymbol{\Gamma}}}_{{\boldsymbol{\xi}}}$ is diagonal with variances proportional to ${{\boldsymbol{\mu}}}_{{\boldsymbol{\xi}}}$, we choose, ${{\boldsymbol{\Gamma}}}_{{\boldsymbol{\xi}}}= {{\rm diag\!}\left( {{\boldsymbol{\mu}}}_{{\boldsymbol{\xi}}}\right)}$; the matrix ${{\bf M}}_{{\boldsymbol{\xi}}}$ is defined accordingly. We compute ORIM updates to ${{\bf P}}$ according to Algorithm \[alg:rank1update\] for the following cases of ${{\bf M}}$: $$\label{eqn:Mmatrices}
{{\bf M}}_{(1)} = \begin{bmatrix}
{{\bf I}}_n & {{\boldsymbol{\mu}}}_{{\boldsymbol{\xi}}}\end{bmatrix}, \quad {{\bf M}}_{(2)} = \begin{bmatrix}
{{\bf M}}_{{\boldsymbol{\xi}}}& {{\bf0}}_{n \times 1}
\end{bmatrix}, \quad \mbox{and} \quad
{{\bf M}}_{(3)} = \begin{bmatrix}
{{\bf M}}_{{\boldsymbol{\xi}}}& {{\boldsymbol{\mu}}}_{{\boldsymbol{\xi}}}\end{bmatrix}.$$ We refer to these matrix updates as $\widehat {{\bf Z}}_{(1)}$, $\widehat {{\bf Z}}_{(2)}$, and $\widehat {{\bf Z}}_{(3)}$ respectively, where $\widehat {{\bf Z}}_{(1)}$ is a rank-1 matrix and $\widehat {{\bf Z}}_{(2)}$ and $\widehat {{\bf Z}}_{(3)}$ are matrices of rank $5$. Image reconstructions were obtained via matrix-vector multiplication, $${{\boldsymbol{\xi}}}_{(j)} = {{\bf P}}{{\bf b}}+ \widehat {{\bf Z}}_{(j)} {{\bf b}}, \quad \mbox{ for } j=1,2,3,$$ and are provided in Figure \[fig:recon\](c)–(e). Corresponding relative reconstruction errors are also provided.
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Initial Tikhonov reconstruction is provided in (a). The mean image, ${{\boldsymbol{\mu}}}$, provided in (b), was taken to be the average of images slices 8-22 of the MRI image stack (omitting slice 15, the image of interest). Image reconstructions in (c)-(e) correspond to ORIM updates to the initial Tikhonov reconstruction, for the various choices for ${{\bf M}}$ provided in . Relative reconstruction errors are provided.[]{data-label="fig:recon"}](Tik "fig:"){width="25.00000%"} ![Initial Tikhonov reconstruction is provided in (a). The mean image, ${{\boldsymbol{\mu}}}$, provided in (b), was taken to be the average of images slices 8-22 of the MRI image stack (omitting slice 15, the image of interest). Image reconstructions in (c)-(e) correspond to ORIM updates to the initial Tikhonov reconstruction, for the various choices for ${{\bf M}}$ provided in . Relative reconstruction errors are provided.[]{data-label="fig:recon"}](Meanimage "fig:"){width="25.00000%"}
\(a) Tikhonov, ${{\bf P}}{{\bf b}}$, ${{\rm rel}}= 0.2247$ \(b) Mean image, ${{\boldsymbol{\mu}}}_{{\boldsymbol{\xi}}}$
\[1ex\]
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Initial Tikhonov reconstruction is provided in (a). The mean image, ${{\boldsymbol{\mu}}}$, provided in (b), was taken to be the average of images slices 8-22 of the MRI image stack (omitting slice 15, the image of interest). Image reconstructions in (c)-(e) correspond to ORIM updates to the initial Tikhonov reconstruction, for the various choices for ${{\bf M}}$ provided in . Relative reconstruction errors are provided.[]{data-label="fig:recon"}](Recon1 "fig:"){width="25.00000%"} ![Initial Tikhonov reconstruction is provided in (a). The mean image, ${{\boldsymbol{\mu}}}$, provided in (b), was taken to be the average of images slices 8-22 of the MRI image stack (omitting slice 15, the image of interest). Image reconstructions in (c)-(e) correspond to ORIM updates to the initial Tikhonov reconstruction, for the various choices for ${{\bf M}}$ provided in . Relative reconstruction errors are provided.[]{data-label="fig:recon"}](Recon2 "fig:"){width="25.00000%"} ![Initial Tikhonov reconstruction is provided in (a). The mean image, ${{\boldsymbol{\mu}}}$, provided in (b), was taken to be the average of images slices 8-22 of the MRI image stack (omitting slice 15, the image of interest). Image reconstructions in (c)-(e) correspond to ORIM updates to the initial Tikhonov reconstruction, for the various choices for ${{\bf M}}$ provided in . Relative reconstruction errors are provided.[]{data-label="fig:recon"}](Recon3 "fig:"){width="25.00000%"}
\(c) ${{\boldsymbol{\xi}}}_{(1)}$, ${{\rm rel}}= 0.1938$ \(d) ${{\boldsymbol{\xi}}}_{(2)}$, ${{\rm rel}}= 0.2179$ \(e) ${{\boldsymbol{\xi}}}_{(3)}$, ${{\rm rel}}= 0.1904$
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Furthermore, absolute error images (in inverted colormap so that black corresponds to larger reconstruction error) in Figure \[fig:errimages\] show that the errors for the ORIM updated solution ${{\boldsymbol{\xi}}}_{(3)}$ have smaller and more localized errors than the initial Tikhonov reconstruction.
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Error images (in inverted colormap where white corresponds to $0$) for the initial Tikhonov reconstruction and the ORIM updated solution ${{\boldsymbol{\xi}}}_{(3)}$ which corresponds to ${{\bf M}}_{(3)}$ (i.e., nonzero mean and covariance matrix for ${{\boldsymbol{\xi}}}$).[]{data-label="fig:errimages"}](ErrTik "fig:"){width=".35\textwidth"} ![Error images (in inverted colormap where white corresponds to $0$) for the initial Tikhonov reconstruction and the ORIM updated solution ${{\boldsymbol{\xi}}}_{(3)}$ which corresponds to ${{\bf M}}_{(3)}$ (i.e., nonzero mean and covariance matrix for ${{\boldsymbol{\xi}}}$).[]{data-label="fig:errimages"}](ErrM3 "fig:"){width=".35\textwidth"}
\(a) Tikhonov \(b) ${{\boldsymbol{\xi}}}_3$
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
We repeated this experiment $20,\!000$ times, each time with a different noise realization in ${{\bf b}}$ and provide the distribution of the corresponding relative reconstruction errors in Figure \[fig:hist\]. Additionally, for each of these approaches, we provide the average reconstruction error, along with the standard deviation over all noise realizations in Table \[tab:recon\]. It is evident from these experiments that ORIM rank-updates to the Tikhonov reconstruction matrix can lead to reconstructions with smaller relative errors and allows users to easily incorporate prior knowledge regarding the distributions of ${{\boldsymbol{\xi}}}$ and ${{\boldsymbol{\delta}}}$.
![Distributions of relative reconstruction errors[]{data-label="fig:hist"}](Errhist.png){width="\textwidth"}
\[tab:recon\]
mean $\pm$ standard deviation
-------------------------------- -----------------------------------------------------
Tikhonov $1.1215\cdot 10^{-5}$ $\pm \, 3.4665\cdot 10^{-8}$
ORIM update, ${{\bf M}}_{(1)}$ $9.6881\cdot 10^{-6}$ $\pm \, 3.2040\cdot 10^{-8}$
ORIM update, ${{\bf M}}_{(2)}$ $1.0880\cdot 10^{-5}$ $\pm \, 3.4402\cdot 10^{-8}$
ORIM update, ${{\bf M}}_{(3)}$ $9.5254\cdot 10^{-6}$ $\pm \, 3.1541\cdot 10^{-8}$
: Comparison of average relative reconstruction error and standard deviation for $1,\!000$ noise realizations.
We then applied our reconstruction matrices, ${{\bf P}}+ \widehat {{\bf Z}}_{(j)},$ to the other images in the MRI stack and provide the relative reconstruction errors in Figure \[fig:mrislices\]. We observe that in general, all of the reconstruction matrices provide fairly good reconstructions, with smaller relative errors corresponding to images that are most similar to the mean image. Some of the true images were indeed included in the mean image. Regardless, our goal here is to illustrate that ORIM update matrices can be effective and efficient, if a good mean image and/or covariance matrix are provided. Other covariance matrices can be easily incorporated in this framework, but comparisons are beyond the scope of this work.
![Reconstructions of different slices from the MRI image stack using the initial Tikhonov reconstruction matrix, as well as the ORIM-updated reconstruction matrices.[]{data-label="fig:mrislices"}](Errdiffimages){width="\textwidth"}
Experiment 3: ORIM updates for perturbed problems {#sub:experiment3}
-------------------------------------------------
Last, we consider an example where ORIM updates to existing regularized inverse matrices can be used to efficiently solve perturbed problems. That is, consider a linear inverse problem such as where a good regularized inverse matrix denoted by ${{\bf P}}$ can be obtained. Now, suppose ${{\bf A}}$ is modified slightly (e.g., due to equipment setup or a change in model parameters), and a perturbed linear inverse problem $$\label{eqn:perturbed}
\widetilde {{\bf b}}= \widetilde {{\bf A}}{{\boldsymbol{\xi}}}+ \widetilde {{\boldsymbol{\delta}}}$$ must be solved. We will show that as long as the perturbation is not too large, a good solution to the perturbed problem can be obtained using low-rank ORIM updates to ${{\bf P}}$. This is similar to the scenario described in Experiment 1, but here we use an example from 2D tomographic imaging, where the goal is to estimate an image or object $f(x,y)$, given measured projection data. The Radon transform can be used to model the forward process, where the Radon transform of $f(x,y)$ is given by $$\label{eq:Radon}
b(\xi,\phi) = \int f(x,y) \delta(x \cos\phi + y \sin\phi - \xi )\,{{\rm d}}x \,{{\rm d}}y$$ where $\delta$ is the Dirac delta function. Figure \[fig:Radon\] illustrates the basic tomographic process.
![Experiment 3: Illustration of 2D tomography problem setup, where $f(x,y)$ is the desired object and projection data is obtained by x-ray transmission at various angles around the object.[]{data-label="fig:Radon"}](Radoncoord.png){width=".8\textwidth"}
The goal of the inverse problem is to compute a (discretized) reconstruction of the image $f(x,y)$, given projection data that is collected at various angles around the object. The projection data, when stored as an image, gives the sinogram. In Figure \[fig:tomo\] (a), we provide the true image which is a $128 \times 128$ image of the Shepp-Logan phantom, and two sinograms are provided in Figure \[fig:tomo\] (b) and (c), where the rows of the image contain projection data at various angles. In particular, for this example, we take $60$ projection images at $3$ degree intervals from $0$ to $177$ degrees (i.e., the sinogram contains $60$ rows). In order to deal with boundary artifacts, we pad the original image with zeros.
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Tomography Problem. The true image is shown in (a), the observed sinogram for the initial problem is given in (b) and the sinogram corresponding to the perturbed problem is given in (c).[]{data-label="fig:tomo"}](tomo_true "fig:"){width=".2\textwidth"} ![Tomography Problem. The true image is shown in (a), the observed sinogram for the initial problem is given in (b) and the sinogram corresponding to the perturbed problem is given in (c).[]{data-label="fig:tomo"}](tomo_b1 "fig:"){width=".35\textwidth"} ![Tomography Problem. The true image is shown in (a), the observed sinogram for the initial problem is given in (b) and the sinogram corresponding to the perturbed problem is given in (c).[]{data-label="fig:tomo"}](tomo_b2 "fig:"){width=".35\textwidth"}
\(a) True image \(b) Sinogram 1 \(c) Sinogram 2
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
The discrete tomographic reconstruction problem can be modeled as where ${{\boldsymbol{\xi}}}$ represents the (vectorized) desired image, ${{\bf A}}$ models the tomographic process, and ${{\bf b}}$ is the (vectorized) observed sinogram. For this example, we construct $${{\bf A}}= \begin{bmatrix}{{\bf R}}{{\bf S}}_{(1)} \\ \vdots\\{{\bf R}}{{\bf S}}_{(60)}
\end{bmatrix},$$ where ${{\bf S}}_{(j)}$ is a sparse matrix that represents rotation of the image for the $j$-th angle, whose entries were computed using bilinear interpolation as described in [@ChHaNa06; @Chung2010b], and ${{\bf R}}$ is a Kronecker product that approximates the integration operation. It is worth mentioning that in typical tomography problems, ${{\bf A}}$ is never created, but rather accessed via projection and backprojection operations [@feeman2015mathematics]. Our methods also work for scenarios where ${{\bf A}}$ represents a function call or object, but our current approach allows us to to build the sparse matrix directly. White noise is added to the problem at relative noise level $0.005$.
Since ${{\bf A}}$ has no obvious structure to exploit, we use iterative reconstruction methods to get an initial reconstruction matrix. This mimics a growing trend in tomography where reconstruction methods have shifted from filtered back projection approaches to iterative reconstruction methods [@hsieh2009computed; @beister2012iterative]. Furthermore, these iterative approaches are ideal for problems such as limited angle tomogography or tomosynthesis, where the goal is to obtain high quality images while reducing the amount of radiation to the patient [@dobbins2003digital; @Chung2010]. In this paper, we define a regularized inverse matrix ${{\bf P}}$ in terms of a partial Golub-Kahan bidiagonalization. That is, given a matrix ${{\bf A}}$ and vector ${{\bf b}},$ the Golub-Kahan process iteratively transforms matrix $[{{\bf b}}\,\,\, {{\bf A}}]$ to upper-bidiagonal form $[\beta_1 {{\bf e}}_1\,\,\, {{\bf B}}^{(k)}]$, with initializations $\beta_1 = {\left\|{{\bf b}}\right\|_{2}}$, ${{\bf w}}_1 = {{\bf b}}/ \beta_1$ and $\alpha_1 {{\bf q}}_1 = {{\bf A}}{^{\top}}{{\bf w}}_1$. After $k$ steps of the Golub-Kahan bidiagonalization process, we have matrices ${{\bf Q}}^{(k)} = \begin{bmatrix}
{{\bf q}}_1 & \ldots& {{\bf q}}_k
\end{bmatrix} \in {\mathbb{R}}^{n\times k}$, ${{\bf W}}^{(k)} = \begin{bmatrix}
{{\bf w}}_1 & \ldots& {{\bf w}}_k
\end{bmatrix} \in {\mathbb{R}}^{m\times k}$, and bidiagonal matrix $${{\bf B}}^{(k)} = \begin{bmatrix} \alpha_1 & & & \\
\beta_2 & \alpha_2 & & \\
& \ddots & \ddots & \\
& & \beta_k & \alpha_k \\
& & & \beta_{k+1}\\
\end{bmatrix} \in {\mathbb{R}}^{(k+1)\times k},$$ such that $${{\bf A}}{{\bf Q}}^{(k)} = {{\bf W}}^{(k+1)} {{\bf B}}^{(k)}.$$ It is worth noting that in exact arithmetic, the $k$-th LSQR [@PaSa82a; @PaSa82b] iterate is given by ${{\bf x}}_{\rm LSQR} = {{\bf Q}}^{(k)} ({{\bf B}}^{(k)})^{\dagger} ({{\bf W}}^{(k+1)}){^{\top}}{{\bf b}}$. Thus, we define ${{\bf P}}= {{\bf Q}}^{(k)} ({{\bf B}}^{(k)})^{\dagger} ({{\bf W}}^{(k+1)}){^{\top}}$ to be a regularized inverse matrix for the original problem, where $k=46$ corresponds to minimal reconstruction error ${\left\|{{\bf x}}_{\rm LSQR} - {{\bf x}}_{\rm true}\right\|_{2}}/ {\left\|{{\bf x}}_{\rm true}\right\|_{2}} = 0.2641$ for the original problem. See Figure \[fig:relerr\] for the relative error plot for the original problem.
![Relative reconstruction errors for LSQR on the original tomography problem, where the bullet $\bullet$ corresponds to minimal reconstruction error.[]{data-label="fig:relerr"}](tomo_Enrm){width="\textwidth"}
The goal of this illustration is to show that a low-rank ORIM update to ${{\bf P}}$ can be used to solve a perturbed problem. Thus, we created a perturbed problem , where $\widetilde {{\bf b}}$ and $\widetilde{{\bf A}}$ were created with slightly shifted projection angles. Again, we take $60$ projection images at $3$ degree intervals, but this time the angles ranged from $1$ to $178$ degrees. The corresponding sinogram is given in Figure \[fig:tomo\](c). A first approach would be to use ${{\bf P}}$ to reconstruct the perturbed data: ${{\bf P}}\widetilde {{\bf b}}$. This reconstruction is provided in the top left corner of Figure \[fig:tomoresults\], and it is evident that this is not a very good reconstruction. After a rank-4 update to ${{\bf P}}$, where ${{\boldsymbol{\mu}}}_{{\boldsymbol{\xi}}}= {{\bf0}}_{n \times 1}$, ${{\bf M}}_{{\boldsymbol{\xi}}}= {{\bf I}}_n$ and $\eta = 0.08$, we get a significantly better reconstruction (middle column of Figure \[fig:tomoresults\]). For comparison purposes, we provide in the last column the best LSQR reconstruction for the perturbed problem (i.e., corresponding to minimal reconstruction error). Relative reconstruction errors are provided, and corresponding absolute error images are presented on the same scale and with inverted colormap.
Initial, ${\rm rel} = 1.438$ ORIM, ${\rm rel} = 0.287$ LSQR, ${\rm rel} = 0.267$\
![Tomographic reconstructions for the perturbed problem, with corresponding error images. The reconstruction in the first column was obtained as ${{\bf P}}\widetilde {{\bf b}}$, the reconstruction in the second column was obtained using a rank-4 ORIM update to P and was computed as $({{\bf P}}+ \widehat {{\bf Z}}) \widetilde {{\bf b}}$. The reconstruction in the last column corresponds to the LSQR reconstruction for the perturbed problem corresponding to minimal reconstruction error.[]{data-label="fig:tomoresults"}](tomo_recon1 "fig:"){width="100.00000%"}\
![Tomographic reconstructions for the perturbed problem, with corresponding error images. The reconstruction in the first column was obtained as ${{\bf P}}\widetilde {{\bf b}}$, the reconstruction in the second column was obtained using a rank-4 ORIM update to P and was computed as $({{\bf P}}+ \widehat {{\bf Z}}) \widetilde {{\bf b}}$. The reconstruction in the last column corresponds to the LSQR reconstruction for the perturbed problem corresponding to minimal reconstruction error.[]{data-label="fig:tomoresults"}](tomo_recon2 "fig:"){width="100.00000%"}
Conclusions {#sec:conclusions}
===========
In this paper, we provide an explicit solution for a generalized rank-constrained matrix inverse approximation problem. We define the solution to be an optimal regularized inverse matrix (ORIM), where we include regularization terms, rank constraints, and a more general weighting matrix. Two main distinctions from previous results are that we can include updates to an existing matrix inverse approximation, and in the Bayes risk minimization framework, we can incorporate additional information regarding the probability distribution of ${{\boldsymbol{\xi}}}.$ For large scale problems, obtaining an ORIM according to Theorem \[thm:mainresult\] can be computationally prohibitive, so we described an efficient rank-update approach that decomposes the optimization problem into smaller rank subproblems and uses gradient-based methods that can exploit linearity. Using examples from image processing, we showed that ORIM updates can be used to compute more accurate solutions to inverse problems and can be used to efficiently solve perturbed systems, which opens the door to new applications and investigations. In particular, our current research is on incorporating ORIM updates within nonlinear optimization schemes such as variable projection methods, as well as on investigating its use for updating preconditioners for slightly changing systems.
[^1]: Department of Mathematics, Virginia Tech, Blacksburg, VA `jmchung@vt.edu www.math.vt.edu/people/jmchung/`
[^2]: Department of Mathematics, Virginia Tech, Blacksburg, VA `mcchung@vt.edu www.math.vt.edu/people/mcchung/`
|
---
abstract: |
Broadcasting is known to be an efficient means of disseminating data in wireless communication environments (such as Satellite, mobile phone networks,...). It has been recently observed that the average service time of broadcast systems can be considerably improved by taking into consideration existing correlations between requests. We study a pull-based data broadcast system where users request possibly overlapping sets of items; a request is served when all its requested items are downloaded. We aim at minimizing the average user perceived latency, [*i.e.*]{} the average flow time of the requests. We first show that any algorithm that ignores the dependencies can yield arbitrary bad performances with respect to the optimum even if it is given arbitrary extra resources. We then design a $(4+\epsilon)$-speed $O(1+1/\epsilon^2)$-competitive algorithm for this setting that consists in 1) splitting evenly the bandwidth among each requested set and in 2) broadcasting arbitrarily the items still missing in each set into the bandwidth the set has received. Our algorithm presents several interesting features: it is simple to implement, non-clairvoyant, fair to users so that no user may starve for a long period of time, and guarantees good performances in presence of correlations between user requests (without any change in the broadcast protocol). We also present a $ (4+\epsilon)$-speed $O(1+1/\epsilon^3)$-competitive algorithm which broadcasts at most one item at any given time and preempts each item broadcast at most once on average. As a side result of our analysis, we design a competitive algorithm for a particular setting of non-clairvoyant job scheduling with dependencies, which might be of independent interest.
#### Keywords:
Multicast scheduling, Pull-based broadcast, Correlation-based, Non-clairvoyant scheduling, Resource augmentation.
author:
- |
Julien Robert and Nicolas Schabanel\
École normale supérieure de Lyon\
[ Laboratoire de l’informatique du parallélisme]{}\
[UMR CNRS ENS-LYON INRIA UCBL n°5668]{}\
46 allée d’Italie, 69364 Lyon Cedex 07, France\
$\texttt{http://perso.ens-lyon.fr/}\{\texttt{julien.robert},
\texttt{nicolas.schabanel}\}$
bibliography:
- 'biblio-soda.bib'
title: |
Pull-Based Data Broadcast with Dependencies:\
Be Fair to Users, not to Items
---
*Omitted proofs, lemmas, notes and figures\
may be found in appendix.[^1]*
Introduction
============
#### Motivations.
Broadcasting is known to be an efficient means of disseminating data in wireless communication environments (such as Satellite, mobile phone networks,...). It has been recently observed in [@HuangChen2004; @HuangChen2003; @CaiLinChen2005] that the average service time of broadcast systems can be considerably improved by taking into consideration existing correlations between requests. Most of the theoretical research on data broadcasting was conduct until very recently under the assumption that user requests are for a single item at a time and are independent of each other. However, users usually request several items at a time which are, to a large extent, correlated. A typical example is a web server: users request web pages that are composed of a lot of shared components such as logos, style sheets, title bar, news headers,..., and all these components have to be downloaded together when any individual page is requested. Note that some of these components, [*e.g.*]{} news header, may constantly vary over time (size and/or content).\
#### Pull-based data broadcast with dependencies.
We study a pull-based data broadcast system where users request possibly overlapping sets of items. We aim at minimizing the average user perceived latency, [*i.e.*]{} the *average flow time* of the requests, where the flow time of a request is defined as the time elapsed between its arrival and the end of the download of the last requested item. We assume that user cannot start downloading an item in the middle of its broadcast. When the broadcast of an item starts, all the outstanding requests asking for this item can start downloading it. Several items may be downloaded simultaneously. We consider the *online* setting where the scheduler is *non-clairvoyant* and discovers each request at the time of its arrival; furthermore, the scheduler does not even know the lengths of the requested items and is aware of the completion of a broadcast only at the time of its completion. Items are however labeled with a unique ID to allow their retrieval. Note that this are the typical requirements of a real life systems where items may vary over time.\
#### Background.
It is well known that preemption is required in such systems in order to achieve reasonable performances. Furthermore, [@EdmondsPruhs2003] proved that even without dependencies, no algorithm can guarantee a flow time less than $\Omega(\sqrt n)$ times the optimal. The traditional approach in online algorithms consists then in penalizing the optimum by increasing the bandwidth given to the algorithm so that its performances can be compared to the optimum. This technique is known as *resource augmentation* and provides interesting insights on the relative performances of different algorithms that could not be compared directly to the optimum cost. In our case, we give to our algorithm a bandwidth $s>1$ and show that it achieves a flow time less than a constant times the optimum cost with a bandwidth $1$. Formally, an algorithm is *$s$-speed $c$-competitive* if when given a bandwidth $s$, its flow time is at most at a factor $c$ of the optimum flow time with bandwidth $1$.
To our knowledge the only positive results [@EdmondsPruhs2003; @EdmondsPruhs2005] in the online setting assume that the requests are independent and ask for one single item. The authors show that without dependencies the algorithms [[[[<span style="font-variant:small-caps;">**[Equi]{}**</span>]{}]{}]{}]{} and [[[[<span style="font-variant:small-caps;">**[LWF]{}**</span>]{}]{}]{}]{} are competitive. [[[[<span style="font-variant:small-caps;">**[Equi]{}**</span>]{}]{}]{}]{} which splits evenly the bandwidth among the alive requested items, is $(4+\epsilon)$-speed $(2+8/\epsilon)$-competitive, and [[[[<span style="font-variant:small-caps;">**[LWF]{}**</span>]{}]{}]{}]{}, which broadcasts the item where the aggregate waiting times of the outstanding requests for that item is maximized, is $6$-speed $O(1)$-competitive (where the bound proved on the competitive ratio is $O(1)=6,\!000,\!000$). In the *offline* setting, where the requests and their arrival times are known at time $t=0$, the problem is already NP-hard but better bounds can be obtained using linear programming [@KalyanasundaramPruhsVelauthapillai2000; @ErlebachHall2002; @GandhiKhullerKimWan2002; @BansalCharikarKhannaNaor2005ondemand; @BansalCoppersmithSviridenko2006]; the latest result, [@BansalCoppersmithSviridenko2006] to our knowledge, is a $O(\log^2(T+n)/\log\log(T+n))$-approximation where $n$ is the number of requests and $T$ the arrival time of the last request. To our knowledge, our results are the first provably efficient algorithms to deal with dependencies in the online setting.
Concerning the push-based variant of the problem, where the requests arrival times follow some Poisson process and the requested sets are identically distributed according to a fixed distribution, constant factor approximations exist in presence of dependencies [@BarnoyShilo2000; @BarnoyNaorSchieber2003; @DeySchabanel2006]. The latest result, [@DeySchabanel2006], obtains a $4$-approximation if the requested sets are drawn according to an arbitrary fixed distribution over a finite number of subsets of items.\
#### Our contribution.
We first show that the performances of any algorithm that ignores the dependencies can be arbitrarily far from the optimal cost even if it is given *arbitrary* extra resources. We then design a $(4+\epsilon)$-speed $O(1+1/\epsilon^2)$-competitive algorithm [[[[<span style="font-variant:small-caps;">**[B-EquiSet]{}**</span>]{}]{}]{}]{} for the non-clairvoyant data broadcast problem with dependencies. [[[[<span style="font-variant:small-caps;">**[B-EquiSet]{}**</span>]{}]{}]{}]{} consists in 1) splitting evenly the bandwidth among each requested set and in 2) broadcasting arbitrarily the items still missing in each set into the bandwidth the set has received. The spirit of the algorithm is that *one should favor the users over the items* in the sense that it splits the bandwidth evenly among the outstanding requested sets and arbitrarily among the outstanding items within each requested set. Our algorithm presents several interesting features: it is simple to implement, non-clairvoyant, fair to users so that no user may starve for a long period of time, and improves performances in presence of correlations between user requests (without any change in the broadcast protocol). Presicely, we prove that:
\[thm:EE\] For all $\delta>0$ and $\epsilon>0$, [[[[[<span style="font-variant:small-caps;">**[B-EquiSet]{}**</span>]{}]{}]{}]{}]{} is a ${(1+\delta)(4+\epsilon)}$-speed ${(2+8/\epsilon)(1+1/\delta)}$-competitive algorithm for the online data broadcast problem with dependencies.
One could object that [[[[<span style="font-variant:small-caps;">**[B-EquiSet]{}**</span>]{}]{}]{}]{} is unrealistic since it can split the bandwidth arbitrarily. But using the same technic as in [@EdmondsPruhs2003], it is easy to modify [[[[<span style="font-variant:small-caps;">**[B-EquiSet]{}**</span>]{}]{}]{}]{} to obtain an other competitive algorithm ${{{{\textsc{\bfseries{B-EquiSet-Edf}}}\/}}}$ (described at the end of section \[sec:EEDF\]) which, with a slight increase of bandwidth, ensures that at most one item is broadcast at any given time and that each broadcast is preempted at most once on average.
\[thm:eedf\] For all $\delta>0$ and $\epsilon>0$, [[[[[<span style="font-variant:small-caps;">**[B-EquiSet-Edf]{}**</span>]{}]{}]{}]{}]{} is a ${(1+\delta)^2(4+\epsilon)}$-speed $(2+8/\epsilon)(1+1/\delta)^2$-competitive algorithm for the online data broadcast problem with dependencies, where each broadcast is preempted at most once on average.
Our analysis takes its inspiration in the methods developed in [@EdmondsPruhs2003]. In order to extend their analysis to our algorithm, we have also designed a new competitive algorithm [[[[<span style="font-variant:small-caps;">**[Equi$\circ$A]{}**</span>]{}]{}]{}]{} for a particular setting of non-clairvoyant job scheduling with dependencies which might be of independent interest (Theorem \[thm:EA\]).
The next section gives a formal description of the problem and shows that it is required to take dependencies into account to obtain a competitive algorithm. Section \[sec:alg:EE\] exposes the algorithm [[[[<span style="font-variant:small-caps;">**[B-EquiSet]{}**</span>]{}]{}]{}]{} and introduces useful notations. Section \[sec:EA\] designs a competitive algorithm [[[[<span style="font-variant:small-caps;">**[Equi$\circ$A]{}**</span>]{}]{}]{}]{} for a variant of job scheduling with dependencies that is used in Section \[sec:EE\] to analyze the competitiveness of our algorithm [[[[<span style="font-variant:small-caps;">**[B-EquiSet]{}**</span>]{}]{}]{}]{}.
Definitions and notations {#sec:not}
=========================
#### The problem.
The input consists of:\
- A set ${{{\ensuremath{\mathscr{I}}}}}$ of $n$ *items* $I_1,\ldots,I_n$ each of length $\ell_1,\ldots,\ell_n$\
- A set ${{{\ensuremath{\mathscr{S}}}}}$ of $q$ *requests* for $q$ non-empty sets of items $S_1,\ldots, S_q\subseteq {{{\ensuremath{\mathscr{I}}}}}$, with arrival times $a_1,\ldots,a_q$.\
#### Schedule.
A *$s$-speed schedule* is an allocation of a bandwidth of size $s$ to the items of ${{{\ensuremath{\mathscr{I}}}}}$ over the time. Formally, it is described by a function $r : {{{\ensuremath{\mathscr{I}}}}}\times [0,\infty) \rightarrow [0,s]$ such that for all time $t$, $\sum_{I \in {{{\ensuremath{\mathscr{I}}}}}} r(I,t) {\leqslant}s$; $r(I,t)$ represents the *rate* of the broadcast of $I$ at time $t$, [*i.e.*]{}, the amount of bandwidth allotted to item $I$ at time $t$. An item $I_i$ is broadcast between $t$ and $t'$ if its broadcast starts at time $t$ and if the total bandwidth allotted to $I_i$ between $t$ and $t'$ sums up to $\ell_i$, [*i.e.*]{}, if $\int_{t}^{t'} r(I_i,t)\,dt = \ell_i$. We denote by $c(I_i,k)$ the date of the completion of the $k$th broadcast of item $I_i$. Formally, it is the first date such that $\int_{0}^{c(I_i,k)} r(I_i,t) dt = k\, \ell_i$ (note that $c(I_i,0) = 0$). We denote by $b(I_i,k)$ the date of the beginning of the $k$th broadcast of item $I_i$, [*i.e.*]{} $b(I_i,k) = \inf\{t {\geqslant}c(I_i,k-1)\,:\, r(I_i,t) > 0\}$.[^2]\
#### Cost.
For all time $t$, let $B(I_i,t)$ be the time of the beginning of the first broadcast of item $I_i$ after $t$, [*i.e.*]{} $B(I_i,t) = \min\{b(I_i,k):b(I_i,k){\geqslant}t\}$. For all time $t$, $C(I_i,t)$ denotes the time of the end of the first broadcast of item $I_i$ starting after $t$, [*i.e.*]{} $C(I_i,t) = \min \{c(I_i,k):b(I_i,k){\geqslant}t\}$. The *completion time* $c_j$ of request $S_j$ is the first time such that every item in $S_j$ has been broadcast (or downloaded) after its arrival time $a_j$, [*i.e.*]{}, ${c_j = \max_{I_i\in S_j} C(I_i,a_j)}$. We aim at minimizing the *average completion time* defined as ${\frac{1}{q} \sum_{S_j \in {{{\ensuremath{\mathscr{S}}}}}} (c_j - a_j)}$, or equivalently the *flow time* defined as the sum of the waiting times, [*i.e.*]{} ${\ensuremath{\operatorname{B-FlowTime}}}= {\sum_{S_j \in {{{\ensuremath{\mathscr{S}}}}}} (c_j - a_j)}$. We denote by ${{\ensuremath{\operatorname{BOPT}}}}_s({{{\ensuremath{\mathscr{S}}}}})$ the flow time of an optimal $s$-speed schedule for a given instance ${{{\ensuremath{\mathscr{S}}}}}$.\
#### $s$-Speed $c$-Competitive Algorithms.
We consider the online setting of the problem, in which the scheduler gets informed of the existence of each request $S_j$ at time $a_j$ and not before. The scheduler is not even aware of the lengths $(\ell_i)_{I_i\in S_j}$ of the requested items in each set nor of the total number $n$ of available items. It is well known ([*e.g.*]{}, see [@EdmondsPruhs2003]) that in this setting, it is impossible to approximate within a factor $o(\!\sqrt{n})$ the optimum flow time for a given bandwidth $s$ even if all items have unit length (independently of any conjecture such as $P=NP$). The traditional approach in online algorithms consists then in penalizing the optimum by increasing the bandwidth given to the algorithm so that its performances can be compared to the optimum. This technique is known as *resource augmentation* and provides interesting insights on the relative performances of different algorithms that could not be compared directly to the optimum cost. In our case, we give to our algorithm a bandwidth $s>1$ and show that it achieves a flow time less than a constant times the optimum cost with a bandwidth $1$. Formally, an algorithm is *$s$-speed $c$-competitive* if when given $s$ times as many resources as the adversary, its cost is no more than $c$ times the optimum cost. In our case the resource is the bandwidth, and we compare the cost $A_s$ of a scheduler $A$ with a bandwidth $s$, to the cost ${{\ensuremath{\operatorname{BOPT}}}}_1$ of an optimal schedule on a unit bandwidth. (We denote by $A_s$ the cost of an algorithm $A$ when given a bandwidth $s$.)
We show below that ignoring existing dependencies can lead to arbitrarily bad solutions.
\[fac:yao\] No algorithm $A$ that ignores dependencies is $s$-speed $c$-competitive for any $c < \frac{2}{3s}\sqrt{n}$ if $A$ is deterministic, and for any $c < \frac1{6s} \sqrt n$ if $A$ is randomized.
Consider first a deterministic algorithm $A$ which is given a bandwidth $s$ and consider the instance where $n$ different items are requested at time $t=0$. Since $A$ ignores the dependencies, we set them after the execution of the algorithm $A$: one request asks for the $n-\sqrt n$ items that have been served the most by $A$ at time $t=(n-\sqrt n)/s$, and $\sqrt{n}$ requests ask for each of the remaining $\sqrt{n}$ items. Then, algorithm $A$ serves each request only after time $t=(n-\sqrt{n})/s$ and its flow time is at least $(\sqrt{n}+1)(n-\sqrt{n})/s \sim n\sqrt n/s$. The optimal solution with bandwidth only $1$ first broadcasts the items corresponding to the $\sqrt{n}$ unit length requests and then broadcasts the $n-\sqrt{n}$ remaining items; the optimal flow time is then $(n+\sum_{k=1}^{\sqrt{n}} k) \sim \frac32 n$. This shows a gap of $\frac2{3s}\sqrt{n}$ between the optimal cost with bandwidth $1$ and every deterministic algorithm with bandwidth $s=O(\sqrt{n})$, which ignores the dependencies. We extend the result to randomized algorithms thanks to Yao’s principle [@Yao1977; @MotwaniRaghavan1995] (Omitted).
The Algorithm [[[[<span style="font-variant:small-caps;">**[B-EquiSet]{}**</span>]{}]{}]{}]{} {#sec:alg:EE}
=============================================================================================
#### Definitions.
A request $S_j$ for a subset of items is said to be *alive* at time $t$ if $t{\geqslant}a_j$ and if the download of at least one item $I_i\in S_j$ is not yet completed at time $t$, [*i.e.*]{}, $t < C(I_i,a_j)$. We say that an item $I_i\in S_j$ whose download is not yet completed ([*i.e.*]{}, such that $a_j {\leqslant}t < C(I_i,a_j)$) is *alive for $S_j$* at time $t$.\
#### The [[[[[<span style="font-variant:small-caps;">**[B-EquiSet]{}**</span>]{}]{}]{}]{}]{} Algorithm.
Consider that we are given a bandwidth $s$. Let $R(t)$ be the set of alive requests at time $t$ during the execution of the algorithm. For all $t$, [[[[<span style="font-variant:small-caps;">**[B-EquiSet]{}**</span>]{}]{}]{}]{} allocates to each alive request the same amount of bandwidth, $s/|R(t)|$; then, for each alive request $S_j$, it splits *arbitrarily* the $s/|R(t)|$ bandwidth allotted to $S_j$ among its alive items. Precisely, it allocates to each item $I_i$ alive for $S_j$ at time $t$, an *arbitrary* amount of bandwidth, $r_{j,i}(t) {\geqslant}0$, such that $\sum_{\text{$I_i$ alive for $S_j$}} r_{j,i}(t) = s/|R(t)|$. [[[[<span style="font-variant:small-caps;">**[B-EquiSet]{}**</span>]{}]{}]{}]{} then broadcasts at time $t$ each item $I_i$ at a rate $r_i(t) = \sum_{S_j\in R(t)\,:\,\text{$I_i$ is alive for $S_j$ at time $t$}} r_{j,i}(t)$.
Figure \[fig:ee:1.5\] illustrates an execution of the algorithm, in which [[[[<span style="font-variant:small-caps;">**[B-EquiSet]{}**</span>]{}]{}]{}]{} chooses for each alive request $S_j$, to divide up the bandwidth allotted to $S_j$ equally among every $S_j$’s alive items.
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --
**The instance** consists of three items $A,B,C$ of length $1.5$ and four requests $S_1=\{A,B,C\}$ (in red), $S_2=\{A\}$ (in green), $S_3=\{B\}$ (in blue), and $S_4=\{C\}$ (in yellow) with arrival times $a_1=0$, $a_2=1$, $a_3=2$, and $a_4=3$. Two schedules are presented: [[[[<span style="font-variant:small-caps;">**[B-EquiSet]{}**</span>]{}]{}]{}]{} with bandwidth $s=1.5$ (to the left) and an optimal schedule with unit bandwidth (to the right). Time flies downwards. Four lines to the right of each schedule represent each request’s lifetime; the bandwidth allotted to each request is outlined in their respective color. [[[[<span style="font-variant:small-caps;">**[B-EquiSet]{}**</span>]{}]{}]{}]{} first allots all the bandwidth to $S_1$ and splits it evenly among its items $A$, $B$ and $C$ (items $A$, $B$, and $C$ get darker and darker as their broadcasts progress). At time $1$, $S_2$ arrives and [[[[<span style="font-variant:small-caps;">**[B-EquiSet]{}**</span>]{}]{}]{}]{} splits the bandwidth
\[-.3em\]
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --
Note that bandwidth adjustments for each item are necessary only when new requests arrive or when the broadcast of some item completes.
As in [@EdmondsPruhs2003], we deduce the performances of our broadcast algorithm [[[[<span style="font-variant:small-caps;">**[B-EquiSet]{}**</span>]{}]{}]{}]{} from the analysis of the performances of an other algorithm, [[[[<span style="font-variant:small-caps;">**[Equi$\circ$A]{}**</span>]{}]{}]{}]{}, for a variant of the non-clairvoyant scheduling problem studied in [@Edmonds1999] which includes dependencies. Section \[sec:EA\] presents this later problem and analyzes the competitiveness of algorithm [[[[<span style="font-variant:small-caps;">**[Equi$\circ$A]{}**</span>]{}]{}]{}]{}. Then, Section \[sec:EE\] deduces the competitiveness of [[[[<span style="font-variant:small-caps;">**[B-EquiSet]{}**</span>]{}]{}]{}]{} by simulating [[[[<span style="font-variant:small-caps;">**[Equi$\circ$A]{}**</span>]{}]{}]{}]{} on a particular instance of non-clairvoyant scheduling built on the execution of [[[[<span style="font-variant:small-caps;">**[B-EquiSet]{}**</span>]{}]{}]{}]{}.
Non-Clairvoyant Seq-Par Batch Scheduling {#sec:EA}
========================================
For the sake of completeness we first sum up the results in [@Edmonds1999], reader may skip this paragraph in a first reading. Edmonds’s non-clairvoyant scheduling problem consists in designing an online algorithm that schedules jobs on $p$ processors without any knowledge of the progress of each job before its completion. An instance of non-clairvoyant job scheduling problem consists in a collection of jobs $(J_k)$ with arrival times $(a_k)$; each job $J_k$ goes through a series of phases $J_k^1,\ldots, J_k^{m_k}$; the amount of work in each phase $J_k^l$ is $w_k^l$; at time $t$, the algorithm allocates to each uncompleted job $J_k$ an amount $\rho_k^t$ of processors (the $(\rho_k^t)$s are arbitrary non-negative real numbers, such that at any time: $\sum_k \rho_k^t {\leqslant}p$); each phase $J_k^l$ progresses at a rate given by a speed-up function $\Gamma_k^l(\rho_k)$ of the amount $\rho_k$ of processors allotted to $J_k$ during phase $J_k^l$, that is to say that the amount of work accomplished between $t$ and $t+dt$ during phase $J_k^l$ is $\Gamma_k^l(\rho_k^t) dt$; let $t_k^l$ denote the completion time of the $l$-th phase of $J_k$, [*i.e.*]{} $t_k^l$ is the first time $t'$ such that $\int_{t_k^{l-1}}^{t'} \Gamma_k^l(\rho_k^t)\,dt = w_k^l$ (with $t_k^0= a_k$). The overall goal is to minimize the *flow time* of the jobs, that is to say the sum of the processing time of each job, [*i.e.*]{} ${\ensuremath{\operatorname{J-FlowTime}}}=\sum_k (t_k^{m_k}-a_k)$. We denote by ${{\ensuremath{\operatorname{JOPT}}}}_s({{{\ensuremath{\mathscr{J}}}}})$ the flow time of an optimal $s$-speed schedule for ${{{\ensuremath{\mathscr{J}}}}}$. The algorithm is *non-clairvoyant* in the sense that it does not know anything about the progress of each job and is only informed that a job is completed at the time of its completion. In particular, it is not aware of the different phases that the job goes through (neither of the amount of work nor of the speed-up function). One of the striking results of [@Edmonds1999] is that in spite of this total lack of knowledge, the algorithm [[[[<span style="font-variant:small-caps;">**[Equi]{}**</span>]{}]{}]{}]{} that allocates an equal amount of processors to each uncompleted job is $(2+\epsilon)$-speed $(2+4/\epsilon)$-competitive when the speed up functions are arbitrary non-decreasing sub-linear functions ([*i.e.*]{}, such that for all $\rho<\rho'$, ${\Gamma_k^l(\rho)}\big/{\rho}{\geqslant}{\Gamma_k^l(\rho')}\big/{\rho'}$, for all $k,l$).
Two particular kinds of phases are of interest for our purposes: sequential and parallel. During a *sequential* phase, $\Gamma(\rho)=1$, that is to say that the job progresses at a unit rate whatever amount of processing power it receives (even if it receives no processor at all, [*i.e.*]{} even if $\rho = 0$)! During a *parallel* phase, the job progresses proportionally to the processing power it receives, [*i.e.*]{} $\Gamma(\rho) = \rho$. Remark that these two kinds of speed-up functions match the requirement of Edmond’s theorem and thus [[[[<span style="font-variant:small-caps;">**[Equi]{}**</span>]{}]{}]{}]{} is $(2+\epsilon)$-speed $(2+4/\epsilon)$-competitive on instances consisting of a collection of jobs composed of sequential and parallel phases.
As in [@EdmondsPruhs2003], we reduce the analysis of our broadcast algorithm [[[[<span style="font-variant:small-caps;">**[B-EquiSet]{}**</span>]{}]{}]{}]{} to the analysis of a non-clairvoyant scheduling algorithm. For that purpose, we need to introduce dependencies between the jobs in Edmonds’s framework. We consider the following variant of the scheduling problem.\
#### Non-Clairvoyant Seq-Par Batches Scheduling.
An instance of this variant consists in a collection ${{{\ensuremath{\mathscr{B}}}}}=\{B_1,\ldots,B_q\}$ of batches $B_j=\{J_{j,1},\ldots,J_{j,u_j}\}$ of jobs with arrival times $a_1,\ldots, a_q$, where each job $J_{j,i}$ is composed of two phases: a sequential phase of work $w_{j,i}^s{\geqslant}0$ followed by a parallel phase of work $w_{j,i}^p{\geqslant}0$. (Note that this problem is different from the classical batch scheduling problem in which *only one* batch has to be treated.) The scheduler is non-clairvoyant and discovers each batch of jobs at the time of its arrival and is in particular *not aware of the amounts of work of each job in each batch*. The scheduler allocates to each job $J_{j,i}$, arrived and uncompleted at time $t$, a certain amount $\rho_{j,i}^t$ of the processors ($\rho_{j,i}^t$ is an arbitrary non-negative real number). Let $t_{j,i}$ denote the completion time of job $J_{j,i}$; $t_{j,i}$ is the first date verifying $\int_{a_j+w_{j,i}^s}^{t_{j,i}} \rho_{j,i}^t\, dt=w_{j,i}^p$. We say that a batch is completed as soon as all its jobs are completed; let $t_j$ denote the completion time of batch $B_j$, $t_j = \max_{i=1,\ldots,u_j} t_{j,i}$. The goal is to minimize the *flow time* of the batches, [*i.e.*]{} ${\ensuremath{\operatorname{\mathscr{B}-FlowTime}}}=\sum_{B_j\in{{{\ensuremath{\mathscr{B}}}}}} (t_j-a_j)$. We denote by ${{\ensuremath{\operatorname{\mathscr{B}OPT}}}}_s({{{\ensuremath{\mathscr{B}}}}})$ the flow time of an optimal $s$-speed schedule for ${{{\ensuremath{\mathscr{B}}}}}$.
Similarly to the broadcast setting, we say that a request $B_j$ (resp., a job $J_{j,i}$) is *alive* at time $t$ if $a_j{\leqslant}t {\leqslant}t_j$ (resp., $a_j {\leqslant}t {\leqslant}t_{j,i}$).\
#### [[[[<span style="font-variant:small-caps;">**[Equi$\circ$A]{}**</span>]{}]{}]{}]{} Algorithms Family.
Given a *job* scheduling algorithm $A$, we define the *batches* scheduling algorithm [[[[<span style="font-variant:small-caps;">**[Equi$\circ$A]{}**</span>]{}]{}]{}]{} as follows. Let $R(t)$ denote the set of batches that are alive at time $t$. [[[[<span style="font-variant:small-caps;">**[Equi$\circ$A]{}**</span>]{}]{}]{}]{} allots to each batch alive at time $t$ an equal amount of processors, [*i.e.*]{}, $p/|R(t)|$; then, it runs algorithm $A$ on each alive batch $B_j$ to decide how to split the amount of processors alloted to $B_j$ among its own alive jobs $J_{j,i}$. In the following, we only require algorithm $A$ to be *fully active*, [*i.e.*]{}, that it *allots at all time all the amount of processors* it is given to the alive jobs ([*i.e.*]{}, never idles on purpose). Under this requirement, our results hold independently of the choice of $A$. Examples of fully active algorithms $A$ are: $A={{{{\textsc{\bfseries{Equi}}}\/}}}$ which equally splits the amount of processors; or $A={{{{\textsc{\bfseries{MinIdx}}}\/}}}$ which allots all the amount of processors to the smallest indexed alive job $J_{j,i}$ in $B_j$, [*i.e.*]{} $i=\min\{i': \text{$J_{j,i'}$ is alive at time $t$}\}$.\
#### Analysis of [[[[<span style="font-variant:small-caps;">**[Equi$\circ$A]{}**</span>]{}]{}]{}]{}.
To analyze the competitiveness of [[[[<span style="font-variant:small-caps;">**[Equi$\circ$A]{}**</span>]{}]{}]{}]{}, we associate to each batches scheduling instance ${{{\ensuremath{\mathscr{B}}}}}$, two instances, ${{{\ensuremath{\mathscr{J}}}}}'$ and ${{{\ensuremath{\mathscr{J}}}}}''$, of job scheduling. We first bound the performances of our algorithm [[[[<span style="font-variant:small-caps;">**[Equi$\circ$A]{}**</span>]{}]{}]{}]{} on ${{{\ensuremath{\mathscr{B}}}}}$ from above by the performances of [[[[<span style="font-variant:small-caps;">**[Equi]{}**</span>]{}]{}]{}]{} on ${{{\ensuremath{\mathscr{J}}}}}'$ (Lemma \[lem:EA:EQ\]). We then use the “harder” job instance ${{{\ensuremath{\mathscr{J}}}}}''$ to show that the job instance ${{{\ensuremath{\mathscr{J}}}}}'$ was in fact “easier” than the batch instance ${{{\ensuremath{\mathscr{B}}}}}$ if one increases slightly the number of processors (Lemmas \[lem:JOPT’:JOPT”\] and \[lem:JOPT”:BOPT\]). Since [[[[<span style="font-variant:small-caps;">**[Equi]{}**</span>]{}]{}]{}]{} is competitive on ${{{\ensuremath{\mathscr{J}}}}}'$, we can then conclude on the competitiveness of [[[[<span style="font-variant:small-caps;">**[Equi$\circ$A]{}**</span>]{}]{}]{}]{} on [[[$\mathscr{B}$]{}]{}]{} (Theorem \[thm:EA\]).
Consider a Seq-Par batches scheduling instance ${{{\ensuremath{\mathscr{B}}}}}=\{B_1,\ldots, B_q\}$ where each batch $B_j=\{J_{j,1},\ldots, J_{j,u_j}\}$ arrives at time $a_j$ and each $J_{j,i}$ in $B_j$ consists of a sequential phase of work $w_{j,i}^s$ followed by a parallel phase of work $w_{j,i}^p$. Consider the $s$-speed schedule obtained by running algorithm [[[[<span style="font-variant:small-caps;">**[Equi$\circ$A]{}**</span>]{}]{}]{}]{} on instance ${{{\ensuremath{\mathscr{B}}}}}$; let $\rho_{j,i}^t$ denote the amount of processors allotted by [[[[<span style="font-variant:small-caps;">**[Equi$\circ$A]{}**</span>]{}]{}]{}]{} to job $J_{j,i}$ at time $t$, and $\rho_j^t = \sum_{J_{j,i}\in B_j} \rho_{j,i}^t$ denote the amount of processors allotted to batch $B_j$ at time $t$; let $t_{j,i}$ (resp., $t_{j}$) be the completion time of job $J_{j,i}$ (resp., batch $B_j$). We define a Seq-Par job scheduling instance ${{{\ensuremath{\mathscr{J}}}}}'=\{J'_1,\ldots,J'_q\}$, where each job $J'_j$ arrives at time $a_j$, and is composed of a sequential phase of work ${w'_j}^s=\max_{J_{j,i}\in B_j} w_{j,i}^s$, followed by a parallel phase of work ${w'_j}^p=\int_{a_j+{w'_j}^s}^{t_j} \rho_j^t\, dt$; intuitively, ${w'_j}^s$ is the length of the longest sequential phase among the jobs in $B_j$ and ${w'_j}^p$ is the total amount of parallel work in $B_j$ to be scheduled by [[[[<span style="font-variant:small-caps;">**[Equi$\circ$A]{}**</span>]{}]{}]{}]{} after the completion of the last sequential phase among the jobs in $B_j$.
The key to the next lemma is that one gets exactly the same job schedule of the jobs in ${{{\ensuremath{\mathscr{J}}}}}'$ by running algorithm [[[[<span style="font-variant:small-caps;">**[Equi]{}**</span>]{}]{}]{}]{} on instance ${{{\ensuremath{\mathscr{J}}}}}'$ as by alloting at all time to each job $J'_j$ the same amount of processors as the jobs in $B_j$ received from [[[[<span style="font-variant:small-caps;">**[Equi$\circ$A]{}**</span>]{}]{}]{}]{}.
\[lem:EA:EQ\] If $A$ is fully active, then $\displaystyle {{{{\textsc{\bfseries{Equi$_s\circ$A}}}\/}}}({{{\ensuremath{\mathscr{B}}}}}) = {{{{\textsc{\bfseries{Equi}}}\/}}}_s({{{\ensuremath{\mathscr{J}}}}}')$.
As long as the longest sequential phase among the jobs in batch $B_j$ is not completed, the batch $B_j$ is alive. By construction, job $J'_j$ is also alive as long as this sequential phase is not completed. Since the amount of processors given to batch $B_j$ in [[[[<span style="font-variant:small-caps;">**[Equi$\circ$A]{}**</span>]{}]{}]{}]{} is given by [[[[<span style="font-variant:small-caps;">**[Equi]{}**</span>]{}]{}]{}]{}, and since [[[[<span style="font-variant:small-caps;">**[Equi]{}**</span>]{}]{}]{}]{} is non-clairvoyant, [[[[<span style="font-variant:small-caps;">**[Equi$\circ$A]{}**</span>]{}]{}]{}]{} allots the same amount of processors to $B_j$ as [[[[<span style="font-variant:small-caps;">**[Equi]{}**</span>]{}]{}]{}]{} allots to $J'_j$ until the completion of the longest sequential phase among the jobs in batch $B_j$. By construction, the longest sequential phase in batch $B_j$ and the sequential phase of $J'_j$ end at the same time and at this moment, all the jobs alive in $B_j$ are in their parallel phase. Thus by construction, the overall amount of remaining parallel work in $B_j$ at that time is equal to the parallel work assigned to $J'_j$. By construction, the amount of processors given to $J'_j$ equals the amount of processors alloted to batch $B_j$ which is in turn equal to the total amount alloted to each of its remaining alive jobs since $A$ is fully active. The overall remaining amount of parallel work is thus identical in $J'_j$ and $B_j$ until they complete at the same time. Their flow times are thus identical in both schedules. We conclude the proof by reasoning inductively on the completion times (sorted in non-decreasing order) of each phase of each job in each batch.
We now define the job instance ${{{\ensuremath{\mathscr{J}}}}}''=\{J''_1,\ldots,J''_q\}$. ${{{\ensuremath{\mathscr{J}}}}}''$ is a kind of worst case instance of the batch instance ${{{\ensuremath{\mathscr{B}}}}}$, where all the parallel work in each batch $B_j$ has to be scheduled after the longest sequential phase in $B_j$. Job $J''_j$ arrives at time $a_j$ and consists of a sequential phase of work ${w''_j}^s = \max_{J_{j,i}\in B_j} w_{j,i}^s$, followed by a parallel phase of work ${w''_j}^p=\sum_{J_{j,i}\in B_j} w_{j,i}^p$.
\[lem:JOPT’:JOPT”\] ${{\ensuremath{\operatorname{JOPT}}}}_s({{{\ensuremath{\mathscr{J}}}}}') {\leqslant}{{\ensuremath{\operatorname{JOPT}}}}_s({{{\ensuremath{\mathscr{J}}}}}'').$
Since for all $j$, the sequential works of jobs $J'_j$ and $J''_j$ are identical and the parallel work in $J'_j$ is bounded from above by the parallel work in $J''_j$, any schedule of ${{{\ensuremath{\mathscr{J}}}}}''$ is valid for ${{{\ensuremath{\mathscr{J}}}}}'$.
\[lem:JOPT”:BOPT\] For all $\delta>0$, $${{{\ensuremath{\operatorname{JOPT}}}}_{1+\delta}({{{\ensuremath{\mathscr{J}}}}}'') {\leqslant}(1+1/\delta) {{\ensuremath{\operatorname{\mathscr{B}OPT}}}}_1({{{\ensuremath{\mathscr{B}}}}})}.$$
The proof consists in showing that when $\delta$ extra processors are given, delaying the completion of each batch $B_j$ by a constant factor, $(1+1/\delta)$, allows to postpone the schedule of all the parallel job phases in $B_j$ after the completion of the last sequential phase in $B_j$, which concludes the proof by construction of ${{{\ensuremath{\mathscr{J}}}}}''$.
Sort the batches of ${{{\ensuremath{\mathscr{B}}}}}$ by non-increasing arrival time, [*i.e.*]{}, assume $a_1 {\geqslant}a_2 {\geqslant}\ldots {\geqslant}a_q$. Consider an optimal schedule ${{\ensuremath{\operatorname{\mathscr{B}OPT}}}}_1$ of batches $B_1,\ldots,B_q$ on one processor. We show by induction that there exists a schedule ${{\ensuremath{\operatorname{\mathfrak{S}}}}}$ of ${{{\ensuremath{\mathscr{J}}}}}''$ on $1+\delta$ processors such that each job $J''_j$ completes before time $t_j+f_j/\delta$, where $t_j$ and $f_j = t_j - a_j$ denote the completion time and the flow time of $B_j$ in ${{\ensuremath{\operatorname{\mathscr{B}OPT}}}}$, respectively. We now show that the parallel phase of each job $J''_j$ can be scheduled between time $t_j$ and $t_j+f_j/\delta$; this concludes the proof since, by construction, the sequential phase of $J''_j$ is necessarily completed before $t_j$. Start with the first job $J''_1$. Clearly, ${w''_1}^p {\leqslant}f_1$. Thus, the total parallel phase of $J''_1$ can be scheduled on the $\delta$ extra processors between time $t_1$ and $t_1+f_1/\delta$. Assume now that the parallel phases of jobs $J''_1,\ldots, J''_{j-1}$ have been scheduled in ${{\ensuremath{\operatorname{\mathfrak{S}}}}}$ during the time intervals $[t_1,t_1+f_1/\delta],\ldots,[t_{j-1},t_{j-1}+f_{j-1}/\delta]$ respectively, and consider job $J''_j$. Since the jobs are considered in non-increasing arrival times, each job $J''_k$ whose parallel phase has been scheduled in ${{\ensuremath{\operatorname{\mathfrak{S}}}}}$ between $t_j$ and $t_j+f_j/\delta$ arrived in the time interval $T=[a_j,t_j+f_j/\delta]$ and furthermore $t_k{\leqslant}t_j+f_j/\delta$. The total parallel work $W$ of all the jobs currently scheduled in ${{\ensuremath{\operatorname{\mathfrak{S}}}}}$ during $T$, is then in fact scheduled completely in ${{\ensuremath{\operatorname{\mathscr{B}OPT}}}}_1$ during $T$. Note that the parallel work of $J''_j$ was also scheduled in ${{\ensuremath{\operatorname{\mathscr{B}OPT}}}}_1$ during this time interval. Since ${{\ensuremath{\operatorname{\mathscr{B}OPT}}}}_1$ uses only one processor, we conclude that $W+{w''_j}^p {\leqslant}t_j+f_j/\delta-a_j = (1+1/\delta)f_j$. As one can schedule up to $(1+\delta)f_j/\delta = (1+1/\delta)f_j$ parallel work between time $t_j$ and $t_j+f_j/\delta$ on $1+\delta$ processors, the parallel work ${w''_j}^p$ of $J''_j$ can be scheduled in ${{\ensuremath{\operatorname{\mathfrak{S}}}}}$ on time.
We can now conclude the analysis of [[[[<span style="font-variant:small-caps;">**[Equi$\circ$A]{}**</span>]{}]{}]{}]{}.
\[thm:EA\] For all $\epsilon>0$ and $\delta>0$, ${{{{\textsc{\bfseries{Equi$\circ$A}}}\/}}}$ is a $(2+\epsilon)(1+\delta)$-speed $(2+4/\epsilon)(1+1/\delta)$-competitive algorithm for the Non Clairvoyant Seq-Par Batches Scheduling problem.
We use the result of [@Edmonds1999] on the competitiveness of [[[[<span style="font-variant:small-caps;">**[Equi]{}**</span>]{}]{}]{}]{} for the non-clairvoyant job scheduling problem to conclude the proof: $
{{{{\textsc{\bfseries{Equi$_{\ensuremath{(2+\epsilon)(1+\delta)}}\circ$A}}}\/}}}({{{\ensuremath{\mathscr{B}}}}})
\underset{\makebox[1cm]{\text{\scriptsize(Lemma~\ref{lem:EA:EQ})}}}{=} {{{{\textsc{\bfseries{Equi}}}\/}}}_{(2+\epsilon)(1+\delta)}({{{\ensuremath{\mathscr{J}}}}}')
\underset{{\text{\scriptsize(Theorem~1 in \cite{Edmonds1999})}}}{{\leqslant}} {(2+4/\epsilon) \, {{\ensuremath{\operatorname{JOPT}}}}_{(1+\delta)}({{{\ensuremath{\mathscr{J}}}}}')}
\underset{\makebox[1cm]{\text{\scriptsize(Lemma~\ref{lem:JOPT':JOPT''})}}}{{\leqslant}} (2+4/\epsilon) \, {{\ensuremath{\operatorname{JOPT}}}}_{(1+\delta)}({{{\ensuremath{\mathscr{J}}}}}'')
\underset{\makebox[1cm]{\text{\scriptsize(Lemma~\ref{lem:JOPT'':BOPT})}}}{{\leqslant}} (2+4/\epsilon)(1+1/\delta) \, {{\ensuremath{\operatorname{\mathscr{B}OPT}}}}_{1}({{{\ensuremath{\mathscr{B}}}}})$.
Competitiveness of [[[[<span style="font-variant:small-caps;">**[B-EquiSet]{}**</span>]{}]{}]{}]{} {#sec:EE}
==================================================================================================
Consider an instance of the online data broadcast problem with dependencies: a set ${{{\ensuremath{\mathscr{S}}}}}=\{S_1,\ldots, S_q\}$ of $q$ requests with arrival times $a_1,\ldots,a_q$, over $n$ items $I_1,\ldots,I_n$ of lengths $\ell_1,\ldots,\ell_n$. Let ${{{\ensuremath{\mathscr{E}}}}}_s$ be the $s$-speed schedule designed by [[[[<span style="font-variant:small-caps;">**[B-EquiSet]{}**</span>]{}]{}]{}]{} on instance ${{{\ensuremath{\mathscr{S}}}}}$, and ${{{{\textsc{\bfseries{B-EquiSet}}}\/}}}_s({{{\ensuremath{\mathscr{S}}}}})$ be its flow time. Let ${{{\ensuremath{\mathscr{O}}}}}_1$ be a $1$-speed optimal schedule of ${{{\ensuremath{\mathscr{S}}}}}$, and ${{\ensuremath{\operatorname{BOPT}}}}_1({{{\ensuremath{\mathscr{S}}}}})$ be its flow time.
Following the steps of [@EdmondsPruhs2003], we define an instance ${{{\ensuremath{\mathscr{B}}}}}$ of non-clairvoyant seq-par batches scheduling from ${{{\ensuremath{\mathscr{E}}}}}_s$ and ${{{\ensuremath{\mathscr{O}}}}}_1$, such that the performances of [[[[<span style="font-variant:small-caps;">**[B-EquiSet]{}**</span>]{}]{}]{}]{} on [[[$\mathscr{S}$]{}]{}]{} can be compared to the performances of [[[[<span style="font-variant:small-caps;">**[Equi$\circ$A]{}**</span>]{}]{}]{}]{} on [[[$\mathscr{B}$]{}]{}]{} for a particular fully-active algorithm $A$. More precisely, we construct [[[$\mathscr{B}$]{}]{}]{} such that 1) the flow time of [[[[<span style="font-variant:small-caps;">**[Equi$\circ$A]{}**</span>]{}]{}]{}]{} on [[[$\mathscr{B}$]{}]{}]{} bounds from above the flow time of [[[[<span style="font-variant:small-caps;">**[B-EquiSet]{}**</span>]{}]{}]{}]{} on [[[$\mathscr{S}$]{}]{}]{} and 2) the (batches) optimal flow time for [[[$\mathscr{B}$]{}]{}]{} is at most the (broadcast) optimal flow time for [[[$\mathscr{S}$]{}]{}]{} if it is given extra resources. Since [[[[<span style="font-variant:small-caps;">**[Equi$\circ$A]{}**</span>]{}]{}]{}]{} is competitive, we can then bound the performances of [[[[<span style="font-variant:small-caps;">**[B-EquiSet]{}**</span>]{}]{}]{}]{} with respect to the (batches) optimal flow time of [[[$\mathscr{B}$]{}]{}]{} which is by 2) bounded by the (broadcast) optimal flow time of [[[$\mathscr{S}$]{}]{}]{}.
The intuition behind the construction of ${{{\ensuremath{\mathscr{B}}}}}$ is the following. A batch of all-new jobs is created for each newly arrived request, with one job per requested item. Each job $J$ stays alive until its corresponding item $I$ is served in ${{{\ensuremath{\mathscr{E}}}}}_s$. $J$ is assigned at most two phases depending on the relative service times of $I$ in ${{{\ensuremath{\mathscr{E}}}}}_s$ and ${{{\ensuremath{\mathscr{O}}}}}_1$. The sequential phase of $J$ lasts until either $I$ is served in ${{{\ensuremath{\mathscr{E}}}}}_s$, or the broadcast of $I$ starts in ${{{\ensuremath{\mathscr{O}}}}}_1$. Intuitively, this means that it is useless to assign processors to $J$ before the optimal schedule does. At the end of its sequential phase, if $J$ is still alive, its parallel phase starts and lasts until the broadcast of $I$ is completed in ${{{\ensuremath{\mathscr{E}}}}}_s$; the parallel work for $J$ is thus defined as the total amount of bandwidth that its corresponding item $I$ received within $J$’s corresponding (broadcast) request in ${{{{\textsc{\bfseries{B-EquiSet}}}\/}}}$. By construction, with a suitable choice of $A$, [[[[<span style="font-variant:small-caps;">**[Equi$\circ$A]{}**</span>]{}]{}]{}]{} constructs the exact same schedule as [[[[<span style="font-variant:small-caps;">**[B-EquiSet]{}**</span>]{}]{}]{}]{} and claim 1) is verified. Concerning claim 2), the key is to consider the jobs corresponding to the broadcast requests for a given item $I$ that are served by a given broadcast of $I$ in ${{{\ensuremath{\mathscr{O}}}}}_1$ starting at some time $t$. The only jobs among them that will receive a parallel phase, are the one for which the broadcast of $I$ in ${{{\ensuremath{\mathscr{E}}}}}_s$ starts just before or just after $t$. By construction, the total amount of parallel work assigned to these jobs corresponds to the bandwidth assigned to the two broadcasts of item $I$ by ${{{\ensuremath{\mathscr{E}}}}}_s$ that start just before and just after time $t$, each of them being bounded by the length of $I$. The total amount of parallel work in the jobs for which the broadcast of the corresponding item $I$ starts in ${{{\ensuremath{\mathscr{O}}}}}_1$ at some time $t$, is then bounded by twice the length of $I$, and can thus be scheduled during the broadcast of $I$ in ${{{\ensuremath{\mathscr{O}}}}}_1$ if one doubles the number of processors, which proves claim 2).
The following formalizes the reasoning exposed above.\
#### The Job Set Instance ${{{\ensuremath{\mathscr{J}}}}}$.
Recall the broadcast instance ${{{\ensuremath{\mathscr{S}}}}}$, and the two broadcast schedules ${{{\ensuremath{\mathscr{E}}}}}_s$ and ${{{\ensuremath{\mathscr{O}}}}}_1$, defined at the beginning of this section, as well as the notations given in Section \[sec:not\]. In particular, let $C^{{{\ensuremath{\mathscr{E}}}}}_s(I_i,t)$ denote the completion time of the broadcast of item $I_i$ that starts just after $t$ in ${{{\ensuremath{\mathscr{E}}}}}_s$, and $B^{{{\ensuremath{\mathscr{O}}}}}_1(I_i,t)$ be the time of the beginning of the first broadcast of item $I_i$ that starts after $t$ in ${{{\ensuremath{\mathscr{O}}}}}_1$ (see Section \[sec:not\]). Recall the description of algorithm [[[[<span style="font-variant:small-caps;">**[B-EquiSet]{}**</span>]{}]{}]{}]{} in Section \[sec:alg:EE\]: at time $t$, let $R(t)$ be the set of alive requests; [[[[<span style="font-variant:small-caps;">**[B-EquiSet]{}**</span>]{}]{}]{}]{} splits equally the bandwidth $s$ among the alive requests and for each alive request $S_j$, it assigns an arbitrary rate $r_{j,i}(t)$ to each alive item $I_i$ in $S_j$, such that $\sum_{\text{$I_i$ alive in $S_j$}} r_{j,i}(t) = s/|R(t)|$; [[[[<span style="font-variant:small-caps;">**[B-EquiSet]{}**</span>]{}]{}]{}]{} broadcasts then each item $I_i$ at a rate $r_i(t) = \sum_{j} r_{j,i}(t)$ at time $t$.
Given [[[$\mathscr{S}$]{}]{}]{}, ${{{\ensuremath{\mathscr{E}}}}}_s$ and ${{{\ensuremath{\mathscr{O}}}}}_1$, we define the non-clairvoyant batches scheduling instance ${{{\ensuremath{\mathscr{B}}}}}=\{B_1,\ldots,B_q\}$, where each batch $B_j$ is released at the same time as $S_j$, [*i.e.*]{} at time $a_j$, and contains one seq-par job $J_{j,i}$ for each item $I_i\in S_j$ (note that the indices $i$ of the jobs $J_{j,i}$ in each batch $B_j$ may not be consecutive depending on the content of $S_j$). Each job $J_{j,i}$ consists of a sequential phase of work ${w_{j,i}^s = (\min\{ C^{{{\ensuremath{\mathscr{E}}}}}_s(I_i,a_j), B^{{{\ensuremath{\mathscr{O}}}}}_1(I_i, a_j)\} - a_j)}$, followed by a parallel phase of work $w_{j,i}^p$. If $C^{{{\ensuremath{\mathscr{E}}}}}_s(I_i,a_j) {\leqslant}B^{{{\ensuremath{\mathscr{O}}}}}_1(I_i, a_j)$, then $w_{j,i}^p = 0$; otherwise, $w_{j,i}^p = \int^{C^{{{\ensuremath{\mathscr{E}}}}}_s(I_i,a_j) }_{B^{{{\ensuremath{\mathscr{O}}}}}_1(I_i, a_j)} r^{{{{\ensuremath{\mathscr{E}}}}}_s}_{j,i}(t)\, dt + \eta$ where $\eta$ is an infinitely small amount of work, [*i.e.*]{} if the download of item $I_i$ in request $S_j$ is completed in ${{{\ensuremath{\mathscr{E}}}}}_s$ after it starts in ${{{\ensuremath{\mathscr{O}}}}}_1$, then the amount of parallel work assigned to $J_{j,i}$ is just slightly higher than the total amount of bandwidth allotted to item $I_i$ within the bandwidth allotted to request $S_j$ by ${{{{\textsc{\bfseries{B-EquiSet$_s$}}}\/}}}$ after the beginning of the corresponding broadcast in ${{{\ensuremath{\mathscr{O}}}}}_1$. Adding an infinitely small amount of work $\eta$ to the parallel phase of $J_{j,i}$ does not change the optimal batches schedule (except on a negligible (discrete) sets of dates) but since the algorithm [[[[<span style="font-variant:small-caps;">**[Equi$\circ$A]{}**</span>]{}]{}]{}]{} is non-clairvoyant, this ensures that the job $J_{j,i}$ remains alive until the broadcast of item $I_i$ completes even if ${{{{\textsc{\bfseries{B-EquiSet$_s$}}}\/}}}$ deliberately chooses not to broadcast item $I_i$ in the bandwidth allotted to request $S_j$ (the introduction of infinitely small extra load can be rigorously formalized by adding an exponentially decreasing extra load $\gamma/2^k$ to the $k$th requested job for a small enough $\gamma$).
\[lem:EE:EA\] There exists a fully-active algorithm $A$ such that: ${{{{\textsc{\bfseries{B-EquiSet}}}\/}}}_s({{{\ensuremath{\mathscr{S}}}}}) {\leqslant}{{{{\textsc{\bfseries{Equi$_s\circ$A}}}\/}}}({{{\ensuremath{\mathscr{B}}}}})$.
The proof follows the lines of [@EdmondsPruhs2003]. Given an amount of processors $\rho$ for an alive batch $B_j$, algorithm $A$ assigns to each alive job $J_{j,i}$ in $B_j$ at time $t$ the same amount of processors as [[[[<span style="font-variant:small-caps;">**[B-EquiSet$_s$]{}**</span>]{}]{}]{}]{} would have assigned at time $t$ to the corresponding alive item $I_i$ of the corresponding alive request $S_j$ which would have been assigned a bandwidth $\rho$. Since [[[[<span style="font-variant:small-caps;">**[B-EquiSet$_s$]{}**</span>]{}]{}]{}]{} allots all the bandwidth available to alive jobs, $A$ is fully-active. Now, since $\eta$ is infinitely small, this extra load does not affect the allocation of processors computed by ${{{{\textsc{\bfseries{Equi$_s\circ$A}}}\/}}}$ except over a negligible (discrete) set of dates. By immediate induction, each job $J_{j,i}$ remains alive in the schedule computed by [[[[<span style="font-variant:small-caps;">**[Equi$_s\circ$A]{}**</span>]{}]{}]{}]{}, as long as item $I_i$ is alive in batch $B_j$ in ${{{\ensuremath{\mathscr{E}}}}}_s$. This is clear as long as $J_{j,i}$ is in its sequential phase. Once $J_{j,i}$ enters its parallel phase, as long as the broadcast of item $I_i$ is not completed, either $I_i$ is broadcast by [[[[<span style="font-variant:small-caps;">**[B-EquiSet$_s$]{}**</span>]{}]{}]{}]{} in batch $B_j$ and $J_{j,i}$ is scheduled by [[[[<span style="font-variant:small-caps;">**[Equi$_s\circ$A]{}**</span>]{}]{}]{}]{} ($A$ copies [[[[<span style="font-variant:small-caps;">**[B-EquiSet$_s$]{}**</span>]{}]{}]{}]{}), or [[[[<span style="font-variant:small-caps;">**[B-EquiSet$_s$]{}**</span>]{}]{}]{}]{} *deliberately* chooses not to broadcast the alive item $I_i$ and since $J_{j,i}$ has an infinitely small amount of extra work, $J_{j,i}$ remains alive in [[[[<span style="font-variant:small-caps;">**[Equi$_s\circ$A]{}**</span>]{}]{}]{}]{} as well. The flow time for each job $J_{j,i}$ is then at least the flow time of the corresponding item $I_i$ in ${{{\ensuremath{\mathscr{E}}}}}_s$; we conclude that each batch $B_j$ completes in [[[[<span style="font-variant:small-caps;">**[Equi$_s\circ$A]{}**</span>]{}]{}]{}]{} no earlier than its corresponding request $S_j$ in [[[[<span style="font-variant:small-caps;">**[B-EquiSet$_s$]{}**</span>]{}]{}]{}]{}.
\[lem:U2\] There exists a $2$-speed batches schedule ${{\ensuremath{\operatorname{\Upsilon\!_2}}}}$ such that: ${{\ensuremath{\operatorname{\Upsilon\!_2}}}}({{{\ensuremath{\mathscr{B}}}}}) {\leqslant}{\ensuremath{\operatorname{B-FlowTime}}}({{{\ensuremath{\mathscr{O}}}}}_1)$.
Again, the proof follows the lines of [@EdmondsPruhs2003]. Consider an item $I_i$. We partition the requests $S_j$ containing item $I_i$ into classes ${{\ensuremath{\mathcal{C}}}}_1,{{\ensuremath{\mathcal{C}}}}_2,\ldots$, one for each broadcast of $I_i$ in ${{{\ensuremath{\mathscr{O}}}}}_1$. The $k$-th class ${{\ensuremath{\mathcal{C}}}}_k$ contains all the requests $S_j$ that download $I_i$ in ${{{\ensuremath{\mathscr{O}}}}}_1$ during its $k$th broadcast, [*i.e.*]{} all requests $S_j$ such that $b^{{{\ensuremath{\mathscr{O}}}}}_1(I_i,k-1) < a_j {\leqslant}b^{{{\ensuremath{\mathscr{O}}}}}_1(I_i,k)$ (see Section \[sec:not\] for notations). We show that for all $k$, the total parallel phases of the jobs $J_{j,i}$ such that $S_j\in {{\ensuremath{\mathcal{C}}}}_k$, can be shoehorned into twice the area of bandwidth allotted by ${{{\ensuremath{\mathscr{O}}}}}_1$ to the $k$th broadcast of item $I_i$. Since this holds for all $i$ and all $k$, we obtain a $2$-speed schedule ${{\ensuremath{\operatorname{\Upsilon\!_2}}}}$ such that ${{\ensuremath{\operatorname{\Upsilon\!_2}}}}({{{\ensuremath{\mathscr{B}}}}}){\leqslant}{\ensuremath{\operatorname{B-FlowTime}}}({{{\ensuremath{\mathscr{O}}}}}_1)$.
Let $t_1 = b^{{{\ensuremath{\mathscr{O}}}}}_1(I_i,k)$ be the time of the beginning of the $k$th broadcast of $I_i$ in ${{{\ensuremath{\mathscr{O}}}}}_1$. Consider a request $S_j$ in class ${{\ensuremath{\mathcal{C}}}}_k$, clearly $a_j{\leqslant}t_1$. By construction, job $J_{i,j}$ is assigned a non-zero parallel work only if $S_j$ completes the download of $I_i$ after $t_1$ in ${{{{\textsc{\bfseries{B-EquiSet$_s$}}}\/}}}$. Since $S_j$ arrives before $t_1$, it downloads $I_i$ during one of the two broadcasts of $I_i$ in ${{{{\textsc{\bfseries{B-EquiSet}}}\/}}}_s$ that start just before or just after $t_1$; let ${{\ensuremath{\mathcal{C}}}}_k^-$ (resp. ${{\ensuremath{\mathcal{C}}}}_k^+$) be the set of requests served by the broadcast that starts just before $t_1$ (resp. just after $t_1$). Let $t_2$ and $t_3$ be the completion times of the broadcast of $I_i$ in [[[[<span style="font-variant:small-caps;">**[B-EquiSet$_s$]{}**</span>]{}]{}]{}]{} that start just before and just after $t_1$ respectively. By construction, the total amounts $W^-$ and $W^+$ of parallel work assigned to the jobs $J_{j,i}$ such that $S_j\in{{\ensuremath{\mathcal{C}}}}_k^-$ and ${{\ensuremath{\mathcal{C}}}}_k^+$ are respectively: $W^- = \displaystyle \sum_{j\,:\,S_j\in{{\ensuremath{\mathcal{C}}}}_k^-} \int_{t_1}^{t_2} r_{j,i}(t)\,dt$ and $W^+ = \displaystyle \sum_{j\,:\,S_j\in{{\ensuremath{\mathcal{C}}}}_k^+} \int_{t_1}^{t_3} r_{j,i}(t)\,dt$. Let us rewrite $W^- + W^+ = R_1 + R_2$ with $R_1 =\int_{t_1}^{t_2} \sum_{j\,:\,S_j\in{{\ensuremath{\mathcal{C}}}}_k} r_{j,i}(t)\,dt {\leqslant}\int_{t_1}^{t_2} r_i(t)\, dt $ and $R_2 = \int_{t_2}^{t_3} \sum_{j\,:\,S_j\in{{\ensuremath{\mathcal{C}}}}_k^+} r_{j,i}(t)\,dt {\leqslant}\int_{t_2}^{t_3} r_i(t)\, dt$. $R_1$ and $R_2$ are thus at most the total area alloted to item $I_i$ by [[[[<span style="font-variant:small-caps;">**[B-EquiSet$_s$]{}**</span>]{}]{}]{}]{} during the broadcasts of $I_i$ that start just before and just after $t_1$; since a broadcast is completed as soon as the rates sum up to the length of the items, $R_1{\leqslant}\ell_i$ and $R_2{\leqslant}\ell_i$, and thus $W^-+W^+ {\leqslant}2\ell_i$. Since ${{{\ensuremath{\mathscr{O}}}}}_1$ allots a total bandwidth of $\ell_i$ to broadcast item $I_i$ after time $t_1$, and since the parallel works of the jobs $J_{j,i}$ such that $S_j\in {{\ensuremath{\mathcal{C}}}}_k$ are released at time $t_1$ and sum up to a total $W^-+W^+{\leqslant}2\ell_i$, one can construct on 2 processors, a $2$-speed schedule ${{\ensuremath{\operatorname{\Upsilon\!_2}}}}$ in which the parallel phases of each of these jobs $J_{j,i}$ completes before the $k$th broadcast of $I_i$ completes in ${{{\ensuremath{\mathscr{O}}}}}_1$.
Since no processor needs to be allotted to the sequential phases, repeating the construction for each item $I_i$ yields a valid $2$-speed schedule ${{\ensuremath{\operatorname{\Upsilon\!_2}}}}$ in which each job $J_{i,j}$ completes before the corresponding request $S_j$ completes the download of $I_i$ in ${{{\ensuremath{\mathscr{O}}}}}_1$. It follows that each batch $B_j$ is completed in ${{\ensuremath{\operatorname{\Upsilon\!_2}}}}$ before its corresponding request $S_j$ is served by ${{{\ensuremath{\mathscr{O}}}}}_1$.
We now conclude with the proof of the main theorem.
[Theorem \[thm:EE\]]{} Setting $s=(4+\epsilon)(1+\delta)$, the competitiveness of [[[[<span style="font-variant:small-caps;">**[Equi$\circ$A]{}**</span>]{}]{}]{}]{} (Theorem \[thm:EA\]) concludes the result: $
{{{{\textsc{\bfseries{B-EquiSet}}}\/}}}_{(4+\epsilon)(1+\delta)}({{{\ensuremath{\mathscr{S}}}}})
\underset{\makebox[1cm]{\text{\scriptsize(Lemma~\ref{lem:EE:EA})}}}{{\leqslant}} {{{{\textsc{\bfseries{Equi$_{\ensuremath{(4+\epsilon)(1+\delta)}}\circ$A}}}\/}}}({{{\ensuremath{\mathscr{B}}}}})
\underset{\makebox{\text{\scriptsize(Theorem~\ref{thm:EA})}}}{{\leqslant}} {(2+8/\epsilon)(1+1/\delta) \, {{\ensuremath{\operatorname{\mathscr{B}OPT}}}}_2({{{\ensuremath{\mathscr{B}}}}})}
{\leqslant}{(2+8/\epsilon)(1+1/\delta) \, {{\ensuremath{\operatorname{\Upsilon\!_2}}}}({{{\ensuremath{\mathscr{B}}}}})}
\underset{\makebox[1cm]{\text{\scriptsize(Lemma~\ref{lem:U2})}}}{{\leqslant}} (2+8/\epsilon)(1+1/\delta) \, {{\ensuremath{\operatorname{BOPT}}}}_1({{{\ensuremath{\mathscr{S}}}}})$.
#### The [[[[<span style="font-variant:small-caps;">**[B-EquiSet-Edf]{}**</span>]{}]{}]{}]{} algorithm. {#sec:EEDF}
We apply the same method as in [@EdmondsPruhs2003]. Let $s = (4+\epsilon)(1+\delta)²$ and $c = (2+8/\epsilon)(1+1\delta)²$. [[[[<span style="font-variant:small-caps;">**[B-EquiSet-Edf]{}**</span>]{}]{}]{}]{} simulates the $s/(1+\delta)$-speed execution of [[[[<span style="font-variant:small-caps;">**[B-EquiSet]{}**</span>]{}]{}]{}]{} and at each time $t$ such that the broadcast of an item $I_i$ in [[[[<span style="font-variant:small-caps;">**[B-EquiSet]{}**</span>]{}]{}]{}]{} is completed, it releases an item $I'_i$ of length $\ell_i$ with a deadline $t+(t-t')/\delta$ where $t'$ is the time of the beginning of the considered broadcast of $I_i$ in [[[[<span style="font-variant:small-caps;">**[B-EquiSet]{}**</span>]{}]{}]{}]{}. Then, [[[[<span style="font-variant:small-caps;">**[B-EquiSet-Edf]{}**</span>]{}]{}]{}]{} schedules on a bandwidth $s$ each item $I'_i$ according the earliest-deadline-first policy. With an argument similar to Lemma \[lem:JOPT”:BOPT\] or [@EdmondsPruhs2003], one can show that a feasible schedule of the items $I'_i$ exists and thus that earliest-deadline-first constructs it which ensures that [[[[<span style="font-variant:small-caps;">**[B-EquiSet-Edf]{}**</span>]{}]{}]{}]{} is $s$-speed $c$-competitive. Since earliest-deadline-first preempts the broadcast of an item only when a new item arrives, [[[[<span style="font-variant:small-caps;">**[B-EquiSet-Edf]{}**</span>]{}]{}]{}]{} preempts each broadcast at most once on average. Note that one can avoid long idle period in [[[[<span style="font-variant:small-caps;">**[B-EquiSet-Edf]{}**</span>]{}]{}]{}]{}’s schedule by broadcasting an arbitrary item $I_i$ alive in [[[[<span style="font-variant:small-caps;">**[B-EquiSet]{}**</span>]{}]{}]{}]{} at time $t$ if no item $I'_i$ is currently alive.\
#### Concluding remarks.
Several directions are possible to extend this work. First, [[[[<span style="font-variant:small-caps;">**[B-EquiSet]{}**</span>]{}]{}]{}]{} does not have precise policy to decide in which order one should broadcast the items within each requested set; deciding on a particular policy may lead to better performances (bandwidth and/or competitive ratio). Second, it might be interesting to design a longest-wait-first greedy algorithm in presence of dependencies; [[[[<span style="font-variant:small-caps;">**[B-EquiSet]{}**</span>]{}]{}]{}]{} shows that the items should not simply receive bandwidth according to the number of outstanding requested sets for this item (the allotted bandwidth depends also on the number of outstanding items within each outstanding set), it is thus a challenging question to design proper weights to aggregate the current waits of the requested sets including a given item.
Omitted proof
=============
[Fact \[fac:yao\]]{} We use Yao’s principle (see [@Yao1977; @MotwaniRaghavan1995]) to extend the result to randomized algorithms. We consider the following probabilistic distribution of requests set over $n$ items: $1+\sqrt n$ requests arrive at time $t=0$; one request asks for an uniform random subset $S_0$ of size $n-\sqrt{n}$ of the $n$ items; and each of the requests $S_1,\ldots, S_{\sqrt n}$ asks for one random distinct item among the $\sqrt n$ remaining items. Consider again any deterministic algorithm $A$ with bandwidth $s$. Since $A$ is deterministic and ignores the dependencies, the schedule designed by $A$ schedule is independent of the random instance. At time $t=n/(2s)$, the broadcast of at least $n/2$ items is not completed. Thus, the probability that request $S_j$, for $j{\geqslant}1$, asks for one of these items is at least $1/2$. Then, the expected number of unsatisfied request at time $t=n/(2s)$ is at least $\sqrt{n}/2$. We conclude that the expected flow time for any deterministic algorithm with bandwidth $s$ under this distribution of request is at least $n\sqrt n/(4s)$. According to Yao’s principle, the worst expected flow time of any randomized algorithm over the collection of all the considered instances is at least $n\sqrt n/(4s)$. But ${{\ensuremath{\operatorname{BOPT}}}}_1 \sim \frac32 n$, which concludes that no randomized algorithm is $s$-speed $c$-competitive, for all $s$ and $c<\sqrt{n}/(6s)$.
[^1]: This work is supported by the CNRS Grant .
[^2]: Remark that this formalization prevents from broadcasting the same item twice at a given time or from aborting the current broadcast of an item. The first point is not restrictive since if two broadcasts of the same item overlap, one reduces the service time by using the beginning of the bandwidth allotted to the second broadcast to complete earlier the first, and then the end of the first to complete the second on time. The second point is at our strict disadvantage since it does not penalize an optimal schedule that would never start a broadcast to abort it later on.
|
---
abstract: 'It has been known for a long time that large-$N$ methods can give invaluable insights into non-perturbative phenomena such as confinement. Lattice techniques can be used to compute quantities at large $N$. In this contribution, I review some recent large-N lattice results and discuss their implications for our understanding of non-perturbative QCD.'
address: 'Physics Department, Swansea University, Singleton Park, Swansea SA2 8PP, UK'
author:
- Biagio Lucini
bibliography:
- 'ichep14bl.bib'
title: 'Non-perturbative results for large-$N$ gauge theories '
---
Lattice Gauge Theories ,Large-$N$ limit ,Meson spectrum ,Glueballs
Introduction and motivations {#sect:introduction}
============================
An analytical determination of observables in Quantum Chromodynamics (QCD) is still an open issue. From the computational point of view, much progress has been achieved by formulating the theory on a spacetime lattice and determining physical quantities using Monte Carlo simulations. Lattice QCD is by now a mature field, which provides a first-principle framework for computing numerically hadronic quantities. However, while the results provide a robust evidence (if still needed at all) that QCD [*is*]{} the theory describing strong interactions, unfortunately our ability to compute the spectrum does not necessarily provides physical insights on the relevant low-energy phenomena, namely confinement and chiral symmetry breaking.
From an analytical perspective, one of the most promising approaches was provided long ago in [@'tHooft:1973jz]. The key observation is that if we consider QCD in the general context of SU($N$) gauge theories and take the limit for the number of colours $N$ going to infinity keeping constant the ’t Hooft coupling $\lambda = g^2 N$ (with $g$ gauge coupling of the SU($N$) theory), the system undergoes a drastic simplification at the diagrammatic level. In fact, it can be easily seen that in this limit only the planar diagrams (i.e. the Feynman diagrams that can be drawn in a plane without crossing lines) survive. In addition, diagrammatic contributions to observables can be arranged in a topological expansion, where the topology of a diagram is reflected by a well-defined power of 1/$N$ weighting its contribution.
While from the qualitative point of view the large-$N$ idea allows us to understand various phenomenological features of QCD, in the strict quantum field theoretical context it has proven to be still difficult to arrive at first-principle determinations of observables even in this simplified framework. Much progress was achieved following the gauge-string duality conjecture [@Maldacena:1997re], which led to the idea of computing non-perturbative quantities in QCD using the supergravity limit of an appropriate string theory. From the analytical point of view, this shifted the game to the construction of a string theory background that is dual to large-$N$ Yang-Mills theory or large-$N$ QCD [@Maldacena:2000yy; @Klebanov:2000hb; @Sakai:2004cn]. However, the evaluation of the size of the 1/$N$ corrections is still out of reach in this framework, since it involves going beyond the supergravity approximation. Besides, string theory naturally embeds supersymmetry. This means that, in addition to gluons and quarks, the gauge theory dual to a string theory will have other fields, whose effects on the infrared spectrum need to be carefully discussed.
Although this approach is one of the most popular, the gauge-string duality is not the only framework that performs analytical calculations in the large $N$ limit of gauge theories. Among other frameworks, we mention the topological string model recently proposed in [@Bochicchio:2013eda].
With these premises, a numerical approach to the large-$N$ limit of SU($N$) gauge theories can serve a twofold purpose: (a) it provides a first-principle quantification of the deviations of QCD observables from their large-$N$ limit in the non-perturbative regime; (b) it can provide a more direct numerical guidance to calculations aiming at identifying appropriate string theory duals of QCD. Moreover, analytical progress can be inputed back into numerical calculations of QCD to inform numerical interpolations or extrapolations. Inspired by these motivations, following earlier attempts, in the past fifteen years a broad lattice programme of numerical simulations has been undertaken with the goal of providing firm quantitative results for QCD observables using lattice techniques (see [@Panero:2012qx; @Lucini:2012gg; @Lucini:2013qja] for recent reviews).
In this contribution, we review the foundations and the most recent developments of lattice calculations in the large-$N$ limit of SU($N$) gauge theories. The rest of the article is organised as follows. In Sect. \[sect:2\] we give a brief overview of the large-$N$ general results that will be used in our lattice calculations, with the lattice formulation of the problem exposed in Sect. \[sect:3\]. Sect. \[sect:4\] will be devoted to the presentation of numerical results for glueballs and mesons. A brief summary with an overview on future perspectives completes this work.
Large $N$ and QCD {#sect:2}
=================
The foundations of large-$N$ gauge theories are the subject of various pedagogical reviews (e.g. [@Lucini:2012gg; @Lucini:2013qja; @Coleman:1980mx; @Manohar:1998xv]). Here we will briefly present an overview of the line of arguments and of the main results, referring to the literature for a more in-depth tractation.
In a SU($N$) gauge theory with $N_f$ fundamental fermion flavours, let us rescale the fields with $N$ in such a way to expose the gauge coupling $\sqrt{\lambda} = g \sqrt{N}$ [@Coleman:1980mx]. This allows us to easily track the factor of $N$ contributions in Feynman diagrams. In particular, one finds that (a) a vertex contributes a factor of $N$; (b) a fermionic loop contributes a factor of $N$; (c) a fermionic propagator contributes a factor of $1/N$. In addition, at large $N$, one can consider a gluon line as a double line with two orientations, corresponding respectively to a fermion and to an antifermion. In this double-line notation, the above considerations about colour contributions provided by fermions naturally extend to gluons. As a result, for a generic connected vacuum amplitude ${\cal A}$ we find $$\begin{aligned}
\label{eq:1}
{\cal A} \propto N^{N_V - N_P + N_L} \ ,\end{aligned}$$ where $N_V$ is the number of interaction vertices, $N_P$ the number of fermionic propagators (considering each gluon propagator as two fermion propagators) and $N_L$ the number of fermion loops (again, considerings the gluonic contributions as due to fermions and antifermions) in the corresponding Feynman diagram. Drawing the latter diagram in the double line notation, one notices that it can be seen as a polygon, or better, as the surface of a three dimensional solid, with the arrows on the fermionic lines giving orientation to its flat faces. In this context, the exponent of the power of $N$ in Eq. (\[eq:1\]) is the Euler characteristic $\chi$: $$\begin{aligned}
\chi = N_V - N_P + N_L = 2 - B - 2H \ ,\end{aligned}$$ where $B$ (the number of holes) and $H$ (the number of handles) are topological invariants of the solid associated with the diagram. Hence, if $B = H = 0$, for instance, we have a polyhedron, which in this context has the topology of a sphere. Likewise, removing one face will pinch a hole in the sphere. Hence, a natural topological classification of vacuum to vacuum connected diagrams emerges. The diagrams with spherical topology are dominant at large $N$, and correspond to vacuum to vacuum processes with only gluonic contributions. Each fermion loop pinches a hole in the sphere, causing a suppression of $1/N$, while more complicated drawings (corresponding to crossing of lines) can be associated to handles, each of which suppresses the diagram by $1/N^2$.
The diagrams for processes involving glueballs and mesons can be obtained from vacuum diagrams considering the operators that create those states as coupled to external sources. One can choose the normalisation so that that two-point functions of glueballs and two-point functions of mesons are of order one in the large-$N$ limit, and hence correlators of those states are finite as $N \to
\infty$. The argument can be further developed, leading to the following considerations:
- in the pure gauge theory, corrections to the large-$N$ limit can be expressed as a power expansion in $1/N^2$, while if fermions are present the power series is in $1/N$;
- quark loop effects are of order $1/N$;
- amplitudes involving three or more glueball or meson operators are zero at $N = \infty$ (i.e. scattering and decays are suppressed at large $N$);
- the mixing between glueballs and mesons is of order $1/\sqrt{N}$;
- processes with initial and final quark states involving annihilation of all initial quarks in intermediate states[^1] are forbidden at $N = \infty$.
Hence, the drastic diagrammatic simplification of the theory is reflected by well-defined physical signatures. Large-$N$ arguments can also be extended to baryons [@Dashen:1993jt], for which a rotor-like spectrum naturally emerges [@Jenkins:1993zu].
A remarkable feature of the large-$N$ limit, which can easily derived from the counting rules provided above, is that in the large-$N$ limit the effect of fermion loops disappear. The resulting theory consists of probe external fermions interacting with the gluons, but not causing any back-reaction on the system. At finite $N$, the approximation that removes all the fermion loops is called the quenched approximation (which is hence exact in the large-$N$ limit). The quenched approximation has played an important role in early numerical simulations of QCD.
The large-$N$ behaviour of SU($N$) gauge theories (including quenching of fermionic matter, observed in lattice simulations of SU(3) for quark masses at which unquenching effects would have been expected to show up clearly) is closely reminiscent of the physics of QCD, and in particular of strong suppressions or long lifetimes with respect to naive evaluations of strengths of couplings and decay widths. However, these diagrammatic arguments do not address crucial questions such as: (1) Can we define rigorously the large-$N$ limit of SU($N$) gauge theories? (2) If this limit exists, how can we quantify how close it is to QCD? (3) Are large-$N$ diagrammatic arguments valid also in the non-perturbative regime of QCD?
Answering those questions requires a first principle approach. In the following section we will show how Lattice Gauge Theory can provide the needed [*ab-initio*]{} framework to address these issues.
Lattice formulation {#sect:3}
===================
The lattice discretisation of an asymptotically free gauge theory provides a non-perturbative gauge invariant regularisation of that theory that can be used to compute (e.g. numerically) $n$-point correlation functions at any value of the coupling. There is a rigorous prescription for removing the ultraviolet cut-off (which in this case is the spacing $a$ of the grid) that allows to define a Quantum Field Theory in continuum Euclidean spacetime. In fact, the lattice prescription can be used as a constructive definition of the Quantum Field Theory. A programme based on these ideas has been carried out for QCD in the past forthy years. The framework has reached a level of maturity such that first principle precision calculations of QCD observables now begin to be possible.
While phenomenology suggests to put a substantial effort in the $N =
3$ case and to specialise tools and techniques for real-world QCD, the arguments exposed in the previous section provide a robust case for investigating generic SU($N$) gauge theories. The lattice action of a SU($N$) gauge theory can be written as $$\begin{aligned}
S = S_g + S_f \ ,\end{aligned}$$ where $S_g$ is the contribution of the gauge fields and $S_f$ contains the fermion contribution. The request for constructing a lattice action is that in the ultraviolet regime (which, by asymptotic freedom, corresponds to weak coupling) it flows to the perturbative Gaussian fixed point of the continuum action. This leaves the freedom to add irrelevant terms, which takes the form of operators of mass dimension $\Delta$ larger than four. At tree level, these operators are suppressed as $a^{\Delta
- 4}$. Asymptotic freedom guarantees that even when loop corrections are taken into account these operators do not spoil the correctness of the continuum limit.
In order to preserve gauge invariance on a lattice, we formulate the gauge fields in terms of parallel transports along links connecting nearest-neighbour lattice points: $$\begin{aligned}
U_{\mu}(i) = \mathrm{P} \exp \left( i g_0 \int_i ^{i + \hat{\mu}}
A_{\mu}(x) \mathrm{d} x^{\mu} \right) \ ,\end{aligned}$$ where $g_0$ is the bare coupling. $i$ is the set of integer coordinates labelling the given point on a grid, $A$ is the gauge field in the continuum and the path ordered exponential is taken along the link stemming from $i$ and ending in $i + \hat{\mu}$, with $\hat{\mu}$ versor in direction $\mu$. Local SU($N$) gauge transformations $G(i)$, which have supports on points $i$, transform the links as follows: $$\begin{aligned}
U_{\mu}(i) \to \left(G(i)\right)^{\dag} U_{\mu}(i) G(i + \hat{\mu}) \ .\end{aligned}$$ The path ordered product of links around the elementary square of the lattice originating from $i$ in positive directions $\hat{\mu}$ and $\hat{\nu}$ is given by the plaquette variable $$\begin{aligned}
U_{\mu \nu}(i) = U_{\mu}(i) U_{\nu}(i + \hat{\mu}) \left( U_{\mu}(i + \hat{\nu})
\right)^\dag \left( U_{\nu}(i) \right)^\dag \ ,\end{aligned}$$ with the negative links connecting $i$ and $i - \hat{\mu}$ identified with the dagger of the positive links connecting $i - \hat{\mu}$ and $i$. The simplest choice for $S_g$ is the Wilson action $$\begin{aligned}
S_g = \beta \sum_{i, \mu < \nu} \left( 1 - {\cal R}\mathrm{e Tr}U_{\mu
\nu}(i) \right) \ ,\end{aligned}$$ which is defined in terms of the real parts of the plaquettes summed over the whole lattice and is weighted by the lattice coupling $\beta
= 2 N/g_0^2$.
Concerning the fermionic action $S_f$, a naive discretisation produces doublers (i.e. 15 unwanted species in the continuum limit). In fact, it has been shown that no fermion discretisation can be performed such that chirality, absence of doublers and ultralocality are preserved at the same time [@Nielsen:1980rz]. The Wilson formulation, which will be used in this work, break explicitly chiral symmetry. In general, we can write the action as a quadratic form in the fermion fields $\psi_{\alpha}(i)$, where $\alpha$ is a spinor index, as follows: $$\begin{aligned}
S_f = \overline{\psi}_{\alpha} (i) M_{\alpha \beta}(ij)
\psi_{\beta}(j) \ .\end{aligned}$$ In the Wilson formulation, the operator $M$ (referred to as the Dirac operator) is given by $$\begin{aligned}
\\ \nonumber
M_{\alpha \beta}(ij) &=& \left(m+ 4r \right) \delta_{ij}
\delta_{\alpha \beta} \\ \nonumber
&-& \frac{1}{2} \left[\left(r - \gamma_{\mu}\right)_{\alpha
\beta}U_{\mu}(i) \delta_{i,j+\mu} \right.\\
&+& \left. \left( r +
\gamma_{\mu}\right)_{\alpha \beta} U_{\mu}^{\dag}(j)\delta_{i,i-\mu}
\right] \ ,\end{aligned}$$ where the explicit chiral symmetry breaking is due to $r \ne 0$. In our simulations, we set $r = 1$. The explicit breaking of chiral symmetry determines a non-zero additive renormalisation for the fermion mass, which needs to be determined as a part of the Monte Carlo simulations (e.g. by tuning the mass of the pseudoscalar to zero). The path integral of the theory reads $$\begin{aligned}
Z = \int \left( {\cal D} U_{\mu}(i)\right)
(\det M(U_{\mu}))^{N_f} e^{-S_g(U_{\mu \nu}(i))} \ ,\end{aligned}$$ where the integration over the fermion fields has been performed and $N_f$ is the number of fermion flavours.
Non-perturbative results for observables in the pure gauge sector (for which, $M = \mathbb{I}$) and in the theory with fermionic matter can be performed at any $N$ and for values of the lattice spacing $a$ for which the theory is close to its continuum limit. At fixed $N$, a controlled $a \to 0$ extrapolation can be performed, giving non-perturbative results for the continuum theory. Finally, each continuum observable can be extrapolated to $N \to \infty$. For the latter extrapolation, we use a power series in $1/N^2$ for the pure gauge theory and theories with non-backreacting fermionic probe matter and in $1/N$ if there are fermion loops. The order of the maximum power that is constrained by our data will give a quantitative characterisation of how close the theory is to its $N = \infty$ limit. Note that this could depend on the observable.
The limits $a \to 0$ and $N \to \infty$ commute [@'tHooft:2002yn]. Due to computational demands, it is not always practical to perform first the limit $a \to 0$ and then the limit $N \to \infty$. In cases where this procedure is unviable, one can still perform the large $N$ limit at some fixed value of the lattice spacing $a$, where the common value is set by fixing the numerical value of a dimensionful operator expressed in units of $a$ (generally, $a \sqrt{\sigma}$, with $\sigma$ the string tension or $a T_c$, with $T_c$ the deconfinement phase transition).
Whether the continuum limit is performed before or after the large-$N$ limit, the practicalities of the simulations and the need to keep finite size discretisation artefacts under control often restrict our maximum $N$ to eight. Hence, the results we will use for the latter extrapolation will be mostly in the interval $2 \le N \le 8$. A different approach is used by other authors (see e.g. [@Narayanan:2004cp; @Hietanen:2009tu; @GonzalezArroyo:2012fx]) that is based on the idea of reduction (at large $N$ the theory can be formulated on a single-point lattice [@Eguchi:1982nm; @Bhanot:1982sh; @GonzalezArroyo:1982hz]) or partial reduction (finite size effects disappear at large $N$, hence the minimal lattice size for which the system is confined is already asymptotic for the spectrum [@Narayanan:2003fc]). Using those ideas allows one to reach larger values of $N$, at the expenses of having different finite-$N$ corrections (in the case of complete reduction) or less control on finite-size effects (for partial reduction). These techniques are complementary to those presented here. Giving a comparison of the methods and providing a discussion of the corresponding results is beyond the scope of this work.
The spectrum {#sect:4}
============
![Masses of the lowest-lying glueballs for $N=2,3,4,6,8$, with their extrapolation to large $N$ [@Lucini:2004my].[]{data-label="fig:sungall"}](FIGS/sun_gall){width="1.1\columnwidth"}
We start from the computationally easier case of the pure Yang-Mills theory. Observables of interest in this case include masses of gauge-invariant states ([*glueballs*]{}). These are obtained from the large time behaviour of correlation functions of the form $$\begin{aligned}
C(t) = \langle O^{\dag}(0) O(t) \rangle \mathop{\propto}^{t \to \infty} e^{- m t} \ ,\end{aligned}$$ where $O$ is a traced product of links along a closed path ${\cal C}$ that transforms in an irreducible representation of angular momentum $J$ of the rotational group. For the definition of $C(t)$, the zero-momentum component (i.e. the spatial average) is taken. If the path is constructed in such a way that it has a definite parity $P$ and a charge conjugation eigenstate is constructed by taking either the real ($C = 1$) or imaginary ($C = -1$) part of the trace, then $m$ is the lowest state in the $J^{PC}$ channel. In practice, after lattice discretisation, the group of rotations is broken to the dihedral group of rotations of the cube. Classifying the lattice states according to the irreducible representations of this group proves to give a cleaner signal in numerical simulations. The full rotational quantum numbers can be reconstructed by looking at the decomposition of the lattice rotational symmetry group under irreducible representations of $SO(3)$. Likewise, the signal over noise ratio is significantly improved if in any channel one measures more than one operator and for each operator its cross-correlations with all others at various $t$. This defines the correlation matrix $$\begin{aligned}
C_{lm}(t) = \langle O^{\dag}_l(0) O_m(t) \rangle \ ,\end{aligned}$$ where $l$ and $m$ label operators associated to two different paths. The eigenvalues of $C^{-1}(0) C(t)$ decay exponentially, with a rate controlled by the masses in the given channel. Hence, ordering the eigenvalues, it is possible to extract the mass of the groundstate and the mass of the first few excitations, in what can be regarded as a variational calculation, in which the eigenstates are found by minimising the hamiltonian over the variational basis provided by the paths. The number of excitations that one can extract and the accuracy of the masses crucially depend on the size of the variational basis and the choice of the operators. The procedure is described in more detail in [@Lucini:2010nv; @Lucini:2014paa].
![Large-$N$ extrapolation of the spectrum of glueballs at fixed lattice spacing corresponding to $a T_c = 1/6$ (from [@Lucini:2010nv]).[]{data-label="fig:glueball_spectrum"}](FIGS/glueball_spectrum){width="0.9\columnwidth"}
The continuum spectrum of the lowest-lying glueball was first extracted in [@Lucini:2001ej]. In Fig. \[fig:sungall\] we show the results of a more recent calculation [@Lucini:2004my]. The results for the $0^{++}$, its first excitation $0^{++*}$ and the $2^{++}$ glueballs can be summarised by the formulae $$\begin{aligned}
\nonumber
0^{++\phantom{*}}: \qquad \frac{m}{\sqrt{\sigma}} &=& 3.28(8)
+ \frac{2.1(1.1)}{N^2} \ , \\
0^{++*}: \qquad \frac{m}{\sqrt{\sigma}} &=& 5.93(17) -
\frac{2.7(2.0)}{N^2} \ , \\
\nonumber
2^{++\phantom{*}}: \qquad
\frac{m}{\sqrt{\sigma}} &=& 4.78(14) + \frac{0.3(1.7)}{N^2} \ .\end{aligned}$$ Remarkably, only the leading correction in $1/N^2$ is needed to describe the data from $N=3$ to $N=8$, with a $\chi^2/\mathrm{dof}$ of order one, in a fit that gives coefficients of order one. This hints towards a well-behaved and convergent large-$N$ expansion. This result provides a quantification of the statement that $N = 3$ is close to $N
= \infty$: a correction $O(1/N^2)$ accounts for the finite value of $N$ all the way down to $N = 3$ with a level of precision of a few percents, which is the accuracy of our numerical data.
In order to gain more insight on the glueball spectrum, a more complete calculation exposing more excitations would be desirable. In order to eliminate possible sources of systematic effects, such a calculation should also be able to identify scattering states and contaminations from finite-size excitations related to loop wrapping around the periodic lattice ([*torelons*]{}). Although both effects disappear in the large-$N$ and large-volume limits, for a typical calculation their footprint can be non-negligible. A calculation of this type would require inserting in the variational basis operators corresponding to the unwanted states, which unavoidably increases the computational demands and the technical difficulties. A solution to the latter practical problems has been proposed in [@Lucini:2010nv], where the construction of the basis operators has been fully automatised and hence it becomes easier to increase their number. The first results for the spectrum (this time at fixed lattice spacing) are reported in Fig. \[fig:glueball\_spectrum\]. An extrapolation to the continuum limit is currently being performed.
![Mesonic observables at various values of $N$ and their large-$N$ extrapolation (from [@Bali:2013kia]).[]{data-label="fig:lnmesonspectrum"}](FIGS/lnmesonspectrum){width="0.9\columnwidth"}
![Comparison between lattice large-$N$ results and meson observables in QCD (from [@Bali:2013fya]).[]{data-label="fig:globalExp"}](FIGS/globalExp){width="0.9\columnwidth"}
![Estimate of lattice size corrections for various meson observables in SU(7) (from [@Bali:2013fya]).[]{data-label="fig:betafit"}](FIGS/betaFit){width="0.9\columnwidth"}
For fermionic observables, all investigations performed so far are in the quenched approximation. Since this approximation is exact at $N =
\infty$, it still allows us to obtain the correct values of observables in that limit. First results for the pseudoscalar and vector mesons at fixed lattice spacing were reported in [@DelDebbio:2007wk; @Bali:2008an]. These studies were then extended to decay constants and other mesonic states in [@Bali:2013kia], with the continuum limit currently in progress (a status update is reported in [@Bali:2013fya]). A state of the art determination of the spectrum at various $N$ showing also the large-$N$ limit is given in Fig. \[fig:lnmesonspectrum\].
In order to provide a qualitative picture of how close the large-$N$ limit is to real-world QCD, in Fig. \[fig:globalExp\] we show our numerical data together with experimental data at a quark mass set to its physical value by imposing that the ratio of the pion mass $m_{\pi}$ and of the pion decay constant $\hat{F}_{\pi}$ at $N = \infty$ is $m_{\pi}/\hat{F}_{\pi}
= 1.6$, i.e. compatible with the observed SU(3) value. In order to convert lattice units into MeV, a remaining ambiguity is the value of the string tension $\sqrt{\sigma}$ (for our calculation, $a
\sqrt{\sigma} = 0.095$). To give a handle on the connected systematics, two values of $\sqrt{\sigma}$ are reported in the figure. The lesson one learns is that the deviation of QCD from its large-$N$ limit is at most 5-7% for the lowest-lying meson spectrum and decay constants, while it can be larger (up to 20%) for excitations. However, concerning the latter remark, one has to consider that our lattice calculation has less control of systematic errors on excited states than it has on groundstates.
Another potential issue to consider are finite lattice spacing corrections. First results of a SU(7) study at various lattice spacings are shown in Fig. \[fig:betafit\]. For most of the states, corrections are of order 5% and below. However, it is interesting to note that the $\rho$ and the scalar meson $a_0$ have larger and opposite corrections that bring their masses close to each other. The degeneracy of the two states at large $N$ has been argued in [@Nieves:2009ez]. This example shows that although in most of the cases we do not expect big surprises arising when taking the continuum limit, removing lattice spacing corrections in some few cases can prove to be crucial.
Conclusions and perspectives {#sect:5}
============================
The large-$N$ limit of SU($N$) gauge theories can help us to understand analytically non-perturbative results in QCD. In order to make progress with analytical calculations, lattice computations can be used as a reference. Two examples in different contexts on how lattice data can be used to inform analytical models are provided in [@Erdmenger:2007cm; @Bochicchio:2013eda] (we refer to those works for further details). By now, there are various lattice calculations that provide increasingly solid results in the large-$N$ limit. In this short review, we have concentrated on the glueball and meson spectrum. For a wider overview of the field, we refer to [@Panero:2012qx; @Lucini:2012gg; @Lucini:2013qja].
Acknowledgements {#acknowledgements .unnumbered}
================
I thank C. N[ú]{}[ñ]{}ez for useful comments on the manuscript and J. Erdmenger and M. Bochicchio for discussions about their respective results. This work is mostly based on recent original results obtained in collaboration with G. Bali, L. Castagnini, M. Panero, A. Rago and E. Rinaldi, whose contributions are gratefully acknowledged. This research has been partially supported by the STFC grant ST/G000506/1. The numerical work benefited from computational resources made available by High Performance Computing Wales and STFC through the DiRAC2 supercomputing facility.
[^1]: In QCD, the suppression of those processes is known as the Okubo-Zweig-Izuka (OZI) rule.
|
---
abstract: 'We study the optimal value function for control problems on Banach spaces that involve both continuous and discrete control decisions. For problems involving semilinear dynamics subject to mixed control inequality constraints, one can show that the optimal value depends locally Lipschitz continuously on perturbations of the initial data and the costs under rather natural assumptions. We prove a similar result for perturbations of the initial data, the constraints and the costs for problems involving linear dynamics, convex costs and convex constraints under a Slater-type constraint qualification. We show by an example that these results are in a sense sharp.'
author:
- 'Martin Gugat$^\dag$, Falk M. Hante$^\dag$'
date: 'December 29, 2016'
title: 'Lipschitz Continuity of the Value Function in Mixed-Integer Optimal Control Problems$^*$'
---
[^1]
Introduction
============
In this paper we address the robustness of solutions to optimal control problems that involve both continuous-valued and discrete-valued control decisions to steer solutions of a differential equation such that an associated cost is minimized. This problem class includes in particular optimal control of switched systems [@Antsaklis2014; @Zuazua2011], but also optimization of systems with coordinated activation of multiple actuators, for example, at different locations in space for certain distributed parameter systems [@IftimeDemetriou2009; @HanteSager2013]. In analogy to mixed-integer programming we call such problems mixed-integer optimal control problems. Algorithms to compute solutions to such problems are discussed in [@Gerdts2006; @Sager2009; @HanteSager2013; @SastryEtAl2013a; @SastryEtAl2013b; @RuefflerHante2016]. From a theoretical point of view, but also for a reliable application of such algorithms, the robustness of the solution with respect to perturbation of data in the problem is essential, for instance, in the case of uncertain initial data. We consider the robustness of the optimal value because this is the criterion determining the control decision. Moreover, we understand robustness in the sense that we consider the regularity of the optimal value as a function of the problem parameters.
For continuous optimization problems many sensitivity results are available, see [@bonnansshapiro; @MR2421286]. In particular certain regularity assumptions and constraint qualifications guarantee the continuity of the optimal value function, see [@MR669727; @gu:onesi]. In the context of mixed-integer programming, in general, the main difficulty is that the admissible set consists of several connected components and jumps in the optimal value as function of the problem parameters can occur if due to parameter changes connected components of the feasible set vanish. In mixed-integer linear programming with bounded feasible sets, the continuity of the value function is therefore equivalent to existence of a Slater-point [@Williams1989]. For mixed-integer convex programs, constraint qualifications are given in [@gu97] which yield the existence of one-sided directional derivatives of the value function and hence its Lipschitz continuity. For optimal control problems in general, it is well known that one cannot expect more regularity of the optimal value function than Lipschitz continuity. The following example is an adaption of a classical one saying that this is also true for integer, and hence mixed-integer controlled systems.
\[ex:nonsmooth\] For some ${t_\mathrm{f}}>0$ and $\lambda \in {\mathbb{R}}$, consider the problem $$\left.\begin{array}{l}
\text{minimize}~y({t_\mathrm{f}})~\text{subject to}\\
\quad \dot{y}(t)=v(t)\,y(t),~\text{for a.\,e.}~t \in (0,{t_\mathrm{f}}),\quad y(0)=\lambda\\
\quad y(t) \in {\mathbb{R}},~v(t) \in \{0,1\}~\text{for a.\,e.}~t \in (0,{t_\mathrm{f}}).
\end{array}\right\}$$ The optimal value function $\nu(\lambda)=\inf\{y({t_\mathrm{f}};\lambda) : v \in L^\infty(0,{t_\mathrm{f}};\{0,1\})\}$ can easily be seen to be $$\nu(\lambda)=\begin{cases}e^{{t_\mathrm{f}}}\lambda, &~\lambda<0,\\ \lambda &~\lambda \geq 0, \end{cases}$$ which is Lipschitz continuous, but not differentiable in $\lambda=0$.
For semilinear mixed-integer optimal control problems, we show below that for parametric initial data as in the example, local Lipschitz continuity of the optimal value function can indeed be guaranteed for a rather general setting without imposing a Slater-type condition. Similar results are well-known in the classical Banach or Hilbert space case without mixed control constraints [@CannarsaFrankowska1992; @BarbuDaPrato1983]. Further, we analyze parametric control constraints and parametric cost functions for convex programs. For this case, we formulate a Slater-type condition guaranteeing again the local Lipschitz continuity of the optimal value function. Finally, for convex programs, we can combine both results to obtain local Lipschitz continuity jointly for parametric initial data, control constraints and cost functions.
Setting and Preliminaries
=========================
Let $Y$ be a Banach space, $U$ be a complete metric space, ${\mathcal{V}}$ be a finite set, and $f{\mathcal{\colon}}[t_0,{t_\mathrm{f}}] \times Y \times U \times {\mathcal{V}}\to Y$. We consider the control system $$\label{eq:controlsys}
\dot{y}(t) = A y(t) + f(t,y(t),u(t),v(t)),~t \in (t_0,{t_\mathrm{f}})~\text{a.\,e.},$$ where $[t_0,{t_\mathrm{f}}]$ is a finite time horizon with $t_0<{t_\mathrm{f}}$, $A{\mathcal{\colon}}D(A) \to Y$ is a generator of a strongly continuous semigroup $\{T(t)\}_{t \geq 0}$ of bounded linear operators on $Y$, and where $u{\mathcal{\colon}}[t_0,{t_\mathrm{f}}] \to U$ and $v{\mathcal{\colon}}[t_0,{t_\mathrm{f}}] \to {\mathcal{V}}$ are two independent measurable control functions. Throughout the paper we consider the Lebesgue-measure. Our main concern will be the confinement that the control $v$ only takes values from a finite set. Without loss of generality, we may identify ${\mathcal{V}}$ with a set of integers $\{0,1,\ldots,N-1\}$ and, in analogy to mixed-integer programming, we refer to as a *mixed-integer control system*, where $u$ represents ordinary controls and $v$ integer controls. Let ${U_{[t_0,{t_\mathrm{f}}]}}$ be a Banach subspace of measurable ordinary control functions $u{\mathcal{\colon}}[t_0,{t_\mathrm{f}}] \to U$ and let ${V_{[t_0,{t_\mathrm{f}}]}}$ be the set of measurable integer control functions $v{\mathcal{\colon}}[t_0,{t_\mathrm{f}}] \to {\mathcal{V}}$. By the assumed finiteness of ${\mathcal{V}}$ we actually have ${V_{[t_0,{t_\mathrm{f}}]}}=L^\infty(t_0,{t_\mathrm{f}};{\mathcal{V}})$.
Let $\Lambda$ be a Banach space and consider subject to a parametric initial condition $$\label{eq:initialcondition}
y(t_0)=y_0(\lambda),$$ where $y_0(\lambda)$ is an initial state in $Y$ parametrized by $\lambda \in \Lambda$.
The separation of the control in $u$ and $v$ and the inherent integer confinement of the latter control lets us formulate parametric control constraints of the mixed form $$\label{eq:controlrestriction}
g_k^v(\lambda,u,t) \leq 0,~k=1,\ldots,M,~t \in [t_0,{t_\mathrm{f}}]$$ where $M\in{\mathbb{N}}$ and, for every $v \in {V_{[t_0,{t_\mathrm{f}}]}}$, the functions $g_1^v,\ldots,g_M^v {\mathcal{\colon}}\Lambda \times {U_{[t_0,{t_\mathrm{f}}]}}\times [t_0,{t_\mathrm{f}}] \to {\mathbb{R}}$ are given. These constraints can for example model anticipating control restrictions, where a decision represented by $v$ at an earlier time limits control decisions for $u$ at different times. We discuss an example in Section \[sec:example\]. In cases without mixed control constraints, we set $M=0$.
\[def:solutionMICP\] For fixed $\lambda \in \Lambda$, let ${W_{[t_0,{t_\mathrm{f}}]}}(\lambda)$ denote the set of all admissible controls $$\label{defWT}
\begin{aligned}
{W_{[t_0,{t_\mathrm{f}}]}}(\lambda):=\{&(u,v) \in {U_{[t_0,{t_\mathrm{f}}]}}\times {V_{[t_0,{t_\mathrm{f}}]}}: \\
&g_k^v(\lambda,u,t) \leq 0,~k=1,\ldots,M,~t \in [t_0,{t_\mathrm{f}}]\}.
\end{aligned}$$ Moreover, we say that $y{\mathcal{\colon}}[t_0,{t_\mathrm{f}}] \to Y$ is a *solution of the mixed-integer control system* if there exists an admissible pair of controls $(u,v)\in{W_{[t_0,{t_\mathrm{f}}]}}(\lambda)$ such that $y \in C([t_0,{t_\mathrm{f}}];Y)$ satisfies the integral equation $$\label{eq:controlsysint}
y(t) = T(t-t_0)y(t_0) + \int_{t_0}^{t} T(t-s)f(s,y(s),u(s),v(s))\,ds,~t\in[t_0,{t_\mathrm{f}}]$$ and holds. Let ${\mathscr{S}}_{[t_0,{t_\mathrm{f}}]}(\lambda)$ denote the set of all such solutions $y$ defined on $[t_0,{t_\mathrm{f}}]$. For any $y \in {\mathscr{S}}_{[t_0,{t_\mathrm{f}}]}(\lambda)$, we denote by $y=y(\cdot;y_0(\lambda),u,v)$ the dependency of $y$ on $y_0(\lambda)$, $u$ and $v$ if needed.
According to Definition \[def:solutionMICP\], ${\mathscr{S}}_{[t_0,{t_\mathrm{f}}]}(\lambda)$ consists of the mild solutions of equation and covers in an abstract sense many evolution problems involving linear partial differential operators [@Pazy1983]. It particular, the mild solutions coincide with the usual concept of weak solutions in case of linear parabolic partial differential equations on reflexive $Y$ with distributed control where $A$ arises from a time-invariant variational problem [@BensoussanDaPratoDelfourMitter1992]. For an example, see Section \[sec:example\].
In conjunction with the mixed-integer control system we consider a cost function $\varphi{\mathcal{\colon}}\Lambda \times C([t_0,{t_\mathrm{f}}];Y) \times {U_{[t_0,{t_\mathrm{f}}]}}\times {V_{[t_0,{t_\mathrm{f}}]}}\to {\mathbb{R}}\cup \{\infty\}$ and define the *mixed-integer optimal control problem* with parameter $\lambda$ as $$\label{eq:miocp}
\left.\begin{array}{l}
\text{minimize}~\varphi(\lambda,y,u,v) \; \text{subject to}\\
\quad \dot{y}(t) = A \, y(t) + f(t,\,y(t),\, u(t), \, v(t)),\; t \in (t_0,\, {t_\mathrm{f}}) \;{\rm a.e.},\\
\quad y(0)= y_0(\lambda),
\\
\quad g_k^v(\lambda,u,t) \leq 0~\text{for all}~t \in [t_0,{t_\mathrm{f}}],~k=1,\ldots,M,\\
\quad y \in C([t_0,{t_\mathrm{f}}];Y),~u \in {U_{[t_0,{t_\mathrm{f}}]}},~v \in {V_{[t_0,{t_\mathrm{f}}]}}.
\end{array}
\right\}$$ We will study the corresponding *optimal value* $\nu(\lambda)\in {\mathbb{R}}\cup \{\pm\infty\}$ given by $$\label{eq:defnu}
\begin{array}{l}
\nu(\lambda)=\inf \bigl\{\varphi(\lambda,y,u,v) :\\
\quad \dot{y}(t) = A \, y(t) + f(t,\,y(t),\, u(t), \, v(t)),\; t \in (t_0,\, {t_\mathrm{f}}) \;{\rm a.e.},\\
\quad y(0)= y_0(\lambda),
\\
\quad g_k^v(\lambda,u,t) \leq 0~\text{for all}~t \in [t_0,{t_\mathrm{f}}],~k=1,\ldots,M,\\
\quad y \in C([t_0,{t_\mathrm{f}}];Y),~u \in {U_{[t_0,{t_\mathrm{f}}]}},~v \in {V_{[t_0,{t_\mathrm{f}}]}}\bigr\}
\end{array}$$ in its dependency on the parameter $\lambda$.
For the mixed-integer control system, we will impose the following assumptions.
\[ass:ControlSys\] The map $f{\mathcal{\colon}}[t_0,{t_\mathrm{f}}] \times Y \times U \times \{v\} \to Y$ is continuous for all $v \in {\mathcal{V}}$. Moreover, there exists a function $k \in L^1(t_0,{t_\mathrm{f}})$ such that for all $(u,v) \in {W_{[t_0,{t_\mathrm{f}}]}}$, $y_1,y_2 \in Y$ and for almost every $t \in (t_0,{t_\mathrm{f}})$ $$\begin{aligned}
{2}
&\text{(i)}\qquad &&|f(t,y_1,u(t),v(t))-f(t,y_2,u(t),v(t))| \leq k(t)|y_1 - y_2|\\
&\text{(ii)}\qquad &&|f(t,0,u(t),v(t))| \leq k(t).\end{aligned}$$
In particular, under these assumptions, the integral in is well-defined in the Lebesgue-Bochner sense and from the theory of abstract Cauchy problems [@Pazy1983] we obtain a solution $y$ in $C([0,{t_\mathrm{f}}];Y)$ for all $y_0 \in Y$, $u \in {U_{[t_0,{t_\mathrm{f}}]}}$ and $v \in {V_{[t_0,{t_\mathrm{f}}]}}$. Moreover, the strong continuity of $T(\cdot)$ and the Gronwall inequality yield the following solution properties.
\[lem:bounds\] Under the Assumptions \[ass:ControlSys\], there exist constants $\gamma \geq 0$ and $w_0 \geq 0$ such that for all $\lambda_1,\lambda_2 \in \Lambda$, setting $y_i=y(\cdot;y_0(\lambda_i),u,v)\in{\mathscr{S}}_{[t_0,{t_\mathrm{f}}]}(\lambda_i)$ for $i \in \{1,2\}$, for all $t \in [t_0,{t_\mathrm{f}}]$ it holds $\|T(t)\| \leq \gamma \exp(w_0(t-t_0 ))$, $$\label{eq:aprioribound}
|y_i(t)| \leq C(t) (1+|y_0(\lambda_i)|),\quad i \in \{1,2\},$$ and $$\label{eq:yLipschitz}
|y_1(t)-y_2(t))| \leq C(t) |y_0(\lambda_1) - y_0(\lambda_2)|$$ with $C(t)=\gamma\exp\left(w_0 (t-t_0)+\gamma\int_{t_0}^{t}k(s)\,ds\right)$.
For the cost function and control constraints, we will impose the following assumptions.
\[ass:CostAndConstraints\] The function $\varphi{\mathcal{\colon}}\Lambda \times C([t_0,{t_\mathrm{f}}];Y) \times {U_{[t_0,{t_\mathrm{f}}]}}\times {V_{[t_0,{t_\mathrm{f}}]}}\to {\mathbb{R}}$ is continuous and, for every $v \in {V_{[t_0,{t_\mathrm{f}}]}}$, the functions $g^v_1,\ldots,g^v_M{\mathcal{\colon}}\Lambda \times {U_{[t_0,{t_\mathrm{f}}]}}\times [t_0,{t_\mathrm{f}}] \to {\mathbb{R}}$ are such that the set of admissible controls ${W_{[t_0,{t_\mathrm{f}}]}}(\lambda)$ is not empty for all $\lambda \in \Lambda$.
In particular, under Assumptions \[ass:ControlSys\] and \[ass:CostAndConstraints\], for every $\lambda \in \Lambda$, the set ${\mathscr{S}}_{[t_0,{t_\mathrm{f}}]}(\lambda)$ is non-empty. Moreover, one obtains local Lipschitz continuity of the value function if the perturbation parameter $\lambda$ acts Lipschitz continuously on $\varphi$ and $y_0$ by similar arguments as in a classical Banach or Hilbert space case [@CannarsaFrankowska1992; @BarbuDaPrato1983].
\[thm:LipInitial\] Under the Assumptions \[ass:ControlSys\] and \[ass:CostAndConstraints\], suppose that the constraint functions $g^v_1,\ldots,g^v_M$ are independent of $\lambda$. Let $\bar\lambda$ be some fixed parameter in $\Lambda$ and assume that for some bounded neighborhood $B(\bar\lambda)$ of $\bar{\lambda}$ and some constant $L_0$ $$\label{eq:y0Lip}
|y_0(\lambda_1)-y_0(\lambda_2)| \leq L_{0} \, |\lambda_1- \lambda_2|,\,~\lambda_1, \lambda_2 \in B(\bar\lambda).$$ Moreover, let $K=\sup_{\lambda \in B(\bar\lambda)}|y_0(\lambda)|$ and assume that for some constant $L_{\varphi}$ $$\label{eq:varphiLip}
|\varphi(\lambda_1,y,u,v)-\varphi(\lambda_2,\bar y,u,v)| \leq L_{\varphi}(|y-\bar{y}|+|\lambda_1-\lambda_2|)$$ for all $(u,v)\in {W_{[t_0,{t_\mathrm{f}}]}}$, $y,\bar{y}$ such that $\max\{|y|,|\bar y|\} \leq C({t_\mathrm{f}})(1+K)$ and $\lambda_1$, $\lambda_2 \in B(\bar\lambda)$, where $C(t)$ is the bound from Lemma \[lem:bounds\]. Then there exists a constant $\hat L_\nu$ such that $$\label{eq:nuLipInitial}
|\nu(\lambda_1)-\nu({\lambda_2})| \leq \hat L_\nu |\lambda_1 - \lambda_2|,\quad~\lambda_1, \lambda_2 \in B(\bar\lambda).$$
Let $\varepsilon>0$ and $\lambda_1$, $\lambda_2 \in B(\bar\lambda)$ be given. Choose $(u_\varepsilon,v_\varepsilon) \in {W_{[t_0,{t_\mathrm{f}}]}}$ such that $$\varphi(\lambda_2,\bar{y}_\varepsilon,u_\varepsilon,v_\varepsilon) \leq \nu(\lambda_2)+\varepsilon,$$ where $\bar{y}_\varepsilon$ denotes the reference solution $y(\cdot;y_0(\lambda_2),u_\varepsilon,v_\varepsilon)\in{\mathscr{S}}_{[t_0,{t_\mathrm{f}}]}$. Let $y_\varepsilon$ denote the perturbed solution $y(\cdot;y_0(\lambda_1),u_\varepsilon,v_\varepsilon)\in{\mathscr{S}}_{[t_0,{t_\mathrm{f}}]}$. Lemma \[lem:bounds\] and the assumptions yield $$|y_\varepsilon(t)| \leq C(t) (1+K),~t \in [t_0,{t_\mathrm{f}}],$$ and $$|y_\varepsilon(t)- \bar{y}_\varepsilon(t)| \leq C(t) L_{0} |\lambda_1 - \lambda_2|,~t \in [t_0,{t_\mathrm{f}}].$$ Hence, $$\begin{aligned}
{1}
\varphi(\lambda_1,y_\varepsilon,u_\varepsilon,v_\varepsilon)
&\leq \varphi(\lambda_2,\bar y_\varepsilon,u_\varepsilon,v_\varepsilon)+|\varphi(\lambda_1,y_\varepsilon,u_\varepsilon,v_\varepsilon) - \varphi(\lambda_2,\bar y_\varepsilon,u_\varepsilon,v_\varepsilon)|\\
&\leq \varphi(\lambda_2,\bar y_\varepsilon,u_\varepsilon,v_\varepsilon)+L_\varphi (C({t_\mathrm{f}}) L_{0}+1) |\lambda_1 - \lambda_2|.\end{aligned}$$ Thus $$\begin{aligned}
\nu(\lambda_1) &\leq \varphi(\lambda_1,y_\varepsilon,u_\varepsilon,v_\varepsilon)
\leq \varphi(\lambda_2,\bar y_\varepsilon,u_\varepsilon,v_\varepsilon)
+L_\varphi (C({t_\mathrm{f}}) L_{0}+1) |\lambda_1 - \lambda_2|
\\
&\leq \nu({\lambda_2})+\varepsilon+L_\varphi (C({t_\mathrm{f}}) L_{0}+1) |\lambda_1 - \lambda_2|.
\end{aligned}$$ Letting $\varepsilon \to 0$ from above gives an upper bound $\nu(\lambda_1) \leq \nu({\lambda_2})+\hat L_\nu |\lambda_1 - \lambda_2|$ with $$\label{eq:hatLnudef}
\hat L_\nu=L_\varphi (C({t_\mathrm{f}}) L_{0}+1).$$ Interchanging the roles of $\lambda_1$ and $\lambda_2$ yields the claim.
In the subsequent section, we will obtain a similar result concerning the perturbation of the functions $g^v_1,\ldots,g^v_M$ and the cost function $\varphi$ under additional structural hypothesis and a constraint qualification.
Perturbation of the constraints for convex problems {#sec:constraints}
===================================================
In this section, we show that under a Slater-type condition the optimal value $\nu(\lambda)$ of the mixed-integer optimal control problem in the case of a convex cost function and linear dynamics is locally Lipschitz continuous as a function of a parameter $\lambda$ acting on the control constraints $g^v_1,\ldots,g^v_M$ and the cost function $\varphi$. We need the following
\[ass:Convex\] The map $(y,u) \mapsto f(t,y,u,v)$ is linear and the map $(y,u) \mapsto \varphi(\lambda,y,u,v)$ is convex. Moreover, the function $\varphi$ is Lipschitz continuous with respect to $\lambda$ in the sense that $$|\varphi(\lambda_1,\, y,\, u,\, v) - \varphi(\lambda_2,\, y,\, u,\, v)|
\leq
L_\varphi(|y|,\, |u|) \,|\lambda_1 - \lambda_2|$$ with a continuous function $L_\varphi{\mathcal{\colon}}[0,\infty)^2 \rightarrow [0,\,\infty)$. For all $k=1,\ldots,M$, the maps $u \mapsto g_k^v(\lambda,\,u,\, t)$ are convex, the maps $(u,t)\mapsto g_k^v(\lambda,\,u,\, t)$ are continuous and the functions $g^v_k$ are Lipschitz continuous with respect to $\lambda$ in the sense that for all $t \in [t_0,{t_\mathrm{f}}]$ $$|g_k^v(\lambda_1,\, u,\, t) - g_k^v(\lambda_2,\, u,\, t)|
\leq
L_g(|u|) \,|\lambda_1 - \lambda_2|$$ with a continuous function $L_g{\mathcal{\colon}}[0,\infty) \rightarrow [0,\,\infty)$.
Under the Assumptions \[ass:ControlSys\]–\[ass:Convex\] and assuming that $y_0$ is independent of $\lambda$, we have for each parameter $\lambda \in \Lambda$ the mixed-integer optimal control problem with $y_0(\lambda)$ replaced by a fixed initial state $y_0 \in Y$. Moreover, in this section, $\nu(\lambda)$ denotes the corresponding optimal value function with fixed initial state $y_0$. The subsequent analysis is based upon the presentation in [@gu97], where for the finite dimensional case the existence of the one sided derivatives of the optimal value function $\nu(\lambda)$ is shown. For a generalization to the above setting, we first introduce a Slater-type constraint qualification, a dual problem and prove a strong duality result.
[**(CQ)**]{}\[ass:CQ\] For some $\bar \lambda \in \Lambda$ and some bounded neighborhood $B(\bar \lambda)\subset \Lambda$ of $\bar \lambda$ there exists a number $\omega>0$ such that for all $v\in {V_{[t_0,{t_\mathrm{f}}]}}$ there is a Slater point $\bar u_v \in U$ such that for all $\lambda \in B(\bar \lambda)$ we have $$\label{eq:slaterpoint}
g_k^v(\lambda, \bar u_v,t) \leq - \omega\quad\mbox{\rm for all } t \in [t_0,{t_\mathrm{f}}],~k=1,\ldots,M,$$ $$\label{30}
\sup_{v\in V_{[t_0,\,{t_\mathrm{f}}]}} \sup_{\lambda \in B(\bar \lambda)} \varphi(\lambda,y(\bar u_v,v),\bar u_v,v)<\infty$$ and that there exists a number $\underline \alpha$ such that for all $\lambda \in B(\bar \lambda)$ we have $$\label{31}
\nu(\lambda) \geq \underline \alpha$$ and that the set $$\label{coercive}
\hspace*{-1em}
\begin{aligned}
\bigcup_{\lambda_1,\lambda_2 \in B(\bar \lambda)}
\biggl\{& (u,v) \in U_{[t_0,{t_\mathrm{f}}]}\times {V_{[t_0,{t_\mathrm{f}}]}}:\\
&~\varphi(\lambda_1,y(u,v),u,v) \leq \varphi(\lambda_1,y(\bar u_v,v),\bar u_v,v) + |\lambda_1 - \lambda_2|^2,\\
&~g_k^v(\lambda_1,u(t),t) \leq 0,~t \in [t_0,{t_\mathrm{f}}],~k=1,\ldots,M\biggr\}=:\bar S(y_0)
\end{aligned}$$ is bounded.
Note that (\[31\]) holds if $\underline \alpha$ is a lower bound for the cost function.
Let $C([t_0,{t_\mathrm{f}}])^\ast_+$ denote the set of positive function of bounded variation on $[t_0,{t_\mathrm{f}}]$. For any controls $v \in {V_{[t_0,{t_\mathrm{f}}]}}$, $u \in {U_{[t_0,{t_\mathrm{f}}]}}$ and any $\mu^\ast \in \left(C([t_0,{t_\mathrm{f}}])^\ast_+\right)^M$ we define the Lagrangian $$\label{lagrangian}
{\mathcal{L}}_v(\lambda,u,\mu^\ast) = \varphi(\lambda,\,y(u,v),\,u,\, v) + \sum_{k=1}^M \int_{t_0}^{{t_\mathrm{f}}} g_k^v(\lambda,u,s)\,\mathrm{d} \mu^\ast_k(s),$$ where the integral is in the Riemann-Stieltjes sense. Further, we define $$\label{lagrangian1}
h_v(\lambda,\mu^\ast) = \inf_{u \in {U_{[t_0,{t_\mathrm{f}}]}}} {\mathcal{L}}_v(\lambda,u,\mu^\ast).$$
Under the constraint qualification , for all fixed $\lambda \in B(\bar \lambda)$ and $v\in {V_{[t_0,{t_\mathrm{f}}]}}$, the classical convex duality theory as presented in [@ekturn] implies the strong duality result (see also [@gu:onesi]) $$\label{duality}
\sup_{\mu^\ast \in \left(C([t_0,{t_\mathrm{f}}])^\ast_+\right)^M} h_v(\lambda,\mu^\ast) = \nu^v(\lambda)$$ where $\nu^v(\lambda)$ denotes the optimal value of the following convex optimal control problem only in the variables $y$ and $u$ $$\label{eq:primalvfix}
\left.
\begin{array}{l}
\text{minimize}~\varphi(\lambda,y,u,v)~\text{subject to}\\
\quad \dot{y}(t) = A \, y(t) + f(t,\,y(t),\, u(t), \, v(t)),\; t \in (t_0,\, {t_\mathrm{f}}) \;{\rm a.e.},
~y(0)= y_0,
\\
\quad g_k^v(\lambda,u,t) \leq 0~\text{for all}~t \in [t_0,{t_\mathrm{f}}],~k \in \{1,\ldots,M\},\\
\quad y \in C([t_0,{t_\mathrm{f}}];Y),~u \in {U_{[t_0,{t_\mathrm{f}}]}},
\end{array}
\right\}$$ see, for example, [@gu:onesi; @ekturn]. Further, we introduce the sets $$F_v(\lambda) =\{\mu^\ast \in \left(C([t_0,{t_\mathrm{f}}])^\ast_+\right)^M : h_v(\lambda,\mu^\ast) > - \infty\}$$ and $$G(\lambda) = \left\{\rho \in \prod_{v \in {V_{[t_0,{t_\mathrm{f}}]}}} F_v(\lambda) : \inf_{v\in {V_{[t_0,{t_\mathrm{f}}]}}} h_v(\lambda,\rho_v) \in {\mathbb{R}}\right\},$$ and, for $(r,\rho) \in \mathbb{R}\times G(\lambda)$, we define the projection $\pi(r,\rho)=r$. Finally, we introduce the following maximization problem as the dual problem of $$\label{eq:dualconvex}
\left.
\begin{array}{l}
\text{maximize}~\pi(r,\rho)~\text{subject to}\\
\quad \rho \in G(\lambda),~r \in \mathbb{R},\\
\quad r \leq h_v(\lambda, \rho_v)~\text{for all}~v\in {V_{[t_0,{t_\mathrm{f}}]}}.
\end{array}\right\}$$ The optimal value of this dual problem is $$\Delta(\lambda) = \sup_{\rho\in G(\lambda)} \inf_{v\in {V_{[t_0,{t_\mathrm{f}}]}}} h_v(\lambda,\rho_v).$$ Now we state a strong duality result. For the convenience of the reader we also present a complete proof. Note however that Theorem \[strongduality\] can also be deduced from Ky Fan’s minimax theorem in [@borweinzhuang86].
\[strongduality\] The constraint qualification implies that $$\nu(\lambda) = \Delta(\lambda),~\text{for all}~\lambda \in B(\bar \lambda),$$ where $\nu(\lambda)$ is the optimal value of with fixed initial state.
Choose $\rho \in G(\lambda)$. Then convex weak duality implies that for all $v\in {V_{[t_0,{t_\mathrm{f}}]}}$ we have $$h_v(\lambda,\rho_v) \leq \nu^v(\lambda).$$ Thus $$\inf_{v\in {V_{[t_0,{t_\mathrm{f}}]}}} h_v(\lambda,\rho_v) \leq \inf_{v\in {V_{[t_0,{t_\mathrm{f}}]}}} \nu^v(\lambda)=\nu(\lambda).$$ This implies that $$\Delta(\lambda) = \sup_{\rho\in G(\lambda)} \inf_{v\in {V_{[t_0,{t_\mathrm{f}}]}}} h_v(\lambda,\rho_v) \leq \nu(\lambda),$$ that is, we have shown the weak duality. Further, due to $\,$ and convex strong duality from , for each $v \in {V_{[t_0,{t_\mathrm{f}}]}}$, we can choose some $\mu^\ast_v \in \left(C([t_0,{t_\mathrm{f}}])^\ast_+\right)^M$ such that $$h_v(\lambda, \mu^\ast_v) = \nu^v(\lambda).$$ Define $\rho^\ast = (\mu^\ast_v)_{v\in {V_{[t_0,{t_\mathrm{f}}]}}}$. Then $\rho^\ast \in G(\lambda)$. This yields $$\begin{aligned}
\Delta(\lambda) &=\sup_{\rho\in G(\lambda)} \inf_{v\in {V_{[t_0,{t_\mathrm{f}}]}}} h_v(\lambda,\rho_v)\\
&\geq \inf_{v \in {V_{[t_0,{t_\mathrm{f}}]}}} h_v(\lambda,\mu^\ast_v)=\inf_{v \in {V_{[t_0,{t_\mathrm{f}}]}}} \nu^v(\lambda) = \nu(\lambda).
\end{aligned}$$ Hence the strong duality follows.
Based upon the above duality concept, we can now show the Lipschitz continuity of the optimal value function in a neighborhood of $\bar\lambda$. To this end, we introduce for any $\varepsilon \geq 0$ the set of $\varepsilon$-optimal points $$\begin{aligned}
P(\lambda,\varepsilon) =~&\bigl\{ u \in U_{[t_0,{t_\mathrm{f}}]} : \text{there exists}~v \in {V_{[t_0,{t_\mathrm{f}}]}}~\text{such that}\\
&\quad g_k^v(\lambda,u,t) \leq 0~\text{for all}~t \in [t_0,{t_\mathrm{f}}],~k=1,\ldots,M,\\
&\quad \varphi(\lambda,\,y(u,v),\,u,\, v) \leq \nu(\lambda) + \varepsilon \bigr\}
\end{aligned}$$ and we set $H(\lambda,\varepsilon) = \{ \rho \in G(\lambda) : \inf_{v\in V} h_v(\lambda,\rho_v) \geq \nu(\lambda)- \varepsilon\}$.
\[bounded\] Under , the set $$\Omega(\bar \lambda ):=\bigcup_{\lambda_1, \lambda_2 \in B(\bar \lambda),\,v\in {V_{[t_0,{t_\mathrm{f}}]}}} \left\{\rho_v : \rho \in H(\lambda_1,|\lambda_1 - \lambda_2|^2) \right\}$$ is bounded.
Due to assumption , for all $v \in {V_{[t_0,{t_\mathrm{f}}]}}$, we have the Slater point $\bar u_v$. Choose $\lambda_1$, $\lambda_2 \in B(\bar \lambda)$ and $\rho \in H(\lambda_1,|\lambda_1 - \lambda_2|^2)$. Then $\inf_{v\in {V_{[t_0,{t_\mathrm{f}}]}}} h_v(\lambda_1,\rho_v) \geq \nu(\lambda_1)- |\lambda_1 - \lambda_2|^2$. Thus by definition of $h_v$, for all $v\in {V_{[t_0,{t_\mathrm{f}}]}}$, we have that ${\mathcal{L}}_v(\lambda_1,\bar u_v,\rho_v) \geq h_v(\lambda_1,\rho_v) \geq \nu(\lambda_1) - |\lambda_1 - \lambda_2|^2$. By definition of ${\mathcal{L}}_v$, this implies $$\varphi(\lambda_1,y(\bar u_v,v),\bar u_v,v) + \sum_{k=1}^M \int_{t_0}^{{t_\mathrm{f}}} g_k^v(\lambda_1,\bar u_v,s)\,\mathrm{d}(\rho_v)_k(s) \geq \nu(\lambda_1) - |\lambda_1 - \lambda_2|^2.$$ Now using that $$g_k^v(\lambda_1,\bar u_v,t) \leq - \omega<0~\text{for all}~t \in [t_0,{t_\mathrm{f}}],~k \in \{1,\ldots,M\},$$ we can divide by $-\omega<0$ and obtain due to (\[31\]) $$\begin{aligned}
\sum_{k=1}^M \int_{t_0}^{{t_\mathrm{f}}} 1\,\mathrm{d} (\rho_v)_k(s) & \leq & \frac{ \nu(\lambda_1) - |\lambda_1 - \lambda_2|^2 - \varphi(\lambda_1,y(\bar u_v,v),\bar u_v, v)}{-\omega}\\
& = & \frac{|\lambda_1 - \lambda_2|^2 + \varphi(\lambda_1,y(\bar u_v,v),\bar u_v,v) - \nu(\lambda_1)}{\omega}\\
& \leq & \frac{|\lambda_1 - \lambda_2|^2 + \varphi(\lambda_1,y(\bar u_v,v),\bar u_v,v) - \underline \alpha}{\omega}\\
& \leq & \frac{|\lambda_1 - \lambda_2|^2 + \sup\limits_{v\in {V_{[t_0,{t_\mathrm{f}}]}}} \sup\limits_{\lambda \in B(\bar \lambda)} \varphi(\lambda,y(\bar u_v,v),\bar u_v,v) - \underline \alpha}{\omega}.\end{aligned}$$ Due to (\[30\]) this yields the assertion.
\[liminf\] Suppose that holds. Then for all $\lambda_1$, $\lambda_2 \in B(\bar \lambda)$ we have $$\nu(\lambda_1) - \nu(\lambda_2)
\geq
-\underline C \, |\lambda_1 - \lambda_2|$$ for some $\underline C$ in ${\mathbb{R}}$.
Let $\lambda_1$, $\lambda_2 \in B(\bar\lambda)$ be given. Choose a solution $u\in P(\lambda_1,|\lambda_1- \lambda_2|^2)$ and $\tilde v \in {V_{[t_0,{t_\mathrm{f}}]}}$ with $g_j^{\tilde v}(\lambda_1, u, t) \leq 0$ for all $t \in [t_0,{t_\mathrm{f}}]$, $j=1, \ldots,M$, $\varphi(\lambda_1, \, y(u,\tilde v),\,u,\, \tilde v)
\leq \nu(\lambda_1) + |\lambda_1- \lambda_2|^2$ and $
\bar \rho \in H( \lambda_2, |\lambda_1- \lambda_2|^2)$. Then we have $$\begin{aligned}
\nu(\lambda_1) - \nu(\lambda_2)
& \geq &
\;\varphi(\lambda_1, y(u,\tilde v),u,\tilde v) - \inf_{v\in {V_{[t_0,{t_\mathrm{f}}]}}} h_v( \lambda_2, \bar \rho_v) - 2 |\lambda_1 - \lambda_2|^2
\\
& \geq &
\;\varphi(\lambda_1, y(u,\, \tilde v),u, \, \tilde v) - h_{\tilde v}( \lambda_2, \bar \rho_{\tilde v}) - 2 |\lambda_1 - \lambda_2|^2
\\
& \geq &
\;\varphi(\lambda_1, y(u,\, \tilde v),\, u, \tilde v) - {\mathcal{L}}_{\tilde v}(\lambda_2, u, \bar \rho_{\tilde v}) - 2 |\lambda_1 - \lambda_2|^2
\\
& \geq &
\;\varphi(\lambda_1, y(u, \tilde v),u, \tilde v)
+ \sum_{j=1}^M \int_{t_0}^{{t_\mathrm{f}}} g_j^{\tilde v}(\lambda_1,u,s)\,\mathrm{d} \bar \rho_{\tilde v}(s)
\\
& ~ &
\qquad~-{\mathcal{L}}_{\tilde v}(\lambda_2, u, \bar \rho_{\tilde v}) - 2 |\lambda_1 - \lambda_2|^2
\\
& = &
{\mathcal{L}}_{\tilde v} (\lambda_1,\, u, \, \bar\rho_{\tilde v}) - {\mathcal{L}}_{\tilde v} (\lambda_2,\, u, \, \bar\rho_{\tilde v}) - 2 |\lambda_1- \lambda_2|^2
\\
& \geq &
- \biggl[ L_\varphi( |y(u,\,\tilde v)|,\, |u|)\\
& ~ & \qquad~+ M \, L_g(|u|) \, \int_{t_0}^{{t_\mathrm{f}}} \, d \bar \rho_{\tilde v}(s) + 2 |\lambda_1 - \lambda_2| \biggr]\, |\lambda_1 - \lambda_2 |.\end{aligned}$$ Due to (CQ), the set $\bar S(y_0)$ from (\[coercive\]) is bounded. Thus our assumptions imply that the set $\{y(\hat u,\, \hat v):\, (\hat u,\,\hat v)\in \bar S(y_0)\}$ is bounded (see (\[eq:aprioribound\])). Due to Lemma \[bounded\], the set $\Omega(\bar\lambda)$ is also bounded. Since $L_\varphi$ and $L_g$ are continuous this allows us to define the real number $$\label{underlinecdefinition}
\begin{split}
\tilde C =
&\sup_{(\hat u,\,\hat v)\in \bar S(y_0)}\, L_\varphi( |y(\hat u,\, \hat v)|,\, |\hat u|)\\
&\quad+ M\, L_g(|\hat u|) \,
\sup_{ \hat \rho_w \in \Omega(\bar \lambda)} \int_{t_0}^{{t_\mathrm{f}}} \, d \hat \rho_{w}(s) \;
+2 \sup_{\lambda_1,\, \lambda_2 \in B(\bar \lambda)}|\lambda_1-\lambda_2|.
\end{split}$$ Due to the definition of $ P(\lambda_1,\, |\lambda_1 - \lambda_2|^2)$ we have $(u,\, \tilde v) \in \bar S(y_0)$. Moreover, we have $\bar \rho_{\tilde v} \in \Omega (\bar\lambda)$. Hence we have $$\begin{aligned}
\nu(\lambda_1) - \nu(\lambda_2)
& \geq &
- \tilde C\; |\lambda_1 - \lambda_2|
\end{aligned}$$ and the assertion follows with $ \underline C = \tilde C$.
Similarly as in Lemma \[liminf\], by interchanging the roles of $\lambda_1 $ and $\lambda_2$, and with the choice $\overline C =\tilde C$ with $\tilde C$ as defined in (\[underlinecdefinition\]) we can prove the following Lemma:
\[limsup\] Suppose that holds. Then, for all $\lambda_1,\, \lambda_2 \in B(\bar \lambda)$, we have $$\nu(\lambda_1) - \nu(\lambda_2)
\leq
\overline C \, |\lambda_1 - \lambda_2|,$$ for some $\overline C$ in ${\mathbb{R}}$.
The above analysis implies our main result about the Lipschitz continuity of the optimal value as a function of the parameter $\lambda$.
\[thm:LipConstraints\] Under the Assumptions \[ass:ControlSys\]–\[ass:Convex\], for any $\bar\lambda \in \Lambda$ and a bounded neighborhood $B(\bar\lambda) \subset \Lambda$ satisfying the constraint qualification it holds $$\label{eq:nuLipCostConstraint}
|\nu(\lambda_1) - \nu(\lambda_2)|
\leq
\tilde C
\,
|\lambda_1 - \lambda_2|\quad\text{for all}~\lambda_1,\,\lambda_2 \in B(\bar \lambda)$$ with $\tilde C$ as defined in , that is, the optimal value function $\nu$ is Lipschitz continuous in a neighborhood of $\bar \lambda$ with Lipschitz constant $\tilde C$.
The result follows from combining the proofs of Lemma \[liminf\] and \[limsup\].
Joint perturbations {#sec:joint}
===================
In this section, we study the joint local Lipschitz continuity of the value function $\nu$ with respect to $\lambda$ acting on the initial data, the constraints and the costs. We consider the mixed-integer optimal control problem . In contrast to Section \[sec:constraints\] the initial state $y_0(\lambda)$ depends on $\lambda$. Also, the constraints and the objective function depend on $\lambda$. The result is obtained by combining Theorem \[thm:LipInitial\] and \[thm:LipConstraints\].
\[thm:JointLip\] Under the Assumptions \[ass:ControlSys\]–\[ass:Convex\], for any $\bar\lambda \in \Lambda$, a bounded neighborhood $B(\bar\lambda) \subset \Lambda$ let $L_0,L_{\varphi}$ be constants such that and hold as in Theorem \[thm:LipInitial\]. Further, suppose that (CQ) holds in the sense that is satisfied and $\cup_{y_0 \in Y_0} \bar S(y_0)$ is bounded with $\bar S(y_0)$ from and $Y_0 = \{y_0(\lambda) : \lambda \in B(\bar\lambda)\}$. Then, there exists a constant $L_\nu$ such that $$\label{eq:nuJointlyLip}
|\nu(\lambda_1)-\nu(\lambda_2)| \leq L_\nu |\lambda_1 - \lambda_2|,\quad\text{for all}~\lambda_1,\,\lambda_2 \in B(\bar \lambda),$$ where $\nu(\lambda)$ is the optimal value of as defined in .
In this proof, for $\lambda \in B(\bar\lambda)$ and $y_0 \in Y$, we use the notation $$\label{eq:defnu2}
\begin{array}{l}
\nu(\lambda,y_0)=\inf \bigl\{\varphi(\lambda,y,u,v) :\\
\quad \dot{y}(t) = A \, y(t) + f(t,\,y(t),\, u(t), \, v(t)),\; t \in (t_0,\, {t_\mathrm{f}}) \;{\rm a.e.},
~y(0)= y_0,
\\
\quad g_k^v(\lambda,u,t) \leq 0~\text{for all}~t \in [t_0,{t_\mathrm{f}}],~k=1,\ldots,M,\\
\quad y \in C([t_0,{t_\mathrm{f}}];Y),~u \in {U_{[t_0,{t_\mathrm{f}}]}},~v \in {V_{[t_0,{t_\mathrm{f}}]}}\bigr\}.
\end{array}$$ Due to and the set $Y_0$ is bounded by the constant $K$ from Theorem \[thm:LipInitial\] and for all $y_0 \in Y_0$, $v \in {V_{[t_0,{t_\mathrm{f}}]}}$, $\lambda \in B(\bar\lambda)$ we have the upper bound $$\begin{split}
\varphi(\lambda,y(y_0,\bar u_v,v),\bar u_v,v) \leq \varphi(\bar\lambda,&\,y(y_0(\bar\lambda),\bar u_v,v),\bar u_v,v)\\
+~L_{\varphi}(|y(y_0,\bar u_v,v),\bar u_v,v)~-&~y(y_0(\bar\lambda),\bar u_v,v),\bar u_v,v)|+|\lambda-\bar\lambda|).
\end{split}$$ Moreover, from Lemma \[lem:bounds\], we obtain $|y(y_0,\bar u_v,v)| \leq C({t_\mathrm{f}})(1+K)$. This implies . Using similar arguments, we get a lower bound $$\nu(\lambda,y_0)\geq\inf_{y_0 \in Y_0} \inf_{v\in V_{[t_0,\,{t_\mathrm{f}}]}} \inf_{\lambda \in B(\bar \lambda)} \varphi(\lambda,y(y_0,\bar u_v,v),\bar u_v,v)=:\underline\alpha>-\infty.$$ This implies with $\underline\alpha$ independent of $\lambda$. Thus Assumptions \[ass:CQ\] holds for all $y_0 \in Y_0$ and the proof of Lemma \[bounded\] shows that the bound of the set $\Omega(\bar\lambda)$ is independent of $y_0$. Due to the function $L_{\varphi}$ in Assumptions \[ass:Convex\] is constant and $\tilde C$ from reduces to $$\begin{split}
\tilde C = L_\varphi~+ &~M \sup_{y_0 \in Y_0} \sup_{(\hat u,\hat v)\in \bar S(y_0)} L_g(|\hat u|) \, \sup_{ \hat \rho_w \in \Omega(\bar \lambda)} \int_{t_0}^{{t_\mathrm{f}}} \, d \hat \rho_{w}(s) \\
&+ 2 \sup_{\lambda_1,\, \lambda_2 \in B(\bar \lambda)}|\lambda_1-\lambda_2|<\infty.
\end{split}$$ Now, let $\lambda_1,\lambda_2 \in B(\bar{\lambda})$. From Theorem \[thm:LipInitial\] with $\lambda_1$ as first argument of $\nu$ fixed we get $|\nu(\lambda_1,y_0(\lambda_1))-\nu(\lambda_1,y_0(\lambda_2))|\leq \hat L_{\nu} |\lambda_1-\lambda_2|$ with $\hat L_{\nu}$ given by . From Theorem \[thm:LipConstraints\] with $\lambda_2$ as an argument of $y_0$ fixed we get $|\nu(\lambda_1,y_0(\lambda_2))-\nu(\lambda_2,y_0(\lambda_2))|\leq \tilde C |\lambda_1-\lambda_2|$. Thus we obtain the inequality $$\begin{aligned}
& |\nu(\lambda_1,y_0(\lambda_1))-\nu(\lambda_2,y_0(\lambda_2))|\\
& \quad \leq |\nu(\lambda_1,y_0(\lambda_1))-\nu(\lambda_1,y_0(\lambda_2))| + |\nu(\lambda_1,y_0(\lambda_2))-\nu(\lambda_2,y_0(\lambda_2))|\\
& \quad \leq (\hat L_{\nu}+\tilde C) |\lambda_1-\lambda_2|\\
\end{aligned}$$ and follows with $L_{\nu}=\hat L_{\nu}+\tilde C$.
Example {#sec:example}
=======
We discuss an academic application concerning the optimal positioning of an actuator motivated from applications in thermal manufacturing [@IftimeDemetriou2009; @HanteSager2013].
Suppose that $\Omega \subset {\mathbb{R}}^2$ is a bounded domain containing two non-overlapping control domains $\omega_1$ and $\omega_2$. For simplicity, we assume that the boundaries of all these domains $\partial \Omega$, $\partial \omega_1$ and $\partial \omega_2$ are smooth. Let $\varepsilon,\delta>0$ be two given parameters, let $\chi_{\omega_i}$ denote the characteristic function of $\omega_i$, let $v|_{[t_1,t_2]^+}$ denote the restriction of $v$ to the non-negative part of the interval $[t_1,t_2]$ and let $\Delta y$ denote the Laplace operator. For a time horizon with ${t_\mathrm{f}}-t_0 > \delta$, we consider the optimal control problem $$\begin{aligned}
&\text{minimize} ~\int_{t_0}^{{t_\mathrm{f}}} \int_\Omega |y(t,x)-\hat{y}(t,x)|^2\,dx\,dt + \int_{t_0}^{{t_\mathrm{f}}} |u(t)|^2\,dt\\
&y_t - \Delta y + v(t)u(t) \chi_{\omega_1} + (1-v(t))u(t) \chi_{\omega_2} = 0\quad \text{on}~(t_0,{t_\mathrm{f}}) \times \Omega\\
&y=0\quad \text{on}~(t_0,{t_\mathrm{f}}) \times \partial\Omega\\
&y=\bar y\quad \text{on}~\{t_0\} \times \Omega\\
&v(t) \in {\mathcal{V}}=\{0,1\}\quad \text{on}~[t_0,{t_\mathrm{f}}]\\
&u(t) \in \begin{cases}[0,1+\varepsilon] & ~\text{if}~v|_{[t-\delta,t]^+} \equiv 1~\text{or}~v|_{[t-\delta,t]^+} \equiv 0 ~\text{a.\,e. on}~[t-\delta,t]^+\\
[0,\varepsilon] &~\text{else}.
\end{cases}
\end{aligned}$$ The combination of the actuator and constraints in this problem model that the continuous control $u$ is restricted to a small uncontrollable disturbance $\varepsilon$ for a dwell-time period of length $\delta$ whenever a decision was taken to change the control region $\omega_1$ to $\omega_2$ or vice versa while the goal is to steer the initial state $\bar{y} \in L^2(\Omega)$ as close as possible to a desired state $\hat{y} \in C([t_0,{t_\mathrm{f}}];L^2(\Omega))$. We consider a perturbation $\lambda=(\bar{y},\varepsilon,\hat{y})$, i.e., a joint perturbation of initial data, the disturbance and the tracking target.
We can consider this problem in the abstract setting with $Y=L^2(\Omega)$, $U={\mathbb{R}}$, ${U_{[t_0,{t_\mathrm{f}}]}}=L^\infty(t_0,{t_\mathrm{f}})$ $$\begin{aligned}
&Ay = {\Delta}y,~y \in D(A)=H^2(\Omega) \cap H^{1}_0(\Omega),\\
&f(y,u,v)=f(u,v)=-(vu \chi_{\omega_1} + (1-v)u \chi_{\omega_2}),\\
&\varphi(\lambda,y,u,v)=\int_{t_0}^{{t_\mathrm{f}}} \int_\Omega |y(t,x)-\hat{y}(t,x)|^2\,dx\,dt + \int_{t_0}^{{t_\mathrm{f}}} |u(t)|^2\,dt,
\end{aligned}$$ defining $$\bar u^v_\varepsilon(t) = \begin{cases}1+\varepsilon & ~\text{if}~v|_{[t-\delta,t]^+} \equiv 1~\text{or}~v|_{[t-\delta,t]^+} \equiv 0 ~\text{a.\,e. on}~[t-\delta,t]^+\\
\varepsilon &~\text{else},
\end{cases}$$ and setting $M=2$ and, for all $v \in {V_{[t_0,{t_\mathrm{f}}]}}$ $$\begin{aligned}
g^v_1(\lambda,\, u,\, t) & = & \displaystyle \operatorname{ess\,sup}_{s \in [t_0,{t_\mathrm{f}}]} (u(s)-\bar u^v_\varepsilon(s)),
\\
g^v_2(\lambda,\, u,\, t) & = & \displaystyle \operatorname{ess\,sup}\limits_{s \in [t_0,{t_\mathrm{f}}]} (-u(s)).\end{aligned}$$ Here the $g^v_i(\lambda,\, u,\, \cdot)$ are constant with respect to $t$ and hence continuous as functions of $t$. Moreover, the maps $u \mapsto g^v_i(\lambda,\, u,\, \cdot)$ are continuous in $L^\infty (t_0, \, {t_\mathrm{f}})$. The objective function is convex with respect to $(y,u)$ and also the maps $u \mapsto g^v_i(\lambda,\, u,\, t)$ are convex. Let $\varepsilon_1$, $\varepsilon_2>0$ be such that without restriction we have $\operatorname{ess\,sup}_{s \in [t_0,{t_\mathrm{f}}]} (u(s)-\bar u^v_{\varepsilon_1}(s))
\geq
\operatorname{ess\,sup}_{s \in [t_0,{t_\mathrm{f}}]} (u(s)-\bar u^v_{\varepsilon_2}(s))$. Then we have $$\begin{aligned}
& &
|
g^v_1(\lambda_1,\, u,\, t)-
g^v_1(\lambda_2,\, u,\, t)
|
\\
&
=
&
\operatorname{ess\,sup}_{s \in [t_0,{t_\mathrm{f}}]} (u(s)-\bar u^v_{\varepsilon_1}(s))
-
\operatorname{ess\,sup}_{s \in [t_0,{t_\mathrm{f}}]} (u(s)-\bar u^v_{\varepsilon_2}(s))
\\
& = &
\operatorname{ess\,sup}_{s \in [t_0,{t_\mathrm{f}}]} (u(s)+\bar u^v_{\varepsilon_2}(s)
-\bar u^v_{\varepsilon_1}(s)-\bar u^v_{\varepsilon_2}(s))+\\
& & \qquad \qquad \qquad \qquad \qquad \qquad
-
\operatorname{ess\,sup}_{s \in [t_0,{t_\mathrm{f}}]} (u(s)-\bar u^v_{\varepsilon_2}(s))
\\
&
\leq
&
\operatorname{ess\,sup}_{s \in [t_0,{t_\mathrm{f}}]}
|\bar u^v_{\varepsilon_2}(s) -\bar u^v_{\varepsilon_1}(s)|
\\
&
\leq
&
|\varepsilon_2 - \varepsilon_1|
.\end{aligned}$$ It is well-known that $(A,D(A))$ is the generator of a strongly continuous (analytic) semigroup of contractions $\{T(t)\}_{t \geq 0}$ on $Y$, see, e.g., [@Pazy1983]. Also, Assumptions \[ass:ControlSys\]–\[ass:Convex\] are easily verified and it is easy to see that and hold. The constraint qualification (CQ) is satisfied with $\omega=\frac{\varepsilon}{2}$ and $\underline \alpha=0$. The control constraints imply that the set $\bar S$ is bounded in ${U_{[t_0,{t_\mathrm{f}}]}}\times {V_{[t_0,{t_\mathrm{f}}]}}$ independently of the initial state $y_0$. Hence we can conclude from Theorem \[thm:JointLip\] that the optimal value function $\nu$ is locally Lipschitz continuous jointly as a function of $\lambda=(\bar{y},\varepsilon,\hat{y})$.
Conclusion
==========
We have studied the optimal value function for control problems on Banach spaces that involve both continuous and discrete control decisions. For control systems of a semilinear type subject to control constraints, we have shown that the optimal value depends locally Lipschitz continuously on perturbations of the initial data and costs under natural assumptions. For problems consisting of linear systems on a Banach space subject to convex control inequality constraints, we have shown that the optimal value of convex cost functions depend locally Lipschitz continuously on Lipschitz continuous perturbations of the costs and the constraints under a Slater-type constraint qualification. The result has been obtained by proving a strong duality for an appropriate dual problem.
By a combination of the above results we have for the linear, convex case obtained local Lipschitz continuity jointly for parametric initial data, control constraints and cost functions. The Example \[ex:nonsmooth\] shows that this result is sharp in the sense that we can, in general, not expect much more regularity than we have proved.
Our analysis currently does not address the stability of the optimal control under perturbations. This is an interesting direction for future work.
Acknowledgements {#acknowledgements .unnumbered}
================
This work was supported by the DFG grant CRC/Transregio 154, projects C03 and A03. The authors thank the reviewers and the editors from *Math. Control Signals Syst.* for the constructive suggestions. The final publication is available at [link.springer.com](http://link.springer.com/article/10.1007/s00498-016-0183-4) and [doi:10.1007/s00498-016-0183-4](http://dx.doi.org/10.1007/s00498-016-0183-4).
[10]{}
Viorel Barbu and Giuseppe Da Prato. , volume 86 of [*Research Notes in Mathematics*]{}. Pitman (Advanced Publishing Program), Boston, MA, 1983.
Alain Bensoussan, Giuseppe Da Prato, Michel C. Delfour, and Sanjoy K. Mitter. . Systems & Control: Foundations & Applications. Birkhäuser Boston Inc., Boston, MA, 1992.
J. Fr[é]{}d[é]{}ric Bonnans and Alexander Shapiro. . Springer Series in Operations Research. Springer-Verlag, New York, 2000.
J. M. Borwein and D. Zhuang. On fan’s minimax theorem. , 26:232–234, 1986.
Piermarco Cannarsa and Halina Frankowska. Value function and optimality conditions for semilinear control problems. , 26(2):139–169, 1992.
Ivar Ekeland and Thomas Turnbull. . Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 1983.
Jacques Gauvin and Fran[ç]{}ois Dubeau. Differential properties of the marginal function in mathematical programming. , (19):101–119, 1982.
Matthias Gerdts. A variable time transformation method for mixed-integer optimal control problems. , 27(3):169–182, 2006.
M. Gugat. One-sided derivatives for the value function in convex parametric programming. , 28(3-4):301–314, 1994.
M. Gugat. Parametric disjunctive programming: one-sided differentiability of the value function. , 92(2):285–310, 1997.
Falk M. Hante and Sebastian Sager. Relaxation methods for mixed-integer optimal control of partial differential equations. , 55(1):197–225, 2013.
Orest V. Iftime and Michael A. Demetriou. Optimal control of switched distributed parameter systems with spatially scheduled actuators. , 45(2):312–323, 2009.
B. S. Mordukhovich, N. M. Nam, and N. D. Yen. Subgradients of marginal functions in parametric mathematical programming. , 116(1-2, Ser. B):369–396, 2009.
A. [Pazy]{}. . Applied Mathematical Sciences Series, Springer-Verlag, New York, 1983.
Fabian Rüffler and Falk M. Hante. Optimal switching for hybrid semilinear evolutions. , 22:215–227, 2016.
Sebastian Sager. Reformulations and algorithms for the optimization of switching decisions in nonlinear optimal control. , 19(8):1238–1247, 2009.
Ramanarayan Vasudevan, Humberto Gonzalez, Ruzena Bajcsy, and S. Shankar Sastry. Consistent approximations for the optimal control of constrained switched systems—part 1: A conceptual algorithm. , 51(6):4463–4483, 2013.
Ramanarayan Vasudevan, Humberto Gonzalez, Ruzena Bajcsy, and S. Shankar Sastry. Consistent approximations for the optimal control of constrained switched systems—part 2: An implementable algorithm. , 51(6):4484–4503, 2013.
A. C. Williams. Marginal values in mixed integer linear programming. , 44(1, (Ser. A)):67–75, 1989.
Feng Zhu and Panos J. Antsaklis. Optimal control of hybrid switched systems: A brief survey. , 25(3):345–364, 2015.
Enrique Zuazua. Switching control. , 13:85–117, 2011.
[^1]: $^*$ The article is published in *Math. Control Signals Syst.* (2017) 29:3.\
$^\dag$ Lehrstuhl für Angewandte Mathematik 2, Department Mathematik, Friedrich-Alexander Universität Erlangen-Nürnberg, [{falk.hante,martin.gugat}@fau.de]({falk.hante,martin.gugat}@fau.de).
|
---
abstract: 'The phenomena of the spin-Hall effect, initially proposed over three decades ago in the context of asymmetric Mott skew scattering, was revived recently by the proposal of a possible intrinsic spin-Hall effect originating from a strongly spin-orbit coupled band structure. This new proposal has generated an extensive debate and controversy over the past two years. The purpose of this workshop, held at the Asian Pacific Center for Theoretical Physics, was to bring together many of the leading groups in this field to resolve such issues and identify future challenges. We offer this short summary to clarify the now settled issues on some of the more controversial aspects of the debate and help refocus the research efforts in new and important avenues. [ I. Adagideli, G. Bauer, M.-S. Choi, Zhong Fang, B. I. Halperin, N. V. Hieu, Jiang-Ping Hu, J. Inoue, H.W. Lee, Minchul Lee, E. Mishchenko, L. Molenkamp, S. Murakami, B. Nikolic, Qian Niu, Junsaku Nitta, M. Onoda, J. Orenstein, C. H. Park, Y.S. Kim, Shun-Qing Shen, D. Sheng, A. Silov, J. Sinova, S. Souma, J. Wunderlich, X. C. Xie, L. P. Zarbo, S.-C. Zhang, Fu-Chun Zhang ]{}'
author:
- Jairo Sinova
- Shuichi Murakami
- 'Shun-Qing Shen'
- 'Mahn-Soo Choi'
title: |
Spin-Hall effect: Back to the Beginning on a Higher Level\
[Summary of the APCTP Workshop on the Spin-Hall Effect and Related Issues]{}\
[Asian Pacific Center for Theoretical Physics, Pohang, South Korea]{}
---
Introduction
============
The spin Hall effect (SHE) is the generation in a paramagnetic system of a spin current perpendicular to an applied charge current leading to a spin accumulation with opposite magnetization at each edge. This effect was first predicted over three decades ago by invoking the phenomenology of the earlier theories of the anomalous Hall effect in ferromagnets, which associated its origin to asymmetric Mott-skew and side-jump scattering from impurities due to spin-orbit coupling.[@Dyakonov:1971_a; @Hirsch:1999_a]
Recently the possibility of an intrinsic (dependent only on the electronic structure) SHE has been put forward [@Murakami:2003_a; @Sinova:2004_a] predicting the presence of a spin current generated perpendicular to an applied electric field in semiconducting systems with strong spin-orbit coupling, with scattering playing a minor role. This proposal has generated an extensive theoretical debate in a very short time motivated by its novel physical concept and potential as a spin injection tool.[@LANL] The interest has also been dramatically enhanced by recent experiments by two groups reporting the first observations of the SHE in n-doped semiconductors[@Kato:2004_d; @Sih:2005_a] and in 2D hole gases (2DHG).[@Wunderlich:2004_a]
These experiments measure directly the spin accumulation induced at the edges of the examples through different optical techniques. On the other hand, most of the early theory has focused on the spin-current generated by an electric field which would drive such spin-accumulation. In most studies this spin current and its associated conductivity has been defined as $j_y^z\equiv\{v_y,s_z\}/2=\sigma^{SHE} E_x$. This choice is a natural one but not a unique one in the presence of spin-orbit coupling since there is no continuity equation for spin density as is the case for charge density. The actual connection between the spin-accumulation and the induced spin-current is [*not*]{} straight forward in the situations where spin-orbit coupling is strong and this relation is the focus of current research and one of the key challenges ahead.
Although two model Hamiltonians with strong spin-orbit coupling have been considered initially, the p-doped 3D valence band system[@Murakami:2003_a] and the 2DEG with Rashba coupling,[@Sinova:2004_a] the one that has attracted the most attention, perhaps due to its simplicity, is the latter one which has the form $H_{\rm R-SO}=\lambda(\sigma_x k_y-\sigma_y k_x)$. In such systems, in a clean sample, where the transport scattering rate $\tau^{-1}$ is small compared to the spin-orbit splitting $\lambda k_F /\hbar$, one finds an intrinsic value $e/8\pi$for the spin Hall conductivity, which is valid at finite frequencies in the range $\tau^{-1} < \omega < \lambda k_F / \hbar $, independent of details of the impurity scattering, in the usual case where both spin-orbit split bands are occupied. The prediction for the dc spin Hall effect in this model has been examined and debated extensively. It was first noticed that contributions to the spin-current from impurity scattering, even in the limit of weak disorder, seemed to cancel exactly the intrinsic contribution.[@Inoue:2004_a; @Mishchenko:2004_a] This lead to speculation that this cancelation destroys the effect in other model as well. On the other hand, it is now understood through recent efforts, culminating in this workshop, that such cancelation only occurs for this *very particular model*, due to the linearity of the spin-orbit coupling and the parabolic dispersion.[@Dimitrova:2004_a; @Chalaev:2004_a]
This motivates the title of this summary: After our initial excitement and our initial worries that such a beautiful effect may not exist, we are back to the original proposal but at a higher level of understanding: that an intrinsic contribution to the SHE in many systems with strong enough spin-orbit coupling is present in general.[@Murakami:2003_a; @Sinova:2004_a] What follows is a summary of the issues agreed upon and debated during the open discussion sessions of the workshop; it is not meant as a summary of all the topics presented in the workshop. Even though feedback from all the speakers in the workshop has been solicited in composing this summary, any ommisions or unnintentional unbalance is ultimately the responsability of the organizers. For further information on this workshop and to view the slides of the talks given and other topics discussed which are not mentioned here we encourage the reader to visit the workshop website.[@site]
Agreement and consensus
=======================
Within the open sessions of this workshop, several key points were discussed and agreement was reached on their conclusions. This is an important and intended result of this workshop, to bring together several of the leading researchers in the field to clarify the now extensive debate in the literature which can be overwhelming to a newcomer.
The agreed upon statements are as follows:
- *The dc spin Hall conductivity, defined through $j_y^z\equiv\{v_y,s_z\}/2=\sigma^{SHE} E_x$, does not vanish in general and it includes both intrinsic and non-intrinsic contributions.*
- *The dc spin Hall conductivity for the model Hamiltonian, ${\cal H_{\rm R}}=\hbar^2 k^2/2m+\lambda (\sigma_x k_y-\sigma_y k_x)$, vanishes in the absence of a magnetic field and spin-dependent scattering, even in the limit of weak scattering. This cancellation is due to the particlar relation in this model between the spin dynamics $d s_y/dt$ and the induced spin-Hall current, i.e. $d s_y/dt=i[{\cal H_R},s_y] \propto j_y^z$, which in a steady state situation indicates a vanishing spin-Hall current. No such relation exists in more complicated models, where the spin-orbit coupling is not simply linear in the carrier momentum.*
The effects of disorder on the induced spin-current, within linear response, come in the form of self-energy lifetime corrections and vertex corrections. The life time corrections only reduce this induced current through a broadening of the bands without affecting its nature. On the other hand, vertex corrections have been the source of important debate since they make the intrinsic SHE vanish in the Rashba 2DEG system for any arbitrary amount of scattering.[@Inoue:2004_a; @Mishchenko:2004_a; @Chalaev:2004_a] For p-type doping in both 3D and 2D hole gases the vertex corrections vanish in the case of isotropic impurity scattering.[@Murakam:2004_a; @Bernevig:2004_c; @Shytov:2005_a; @Khaetskii:2005_a] This result is now understood in the context of the specific relation of the spin-dynamics within this particular model as stated above.[@Dimitrova:2004_a; @Chalaev:2004_a] This spin-dynamics are linked to the magneto-electric effect producing a homogeneous in-plane spin polarization by an electric field in a Rashba 2DEG.[@Edelstein:1990_a; @Inoue:2003_a] These results have recently been found to be consistent with numerical treatments of the disorder through exact diagonalization finite size scaling calculations.[@Nomura:2005_b; @Nomura:2005_a; @Sheng:2005_a]
It is important to point out however that in the mesoscopic regime, where spin Hall conductance of finite size systems rather than conductivity of infinite size systems is considered and the finite width can lead to spin-Hall edge states,[@Adagideli:2005_a] the SHE seems to also be present and robust against disorder even in the 2DEG Rashba system although its link to the bulk regime is still unclear. [@Hankiewicz:2004_b; @Nikolic:2004_a; @Sheng:2004_a; @Adagideli:2005_a]
Semantics
=========
Given the extensive literature it was deemed useful to agree upon several semantics and notations in order not to create confusion from a lack of communication. With this in mind it was agreed that:
- The spin Hall effect is the antisymmetric spin accumulation in a finite width system driven by an applied electric field.
- The word *intrinsic* is reserved for the intrinsic contribution to the spin-current generated in the absence of scattering. This contribution can be calculated through the single bubble diagram within the diagrammatic technique and corresponds to the ac-limit of $\omega \tau
\rightarrow \infty$ where scattering does not play a role. For example, the intrinsic spin Hall conductivity of the Rashba model is $e/(8\pi)$ and for the p-doped valence system it is $(e/6\pi^2)(k_F^{h.h}-k_F^{l.h.})(1+\gamma_1/(2\gamma_2))$.
Future challenges
=================
Theoretical
-----------
Although there is wide agreement within the theoretical community that a spin Hall effect similar in magnitude to the predicted intrinsic contribution should occur in p-doped and in mesoscopic samples, there are still many remaining challenges in order to fully understand this novel effect and related effects in spintronics within strongly spin-orbit coupled systems. At the top of the agenda seems to be a need to better understand the spin-accumulation induced by the spin-Hall effect at a more quantitative level and its relation to the spin-current generated. Some of the issues raised during these open session were:
- What is the effect of the scattering on the induced spin-currents and spin coherence in a strongly spin-orbit coupled system in general and in specific model at a quantitative level (including the sign of the effect in the several experimental set-ups)?
- Can the spin-current density seemingly arising from the Fermi sea lead to spin-accumulation and/or spin transport?
- A clearer understanding of the different contributions and their scaling with respect to disorder (strength, types, range, etc.) to the induced spin current is needed.
- How does spin relax in relation to scattering and to the fact that spin is not a conserved quantity in the strongly spin-orbit coupled regime? How does spin relax near the baoundry?
- Is the effect more readily observable at mesoscopic scales and is there a relation between the mesoscopic and bulk regime?
- Are there other spin-current definitions which give a clearer picture and can be more readily connected to spin-accumulation?
- There is a need for a full theory of spin-accumulation (and detection) in strongly spin-orbit coupled systems.
These are some of the key issues and questions raised but not by all means the only ones that are being considered in current research. It is important to realize that besides the SHE, there is a plethora of effects, linked to spin-transport dynamics in semiconductors, which are important to understand in the context of strongly spin-orbit coupled systems. One in particular is the spin Coulomb drag,[@Damico:2002_a] which is an intrinsic friction mechanism between opposite spin populations studied in non-spin-orbit coupled systems, and is important in degenerate systems where electron-electron interactions are relevant.
Experimental
------------
On of the clear achievements on the spintronics in recent years has been the experimental observation of this novel effect through optical means. Spin transport in spin-orbit coupled systems is governed by characteristic length scales (mean free path, $l=v_F \tau$, spin precession length $l_{so}=\hbar v_F/\Delta_{so}$), time scales (lifetime,$\tau$ , spin coherence time, $\tau_s$) and by the relative strength of spin-orbit coupling, $\Delta_{so}$ and disorder. From these scales it is generally believed that the SHE observed by Awschalom et al. [@Kato:2004_d] is in the extrinsic regime and the one observed by Wunderlich et al. [@Wunderlich:2004_a] in 2DHG is in the intrinsic regime.
Some of the experimental issues raised during the open dicussion session were:
- A key remaining experimental challenge is the detection of the effect through electrical means which could lead to actual useful devices. This detection has to be done in coordination with careful realistic theoretical modeling of particular devices.
- It is important to understand and model in further detail the effects of edge electric field induced spin-polarization vs. the spin-Hall effect, and the angle dependence of the luminescence induced in the present set-ups and their relation to the spin magnetization.
- Is it possible to measure spin current in the bulk; i.e. not indirectly through spin accumulation?
Outlook
=======
The past two years have seen a tremendous amount of research achievements and advances in the area of spintronics which continuous to generate many novel ideas and phenomena. Besides a good and healthy competitiveness in the field, it has been a field, as it is demonstrated by organizing this conference, which moves forward in unison to clarify debates rather than allow them to linger for many years, helping it to move forward to explore interesting new physics.
As illustrated by the topics debated throughout the workshop, there are many remaining challenges and a very healthy outlook of the field, and not just simply of the spin-Hall effect which is a very small part of the whole of the spintronics field. The organizers are grateful for the sponsorship of the Asian Pacific Center for Theoretical Physics and the National Science Foundation (OISE-0527227) which have made this workshop possible.
[27]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{}
, p. ().
, ****, ().
, , , ****, (), .
, , , , , , ****, (), .
, , , , ****, ().
, , , , , , ****, (), .
, , , , ****, (), .
, , , ****, (), .
, , , ****, (), .
, ****, (), .
, ****, (), .
.
, ****, (), .
, ****, (), .
, , (), .
(), .
, ****, ().
, , , ****, ().
, , , , , (), .
, , , , ****, (), .
, , , , ****, (), .
(), .
, , , , ****, (), .
, , , ****, (), .
, , , ****, (), .
, ****, ().
|
---
abstract: 'We calculate the characteristic energies of fusion between planar bilayers as a function of the distance between them, measured from the hydrophobic/hydrophilic interface of one of the two nearest, cis, leaves to the other. The two leaves of each bilayer are of equal composition; 0.6 volume fraction of a lamellar-forming amphiphile, such as dioleoylphosphatidylcholine, and 0.4 volume fraction of a hexagonal-forming amphiphile, such as dioleoylphosphatidylethanolamine. Self-consistent field theory is employed to solve the model. We find that the largest barrier to fusion is that to create the metastable stalk. This barrier is the smallest, about 14.6 $k_BT$, when the bilayers are at a distance about 20 percent greater than the thickness of a single leaf, a distance which would correspond to between two and three nanometers for typical bilayers. The very size of the protein machinery which brings the membranes together can prevent them from reaching this optimum separation. For even modestly larger separations, we find a linear rate of increase of the free energy with distance between bilayers for the metastable stalk itself and for the barrier to the creation of this stalk. We estimate these rates for biological membranes to be about 7.1 $k_BT$/nm and 16.7 $k_BT$/nm respectively. The major contribution to this rate comes from the increased packing energy associated with the hydrophobic tails. From this we estimate, for the case of hemagglutinin, a free energy of 38 $k_BT$ for the metastable stalk itself, and a barrier to create it of 73 $k_BT$. Such a large barrier would require that more than a single hemagglutinin molecule be involved in the fusion process, as is observed.'
author:
- |
J.Y. Lee and M. Schick\
Department of Physics\
University of Washington, Box 351560, Seattle, WA 98195-1560
bibliography:
- '014731JCP.bib'
title: 'Dependence of the energies of fusion on the inter-membrane separation: optimal and constrained'
---
Introduction
============
Although it is essential to a host of biological processes in which material enters, exits, or changes location within the cell, ([*e.g.*]{} viral entry, exocytosis, and intracellular trafficking) the process of membrane fusion is not well understood. Some basic concepts, however, are clear. The membranes to be fused must be put under tension, [*i.e.*]{} their free energy per unit area must be increased, so that the fused state with smaller area has a lower free energy than the unfused system. This tension is brought about by bringing the membranes to be fused in close proximity to one another, on the order of a few nanometers, thereby removing some water from the hydrophilic headgroups of the amphiphiles comprising the membrane and consequently raising the system free energy. This additional energy is supplied by fusion proteins. Even though the free energy of the system is reduced by fusion, the rearrangement of lipids required by the process can only occur if the system surmounts free energy barriers. The calculation of these barriers has been the subject of much attention [@Kozlov83; @Siegel93; @Kuzmin01; @Lentz00; @Chernomordik03].
From the above argument, it follows that the barrier to the fusion process must be a function of the tension. It also depends on the pathway to fusion that the system takes [@Katsov04; @Katsov06], as well as several other factors. Among these are the average compositions of the different amphiphiles comprising the membrane and, in particular, their composition in the cis, or proximal, leaves [@Lee07; @Kasson07]. We have examined each of these factors, and the upshot is, that for bilayers in which the relative fraction of hexagonal-forming and lamellar-forming amphiphiles in the cis leaves are similar to that in biological membranes, the largest barrier to fusion, in either the standard [@Kozlov83] or non-standard [@Noguchi01; @Mueller02; @Mueller03] pathways, is that to form the initial stalk. This is the initial local junction formed by the rearrangement of lipids in the two apposing cis leaves [@Kozlov83]. Further, this barrier is not large; it was estimated [@Lee07] to be on the order of 13$k_BT$, with $T$ the absolute temperature and $k_B$ Boltzmann’s constant. That the rate-limiting barrier to fusion should be so small led us to conclude that fusion should occur rapidly once the two membranes were brought sufficiently close to initiate the process.
This conclusion highlights the question of what is “sufficiently close”, [*i.e.*]{} the issue of the dependence of the fusion barrier on the distance between the membranes to be fused. It is an interesting issue which speaks to the interplay between the lipids and a fusion protein. An example is provided by hemagglutinin, the fusion protein associated with the influenza virus [@Eckert01]. It is anchored in the viral membrane. A cluster of between three and six of them around the eventual site of fusion are required [@Blumenthal96]. A first conformational change of hemagglutinin is accompanied by removal of the receptor binding domains. A second conformational change exposes the hydrophobic fusion peptide which anchors in the target membrane. At this point the conformation of the several hemagglutinins, which are essentially normal to the membranes, keep the viral and target membranes at a distance of 13.5 nm from one another [@Wiley87]. A final conformational change brings the membranes much closer, on the order of 4 nm, with the hemagglutinin now parallel to the membranes and pointing away from the fusion site [@Bentz93a]. This conformational change releases a great deal of energy, on the order of 60$k_BT$ per hemagglutinin [@Kozlov98], which presumably is expended in pulling the membranes to this distance and in bringing about the formation of the stalk. The question is why this distance is what it is. Is it because a smaller distance between membranes would cause fusion to be energetically less expensive, but the very size of the hemagglutinin prevents a closer approach, or is it that the machinery is such that it does bring the membranes to the optimal separation? Just what is the competition that sets the distance at which fusion occurs? Similar questions apply to the SNARE machinery which promotes fusion [@Sollner93].
There has been little theoretical work on the distance dependence of the barrier to fusion [@Kozlovsky02; @Knecht03; @Kasson06]. It was considered explicitly by Kozlovsky and Kozlov [@Kozlovsky02] using a phenomenological model. They found that the energy of an isolated stalk was practically independent of the distance between membranes, and approached a value of about 43$k_BT$ as the distance between membranes increased without limit. This result can be traced to a few assumptions. First the membranes are assumed to be tensionless. Hence, the additional membrane area needed to create a stalk between two membranes at a large distance costs no free energy by assumption. This assumption is presumably quite good when the distance between membranes is greater than that of the hydrophobic repulsion, on the order of a few nm [@Israelachvili82]. The second assumption is that the membranes can bend to take a shape which minimizes the curvature energy of the system. Given the constraints on the membrane separation placed by the presence of the fusion proteins, this is probably not the case. Finally, the phenomenological free energy employed does not capture the energy associated with packing the tails efficiently into the axially symmetric stalk structure, a structure very different from the planar bilayer, the membrane configuration of lowest free energy.
In order to clarify these issues, particularly that of the packing, we employ a microscopic model to study the dependence of the barriers to fusion on the distance, $H$, between the hydrophobic/hydrophilic interfaces of the apposed leaves of planar membranes. The membranes are under either zero or a small tension. The membranes are composed of a mixture of two amphiphiles, one lamellar- and the other hexagonal-forming. The leaves are of equal composition, one that mimics the mix of these two classes of amphiphiles in the cis leaves of red blood cell membranes. This choice is made because previous work [@Kozlovsky02; @Kasson07] shows that the free energy of fusion intermediates is most sensitive to the composition of the cis leaves, and rather insensitive to that of the trans leaves. Only the standard fusion mechanism is considered. We do this because we have found very little difference in the barriers of the two different mechanisms when membranes with a mix of hexagonal and lamellar formers were considered [@Lee07]. In addition, this restriction significantly simplifies the calculation.
We find, once again, that the largest barrier to fusion is that associated with the formation of the initial stalk. We also can understand the dependence of this barrier on the separation between membranes as follows. When the membranes are very close, the barrier to fusion increases with decreasing distance for two reasons. Not only does the repulsive, hydrophobic, interaction, essentially a depletion force, increase with decreasing separation, but also the energy required for the amphiphiles to rearrange into a stalk of such short extent becomes larger with smaller membrane separation. Due to this effect, the stalk is not a metastable structure. As a consequence fusion would have to proceed directly to a fusion pore without a stalk intermediate, an absence which would make the process much less likely. When the membranes are farther apart, the stalk becomes a stable intermediate, and the barrier to fusion decreases. As the distance between membranes increases still further, the barrier to fusion now increases rapidly with increasing distance due to the packing energy of the initial stalk connecting the membranes, an energy which scales with the length of the stalk. We find this rate of increase to be about 7 $k_BT$ per nm. Consequently the lowest barrier to fusion occurs when the two membranes are at a distance large enough that membrane repulsion is not too great, and the stalk is metastable, but small enough that the stalk is relatively short and energetically inexpensive. In our system we find the optimum distance to be about twenty percent greater than the thickness of a single leaf of our bilayer, a distance which would correspond to between two and three nanometers for typical membranes. This is in reasonable agreement with the observed distance to which laboratory membranes must be brought in order to fuse [@Weinreb07]. The lowest barrier to fusion corresponds to about $14.6k_BT$ for a biological membrane. To fuse membranes which are at a somewhat larger distance, as in the case when the very size of hemagglutinin prevents a closer approach, requires traversing a larger barrier. At a distance between headgroups of 4 nm applicable to the case of hemagglutinin, we estimate that the barrier is on the order of 73$k_BT$. It is not surprising, then, that more than a single fusion protein would be required.
The model
=========
To investigate the effect of the distance between planar membranes on the free energy barrier to fuse them, we extend the application of self-consistent field theory to microscopic models of membranes initiated earlier [@Katsov04; @Katsov06; @Lee07]. The basic assumption of this approach is that the self-assembly into bilayer vesicles and the processes which these vesicles can undergo, such as fusion, are common to systems of amphiphiles, of which lipids are but one example. Recent work on vesicles which consist of diblock copolymers serves to illustrate this point [@Discher99]. It follows that these processes can be explored in whatever system of amphiphiles proves to be most convenient. For the application of self-consistent field theory, that system is one of block copolymers in a homopolymer solvent. While the processes that amphiphiles undergo are presumably universal, the energy scales of these processes are system-dependent, and thus it is necessary to be able to compare the energy scale in a biological bilayer with the energy scale in our system of block copolymers. This will be done below.
Here we consider a system of two bilayers each composed of two different amphiphiles that resemble dioleoylphosphatidylcholine, (DOPC), and dioleoylphosphatidylethanolamine, (DOPE), in their hydrophobic/hydrophilic ratios. The two leaves of each bilayer are of the same composition. The system is incompressible and occupies a volume $V$. The two amphiphiles are each AB diblock copolymers. Type 1, a lamellar-former, consists of $N$ monomers and has a molecular volume $Nv$. The fraction of hydrophilic monomers, arbitrarily chosen to be of type $A$, is denoted $f_1$ and is assigned the value $f_1=0.4$ as such a diblock has a “spontaneous curvature" similar to that of DOPC [@Katsov04]. The amphiphile of type 2 consists of $N\tilde\alpha$ monomers and has a molecular volume of $\tilde\alpha Nv$. The fraction of hydrophilic monomers, $f_2$ is chosen to be $f_2=0.294$ as this produces a spontaneous curvature similar to that of DOPE. We set $(1-f_1)Nv=(1-f_2)\tilde{\alpha} Nv$ such that hydrophobic tails of different types of amphiphiles have equal length. For our chosen $f_1=0.4$ and $f_2=0.294$, $\tilde\alpha = 0.85$. The solvent is an A homopolymer with volume $Nv$.
We denote the local volume fraction of hydrophilic elements of amphiphile 1 to be $\phi_{A,1}({\bf r})$, of amphiphile 2 to be $\phi_{A,2}({\bf r})$, and of the solvent to be $\phi_{A,s}({\bf r})$. The total local volume fraction of hydrophilic elements is denoted $$\phi_A({\bf r})=\phi_{A,1}({\bf r})+\phi_{A,2}({\bf r})+\phi_{A,s}({\bf
r}).$$ Similarly the total local volume fraction of hydrophobic elements is $$\phi_B({\bf r})=\phi_{B,1}({\bf r})+\phi_{B,2}({\bf r}).$$ The amounts of each of the components are controlled by activities, $\zeta_1$, $\zeta_2$, and $\zeta_s$. Because of the incompressibility constraint, only two of the activities are independent. Cylindrical coordinates, $(\rho,\theta,z)$, are employed.
Within the self-consistent field approximation, the free energy, $\Omega(T,V,{\cal A},\zeta_1,\zeta_2,\zeta_s)$, of the system containing a bilayer, or bilayers, each of area ${\cal A}$, is given by the minimum of the functional $$\begin{aligned}
\label{free1}
\frac{Nv}{k_BT}{\tilde\Omega}&=&-\zeta_1Q_1-\zeta_2Q_2-\zeta_sQ_s\nonumber \\
&&+\int d{\bf r}[\chi N\phi_A({\bf r})\phi_B({\bf r})-
w_A({\bf r})\phi_A({\bf r})-w_B({\bf r})\phi_B({\bf r})\nonumber \\
&&-\xi({\bf r})(1-\phi_A({\bf r})-\phi_B({\bf r}))],\end{aligned}$$ where $Q_1(T,[w_A,w_B])$, $Q_2(T,[w_A,w_B]),$ and $Q_s(T,[w_A])$ are the configurational parts of the single chain partition functions of amphiphiles 1 and 2 and of solvent. They have the dimensions of volume, and are functions of the temperature, $T$, which is inversely related to the Flory interaction $\chi$, and functionals of the fields $w_A$ and $w_B$. These fields, and the Lagrange multiplier $\xi({\bf r})$, which enforces the local incompressibility condition, are determined by the self-consistent equations which result from minimizing the free energy functional. Insertion of these fields into the free energy functional, Eq. (\[free1\]), yields the free energy within the self-consistent field approximation: $$\begin{aligned}
\label{scsymfree}
\frac{Nv}{k_BT}\Omega(T,V,{\cal A},\zeta_1,\zeta_2,\zeta_s)
&=&-\zeta_1Q_1(T,[w_A,w_B])-\zeta_2Q_2(T,[w_A,w_B])-\zeta_sQ_s(T,[w_A])\nonumber\\
&-&\int\ d{\bf r}\chi N\phi_A({\bf r})\phi_B({\bf r}),\end{aligned}$$ The free energy of the system without the bilayer, i.e. a homogeneous solution, is denoted $\Omega_0(T,V,\zeta_1,\zeta_2,\zeta_s)$. The difference between these two free energies, in the thermodynamic limit of infinite volume, defines the excess free energy of the system with one, or more, membrane: $$\delta\Omega(T,{\cal A},\zeta_1,\zeta_2,\zeta_s)\equiv\lim_{V\rightarrow\infty}
[\Omega(T,V,{\cal A},\zeta_1,\zeta_2,\zeta_s)-\Omega_0(T,V,\zeta_1,\zeta_2,\zeta_s)].$$ With the excess free energy known, the surface free energy per unit area, or equivalently, the surface tension, $\gamma$, is obtained from the excess free energy of a single, flat, bilayer $\delta\Omega_{bilayer}$ $$\gamma(T,\zeta_1,\zeta_2,\zeta_s)\equiv\lim_{{\cal A}\rightarrow\infty}
\frac{\delta\Omega_{bilayer}(T,{\cal A},\zeta_1,\zeta_2,\zeta_s)}{{\cal A}}.$$
In order to calculate the free energy of stalk or hemifusion intermediates as a function of their radius, that radius must be fixed [@Katsov04; @Matsen99] by a local Lagrange multiplier $\psi({\bf r}).$ Similarly, to constrain the membranes to be separated by a specified distance, $H$ at some point ${\bf r}$, we must introduce an additional Lagrange multiplier, $\lambda({\bf r})$. The distance $H$ is chosen to be the distance between the hydrophilic/hydrophobic interfaces of the contacting, cis, leaflets as shown in Fig. 1. With these additional constraints, the free energy functional to be minimized now reads$$\begin{aligned}
\label{eq:systemenergy2}
\frac{Nv{\tilde\Omega}}{k_{B}T} &=& -\zeta_1Q_{1}-\zeta_{2}Q_{2}-\zeta_sQ_{s}
+\int dV [\chi N \phi_{A}({\bf r})\phi_{B}({\bf r}) \nonumber \\
& & -w_{A}({\bf r})\phi_{A}({\bf r})-w_{B}({\bf r})\phi_{B}({\bf r})
-\xi({\bf r})(1-\phi_{A}({\bf r})-\phi_{B}({\bf r})) \\
& & -\psi \delta(\rho-R)\delta(z)(\phi_{A}({\bf r})-\phi_{B}({\bf r})) \\
& &
-\lambda [\delta(z-H/2)+\delta(z+H/2)](\phi_{A}({\bf r})-\phi_{B}({\bf r}))].\nonumber\end{aligned}$$ It is clear that one cannot constrain the bilayers to be a distance $H$ apart at a position at which the stalk or hemifusion diaphragm come in contact. Consequently in the last integral in Eq. (\[eq:systemenergy2\]), the region of integration over $\rho$ is restricted to be greater than $R+R_c$, where $R$ is the radius of the fusion intermediate, and $R_c$ is positive and at least as large as the hydrophilic thickness of the bilayer. The condition that the free energy functional of Eq. (\[eq:systemenergy2\]) be minimized yields a set of self-consistent equations that we solve in real space. A detailed description on the derivation of Eq. (\[free1\]) and the real space solution algorithm can be found elsewhere [@Katsov04; @Fredrickson06; @Mueller06].
Finally we need to compare the energies in a biological system with those in our homopolymer system. There are various choices for the energy of the biological system. One could choose a property of a single bilayer, such as the energy per unit area of a hydrophobic, hydrophilic interface. Alternatively a property of two interacting bilayers coud be chosen, such as the attractive energy per unit area between them. As the former is so well known, we shall employ it, but will show below that this gives essentially the same result had we chosen the latter. We consider the dimensionless quantity $\gamma_{ow}D^2/k_BT$, where $\gamma_{ow}=40\times 10^{-3}$N/m is the oil, water interfacial tension, and $D=4\times 10^{-9}$m is a typical bilayer thickness. With $k_BT=4.3\times 10^{-21}$Nm, this ratio is about 150 for a biological system. The analogous quantity in the polymer system is $\gamma_0d^2/k_BT$, where $\gamma_0$ is the surface tension between coexisting solutions of hydrophobic and hydrophilic homopolymers, and $d$ is the thickness of our bilayers. We calculate $\gamma_0d^2/k_BT=56.7$, so that energy scales in a biological system are about a factor of 150/56.7=2.6 greater than in our polymer model.
Results and Discussion
======================
We first consider some properties of a single bilayer composed of lamellar-forming amphiphiles, chosen to mimic DOPC, whose volume fraction is 0.6, and hexagonal-forming amphiphiles, chosen to mimic DOPE, whose volume fraction is 0.4. The leaves are of equal composition. As in our previous work, we have chosen the volume of amphiphile 1 to be $Nv=1.54 R_g^3$, where $R_g$ is the radius of gyration of the polymer. The bilayer thickness, measured between the planes at which the volume fractions of the hydrophilic part of the amphiphiles and that of the solvent are equal, is $4.3R_g$. The hydrophobic thickness, measured between the planes at which the volume fractions of hydrophilic and hydrophobic parts of the amphiphiles are equal, is $2.7 R_g.$
Two such bilayers have a weak attraction between them due to depletion forces induced by expulsion of some solvent when they are brought together. To see this, we calculate the excess free energy of a system of two flat bilayers a distance $H$ apart, $\delta\Omega_{2bilayers}(H)$ and define the free energy per unit area $$\label{fofh}
F(H)\equiv \frac{\delta\Omega_{2bilayers}(H)}{{\cal A}}-2\gamma.$$ By definition, this quantity asymptotes to zero for large $H$, and is negative when the bilayers attract one another. For the case of bilayers under zero tension, the dimensionless quantity $F(H)R_g^2/k_BT$ is plotted in Fig. 2.
This energy of attraction per unit area can be compared with those measured between phospholipid bilayers provided we know the length scale given by $R_g$, the radius of gyration of the polymers in our system. To obtain this we note that the thickness of our bilayers is approximately 4.3$R_g$. If we take a typical bilayer thickness to be 4 nm, then $R_g\sim 0.93$ nm. With this and $k_BT=4.1\times 10^{-21}$ J, our calculated value of the free energy per unit area at the equilibrium distance between membranes corresponds to 0.07 mJ/m$^2$. This should be increased by the factor of 2.6 if the energy scale we obtained by comparison with the hydrophilic, hydrophobic repulsion, is correct. Thus we expect that the energies of attraction per unit area between two phospholipid bilayers should be approximately 0.18 mJ/m$^2$. This agrees extremely well with the results presented by Marra and Israelachvili [@Marra85] in their Fig. 2. It shows that we could have obtained our energy scale equally well from the interaction energy of two bilayers.
The excess free energy of an intermediate, such as a stalk, is calculated as follows. We compute the excess free energy, $\delta\Omega(H)$, of the system of two bilayers which are connected by the intermediate, and which, far from it, are separated by a distance $H$. The excess free energy of the intermediate is, then $$\delta\Omega_{int}(H)= \lim_{{\cal A}\rightarrow\infty}
\left\{\delta\Omega(H)-[F(H)+2\gamma]{\cal A}\right\}.$$ In Figure 3, we show the excess surface energy of the stalk as a function of its radius, $R$, at different bilayer separations $H$. Again, the tension of the bilayer is zero. Each leaf of the bilayers shown here have compositions, $\phi_1=0.60$ and $\phi_2=0.40$, which are almost the same as the cis leaves of the asymmetric membranes we considered previously [@Lee07]. We note that for stalk radii which are quite small, less than about 0.5 $R_g$, we find no solution for a stalk-intermediate. This reflects the fact that the process by which the stalk initially forms cannot necessarily be thought of as one which produces a stalk of infinitesimal radius which then expands. At large radii, the stalk expands into a hemifusion diaphragm. We find that as the membrane separation $H$ increases, this hemifusion diaphragm becomes indistinguishable from a single bilayer membrane. Hence for all large $H$ the free energy increases linearly with $R$ with a slope directly related to a line tension, one which arises from the junction of the hemifusion diaphragm with the two bilayer membranes.
The most important result in Fig. 3 is that the increase of separation between fusing bilayers causes the energy of the metastable stalk to increase significantly. It follows that the barrier to the formation of this stalk also increases significantly with separation. As an estimate to this barrier, we take the energy of the stalk with the smallest radius for which we find a solution of our equations. This should be considered an upper bound, as there may be less expensive paths to the creation of the stalk. A second result of note is that there is no metastable stalk if the bilayers are too close to one another. This is because the energy associated with the rearrangement of amphiphiles needed to make the stalk is simply too large at small membrane separations. As the intermembrane distance increases, the stalk does become metastable with a radius on the order of 1.3$R_g$. This is reasonable as the diameter of this stalk is about the same as the hydrophobic thickness of our bilayers, $2.7R_g$, so that amphiphiles that make up the stalk can take configurations somewhat similar to those of amphiphiles in the unperturbed bilayers.
The importance of the stalk being metastable can be seen in Fig. 4, which summarizes the results of our calculation. We have plotted, as a function of separation, $H$, the energy of the stalk with the smallest radius for which we find a solution, (squares), the energy of the metastable stalk (circles), and the barrier (triangles) which is associated with the expansion of the stalk into a hemifusion diaphragm before pore formation. For the smallest two interbilayer separations shown, $H=1.96 R_g$ and $H=2.20
R_g$, there is no metastable stalk. Consequently one large activation energy of approximately $11 k_BT$ (corresponding to 29 $k_BT$ for a biological membrane) is required before a fusion pore can form. However, for separations for which there is a metastable stalk, $H \geq 2.5 R_g$, fusion can occur in two steps: formation of the initial stalk which relaxes to the metastable stalk, and expansion into a hemifusion diaphragm with formation of a fusion pore. An additional activation energy is required for this second step, and is given by the difference between the energy of the second barrier and that of the metastable stalk. A third point of interest concerns the range from $H/R_g>2.49$, at which the stalk first becomes metastable, to $H/R_g<3.05$ at which the barrier to make the hemifusion diaphragm (triangles in Fig. 4) is no longer larger than the barrier to make the initial stalk, (squares in Fig. 4). Within this range, the additional energy needed by the metastable stalk to surmount the second barrier and go forward to the hemifusion diaphragm is larger than that required for the process to reverse itself by means of the disappearance of the stalk. In other words, in this range successful fusion is a less likely outcome of stalk formation than the simple disappearance of the stalk. The probabilities of these outcomes are not reversed until $H/R_g$ exceeds 3.25. But at this larger separation, the barrier to form the initial stalk is also larger. Thus we expect that most of the time a metastable stalk actually forms, it does not lead to successful fusion.
A fourth point we wish to make is the following: once a metastable stalk becomes possible, the additional activation energy needed to pass to the hemifusion diaphragm is always less than the barrier to create the initial stalk. Hence this barrier to create the initial stalk, whose magnitude is shown by the squares in Fig. 4, becomes the largest barrier to fusion. Its magnitude is the smallest when the stalk first becomes metastable, which occurs when the bilayers are at a distance $H\sim 2.5R_g$ which exceeds by about 20% a distance equal to half the hydrophobic thickness of our bilayers. This small membrane separation, again defined between the hydrophilic/hydrophobic interfaces of the apposed cis leaflets can be compared with the results of Weinreb and Lentz [@Weinreb07] who found optimum fusion at a distance between hydrophobic/hydrophilic interfaces that was comparable to half the hydrophobic thickness of their bilayers. The value of this smallest barrier for stalk formation is, from Fig. 4, about 5.6$k_BT$ for the copolymer membranes, which corresponds to about 14.6 $k_BT$ for a biological membrane.
We note from Fig. 4 that the free energies of the metastable stalk and of the barrier to its creation become linear functions of $H$ even for values of $H$ which are not too large. The rate of increase of the free energy of the metastable stalk with intermembrane distance, (circles in Fig. 4), is 2.5 $k_BT/(H/R_g)$. We can convert this rate of change of free energy with distance to practical units as follows. We increase the energy by a factor of 2.6 to account for the difference between our amphiphilic bilayers and those composed of lipids and utilize the length scale $R_g\sim 0.93$ nm obtained earlier. From these we find that the above rate of increase of the metastable stalk free energy with thickness becomes $$\frac{d\delta\Omega_{stalk}}{dH}\sim 7.1\ k_BT/{\rm nm},$$ As we have set the tension of the bilayers to zero, this increase in stalk free energy does not arise simply from the additional surface area of a longer stalk. We have repeated our calculations taking a surface tension equivalent to 2.68 mN/m, a value in the range of tensions which can cause rupture [@Evans03], and found that the rate of change of metastable stalk energy with membrane separation increased from the value of 7.1 k$_B$T/nm only to 9.4 k$_B$T/nm. Therefore we conclude that the increased area associated with a stalk of greater length is not the major contribution to the stalk free energy. Rather it is plausible that the dominant contribution to the length dependence of the metastable stalk free energy comes from the packing of the hydrophobic tails. That is, although the stalk has a diameter comparable to the hydrophobic thickness of the bilayer, the axially symmetric configuration is very different from the planar bilayer. If the density of headgroups in the stalk is comparable to that in the bilayer, then the tails become crowded near the center of the stalk. Conversely, if the tail density at the center is comparable to that of the interior of the bilayer, then the density of headgroups must be considerably less than that of the bilayer causing a significant energy penalty of contact between solvent and tails. This conjecture is strengthened by the observation, from Fig. 4, that the rate of increase with distance, $H$, of the barrier to stalk formation is greater than that for the metastable stalk itself. This is reasonable as the intermediate that we consider, and which corresponds to the barrier, is a stalk of diameter smaller than that of the metastable stalk, and also smaller than the thickness of an unperturbed bilayer. Hence the hydrophobic tails are packed quite densely. From Fig. 4, (squares), this slope is $d\delta\Omega_{barrier}=6.0\ d(H/R_g)$ which for a biological membrane translates to $$\frac{d\delta\Omega_{barrier}}{d H}\sim 16.7\ k_BT/{\rm nm}.$$
These results permit us to discuss the interesting case which arises when the apposing membranes cannot be brought to the optimum, small, distance which the amphiphiles would like simply because of the very size of the protein machinery which brings the membranes together. This is the case with hemagglutinin whose approximate 4nm width [@Wilson81] keeps the head groups of apposing membranes this distance apart. If we assume a headgroup of 1 nm [@Marra85], then the minimum distance between hydrophilic/hydrophobic interfaces is on the order of 6nm. The free energy of the metastable stalk and the barrier to its creation when the apposing bilayers are constrained to be at such a distance can be estimated from Fig. 4 and the linear behavior at large distances given above. We find the metastable stalk to have an excess free energy of 38 $k_BT$. The barrier to be overcome to create this metastable stalk is about 73 $k_BT$. It is understandable that more than a single hemagglutinin molecule is required to bring about the amphiphile reorganization needed to produce a stalk linking membranes at such a distance, one imposed by the very machinery of fusion itself.
Acknowledgments
===============
This work was supported by the National Science Foundation under Grant No. DMR-0503752.
![[]{data-label="distance"}](fig1.eps){width="5.25in"}
{width="5.25in"}
{width="5.25in"}
{width="5.25in"}
Figure Captions
===============
- [Figure 1]{} Apposed bilayers separated by distance $H$. Circles represent hydrophobic head groups and curved lines hydrophobic tails. The separation $H$ is measured between the hydrophilic/hydrophobic interfaces of the contacting leaflets.
- [Figure 2]{} Free energy per unit area of apposed bilayers, $F(H)$ of eq (\[fofh\]), in units of $k_BT/R_g^2$ as a function of separation distance, $H/R_g$, between bilayers composed of 60% lamellar formers and 40% hexagonal formers and under zero tension.
- [Figure 3]{} Excess surface energy of stalk-like fusion intermediates as a function of stalk radius, $R$ for $H=2.2 R_g$ (solid), $H=2.7 R_g$ (dotted), $H=3.2 R_g$ (dashed), $H=3.7 R_g$ (dot-dashed), and $H=4.0
R_g$ (dot double-dashed) for systems composed of 60% DOPC-like and 40% DOPE-like diblocks under zero tension.
- [Figure 4]{} Various energies related to fusion in the standard mechanism as a function of separation $H$ for bilayers shown in Fig. 3. Squares represent the initial barrier to create a stalk, circles the metastable stalk energy, and triangles the second barrier as the stalk expands to a hemifusion diaphragm. For the lowest two values of separation ($H=1.96
R_g$ and $2.20 R_g$), metastable stalks do not exist.
|
---
abstract: 'We present a new class of black hole solutions in third-order Lovelock gravity whose horizons are Einstein space with two supplementary conditions on their Weyl tensors. These solutions are obtained with the advantage of higher curvature terms appearing in Lovelock gravity. We find that while the solution of third-order Lovelock gravity with constant-curvature horizon in the absence of a mass parameter is the anti de Sitter (AdS) metric, this kind of solution with nonconstant-curvature horizon is only asymptotically AdS and may have horizon. We also find that one may have an extreme black hole with non-constant curvature horizon whose Ricci scalar is zero or a positive constant, while there is no such black hole with constant-curvature horizon. Furthermore, the thermodynamics of the black holes in the two cases of constant- and nonconstant-curvature horizons are different drastically. Specially, we consider the thermodynamics of black holes with vanishing Ricci scalar and find that in contrast to the case of black holes of Lovelock gravity with constant-curvature horizon, the area law of entropy is not satisfied. Finally, we investigate the stability of these black holes both locally and globally and find that while the black holes with constant curvature horizons are stable both locally and globally, those with nonconstant-curvature horizons have unstable phases.'
author:
- 'N. Farhangkhah $^{1}$and M. H. Dehghani $^{2,3}$[^1]'
title: Lovelock black holes with nonmaximally symmetric horizons
---
Introduction
============
Higher-order curvature theories of gravity have gained a lot of attention. Not surprisingly, the mere extension of general relativity in higher dimensions can immediately lead to a wide variety of alternative theories of gravity whose actions contain higher-order curvature terms. The inclusion of higher curvature terms in the gravitational action increases further the diversity of the models available and gives rise to a rich phenomenology, which is actively investigated these days. Amongst the higher curvature theories of gravity, Lovelock theory [@Lovelock] which is the most general second-order gravity theory in higher-dimensional spacetimes has attracted a lot of attention. The action imposed in this theory is consistent with the corrections inspired by string theory to Einstein-Hilbert action [@string]. The most extensively researches are done on Einstein-Gauss-Bonnet gravity with second-order curvature corrections [@GB1; @GB2; @GB3]. Although the Lagrangian and field equations look complicated in third-order Lovelock gravity, there are a large number of works on introducing and discussing various exact black hole solutions of third-order Lovelock gravity [@Lovelockex1; @Lovelockex2; @Lovelockex3]. It is known that the second-order Lovelock gravity admits supersymmetric extension [@SupGB], while all the higher orders of this theory have only the necessary condition of supersymmetric extension [SupLg]{}. Throughout the recent years, most of the interesting holographic aspects of Lovelock gravity have been studied [@HolLg3]. Recently, some works have been extended to general Lovelock gravity to investigate the solutions and their properties [@Lovelockg1; @Lovelockg2; @Lovelockg3].
Although most of the known black hole solutions of Lovelock gravity are those with curvature constant horizons, one may raise the question of having black hole solutions with nonconstant curvature. Here, specially, we investigate black hole solutions with Einstein horizon. In four dimensions, the first explicit inhomogeneous compact Einstein metric was constructed by Page [@Page] and a higher-dimensional version of the method of Page was given in [@Hashimoto]. Bohm constructed an infinite family of inhomogeneous metrics with positive scalar curvature on products of spheres [@Bohm]. After that, examples in higher-dimensional spacetimes have been worked in Refs. [@Gibbons1; @LuPa; @Gauntlett]. The properties of such Einstein manifolds are investigated in five and higher dimensions in Refs. [@Gibbons2] and [@Gibbons3; @Gibbons4], respectively.
In this paper we are interested in black hole solutions of third-order Lovelock gravity whose horizons are Einstein manifolds of nonconstant curvature. In Einstein gravity, no new solution can be obtained with nonconstant-curvature boundary. This is due to the fact that the Einstein equation deals with the Ricci tensor and therefore the Weyl tensor does not appear in the field equation. On the other hand, the Riemann tensor has direct contribution in the field equation of Lovelock gravity, and therefore the Weyl tensor appears in the field equation of Lovelock gravity. In Ref. [Dotti]{} Dotti and Gleiser obtained a condition on an invariant built out of the Weyl tensor in Gauss-Bonnet gravity when the horizon is an Einstein manifold. This constraint appears in the metric and consequently changes the properties of the spacetime. The properties of such static and dynamical solutions in Einstein-Gauss-Bonnet (EGB) gravity have been investigated in [@Maeda]. Also the magnetic black hole that has space with such specific condition was obtained in [@Maeda2]. While the base manifold of black hole solutions in EGB gravity with a generic value of the coupling constant must be necessarily Einstein, the boundary admits a wider class of geometries in the special case when the coupling constant is such that the theory admits a unique maximally symmetric solution [@Dot2]. The Birkhoff’s theorem in six-dimensional EGB gravity for the case of nonconstant-curvature horizons with various features has been investigated in [@Bog]. In Ref. [@Oliv1], it is shown that the horizons of black holes of Lovelock gravity in the Chern-Simons case [@LBI] in odd dimensions are not restricted. Some specific examples of black holes of Lovelock-Born-Infeld gravity [@LBI] with non-Einstein horizons in even dimensions were found in [@Can]. In Ref. [@Oliv2], it is shown that the base manifolds of these black hole solutions possess more than one curvature scale provided avoiding tensor restrictions on the base manifold and allowing at most a reduced set of scalar constraints on it. While all the Lovelock coefficients in Lovelock-Chern-Simons and Lovelock-Born-Infeld gravity are given in term of the cosmological constant, here we do not impose any condition on the coupling constants of Lovelock gravity and generalize the idea of Ref. [@Dotti] to the case of third-order Lovelock gravity with arbitrary Lovelock coefficients. That is, we like to obtain the black hole solutions of third-order Lovelock gravity with arbitrary coupling constants and nonconstant-curvature horizons. We predict that the appearing higher-curvature terms in third-order Lovelock gravity, even more sharply, may cause novel changes in the properties of the spacetime. This is the motivation for obtaining new black hole solutions in third-order Lovelock gravity with nonconstant-curvature horizons and investigating their thermodynamic properties.
The paper is organized as follows. In the following section we begin with a brief review of the field equation in third-order Lovelock gravity and obtain the equations getting use of the expressions in warped geometry for our spacetime ansatz. In Sec. \[Bla\] we obtain the black hole solutions and discuss their properties. In Sec. \[The\], we calculate the thermodynamic quantities of the solutions and investigate the first law of thermodynamics. Section \[Stab\] is devoted to the analysis of local and global stabilities by considering the variation of temperature versus entropy and the free energy for the special case of $\kappa=0$. We finish our paper with some concluding remarks.
Field Equations
===============
The most fundamental assumption in standard general relativity is the requirement that the field equations should be generally covariant and contain at most a second-order derivative of the metric. Based on this principle, the most general classical theory of gravitation in $n$ dimensions is Lovelock gravity [@Lovelock]. The Lovelock equation up to third-order terms in vacuum may be written as $$\mathcal{G}_{\mu \nu } \equiv -\Lambda g_{\mu \nu }+G_{\mu \nu
}^{(1)}+\sum_{p=2}^{3}\alpha _{i}\left( H_{\mu \nu }^{(p)}-\frac{1}{2}g_{\mu
\nu }\mathcal{L}^{(p)}\right) =0, \label{Geq}$$ where $\Lambda $ is the cosmological constant, $\alpha _{p}$’s are Lovelock coefficients,$G_{\mu \nu }^{(1)}$ is just the Einstein tensor, $\mathcal{L}%
^{(2)}=R_{\mu \nu \gamma \delta }R^{\mu \nu \gamma \delta }-4R_{\mu \nu
}R^{\mu \nu }+R^{2} $ is the Gauss-Bonnet Lagrangian, $$\begin{aligned}
\mathcal{L}^{(3)} &=&2R^{\mu \nu \sigma \kappa }R_{\sigma \kappa \rho \tau
}R_{\phantom{\rho \tau }{\mu \nu }}^{\rho \tau }+8R_{\phantom{\mu
\nu}{\sigma \rho}}^{\mu \nu }R_{\phantom {\sigma \kappa} {\nu \tau}}^{\sigma
\kappa }R_{\phantom{\rho \tau}{ \mu \kappa}}^{\rho \tau }+24R^{\mu \nu
\sigma \kappa }R_{\sigma \kappa \nu \rho }R_{\phantom{\rho}{\mu}}^{\rho }
\notag \\
&&+3RR^{\mu \nu \sigma \kappa }R_{\sigma \kappa \mu \nu }+24R^{\mu \nu
\sigma \kappa }R_{\sigma \mu }R_{\kappa \nu }+16R^{\mu \nu }R_{\nu \sigma
}R_{\phantom{\sigma}{\mu}}^{\sigma }-12RR^{\mu \nu }R_{\mu \nu }+R^{3}
\label{Lag3}\end{aligned}$$ is the third-order Lovelock Lagrangian, and $H_{\mu \nu }^{(2)}$ and $H_{\mu
\nu }^{(3)}$ are $$H_{\mu \nu }^{(2)}=2(R_{\mu \sigma \kappa \tau }R_{\nu }^{\phantom{\nu}%
\sigma \kappa \tau }-2R_{\mu \rho \nu \sigma }R^{\rho \sigma }-2R_{\mu
\sigma }R_{\phantom{\sigma}\nu }^{\sigma }+RR_{\mu \nu }), \label{Love2}$$ $$\begin{aligned}
H_{\mu \nu }^{(3)} &=&-3(4R^{\tau \rho \sigma \kappa }R_{\sigma \kappa
\lambda \rho }R_{\phantom{\lambda }{\nu \tau \mu}}^{\lambda }-8R_{%
\phantom{\tau \rho}{\lambda \sigma}}^{\tau \rho }R_{\phantom{\sigma
\kappa}{\tau \mu}}^{\sigma \kappa }R_{\phantom{\lambda }{\nu \rho \kappa}%
}^{\lambda }+2R_{\nu }^{\phantom{\nu}{\tau \sigma \kappa}}R_{\sigma \kappa
\lambda \rho }R_{\phantom{\lambda \rho}{\tau \mu}}^{\lambda \rho } \notag \\
&&-R^{\tau \rho \sigma \kappa }R_{\sigma \kappa \tau \rho }R_{\nu \mu }+8R_{%
\phantom{\tau}{\nu \sigma \rho}}^{\tau }R_{\phantom{\sigma \kappa}{\tau \mu}%
}^{\sigma \kappa }R_{\phantom{\rho}\kappa }^{\rho }+8R_{\phantom
{\sigma}{\nu \tau \kappa}}^{\sigma }R_{\phantom {\tau \rho}{\sigma \mu}%
}^{\tau \rho }R_{\phantom{\kappa}{\rho}}^{\kappa } \notag \\
&&+4R_{\nu }^{\phantom{\nu}{\tau \sigma \kappa}}R_{\sigma \kappa \mu \rho
}R_{\phantom{\rho}{\tau}}^{\rho }-4R_{\nu }^{\phantom{\nu}{\tau \sigma
\kappa }}R_{\sigma \kappa \tau \rho }R_{\phantom{\rho}{\mu}}^{\rho
}+4R^{\tau \rho \sigma \kappa }R_{\sigma \kappa \tau \mu }R_{\nu \rho
}+2RR_{\nu }^{\phantom{\nu}{\kappa \tau \rho}}R_{\tau \rho \kappa \mu }
\notag \\
&&+8R_{\phantom{\tau}{\nu \mu \rho }}^{\tau }R_{\phantom{\rho}{\sigma}%
}^{\rho }R_{\phantom{\sigma}{\tau}}^{\sigma }-8R_{\phantom{\sigma}{\nu \tau
\rho }}^{\sigma }R_{\phantom{\tau}{\sigma}}^{\tau }R_{\mu }^{\rho }-8R_{%
\phantom{\tau }{\sigma \mu}}^{\tau \rho }R_{\phantom{\sigma}{\tau }}^{\sigma
}R_{\nu \rho } \notag \\
&&-4RR_{\phantom{\tau}{\nu \mu \rho }}^{\tau }R_{\phantom{\rho}\tau }^{\rho
}+4R^{\tau \rho }R_{\rho \tau }R_{\nu \mu }-8R_{\phantom{\tau}{\nu}}^{\tau
}R_{\tau \rho }R_{\phantom{\rho}{\mu}}^{\rho }+4RR_{\nu \rho }R_{%
\phantom{\rho}{\mu }}^{\rho }-R^{2}R_{\mu \nu }), \label{Love3}\end{aligned}$$ respectively.
We take the $n$-dimensional manifold $\mathcal{M}^{n}$ to be a warped product of a two-dimensional Riemannian submanifold $\mathcal{M}^{2}$ with the following line element
$$ds^{2}=-f(r)dt^{2}+g(r)dr^{2}. \label{metric1}$$
and an $(n-2)$-dimensional submanifold $\mathcal{K}^{(n-2)}$ with the metric
$$ds^{2}=r^{2}\gamma _{ij}(z)dz^{i}dz^{j}. \label{metric2}$$
We assume the submanifold $\mathcal{K}^{(n-2)}$ with the unit metric $\gamma
_{ij}$ to be an Einstein manifold with nonconstant curvature and volume $%
V_{n-2}$, where $i,j=2...n-1$. We use tilde for the tensor components of the submanifold $\mathcal{K}^{(n-2)}$ through the paper. The Ricci tensor, Ricci scalar and Einstein tensor of the Einstein manifold $\mathcal{K}^{(n-2)}$ are
$$\begin{aligned}
\tilde{R}{_{ij}} &=&\kappa (n-3)\gamma _{ij},\text{ \ \ \ }\tilde{R}=\kappa
(n-2)(n-3), \\
\widetilde{G}{_{ij}} &{=}&\kappa (n-3)\left( 1-\frac{n-2}{2}\right) \gamma
_{ij},\end{aligned}$$
respectively. It is worth mentioning that Einstein metrics are vacuum solutions of Einstein’s theory of gravity only in three and four dimensions. The Riemann tensor of the Einstein manifolds should satisfy $$\tilde{R}{_{ij}}^{kl}=\tilde{C}{_{ij}}^{kl}+\kappa ({\delta _{i}}^{k}{\delta
_{j}}^{l}-{\delta _{i}}^{l}{\delta _{j}}^{k}) \label{Riem Ten}$$with $\kappa $ being the sectional curvature and $\tilde{C}{_{ij}}^{kl}$ is the Weyl tensor of $\mathcal{K}^{(n-2)}$. Using the expressions in warped geometry, the sectional components of the field equation (\[Geq\]) are calculated to be $$\mathcal{G}_{i}^{j}=\frac{2\hat{\alpha}_{2}{\tilde{C}_{ki}}^{nl}{\tilde{C}%
_{nl}}^{kj}}{r^{4}}-\frac{3\hat{\alpha}_{3}(4\tilde{C}^{nmkl}\tilde{C}_{klpm}%
{\tilde{C}^{pj}}_{ni}-8{\tilde{C}^{nm}}_{pk}{\tilde{C}^{kl}}_{ni}{\tilde{C}%
^{pj}}_{ml}+2\tilde{C}^{jnkl}\tilde{C}_{klpm}{\tilde{C}^{pm}}_{ni})}{2r^{6}},%
\text{ \ \ \ }(i\neq j) \label{Gij}$$$$\begin{aligned}
\mathcal{G}_{i}^{i} &=&\frac{(n-2)}{4g^{4}f^{2}r^{5}}\{2fgr[r^{4}g^{2}+2\hat{%
\alpha}_{2}r^{2}(kg-1)g+\hat{\alpha}_{3}(kg-1)^{2}]f^{\prime \prime
}-gr[r^{4}g^{2}+2\hat{\alpha}_{2}r^{2}(kg-1)g \notag \\
&&+\hat{\alpha}_{3}(kg-1)^{2}]f^{\prime }{}^{2}-f[((r^{4}+2k\hat{\alpha}%
_{2}r^{2}+\hat{\alpha}_{3})g^{2}+(-6\hat{\alpha}_{2}r^{2}-6k\hat{\alpha}%
_{3})g+5\hat{\alpha}^{3})rg^{\prime } \notag \\
&&-g(2(n-3)r^{4}g^{2}+4(n-5)\hat{\alpha}_{2}r^{2}(kg-1)g+2(n-7)\hat{\alpha}%
_{3}(kg-1)^{2})]f^{\prime }-2(n-3) \notag \\
&&[g^{\prime }(r^{4}g^{2}+2g\hat{\alpha}_{2}\frac{n-5}{n-3}r^{2}(kg-1)+\frac{%
n-7}{n-3}\hat{\alpha}_{3}(kg-1)^{2})+(n-4)g(kg-1) \notag \\
&&(r^{3}g^{2}+\frac{n-5}{10}\hat{\alpha}_{2}r(kg-1)g+\frac{n-8}{n(n-4)}\hat{%
\alpha}_{3}r^{(8-n)}(kg-1)^{2})]f^{2}\} \notag \\
&&-\left\{ \frac{(n-1)(n-2)\hat{\alpha}_{0}}{2}+\frac{\hat{\alpha}_{2}{%
\tilde{C}_{km}}^{ln}{\tilde{C}_{ln}}^{km}}{2r^{4}}-\frac{\hat{\alpha}_{3}(%
\tilde{C}^{nqkl}\tilde{C}_{klpm}{\tilde{C}^{pm}}_{nq}+4{\tilde{C}^{nm}}_{pk}{%
\tilde{C}^{kl}}_{nr}{\tilde{C}^{pr}}_{ml})}{r^{6}}\right\} \notag \\
&&+\frac{2\hat{\alpha}_{2}\sum_{kln}{\tilde{C}_{ki}}^{ln}{\tilde{C}_{ln}}%
^{ki}}{r^{4}}+\frac{8\hat{\alpha}_{3}\sum_{klmnp}(\tilde{C}^{inkl}\tilde{C}%
_{klpm}{\tilde{C}^{pm}}_{in}+4{\tilde{C}^{nm}}_{pk}{\tilde{C}^{kl}}_{ni}{%
\tilde{C}^{pi}}_{ml})}{r^{6}};\text{ no sum on }i, \notag\end{aligned}$$where $\hat{\alpha}_{p}$ are defined as $\hat{\alpha}_{0}\equiv -2\Lambda
/(n-1)(n-2)$, $\hat{\alpha}_{2}\equiv (n-3)(n-4)\alpha _{2}$ and $\hat{\alpha%
}_{3}\equiv (n-3)!\alpha _{3}/(n-7)!$ for simplicity. In vacuum, $\mathcal{G}%
_{i}^{j}=0$ and $\mathcal{G}_{i}^{i}-\mathcal{G}_{j}^{j}=0$ and therefore one obtains the following constraints on the Weyl tensor: $$0=\frac{2\hat{\alpha}_{2}{\tilde{C}_{ki}}^{nl}{\tilde{C}_{nl}}^{kj}}{r^{4}}-%
\frac{3\hat{\alpha}_{3}(2\tilde{C}^{nmkl}\tilde{C}_{klpm}{\tilde{C}^{pj}}%
_{ni}-4{\tilde{C}^{nm}}_{pk}{\tilde{C}^{kl}}_{ni}{\tilde{C}^{pj}}_{ml}+%
\tilde{C}^{jnkl}\tilde{C}_{klpm}{\tilde{C}^{pm}}_{ni})}{r^{6}},\text{ \ \ \ }%
(i\neq j) \label{Gij0}$$$${\tilde{C}_{ki}}^{nl}{\tilde{C}_{nl}}^{kj}=\frac{1}{n}{\delta _{i}}^{j}{%
\tilde{C}_{km}}^{pq}{\tilde{C}_{pq}}^{km}\equiv \eta _{2}{\delta _{i}}^{j},
\label{theta}$$$$\begin{aligned}
&&2(4{\tilde{C}^{nm}}_{pk}{\tilde{C}^{kl}}_{ni}{\tilde{C}^{pj}}_{ml}+{\tilde{%
C}^{pm}}_{in}\tilde{C}^{jnkl}\tilde{C}_{klpm}) \notag \\
&=&\frac{2}{n}{\delta _{i}}^{j}\left( 4{\tilde{C}^{qm}}_{pk}{\tilde{C}^{kl}}%
_{qr}{\tilde{C}^{pr}}_{ml}+{\tilde{C}^{pm}}_{qr}\tilde{C}^{qrkl}\tilde{C}%
_{klpm}\right) \notag \\
&\equiv &\eta _{3}{\delta _{i}}^{j}. \label{eta}\end{aligned}$$Here, we pause to add some comments about the expected patterns of conditions in $k$th-order Lovelock gravity. Comparing conditions ([theta]{}) and (\[eta\]) with the second- and third-order Lovelock Lagrangians, respectively and using the expression of Lovelock Lagrangian [@Lovelock], one may expect that the conditions on the Weyl tensor of base manifold are:$$\delta _{j_{1}j_{2}...j_{2p-1}j_{2p}}^{i_{1}i_{2}...i_{2p-1}i_{2p}}{\tilde{C}%
}_{i_{1}i_{2}}^{j_{1}j_{2}}...{\tilde{C}}_{i_{2p-1}i_{2p}}^{j_{2p-1j_{2p}}}%
\propto \eta _{p}, \text{\ \ \ } p=2...k.$$Of course, one should note that $\tilde{C}_{jk}^{ik}=\tilde{C}%
_{ij}^{ij}=0$. For instance, the only term in the Gauss-Bonnet Lagrangian which is nonzero is $\tilde{C}_{ki}^{nl}\tilde{C}_{nl}^{kj}$and the nonvanishing part of the third-order Lovelock is $8\tilde{C}_{pk}^{qm}\tilde{C}%
_{qr}^{kl}\tilde{C}_{ml}^{pr}+2\tilde{C}_{qr}^{pm}\tilde{C}^{qrkl}\tilde{C}%
_{klpm}$.
Getting use of these definitions, the $tt$ and $rr$ components of field equation (\[Geq\]) in vacuum reduce to $$\begin{aligned}
0 &=&{\mathcal{G}_{t}}^{t}=\frac{(n-2)}{2r^{6}g^{4}}\{[r^{4}g^{2}+3\hat{%
\alpha}_{3}\hat{\eta}_{2}g^{2}+2\hat{\alpha}_{2}r^{2}(kg-1)g+3\hat{\alpha}%
_{3}(kg-1)^{2}]rg^{^{\prime }}+(kg-1)[(n-3)r^{4}g^{2} \notag \\
&&+3(n-7)\hat{\alpha}_{3}\hat{\eta}_{2}g^{2}+(n-5)\hat{\alpha}%
_{2}r^{2}(kg-1)g+(n-7)\hat{\alpha}_{3}(kg-1)^{2}]g \notag \\
&&+\left( (n-1)\hat{\alpha}_{0}+\frac{(n-5)\hat{\alpha}_{2}\hat{\eta}_{2}}{%
r^{4}}+\frac{(n-7)\hat{\alpha}_{3}\hat{\eta}_{3}}{r^{6}}\right) r^{6}g^{4}\},
\label{Gtt}\end{aligned}$$$$\begin{aligned}
0 &=&{\mathcal{G}_{r}}^{r}=\frac{(n-2)}{2r^{6}fg^{3}}\{[r^{4}g^{2}+3\hat{%
\alpha}_{3}\hat{\eta}_{2}g^{2}+2\hat{\alpha}_{2}r^{2}(kg-1)g+3\hat{\alpha}%
_{3}(kg-1)^{2}]rf^{\prime }-(kg-1)[(n-3)r^{4}g^{2} \notag \\
&&+3(n-7)\hat{\alpha}_{3}\hat{\eta}_{2}g^{2}+(n-5)\hat{\alpha}%
_{2}r^{2}(kg-1)g+(n-7)\hat{\alpha}_{3}(kg-1)^{2}]f \notag \\
&&+\left( (n-1)\hat{\alpha}_{0}+\frac{\hat{\alpha}_{2}(n-5)\hat{\eta}_{2}}{%
r^{4}}+\frac{(n-7)\hat{\alpha}_{3}\hat{\eta}}{r^{6}}\right) r^{6}g^{4}\},
\label{Grr}\end{aligned}$$where we have used the definition $\hat{\eta}_{2}=(n-6)!\eta _{2}/(n-2)!$ and $\hat{\eta}_{3}=(n-8)!\eta _{3}/(n-2)!$ for simplicity. It is notable to mention that for these kinds of Einstein metrics $\hat{\eta}_{2}$ is always positive, but $\hat{\eta}_{3}$ can be positive or negative relating to the metric of the spacetime. As an example, the manifolds that are cross-products of $p$ $(p\geq 3)$ of two-hyperbola ($\mathcal{H}^{2}$) are Einstein manifolds with negative $\hat{\eta}_{3}$. The vacuum equation $%
\mathcal{G}_{t}^{t}-\mathcal{G}_{r}^{r}=0$ implies that $d(fg)/dr=0,$ and therefore one can take $g(r)=1/f(r)$ by rescaling the time coordinate $t$. Introducing
$$\psi (r)=\frac{\kappa -f(r)}{r^{2}}, \label{Psi}$$
we find that the remaining equations admit a solution if $\hat{\eta}_{2}$ and $\hat{\eta}_3$ defined in Eqs. (\[theta\]) and (\[eta\]) are constant and $\psi (r)$ satisfies $$\left\{ r^{n-1}\left[ \hat{\alpha}_{3}\psi ^{3}+\hat{\alpha}_{2}\psi
^{2}+\left( 1+\frac{3\hat{\alpha}_{3}\hat{\eta}_{2}}{r^{4}}\right) \psi +%
\hat{\alpha}_{0}+\frac{\hat{\alpha}_{2}\hat{\eta}_{2}}{r^{4}}+\frac{\hat{%
\alpha}_{3}\hat{\eta}_3}{r^{6}}\right] \right\}^{\prime}=0.$$Integrating the above equation, one obtains
$$\left( 1+\frac{3\hat{\alpha}_{3}\hat{\eta}_{2}}{r^{4}}\right) \psi +\hat{%
\alpha}_{2}\psi ^{2}+\hat{\alpha}_{3}\psi ^{3}+\hat{\alpha}_{0}+\frac{\hat{%
\alpha}_{2}\hat{\eta}_{2}}{r^{4}}+\frac{\hat{\alpha}_{3}\hat{\eta}_3}{r^{6}}-%
\frac{m}{r^{n-1}}=0, \label{Eq3}$$
where $m$ is the integration constant known as the mass parameter. One may note that Eq. (\[Eq3\]) reduces to the algebraic equation of Lovelock gravity for constant-curvature horizon when $\hat{\eta}_{2}=\hat{\eta}_3=0$.
The mass density, the mass per unit volume $V_{n-2}$, associated to the spacetime may be written as $$M=\frac{(n-2)r_{h}^{n-1}}{16\pi }\left\{ \hat{\alpha}_{0}+\frac{\kappa }{%
r_{h}^{2}}+\frac{\hat{\alpha}_{2}}{r_{h}^{4}}(\kappa ^{2}+\hat{\eta}_{2})+%
\frac{\hat{\alpha}_{3}}{r_{h}^{6}}(\kappa ^{3}+3\kappa \hat{\eta}_{2}+\hat{%
\eta}_{3})\right\} . \label{massP}$$
Black Hole Solutions \[Bla\]
============================
One may note that in order to have the effects of nonconstancy of the curvature of the horizon in third-order Lovelock gravity, $n$ should be larger than $7$. This can be seen in the definition of $\hat{\eta}_{3}$, which is zero for $n\leq 7$. A general solution of this equation can be written as $$\begin{aligned}
f(r) &=&\kappa +\frac{\hat{\alpha}_{2}r^{2}}{3\hat{\alpha}_{3}}\left\{
1+\left( j(r)\pm \sqrt{h+j^{2}(r)}\right) ^{1/3}-h^{1/3}\left( j(r)\pm \sqrt{%
h+j^{2}(r)}\right) ^{-1/3}\right\} , \notag \\
j(r) &=&1-\frac{9\hat{\alpha}_{3}}{2\hat{\alpha}_{2}^{2}}+\frac{27\hat{\alpha%
}_{3}^{2}}{2\hat{\alpha}_{2}^{3}}\left( \hat{\alpha}_{0}-\frac{m}{r^{n-1}}%
+\frac{\hat{\alpha}_{3}\hat{\eta}_{3}}{r^{6}}\right) ,\text{ \ } \notag \\
\text{\ \ \ \ }h &=&\left( -1+\frac{3\hat{\alpha}_{3}}{\hat{\alpha}_{2}^{2}}+%
\frac{9\hat{\alpha}_{3}^{2}\hat{\eta}_{2}}{\hat{\alpha}_{2}^{2}r^{4}}\right)
^{3}, \label{fstat}\end{aligned}$$These are the most general solutions of the third-order Lovelock equation in vacuum with the conditions (\[theta\]) and (\[eta\]) on their boundaries $\mathcal{K}^{(n-2)}$.
First, we investigate the asymptotic behavior of this solution. The asymptotic behavior of the solutions is the same as those with constant-curvature horizons. This is due to the fact that Eq. (\[Eq3\]) at very large $r$ reduces to $$\hat{\alpha}_{3}\psi _{\infty }^{3}+\hat{\alpha}_{2}\psi _{\infty }^{2}+\psi
_{\infty }+\hat{\alpha}_{0}=0, \label{Asym}$$which is exactly the same as third-order Lovelock or quasitopological cubic gravity [@Myers]. One may note that in the absence of the cosmological constant ($\hat{\alpha}_{0}=0$), the solution is asymptotically flat provided $%
\kappa =1$. This can be noted by considering Eq. (\[Asym\]) which has a zero root for $\hat{\alpha}_{0}=0$. For $\hat{\alpha}_{0}=1$, the solution is asymptotically AdS if Eq. (\[Asym\]) has positive real roots. For more details on the asymptotic behavior see [@Myers]. As in the case of black hole solutions wdith constant-curvature horizon, the Kretschmann scalar $%
R_{\mu \nu \rho \sigma }R^{\mu \nu \rho \sigma }$ diverges at $r=0$. Since the dominant term as $r$ goes to zero is $m/r^{n-1}$ for $n>7$, as in the case of third-order Lovelock gravity with constant-curvature horizon, there is an essential singularity located at $r=0$ which is spacelike. Note that the radius of horizon is given by the largest real root of $$\hat{\alpha}_{0}r_{h}^{n-1}+\kappa r_{h}^{n-3}+\hat{\alpha}_{2}(\kappa ^{2}+%
\hat{\eta}_{2})r_{h}^{n-5}+\hat{\alpha}_{3}(\kappa ^{zz3}+3\kappa \hat{\eta}%
_{2}+\hat{\eta}_{3})r_{h}^{n-7}-m=0, \label{horizon}$$where $r_{h}$ is the radius of horizon.
Here, we pause to give a few comments on the differences of the solutions of third-order Lovelock gravity with constant and nonconstant-curvature horizons. While the solutions of third order Lovelock gravity with constant curvature horizon and $m=0$ is the AdS metric with no horizon, the solutions of Lovelock gravity with nonconstant-curvature horizon and $m=0$ are only asymptotically AdS and may have horizon. This is due to the fact that $\hat{%
\eta}_{3}$ can be negative and therefore Eq. (\[horizon\]) can have a real positive root. Moreover, in third-order Lovelock gravity with constant- curvature horizon $h$ can be zero for $\hat{\alpha}_{3}=\hat{\alpha}%
_{2}^{2}/3$ and therefore the solution may be written in the simpler form:$$\begin{aligned}
f(r) &=&\kappa +\frac{r^{2}}{\hat{\alpha}_{2}}\left\{
1-[2j(r)]^{1/3}\right\} , \notag \\
j(r) &=&-\frac{1}{2}+\frac{3\hat{\alpha}_{2}}{2}\left( \hat{\alpha}_{0}-%
\frac{m}{r^{n-1}}\right) , \label{Spe}\end{aligned}$$while for the case of nonconstant curvature, $h$ cannot be zero and therefore we cannot have this special kind of solution.
Thermodynamics of The black hole solutions \[The\]
==================================================
Using the relation between the temperature and surface gravity, the Hawking temperature of the black hole is obtained to be $$T=\frac{f^{\prime }(r_{h})}{4\pi }=\frac{(n-1)r_{h}^{6}\hat{\alpha}%
_{0}+(n-3)\kappa r_{h}^{4}+(n-5)\hat{\alpha}_{2}(\hat{\eta}_{2}+\kappa
^{2})r_{h}^{2}+(n-7)\hat{\alpha}_{3}(\hat{\eta}_{3}+3\kappa \hat{\eta}%
_{2}+\kappa ^{3})}{4\pi r_{h}[r_{h}^{4}+2\kappa \hat{\alpha}_{2}r_{h}^{2}+3%
\hat{\alpha}_{3}(\hat{\eta}_{2}+\kappa ^{2})]}. \label{Temp}$$Due to the fact that $\hat{%
\eta}_{3}$ can be negative, it is apparent from Eq. (\[Temp\]) that a degenerate Killing horizon can exist for $\kappa \geq 0$ and therefore one may have an extreme black hole. This feature does not happen for the solutions of third-order Lovelock gravity with constant-curvature horizons [Dehghani]{} or second-order Lovelock gravity with constant or nonconstant- curvature horizons [@Maeda].
In higher curvature gravity the area law of entropy, which states that the black hole entropy equals one-quarter of the horizon area [@Beckennstein], is not satisfied [@Lu]. One approach to calculate the entropy is through the use of the Wald prescription which is applicable for any black hole solution whose event horizon is a Killing one [@Wald]. The Wald entropy may be written as $$S=-2\pi \oint d^{n-2}x\sqrt{\gamma }\sum_{p=1}^{3}Y_{p},\text{ \ \ \ \ \ }%
Y_{p}=Y_{p}^{\mu \nu \rho \sigma }\hat{\varepsilon}_{\mu \nu }\hat{%
\varepsilon}_{\rho \sigma },\text{\ \ \ \ \ \ }Y_{p}^{\mu \nu \rho \sigma }=%
\frac{\partial \mathcal{L}^{(p)}}{\partial R_{\mu \nu \rho \sigma }},
\label{entropy}$$where $\hat{\varepsilon}_{\mu \nu }$ is the binormal to the horizon and $%
\mathcal{L}^{(p)}$ is the $p$th-order Lovelock Lagrangian. Following the given description, $Y_{1}$ and $Y_{2}$ are [@Myers] $$Y_{1}=-\frac{1}{8\pi } \label{Ein-entropy}$$$$Y_{2}=-\frac{\hat{\alpha}_{2}}{4\pi }[R-2(R_{t}^{t}+R_{r}^{r})+2R_{tr}^{tr}]
\label{Gauss-entropy}$$Also we calculate $Y_{3}$ to be $$\begin{aligned}
Y_{3} &=&-\frac{3\hat{\alpha}_{3}}{4\pi }\{-12(R{^{tm}}_{tn}R{^{rn}}_{rm}-R{%
^{tm}}_{rn}R{{^{r}}_{mt}}^{n})+12R^{trmn}R_{trmn}-24[R{^{tr}}_{tm}R{_{r}}%
^{m}-R{^{tr}}_{rm}{R_{t}}^{m} \notag \\
&&+\frac{1}{4}(R_{mnpr}R^{mnpr}+R_{mnpt}R^{mnpt})]+3(2R{R^{tr}}_{tr}+\frac{1%
}{2}R_{mnpq}R^{mnpq}) \notag \\
&&+12({R^{t}}_{t}{R^{r}}_{r}-{R^{t}}_{r}{R^{r}}_{t}+{R^{r}}_{mrn}R^{mn}+{%
R^{t}}_{mtn}R^{mn})+12(R^{rm}R_{rm}+R^{tm}R_{tm}) \notag \\
&&-6[R_{mn}R^{mn}+R({R^{r}}_{r}+{R^{t}}_{t})]+\frac{3}{2}R^{2}\}.
\label{Lov-entropy}\end{aligned}$$Using Eq. (\[Riem Ten\]), one can calculate the entropy density for nonconstant-curvature manifold in third-order Lovelock gravity to be $$S=-2\pi \{Y_{1}+Y_{2}+Y_{3}\}=\frac{r_{h}^{n-2}}{4}\left\{ 1+\frac{2\kappa
\hat{\alpha}_{2}(n-2)}{r_{h}^{2}(n-4)}+\frac{3\hat{\alpha}_{3}(n-2)(\hat{\eta%
}_{2}+\kappa ^{2})}{r_{h}^{4}(n-6)}\right\} . \label{Entro}$$We see that $\hat{\eta}_{2}$ appears in the entropy and therefore the the nonconstancy of the horizon affects the entropy of the black hole. This does not happen for the Gauss-Bonnet solution. One should notice that the entropy calculated in Eq. (\[Entro\]) could also be obtained using the relation $$S=\frac{1}{4}\sum\limits_{q=1}^{p}p\hat{\alpha}_{p}\int d^{n-2}x\sqrt{\gamma
}\mathcal{\tilde{L}}^{(p-1)}$$introduced in [@Jacobson], where $\gamma $ is the determinant of induced metric and $\mathcal{\tilde{L}}^{(p)}$ is the $p$th-order Lovelock Lagrangian of the metric $\gamma _{ij}$. Getting use of Eqs. (\[massP\]), (\[Temp\]) and (\[Entro\]), we obtain $\partial {M}=T\partial {S}$ and therefore the first law of black hole thermodynamics is satisfied.
Stability of Black holes with $\protect\kappa =0$ \[Stab\]
==========================================================
It is known that the black holes of Lovelock gravity with zero curvature horizon are stable [@Dehghani]. Here, we investigate the stability of black holes of Lovelock gravity with nonconstant-curvature horizons and give some special features of $\kappa =0$ black hole solutions with non-constant curvature horizon, which are drastically different from $\kappa
=0$ solutions with constant-curvature horizon. In the case of $\kappa =0$, the entropy density of the black hole reduces to $$S_{0}=\frac{r_{h}^{n-2}}{4}\left\{ 1+\frac{3\hat{\alpha}_{3}(n-2)\hat{\eta}%
_{2}}{r_{h}^{4}(n-6)}\right\} . \label{Sn0}$$Since $\hat{\eta}_{2}\neq 0$, the black holes with nonconstant-curvature horizon do not obey the area law of entropy, while the entropy of $\kappa =0$ black holes with constant-curvature horizon obey the area law. The temperature of such a black hole is $$T_{0}=\frac{(n-1)r_{h}^{6}\hat{\alpha}_{0}+(n-5)\hat{\alpha}_{2}\hat{\eta}%
_{2}r_{h}^{2}+(n-7)\hat{\alpha}_{3}\hat{\eta}_{3}}{4\pi r_{h}(r_{h}^{4}+3%
\hat{\alpha}_{3}\hat{\eta}_{2})}, \label{Tn0}$$where $\hat{\alpha}_{0}=1$ and $0$ for asymptotically AdS and flat solutions, respectively. It is worth noting that for $\hat{\eta}_3 =\hat{\eta%
}_{3ext}$: $$\hat{\eta}_{3ext}=-\frac{(n-1)r_{h}^{6}\hat{\alpha}_{0}+(n-5)\hat{\alpha}_{2}%
\hat{\eta}_{2}r_{h}^{2}}{(n-7)\hat{\alpha}_{3}}$$the temperature can be zero, and therefore in contrast to the case of $%
\kappa =0$ of Lovelock black holes, extreme black holes may exist. This can be seen in Fig. \[extreme\].
The local stability of a thermodynamic system may be performed by analyzing the curve of $T$ versus $S$. Figure \[Lstab\] depicts $\log T$ versus $%
\log S$ and shows that small black holes are unstable for positive $\hat{\eta}_3>%
\hat{\eta}_{3ext}$, while the very large black holes with nonconstant- curvature horizon are the same as black holes with constant curvature. To analyze the global stability, we should check the free energy of the black hole which is defined by $F\equiv M-TS$, whereby negative value ensures global stability [@Hawking]. Substituting the expressions for mass, temperature and entropy from Eqs. (\[massP\]), (\[Tn0\]) and ([Sn0]{}), one can perform the analysis of global stability. We plot the free energy versus the radius of black holes for $\hat{\eta}_3>\hat{\eta}_{3ext}$ in Fig. \[Gstab\] which shows that small black holes are unstable both locally and globally, while there are medium black holes which may be locally stable, but they are globally unstable. So, in contrast to the case of $\kappa =0$ black holes with constant-curvature horizon which are stable [@Dehghani], the black hole solutions here may have unstable phases both locally and globally.
Concluding Remarks
==================
In this paper, we assumed that the $n$-dimensional spacetime is a cross product of the two-dimensional Lorentzian spacetime and an $(n-2)$-dimensional nonconstant space. We found that the nontrivial Weyl tensor of such exotic horizons is exposed to the bulk dynamics through the higher- order Lovelock terms, severely constraining the allowed horizon geometries and adding a novel chargelike parameter to the black hole potential. Indeed, we found that the third-order Lovelock gravity can have a new class of black hole solutions with nonconstant-curvature horizons provided one imposes two conditions on the Weyl tensor. The first condition is the one which has been introduced by Dotti and Gleiser [@Dotti] in Gauss-Bonnet gravity, while the second one is an additional condition involving the Weyl tensor of the horizon manifold with the advantage of higher curvature terms appearing in third-order Lovelock equations. This leads to a new class of static asymptotically flat and (A)dS black hole solutions. It is worth comparing our result with the already existing results in the literature. First, while in EGB gravity with an arbitrary Gauss-Bonnet coefficient only one condition is imposed on the Weyl tensor of Einstein horizon [@Dotti], here we faced with two conditions on the Weyl tensor of Einstein horizon. Second, as in the case of Gauss-Bonnet gravity with an arbitrary coupling constant we found that the horizon should be an Einstein manifold. However, for the cases when there is a unique maximally symmetric solution, the base manifold acquires more freedom [@Dot2; @Bog; @Oliv1; @Can; @Oliv2]. Third, while only one parameter appears in the solutions of Lovelock gravity in the Chern-Simons case [@Oliv1] or third- and higher-order Lovelock Born-Infeld gravity [@Can; @Oliv2], here we encountered with two new different chargelike parameters $\hat{\eta}_{2}$ and $\hat{\eta}_{3}$. Thus, one may expect that in the case of Lovelock gravity with arbitrary Lovelock coefficients the number of charge-like parameters $\hat{\eta}_{p}$’s will increase as the order of Lovelock gravity becomes larger.
The thermodynamics of these black hole solutions have been investigated by calculating the temperature and the entropy through the use of the Wald formula. We found that the thermodynamic quantities satisfy the first law of thermodynamics. In contrast to the black holes of Lovelock gravity with constant curvature horizons, we found that there may exist extreme black holes with nonconstant-curvature horizon. We also found that the effect of the Weyl tensor in the metric and the expressions for temperature and entropy, lead to new features of the solutions that do not appear for the solutions of second-order Lovelock gravity with nonconstant-curvature horizons or in higher-order Lovelock gravity with constant-curvature ones. In order to show the main differences of black holes with constant- and nonconstant-curvature horizons, we went through the details of thermodynamics and stability analysis of black holes with $\kappa =0$. First, we found that the entropy does not obey the area law as a consequence of $\hat{\eta}_{2}$ appearing in the entropy expression. Second, we found that there may exist extreme black hole. Moreover, black holes with nonconstant-curvature horizons have an unstable phase both locally and globally, while those with constant-curvature horizons are stable both locally and globally [@Dehghani]. But, it is worth mentioning that very large black holes with constant- and nonconstant-curvature horizons have the same features.
[99]{} D. Lovelock, J. Math. Phys. **12**, 498 (1971).
M. B. Greens, J. H. Schwarz, and E. Witten, *Superstring Theory* (Cambridge University Press, Cambridge, England, 1987); D. Lust and S. Theusen, *Lectures on String Theory* (Springer, Berlin, 1989); J. Polchinski, *String Theory* (Cambridge University Press, Cambridge, England, 1998).
D. G. Boulware and S. Deser, Phys. Rev. Lett. **55**, 2656 (1985); J. T. Wheeler, Nucl. Phys. **B268**, 737 (1986); R. G. Cai, Phys. Rev. D **65**, 084014 (2002).
Y. Brihaye and E. Radu, Phys. Lett. **B 661**, 167 (2008); S. Nojiri, S. D. Odintsov, and S. Ogushi, Phys. Rev. D **65**, 023521 (2002); Y. Brihaye and E. Radu, J. High Energy Phys. **09** (2008) 006.
M. H. Dehghani and R. B. Mann, Phys. Rev. D **72**, 124006 (2005); P. Mora, R. Olea, R. Troncoso, and J. Zanelli, J. High Energy Phys. **06** (2004) 036.
M. H. Dehghani and R. B. Mann, Phys. Rev. D **72**, 124006 (2005); M. H. Dehghani and M. Shamirzaie, Phys. Rev. D **72**, 124015 (2005); M. H. Dehghani and R. B. Mann, Phys. Rev. D **73**, 104003 (2006).
R. G. Cai, L.M. Cao, Y. P. Hu, and S. P. Kim, Phys. Rev D **78**, 124012 (2008); S.H. Mazharimousavi and M. Halilsoy, Phys. Lett. **B 665**, 125 (2008).
M. H. Dehghani and N. Farhangkhah, Phys. Rev. D **78**, 064015 (2008); M. H. Dehghani and N. Farhangkhah, Phys. Lett. **B 674**, 243 (2009).
M. Ozkan and Y. Pang, J. High Energy Phys. **03** (2013) 158; 07 (2013) 152(E).
M. Kulaxizi and A. Parnachev, Phys. Rev. D **82**, 066001 (2010); X.O. Camanho, J. D. Edelstein and M. F. Paulos, J. High Energy Phys. **05** (2011) 127.
X.O. Camanho, J. D. Edelstein, and M. F. Paulos, J. High Energy Phys. **05** (2011) 127; M. Brigante, H. Liu, R. C. Myers, S. Shenker, and S. Yaida, Phys. Rev. Lett. **100**, 191601 (2008) ; Phys. Rev. D **77**, 126006 (2008); J. de Boer, M. Kulaxizi, and A. Parnachev, J. High Energy Phys. **03** (2010) 087; X.O. Camanho and J. D. Edelstein, J. High Energy Phys. **06** (2010) 099; X.O. Camanho and J. D. Edelstein, J. High Energy Phys. **04** (2010) 007; A. Buchel, J. Escobedo, R. C. Myers, M. F. Paulos, A. Sinha, and M. Smolkin, J. High Energy Phys. **03** (2010) 111; J. de Boer, M. Kulaxizi and A. Parnachev, J. High Energy Phys. **06** (2010) 008.
R. G. Cai, Phys. Lett. **B 582**, 237 (2004); C. Garraffo and G. Giribet, Mod. Phys. Lett. **A 23**, 1810 (2008); C. Charmousis, Lect. Notes Phys. **769**, 299 (2009); R. G. Cai and S. P. Kim, J. High Energy Phys. **02** (2005) 050; T. Padmanabhan and A. Paranjape, Phys. Rev. D **75**, 064004 (2007).
H. Maeda, S. Willison, and S. Ray, Classical Quantum Gravity **28**, 165005 (2011).
D. Kastor, S. Ray, and J. Traschen, Classical Quantum Gravity **27** 235014 (2010).
D. N. Page, Phys. Lett. **B 79**, 235 (1978).
Y. Hashimoto, M. Sakaguchi and Y. Yasui, Commun. Math. Phys. **257**, 273 (2005).
C. Bohm, Invent. Math. **134**, 145 (1998).
G. Gibbons and S. A. Hartnoll, Phys. Rev. D **66**, 064024 (2002).
H. Lu, D. N. Page, and C. N. Pope, Phys. Lett. **B 593**, 218 (2004).
J. P. Gauntlett, D. Martelli, J. F. Sparks, and D. Waldram, Adv. Theor. Math. Phys. **8**, 987 (2006).
G. W. Gibbons, S. A. Hartnoll, and Y. Yasui, Classical Quantum Gravity **21**, 4697 (2004).
G. W. Gibbons, S. A. Hartnoll, and C. N. Pope, Phys. Rev. D **67**, 084024 (2003).
G. W. Gibbons and S. A. Hartnoll, Phys. Rev. D **66**, 064024 (2002).
G. Dotti and R. J. Gleiser, Phys. Lett. **B 627**, 174 (2005).
H. Maeda, Phys. Rev. D **81**, 124007 (2010).
H. Maeda, M. Hassaine, and C. Martinez, J. High Energy Phys. **08** (2010) 123.
G. Dotti, J. Oliva, and R. Troncoso, Phys. Rev. D **75**, 024002 (2007); G. Dotti, J. Oliva, and R. Troncoso, Phys. Rev. D **76**, 064038 (2007); G. Dotti, J. Oliva, and R. Troncoso, Int. J. Mod. Phys. A **24**, 1690 (2009); G. Dotti, J. Oliva, and R. Troncoso, Phys. Rev. D **82**, 024002 (2010).
C. Bogdanos, C. Charmousis, B. Gouteraux, and R. Zegers, J. High Energy Phys. **10** (2009) 037.
J. Oliva, J. Math. Phys. **54**, 042501 (2013).
R. Troncoso and J. Zanelli, Classical Quantum Gravity **17**, 4451 (2000); R. Aros, M. Contreras, R. Olea, R. Troncoso, and J. Zanelli, Phys. Rev. D **62**, 044002 (2000).
F. Canfora and A. Giacomini, Phys. Rev. D **82**, 024022 (2010).
A. Anabalon, F. Canfora, A. Giacomini, and J. Oliva, Phys. Rev. D **84**, 084015 (2011).
M. H. Dehghani and R. Pourhasan, Phys. Rev. D **79** 064015 (2009).
J. D. Beckenstein, Phys. Rev. D **7**, 2333 (1973); W. Hawking, Nature (London) **248**, 30 (1974); G. W. Gibbsons and S. W. Hawking, Phys. Rev D **15**, 2738 (1977).
M. Lu and M. B. Wise, Phys. Rev. D **47**, R3095 (1993); M. Visser, Phys. Rev. D **48**, 583 (1993).
R. M. Wald, Phys. Rev. D **48**, 3427 (1993); V. Iyer and R. M. Wald, Phys. Rev. D **50**, 846 (1994); T. Jacobson, G. Kang, and R. C. Myers, Phys. Rev. D **49**, 6587 (1994).
R. C. Myers and B. Robinson, J. High Energy Phys. **08** (2010) 067.
T. Jacobson and R. C. Myers, Phys. Rev. Lett. **70**, 3684 (1993); M. Visser,Phys. Rev. D **48**, 5697 (1993); (1994).
S. W. Hawking and D. N. Page, Commun. Math. Phys. **87**, 577 (1983).
[^1]: mhd@shirazu.ac.ir
|
---
bibliography:
- 'hybrid\_manuscript\_bibliography.bib'
---
[**Hybrid scheme for modeling local field potentials from point-neuron networks** ]{}\
**Espen Hagen**$^{1,2,\dagger,\ast}$, **David Dahmen**$^{1,\dagger}$, **Maria L. Stavrinou**$^{2}$, **Henrik Lind[é]{}n**$^{3,4}$, **Tom Tetzlaff**$^{1}$, **Sacha J. van Albada**$^{1}$, **Sonja Gr[ü]{}n**$^{1,5}$, **Markus Diesmann**$^{1,6,7}$, **Gaute T. Einevoll**$^{2,8,\ast}$\
Inst. of Neuroscience and Medicine (INM-6) and Inst. for Advanced Simulation (IAS-6) and JARA BRAIN Inst. I, Jülich Research Centre, Jülich, Germany\
[**[2]{}**]{} Dept. of Mathematical Sciences and Technology, Norwegian University of Life Sciences, [Å]{}s, Norway\
[**[3]{}**]{} Dept. of Neuroscience and Pharmacology, University of Copenhagen, Copenhagen, Denmark\
[**[4]{}**]{} Dept. of Computational Biology, School of Computer Science and Communication, Royal Institute of Technology, Stockholm, Sweden\
[**[5]{}**]{} Theoretical Systems Neurobiology, RWTH Aachen University, Aachen, Germany\
[**[6]{}**]{} Dept. of Psychiatry, Psychotherapy and Psychosomatics, Medical Faculty, RWTH Aachen University, Aachen, Germany\
[**[7]{}**]{} Dept. of Physics, Faculty 1, RWTH Aachen University, Aachen, Germany\
[**[8]{}**]{} Dept. of Physics, University of Oslo, Oslo, Norway\
[**$\dagger$**]{} Equal contribution\
$\ast$ E-mail: Corresponding authors: <e.hagen@fz-juelich.de>, <gaute.einevoll@nmbu.no>
Abstract {#section:abstract .unnumbered}
========
Due to rapid advances in multielectrode recording technology, the local field potential (LFP) has again become a popular measure of neuronal activity in both basic research and clinical applications. Proper understanding of the LFP requires detailed mathematical modeling incorporating the anatomical and electrophysiological features of neurons near the recording electrode, as well as synaptic inputs from the entire network. Here we propose a hybrid modeling scheme combining the efficiency of commonly used simplified point-neuron network models with the biophysical principles underlying LFP generation by real neurons. The scheme can be used with an arbitrary number of point-neuron network populations. The LFP predictions rely on populations of network-equivalent, anatomically reconstructed multicompartment neuron models with layer-specific synaptic connectivity. The present scheme allows for a full separation of the network dynamics simulation and LFP generation. For illustration, we apply the scheme to a full-scale cortical network model for a $\sim$1 mm$^2$ patch of primary visual cortex and predict laminar LFPs for different network states, assess the relative LFP contribution from different laminar populations, and investigate the role of synaptic input correlations and neuron density on the LFP. The generic nature of the hybrid scheme and its publicly available implementation in `hybridLFPy` form the basis for LFP predictions from other point-neuron network models, as well as extensions of the current application to larger circuitry and additional biological detail.
Author summary {#section:author .unnumbered}
==============
The recording of extracellular potentials inside the brain is among the most commonly used measures of neural activity. While the high-frequency part of the signal measures neural action potentials, the low-frequency part (local field potential, LFP) carries information from thousands of neurons and is difficult to interpret. The interpretation of the LFP has been hampered by the lack of a good ‘forward modeling scheme’, that is, a scheme providing a link between activity in candidate network models and the resulting LFP signal. While many models of neural network dynamics are based on simplified point neurons such as the leaky integrate-and-fire (LIF) neuron model, point neurons do not generate LFPs per se.
Here we describe a new hybrid modeling scheme overcoming this limitation, where the network spiking dynamics is modeled by means of point-neuron networks, while the LFP is subsequently computed from the resulting spike trains according to the biophysical principles underlying LFP generation in real neurons. For illustration, we apply the scheme to a full-scale cortical network model of a 1 mm$^2$ patch of primary visual cortex comprising 78,000 neurons and explore how the different cortical populations contribute to the LFP and how the signal depends on network state and other system properties.
Introduction {#sec:introduction}
============
The local field potential (LFP), the low-frequency component ($\lesssim$500 Hz) of the extracellular potential recorded in the brain, is commonly used as a measure of neuronal activity [@Buzsaki2012; @Einevoll2013]. The LFP originates from transmembrane currents [@Nicholson1975], and at the single-cell level the biophysical origin of such extracellular potentials is well understood (see, e.g., @Rall1968 [@Holt1999; @Buzsaki2012; @Einevoll2013]). However, the interpretation of the LFP remains difficult due to the large number of neurons contributing to the recorded signal. In neocortex, for example, the measured LFP is typically generated by thousands or even millions of neurons near the recording electrode [@kajikawa11_847; @Linden2011; @Leski2013]. Moreover, the LFP reflects synaptic input also generated by remote populations, e.g., inputs from other cortical or subcortical areas in addition to local network interactions [@Herreras2015]. A thorough theoretical description of the LFP therefore needs to account not only for the anatomical and electrophysiological features of neurons in the vicinity of the recording electrode, but also for the entire large-scale neuronal circuitry generating synaptic input to these cells.
Modeling large-scale neural-network dynamics with individual spiking neurons is challenging due to the memory required to represent the large number of synapses. With current technology and using the largest supercomputers available today, simulations of neural networks comprising up to $10^9$ neurons and $10^{13}$ synapses (roughly corresponding to the size of a cat brain) are feasible for simplified model neurons [@Diesmann13_8; @Kunkel2014]. Typically, these simplified models neglect the spatial aspects of neuronal morphologies and describe neurons as points in space (point-neuron models). Despite their simplicity, point-neuron-network models explain a variety of salient features of neural activity observed *in vivo*, such as spike-train irregularity [@Softky1993; @Vreeswijk1996; @Amit1997; @Shadlen1998], membrane-potential fluctuations [@Destexhe1999], asynchronous firing [@Ecker2010; @Renart2010; @Ostojic2014], correlations in neural activity [@Gentet2010; @Okun2008; @Helias2013], self-sustained activity [@Ohbayashi2003a; @Kriener2014] and realistic firing rates across laminar cortical populations [@Potjans2014]. Point-neuron networks are amenable to mathematical analysis (see, e.g., @Brunel2000 [@Deco2008; @Tetzlaff2012; @Helias2013; @Kamps2013; @Schuecker2015]) and can be efficiently evaluated numerically [@Plesser07_672; @Brette07_349; @Helias2012; @Kunkel2014]. The mechanisms governing networks of biophysically detailed multicompartment model neurons, in contrast, are less accessible to analysis and these models are more prone to overfitting. Existing multicompartment neuron network models accounting for realistic cell morphologies are restricted to sizes of $\sim10^4$–$10^5$ neurons [@Hines2008j; @Reimann2013; @Migliore2014; @Markram2015]. Large-scale models are, however, necessary to include contributions to the LFP from distant populations in situations where the spatial reach of the LFP is known to be large [@Linden2011; @Leski2013].
Although point-neuron networks capture many features of *in vivo* spiking activity, they fail to predict extracellular potentials that result from transmembrane currents distributed across the cell surface. According to Kirchhoff’s law of current conservation, the sum of all transmembrane currents, including all ionic and capacitive currents, must be zero for each neuron. In a point-neuron model, all transmembrane currents are collapsed in a single point in space. The net transmembrane current, and hence the extracellular potential, therefore vanishes. Only the spatial separation between current sinks and sources leads to a non-zero extracellular potential [@Pettersen2012; @Einevoll2013]. *A priori*, the prediction of extracellular potentials therefore requires spatially extended neuron models accounting for the spatial distribution of transmembrane currents, commonly handled using multicompartment neuron models [@DeSchutter2009]. In several previous studies [@Bazhenov2001; @Hill2005; @Ursino2006; @Mazzoni2008; @Mazzoni2010; @Mazzoni2011], the activity of point-neuron networks (e.g. population firing rates, synaptic currents, membrane potentials) has nevertheless been used as a proxy for the LFP when comparing with experiments. In a recent study comparing different candidate proxies, it was found that a suitably chosen sum of synaptic currents could provide a good LFP proxy, but only for the case when the LFP is generated from transmembrane currents of a single population of pyramidal neurons [@Mazzoni2015]. In cortex, however, several populations in general contribute to the LFP, and there are spatial cancellation effects when positive LFP contributions from one population overlap in space with negative LFP contributions from other populations. This effect cannot be accounted for by a simple LFP proxy.
In this article, we present a hybrid modeling scheme which combines the simplicity and efficiency of point-neuron network models and the biophysical principles underlying LFP generation captured by multicompartment neuron models with anatomically reconstructed morphologies. The scheme allows for arbitrary numbers of LFP-contributing populations, and directly incorporates spatial cancellation effects. Further, the spatially extended LFP-generating neurons assure that effects from intrinsic dendritic filtering of synaptic inputs are included in the predicted LFP [@Linden2010]. The scheme assumes that the spiking activity of the neural network (B) generating the synaptic input reflected in the LFP is well described by a point-neuron network model (A). The network spiking activity serves as synaptic input to a population of mutually unconnected multicompartment model neurons with realistic morphologies positioned in 3D space (C) and is thereby translated into a distribution of transmembrane currents and, hence, an LFP (D). Thus each multicompartment model neuron has its equivalent in the point-neuron network and receives input spikes from the same presynaptic neurons as this point-neuron equivalent.
![**Overview of the hybrid LFP modeling scheme for a cortical microcircuit model.** **A)** Sketch of the point-neuron network representing a 1 mm$^2$ patch of early sensory cortex (adapted from @Potjans2014). The network consists of 8 populations of leaky integrate-and-fire (LIF) neurons, representing excitatory (E) and inhibitory neurons (I) in cortical layers 2/3, 4, 5 and 6. External input is provided by a population of thalamo-cortical (TC) neurons and cortico-cortical afferents. The color coding of neuron populations is used consistently throughout this paper. Red arrows: excitatory connections. Blue arrows: inhibitory connections. See -\[tab:2\], \[tab:5\]-\[tab:6\] for details on the network model. **B)** Spontaneous ($t<900$ ms) and stimulus-evoked spiking activity (synchronous firing of TC neurons at time $t=900$ ms, denoted by thin vertical line) generated by the point-neuron network model shown in panel A, sampled from all neurons in each population. Each dot represents the spike time of a particular neuron. **C)** Populations of LFP-generating multicompartment model neurons with reconstructed, layer- and cell-type specific morphologies. Cells are distributed within a cylinder spanning the cortex. Layer boundaries are marked by horizontal black lines (at depths $z$ relative to cortex surface $z=0$). Only one representative neuron for each population is shown (see Fig. 4 for a detailed overview of cell types and morphologies). Sketch of a laminar recording electrode (gray) with 16 contacts separated by $100~\mu$m (black dots). **D)** Depth-resolved LFP traces predicted by the model (cf. -\[tab:4\]). Note that channel 1 is at the pial surface, so that channel 2 corresponds to a cortical depth of 100 $\mu$m and so forth. []{data-label="fig:1"}](figures/figure_01){width="\textwidth"}
In the proposed hybrid modeling scheme, the LFP stems from the presynaptic spiking activity, but does not affect the spike-generation dynamics. Thus, the modeling of the spike trains and the LFP generation are separated so that the effects of the spatial and electrophysiological properties of the postsynaptic (multicompartment) neurons on the LFP can be investigated independently of the spike-generation dynamics. Due to the linearity of Maxwell’s equations and volume conduction theory linking transmembrane currents to extracellular potentials [@Pettersen2012; @Einevoll2013], the compound LFP results from the linear superposition of all single-cell LFPs generated by the collection of neurons in the multicompartment model neuron population [@Einevoll2013a]. Note that this linear superposition principle applies even for nonlinear cell dynamics (e.g., nonlinear synaptic integration, action-potential generation, active conductances) as in [@Reimann2013]. As ephaptic interactions [@Anastassiou2011] are neglected, the LFP contribution from each multicompartment model neuron can be treated independently from the others. The computational hybrid LFP scheme proposed here exploits the methodological and conceptual advantages due to the independence of the contributions to the LFP from each multicompartment model neuron: The evaluation of the LFPs becomes ‘embarrassingly parallel’ (see [@Foster1995]) and simulations of the multicompartment model neuron dynamics can be easily distributed in parallel across many compute units (i.e., CPUs). Although tailored towards use on high-performance computing facilities, the hybrid simulation can in principle be run on a single laptop.
The hybrid scheme predicts spatially and temporally resolved neural activity at various scales: spikes, synaptic currents, membrane potentials, current-source densities (CSD, see e.g., @Nicholson1975 [@Pettersen2006; @Pettersen2008]), and LFPs. It therefore allows for investigation of relationships between different measures of neural activity. Thus, although point-neuron networks until now only have connected to *in vivo* experiments via measurement of spikes, single-neuron membrane potentials and currents, the present hybrid scheme allows for comparison of model predictions also with measured LFPs (and associated CSDs).
As an illustration, we apply the hybrid scheme to a multi-layered point-neuron network model of an early sensory cortical microcircuit [@Potjans2014]. We thereby demonstrate how to obtain LFP predictions from point-neuron network models using additional spatial connectivity information from anatomical data [@Binzegger2004; @Izhikevich2008]. The example illustrates how the hybrid scheme can be used to examine the relation between single-neuron and population signals, i.e., spikes and LFPs, the effect of network dynamics on the LFP, and the interpretation of the LFP in terms of underlying laminar neuron populations. We further use the example to demonstrate that synaptic-input correlations result in a non-trivial dependence of the LFP on the neuron density. Correct LFP predictions can therefore only be obtained by accounting for realistic neuron densities.
The network model of @Potjans2014 is chosen here since it has a minimum level of detail in the sense that individual neurons have simplified leaky integrate-and-fire (LIF) dynamics, but still represents a cortical column with full density of neurons and connections. The connectivity in such a full-scale circuit alone suffices to explain realistic firing rates across populations as well as propagation of activity through layers [@Potjans2014]. Applicability of the scheme is, however, not restricted to this model as it in principle can be used for all network models generating spikes.
In Methods (), we detail the components of the hybrid scheme and their application to the cortical microcircuit model: The point-neuron-network model (), the populations of multicompartment neurons (), the synaptic connectivity of the point-neuron network and the multicompartment model neuron populations (), and the biophysical forward-modeling scheme of extracellular potentials (). We further describe the analysis of the data generated by the simulations, as well as the `hybridLFPy` software implementation (). In Results (), we apply the hybrid scheme to the cortical microcircuit model of Potjans & Diesmann [@Potjans2014] and study the effects of network dynamics on the LFP, the contributions of individual cortical subpopulations to the LFP, the role of correlations and neuron density, and how well the LFP can be predicted from population firing rates (rather than from individual spikes). In Discussion (), we outline implications of our work, and in particular future applications and extensions of the hybrid LFP modeling scheme.
Methods: Hybrid LFP modeling scheme {#sec:methods}
===================================
Point-neuron network model {#sec:2.1}
--------------------------
The point-neuron network model is a key component of the hybrid scheme. The hybrid scheme enables LFP predictions from network models with an arbitrary number of populations and thus permits application to a large class of networks with arbitrarily complex single-neuron and synapse dynamics. The example network of ’spike-generators’ used here is, except for some minor adjustments (see below), the multi-layered model of a cortical microcircuit published by @Potjans2014. The model is implemented and included in `NEST` (<http://www.nest-simulator.org>, @eppler_2015_32969) and was recently made freely available (<http://www.opensourcebrain.org/projects/potjansdiesmann2014>).
The network model describes $1$ mm$^2$ of primary sensory cortex and consists of four layers with one excitatory (E) and one inhibitory (I) neuron population each, as illustrated in A. The network receives modulated thalamic input in addition to stationary external input. Whereas the neuron (leaky-integrate-and-fire) and synapse (static, exponential-current-based) model are intentionally left simple, the focus of this network implementation is on the complex connectivity (see ) which integrates multiple sources of anatomical and electrophysiological data [@Potjans2014 and references therein]. Apart from the layer identity, the model does not explicitly account for cell positions. For the full network description, see ,\[tab:2\] and \[tab:5\]. The microcircuit model reproduced experimentally observed distributions of firing rates across populations and propagation of activity across layers [@Potjans2014]. It thus forms a suitable starting point for LFP predictions in a cortical column.
The stationary thalamic Poisson input and cortico-cortical input to the microcircuit present in the original model of @Potjans2014, are here replaced by DC currents. DC input slightly increases the degree of synchrony (see, e.g., @Brunel2000 [Fig. 5]), but retains network dynamics and firing rate distributions across populations as in @Potjans2014. The Potjans & Diesmann network shows slightly synchronous behavior due to the E-I network of layer 4 being close to the synchronous irregular (SI) regime [@Brunel2000]. In order to reduce synchrony, we here increased the average synaptic weight from neurons in population L4I (inhibitory) to L4E (excitatory) neurons by 12.5%, resulting in attenuated oscillations in layer 4. Taking advantage of the fact that point-neuron networks are amenable for theoretical analysis, we derived modified weights based on predictions from dynamical mean-field theory applied to the microcircuit model [@Bos2015]. Moreover, we found that high-frequency network oscillations seen for Gaussian synaptic weight distributions are reduced when using lognormally distributed synaptic weights [@Sarid2007; @Iyer2013; @Teramae2014]. This made the dynamics more similar to experimental observations [@Song2005; @Buzsaki2014], and we thus also chose this for our network. Henceforth, we refer to our modified network as the ‘reference network’. Modulated activity of each thalamo-cortical (TC) neuron in the external thalamic population was modeled as synchronous spikes or as independent non-stationary Poisson processes with sinusoidally oscillating rate profiles (cf. ).
Populations of multicompartment model neurons {#sec:2.2}
---------------------------------------------
Cancellation effects from positive and negative contributions to extracellular potentials and effects of intrinsic dendritic filtering can only be captured with spatially extended multicompartment neuron models [@Einevoll2013]. In the hybrid scheme, extracellular potentials are estimated from the spiking activity in the point-neuron network (cf. ) through synaptic activation of populations of multicompartment model neurons (‘LFP generators’). In principle, these mutually unconnected model neurons mirror their network counterparts and receive inputs from exactly the same point neurons. In addition to the description of the point-neuron network model, different types of spatial information are thus needed to predict LFPs. For one, detailed dendritic morphologies are required for each individual network population (). Further, the positions of neurons and synaptic connections must also be specified, as well as the separation of network populations into morphologically distinct cell types ( and \[fig:3\]).
Availability of detailed cell-type specific connectivity of neural circuits, especially including information about synapse positions, is limited due to the substantial experimental effort involved. However, several ongoing large-scale neuroscience projects [@Kandel2013] address this issue and detailed connectomes are beginning to become publicly available [@Jiang2015; @Reimann2015; @Markram2015]. In the present example application we used the connection probabilities as given by @Izhikevich2008 derived from @Binzegger2004. Note that the point-neuron network connectivity was partially derived from the same data [@Potjans2014]. Quantitative data was provided for the number of connections in five cortical layers (layer 1 (L1), layers 2 and 3 grouped into a joint layer 2/3 (L2/3), and layers 4 (L4), 5 (L5) and 6(L6)) between 17 cortical cell types, cortico-cortical connections from other areas, and two thalamo-cortical relay cell types. We follow the nomenclature of @Izhikevich2008, where $y=\text{p23}$ denotes pyramidal cell types in layer 2/3, $y=\text{b23}$ and $y=\text{nb23}$ basket interneurons and non-basket interneurons within the same layer, $y=\text{ss4(L23)}$ spiny stellate cells in layer 4 with targets mainly within layer 2/3, $y=\text{p4}$ layer 4 pyramidal cells and so forth. Out of the 17 covered intracortical cell types only the $y=\text{nb1}$ cell type is not associated with any point-neuron network population in our scheme. To account for the lack of layer 1 in our model, we renormalized the connection probabilities for the remaining 16 cortical cell types including the two thalamo-cortical (TC) relay-cell types, such that the occurrences $F_y$ of all cell types $y$ summed to 100% as given in . Further, we assumed that the excitatory point-neuron network populations within one layer are composed of pyramidal cells and spiny stellate cells if both are present in the layer, and that inhibitory network populations encompass both types of interneurons. This results in the grouping of cell types $y$ into postsynaptic populations $Y$ illustrated in . The neuron count $N_y$ of each cell type is then trivially computed from the frequency of occurrence $F_y$ as given in and .
Inclusion of cell-type and layer-specific connections in the present hybrid scheme has some implications for how we proceed with setting up equivalent populations consisting of morphologically detailed model neurons. Different cell types belonging to a particular population may have different spatial distributions of synapses, or the populations may consist of different morphological classes of neurons [@Kisvarday1992; @Nowak2003; @Stepanyants2008]. An example is layer 4 in which spiny stellate cells lack apical dendrites, while pyramidal cells have apical dendrites extending into layer 1. To incorporate some of this morphological diversity we considered altogether 16 cell types for the 8 cortical network populations. These cell types are described below, with cell- and layer-specific connectivity derived in .
For each of the 16 cell types, we acquired representative morphological reconstructions of predominantly cat visual cortex neurons from several sources [@Contreras1997; @Kisvarday1992; @Mainen1996; @Stepanyants2008] (cf. Fig. \[fig:4\], Tab. \[tab:7\])). Morphology files were obtained either from [NeuroMorpho.org](http://neuromorpho.org) [@Ascoli2007] or through personal communication with the authors. Constrained by layer boundary depths [@Stepanyants2008 Fig. 3] and laminar connectivities (cf. ) we applied an intermediate preprocessing step to our pyramidal cell morphologies. Assuming that the soma compartments of each cell type were centered in their corresponding layer, and noting that the layer-specific connectivity (cf. ) implies connections to layer 1, we stretched the apical dendrites along the axis perpendicular to the cortical surface such that they reached the pial surface. The only exception was the p6(L4) morphology, which we extended to reach the center of layer 2/3 in accordance with @Stepanyants2008 and the observation that Tab. \[tab:8\] predicts zero connections within layer 1 and very few connections in layer 2/3 to the p6(L4) cell type. Due to lack of available morphologies of sufficient reconstruction quality, certain cell types were represented by the same neuron morphology. Interneuron types and spiny stellate cells in a given layer shared morphologies, the same interneuron morphology was reused in both layer 5 and 6, and finally the p5(L23) and p6(L56) cell morphology were similar except for the stretching of the apical dendrites.
Preserving the laminar cell density under 1 mm$^2$ surface area of the point-neuron network model, we created for each postsynaptic cell type $y$ model populations where somas were assigned random locations within cylindrical slabs with radius $r$=564 $\mu$m and thickness $h$=50 $\mu$m, each centered in their respective layer (illustrated in D). Regardless of the vertical offset of the soma of pyramidal cells, postsynaptic target dendrites were therefore present within the $\sim$80 $\mu$m thick [@Stepanyants2008] uppermost layer 1 except for cell type p6(L4). For simplicity, each cell type was represented by a single reconstructed morphology in the present application. The full specification of the populations is given in .
Each neuron is modeled using the multicompartmental, passive cable formalism [@Rall1964; @Rall2009; @DeSchutter2009], describing the changes in membrane voltage and the associated transmembrane currents throughout all parts of the neuron geometry (cf. and ). We used (non-plastic) exponential current-based synapses as in the point-neuron network model (cf. and ). -\[tab:6\] summarize parameters relevant for the synapse models and passive parameters of the multicompartment models.
![**Cell types and morphologies of the multicompartment-neuron populations.** The 8 cortical populations $Y$ of size $N_Y$ in the microcircuit network model are represented by 16 subpopulations of cell type $y$ with detailed morphologies $M_y$ [@Binzegger2004; @Izhikevich2008]. Neuron reconstructions are obtained from cat visual cortex and cat somatosensory cortex (source: NeuroMorpho.org [@Ascoli2007], @Contreras1997 [@Mainen1996; @Kisvarday1992; @Stepanyants2008], cf. ). Each morphology $M_y$ is here shown in relation to the layer boundaries (horizontal lines). Colors distinguish between network populations as in . The number of compartments ($n_\text{comp}$), frequencies of occurrence ($F_y$), relative occurrence ($F_{yY}$) and cell count ($N_y$) is given for each cell type $y \in Y$. []{data-label="fig:4"}](figures/figure_04){width="\textwidth"}
Spatial synaptic connectivity {#sec:2.3}
-----------------------------
A full description of the connectivity in networks of multicompartment model neurons requires a 3-dimensional (3D) representation, for example in the form of sparse $N_X \times N_Y \times n_\text{comp}$ matrices of synaptic weights and spike-transmission delays between presynaptic neurons $i\in[1,N_X]$ and compartments $n\in[1,n_\text{comp}]$ of postsynaptic cells $j\in[1,N_Y]$. Here, $N_X$ and $N_Y$ denote the number of pre- and postsynaptic neurons in population $X$ and $Y$, respectively, and $n_\text{comp}$ the number of compartments of the postsynaptic cell. In point-neuron networks, in contrast, connectivity is by definition only 2-dimensional (2D) as the cell morphology is collapsed into a single point and, consequently, the specificity of synapse locations on the postsynaptic morphology is ignored. In the proposed hybrid modeling scheme, the connectivity within the point-neuron network is consistent with the connectivity between point neurons and multicompartment model neurons. Ideally, each multicompartment model neuron has its equivalent in the point-neuron network and receives inputs from exactly the same presynaptic sources as its point-neuron counterpart. Synapses should be positioned on the dendritic tree according to anatomical data, and synaptic weights and time constants should be adapted such that the somatic membrane potential or somatic current match the point-neuron counterparts. Such mapping between point neurons and passive multicompartment neurons is feasible [@Koch1985; @Wybo2013; @Wybo2015].
In the current application of the hybrid scheme to the cortical microcircuit model, we make the simplest approximation to the mapping problem and fixed the current amplitudes $I_{ji,\text{max}}$ and synaptic time constants as in the network model, with compartment specificity of connections dependent on compartment surface area (see ). We further preserve only the statistics of connections (average number of inputs, distribution of spike-transmission delays) for each pair of pre- and postsynaptic neuron populations, exploiting that connections between network populations are drawn randomly with fixed probabilities. Finally, we simplify the positioning of synapses to a layer-specificity of connections. The activation times of each synapse are then given by the spike train of a randomly drawn point neuron in the network model, with random delays consistent with the delay distribution in the network ( and \[tab:4\]).
In , we first show how to derive a 2D point-neuron connectivity from a given 3D multicompartment-neuron connectivity and describe the case where the complexity of the point-neuron network is further reduced by pooling cell types. In we describe the opposite procedure, connecting an existing (published) point-neuron network with a predefined 2D connectivity to a population of multicompartment model neurons such that the resulting 3D connectivity is as consistent as possible with anatomical datasets accounting for the compartment specificity of connections (for example, the layer specificity of connections as in the anatomical data published by @Binzegger2004). The procedures outlined below, allow a reduction of complexity within the point-neuron network while accounting for the full diversity in cell types and synapse locations for multicompartment-neuron populations which is essential for predicting extracellular potentials ().
![ **Example LFP responses from single-synapse activations of layer 4 neurons.** **A**) Illustration of the non-trivial relationship between apical synaptic input (red circle) onto a reconstructed morphology (black) of a pyramidal cell in layer 4 and the corresponding extracellular potential. The exponential synaptic input current $I_{i, j}(t)$ (upper inset) results in deflections in the extracellular potential $\phi(\mathbf{r},t)$ here shown as time courses at two locations in proximity to the input site and the basal dendrites (green and blue circle, respectively; lower inset). The color-coded isolines show the magnitude of the scalar extracellular potential at $t=2$ ms (vertical black line in insets) in the vicinity of the cell. Negative signs are indicated by dashed lines. **B**) Same as in panel A, however with the synaptic input current relocated to a basal dendrite, resulting in an extracellular potential with a different spatiotemporal signature less dependent on the geometry of the apical dendritic tree. At the location denoted by the blue circle, the extracellular potential changes sign with time due to interactions between signal propagation in the passive model neuron and volume conduction. **C**) Same as panels B and C for a spiny stellate cell in layer 4 receiving an excitatory synaptic input on a basal dendrite. []{data-label="fig:2"}](figures/figure_02_a "fig:"){width="33.00000%"} ![ **Example LFP responses from single-synapse activations of layer 4 neurons.** **A**) Illustration of the non-trivial relationship between apical synaptic input (red circle) onto a reconstructed morphology (black) of a pyramidal cell in layer 4 and the corresponding extracellular potential. The exponential synaptic input current $I_{i, j}(t)$ (upper inset) results in deflections in the extracellular potential $\phi(\mathbf{r},t)$ here shown as time courses at two locations in proximity to the input site and the basal dendrites (green and blue circle, respectively; lower inset). The color-coded isolines show the magnitude of the scalar extracellular potential at $t=2$ ms (vertical black line in insets) in the vicinity of the cell. Negative signs are indicated by dashed lines. **B**) Same as in panel A, however with the synaptic input current relocated to a basal dendrite, resulting in an extracellular potential with a different spatiotemporal signature less dependent on the geometry of the apical dendritic tree. At the location denoted by the blue circle, the extracellular potential changes sign with time due to interactions between signal propagation in the passive model neuron and volume conduction. **C**) Same as panels B and C for a spiny stellate cell in layer 4 receiving an excitatory synaptic input on a basal dendrite. []{data-label="fig:2"}](figures/figure_02_b "fig:"){width="33.00000%"} ![ **Example LFP responses from single-synapse activations of layer 4 neurons.** **A**) Illustration of the non-trivial relationship between apical synaptic input (red circle) onto a reconstructed morphology (black) of a pyramidal cell in layer 4 and the corresponding extracellular potential. The exponential synaptic input current $I_{i, j}(t)$ (upper inset) results in deflections in the extracellular potential $\phi(\mathbf{r},t)$ here shown as time courses at two locations in proximity to the input site and the basal dendrites (green and blue circle, respectively; lower inset). The color-coded isolines show the magnitude of the scalar extracellular potential at $t=2$ ms (vertical black line in insets) in the vicinity of the cell. Negative signs are indicated by dashed lines. **B**) Same as in panel A, however with the synaptic input current relocated to a basal dendrite, resulting in an extracellular potential with a different spatiotemporal signature less dependent on the geometry of the apical dendritic tree. At the location denoted by the blue circle, the extracellular potential changes sign with time due to interactions between signal propagation in the passive model neuron and volume conduction. **C**) Same as panels B and C for a spiny stellate cell in layer 4 receiving an excitatory synaptic input on a basal dendrite. []{data-label="fig:2"}](figures/figure_02_c "fig:"){width="33.00000%"}
### Construction of point-neuron network connectivity {#sec:2.3.1}
For our example point-neuron network model, the cortical microcircuit model by @Potjans2014, the connectivity is to a large extent based on anatomical data from cat visual cortex [@Binzegger2004; @Izhikevich2008] (cf. ). From we obtain (i) the number $N_y$ of neurons belonging to cell type $y$, (ii) the average total number $k_{yL}$ of synapses on all compartments in layer $L$ (input layer) of a single postsynaptic neuron of type $y$, and (3) the fraction $p_{yxL}$ of the $k_{yL}$ synapses formed with presynaptic neurons of cell type $x$. The quantity $$k_{yxL} = p_{yxL} \, k_{yL}
\label{eq:kyxL}$$ defines the number of synapses between all presynaptic cells of type $x$ and a single postsynaptic cell of type $y$ in input layer $L$ (cf. network connectivity in @Izhikevich2008). The number of synapses between all neurons in $x$ and all neurons in $y$, irrespective of the input layer $L$, is given by $$K_{yx} = N_y \sum_L k_{yxL}~.
\label{eq:Kyx}$$ The number $K_{yx}$ of connections in combination with a chosen connectivity model (e.g., random graphs with binomially distributed [@Erdos59], fixed in-/out-degree [@Newman2003] or random graphs with defined higher-order statistics [@Song2005; @Zhao2011]) is sufficient for setting up the point-neuron network. Assuming independently drawn synapses (allowing multiple connections between neurons), the probability $C_{yx}$ of at least one connection between a neuron of type $x$ and a neuron of type $y$ can be obtained from $K_{yx}$ as [@Potjans2014] $$C_{yx}=1-\left(1-\frac{1}{N_xN_y}\right)^{K_{yx}}~.
\label{eq:Cyx}$$ In our case, the point-neuron microcircuit model consists of excitatory and inhibitory populations $X,Y$ (see -\[tab:2\]) pooling different pre- and postsynaptic cell types $x\in X$ and $y \in Y$ (cf. ). Given a single multicompartment model neuron of type $y$ we compute the number $k_{yXL}$ of incoming connections (in-degree) from cell types $x$ in each presynaptic population $X$ in a given layer $L$ by pooling all connections as illustrated in A as $$k_{yXL} = \sum_{x\in X} k_{yxL}~.
\label{eq:kyXL}$$ The total number of connections onto postsynaptic cells $y$ from cells in $X$ is then $$K_{yXL} = N_y k_{yXL}~.
\label{eq:KyXL}$$ The layer-specific connection probability $C_{yXL}$ (B) can be derived from analogously to for a presynaptic population size $N_X$ (here, $N_X=\sum_{x \in X} N_x$). In order to obtain the connectivity within the point-neuron network, i.e., between populations $X,Y$, we also need to pool over all synapses of input layers $L$ and cell types $y$ within the postsynaptic population $Y$ (dashed/dotted lines in B). Thus $$K_{YX} = \sum_{y \in Y} K_{yX} = \sum_{y \in Y} \sum_{L} K_{yXL}~,
\label{eq:KYX}$$ which yields the connectivity of the simplified network structure $C_{YX}$ (cf. , A).
![ **Constructing spatial synaptic connectivity for the cortical microcircuit model.** **A)** Illustration of pooling of presynaptic cell types. Presynaptic populations $X$ in the point-neuron model (left box; here $X=$ L4E) consist of multiple cell types $x$ (here $x\in\{$p4, ss4(L4), ss4(L23)$\}$). The layer-specific number of synapses $k_{yXL}$ (dash-dotted lines) formed between one cell of postsynaptic cell type $y$ (right part of panel A: morphology projected onto cortical layers 1–6; here $y=$ p5(L56)) and a presynaptic population $X$ is given by the sum of all individual cell-type resolved synapse counts $k_{yxL}$ (dotted or dash-dotted lines). **B)** Bi-directional cell- and layer-specific pooling and dispersing of synapses between pre- and postsynaptic cell types. Postsynaptic populations $Y$ (right box; here $Y=$ L5E) in the point-neuron model consist of multiple cell types $y$ (here $y\in\{$p5(L56), p5(L23)$\}$). A given presynaptic population $X$ (left box; here $X=$ L4E) containing cell types $x$ (here $x\in\{$p4, ss4(L4), ss4(L23)$\}$) forms cell-type and layer-specific connections within $Y$ (black connection tree). For the number of synapses $K_{yXL}$ between population $X$ and cells of type $y$ in layer $L$ (right-most branching of connection tree) the synapse count $K_{YX}$ between all cells in $X$ and $Y$ can be obtained by pooling all synapses onto cell types $y\in~Y$ and input layers $L$. Conversely, for a given total number of synapses $K_{YX}$ between all cells in $X$ and $Y$, the number of synapses $K_{yXL}$ onto a specific cell type $y$ and layer $L$ can, as described by , be obtained by calculating the cell-type and layer specificity of connections $\mathcal T_{yX}$ and $\mathcal L_{yXL}$ (see ) from anatomical data (). []{data-label="fig:3"}](figures/figure_03){width="\textwidth"}
![ **Connectivity of the cortical microcircuit model**. **A**) Connection probability $C_{YX}$ between presynaptic population $X$ and postsynaptic population $Y$ of the cortical microcircuit model by @Potjans2014 given in . Zero values are shown as gray here and in subsequent plots. **B**) Layer- and cell-type specific connectivity map $C_{yXL}$, where $X$, $y$ and $L$ denote presynaptic populations, postsynaptic cell types and the synapse location (layer), respectively. This map is computed from the connectivity of the point-neuron network (panel A) and cell-type (panel C) and layer specificity (panel D) of connections. **C**) Cell-type specificity $\mathcal{T}_{yX}$ of connections quantified as the fraction of synapses between pre- and postsynaptic populations $X$ and $Y$ formed with a specific postsynaptic cell type $y$. **D**) Layer specificity $\mathcal{L}_{yXL}$ of connections denoting the fraction of synapses between population $X$ and cell type $y$ formed in a particular layer $L$. Both $\mathcal{T}_{yX}$ and $\mathcal{L}_{yXL}$ in panels C and D respectively are calculated from anatomical data [@Binzegger2004; @Izhikevich2008], cf. . []{data-label="fig:5"}](figures/figure_05){width="\textwidth"}
### From pooled to specific network connectivity {#sec:2.3.2}
In case of an already existing point-neuron network model such as introduced in , the reverse task of creating a spatial connectivity $C_{yXL}$ from a given point-neuron network connectivity $C_{YX}$ is necessary. This inverse procedure compared to pooling over cell types and input layers entails introducing the cell-type specificity $$\mathcal T_{yX} = \frac{K_{yX}}{K_{YX}}~,
\label{eq:TyX}$$ which describes the fraction of synapses between populations $X$ and $Y$ that are formed with a specific postsynaptic cell type $y$ (C), and the layer specificity of connections $$\mathcal L_{yXL} = \frac{K_{yXL}}{K_{yX}},
\label{eq:LyXL}$$ denoting the fraction of synapses between population $X$ and all cells of cell type $y$ formed in a particular layer $L$ (D). The product $\mathcal T_{yX} \mathcal L_{yXL}$ defines the probability of a synapse between populations $X$ and $Y$ formed with a specific postsynaptic cell type $y$ in a particular layer $L$ (B). Thus, if $K_{YX}$ is given, the total number of connections in layer $L$ onto postsynaptic cells $y$ from cells in $X$ is $$K_{yXL} = K_{YX} \mathcal T_{yX} \mathcal L_{yXL}~.
\label{eq:KyXL2}$$ If $K_{YX}$ is constructed from the same data as $\mathcal T_{yX}$ and $\mathcal L_{yXL}$, Equations \[eq:TyX\]-\[eq:KyXL2\] are fully consistent. However, $K_{YX}$ can also be computed from any given point-neuron network connectivity $C_{YX}$. This is particularly relevant for the network connectivity $C_{YX}$ (A) of @Potjans2014 that includes additional data sets for which spatial information on synapse locations is not available. Here the number of synapses $k_{yXL}$ from population $X$ established in layer $L$ on each multicompartment model neuron of type $y$ is obtained from (9) as $k_{yXL}=K_{yXL}/N_y$.
Forward modeling of extracellular potentials {#sec:2.4}
--------------------------------------------
The LFP signal reflects transmembrane currents weighted according to the distance from the source to the measurement location [@Einevoll2013], and here we compute the LFP from the model neurons using a now well established forward-modeling scheme combining multicompartment neuron modeling and electrostatic (volume-conduction) theory [@Holt1999; @Gold2006; @Pettersen2008; @Linden2010; @Linden2011; @Reimann2013; @Linden2014; @Tomsett2014]. Each morphology was spatially discretized into compartments using the `d_lambda` rule [@Hines2001] with electrotonic length constants computed at $f=100$ Hz. In this forward modeling scheme, localized synaptic activation of a morphologically detailed neuron results in spatially distributed transmembrane currents across the neuronal membrane as calculated using standard cable theory, see e.g., @DeSchutter2009. The extracellular potentials, including the LFP, are in turn given as a weighted sum of transmembrane currents as described by volume conduction theory [@Holt1999; @Einevoll2013].
The cable-equation description is summarized in box D in . relates synaptic input currents $I_{jin}$ onto compartment $n$ in a neuron $j$ from presynaptic neurons $i$, the membrane voltages $V_{\text{m}jn}$, and transmembrane currents $I_{\text{m}jn}$ and is derived from the assumption (Kirchoff’s law) of current balance in the intracellular node of the equivalent electrical circuit of a cylindrical compartment $n$ with $m$ neighboring compartments. We use the standard convention that a positive membrane current is a positive current from the intracellular to the extracellular space across the membrane. $I_{\text{m}jn}$ is assumed to be homogeneously distributed across the outer surface of the cylindrical compartment, and the calculation assumes that the electrical potential on the outside boundary of the membrane is zero at all times. Hence, there are no mutual interactions (i.e., ephaptic coupling [@Anastassiou2011]) between the extracellular potential estimated using volume conduction theory, the transmembrane currents, and intracellular potentials.
The associated extracellular potential resulting from the transmembrane currents is calculated based on volume conduction theory [@Nunez2006; @Einevoll2013]. In the present application where the signal frequencies are well below 1,000 Hz, this calculation is simplified by applying the quasistatic approximation to Maxwell’s equations, i.e., terms with time derivatives of the electrical and magnetic fields are omitted, cf. @Hamalainen1993 [p. 426]. Further, we assume the extracellular medium to be linear, isotropic, homogeneous and ohmic [@Pettersen2012; @Einevoll2013] and represented by a scalar extracellular conductivity $\sigma_\text{e}$.
Given a time-varying point current source with magnitude $I(t)$ at position $\mathbf{r'}$, the scalar extracellular potential $\phi(\mathbf{r}, t)$ at position $\mathbf{r}$ and time $t$ is then given by [@Nunez2006; @Linden2014] $$\phi(\mathbf{r},t) = \frac{I(t)}{4\pi\sigma_\text{e}|\mathbf{r} - \mathbf{r'}|}~.
\label{eq:extracellular1}$$ Contributions to the extracellular potential from multiple current sources, i.e., transmembrane currents of all individual compartments $n$ from all cells $j$ in a population of $N$ cells sum linearly. In accordance with the assumed homogeneous current distribution along each cylindrical compartment, the *line-source* approximation is used for dendritic compartments [@Holt1999]. The line-source forward-modeling formula is obtained by integrating along the cylindrical axis of each compartment $n$, and summing the contributions from all $n_\text{comp}$ compartments [@Holt1999; @Pettersen2008; @Linden2014]: $$\phi(\mathbf{r}, t) = \sum^N_{j=1} \sum^{n_\text{comp}}_{n=1} \frac{I_{\text{m}jn}(t)}{4 \pi \sigma_\text{e}}\int \frac{1}{|\mathbf{r} - \mathbf{r}_{jn}|}d\mathbf{r}_{jn}~.
\label{eq:extracellular2}$$ Presently, we approximate the thick soma compartments as spherical current sources, and thus combine the point-source equation () with the line-source formula () for dendrite compartments, obtaining [@Linden2014]: $$\begin{aligned}
\phi(\mathbf{r}, t)
&= \sum^N_{j=1} \frac{1}{4 \pi \sigma_\text{e}}
\left(\frac{I_{\text{m}j,\text{soma}}(t)}{|\mathbf{r} - \mathbf{r}_{j, \text{soma}}|}
+ \sum^{n_\text{comp}}_{n=2} \int \frac{I_{\text{m}jn}(t)}{|\mathbf{r} - \mathbf{r}_{jn}|}d\mathbf{r}_{jn} \right) \nonumber \\
&= \sum^N_{j=1} \frac{1}{4 \pi \sigma_\text{e}}
\left( \frac{I_{\text{m}j,\text{soma}}(t)}{|\mathbf{r} - \mathbf{r}_{j,\text{soma}}|}
+ \sum_{n=2}^{n_\text{comp}} \frac{I_{\text{m}jn}(t)}{\Delta s_{jn}}
\ln \left| \frac{\sqrt{h_{jn}^2+r_{\perp jn}^2}-h_{jn}}{\sqrt{l_{jn}^2+r_{\perp jn}^2}-l_{jn}} \right| \right) ~.
\label{eq:extracellular3}\end{aligned}$$ Here, $\Delta s_{jn}$ denotes compartment length, $r_{\perp jn}$ the perpendicular distance from the electrode point contact to the axis of the line compartment, $h_{jn}$ the longitudinal distance measured from the start of the compartment, and $l_{jn} = \Delta s_{jn}+h_{jn}$ the longitudinal distance from the other end of the compartment. If the distance between electrode contacts and dendritic current sources becomes smaller than the radius of the dendritic segment, an unphysical singularity in our extracellular potential may occur. In these cases singularities are avoided by setting $|{\mathbf r}-{\mathbf r}_{j,\text{soma}}|$ or $r_{\perp jn}$ equal to the compartment radius. Electrode contacts of real recording devices have finite spatial extent and are not point contacts as assumed above. However, the recorded signal can be well approximated as the mean of the potential averaged across the uninsulated surface [@Robinson1968; @Nelson2008; @Nelson2010; @Ness2015], at least for current sources positioned further away than an electrode radius or so [@Ness2015]. Here we employed the *disc-electrode* approximation to the potential [@Camunas-Mesa2013; @Linden2014; @Ness2015]: $$\phi_{\text{disc}}(\mathbf{u},t) = \frac{1}{A_S} \iint_{S} \phi(\mathbf{u},t) \,d^2 r \approx \frac{1}{m} \sum_{h=1}^m \phi({\bf u}_h,t) ~.
\label{eq:extracellular4}$$ We further considered circular electrode contacts with a radius of $r_{\text{contact}}$=7.5 $\mu$m, and we averaged the point-contact potential in over $m=50$ random locations $\mathbf{u}_h$ across the contact surface $S$, $A_S$ being the surface area. The chosen locations were distributed with uniform probability on circular discs representing each contact surface, with surface vectors oriented perpendicular to the electrode axis [@Linden2014]. Calculations of extracellular potentials were facilitated by `LFPy` (<http://LFPy.github.io>) [@Linden2014], in which `NEURON` simulation software is used for calculations of transmembrane currents (i.e., solving ) [@Hines2001; @Hines2009; @Carnevale2006].
Data analysis and software {#sec:2.5}
--------------------------
### Model measurements {#sec:2.5.1}
The main simulation output of the hybrid scheme consists of spike trains of each neuron in the point-neuron network, ‘ground-truth’ current-source density (CSD) and local field potentials (LFP) of each neuron in the morphologically detailed postsynaptic model populations (see and \[tab:4\]). Here the term ’ground-truth’ refers to the fact that the CSD is computed from transmembrane currents rather than estimated from the LFP.
As the transmembrane current of each compartment () is known at each simulation time step, we follow the procedure of @Pettersen2008 to compute the ‘ground-truth’ CSD in addition to the LFP. From $N$ model neurons with $n_\text{comp}$ compartments having membrane currents $I_{\text{m}jn}(t)$ and lengths $\Delta s_{jn}$, we calculate the CSD signals $\rho(\mathbf{r},t)$ inside cylinder elements $V_\rho(\mathbf{r})$ around each electrode contact as: $$\rho(\mathbf{r},t) = \frac{1}{\pi r^2 h_\text{elec}}\sum_{j=1}^N \sum_{n=1}^{n_\text{comp}} I_{\text{m}jn}(t)\frac{\Delta s_{jn, \text{inside}}(\mathbf{r})}{\Delta s_{jn}}~.
\label{eq:csd}$$ $\Delta s_{jn, \text{inside}}(\mathbf{r})$ denotes the length of the line source contained within $V_\rho(\mathbf{r})$. In contrast to @Pettersen2008, we do not apply a spatial filter to the CSD. The volumes have radii equal to the population radius $r$ and heights equal to the electrode separation $h_\text{elec}$ (cf. ).
In the present example application, the extracellular potential is computed at locations corresponding to a laminar multi-electrode array with 16 recording electrodes with an inter-electrode distance of $h_\text{elec}=100~\mu$m, positioned at the cylindrical axis of the model column with the topmost contact at the pial surface (cf. D, see for details). Each electrode contact is set to have a radius of $7.5~\mu$m (cf. in ). In the network we also record membrane voltages and input currents from a subset of cells in each of the eight cortical populations (see ).
We ran our simulations for a total duration of $T=5{,}200$ ms using a temporal resolution of $dt=0.1$ ms (cf. and \[tab:11\]). However, LFP and CSD signals were resampled prior to file storage to a temporal resolution of $dt_{\psi}=$1 ms by (i) applying a 4th-order Chebyshev type I low-pass filter with critical frequency $f_\text{c}=400$ Hz and 0.05 dB ripple in the passband using a forward-backward linear filter operation, and (ii) then selecting every 10th time sample.
### Post-processing and data analysis {#sec:2.5.2}
As the contributions to the CSD and LFP of the different cells sum linearly (cf. ), we compute population-resolved signals as the sum over contributions from all cells in a population, and the full compound signals as the sum over all population signals (see ). In we derive a rescaled ’low-density predictor’ $\phi^{\gamma \xi}(\mathbf{r},t)$ of the LFP from random subsets of neurons in all populations. Thereby, we make a downscaled LFP-generating model setup with the same column volume, but with neuron density reduced to a factor $\gamma \in (0,1)$ of the original density, while preserving the in-degrees, i.e., the number of synaptic connections onto individual neurons. The LFP from the downscaled setup is multiplied by an overall scaling factor $\xi$ chosen to roughly preserve the LFP from the full-scale model. For analysis and plotting, the initial 200 ms of results after simulation onset was removed, and the signal mean was subtracted from LFP and CSD traces emulating DC filtering during experimental data acquisition. The contribution from each population to the overall LFP signal was assessed by computing and comparing the signal variances (see ).
Cross-correlations between single-cell LFPs $\phi_i(\mathbf{r},t)$ were quantified by analyzing the power spectrum of the compound extracellular signals. Given that the compound LFP/CSD is a linear superposition of single-cell LFP/CSD contributions, the power spectrum of the compound signal $P_{\phi}(\mathbf{r},f)$ can be obtained as the sum of all single-cell power spectra $P_{\phi_i}(\mathbf{r},f)$ and all pairwise cross-spectra $C_{\phi_i \phi_j}(\mathbf{r},f)$, or equivalently from the average single-cell power spectrum $\overline{P_{\phi}}(\mathbf{r},f)$, the average pairwise cross-spectrum $\overline{C_{\phi}}(\mathbf{r},f)$, and the total cell count $N$, as shown in . Note that $\overline{C_{\phi}}(\mathbf{r},f)$ and hence the average pairwise single-cell LFP coherence $\overline{\kappa_{\phi}}(\mathbf{r},f)$ () are real, while $C_{\phi_i\phi_j}(\mathbf{r},f)$ is complex. Note that this definition allows for negative values of $\overline{\kappa_{\phi}}(\mathbf{r},f)$. In the sum over all $i$ and $j$ in , the imaginary parts of $C_{\phi_i\phi_j}(\mathbf{r},f)$ and $C_{\phi_j\phi_i}(\mathbf{r},f)$ cancel because $C_{\phi_i\phi_j}(\mathbf{r},f)=C_{\phi_j\phi_i}(\mathbf{r},f)^*$ (${}^*$ denotes the complex conjugate). The power spectrum $P_{\phi^{\gamma \xi}}(\mathbf{r},f)$ of the compound signals of the ‘downscaled’ network (i.e., low-density LFP predictor, see above) is given by the reduced cell count $\gamma N$, the average single-cell power spectrum $\overline{P_{\phi^{\gamma \xi}}}(\mathbf{r},f)$, and the average pairwise single-cell cross-spectrum $\overline{C_{\phi^{\gamma \xi}}}(\mathbf{r},f)$ calculated from that subset of neurons (see ). These single-cell averages are the same as the respective single-cell averages of the full-scale model setup, apart from variability due to subsampling.
Throughout this paper, signal power spectra are estimated using Welch’s average periodogram method [@Welch1967] (with the `matplotlib.mlab.psd` implementation in `Python`, see ). Temporal cross-correlations are quantified as the zero time-lag correlation coefficient ( in ). Spike-triggered average LFP (staLFP) signals are computed as the cross-covariance between the time-resolved population spike rate $\nu_X(t)$ and the compound LFP $\phi(\mathbf{r}, t)$, divided by the total number of spikes (i.e., $\int_0^T \nu_X(t) dt$).
### The `hybridLFPy` `Python` package {#sec:2.5.3}
To facilitate usage of the hybrid scheme by other users, a novel `Python` software package, `hybridLFPy`, has been made publicly available under the General Public License version 3 (GPLv3, <http://www.gnu.org/licenses/gpl-3.0.html>) on GitHub (<http://github.com/INM-6/hybridLFPy>). Compatibility with a host of different machine architectures and operating systems ($\ast$nix, OSX, Windows) is ensured with the freely available, object-oriented programming language `Python` (<http://www.python.org>). `Python` adds tremendous flexibility in terms of interfacing a large number of packages and libraries for, e.g., performing numerical analysis and data visualization, such as `numpy` (<http://www.numpy.org>) and `matplotlib` (<http://www.matplotlib.org>), while several other neural simulation softwares also come with their own `Python` interfaces, such as `NEST` (<http://www.nest-initiative.org>) [@Eppler2008] and `NEURON` (<http://www.neuron.yale.edu>) [@Hines2009].
The source code release of `hybridLFPy` provides a set of classes implementing the hybrid scheme, as well as example network simulation codes implemented with `NEST` [@eppler_2015_32969] (at present a simplified two-population network [@Brunel2000] and the full cortical microcircuit model of @Potjans2014 adapted from public codes (see )). The class `hybridLFPy.CachedNetwork` uses an efficient `sqlite3` database implementation for reading in all point-neuron network spike events and interfacing network spike events with the main simulation in which the LFP and CSD are calculated. The class `hybridLFPy.Population` defines populations of multicompartment model neurons representing each cell type, assigns synapse locations across the laminae, selects spike trains for each synapse location from the appropriate presynaptic population, and calculates the LFP and CSD. The single-cell calculations are handled using `LFPy` (<http://LFPy.github.io>) [@Linden2014] which builds on `NEURON` [@Carnevale2006; @Hines2009]. As there are no mutual interactions between the multicompartment model neurons in the calculation of LFPs, these calculations remain embarrassingly parallel operations. Finally, the class `hybridLFPy.PostProcess` constructs the full compound signals in terms of LFPs and CSDs created by multiple instances of the `Population` class, and performs the main analysis steps as described in . Reproducible simulation and data analysis are assured by tracking code revisions using `git` and by fixing random number generation seeds and the number of parallel processes. Further documentation and information on installing and using `hybridLFPy` is provided online, see <http://github.com/INM-6/hybridLFPy>.
The present implementation of `hybridLFPy` and corresponding simulations were made possible by `Open MPI` (v.1.6.2), `HDF5` (v.1.8.13), `sqlite3` (v.3.6.20), `Python` (v.2.7.3) with modules `Cython` (v.0.23dev), `NeuroTools` (v.0.2.0dev), `SpikeSort` (v.0.13), `h5py` (v.2.5.0a0), `ipython` (v.0.13), `matplotlib` (v.1.5.x), `mpi4py` (v.1.3), `numpy` (v.1.10.0.dev-c63e1f4), `pysqlite` (v.2.6.3) and `scipy` (v.0.17.0.dev0-357a1a0). Point-neuron network simulations were performed using `NEST` (v.2.8.0 ff71a29), and simulations of multicompartment model neurons using `NEURON` (v.7.4 1186:541994f8f27f) through `LFPy` (dev. v.3761c4). All software was compiled using `GCC` (v.4.4.6). Simulations were performed in parallel (256 threads) on the Stallo high-performance computing facilities (NOTUR, the Norwegian Metacenter for Computational Science) consisting of 2.6 GHz Intel E5-2670 CPUs running the Rocks Cluster Distribution (Linux) operating system (v.6.0).
Results: LFP generated by a cortical microcircuit {#sec:results}
=================================================
To illustrate the application of the hybrid scheme for predictions of LFPs from point-neuron networks, we here present results for a modified version of the point-neuron network model of @Potjans2014 with $\sim 78{,}000$ neurons mimicking a 1 mm$^2$ patch of cat primary visual cortex (see ). The microcircuit model has realistic cell density and deliberately neglects many biological details on the single-cell level, focussing on the effect of the connectivity on the local network dynamics in such circuits. Despite this simplicity, the model displayed firing rates across populations in agreement with experimental observations @Potjans2014, as well as propagation of spiking activity across layers. Likewise, the microcircuit model in conjunction with simplified, passive multicompartment populations is used here to study the effect of the (spatial) connectivity on the laminar pattern of spontaneous and stimulus-evoked CSD and LFP signals. For this, CSD and LFP signals for a laminar multielectrode recording at different cortical depths are computed. The large network size of the model is further used to illustrate the effect of correlations and neuron density on CSD and LFP predictions.
Spontaneous vs. stimulus-evoked LFP {#sec:3.1}
-----------------------------------
We first consider the LFP generated by spontaneous network activity. The output of our hybrid scheme covers various scales and measurement modalities, from spikes of each neuron (A), population-averaged firing rates (B), excitatory, inhibitory, and total synaptic input currents (C), and membrane voltages (D), to the compound CSD and LFP stemming from all populations of different cell types (E-G). For spontaneous activity (cf. , ), i.e., no modulated thalamic input, we observe asynchronous irregular spiking in all populations (A) and firing rates similar to the original model [@Potjans2014] (B). In particular, layer 2/3 exhibits low firing rates, and generally inhibitory neurons fire with higher rates than excitatory neurons of the respective layers. The network is in a balanced regime [@Brunel2000], reflected by the substantial cancellation of population-averaged excitatory and inhibitory input currents (C). The population-averaged membrane potential fluctuates below the fixed firing threshold $\theta=-50$ mV down to $\sim-80$ mV (D).
The corresponding compound CSD and LFP signals with contributions from all cortical populations are shown in F and G, respectively. As expected for spontaneous cortical network activity, the LFP signal amplitudes are small, $\simeq$0.1 mV (intriguingly close to what has been seen in experiments, see, e.g., Fig. 1 in @Maier2010 for macaque visual cortex, Fig. 7 in @Hagen2015 for mouse visual cortex), and exhibit strong across-channel covariance in line with experimental observations, e.g., [@Einevoll2007; @Riehle2013; @Hagen2015]. In the model these correlations stem from dendritic cable properties and volume conduction effects [@Pettersen2008; @Linden2011; @Leski2013]. The correlations across channels are, as expected, generally less visible in the more localized CSD signal where volume conduction effects are absent [@Nicholson1975; @Pettersen2006; @Pettersen2008].
LFP and associated CSD studies have commonly been used to investigate stimulus-evoked responses in sensory cortices, see, e.g., @Mitzdorf1979 [@Mitzdorf1985; @Di1990; @Schroeder1998; @Swadlow2002; @Einevoll2007; @Sakata2009; @Szymanski2009; @Maier2010; @Jin2011; @Szymanski2011], as well as @Einevoll2013 and references therein. To model the situation with a sharp onset of a visual stimulus (or direct electrical stimulation of the thalamocortical pathway [@Mitzdorf1979]) we drive the network with a short thalamic pulse mimicking a volley of incoming spikes onto primary visual cortex from the visual thalamus (lateral geniculate nucleus, LGN). The activation targets populations in layers 4 and 6 (see A or A,B) and propagates in the network to populations in layers 2/3 and 5 (A,B). At the level of spiking activity, the results match the behavior of the original model [@Potjans2014 Fig. 10A,B] and even agree qualitatively with experimental findings in rodents from stimulus-evoked activation of auditory cortex [@Sakata2009] and somatosensory cortex [@Armstrong-James1992; @Einevoll2007; @Reyes-Puerta2015]. This points to a general biological plausibility of this generic network model, based largely on data from cat V1.
The corresponding CSD and LFP profiles across depth associated with this spiking activity are determined by the synapse locations and dendritic filtering by cell-type specific morphologies (see , D). For the CSD (C) a complex alternating spatiotemporal pattern of current sinks (negative CSD) and sources (positive CSD) is observed. Due to volume conduction this detailed spatial pattern is largely smeared out in the LFP profile (D), which displays a strong positivity across the middle layer around $t$=910 ms. We also note the different spatiotemporal profiles of the spiking activity (A,B) compared to CSD and LFP laminar profiles. Not only do the CSD and LFP signals typically fade out 5–10 ms later than the spiking, the spatial profiles are also very different. For example, the LFP signal is very weak between channels 11 and 12 (i.e., between 1,100 and 1,200 $\mu$m depth), even if the firing rate of layer 5 positioned between these channels is very high. We also note that the predicted LFP magnitudes are an order of magnitude larger compared to the LFP predicted for spontaneous activity, that is, $\sim$1 mV for stimulus-evoked versus $\sim$0.1 mV for spontaneous activity.
Although the present example has not been tuned to address specific experiments, we nevertheless observe that the model predictions display several features seen in experiments. For example, stimulus-evoked LFP amplitudes on the order of 1 mV are similar to the maximal amplitudes ($\sim$1–3 mV) observed in cat V1 following electric stimulation of thalamocortical axons (optical radiation) [@Mitzdorf1985], in visually evoked LFPs in monkey V1 [@Schroeder1998], and in rat somatosensory (barrel) cortex following whisker flicks [@Di1990; @Einevoll2007; @Reyes-Puerta2015]. Also some qualitative features of the spatiotemporal LFP and CSD patterns from the cat visual cortex experiments of @Mitzdorf1985 can be recognized. One example is the early CSD sink around layer 4 (i.e., at channel 6 close to the boundary between layers 2/3 and 4), another is the large positivity in LFP extending through most of the layers following the initial response to the thalamic input volley.
The observed time courses of the LFP and CSD in our simulations with $\delta$-pulse thalamic activations are not as directly comparable with experimental stimulus-evoked activity, where the input is temporally filtered by several cell populations before reaching thalamus on the way to cortex. However, we note that very swift stimulus-evoked responses lasting not much longer than the $~\sim$10 ms response volleys seen in our simulations also are observed in the somatosensory system [@Di1990; @Einevoll2007; @Reyes-Puerta2015].
![ **Overview of output signals obtained from application of the hybrid scheme to a cortical microcircuit (spontaneous activity).** *Point-neuron network:* **A**) Spiking activity. Each dot represents the spike time of a point neuron (color coding as in ). **B**) Population-averaged firing rates for each population. **C**) Population-averaged somatic input currents (red: excitatory, blue: inhibitory, black: total). **D**) Population-averaged somatic voltages. Averaged somatic input currents and voltages are obtained from $100$ neurons in each population. *Multicompartment model neurons:* **E**) Somas of excitatory (triangles) and inhibitory (stars) multicompartment cells and layer boundaries (gray/black ellipses). Illustration of a laminar electrode (gray) with 16 recording channels (black circles). **F**) Depth-resolved current-source density (CSD) obtained from summed transmembrane currents in cylindrical volumes centered at each contact. **G**) Depth-resolved local field potential (LFP) calculated at each electrode contact from transmembrane currents of all neurons in the column. Channel 1 is at pial surface, channel 2 at 100 $\mu$m depth, etc. []{data-label="fig:6"}](figures/figure_06_a "fig:"){width="\textwidth"} ![ **Overview of output signals obtained from application of the hybrid scheme to a cortical microcircuit (spontaneous activity).** *Point-neuron network:* **A**) Spiking activity. Each dot represents the spike time of a point neuron (color coding as in ). **B**) Population-averaged firing rates for each population. **C**) Population-averaged somatic input currents (red: excitatory, blue: inhibitory, black: total). **D**) Population-averaged somatic voltages. Averaged somatic input currents and voltages are obtained from $100$ neurons in each population. *Multicompartment model neurons:* **E**) Somas of excitatory (triangles) and inhibitory (stars) multicompartment cells and layer boundaries (gray/black ellipses). Illustration of a laminar electrode (gray) with 16 recording channels (black circles). **F**) Depth-resolved current-source density (CSD) obtained from summed transmembrane currents in cylindrical volumes centered at each contact. **G**) Depth-resolved local field potential (LFP) calculated at each electrode contact from transmembrane currents of all neurons in the column. Channel 1 is at pial surface, channel 2 at 100 $\mu$m depth, etc. []{data-label="fig:6"}](figures/figure_06_b "fig:"){width="75.00000%"}
![ **Network activity following transient activation of thalamocortical afferents**. **A**) Raster plot of spiking activity before and after $\delta$-shaped thalamic stimulus presented at $t=900$ ms (vertical black line in panels A, C and D). **B**) Population-averaged firing rate histogram for each population (color coding as in panel A). **C**) Depth-resolved compound current-source density (CSD) of all populations (shown both in color and by the black traces). **D**) Depth-resolved compound local field potential (LFP, shown both in color and by the black traces) at each electrode channel as generated by all populations. Channel 1 is at pial surface, channel 2 at 100 $\mu$m depth, etc. []{data-label="fig:7"}](figures/figure_07){width="\textwidth"}
Effect of network dynamics on LFP {#sec:3.2}
---------------------------------
The spiking activity in the microcircuit model is highly sensitive to modification of intrinsic model parameters and external input [@Bos2015]. In general LFPs reflect synaptic input both from local and distant neurons [@Herreras2015], and also depend on network state [see, e.g., @Kelly2010; @Gawne2010]. With the hybrid scheme, we first illustrate as an example the dependence of the spontaneous LFP, i.e., the LFP without thalamic input, on local network dynamics as determined by intrinsic network parameters. In particular we compare our reference network model with the original model proposed by @Potjans2014 where the strength of connections from L4I to L4E neurons is weaker and the synaptic weights are drawn from a Gaussian distribution. We next investigate the LFP when the network is stimulated by sinusoidally modulated thalamic input.
For spontaneous activity (A-E), the spiking (A) is asynchronous irregular in all populations. The firing-rate power spectra (B) vary from relatively flat to more band-pass-like with a maximum power around 80 Hz. The suppressed power at lower frequencies arises from active decorrelation due to inhibitory feedback [@Tetzlaff2012]. In combination with the low-pass filtering involved in the generation of LFPs from spiking activity [@Linden2010; @Leski2013], the firing-rate spectra translate into an LFP power spectrum that, depending on recording depth, has either low-pass or band-pass filter characteristics (D). The notably sharper attenuation of the LFP power spectra compared to the firing-rate power spectra above $\gtrsim$ 100 Hz is expectedly due to the intrinsic dendritic filtering effect [@Linden2010] (however, the frequencies $\gtrsim$ 400 Hz are sharply attenuated due to our anti-aliasing filter, cf. ). As the effect of this filtering depends on the position of the electrode compared to the neuronal morphology [@Linden2010], the result is a variable LFP power spectral density (PSD) profile across cortical depths, even in the absence of any structured external input (E).
There is experimental evidence that cortical microcircuits receive oscillating input at various frequencies from remote areas and subcortical structures [@Bastos2015; @Kerkoerle2014; @Ito2014]. To illustrate the effect of an oscillatory external input in our model, we model thalamic input as independent realizations of non-stationary Poisson processes with a sinusoidal rate profile (see , F-J). The spiking activity of all populations as well as the CSD and LFP across depth are sensitive to thalamic input, showing that the network response goes beyond the layers receiving thalamic input, i.e., layers 4 and 6. The stimulus-evoked spiking activity follows the $15$ Hz modulated rate of the thalamic input (F). The 15 Hz oscillation is reflected in the firing-rate spectrum as a peak around 15 Hz (G) and is robustly transferred to the LFP (H-J). The LFP oscillation strength varies with depth and is greatest in channels 1–2 and channels 8–14, while the oscillation is barely seen in channels 3–6. Populations L4E/I and L6E/I receive thalamic inputs around the depths of channels 8–10 and channels 14–15, and these channels are strongly affected by the stimulus. However, recurrent connections between populations, dendritic propagation of currents, and volume conduction produce strong LFP oscillations also in other channels.
The LFP amplitudes are not only influenced by the temporal structure of the external input, but also by synaptic weights in two ways: first via their influence on the spiking dynamics in the point-network simulation and second via the influence on the size of the synaptic currents setting up the transmembrane currents in the multicompartment models in the LFP-computing step. In order to illustrate the weight dependence, we compare the LFP under spontaneous activity in our reference network model (A-E) with the corresponding spontaneous LFP in the original model by @Potjans2014 (K-O). The lower inhibition from population L4I onto L4E and the narrow weight distribution compared to our model gives a higher degree of spike synchrony in all populations. Although our model exhibits asynchronous irregular activity (A), the original model is closer to a synchronous irregular regime [@Brunel2000] (K), resulting in high-frequency oscillations around 80 Hz and in the 300-400 Hz band (L), both associated with delay loops in the multi-layered network [@Brunel2000]. The 80 Hz oscillation also appears in the LFP and its corresponding power spectra (M-O), but the magnitude of the peak is not constant across depth. The lowest magnitudes are located in the vicinity of layers 2/3 and 5 (in channels 3–4 and channels 11–13).
![ **Effect of network dynamics on LFP.** Comparison of two different thalamic input scenarios and two different networks. [**Top:**]{} Reference network, spontaneous activity. [**Center:**]{} Reference network, oscillatory thalamic activation. [**Bottom:**]{} Original model by @Potjans2014, spontaneous activity. [**A,F,K**]{}) Population-resolved spiking activity. [**B,G,L**]{}) Population-averaged firing rate spectra. [**C,H,M**]{}) Depth-resolved LFP. [**D,I,N**]{}) LFP power spectra in layer 1 and at typical somatic depths of network populations. [**E,J,O**]{}) LFP power spectra across all channels. Channel 1 is at pial surface, channel 2 at 100 $\mu$m depth, etc. []{data-label="fig:8"}](figures/figure_08_a "fig:"){width="\textwidth"} ![ **Effect of network dynamics on LFP.** Comparison of two different thalamic input scenarios and two different networks. [**Top:**]{} Reference network, spontaneous activity. [**Center:**]{} Reference network, oscillatory thalamic activation. [**Bottom:**]{} Original model by @Potjans2014, spontaneous activity. [**A,F,K**]{}) Population-resolved spiking activity. [**B,G,L**]{}) Population-averaged firing rate spectra. [**C,H,M**]{}) Depth-resolved LFP. [**D,I,N**]{}) LFP power spectra in layer 1 and at typical somatic depths of network populations. [**E,J,O**]{}) LFP power spectra across all channels. Channel 1 is at pial surface, channel 2 at 100 $\mu$m depth, etc. []{data-label="fig:8"}](figures/figure_08_b "fig:"){width="\textwidth"} ![ **Effect of network dynamics on LFP.** Comparison of two different thalamic input scenarios and two different networks. [**Top:**]{} Reference network, spontaneous activity. [**Center:**]{} Reference network, oscillatory thalamic activation. [**Bottom:**]{} Original model by @Potjans2014, spontaneous activity. [**A,F,K**]{}) Population-resolved spiking activity. [**B,G,L**]{}) Population-averaged firing rate spectra. [**C,H,M**]{}) Depth-resolved LFP. [**D,I,N**]{}) LFP power spectra in layer 1 and at typical somatic depths of network populations. [**E,J,O**]{}) LFP power spectra across all channels. Channel 1 is at pial surface, channel 2 at 100 $\mu$m depth, etc. []{data-label="fig:8"}](figures/figure_08_c "fig:"){width="\textwidth"}
Contributions from individual populations to CSD and LFP {#sec:3.4}
--------------------------------------------------------
The direct interpretation of CSD and LFP signals in terms of the underlying activity of different populations or input pathways is inherently ambiguous and thus difficult: For example, a CSD sink observed in cortical layer 2/3 can alternatively stem from excitatory synaptic inputs to the basal dendrites of layer 2/3 cells, similar inputs into the apical dendrites of layer 5 cells, or even return currents from appropriately placed inhibitory inputs onto the same cells [@Linden2010; @Einevoll2013]. Several schemes for decomposition of CSD and LFP data into contributions from cortical populations have thus been proposed: principal component analysis (PCA) [@Di1990], laminar population analysis [@Einevoll2007], and independent component analysis (ICA) [@Leski2010; @Makarov2010; @Glabska2014; @Herreras2015]. In our modeling world we have the benefit of having the contributions from the various populations, connections or different synapse types to the CSD and LFP signals directly accessible.
Here, we focus on the LFP and CSD contributions from individual populations and different synapse types. To quantify the contributions from each postsynaptic population, i.e., the CSD and LFP stemming from the transmembrane currents of a population, we simply summed all single-cell CSD and LFP contributions from all neurons in the population. Results for spontaneous network activity are shown in . As different cell types are assigned appropriate morphologies and cortical depths based on available anatomical data, a trivial consequence is that the neurons in each population make their main contributions to the CSD and LFP at depths spanned by their dendrites (B,C for post-synaptic populations L23E and L6E, respectively). For example, L6E neurons have a high density of afferent synapses on apical dendrites in layer 4, and as a consequence, the amplitude of CSDs and LFPs generated by population L6E (C) is large in the vicinity of layer 4 and not just in layer 6.
To quantify the relative contributions from the various populations we show in D,E the CSD and LFP variances across time (corresponding to power spectral densities summed over all non-zero frequencies) for all depths [@Linden2011; @Leski2013]. For the compound CSD, only the L23E neurons contribute substantially in the superficial channels (ch. 1–4) (D). At deeper contacts the main contributing population is L6E. L4E and L5E populations make sizable contributions only in the channels closest to their somatic location reflecting that the net associated return currents of their distributed synaptic inputs are largely restricted to somatic regions [@Linden2010]. The bulk of the variance of the CSD in layer 5 arises in about equal parts from the L6E and relatively sparse L5E populations. The magnitude of the depth-resolved CSD variances of each layer’s inhibitory population (L23I, L4I, ...) is consistently one order of magnitude or more smaller than that of the corresponding excitatory cell populations. Moreover, they span comparatively small depth ranges as determined by the maximum extent of the dendrites.
The depth-resolved LFP variance (E) has similar features as the CSD variance, as expected from their common biophysical origin. However, volume conduction has some qualitative effects, such as the reduced relative contribution from the L6E population in layer 2/3. As correlated synaptic inputs are known to amplify and increase the spread of the LFP generated from a cortical population [@Linden2011; @Leski2013] (see also ), this may reflect that the synaptic inputs to the L23E population are more correlated than to the L6E population. Overall, we conclude that for the spontaneous activity in our network, the L23E and L6E populations dominate the compound LFP and CSD with only smaller contributions from the other excitatory populations. The signal variances from the inhibitory populations are typically much smaller than the contributions from the excitatory populations, suggesting that they can be safely neglected, in line with recent findings of @Mazzoni2015.
Although transmembrane currents in inhibitory neurons provide little of the observed CSD and LFP, the inhibitory synaptic inputs onto excitatory neurons provide a substantial contribution. In the present network, inhibitory synaptic currents have a four-fold larger amplitude compared to most excitatory synapses (), inhibitory neurons have higher overall firing rates compared to excitatory neurons [@Potjans2014], and inhibition specifically targets soma-proximal sections [@Markram2004]. Since our LFP-generating model is linear (passive cable formalism, linear synapse model), we can decompose the compound signal into contributions from each synapse type. For example, selective removal of either inhibitory or excitatory synaptic currents in the CSD and LFP modeling (F,G) shows that the CSD and LFP signals (H) are dominated by inhibitory synaptic currents and their associated return currents (I,J) when the network operates in the spontaneous asynchronous irregular firing regime. Further, at most depths the variance of the signal arising from inhibitory input exceeds the compound variance, implying that the inhibitory component is generally negatively correlated with the excitatory component (I,J). Visual inspection of F-H also reveals that the inhibitory dominance appears particularly strong at high frequencies, in accordance with the firing-rate PSDs in B showing less power of inhibitory spiking, and thus inhibitory synaptic input currents, at low frequencies.
The relative contribution from excitation and inhibition to the CSD and LFP depends on thalamic input and network state, however. For oscillatory thalamic input, the CSDs and LFPs from excitatory (A) or inhibitory synapses (B) alone show much stronger oscillations than the compound signals (C). This follows from the observations that the contributions to the CSD and LFP from excitatory and inhibitory synapses are anticorrelated, which, in turn, is a consequence of the dynamical balance between excitation and inhibition in asynchronous-irregular states of balanced random networks [@Hertz10_427; @Renart2010; @Tetzlaff2012].
For transient thalamic activation, the same cancellation of excitation and inhibition can be observed (F-H), although it is more pronounced at some depths (layers 5 and 6, i.e., channels 11-13) than at others, depending on whether excitation and inhibition are in phase or not (I,J). For such strong and transient thalamic input, the network activity is briefly imbalanced as inhibition cannot keep up with thalamic excitation on the short time scales. Since thalamic input is most prominent in layer 4 (channels 7–10), fewer cancellation effects are present here: in channel 9 in I the total CSD variance is, e.g., seen to be larger than the individual contributions from inhibitory and excitatory synapses.
![[**Composition of CSD and LFP during spontaneous activity.**]{} [**A**]{}) Representative morphologies of each population $Y$ illustrating dendritic extent. [**B**]{}) LFP (black traces) and CSD (color plot) produced by the superficial population L23E for spontaneous activity in the reference network. [**C**]{}) Similar to panel B for population L6E (summing over contributions of $y\in\{$p6(L4), p6(L56)$\}$). [**D**]{}) CSD variance as function of depth for each individual subpopulation (colored) and for the full compound signal (black line). [**E**]{}) Same as in panel D, but for LFPs. [**F**]{}) Compound LFP (red traces) and CSD (color plot) resulting from only excitatory input to the LFP-generating multicompartment model neurons. [**G**]{}) Conversely, LFP (blue traces) and CSD (color plot) resulting from only inhibitory input to the neurons. [**H**]{}) Full compound LFP (black traces) and CSD (color plot) resulting from both excitatory and inhibitory synaptic currents. [**I**]{}) Compound CSD variance as a function of depth with all synapses intact (black), or having only excitatory (red) or inhibitory synapse input (blue). [**J**]{}) Same as in panel I, but for the LFP signal. []{data-label="fig:9"}](figures/figure_09_a "fig:"){width="\textwidth"} ![[**Composition of CSD and LFP during spontaneous activity.**]{} [**A**]{}) Representative morphologies of each population $Y$ illustrating dendritic extent. [**B**]{}) LFP (black traces) and CSD (color plot) produced by the superficial population L23E for spontaneous activity in the reference network. [**C**]{}) Similar to panel B for population L6E (summing over contributions of $y\in\{$p6(L4), p6(L56)$\}$). [**D**]{}) CSD variance as function of depth for each individual subpopulation (colored) and for the full compound signal (black line). [**E**]{}) Same as in panel D, but for LFPs. [**F**]{}) Compound LFP (red traces) and CSD (color plot) resulting from only excitatory input to the LFP-generating multicompartment model neurons. [**G**]{}) Conversely, LFP (blue traces) and CSD (color plot) resulting from only inhibitory input to the neurons. [**H**]{}) Full compound LFP (black traces) and CSD (color plot) resulting from both excitatory and inhibitory synaptic currents. [**I**]{}) Compound CSD variance as a function of depth with all synapses intact (black), or having only excitatory (red) or inhibitory synapse input (blue). [**J**]{}) Same as in panel I, but for the LFP signal. []{data-label="fig:9"}](figures/figure_09_b "fig:"){width="\textwidth"}
![**Decomposition of CSD and LFP into contributions due to excitatory and inhibitory inputs for thalamic activation.** **A–E)** Oscillatory thalamic activation ($f=15$ Hz). **F–J)** Transient thalamic activations at $t=900+n \cdot 1000$ ms for $n = 0,1,2,3,4$. Same row-wise figure arrangement as in F-J. []{data-label="fig:10_2"}](figures/figure_10_a "fig:"){width="\textwidth"} ![**Decomposition of CSD and LFP into contributions due to excitatory and inhibitory inputs for thalamic activation.** **A–E)** Oscillatory thalamic activation ($f=15$ Hz). **F–J)** Transient thalamic activations at $t=900+n \cdot 1000$ ms for $n = 0,1,2,3,4$. Same row-wise figure arrangement as in F-J. []{data-label="fig:10_2"}](figures/figure_10_b "fig:"){width="\textwidth"}
Effect of input correlations {#sec:correlations}
----------------------------
Synaptic inputs to two neighboring cells are typically correlated because (i) they receive, to some extent, inputs from the same presynaptic sources (’shared-input correlation’), and (ii) the spike trains of the presynaptic neurons may be correlated (’spike-train correlation’). The net synaptic-input correlation is determined by the interplay between these two contributions, shared-input correlations and spike-train correlations [@Renart2010; @Tetzlaff2012]. As the LFP is largely generated by synaptic inputs, synaptic-input correlations result in correlated single-cell LFP contributions $\phi_i(\mathbf{r},t)$ [for details, see @Linden2011; @Leski2013]. As outlined in the following, these single-cell-LFP correlations play a dominating role for the spectrum of the compound LFP.
The power spectrum $P_{\phi}(\mathbf{r},f)$ () of the compound LFP $\phi(\mathbf{r},t) = \sum_{i=1}^N\phi_i(\mathbf{r},t)$ of a population of $N$ neurons is given by $$P_{\phi}(\mathbf{r},f) =
N \overline{P_{\phi}}(\mathbf{r},f) + N(N-1)\overline{P_{\phi}}(\mathbf{r},f)\overline{\kappa_{\phi}}(\mathbf{r},f)
\label{eq:psd_compoundLFP}$$ (see and ). Here $\overline{P_{\phi}}(\mathbf{r},f)$ is the average single-cell LFP power spectrum () and $\overline{\kappa_{\phi}}(\mathbf{r},f)$ the average pairwise single-cell LFP coherence (), a measure for cross-correlations, across all cells. Note that, while the first term in scales linearly with the number of neurons $N$, the second term is proportional to $N(N-1)\approx N^2$ for large $N$. Hence, for large $N$, even small cross-correlations may dominate the spectrum of the compound LFP. Here, we investigate this situation by calculating the power spectrum $P^{0}_{\phi}(\mathbf{r},f)$ of the compound LFP under the assumption of zero cross-correlation (where it simply reduces to a sum over single-cell spectra $P_{\phi_i}(\mathbf{r},f)$), and compare to the true spectrum $P_{\phi}(\mathbf{r},f)$. The ratio between these quantities is given by $$\label{eq:P_ratio}
\frac{P_{\phi}(\mathbf{r},f)}{P^{0}_{\phi}(\mathbf{r},f)} = 1+(N-1) \overline{\kappa_{\phi}}(\mathbf{r},f)\;.$$ With weak or no cross-correlations, i.e., $(N-1)\overline{\kappa_{\phi}}(\mathbf{r},f) \ll 1$, the ratio approaches unity, and the power of the compound LFP is essentially the sum of the power of the single-cell LFPs. For $N\overline{\kappa_{\phi}}(\mathbf{r},f) \gg 1$, i.e., in the correlation-dominated regime, this ratio is instead proportional to the number of neurons $N$. Note also that anti-correlated signals ($\overline{\kappa_{\phi}}(\mathbf{r},f)<0$) may lead to a ratio $P_{\phi}(\mathbf{r},f)/P^{0}_{\phi}(\mathbf{r},f) < 1$.
In the example application, both for spontaneous (A,B) and for evoked activity (C,D) the compound power spectra $P_{\phi}(\mathbf{r},f)$ are systematically (across channels and frequencies) larger than $P^{0}_{\phi}(\mathbf{r},f)$, demonstrating the importance of cross-correlations in the present network. Depending on the recording depth and frequency, the ratio varies from $\sim 1$ to $10^3$ (see B,D). For spontaneous activity (see , A-E), the largest effects of cross-correlations are typically found at higher frequencies (A,B). At low frequencies, cross-correlations are suppressed by inhibitory feedback (cf. B, @Tetzlaff2012). The thalamic sinusoidally modulated input to the network (, F-J) synchronizes single-cell CSDs and LFPs at the stimulus frequency and gives a large boost of the LFP power at this frequency (see peak in C,D). Close inspection reveals that there is also in fact a boost of the power at around 80 Hz, but much less so than around 15 Hz. Note that the external activation hardly affects the single-cell spectra (see red curves in A,C). LFP synchronization is thus mainly encoded in the phase of $\phi_i(\mathbf{r},t)$.
In conclusion, cross-correlations between single-cell LFP contributions play a pivotal role in shaping the compound LFP spectra (similar for CSD spectra, results not shown). To account for the dominant features of the LFP (CSD) in such models, it is therefore essential to include the main factors determining the synaptic-input correlations, i.e., realistic correlations in presynaptic spike trains and shared-input structure. The findings presented here for the cortical microcircuit model hold in general in the presence of correlated activity. Only the details of the spectra depend on the specific underlying network dynamics.
![ **Effect of single-cell-LFP cross-correlations on compound-LFP power spectra during spontaneous activity (A,B) and for oscillatory thalamic input (C,D).** **A,C)** Compound-LFP power spectra $P_{\phi}(\mathbf{r},f)$ (black traces) and compound spectra $P^{0}_{\phi}(\mathbf{r},f)$ obtained when omitting cross-correlations between single-cell LFPs (red traces; see main text, , computed for 10% of the cells and multiplied by a factor 10) at recording channels corresponding to the centers of layers 1, 2/3, 4, 5 and 6. **B,D)** Depth and frequency-resolved ratio $P_{\phi}(\mathbf{r},f)/P^{0}_{\phi}(\mathbf{r},f)$ of LFP power spectra, cf. . []{data-label="fig:11"}](figures/figure_11){width="\textwidth"}
Network downscaling {#sec:downscaling}
-------------------
Due to the computational cost associated with modeling LFPs, it would be desirable to downsize the postsynaptic populations of multicompartment model neurons to a fraction $\gamma N$ ($\gamma\in\left(0,1\right)$ ) while leaving the point-neuron network at full size ($N$) and at the same time preserve the in-degrees of each postsynaptic cell. The power spectrum of the full-scale LFP can indeed be estimated from the population-averaged single-cell power spectra $\overline{P_{\phi^{\gamma}}}(\mathbf{r},f) \approx \overline{P_{\phi}}(\mathbf{r},f)$ and coherences $\overline{\kappa_{\phi^{\gamma}}}(\mathbf{r},f) \approx \overline{\kappa_{\phi}}(\mathbf{r},f)$ computed for downsized networks by means of . These quantities are preserved except for deviations due to smaller sampling size $\gamma N$ ( in ). However, due to lack of phase information in the power spectra, one cannot estimate the LFP time course.
One could attempt to obtain a time-course estimate $\phi^{\gamma \xi}(\mathbf{r},t)$, i.e., a ‘low-density LFP prediction’, of the full-scale signal $\phi(\mathbf{r},t)$ by upscaling single-cell LFPs $\phi_i(\mathbf{r},t)$ computed in the downsized setup by a scalar factor $\xi$ (cf. ). Such a naive upscaling can grossly recover the amplitude of the full-scale LFP $\phi(\mathbf{r},t)$, but it still only partially reconstructs its detailed time course (A,E). Also, this approach does not generally give accurate power spectra as the two terms in scale differently with $\xi$: The rescaling introduces a prefactor $\xi^2$ in the population-averaged single-cell power spectra $\overline{P_{\phi^{\gamma \xi}}}(\mathbf{r},f)\approx \xi^2 \overline{P_{\phi}}(\mathbf{r},f)$, while the coherences $\overline{\kappa_{\phi^{\gamma \xi}}}(\mathbf{r},f) \approx \overline{\kappa_{\phi}}(\mathbf{r},f)$ are unchanged. Thus the compound spectra $P_{\phi}(\mathbf{r},f)$ and $P_{\phi^{\gamma \xi}}(\mathbf{r}, t)$ of the full-size LFP and the low-density LFP predictor, respectively, differ. Their ratio $$\label{eq:P_ratio2}
\frac{P_{\phi}(\mathbf{r},f)}{P_{\phi^{\gamma \xi}}(\mathbf{r},f)}
= \frac{1+(N-1) \overline{\kappa_{\phi}}(\mathbf{r},f)}
{\gamma\xi^2+ \gamma\xi^2(\gamma N-1) \overline{\kappa_{\phi}}(\mathbf{r},f)}
= \begin{cases} 1/(\gamma\xi^2) & \text{ for } \overline{\kappa_{\phi}}(\mathbf{r},f)=0 \\
1/(\gamma^2\xi^{2}) & \text{ for } \overline{\kappa_{\phi}}(\mathbf{r},f)=1 \end{cases}$$ demonstrates that in the general case there is no scaling factor $\xi$ which allows for the recovery of the full-size compound LFP power, i.e., makes the ratio in equal to one for all spatial positions $\mathbf{r}$ and frequencies $f$. This can only be done in the special case where $\overline{\kappa_{\phi}}(\mathbf{r},f$) is a constant $c$ ($0 \leq c \leq 1$). Here the two extreme cases correspond to no correlation ($\overline{\kappa_{\phi}}(\mathbf{r},f)=0$ with $\xi=1/\sqrt{\gamma}$) and full correlation between all single-cell signals ($\overline{\kappa_{\phi}}(\mathbf{r},f)=1$ with $\xi=1/\gamma$).
The substantial scaling effects observed for our microcircuit model in the asynchronous state (A–D) suggest that correlations cannot be neglected even when modeling the LFP for spontaneous network activity. Choosing the scaling factor $\xi=1/\sqrt{\gamma}$ corresponding to $\overline{\kappa_{\phi}}(\mathbf{r},f)=0$ (red lines in A,C) leads to a severe underestimation of the full-size compound power spectrum (C,D). Even though the correlation (i.e., Pearson’s correlation coefficients) between the full-size full-size LFP signals and low-density LFP predictions are quite high (B), the power ratios (D) reveal that the rescaled signals are systematically wrong in frequency bands where single-cell LFPs are most strongly correlated (i.e., the frequencies for which the compound spectra are much larger than the predictions when omitting cross-correlations (A,B)). Assuming the full-correlation scaling factor $\xi=1/\gamma$, on the other hand, typically overestimates the full-size compound power spectrum (cf. gray spectra in C), particularly at low frequencies.
The results for the sinusoidally stimulated network (E–H) are quite similar to the spontaneous-activity results, except around the stimulation frequency 15 Hz where the modulated input leads to strongly correlated single-cell LFP contributions and a strong boost of the compound LFP. The approximate downscaling procedure assuming the full-correlation scaling factor $\xi=1/\gamma$ thus essentially agrees with the full-size compound spectrum for 15 Hz (while giving a strong overestimation for frequencies other than $\sim$15 and $\sim$80 Hz).
We note in passing that in contrast to the power spectra, the computation of the *spike-triggered averaged LFP* (staLFP) [@Swadlow2002; @Nauhaus2009; @Jin2011; @Denker2011] in downsized networks do not have a principled problem due to cross-correlations between single-cell LFPs. As staLFPs are linearly dependent on the single-neuron LFP contributions, the only principled problem with downsizing is increased noise in the estimates due to sampling over fewer postsynaptic neurons.
![**Prediction of LFPs from downsized networks.** Top row: Spontaneous activity. Bottom row: Oscillatory thalamic activation. [**A,E)**]{} Full-scale LFP traces $\phi(\mathbf{r},t)$ (black) and low-density predictors $\phi^{\gamma \xi}(\mathbf{r},t)$ (red) obtained from a fraction $\gamma=0.1$ of neurons in all populations and upscaling by a factor $\xi=\gamma^{-\frac{1}{2}}$. [**B,F**]{}) Correlation coefficients between full-scale LFP and low-density predictor shown in panels A and E, respectively. [**C,G**]{}) Power spectra $P_{\phi}(\mathbf{r},f)$ and $P_{\phi^{\gamma \xi}}(\mathbf{r},f)$ of full-scale LFPs (black) and low-density predictors with $\gamma=0.1$ and $\xi=\gamma^{-\frac{1}{2}}$ (red) or $\xi=\gamma^{-1}$ (gray). [**D,H**]{}) Ratio $P_{\phi}(\mathbf{r},f)/P_{\phi^{\gamma \xi}}(\mathbf{r},f)$ between power spectra of full-scale LFP and low-density predictor with $\gamma=0.1$ and $\xi=\gamma^{-\frac{1}{2}}$ (cf. ). []{data-label="fig:12"}](figures/figure_12_a "fig:"){width="\textwidth"} ![**Prediction of LFPs from downsized networks.** Top row: Spontaneous activity. Bottom row: Oscillatory thalamic activation. [**A,E)**]{} Full-scale LFP traces $\phi(\mathbf{r},t)$ (black) and low-density predictors $\phi^{\gamma \xi}(\mathbf{r},t)$ (red) obtained from a fraction $\gamma=0.1$ of neurons in all populations and upscaling by a factor $\xi=\gamma^{-\frac{1}{2}}$. [**B,F**]{}) Correlation coefficients between full-scale LFP and low-density predictor shown in panels A and E, respectively. [**C,G**]{}) Power spectra $P_{\phi}(\mathbf{r},f)$ and $P_{\phi^{\gamma \xi}}(\mathbf{r},f)$ of full-scale LFPs (black) and low-density predictors with $\gamma=0.1$ and $\xi=\gamma^{-\frac{1}{2}}$ (red) or $\xi=\gamma^{-1}$ (gray). [**D,H**]{}) Ratio $P_{\phi}(\mathbf{r},f)/P_{\phi^{\gamma \xi}}(\mathbf{r},f)$ between power spectra of full-scale LFP and low-density predictor with $\gamma=0.1$ and $\xi=\gamma^{-\frac{1}{2}}$ (cf. ). []{data-label="fig:12"}](figures/figure_12_b "fig:"){width="\textwidth"}
LFP prediction from population firing rates {#sec:3.3}
-------------------------------------------
An important question in systems neuroscience is to what extent the dynamics of networks of thousands or millions of neurons can be described by much simpler mathematical descriptions in terms of neural *populations* [@Deco2008; @Blomquist2009]. Likewise, we here ask the question of whether LFPs can be predicted from knowledge of the population firing rate [@Einevoll2007; @Moran2008; @Einevoll2013]. The hybrid scheme is excellently suited for testing and development of simplified numerical schemes for LFP prediction as the ground truth, i.e., the LFP from the full network, is available as benchmarking data.
The use of current-based synapses and passive dendrites in the present application of the hybrid scheme, renders synaptic events independent of each other in the LFP prediction. This inherent linearity results in a unique spatio-temporal relation $H_{X}^{i}(\mathbf{r},\tau)$ for $\tau \in [-\infty, \infty]$ between a spike event of a point neuron $i$ in population $X$ and its contribution to the compound LFP $\phi(\mathrm{r},t)$ from all its postsynaptic multicompartment model neurons. In this scheme the link is causal, i.e., the spikes drive the LFP, so that $H_{X}^{i}(\mathbf{r},\tau)=0$ for $\tau<0$ (as in laminar population analysis (LPA) [@Einevoll2007]). $H_{X}^{i}(\mathbf{r},\tau)$ encompasses connectivity, spike transmission delays and all postsynaptic responses including effects of synaptic input currents and passive return currents. With a linear, current-based model such as our example cortical column, it is in principle possible by linear superposition to fully reconstruct the compound LFP if $H_{X}^{i}(\mathbf{r},\tau)$ and the spike times $t^i_l$ are known for all neurons $i$ in each population of the network. It is, however, in the case of large networks impractical to assess each $H_{X}^{i}(\mathbf{r},\tau)$, as the LFP response needs to be determined for every neuron separately. In contrast, a large reduction in dimensionality can be achieved by determining the population-averaged LFP responses $\overline{H}_X(\mathbf{r},\tau)$ of a spike within each population $X$. We thereby ignore heterogeneity in kernels $H_{X}^{i}(\mathbf{r},\tau)$ due to the variability in the connections from neurons in population $X$. An approximate compound LFP $\phi^\ast(\mathbf{r},t)$ based on population firing rates [@Einevoll2007] can be computed from these extracted population kernels by means of the convolution $\phi^\ast(\mathbf{r},t) = \sum_X \left(\nu_X \ast \overline{H}_X\right)(\mathbf{r}, t)$, where $\nu_X(t)$ are the instantaneous population firing rates.
Here we estimate the population LFP kernels by computing the response to synchronous activation of all neurons in a population (). The spatio-temporal kernels $\overline{H}_X(\mathbf{r},\tau)$ are extracted from time slices $[t_X -20~\text{ms}, t_X+20~\text{ms}]$ of the compound LFP response $\phi(\mathbf{r},t) / N_X$, where $N_X$ is the number of neurons in a presynaptic population. The procedure results in unique kernels $\overline{H}_X(\mathbf{r},\tau)$ for each excitatory and inhibitory population in the network (A).
In the example application, the population kernels $\overline{H}_X(\mathbf{r},\tau)$ differ significantly between populations. Excitatory spike events result in prominent LFP negativities at depths where most connections are made, such as in layer 4 (channels 7–10) for thalamocortical connections ($\overline{H}_\text{TC}(\mathbf{r},\tau)$, column 1 in A, cf. C). In contrast, spikes of inhibitory point neurons on average produce prominent LFP positivities in their corresponding layer, such as in layer 2/3 (channels 3–6) for population L23I (column 2 in A). In all cases, the signatures of opposite-sign return currents and also other, weakly connected populations are seen across depth.
As seen in C,F the population-rate predictions are in good qualitative agreement with the ground-truth LFP for both spontaneous and sinusoidally modulated network activity with correlation coefficients ($cc$) between 0.48 and 0.94 for spontaneous activity (D) and 0.52 and 0.98 for thalamically evoked oscillations (G). Overall, the population-rate predictions appear to be best for the lower frequencies (E,H), while the inherent variability in the individual point-neuron kernels $H_{X}^{i}(\mathbf{r},\tau)$ (which is not accounted for in the population approximation) has a larger effect on the higher frequencies. This can be understood on biophysical grounds, as the lower frequencies are expected to mainly reflect the gross anatomical features of the postsynaptic populations and their presynaptic connections patterns where the individual variability plays a lesser role [@Linden2010; @Pettersen2012]. The correlation coefficients and power spectra of the population-rate prediction thus show that the population-rate LFP predictor is more accurate than the low-density LFP predictors (C,F) in case of substantial downscaling.
Although the spike-triggered average LFP (staLFP) [@Swadlow2002; @Nauhaus2009; @Jin2011; @Denker2011], calculated as the cross-covariance between the population spike rate $\nu_X(t)$ and the compound LFP $\phi(\mathbf{r}, t)$ divided by the total number of spikes (i.e., $\int_0^T \nu_X(t) dt$), is related to our LFP population kernels, it measures very different aspects of cortical dynamics. The population kernels $\overline{H}_X(\mathbf{r},\tau)$ are causal and independent of effects of spike-train correlations. The staLFP, on the other hand, is non-causal and strongly depends on spike-train correlations, and thus also network state [@Einevoll2013]. The staLFP is thus not only very different from $\overline{H}_X(\mathbf{r},\tau)$, it also varies strongly between the spontaneous and sinusoidally modulated network state (see example for L5E neurons in B).
![ **Linear prediction of LFPs from population firing rates.** [**A**]{}) LFP responses $\overline{H}_X(\mathbf{r},\tau)$ (kernels) to simultaneous firing of all neurons in a single presynaptic population $X$ (see subpanel titles) at time $\tau=0~$ms, normalized by size $N_X$ of the presynaptic population (red/blue: responses to firing of excitatory/inhibitory presynaptic populations). [**B**]{}) LFPs triggered on spikes of L5E neurons during spontaneous activity (left) and oscillatory thalamic network activation (right), averaged across all L5E spikes ($T$=5 s simulation time). [**C,F**]{}) LFP traces of the full model (black) compared to predictions (red) obtained from superposition of linear convolutions of population firing rates $\nu_X$ with LFP kernels $\overline{H}_X(\mathbf{r},\tau)$ shown in A. [**D,G**]{}) Correlation coefficients between LFPs and population-rate predictors shown in C and F. [**E,H**]{}) Power spectra of LFPs (black) and the population-rate predictors (red) for different recording channels. Panels C-E and F-H show results for spontaneous activity and oscillatory thalamic activation, respectively. []{data-label="fig:13"}](figures/figure_13){width="\textwidth"}
Discussion {#sec:discussion}
==========
We have here described a hybrid modeling scheme for computing the local field potential (LFP) incorporating both large-scale neural network dynamics and the biophysics underlying LFP generation on the single-neuron level. The hybrid modeling scheme was illustrated with a full-scale network model of a cortical column in early sensory cortex [@Potjans2014], and the impact of individual populations, network dynamics and cell density on the mesoscopic LFP signal was investigated.
The hybrid LFP modeling scheme
------------------------------
The hybrid scheme combines the simplicity and efficiency of point-neuron network models with the biophysics-based modeling of LFP by means of multicompartment model neurons with detailed dendritic morphologies. The neuronal network dynamics are governed by the point-neuron network model independent of LFP predictions. The spikes of the point-neuron network are distributed to the synapses of the multicompartment model neurons with realistic cell-type and layer-specific connectivity. Synapse activation results in spatially distributed transmembrane currents, which are mapped to an LFP signal according to well-established volume-conduction theory. A main motivation for developing the hybrid LFP modeling scheme was to obtain the ability to compute LFPs for a key class of network models that are amenable to mathematical analysis and can provide intuitive understanding of emerging network dynamics, namely point-neuron models. Similar to networks of anatomically and biophysically detailed neuron models, point-neuron networks can generate realistic spiking activity. In addition, the hybrid modeling scheme brings a substantial computational advantage: With present-day computing and software technologies, point-neuron networks with $\sim$100,000 neurons can be modeled with laptop computers, and networks comprising millions of point neurons can be routinely simulated on high-performance compute facilities [@Helias2012; @Kunkel2014]. Until now, the largest simulation of LFPs based on networks of multicompartmental neuron models with reconstructed morphologies, in contrast, comprised about 12,000 neurons and was done on a Blue Gene/P supercomputer with 4096 CPUs [@Reimann2013]. The linearity of electromagnetic theory allows for the implementation of the LFP hybrid modeling scheme as an “embarrassingly” parallel operation [@Foster1995]. Therefore, the results for the cortical microcircuit application with $\sim$78,000 neurons were obtained with only 256 CPUs, and could even be acquired with much smaller computing architectures.
A full implementation of the hybrid scheme is provided by the freely available `Python` module `hybridLFPy` (<http://github.com/INM-6/hybridLFPy>). Our model implementation in `hybridLFPy` relies on the publicly available `NEURON` software as a simulation backend through `Python` with `LFPy` [@Linden2014] for calculating single-cell LFP contributions. This ensures flexibility and compatibility with a large library of existing neuron models, with or without active channels and with morphologies of arbitrary levels of detail, obtained from ModelDB [@Hines2004], NeuroMorpho.org [@Ascoli2007] or other resources. While we did use `NEST` [@eppler_2015_32969] for simulating our reference network, the `hybridLFPy` module can be used in combination with any other neural-network simulation software. The present hybrid LFP scheme involves several assumptions with respect to (i) the generation of realistic spiking activity, (ii) forward modeling of extracellular potentials, and (iii) the combined use of point-neuron networks and multicompartment modeling. In the following we review the main assumptions and discuss potential extensions:
\(i) *Spike-train generation by point-neuron networks:* Although highly simplified, single-compartment models of individual neurons (point-neuron models) can mimic realistic spiking for a variety of cell types [@Izhikevich2008; @Kobayashi2009; @Yamauchi2011] and can make accurate predictions of single-cell firing responses under in-vivo like conditions [@Jolivet2008; @Gerstner2009]. Moreover, networks of point neurons can reproduce a number of activity features observed *in vivo*, such as spike-train irregularity [@Softky1993; @Vreeswijk1996; @Amit1997; @Shadlen1998], membrane-potential fluctuations [@Destexhe1999], asynchronous firing [@Ecker2010; @Renart2010; @Ostojic2014], correlations in neural activity [@Gentet2010; @Okun2008; @Helias2013], self-sustained activity [@Ohbayashi2003a; @Kriener2014] and realistic firing rates across laminar cortical populations [@Potjans2014]. Note that the hybrid LFP modeling scheme is not necessarily restricted to point-neuron networks as generators of spiking activity. In principle, they could, for example, be replaced by statistical models of spike generation [@Linden2011; @Leski2013], or even experimentally measured spiking activity.
\(ii) *Biophysical forward modeling of LFPs:* The biophysical forward model described by , implemented in `LFPy` [@Linden2014], underlies the presently used computational scheme for LFPs of point-neuron networks. This forward model is based on well-established volume conductor theory [@Rall1968; @Holt1999] and assumes an *infinite*, *isotropic* (same in all directions), *homogeneous* (same in all positions) and *ohmic* (frequency-independent) extracellular medium represented by a scalar conductivity $\sigma_\text{e}$. However, one could generalize the forward model in a straightforward manner to account for anisotropy [@Nicholson1975; @Logothetis2007; @Goto2010], or jumps in conductivities at tissue interfaces [@Pettersen2006; @Gold2006; @Hagen2015; @Ness2015]. For even more complicated geometrical spatial variations of the conductivity, the forward modeling problem can always be solved by means of Finite Element Modeling (FEM) [@Ness2015]. Recent experiments have only found a small frequency dependence of the extracellular conductivity $\sigma_\text{e}$ at LFP frequencies ($f\lesssim$ 500 Hz, @Logothetis2007 [@Wagner2014]), but see @Gabriel1996 [@Gabriel2009; @Bedard2009]. In any case the forward model could still be applied with frequency-dependent conductivity by means of Fourier decomposition where each frequency component of the LFP signal is considered separately. For more information on possible generalizations of the biophysical forward-modeling scheme, see @Pettersen2012. Finally, we assumed the so-called disc-electrode approximation and averaged the computed LFP signal across the electrode surface [@Moulin2008; @Linden2014; @Ness2015]. Although electrode impedance will affect the measurement, it appears that confounding effects from this can easily be avoided with present-day LFP recording techniques [@Nelson2010], hence few compelling reasons exist to incorporate additional temporal filters.
\(iii) *Combined use of point-neuron and multicompartmental models:* The key approximation in the hybrid LFP scheme comes from the combined use of point-neuron (single-compartment) and multicompartmental neuron models. The multicompartment neurons are mutually unconnected, have no outgoing (efferent) connections, and are solely used to compute the LFP. Further, due to dendritic filtering, the somatic postsynaptic potentials in the multicompartment model neurons are not identical to those of their point-neuron counterparts. This inconsistency could, at least partially, be resolved by adjusting the amplitudes and temporal shapes of the synaptic currents in either the multicompartment neurons or the point neurons [@Koch1985; @Wybo2013; @Wybo2015].
Applications of the hybrid LFP scheme
-------------------------------------
The hybrid scheme is not limited to the example point-neuron network model and the particular multicompartment neuron models chosen here. It can be applied to networks (i) of arbitrary topology (graph structure, distance dependencies, dimensions), (ii) with any number of populations, (iii) with arbitrarily complex point-neuron (e.g. LIF, Izhikevich, MAT, Hodgkin-Huxley) and synapse dynamics (e.g., current-based, conductance-based, static, plastic), and (iv) any level of biophysical detail in multicompartment neuron models (e.g., morphologies, active channels).
For illustration, we used the hybrid scheme to compute LFPs along a virtual laminar multielectrode from activity in a multilayered spiking point-neuron network, modeling signal processing in a patch of primary visual cortex. The network consisted of $\sim$78,000 neurons organized in four layers, each with an excitatory and an inhibitory population, representing a cortical patch of $\sim$1 mm$^2$ [@Potjans2014]. Altogether 16 different cell types and 10 different morphologically reconstructed neurons were used in the LFP calculation. This cortical microcircuit model is well-suited for the illustration of the hybrid scheme due to its (i) minimum level of detail in single-neuron dynamics of both point neurons (LIF) and multicompartment neurons (passive membranes), (ii) realistic neuron density allowing investigation of the effects of correlations and scaling of network size, and (iii) its spatial organization of multiple populations across cortical layers which yields cancellation effects not captured by LFP proxies such as in @Mazzoni2015.
Even though the example application was based on a generic network model biased towards cat visual cortex and not tuned to address specific experiments, its spiking activity nevertheless matched experimental findings [@Potjans2014]. We even observed the predicted LFP to be in qualitative accordance with LFP measurements in primary sensory cortices from a variety of animal species and sensory modalities in terms of (i) LFP amplitude, for both spontaneous ($\simeq$0.1 mV [@Maier2010; @Hagen2015]) and stimulated activity ($\sim$1–3 mV) [@Mitzdorf1979; @Mitzdorf1985; @Castro-Alamancos1996; @Schroeder1998; @Di1990; @Einevoll2007], and (ii) stimulus-evoked spatiotemporal LFP and CSD patterns [@Mitzdorf1985; @Einevoll2007; @Reyes-Puerta2015]. This supports the overall biological plausibility of the hybrid LFP scheme.
For the present example the LFP was dominated by synaptic inputs, and their associated return currents, on excitatory neurons, in particular onto pyramidal cells in layers 2/3 and 6. Further, contributions from inhibitory synaptic inputs typically dominated the contributions from the excitatory inputs, particularly for LFPs stemming from spontaneous network activity. Although the main point of employing the present example was to illustrate the use of the hybrid LFP scheme and not to make predictions for specific neural systems, we note in passing that a dominance of inhibitory synaptic inputs appears to be in agreement with LFPs generated in the CA3 region of hippocampus as observed in an *in vitro* setting [@Bazelot2010]. In accordance with a previous study [@Linden2011] we found that correlations in synaptic input play a major role in determining the CSD and LFP stemming from the network activity, for both spontaneous and stimulus-evoked activity. We further showed that due to inevitable correlations between synaptic input currents the main features of the LFP can only be correctly predicted by a full-scale model. As our ambition is to compute LFPs also for extended point-neuron networks with millions of neurons covering, for example, entire cortical areas, we finally demonstrated how the hybrid modeling scheme can predict LFPs from population firing rates rather than from spikes of individual neurons [@Einevoll2007].
The present microcircuit model application involving simplified, passive multicompartment populations was used here to study the effect of the spatial connectivity on the laminar pattern of spontaneous and stimulus-evoked CSD and LFP signals. The application thus represented a minimal approach incorporating spatial features in LFP predictions of multilaminar point-neuron networks. However, several of the simplifying model assumptions made in the present example application can straightforwardly be generalized. In particular, such generalizations concern (i) the synaptic connectivity between point neurons and the equivalent multicompartment neurons, (ii) the absence of active conductances, (iii) the positioning of the cells, (iv) the reconstructed morphologies, (v) the representation of external inputs, and (vi) the fact that the model only encompassed the local circuitry of a $\sim$1 mm$^2$ patch of cortex.
\(i) *Synaptic connectivity:* While the population-specific connection probabilities, delay distributions, synapse time constant and mean synaptic weights were identical for the connections in the point-neuron network and those between point neurons and LFP-generating multicompartment neurons, the exact realizations of the two types of connectivities were different. In contrast to the point-neuron network, each cell of a particular type had a fixed in-degree, i.e., a fixed number of synaptic inputs, and a fixed synaptic current amplitude in the LFP modeling step. We positioned the synapses randomly with the prescribed layer specificity, i.e., without clustering onto specific dendrites. The use of the hybrid LFP scheme, however, is not restricted to these or any other specific assumptions about the synaptic connectivity patterns. One could, for example, gather all point-neuron network connections and corresponding weights and delays for use with the LFP-generating multicompartment neuron populations. However, for large networks this would require additional computing and memory resources as the number of recorded connection weights and delays grow proportionally to $N^2$ in a network of $N$ neurons with fixed connection probabilities.
\(ii) *Active conductances:* In the present study we neither included the active channels underlying spike generation, nor active dendritic conductances in the multicompartment neuron models [@Remme2011]. Experiments suggest that the contribution to the LFP from the former is small in stimulus-evoked recordings from sensory cortex, at least for the low frequencies of the LFP [@Pettersen2008], but see @Ray2011a. In any case the contribution of the spikes to the extracellular potentials, including the LFP, could be included in the present scheme by giving each spike produced in the point-neuron network simulations a cell-type-specific spatiotemporal signature in the computation of the extracellular potential (e.g., as calculated in @Holt1999 [@Hagen2015]). Substantial effects of active dendritic conductances on the LFP were observed in a two-layered model by @Reimann2013, but this should be further explored. Recent modeling results [@Ness2015b] suggest that for the purposes of LFP prediction, active dendritic conductances can, at least for subthreshold potentials, be effectively described by means of “quasi-active linearized” theory [@Sabah1969; @Koch1984a; @Remme2011]. This simplifies the LFP modeling substantially and makes the computation similar in complexity to the present case with purely passive membranes.
\(iii) *Soma positioning:* For model conciseness, the somas of all neurons belonging to a specific cortical populations were set to have the same cortical depth, cf. E. By instead assuming a biologically more plausible, distribution of soma depths, the CSD profiles are expected to be spatially smoothed compared to the present profiles, e.g., F. Also the LFP profiles will be affected, but to a lesser degree since the LFP profiles already are spatially smoothed due to volume conduction effects.
\(iv) *Reconstructed morphologies:* We further chose to rely on a small number of highly detailed reconstructed dendritic morphologies from experimental preparations, and partly reuse morphologies across neural populations. Obviously, larger sets of distinct reconstructed dendritic morphologies can be used in future studies as they become available. The effect of dendritic morphologies on generated LFP can be further assessed by use of stylized [@Tomsett2014; @Glabska2014] or artificially grown morphologies resembling real neurons [@Cuntz2010; @Cuntz2011; @Torben-Nielsen2014; @Mazzoni2015].
\(v) *External input:* In our point-neuron network modified from @Potjans2014, we assumed the depolarizing input from surrounding cortex and remote areas to be represented by deterministic, fixed-amplitude input currents (DC inputs) rather than independent Poisson spike trains. We thus avoided the generation and storage of $O(10^5)$ high-rate uncorrelated Poisson spike trains and distributing the corresponding spike events onto the multicompartment model neurons, but this can be introduced in future applications.
\(vi) *Scale:* Our network model represents only an isolated cortical column under $\sim$1 mm$^2$ of pial surface [@Potjans2014], but work is underway to extend the network to a larger scale, e.g., by incorporating additional cortical areas [@Schmidt2015] or extending the network size in the lateral directions [@Senk2015]. In addition to allowing for predictions of LFPs at several lateral positions as measured by multishank electrodes, the anticipated outcome is also an altered network dynamics and consequently altered LFPs. The present columnar model lacks structured input from other parts of cortex which corresponds to about 50% of all excitatory synapses (see @Potjans2014 and references therein). For example, accounting for different cortical areas and their interactions would allow for the detailed investigation of pathway-specific LFP contributions as recently reviewed in hippocampus by @Herreras2015.
Effects of correlations and network size
----------------------------------------
Synaptic inputs to neurons are typically correlated, and as shown in this article and in earlier studies [@Linden2011; @Leski2013], these correlations have a major impact on the properties of the generated LFP (and corresponding CSD). There are different contributions to these input correlations, shared presynaptic neurons and correlations in the presynaptic spiking activity, see, e.g., [@Renart2010; @Tetzlaff2012], and correct LFP predictions require that both effects are properly taken into account. The generation of presynaptic spike trains with realistic correlation structure requires networks of realistic size [@Albada2015]. The contribution from shared presynaptic input can only be accounted for by using realistic statistics of inputs to the LFP-generating multicompartment model neurons, i.e., realistic synaptic connection probabilities resulting in realistic statistics of shared inputs. If these two requirements are fulfilled, the properties of single-cell LFPs as well as the correlations between pairs of single-cell LFPs, will be correctly accounted for.
Given synaptic inputs with realistic statistics from sufficiently large point-neuron networks, do we actually need to represent the full population of LFP generating neurons in order to predict a realistic compound LFP? Or can we alternatively get a good estimate of the compound population-LFP signal from a downscaled population of neuronal LFP generators if we know the correct single-cell and pairwise LFP statistics, thereby reducing computational costs? As shown in this article (), the answer is negative: In the presence of (even tiny) synaptic-input correlations, a realistic compound LFP can only be generated by multicompartmental neuron populations with realistic size and cell densities.
The hybrid modeling scheme allows one to account for both, i.e., realistic sizes of networks to generate spike trains with correct correlation structure and realistic sizes and cell densities of the LFP-generating multicompartmental neuron populations. This can be achieved since point-neuron networks can be simulated very efficiently, and the multicompartmental neurons are independent and can be simulated serially (or in an embarrassingly parallel manner).
Outlook
-------
While we here have focused on the computation of the LFP based on output from spiking point-neuron networks, a similar hybrid approach could be used when the network dynamics is rather modeled in terms of firing rates or even neural fields [@Deco2008]. For our example case of a four-layered cortical network with an excitatory and an inhibitory population in each layer, the present scheme could be adapted directly by replacing the set of spike trains for each population with the corresponding population firing rates [@Schuecker2015] in the LFP-generating step. However, the feasibility and prediction accuracy of such a scheme would have to be investigated in detail.
Another natural development would be to consider other measurement modalities. The present LFP scheme already incorporates the prediction of ECoG (electrocorticography) signals, i.e., the electrical signals recorded at the cortical surface, although the LFP forward-modeling scheme may have to be adjusted to account for the discontinuity in electrical conductivity at the cortical surface [@Nunez2006]. An extension to EEG (electroencephalography) and MEG (magnetoencephalography) would in principle also be straightforward as the key variable linking single-neuron activity and the measured signal is the single-neuron current dipole moment [@Hamalainen1993; @Nunez2006]. This dipole moment can be computed from multicompartmental neuron models when the transmembrane or axial currents are known [@Linden2010; @Ahlfors2015]. Given the magnitude and orientation of the current dipole moments for all contributing neurons, the EEG and MEG signal can be computed by a linear superposition of single-cell contributions (given an appropriate extracellular volume conductor model for the EEG signal). Another measurement that could be modeled is voltage-sensitive dye imaging (VSDi) where the signal largely reflects average membrane potentials of dendrites close to the cortical surface [@Chemla2010]. The spatial profile of the weights in the averaging procedure of the VSDi forward-model will be determined by the spatial distribution of dye and the propagation of light in the neural tissue [@Chemla2010a; @Tian2011].
Even though the LFP has been measured for more than half a century, the interpretation of the recorded data has so far largely been qualitative [@Einevoll2013]. The biophysical origin of the signal on the single-cell level appears well understood [@Rall1968; @Holt1999; @Gold2006; @Linden2010; @Buzsaki2012; @Pettersen2012; @Einevoll2013], and several modeling studies have explored the link between neuron and network activity [@Pettersen2008; @Linden2010; @Linden2011; @Leski2013; @Reimann2013; @Tomsett2014; @Glabska2014]. However, the computation of LFPs from network activity has until now been too cumbersome and computer-intensive to allow for practical exploration of the links between different types of network dynamics and the resulting LFP. Thus a validation of network models against measured LFP data has essentially been absent. With the present hybrid LFP scheme, accompanied by the release of the simulation tool `hybridLFPy`, we believe that a significant step has been taken towards the goal of making combined modeling and measurement of the LFP signal a practical research tool for probing neural circuit activity.
Acknowledgments {#acknowledgments .unnumbered}
===============
The research leading to these results has received funding from the European Union Seventh Framework Programme (FP7/2007-2013) under grant agreement 604102 (Human Brain Project, HBP) and grant agreement 269912 (BrainScaleS), the Helmholtz Association through the Helmholtz Portfolio Theme “Supercomputing and Modeling for the Human Brain” (SMHB), Juelich Aachen Research Alliance (JARA), the Danish Council for Independent Research and FP7 Marie Curie Actions COFUND (grant id: DFF-1330–00226) and the Research Council of Norway (NFR, through ISP, NOTUR -NN4661K). In addition, we would thank Zoltán F. Kisvárday and Armen Stepanyants for useful discussions in the early phase of this study and providing us with experimentally reconstructed morphologies of cat visual cortex neurons.
Statement on competing interests {#statement-on-competing-interests .unnumbered}
================================
The authors have no financial or non-financial competing interests.
[|p[0.15]{}|X|]{} [ ]{}\
**Structure & Multi-layered excitatory-inhibitory (E-I) network\
**Populations & 8 cortical in 4 layers, 1 thalamic (TC)\
**Connectivity & Random, independent, population-specific, fixed number of connections\
**External input & Cortico-cortical: constant current with population-specific strength\
**Neuron model & Cortex: leaky integrate-and-fire (LIF); TC: point process\
**Synapse model & Exponential postsynaptic currents, static weights, population-specific weight distributions\
**Measurements & Spike activity, input currents, membrane potential of each neuron\
**************
[|p[0.15]{}|X|]{} [ ]{}\
**Connectivity & Connection probability $C_{YX}$ ($X,Y \in \{\text{L2/3, L4, L5, L6}\} \times \{\text{E,I}\} \cup \text{TC}$, $C_{YX}=0$ for $Y=\text{TC}$)\
&- Fixed number of synapses $K_{YX}$ between populations $X$ and $Y$\
&- Binomial in-/outdegrees\
**Input & Cortico-cortical direct current $I_Y^\text{ext}$\
****
[|p[0.15]{}|X|]{} [ ]{}\
\
**Type & Leaky integrate-and-fire neuron\
**Description & Dynamics of membrane potential $V_i(t)$ (neuron $i\in[1,N]$):****
- Spike emission at times $t^i_l$ with $V_i(t^i_l)\ge\theta$
- Subthreshold dynamics:
- $\tau_\text{m}\dot{V_i}=-V_i+ R_\text{m} I_i(t)$ if $\forall l:\,t\notin(t^i_l,t^i_l+\tau_\text{ref}]$
- Reset + refractoriness: $V_i(t)= V_\text{reset}$ if $\forall l:\,t\in(t^i_l,t^i_l+\tau_\text{ref}]$
\
& Exact integration with temporal resolution $dt$ [@Rotter99a]\
& Uniform distribution of membrane potentials at $t=0$\
\
**Type &- DC current for constant background input\
&- Nonstationary Poisson process for modulation\
**Description & DC current included in external DC input\
&\
& Types of thalamic input modulation:\
&- Spontaneous activity: no modulation in activation of thalamic neurons\
&- Thalamic pulses: fixed-interval coherent activation of all $N_\text{TC}$ thalamic neurons\
&- AC modulation: Poisson spike trains with sinusoidally modulated rate profile (discretized with time resolution $dt$):\
& $$\nu_\text{th}(t)
= \overline\nu_\text{TC} + \Delta\nu_\text{TC} \sin{(2\pi t f_\text{TC})}
\label{eq:ac_modulation}$$\
****
[|p[0.15]{}|X|]{} [ ]{}\
**Type & Exponential postsynaptic currents, static weights\
**Description & Input current of neuron $j$ of synapses formed with presynaptic neurons $i$:\
& $ I_j(t) = \sum_i J_{ji}\sum_l \exp{(-(t-t^i_l-d_{i})/\tau_\text{s})}~ H(t-t^i_l-d_{i}) + I_{j}^\text{ext}
$\
&****
- Static synaptic weights $J_{ji}=\text{sgn}(X) \left|J_{YX}\right|$ ($i \in X$, $j \in Y$); $\text{sgn}(X)=1$ for $X~\in~\{\text{L2/3E, L4E, L5E, L6E, TC}\}$, $-1$ otherwise
- Absolute weights $\left|J_{YX}\right|$ drawn from lognormal distribution $$p(\left|J_{YX}\right|)=\frac{1}{\sqrt{2\pi}\sigma_{YX} \left|J_{YX}\right|}\exp{\left(-\frac{(\ln{\left|J_{YX}\right|}-\mu_{YX})^2}{2 \sigma_{YX}^2}\right)}$$ or normal distribution $$p(\left|J_{YX}\right|)=\frac{1}{\sqrt{2\pi}\sigma_{YX}}\exp{\left(-\frac{(\left|J_{YX}\right|-\mu_{YX})^2}{2 \sigma_{YX}^2}\right)}$$ with $\mu_{YX}=g_{YX} J$ and $\sigma_{YX}=\sigma_{J,\text{rel}} \, \mu_{YX}$
- Delays $d_{i}=d_{X}$ ($i \in X$) drawn from (left-clipped) Gaussian distribution $$p(d_{X})=\frac{1}{\sqrt{2\pi}\sigma_{X}}\exp{\left(-\frac{(d_{X}-\mu_{X})^2}{2 \sigma_{X}^2}\right)}$$ with mean $\mu_{X}=d_\text{E},d_\text{I}$ for $X$ exc., inh., stdev $\sigma_{X}=\sigma_{d,\text{rel}} \, \mu_{X}$ and $d_X \in [dt,\infty)$
- $ H(t)= 1 \text{ for } t \geq{0}~, \text{ and } 0 \text{ elsewhere.}$
- External DC input $I_{j}^\text{ext} = I_Y^\text{ext}= I^\text{ext} k^\text{ext}_Y$ ($j \in Y$)
\
[|p[0.15]{}|X|]{} [ ]{}\
**Topology & Cortical column under $1\,\text{mm}^2$ of cortical surface\
**Populations & 8 excitatory and 8 inhibitory cell types\
**Input & Spiking activity of thalamic and cortical populations as modeled by point-neuron network\
**Neuron model & Multicompartment, passive cable formalism\
**Synapse model & Exponential postsynaptic current, static weights\
**Measurements & Current source density (CSD), local field potential (LFP)\
************
[|p[0.15]{}|X|]{} [ ]{}\
**Type & Cylindrical volume with layer-specific distribution of cell types and synapses\
**Description & Cylinder radius $r$\
& Laminar, defining upper/lower boundaries of layers 1, 2/3, 4, 5, 6\
****
[|p[0.15]{}|X|]{} [ ]{}\
**Type & Each cell type $y$ assigned to population $Y$, $y \in Y$\
**Description & Populations $Y \in \{\text{L2/3, L4, L5, L6}\} \times \{\text{E,I}\}$ (population size $N_Y$, cell types $y \in Y$)\
& (e.g., $\text{L4E}=\{\text{p4},~\text{ss4(L23)},~\text{ss4(L4)}\}$, cf. ).\
& Cell types $y$:\
& - Size $N_y = F_y \sum_{Y} N_Y$, $F_y$ is the occurrence of cell type $y$ in the full model\
& - Morphology $M_y$\
& - Extrapolated according to spatial connectivity data ()\
& Somatic placement, population $Y$:\
& - Random soma placement in cylindrical volumes with radius $r$, thickness $h$\
& - Volumes centered between boundaries of layers 2/3–6\
\
**Type & 3D histological reconstructions from slice preparations (see @Jacobs2009 [@DeSchutter2009]) of cat visual and somatosensory cortices\
**Description & One morphology $M_y$ per cell type:\
& - Excitatory and inhibitory cells in layers 2/3–6\
& - For all cells $j \in y$: $M_j = M_y$\
& - For some cell types $y,y'$: $M_y = M_{y'}$ (limited availability)\
& Orientations:\
& - Pyramidal cells: apical dendrites oriented along depth axis with random depth-axis rotation\
& - Interneurons, stellate cells: random rotation around all axes\
& Corrections:\
& - Apical dendrites of pyramidal cells elongated to accommodate spatial connectivity\
& - Axons removed if present\
& Reconstructed morphologies (cf. ):\
& - Cat visual cortex [@Kisvarday1992; @Mainen1996; @Contreras1997; @Stepanyants2008]\
& - Cat somatosensory cortex from NeuroMorpho.org [@Contreras1997; @Ascoli2007].\
********
[|p[0.15]{}|X|]{} [ ]{}\
**Type & Passive, multicompartment, reconstructed morphologies\
**Description & Compartment $n$ membrane potential $V_{\text{m}jn}$ of cell $j$ having length $l_{jn}$, diameter $d_{jn}$ and surface area $A_{jn}$: $$\begin{aligned}
C_{\text{m}jn} \frac{dV_{\text{m}jn}}{dt} &=& \sum_{k=1}^m I_{\text{a}jkn} - G_{\text{L}jn}(V_{\text{m}jn} - E_\text{L}) - \sum_i I_{jin} ~, \label{eq:cable} \\
C_{\text{m}jn} &=& c_\text{m} A_{jn} ~, \\
I_{\text{a}jkn} &=& G_{\text{a}jkn}\left(V_{\text{m}jk} - V_{\text{m}jn}\right) ~, \\
G_{\text{a}jkn} &=& \pi(d_{jk}^2 + d_{jn}^2) / 4r_\text{a}(l_{jk} + l_{jn})~,\\
G_{\text{L}jn} &=& A_{jn} / r_\text{m} ~, \\
I_{\text{m}jn} &=& C_{\text{m}jn} \frac{dV_{\text{m}jn}}{dt} + G_{\text{L}jn}(V_{\text{m}jn} - E_\text{L}) - \sum_i I_{jin}~. \label{eq:imem}
\end{aligned}$$ $C_{\text{m}jn}$ is compartment capacitance, $G_{\text{L}jn}$ its passive leak conductance, $E_\text{L}$ the passive leak reversal potential, $I_{\text{a}jkn}$ axial current between compartment $n$ and neighboring compartment $k$ (out of $m$ compartments), $G_{\text{a}jkn}$ axial conductance between $n$ and $k$, $I_{jin}$ synaptic currents, and $I_{\text{m}jn}$ transmembrane current of compartment $n$. For specific parameter values, see . Membrane potentials and transmembrane currents are computed using `NEURON` through `LFPy` [@Carnevale2006; @Linden2014], assuming the extracellular potential to be zero everywhere on the outside of the neuron, that is, an infinite extracellular conductivity.\
****
[|p[0.15]{}|X|]{} [ ]{}\
**Type & Exponential postsynaptic current, static weights\
**Description & Neuron $j$ input current of synapse formed with presynaptic neuron $i$: $$\begin{aligned}
I_{ji}(t) &=& I_{ji,\text{max}} \sum_l \exp{(-(t-t^i_l-d_i)/\tau_\text{s})} ~ H(t-t^i_l-d_i) ~, \\
I_{ji,\text{max}} &=& C_\text{m}\,\mu_{YX} ~ \text{of point-neuron network,}\\
H(t) &=& 1 \text{ for } t \geq{0}~, \text{ and } 0 \text{ elsewhere.}
\end{aligned}$$ - Static synaptic weights $J_{ji} = \mu_{YX}$ ($j \in Y$, $i \in X$) (see )\
& - Delays $d_{i}$ from Gaussian distribution with mean $d_{X}$ ($i \in X$), relative standard deviation $\sigma_{d,\text{rel}}$\
& - Synapse activation times: network spike trains plus delay\
& - No cortico-cortical connections: $I^\text{ext} = 0$ (cf. )\
****
[|p[0.15]{}|X|]{} [ ]{}\
**Type & Spike times $t^i_l$ of spiking neuron network (including thalamic input spikes), no cortico-cortical input\
**Description & Synapse placement, postsynaptic cell $j \in y,~y \in Y$ ():\
& - Number of synapses from presynaptic population $X$ in layer $L$: $k_{yXL}$ ()\
& - Compartment specificity of connections: $A_{jn} / \sum_{n \in L}A_{jn}$, compartment $n \in L$\
& - Synapse locations within layers are chosen randomly among dendritic compartments only\
****
[|p[0.15]{}|X|]{} [ ]{}\
**Type & Local field potential (LFP) and current source density (CSD)\
**Description & Laminar multielectrode, see parameter values in :\
& - Axis perpendicular to pial surface\
& - $n_\text{contacts}$: number of contacts\
& - $h_\text{contacts}$: intercontact distance\
& - $r_\text{contact}$: contact surface radius\
****
=0.11cm
[|p[0.10]{}|p[0.20]{}|X|]{} [ ]{}\
**Symbol & **Value & **Description\
$T$ & 5,200 ms & simulation duration\
$dt$ & 0.1 ms & temporal resolution\
******
=0.11cm
[|p[0.1]{}|ccccccccc|X|]{} [ ]{}\
[ ]{}\
**Symbol & & **Description\
$X$ & L23E & L23I & L4E & L4I & L5E & L5I & L6E & L6I & TC & Name\
$N_X$ & 20,683 & 5,834 & 21,915 & 5,479 & 4,850 & 1,065 & 14,395 & 2,948 & 902 & Size\
$k_X^\text{ext}$ & 1,600 & 1,500 & 2,100 & 1,900 & 2,000 & 1,900 & 2,900 & 2,100 & & Ext. in-degree per neuron\
$I^{\text{ext}}$ & $\tau_\text{syn}\nu_\text{bg}J$, & $\nu_\text{bg}=$ &8 Hz&&&&&&& DC ampl. per ext. input\
****
[|p[0.1]{}|XXXXXXXXXX|]{} [ ]{}\
$C_{YX}$ &\
& & L23E & L23I & L4E & L4I & L5E & L5I & L6E & L6I & TC\
to $Y$ & L23E & 0.101 & 0.169 & 0.044 & 0.082 & 0.032 & 0.0 & 0.008 & 0.0 & 0.0\
& L23I & 0.135 & 0.137 & 0.032 & 0.052 & 0.075 & 0.0 & 0.004 & 0.0 & 0.0\
& L4E & 0.008 & 0.006 & 0.050 & 0.135 & 0.007 & 0.0003 & 0.045 & 0.0 & 0.0983\
& L4I & 0.069 & 0.003 & 0.079 & 0.160 & 0.003 & 0.0 & 0.106 & 0.0 & 0.0619\
& L5E & 0.100 & 0.062 & 0.051 & 0.006 & 0.083 & 0.373 & 0.020 & 0.0 & 0.0\
& L5I & 0.055 & 0.027 & 0.026 & 0.002 & 0.060 & 0.316 & 0.009 & 0.0 & 0.0\
& L6E & 0.016 & 0.007 & 0.021 & 0.017 & 0.057 & 0.020 & 0.040 & 0.225 & 0.0512\
& L6I & 0.036 & 0.001 & 0.003 & 0.001 & 0.028 & 0.008 & 0.066 & 0.144 & 0.0196\
[|p[0.1]{}|p[0.20]{}|X|]{} [ ]{}\
**Symbol & **Value & **Description\
$J$ & $87.81$ pA & Reference synaptic strength. All synapse weights are measured in units of $J$.\
$\sigma_{J,\text{rel}} $ & & Relative width of synaptic strength distribution\
& 3 & - for lognormal distribution\
& 0.1 & - for Gaussian distribution\
$g_{YX}$ & & Relative synaptic strength:\
& 1 & $X \in \{\text{TC, L23E, L4E, L5E, L6E}\}, $\
& $-4$ & $X \in \{\text{L23I, L4I, L5I, L6I}\}$, except for\
& 2 & $(X,Y)= (\text{L4E, L23E})$\
& $-4.5$ & $(X,Y)= (\text{L4I, L4E})$\
$d_\text{E}$ & 1.5 ms & Mean excitatory spike transmission delay\
$d_\text{I}$ & 0.75 ms & Mean inhibitory spike transmission delay\
$\sigma_{d,\text{rel}}$ & 0.5 & Relative width (stdev/mean) of transmission delay distributions\
[ ]{}\
**Symbol & **Value & **Description\
$R_\text{m}$ & 40 M$\Omega$ & Membrane resistance\
$C_\text{m}$ & 250 pF & Membrane capacitance\
$\tau_\text{m}$ & $R_\text{m}C_\text{m}$ (10 ms) & Membrane time constant\
$E_\text{L}$ & $-65$ mV & Resting potential\
$\theta$ & $-50$ mV & Fixed firing threshold\
$V_\text{m}(t=0)$ & $[-65, -50]$ mV & Uniformly distributed initial membrane potential\
$V_\text{reset}$ & $E_\text{L}$ & Reset potential\
$\tau_\text{ref}$ & 2 ms & Absolute refractory period\
$\tau_\text{syn}$ & 0.5 ms & Postsynaptic current time constant\
[ ]{}\
**Symbol & **Value & **Description\
$\overline\nu_\text{TC}$& 30 s$^{-1}$& Mean firing rate per thalamocortical neuron\
$ \Delta\nu_\text{TC}$ & 30 s$^{-1}$ & Firing-rate modulation amplitude per thalamocortical neuron\
$f_\text{TC}$ & 15 Hz& Frequency of sinusoidal firing-rate modulation\
******************
[|p[0.1]{}|p[0.20]{}|X|]{} [ ]{}\
**Symbol & **Value & **Description\
$c_\text{m}$ & 1.0 $\mu$Fcm$^{-2}$ & Membrane capacity\
$r_\text{m}$ & $\tau_\text{m} / c_\text{m}$ & Membrane resistivity\
$r_\text{a}$ & 150 $\Omega$cm & Axial resistivity\
$E_\text{L}$ & $E_\text{L}$ & Passive leak reversal potential\
$V_\text{init}$ & $E_\text{L}$ & Membrane potential at $t=0$ ms\
$\lambda_f$ & 100 Hz & Frequency of AC length constant\
$\lambda_d$ & 0.1 & Factor for `d_lambda` rule [@Hines2001]\
$\sigma_\text{e}$ & 0.3 Sm$^{-1}$ & Extracellular conductivity\
$r$ & $\sqrt{1,000^2/\pi} ~\mu$m & Population radius\
$h$ & $50 ~\mu$m & Soma layer thickness\
$n_\text{contact}$ & 16 & Number of electrode contacts\
$h_\text{elec}$ & 100 $\mu$m & Laminar-electrode intercontact distance\
$r_\text{contact}$ & 7.5 $\mu$m & Electrode contact-point radius\
******
[|X|X|X|X|X|]{} [ ]{}\
**Cell type $y$ & **Morphology $M_y$ & **File & **Source & **Online source\
p23 & p23 & oi24rpy1.hoc & [@Kisvarday1992] & \#NMO\_00851 (\#NMO\_10045)\
b23 & i23 & oi38lbc1.hoc & [@Stepanyants2008] & -\
nb23 & i23 & oi38lbc1.hoc & [@Stepanyants2008] & -\
p4 & p4 & oi53rpy1.hoc & [@Kisvarday1992] & \#NMO\_00855 (\#NMO\_10040)\
ss4(L23) & ss4 & j7\_L4ste.hoc & [@Mainen1996] & \#MDB\_2488, \#NMO\_00905\
ss4(L4) & ss4 & j7\_L4ste.hoc & [@Mainen1996] & \#MDB\_2488, \#NMO\_00905\
b4 & i4 & oi26rbc1.hoc & [@Stepanyants2008] & -\
nb4 & i4 & oi26rbc1.hoc & [@Stepanyants2008] & -\
p5(L23) & p5v1 & oi15rpy4.hoc & [@Kisvarday1992] & \#NMO\_00850 (\#NMO\_10046)\
p5(L56) & p5v2 & j4a.hoc & [@Mainen1996] & \#MDB\_2488\
b5 & i5 & oi15rbc1.hoc & [@Stepanyants2008] & -\
nb5 & i5 & oi15rbc1.hoc & [@Stepanyants2008] & -\
p6(L4) & p6 & 51-2a.CN.hoc & [@Contreras1997] & \#NMO\_00879\
p6(L56) & p5v1 & oi15rpy4.hoc &[@Kisvarday1992] & \#NMO\_00850 (\#NMO\_10046)\
b6 & i5 & oi15rbc1.hoc & [@Stepanyants2008] & -\
nb6 & i5 & oi15rbc1.hoc & [@Stepanyants2008] & -\
**********
[|p[0.1]{}|p[0.45]{}|X|]{} [ ]{}\
[ ]{}\
**Symbol & **Description & **Number of recorded units\
$I^{\text{ex}}_i(t)$ & Excitatory synaptic input current of neuron $i$ & $100$ per population $X$\
$I^{\text{in}}_i(t)$ & Inhibitory synaptic input current of neuron $i$ & $100$ per population $X$\
$V_i(t)$ & Somatic voltage of neuron $i$ & $100$ per population $X$\
$s_i(t)$ & Spike train of neuron $i$ & $N_X$ neurons\
$\phi_i(\mathbf{r},t)$ & Single-cell LFP generated by neuron $i$ & $N_X$ neurons\
$\rho_i(\mathbf{r},t)$ & Single-cell CSD generated by neuron $i$ & $N_X$ neurons\
[ ]{}\
**Symbol & **Definition & **Description\
$I_i(t)$ & $I^\text{ex}_i(t) + I^\text{in}_i(t)$ & Total synaptic input current of neuron $i$\
$\overline{I}^{\text{ex}}_X(t)$ & $\frac{1}{n_\text{av}}\sum \limits_{i\in X}^{n_\text{av}} I^\text{ex}_i(t)$ & Average excitatory synaptic input current of population $X$ ($n_\text{av}=100$)\
$\overline{I}^{\text{in}}_X(t)$ & $\frac{1}{n_\text{av}}\sum \limits_{i\in X}^{n_\text{av}} I^\text{in}_i(t)$ & Average inhibitory synaptic input current of population $X$ ($n_\text{av}=100$)\
$\overline{I}_X(t)$ & $\frac{1}{n_\text{av}}\sum \limits_{i\in X}^{n_\text{av}} I_i(t)$ & Average total synaptic input current of population $X$ ($n_\text{av}=100$)\
$\overline{V}_X(t)$ & $\frac{1}{n_\text{av}}\sum \limits_{i\in X}^{n_\text{av}} V_i(t)$ & Average membrane voltage of population $X$ ($n_\text{av}=100$)\
$\nu_X(t)$ & $\frac{n^\text{s}_{X}(t)}{t_\text{bin}}$ & Instantaneous population ($X$) firing rate, with $n^\text{s}_{X}(t)$ being the number of spikes in $[t,t+t_{\text{bin}})$ of all cells in population $X$, $t_{\text{bin}}=1$ ms\
$\overline{\nu}_X(t)$ & $\frac{\nu_X(t)}{N_X}$ & Average instantaneous firing rate of population $X$\
$\phi_X(\mathbf{r},t)$ & $\sum_{i\in X} \phi_i(\mathbf{r},t)$ (cf. & ) & Population LFP of population $X$\
$\rho_X(\mathbf{r},t)$ & $\sum_{i\in X} \rho_i(\mathbf{r},t)$ (cf. & ) & Population CSD of population $X$\
$\phi(\mathbf{r},t)$ & $\sum_{X} \phi_X(\mathbf{r},t)$ & Compound LFP of all cells\
$\rho(\mathbf{r},t)$ & $\sum_{X} \rho_X(\mathbf{r},t)$ & Compound CSD of all cells\
[ ]{}\
$\phi^{\gamma}(\mathbf{r},t)$ & $\sum_{X}\sum_{i\in X^\prime \subset X} \phi_i(\mathbf{r},t)$ & Compound LFP signal from subset of neurons $i \in X^\prime \subset X$ with $N_{X^\prime}=\gamma N_X$, $\gamma \in [0,1]$\
$\phi^{\gamma \xi}(\mathbf{r},t)$ & $\xi \phi^{\gamma}$$(\mathbf{r},t)~\refstepcounter{equation}(\theequation)\label{eq:phi_gamma_xi}$ & Compound LFP signal from subset of neurons $i \in X^\prime \subset X$ with $N_{X^\prime}=\gamma N_X$, $\gamma \in [0,1]$, rescaled by factor $\xi$\
************
[|p[0.15]{}|p[0.45]{}|X|]{} [ ]{}\
**Symbol & **Definition & **Description\
$\psi,\psi^\prime$ & $\psi,\psi^\prime \in \{\phi_i,\phi_X,\phi,\rho_i,\rho_X,\rho,\nu_X\}$ & Signal (LFPs, CSDs, firing rates)\
$\mu_{\psi}(\mathbf{r})$ & $\frac{dt}{T}\sum_{h=1}^{T/dt} \psi(\mathbf{r},h\,dt) $ & Temporal mean of signal $\psi(\mathbf{r},t)$\
$\mathrm{cov}_{\psi\psi^\prime}(\mathbf{r})$ & $\frac{dt}{T}\sum_{h=1}^{T/dt} \psi(\mathbf{r},h\,dt)\psi^\prime(\mathbf{r},h\,dt) - \mu_{\psi}(\mathbf{r})\mu_{\psi^\prime}(\mathbf{r})$& Temporal covariance of signals $\psi(\mathbf{r},t),\psi^\prime(\mathbf{r},t)$\
$\sigma^2_{\psi}(\mathbf{r})$ & $\mathrm{cov}_{\psi\psi}(\mathbf{r})$& Temporal variance of signal $\psi(\mathbf{r},t)$\
$cc_{\psi\psi^\prime}(\mathbf{r})$ & $ \mathrm{cov}_{\psi\psi^\prime}(\mathbf{r})/\sqrt{\sigma^2_{\psi}(\mathbf{r}) \sigma^2_{\psi^\prime}(\mathbf{r})}\hfill\refstepcounter{equation}(\theequation)\label{eq:cc}$ & Zero time-lag correlation coefficient of signals $\psi(\mathbf{r},t),\psi^\prime(\mathbf{r},t)$\
$C_{\psi\psi^\prime}(\mathbf{r},f)$ & $\mathcal{F}[\psi](\mathbf{r},f)^* \mathcal{F}[\psi^\prime](\mathbf{r},f)$ (implemented using Welch’s method) & Pairwise cross-spectral density of signals $\psi(\mathbf{r},t),\psi^\prime(\mathbf{r},t)$, $\mathcal{F}[\psi]$: Fourier transform of $\psi$\
$P_{\psi}(\mathbf{r},f)$ & $C_{\psi\psi}(\mathbf{r},f)$ & Power spectral density (PSD) of signal $\psi(\mathbf{r},t)$\
$\overline{P_{\phi}}(\mathbf{r},f)$ & $\frac{1}{N}\sum \limits_{i=1}^N P_{\phi_i}(\mathbf{r},f)\hfill\refstepcounter{equation}(\theequation)\label{eq:psd_mean_i}$ & Average single-cell LFP power spectrum\
$\overline{C_{\phi}}(\mathbf{r},f)$ & $\frac{1}{N(N-1)}\sum\limits_{i=1}^N\sum \limits_{\substack{j=1 \\ j\neq i}}^{N}
C_{\phi_i \phi_j}(\mathbf{r},f)~\hfill\refstepcounter{equation}(\theequation)\label{eq:mean-cross-spectrum}$ & Average cross-spectrum between single-cell LFPs\
$\overline{\kappa_{\phi}}(\mathbf{r},f)$ & $ \overline{C_{\phi}}(\mathbf{r},f)/\overline{P_{\phi}}(\mathbf{r},f)~\hfill\refstepcounter{equation}(\theequation)\label{eq:coherence}$ & Average LFP coherence between cells\
$P_{\phi}(\mathbf{r},f)$ & $N \overline{P_{\phi}}(\mathbf{r},f)+ N(N-1)\overline{C_{\phi}}(\mathbf{r},f)\hfill\refstepcounter{equation}(\theequation)\label{eq:psd}$ & Power spectral density (PSD) of the compound LFP\
$P^{0}_{\phi}(\mathbf{r},f)$ & $N \overline{P_{\phi}}(\mathbf{r},f)\hfill\refstepcounter{equation}(\theequation)\label{eq:psd_uncorrelated}$ & Power spectral density (PSD) of the compound LFP signal omitting pairwise cross-correlations\
******
[|p[0.15]{}|p[0.25]{}|X|]{} [ ]{}\
[ ]{}\
**Symbol & **Value & **Description\
$T_{\text{trans}}$ & $200\,$ms & Start-up transient\
$dt_{\psi}$ & $1\,$ms & Signal resolution\
[ ]{}\
**Method & `plt.mlab.psd^\ast` & Welch’s average periodogram\
**Symbol & **Value & **Description\
$T_\psi$ & $5{,}000\,$ms & Signal length\
NFFT & $256$ & Number of data points used in each block for the FFT\
Fs & $1\,$kHz & Sampling rate\
noverlap & $128$ & Number of overlapping data points between blocks\
window & `plt.mlab.window_hanning^\ast` & Window filter (`^\ast plt` denote `matplotlib`)\
**************
|
---
abstract: 'A Lagrangian for flat domain walls in spaces with Cartan torsion and electromagnetic fields is proposed.The Lagrangian is very similar to a recently proposed Lagrangian for domain walls in a Chern-Simons electrodynamics in 2+1 dimensions.We show that in the first approximation of the torsion scalar potential the field equations are reduced to a Klein-Gordon type field equation for the torsion potential and the electromagnetic wave equation.A planar symmetric solution representing a parallel plates electric capacitor interacting with the electric field is given.The photon mass is proportional to the torsion potential and in the time dependent case the angular momentum is computed and is shown to be connected with torsion in analogy with the spin-torsion relation which appears in Einstein-Cartan gravity.When the curvature Ricci scalar is introduced we are able to show that the torsion potential can be associated with Higgs massive vectorial bosons.'
---
**[Domain walls and torsion potentials.]{}**
[L.C. Garcia de Andrade[^1]]{}
Recently vortices and domain walls in a Chern-Simons (CS) theory with magnetic moment interactions were investigated by Antillon, Escalona and Torres [@1].In their paper they work in 2+1 dimensional spacetime.Here following recently investigation on non-Riemannian domain walls in space-times with torsion [@2; @3; @4] we worked out avery similar Lagrangian like the one of reference one with the exception that dual field in the (CS) Lagrangian is replaced by torsion and we work in the full four-dimensional spacetime.The equations obtained for the torsion potential and the electromagnetic field are reduced to non-linear Klein-Gordon field equation and to the Proca field equation in the case of the electromagnetic vector potetencial.The controversy concerning the interaction of the electromagnetic fields and torsion [@5; @6; @7] here reachs a slightly different approach since the interaction between photons and torsion appears intermediated by a spin-0 massive boson given by the torsion potential ,besides in the first approximation of the torsion potential the Proca equation is reduced to the Maxwell equation and the photon is massless and the theory is gauge invariant in first approximation.As an application we solve the field equations in the first order in the torsion potential for the case of the parallel plates Capacitor.This simple example is analogous to the case of the Casimir effect [@8] with torsion to be investigated in a future research.Of course throughout the paper the metric is considered to be flat.Let us assume that our Lagrangian is given by $$L=-\frac{1}{4}F^{2}+\frac{1}{2}|D_{\mu}{\phi}|^{2}-V({\phi})
\label{1}$$ where the bars denotes the modulus since we considering that the torsion potential ${\phi}$ is real but the covariant derivative defined by $$D_{\mu}={\partial}_{\mu}-ifA_{\mu}-igS_{\mu}
\label{2}$$ is complex.Here $F^{2}=F_{\mu\nu}F^{\mu\nu}$ is the electromagnetic invariant and $F_{\mu\nu}={\partial}_{\mu}A_{\nu}-{\partial}_{\nu}A_{\mu}$ is the electromagnetic field tensor.The torsion potential generates the torsion vector through the relation $S_{\mu}={\partial}_{\mu}{\phi}$.Substitution of the definition (\[2\]) into the Lagrangian (\[1\]) reduces the previous Lagrangian to $$L=\frac{1}{2}({\partial}_{\mu}{\phi})^{2}-\frac{1}{4}F^{2}-\frac{1}{2}(f^{2}A^{2}+g^{2}S^{2}){\phi}^{2}-V({\phi})
\label{3}$$ or $$L=\frac{1}{2}(1+f^{2})({\partial}{\phi})^{2}+\frac{1}{2}g^{2}A^{2}{\phi}^{2}-\frac{1}{4}F^{2}-V({\phi})
\label{4}$$ Variation of this Lagrangian with respect to torsion potential and the electromagnetic potential vector $A_{\mu}$ yields the following non-linear field equations $${\nabla}^{2}A^{\mu}+\frac{1}{2}g^{2}{\phi}^{2}A^{\mu}=0
\label{5}$$ and $${\partial}^{2}{\phi}-\frac{2f^{2}{\phi}}{(1+f^{2}{\phi}^{2})}
({\partial}{\phi})^{2}=\frac{g^{2}A^{2}{\phi}}{1+f^{2}{\phi}^{2}}
\label{6}$$ where we have supressed the indices and have already considered the domain wall potential $V({\phi})$ as zero since we will not need it in our next application.Notice that the equation (\[5\]) yields a mass for the photon like $ m_{\gamma}=g{\phi} $.Thus the photon mass can be expressed in terms of the torsion potential.This situation is similar to the photon mass dependence on torsion developed some years ago by Sivaram and myself [@9].One should notice that equation (\[4\]) is the Proca equation for the massive photon while the second is a non-linear field equation which reduces to the Klein-Gordon type field equation in the case that the terms quadratic in the torsion potential are dropped out.In this case also we note that the Proca equation is reduced to the massless wave equation which in the time independent case reads $${\nabla}^{2}A^{\mu}=0
\label{7}$$ As an example let us solve these equations inthe static case of a parallel plate capacitor interacting with the scalar torsion field. Since the capacitor electrostatic potential is given by ${\phi}^{el}=qd $ where d is the separation between the plates and q is the electric charge.Here we have taken the Parallel plate orthogonal to the z-coordinate axis,the Klein-Gordon-Cartan equation is $${\phi}"-g^{2}d^{2}q^{2}{\phi}=0
\label{8}$$ where the double lines denote second derivative in $z$.To solve this Klein-Gordon field equation for torsion potential,let us use the following ansatz $${\phi}=e^{-{\alpha}z}
\label{9}$$ Since we assume that the torsion field vanishes at infinity. Substitution of (\[9\]) into (\[8\]) yields $${\alpha}=gqd
\label{10}$$ which yields the following solution for the torsion potential ${\phi}=e^{-gqdz}$ which displays the coupling between the electric charge and the torsion coupling constant.In the time independent case we have an expression for the electromagnetic vector potential in terms of the scalar field as $${A}^{0}=\frac{({\nabla}.{S})^{\frac{1}{2}}{( 1+f^{2}{\phi}^{2})}^{\frac{1}{2}}}{g{\phi}}
\label{11}$$ From this expression it is very easy to compute the electrostatic field as $$E=-{\nabla}A^{0}=-({\nabla}.S)^{\frac{1}{2}}\frac{(1-\frac{1}{2}{\phi})}{g{\phi}^{2}}
\label{12}$$ where to simplify matters we have considered that the divergence of the torsion is constant.In the static case the energy can be computed from the Lagrangian above as $$T_{\mu\nu}=(1+f^{2}){\partial}_{\mu}{\phi}{\partial}_{\nu}{\phi}-{\eta}_{\mu\nu}L
\label{13}$$ The energy is $${\epsilon}=\frac{1}{4}(E^{2}-B^{2})-\frac{1}{2}g^{2}A^{2}{\phi}^{2}
\label{14}$$ In the time dependent case a relation between the angular momentum of the scalar field and torsion is obtained by using the definition of the angular momentum $$J_{ij}={\int}(T_{0i}x_{j}-T_{0j}x_{i})d^{4}x
\label{15}$$ By making use of equation (\[13\]) and the definition of the torsion as the gradient of the scalar field yields $$J_{ij}=(1+f^{2}){\int}S_{0}(S_{i}x_{j}-S_{j}x_{i})d^{4}x
\label{16}$$ One must notice that this expression is similar to the relation between spin and torsion which appears in Einstein-Cartan gravity and to the 2+1 Kerr solution discovered by Jackiw [@11]. It is interesting to note that when the curvature scalar is introduced into the Lagrangian one is able to show that the torsion potential is in fact a massive Higgs field.This can be show by considering the following lagrangian $$L=R(S)-m^{2}{\phi}^{2}-V({\phi})
\label{17}$$ where in considering just the flat metric terms and remiding that the Ricci scalar is given by $R(S)={\partial}S+S^{2}$ where S represents the torsion vector,the Lagrangian (\[13\]) reduces to $$L={\partial}^{2}{\phi}-({\partial}{\phi})^{2}-m^{2}{\phi}^{2}-V({\phi})
\label{18}$$ The Euler-Lagrange equation for this Lagrangian yields $${\partial}_{\mu}{\partial}^{\mu}{\phi}+m^{2}{\phi}=-\frac{{\partial}V}{{\partial}{\phi}}
\label{19}$$ Which is the equation for a massive Higgs field given by the torsion potential.Recently in a similar approach Kleinert [@10] discussed the spontaneous symmetry breaking for the electroweak bosons induced by the torsion potential,nevertheless in his case the Higgs potential did not coincide with the torsion potential.A more detailed investigation of our model as well a investigation of the Casimir effect in spacetimes with torsion may appear elsewhere.
Acknowledgements {#acknowledgements .unnumbered}
================
I am very much indebt to Professor A.Wang,I.D. Soares and H.Kleinert for helpful discussions on the subject of this paper. Thanks are also due to Universidade do Estado do Rio de Janeiro (UERJ) for financial Support.
[11]{} A.Antillon,J.Escalona and M.Torres,Phys.Rev.D55,10,(1997),6327. L.C.Garcia de Andrade,J.Math.Phys.32,(1998)372. L.C.Garcia de Andrade,Gen.Rel.and Grav.30,11,(1998)1629. L.C.Garcia de Andrade,Mod.Phys.Lett.A27,(1997),2005. A.Unzicker,Teleparallel Space-time with Defects yields Geometrization of Electrodynamics with quantized charges,Los Alamos Electronic archives,gr-qc/9612061. L.C.Garcia de Andrade,Gen.Rel.and Grav,(1990),8,882. R.A.Puntigam,C.Lammerzhal and F.W.Hehl,Class.and Quantum Grav.14,5(1997),1347. Mostepanenko,The Casimir Effect. L.C.Garcia de Andrade and C.Sivaram,Spontaneous Symmetry Breaking in massive electrodynamics induced by torsion,dft-UERJ internal reports(1992). H.Kleinert,Spontaneous Generation of Torsion Coupling of Electroweak Massive Gauge Bosons,(1998),Institut fur theoretische Physik,Freie University Berlin. R.Jackiw in SILARG VII-Relativity and Gravitation:Classical and Quantum,(1990),Simposio Latino Americano de Relatividade Geral e Gravitacion,World Scientific.
[^1]: Departamento de Física Teorica - UERJ. Rua São Fco. Xavier 524, Rio de Janeiro, RJ Maracanã, CEP:20550-003 , Brasil. [E-Mail.: GARCIA@SYMBCOMP.UERJ.BR]{}
|
---
abstract: 'Monte Carlo methods are essential tools for Bayesian inference. Gibbs sampling is a well-known Markov chain Monte Carlo (MCMC) algorithm, extensively used in signal processing, machine learning, and statistics, employed to draw samples from complicated high-dimensional posterior distributions. The key point for the successful application of the Gibbs sampler is the ability to draw efficiently samples from the full-conditional probability density functions. Since in the general case this is not possible, in order to speed up the convergence of the chain, it is required to generate auxiliary samples whose information is eventually disregarded. In this work, we show that these auxiliary samples can be recycled within the Gibbs estimators, improving their efficiency with no extra cost. This novel scheme arises naturally after pointing out the relationship between the standard Gibbs sampler and the chain rule used for sampling purposes. Numerical simulations involving simple and real inference problems confirm the excellent performance of the proposed scheme in terms of accuracy and computational efficiency. In particular we give empirical evidence of performance in a toy example, inference of Gaussian processes hyperparameters, and learning dependence graphs through regression. [ **Keywords:**]{} Bayesian inference, Markov Chain Monte Carlo (MCMC), Gibbs sampling, Metropolis within Gibbs, Gaussian Processes (GP), automatic relevance determination (ARD).'
author:
- |
L. Martino$^{\star\Diamond}$, V. Elvira$^\top$, G. Camps-Valls$^\star$\
[$^\star$ Image Processing Laboratory (IPL), Universitat de Valencia, (Spain). ]{}\
[$^\top$ Télécom ParisTech, Université Paris-Saclay. (France),]{}\
[$^\Diamond$ Universidad Carlos III de Madrid, Leganés (Spain).]{}
bibliography:
- 'References.bib'
title: The Recycling Gibbs Sampler for Efficient Learning
---
Introduction
============
[*‘Reduce, Reuse, Recycle’*]{}\
[*The Greenpeace motto*]{}
Monte Carlo algorithms have become very popular over the last decades [@Liu04b; @Robert04]. Many practical problems in statistical signal processing, machine learning and statistics, demand fast and accurate procedures for drawing samples from probability distributions that exhibit arbitrary, non-standard forms [@Andrieu2003; @Fitzgerald01; @ReadLuca2014], [@Bishop Chapter 11]. One of the most popular Monte Carlo methods are the families of Markov chain Monte Carlo (MCMC) algorithms [@Andrieu2003; @Robert04] and particle filters [@Bugallo07; @Djuric03]. The MCMC techniques generate a Markov chain (i.e., a sequence of correlated samples) with a pre-established target probability density function (pdf) as invariant density [@Liu04b; @Liang10]. The Gibbs sampling technique is a well-known MCMC algorithm, extensively used in the literature in order to generate samples from multivariate target densities, drawing each component of the samples from the full-conditional densities [@Chen16; @Koch07; @Kotecha99; @Goudie16; @Lucka16; @Zhang16].[^1] In order to draw samples from a multivariate target distribution, the key point for the successful application of the standard Gibbs sampler is the ability to draw efficiently from the univariate conditional pdfs [@Liu04b; @Robert04]. The best scenario for Gibbs sampling occurs when specific direct samplers are available for each full-conditional, e.g. inversion method or, more generally, some transformation of a random variable [@Devroye86; @Robert04]. Otherwise, other Monte Carlo techniques, such as rejection sampling (RS) and different flavors of the Metropolis-Hastings (MH) algorithms, are typically used [*within*]{} the Gibbs sampler to draw from the complicated full-conditionals. The performance of the resulting Gibbs sampler depends on the employed [*internal*]{} technique, as pointed out for instance in [@Cai08; @Gilks95; @MartinoA2RMS; @FUSS].
In this context, some authors have suggested to use more steps of the MH method within the Gibbs sampler [@Muller91; @Gelfand93; @Fox12]. Moreover, other different algorithms have been proposed as alternatives to the MH technique [@Cai08; @Koch07; @Shao13]. For instance, several automatic and self-tuning samplers have been designed to be used primarily [*within-Gibbs*]{}: the adaptive rejection sampling (ARS) [@Gilks92derfree; @Gilks92], the griddy Gibbs sampler [@ritter1992griddyGibbs], the FUSS sampler [@FUSS], the Adaptive Rejection Metropolis Sampling (ARMS) method [@Gilks95; @CorrGilks97; @Meyer08; @Zhang16], and the Independent Doubly Adaptive Rejection Metropolis Sampling (IA$^2$RMS) technique [@MartinoA2RMS], just to name a few.
Most of the previous solutions require performing several MCMC steps for each full-conditional in order to improve the performance, although only one of them is considered to produce the resulting Markov chain because the rest of samples play the mere role of auxiliary variables. Strikingly, they require an increase in the computational cost that is not completely paid off: several samples are drawn from the full-conditionals, but only a subset of these generated samples is employed in the final estimators. In this work, we show that the rest of generated samples can be directly incorporated within the corresponding Gibbs estimator. We call this approach the [*Recycling Gibbs (RG) sampler*]{} since [*all*]{} the samples drawn from each full-conditional can be used also to provide a better estimation, instead of discarding them.
The consistency of the proposed RG estimators is guaranteed, as will be noted after considering the connection between the Gibbs scheme and the chain rule for sampling purposes [@Devroye86; @Robert04]. In particular, we show that the standard Gibbs approach is equivalent (after the burn-in period) to the standard chain rule, whereas RG is equivalent to an alternative version of the chain rule presented in this work as well. RG fits particularly well combined with adaptive MCMC schemes where different internal steps are performed also for adapting the proposal density, see e.g. [@Gilks95; @MartinoA2RMS; @Meyer08; @Zhang16]. The novel RG scheme allows us to obtain better performance without adding any extra computational cost. This will be shown through intensive numerical simulations. First, we test RG in a simple toy example with a bimodal bivariate target. We also include experiments for hyper-parameter estimation in Gaussian Processes (GPs) regression problems with the so-called [*automatic relevance determination*]{} (ARD) kernel function [@Bishop]. Finally, we apply the novel scheme in real-life geoscience problems of dependence estimation among bio-geo-physical variables from satellite sensory data. The MATLAB code of the numerical examples is provided at <http://isp.uv.es/code/RG.zip>.
The remainder of the paper is organized as follows. Section \[BaySect\] fixes notation and recalls the problem statement of Bayesian inference. The standard Gibbs sampler and the chain rule for sampling purposes are summarized in Section \[SGsect\], highlighting their connections. In the same section, we then introduce an alternative chain rule approach, which is useful for describing the novel scheme. The RG technique proposed here is formalized in Section \[NovelSect\]. Sections \[SIMU\] provides empirical evidence of the benefits of the proposed scheme, considering different multivariate posterior distributions. Finally, Section \[ConclSect\] concludes and outlines further work.
Bayesian inference {#BaySect}
==================
Machine learning, statistics, and signal processing often face the problem of inference through density sampling of potentially complex multivariate distributions. In particular, Bayesian inference is concerned about doing inference about a variable of interest exploiting the Bayes’ theorem to update the probability estimates according to the available information. Specifically, in many applications, the goal is to infer a variable of interest, ${\bf x}=[x_1,\ldots,x_{D}]\in \mathbb{R}^{D}$, given a set of observations or measurements, ${\bf y}\in \mathbb{R}^{P}$. In Bayesian inference all the statistical information is summarized by means of the posterior pdf, i.e., $$\bar{\pi}({\bf x})= p({\bf x}| {\bf y})= \frac{\ell({\bf y}|{\bf x}) g({\bf x})}{Z({\bf y})},
\label{eq:posterior}$$ where $\ell({\bf y}|{\bf x})$ is the likelihood function, $g({\bf x})$ is the prior pdf and $Z({\bf y})$ is the marginal likelihood (a.k.a., Bayesian evidence). In general, $Z({\bf y})$ is unknown and difficult to estimate in general, so we assume to be able to evaluate the unnormalized target function, $$\pi({\bf x})=\ell({\bf y}|{\bf x}) g({\bf x}).
\label{eq:target}$$ The analytical study of the posterior density $\bar{\pi}({\bf x}) \propto \pi({\bf x})$ is often unfeasible and integrals involving $\bar{\pi}({\bf x})$ are typically intractable. For instance, one might be interested in the estimation of $$\label{MainInt}
I=\int_{\mathbb{R}^D} f({\bf x})\bar{\pi}({\bf x}) d{\bf x},$$ where $f({\bf x})$ is a squared integrable function (with respect to $\bar{\pi}$). In order to compute the integral $I$ numerical approximations are typically required. Our goal here is to approximate this integral by using Monte Carlo (MC) quadrature [@Liu04b; @Robert04]. Namely, considering $T$ independent samples from the posterior target pdf, i.e., ${\bf x}^{(1)},\ldots,{\bf x}^{(T)} \sim \bar{\pi}({\bf x})$, we can write $${\widehat I}_T=\frac{1}{T} \sum_{t=1}^T f({\bf x}^{(t)})
\overset{p}{\longrightarrow} I.
%\xrightarrow[]{p}$$ This means that for the weak law of large numbers, ${\widehat I}_T$ converges in probability to $I$: that is, for any positive number $\epsilon>0$, we have $\lim\nolimits_{T\rightarrow \infty}\mbox{Pr}( |{\widehat I}_T-I |> \epsilon)=0 $. In general, a direct method for drawing independent samples from $\bar{\pi}({\bf x})$ is not available, and alternative approaches, e.g., MCMC algorithms, are needed. An MCMC method generates an ergodic Markov chain with invariant density $ \bar{\pi}({\bf x})$ (a.k.a., stationary pdf). Even though, the generated samples $\{{\bf x}^{(1)},\ldots,{\bf x}^{(T)}\}$ are then correlated in this case, ${\widehat I}_T$ is still a consistent estimator.
Within the MCMC framework, we can consider a block approach working directly into the $D$-dimensional space, e.g., using a Metropolis-Hastings (MH) algorithm [@Robert04], or a component-wise approach [@HaarioCW; @Johnson13; @Levine05] working iteratively in different uni-dimensional slices of the entire space, e.g., using a Gibbs sampler [@Liu04b; @Liang10].[^2] In many applications, and for different reasons, the component-wise approach is the preferred choice. For instance, this is the case when the full-conditional distributions are directly provided or when the probability of accepting a new state with a complete block approach becomes negligible as the dimension of the problem $D$ increases. In the following section, we outline the standard Gibbs approach, and remark its connection with the chain rule method. The main notation and acronyms of the work are summarized in Table \[tab:notation\].
------------------------------------- ------------------------------ ------------ ---------------------------
$D$
$T$
$M$
$t_b$
${\bf x}$
${\bf y}$
${\bf x}_{\neg d}$
$x_{a:b}$
$\bar{\pi}({\bf x})$
$\pi({\bf x})$
$\bar{\pi}_d(x_d|{\bf x}_{\neg d})$ $d$-th full-conditional pdf. $p_d(x_d)$ $d$-th marginal pdf.
SG Standard Gibbs. TRG Trivial Recycling Gibbs.
MH Metropolis-Hastings. MRG Multiple Recycling Gibbs.
------------------------------------- ------------------------------ ------------ ---------------------------
: Main notation and acronyms of the work.
\[tab:notation\]
Gibbs sampling and the chain rule method {#SGsect}
========================================
This section reviews the fundamentals about the standard Gibbs sampler, reviews the recent literature on Gibbs sampling when complicated full-conditional pdfs are involved, and points out the connection between GS and the chain rule. A variant of the chain rule is also described, which is related to the novel scheme introduced in the next section.
The Standard Gibbs (SG) sampler
-------------------------------
The Gibbs sampler is perhaps the most widely used algorithm for inference in statistics and machine learning [@Chen16; @Koch07; @Goudie16; @Robert04]. Let us define ${\bf x}_{\neg d} := [x_1,\ldots,x_{d-1},x_{d+1},\ldots,x_D]$ and introduce the following equivalent notations $$\begin{aligned}
{\bar \pi}_d(x_d|x_1,\ldots,x_{d-1},x_{d+1},\ldots,x_D) = {\bar \pi}_d(x_d|x_{1:d-1},x_{d+1:D}) = {\bar \pi}_d(x_d|{\bf x}_{\neg d}).
\end{aligned}$$ In order to denote the unidimensional full-conditional pdf of the component $x_d\in \mathbb{R}$, $d\in\{1,\ldots,D\}$, given the rest of variables ${\bf x}_{\neg d}$, i.e. $$\begin{aligned}
{\bar \pi}_d(x_d|{\bf x}_{\neg d}) = \frac{\bar{\pi}({\bf x})}{\bar{\pi}_{\neg d}({\bf x}_{\neg d})} = \frac{\bar{\pi}({\bf x})}{\int_{\mathbb{R}} \bar{\pi}({\bf x}) dx_{d}}.
\end{aligned}$$ The density $\bar{\pi}_{\neg d}({\bf x}_{\neg d})=\int_{\mathbb{R}} \bar{\pi}({\bf x}) dx_{d}$ is the joint pdf of all variables but $x_d$. The Gibbs algorithm generates a sequence of $T$ samples, and is formed by the steps in Algorithm \[alg:gibbs\]. Note that the main assumption for the application of Gibbs sampling is being able to draw efficiently from these univariate full-conditional pdfs $\bar{\pi}_d$. However, in general, we are not able to draw directly from any arbitrary full-conditional pdf. Thus, other Monte Carlo techniques are needed for drawing from all the $\bar{\pi}_d$.
- For $t=1,\ldots,T$:
1. For $d=1,\ldots,D$:
1. Draw $x_d^{(t)}\sim\bar{\pi}_d(x_d|x_{1:d-1}^{(t)},x_{d+1:D}^{(t-1)})$.
2. Set ${\bf x}^{(t)}=[x_1^{(t)},x_2^{(t)},\ldots,x_D^{(t)}]$.
Monte Carlo-within-Gibbs sampling schemes
-----------------------------------------
In many cases, drawing directly from the full-conditional pdf is not possible, hence the use of another Monte Carlo scheme is needed. Figure \[FigMonteCarloGibbs\] summarizes the main techniques proposed in literature for this purpose. In some specific situations, rejection samplers [@Caffo02; @Hormann02; @Hormann07; @Marrelec04; @Tanizaki99] and their adaptive version, as the [*adaptive rejection sampler*]{} (ARS) [@Gilks92], are employed to generate one sample from each $\bar{\pi}_d$ per iteration. Since the standard ARS can be applied only to log-concave densities, several extensions have been introduced [@Hoermann95; @Gorur08rev; @MartinoStatCo10]. Other variants or improvements of the standard ARS scheme can be found [@PARS; @CARS]. The ARS algorithms are very appealing techniques since they construct a non-parametric proposal to mimic the shape of the target pdf, yielding in general excellent performance (i.e., independent samples from $\bar{\pi}_d$ with a high acceptance rate).
![Summary of the main Monte Carlo algorithms which have been employed within the Gibbs sampling technique. []{data-label="FigMonteCarloGibbs"}](FigLOCA.pdf){width="15.4cm"}
However, the range of application of the ARS samplers is limited to some specific classes of densities. Thus, in general, other approaches are required. For instance in [@Koch07] an approximated strategy is used, considering the application of the importance sampling (IS) scheme within the Gibbs sampler. A more common approach is to apply an additional MCMC sampler to draw samples from $\bar{\pi}_d$ [@Gelfand93]. Therefore, in many practical scenarios, we have an MCMC (e.g., an MH sampler) inside another MCMC scheme (i.e., the Gibbs sampler) as shown in Figures \[FigMonteCarloGibbs\]-\[FigMCMCGibbs\]. In the so-called [*MH-within-Gibbs*]{} approach[^3], only one MH step is often performed within each Gibbs iteration to draw samples from each full-conditional. This hybrid approach preserves the ergodicity of the Gibbs sampler [@Robert04 Chapter 10], and provides good performance in many cases. However, several authors have noted that using a single MH step for the internal MCMC is not always the best solution in terms of performance, c.f. [@Brewer93].
![Graphical representation of a generic MCMC-within-Gibbs scheme, where $M$ steps of the internal MCMC algorithm are applied for each full-conditional pdf (see Algorithm \[tab:MCMC\_GIBBS\]). Different internal MCMC methods have been proposed in literature. []{data-label="FigMCMCGibbs"}](MCMCwithinGibbs.pdf){height="5.5cm"}
Using a larger number of iterations of the MH algorithm within-Gibbs can improve the performance [@Muller91; @Gelfand93; @Fox12]. This is the scenario graphically represented in Figure \[FigMCMCGibbs\]. Moreover, different more sophisticated MCMC algorithms to be applied within-Gibbs have been proposed [@Gilks95; @Cai08; @HaarioCW; @ritter1992griddyGibbs]. Some of these techniques employ an automatic construction of the proposal density tailored to the specific full-conditional [@ritter1992griddyGibbs; @Shao13; @FUSS]. Other methods use an adaptive parametric proposal pdf [@HaarioCW; @Levine05], while other ones employ adaptive non-parametric proposals [@Gilks95; @Meyer08; @MartinoA2RMS; @Sticky13], in the same fashion of the ARS schemes. It is important to remark here that performing more steps of an adaptive MH method within a Gibbs sampler can provide better results than a longer Gibbs chain applying only one step of a standard MH method [@Gilks95]. Algorithm \[tab:MCMC\_GIBBS\] describes a generic MCMC-within-Gibbs sampler considering $M$ steps of the internal MCMC at each Gibbs iteration.
While these algorithms are specifically designed to be applied “within-Gibbs” and provide very good performance, they still require an increase in the computational cost that is not completely exploited: several samples are drawn from the full-conditionals and used to adapt the proposal pdf, but only a subset of them is employed within the resulting Gibbs estimator. In this work, we show how they can be incorporated within the corresponding Gibbs estimator to improve performance without jeopardizing its consistency. In the following, we show the relationships between the standard Gibbs scheme and the chain rule. Then, we describe an alternative formulation of the chain rule useful for introducing the novel Gibbs approach described in Section \[NovelSect\].
Perform $M$ steps of an MCMC algorithm with initial state $v_{d,0}^{(t)}=x_d^{(t-1)}$, and target pdf $\bar{\pi}_d(x_d|x_{1:d-1}^{(t)},x_{d+1:D}^{(t-1)})$, yielding the sequence of samples $v_{d,1}^{(t)},\ldots, v_{d,M}^{(t)}$. Set $x_d^{(t)}=v_{d,M}^{(t)}$. Set ${\bf x}^{(t)}=x_{1:D}^{(t)}=[x_1^{(t)},x_2^{(t)},\ldots,x_D^{(t)}]$. Return $\{{\bf x}^{(1)},\ldots,{\bf x}^{(T)}\}$
Chain rule and the connection with Gibbs sampling {#EstoAltChainRule0}
-------------------------------------------------
Let us highlight an important consideration for the derivation of the novel Gibbs approach we will introduce in the following section. For the sake of simplicity, let us consider a bivariate target pdf that can be factorized according to the chain rule, $$\begin{aligned}
{\bar \pi}(x_1,x_2)&=&{\bar \pi}_2(x_2|x_1)p_1(x_1) \\
&=&{\bar \pi}_1(x_1|x_2)p_2(x_2), \end{aligned}$$ where we have denoted with $p_1$, $p_2$, the marginal pdfs of $x_1$ and $\bar\pi_2$, $\bar\pi_1$, are the conditional pdfs. Let us consider the first equality. Clearly, if we are able to draw from the marginal pdf $p_1(x_1)$ and from the conditional pdf ${\bar \pi}_2(x_2|x_1)$, we can draw samples from ${\bar \pi}(x_1,x_2)$ following the chain rule procedure in Algorithm \[alg:gibbs2\]. Note that, consequently, the $T$ independent random vectors $[x_1^{(t)},x_2^{(t)}]$, with $t=1,\ldots,T$, are all distributed as ${\bar \pi}(x_1,x_2)$.
### Standard Gibbs sampler as the chain rule
Let us consider again the previous bivariate case where the target pdf is factorized as ${\bar \pi}({\bf x})={\bar \pi}(x_1,x_2)$. The standard Gibbs sampler in this bivariate case consists of the steps in Algorithm \[alg:gibbs3\]. After the burn-in period, the chain converges to the target pdf, i.e., ${\bf x}^{(t)} \sim {\bar \pi}({\bf x})$. Therefore, recalling that ${\bar \pi}(x_1,x_2)={\bar \pi}_2(x_2|x_1)p_1(x_1)={\bar \pi}_1(x_1|x_2)p_2(x_2)$ for $t \geq t_b$, each component of the vector ${\bf x}^{(t)}=[x_1^{(t)},x_2^{(t)}]$ is distributed as the corresponding marginal pdf, i.e., $x_1^{(t)} \sim p_1(x_1)$ and $x_2^{(t)} \sim p_2(x_2)$. Therefore, after $t_b$ iterations, the standard Gibbs sampler can be interpreted as the application of the chain rule procedure in Algorithm \[alg:gibbs2\]. Namely, for $t \geq t_b$, Algorithm \[alg:gibbs3\] is equivalent to generate $x_1^{(t)}\sim p_1(x_1)$, and then draw $x_2^{(t)}\sim\bar{\pi}_1(x_2|x_1^{(t)})$.
Draw $x_2^{(t)}\sim\bar{\pi}_2(x_2|x_1^{(t-1)})$. Draw $x_1^{(t)}\sim\bar{\pi}_1(x_1|x_2^{(t)})$. Set ${\bf x}^{(t)}=[x_1^{(t)},x_2^{(t)}]$.
### Alternative chain rule procedure {#EstoAltChainRule}
An alternative procedure is shown in Algorithm \[alg:chain1\]. This chain rule draws $M$ samples from the full conditional $\bar{\pi}_2(x_2|x_1)$ at each $t$-th iteration, and generates samples from the joint pdf $\bar{\pi}(x_1,x_2)$.
Draw $x_1^{(t)}\sim p_1(x_1)$. Draw $x_{2,m}^{(t)}\sim\bar{\pi}_2(x_2|x_1^{(t)})$, with $m=1,\ldots.M$.
Note that all the $TM$ vectors, $[x_{1}^{(t)},x_{2,m}^{(t)}]$, with $t=1,\ldots,T$ and $m=1,\ldots,M$, are samples from ${\bar \pi}(x_1,x_2)$. This scheme is valid and, in some cases, can present some benefits w.r.t. the traditional scheme in terms of performance, depending on some characteristics contained in the joint pdf ${\bar \pi}(x_1,x_2)$. For instance, the correlation between variables $x_1$ and $x_2$, and the variances of the marginal pdfs $p_1(x_1)$ and $p_2(x_2)$. Figure \[FigChainRule\] shows the graphical representation of the standard chain rule sampling scheme (with $T=3$ and $M=1$), and the alternative chain rule sampling procedure described before (with $T=3$, $M=4$).
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
\(a) Standard sampling \(b) Alternative sampling
![Graphical representation of the (a) standard chain rule sampling ($M=1$), and (b) the alternative chain rule sampling ($M=4$). In both cases, $N=3$. The total number of drawn vectors $[x_1^{(t)},x_{2,m}^{(t)}] \sim {\bar \pi}(x_1,x_2)={\bar \pi}_2(x_2|x_1)p_1(x_1)$ is $NM=3$ and $NM=12$, respectively.[]{data-label="FigChainRule"}](FigStandCR.pdf "fig:"){height="3cm"} ![Graphical representation of the (a) standard chain rule sampling ($M=1$), and (b) the alternative chain rule sampling ($M=4$). In both cases, $N=3$. The total number of drawn vectors $[x_1^{(t)},x_{2,m}^{(t)}] \sim {\bar \pi}(x_1,x_2)={\bar \pi}_2(x_2|x_1)p_1(x_1)$ is $NM=3$ and $NM=12$, respectively.[]{data-label="FigChainRule"}](FigAltCR.pdf "fig:"){height="3cm"}
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
At this point, a natural question arises: is it possible to design a Gibbs sampling scheme equivalent to the alternative chain rule scheme described before? In the next section, we introduce the Multiple Recycling Gibbs Sampler (MRG), which corresponds to the alternative chain rule procedure, as summarized in Fig. \[FigTeo\].
![Graphical representation of the relationships between chain rules and Gibbs schemes.[]{data-label="FigTeo"}](FigTeoGibbsChain.pdf){width="60.00000%"}
The Recycling Gibbs sampler {#NovelSect}
===========================
The previous considerations suggest that we can benefit from some previous intermediate points produced in the Gibbs procedure. More specifically, let us consider the following [*Trivial Recycling Gibbs*]{} (TRG) procedure in Algorithm \[alg:trg1\].
Draw $x_d^{(t)}\sim\bar{\pi}_d(x_d|x_{1:d-1}^{(t)},x_{d+1:D}^{(t-1)})$. Set ${\bf x}_d^{(t)}=[x_{1:d-1}^{(t)},x_d^{(t)},x_{d+1:D}^{(t-1)}]=[x_{1:d}^{(t)},x_{d+1:D}^{(t-1)}]$.
The procedure generates $DT$ samples ${\bf x}_d^{(t)}$, with $d=1,\ldots,D$ and $t=1,\ldots,T$, shown in Figure \[FigTodo\](b) with circles and squares. Note that if we consider only the subset of generated vectors $${\bf x}_{D}^{(t)}, \quad t=1,\ldots, T,$$ by setting $d=D$, we obtain the outputs of the standard Gibbs (SG) sampler approach in Algorithm \[alg:gibbs\]. Namely, the samples generated by a SG procedure can be obtained by subsampling the samples obtained by the proposed RG. Figure \[FigTodo\](a) depicts with circles $T+1$ vectors (considering also the starting point) corresponding to a run of SG with $T=4$. Figure \[FigTodo\](b) shows with squares the additional points used in TRG.
Let us consider the estimation by SG and TRG of a generic moment, i.e., given a function $f(x_d)$, of $d$-th marginal density, i.e., $p_d(x_d)=\int_{\mathbb{R}^{D-1}} \bar{\pi}({\bf x}) d{\bf x}_{\neg d} $. After a closer inspection, we note that both estimators corresponding to the SG and TRG coincide: $$\begin{aligned}
\int_{\mathbb{R}^{D}} f(x_d) \bar{\pi}({\bf x}) d{\bf x}=\int_{\mathbb{R}} f(x_d) p_d(x_d) dx_d\approx \frac{1}{DT} \sum_{t=1}^{T} \sum_{d=1}^{D} f(x_{d}^{(t)})=\frac{1}{T} \sum_{t=1}^{T} f(x_{D}^{(t)}), \end{aligned}$$ where for the last equality we are assuming (for the sake of simplicity) that the $d$-th component is the second variable in the Gibbs scan and $T=kD$, $k\in \mathbb{N}$. This is due to the fact that, in TRG, each component $x_{d}^{(t)}$ is repeated exactly $D$ times (inside different consecutive samples) and we have $D$ times more samples in TRG than in a standard SG. Hence, in such situation, there are no apparent advantages of using TRG w.r.t. a SG approach. Namely, TRG and SG are equivalent schemes in the approximation of the marginal densities (we remark that the expression above is valid only for marginal moments). The advantages of a RG strategy appear clear when more than one sample is drawn from the full-conditional, $M>1$, as discussed below.
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
\(a) SG \(b) TRG \(c) MRG
![We consider $T=4$ iterations of a Gibbs sampler and $M=5$ iterations of the MH for drawing from each full-conditional pdfs. [**(a)**]{} With the circles we denote the $T+1$ points (considering the starting point) used in the standard Gibbs estimators. [**(b)**]{} The vectors (denoted with circles and squares) used in the TRG estimators. [**(c)**]{} The vectors (denoted with circles, squares and diamonds) used in the MRG estimators.[]{data-label="FigTodo"}](FigStandG.pdf "fig:"){height="3.8cm"} ![We consider $T=4$ iterations of a Gibbs sampler and $M=5$ iterations of the MH for drawing from each full-conditional pdfs. [**(a)**]{} With the circles we denote the $T+1$ points (considering the starting point) used in the standard Gibbs estimators. [**(b)**]{} The vectors (denoted with circles and squares) used in the TRG estimators. [**(c)**]{} The vectors (denoted with circles, squares and diamonds) used in the MRG estimators.[]{data-label="FigTodo"}](FigRG.pdf "fig:"){height="3.8cm"} ![We consider $T=4$ iterations of a Gibbs sampler and $M=5$ iterations of the MH for drawing from each full-conditional pdfs. [**(a)**]{} With the circles we denote the $T+1$ points (considering the starting point) used in the standard Gibbs estimators. [**(b)**]{} The vectors (denoted with circles and squares) used in the TRG estimators. [**(c)**]{} The vectors (denoted with circles, squares and diamonds) used in the MRG estimators.[]{data-label="FigTodo"}](FigConTodoPrueba.pdf "fig:"){height="3.8cm"}
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Based on the previous considerations, we design the [*Multiple Recycling Gibbs*]{} (MRG) sampler which draws $M>1$ samples from each full conditional pdf, as shown in Algorithm \[tab:MRG\]. Figure \[FigTodo\](c) shows all the samples (denoted with circles, squares and diamonds) used in the MRG estimators. Thus, given a specific function $f({\bf x})$ in the integral in Eq. , the MRG estimator is eventually formed by $TDM$ samples, without removing any burn-in period, $$\widehat{I}_T=\frac{1}{TDM} \sum_{t=1}^{T} \sum_{d=1}^{D} \sum_{m=1}^{M} f({\bf x}_{d,m}^{(t)}).$$ Observe that in order to go forward to sampling from the next full-conditional, we only consider the last generated component, i.e., $z_d^{(t)}=x_{d,M}^{(t)}$. However, an alternative to step \[ThisStep\] of Algorithm \[tab:MRG\] is: (a) draw $j\sim \mathcal{U}(1,\ldots,M)$ and (b) set $z_d^{(t)}=x_{d,j}^{(t)}$. Note that choosing the last sample $x_{d,M}^{(t)}$ is more convenient for an MCMC-within-MRG scheme.
Choose a starting point $[z_1^{(0)},\ldots,z_D^{(0)}]$. Draw $x_{d,m}^{(t)}\sim\bar{\pi}_d(x_d|z_{1:d-1}^{(t)},z_{d+1:D}^{(t-1)})$. Set ${\bf x}_{d,m}^{(t)}=[z_{1:d-1}^{(t)}, x_{d,m}^{(t)},z_{d+1:D}^{(t-1)}]$. \[ThisStep\] Set $z_d^{(t)}=x_{d,M}^{(t)}$.
As shown in Figure \[FigTeo\], MRG is equivalent to the alternative chain rule scheme described in the previous section, so that the consistency of the MRG estimators is guaranteed. The ergodicity of the generated chain is also ensured since the dynamics of the MRG scheme is identical to the dynamics of the SG sampler (they differ in the construction of final estimators). Note that with $M=1$, we go back to the TRG scheme.
The MRG approach is convenient in terms of accuracy and computational efficiency, as also confirmed by the numerical results in Section \[SIMU\]. MRG is particularly advisable if an adaptive MCMC is employed to draw from the full-conditional pdfs, i.e., when several MCMC steps are performed for sampling from each full-conditional and adapting the proposal. We can use all the sequence of samples generated by the internal MCMC algorithm in the resulting estimator. Algorithm \[tab:MCMC\_MRG\] shows the detailed steps of an MCMC-within-MRG algorithm, when a direct method for sampling the full-conditionals is not available.
Choose a starting point $[z_1^{(0)},\ldots,z_D^{(0)}]$. Perform $M$ steps of an MCMC algorithm with target pdf $\bar{\pi}_d(x_d|x_{1:d-1}^{(t)},x_{d+1:D}^{(t-1)})$, yielding the sequence of samples $x_{d,1}^{(t)},\ldots, x_{d,M}^{(t)}$, with initial state $x_{d,0}^{(t)}=x_d^{(t-1)}$. Set ${\bf x}_{d,m}^{(t)}=[z_{1:d-1}^{(t)}, x_{d,m}^{(t)},z_{d+1:D}^{(t-1)}]$, for $m=1,\ldots,M$. Set $z_d^{(t)}= x_{d,M}^{(t)}$.
Figure \[FigTodo3\](a) depicts the random vectors obtained with one run of an MH-within-Gibbs procedure, with $T=10^3$ and $M=5$. Figure \[FigTodo3\](b) illustrates all the outputs of the previous run, including all the auxiliary samples generated by the MH algorithm. Hence, these vectors are the samples obtained with a MH-within-MRG approach. The histogram of the samples in Figure \[FigTodo3\](b) is depicted Figure \[FigTodo3\](c). Note that the histogram of the MH-within-MRG samples reproduces adequately the shape of the target pdf shown in Figure \[FigTodo\]. This histogram was obtained with one run of MH-within-MRG fixing $T=10^4$ and $M=5$.
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
\(a) SG \(b) MRG \(c) Histogram - MRG
![[**(a)**]{} Outputs of one MH-within-Gibbs run with $T=10^3$ and $M=5$, considering the target with contour plot shown in Fig. \[FigTodo\]. [**(b)**]{} Outputs of one MH-within-MRG run with $T=10^3$ and $M=5$. [**(c)**]{} Histograms obtained using all the points in Figure (b), i.e., the MRG outputs with $T=10^4$ and $M=5$.[]{data-label="FigTodo3"}](FigStandG_points.pdf "fig:"){height="3.8cm"} ![[**(a)**]{} Outputs of one MH-within-Gibbs run with $T=10^3$ and $M=5$, considering the target with contour plot shown in Fig. \[FigTodo\]. [**(b)**]{} Outputs of one MH-within-MRG run with $T=10^3$ and $M=5$. [**(c)**]{} Histograms obtained using all the points in Figure (b), i.e., the MRG outputs with $T=10^4$ and $M=5$.[]{data-label="FigTodo3"}](FigTODO_points.pdf "fig:"){height="3.8cm"} ![[**(a)**]{} Outputs of one MH-within-Gibbs run with $T=10^3$ and $M=5$, considering the target with contour plot shown in Fig. \[FigTodo\]. [**(b)**]{} Outputs of one MH-within-MRG run with $T=10^3$ and $M=5$. [**(c)**]{} Histograms obtained using all the points in Figure (b), i.e., the MRG outputs with $T=10^4$ and $M=5$.[]{data-label="FigTodo3"}](FigHistTODO_points.pdf "fig:"){height="3.8cm"}
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
[|p[0.95]{}|]{}\
- Choose a starting point $[z_1^{(0)},\ldots,z_D^{(0)}]$.
1. For $t=1,\ldots,T$:
1. For $d=1,\ldots,D$:
1. For $m=1,\ldots,M$:
1. Draw $x_{d,m}^{(t)}\sim\bar{\pi}_d(x_d|z_{1:d-1}^{(t)},z_{d+1:D}^{(t-1)})$.
2. Set ${\bf x}_{d,m}^{(t)}=[z_{1:d-1}^{(t)}, x_{d,m}^{(t)},z_{d+1:D}^{(t-1)}]$.
2. \[ThisStep\] Set $z_d^{(t)}=x_{d,M}^{(t)}$.
2. Return $\{{\bf x}_{d,m}^{(t)}\}$ for all $d$, $m$ and $t$.
\
\[tab:MRG\]
Experimental Results {#SIMU}
====================
This section gives experimental evidence of performance of the proposed scheme. [ First of all, we study the efficiency of the proposed scheme in three bi-dimensional toy examples with different target densities: a unimodal Gaussian target presenting linear correlation among the variables, a bimodal target (see Fig. \[FigTodo\]) and an “elliptical” target which presents a strong nonlinear correlation between the variables, as shown in Fig. \[FigSIMUdonut\_2\](b).]{} Then, we show its use in a hyper-parameter estimation problem using Gaussian process (GP) regression [@rasmussen2006gaussian] with a kernel function usually employed for automatic relevance determination (ARD) of the input features, considering different dimension of the inference problem. Results show the advantages of the MRG scheme in all the experiments. Furthermore, we apply MRG in a dependence detection problem using both real and simulated remote sensing data. For the sake of reproducibility, the interested reader may find related source codes in <http://isp.uv.es/code/RG.zip>.
Experiment 1: A first analysis of the efficiency {#ExFirstAn}
------------------------------------------------
Let us consider two Gaussian full-conditional densities, $$\begin{aligned}
\label{FullEx0}
\bar{\pi}_1(x_1|x_2) & \propto \exp\left(-\frac{(x_1-0.5x_2)^2}{2\delta^2}\right), \\
\bar{\pi}_2(x_2|x_1) & \propto \exp\left(-\frac{(x_2-0.5x_1)^2}{2\delta^2}\right),\end{aligned}$$ with $\delta=1$. The joint target pdf $\bar{\pi}(x_1,x_2)=\mathcal{N}(x_1,x_2|\boldsymbol\mu,\boldsymbol\Sigma)$ is a bivariate Gaussian pdf with mean vector $\boldsymbol\mu = [0,0]^{\top}$ and covariance matrix $\boldsymbol\Sigma = [1.33 \ 066; \ 0.66 \ 1.33]$. Hence, note that $x_1$ and $x_2$ are linearly correlated. We apply a Gibbs sampler with $T$ iterations to estimate both the mean and the covariance matrix of the joint target pdf. Then, we estimate $5$ values ($2$ for the mean $\boldsymbol\mu$ and $3$ for the covariance matrix $\boldsymbol\Sigma$ ) and takes the average Mean Square Error (MSE). The results are averaged over $2000$ independent runs.
In this toy example, it is possible to draw directly from the conditional pdfs in Eq. since they are both Gaussian densities. Thus, we can show the performance of the two ideal Gibbs schemes: the Ideal SG method (see Alg. \[alg:gibbs\]) and the Ideal MRG technique (see Alg. \[tab:MRG\]). Furthermore, we test a standard MH method and an Adaptive MH (AMH) technique [@Haario01] within SG and MRG. For both MH and AMH, we use a Gaussian random walk proposal, $q(x_{d,m}^{(t)}|x_{d,m-1}^{(t)}) \propto \exp\left(-\frac{(x_{d,m}^{(t)}-x_{d,m-1}^{(t)})^2}{2\sigma^2}\right)$, for $d \in \{1,2\}$, $1 \le m \le M$ and $1 \le t \le T$.[^4] We test different values of the scale parameter $\sigma\in\{0.5,1\}$. At each iteration, AMH adapts the value of $\sigma$ as $m$ grows, using the generated samples from the corresponding full-conditional.
First, we set $T=1000$ and vary $M$. The results are given in Figure \[FigSIMU0\_1\](a). Then, we keep fixed $M=20$ and vary $T$, as shown Figure \[FigSIMU0\_1\](b). In both figures, all the MRG schemes are depicted with solid lines whereas all the SG methods are shown with dashed lines. We can observe that the MRG schemes always provide smaller MSE values. The performance of the MH-within-Gibbs methods depends sensibly on the choice of $\sigma$, showing the importance of using an adaptive technique. Obviously, the MSE of Ideal SG is constant in Figure \[FigSIMU0\_1\](a) (as function of $M$), and the Ideal MRG give a considerable improvement in both scenarios (varying $M$ or $T$). Note that, in Figure \[FigSIMU0\_1\](b), the difference between the Ideal SG and Ideal MRG techniques increases as $T$ grows. As expected, in both figures we can observe that the MH-within-Gibbs methods need to increase $M$ in order to approach the performance of the corresponding Ideal Gibbs schemes.
In Figure \[FigSIMU0\_2\](a), we show the MSE of AMH as function of the number of target evaluation per full-conditional $E=MT$.[^5] AMH adapts the scale parameter $\sigma$ as $M$ grows (starting with $\sigma=1$). We consider $M=1$ and $M=10$ and vary $T$ in order to provide the same value of $E$. Clearly, the chain corresponding to AMH-within-Gibbs with $M=10$ is always shorter than the chain of AMH-within-Gibbs with $M=1$. We can see that AMH with $M=10$ takes advantage of the adaptation and provides smaller MSE value. Furthermore, with a Matlab implementation, the use of a Gibbs sampler with a greater $M$ and smaller $T$ seems a faster solution, as shown in Figure \[FigSIMU0\_2\](b).
Experiment 2: A second analysis of the efficiency {#ExSecAn}
-------------------------------------------------
We test the new MRG scheme in a simple numerical simulation involving a bi-modal, bi-dimensional target pdf: $${\bar \pi}(x_1,x_2)\propto \exp\left(-\frac{(x_1^2-\mu_1)^2}{2\delta_1^2}-\frac{(x_2-\mu_2)^2}{2\delta_2^2}\right),$$ with $\mu_1=4$, $\mu_2=1$, $\delta_1=\sqrt{\frac{5}{2}}$ and $\delta_2=1$. Figure \[FigTodo\] shows the contour plot of ${\bar \pi}(x_1,x_2)$ and Figures \[FigTodo\](a)-(b) depicts some generated samples by MH-within-SG and MH-within-MRG, respectively. Figure \[FigTodo\](c) the corresponding histogram obtained by the MRG samples. Our goal is to approximate via Monte Carlo the expected value, ${\mathbb E}[{\bf X}]$ where ${\bf X}=[X_1,X_2] \sim {\bar \pi}(x_1,x_2) $. We test different Gibbs techniques: the MH [@Robert04] and IA$^2$RMS [@MartinoA2RMS] algorithms within Standard Gibbs (SG) and within MRG sampling schemes. For the MH method, we use a Gaussian random walk proposal, $$q(x_{d,m}^{(t)}|x_{d,m-1}^{(t)}) \propto \exp\left(-\frac{(x_{d,m}^{(t)}-x_{d,m-1}^{(t)})^2}{2\sigma^2}\right),$$ for $d \in \{1,2\}$, $1 \le m \le M$ and $1 \le t \le T$. We test different values of the $\sigma$ parameter. For IA$^2$RMS, we start with the set of support points $\mathcal{S}_0=\{ -10,-6,-2,2,6,10\}$, see [@MartinoA2RMS] for further details. We averaged the Mean Square Error (MSE) over $10^5$ independent runs for each Gibbs scheme. Figure \[FigSIMU1\](a) shows the MSE (in log-scale) of the MH-within-SG scheme as function of the standard deviation $\sigma$ of the proposal pdf (we set $M=1$ and $T=1000$, in this case). The performance of the Gibbs samplers depends strongly on the choice of $\sigma$ of the [*internal*]{} MH method. The optimal value is approximately $\sigma^*\approx 3$. The use of an adaptive proposal pdf is a possible solution, as shown in Figure \[FigSIMU2\](a). Figure \[FigSIMU1\](b) depicts the MSE (in log-scale) as function of $T$ with $M=1$ and $M=20$ (for MH-within-SG we also show the case $\sigma=1$ and $\sigma=3$). Again we observe the importance of using the optimal value $\sigma^*\approx 3$ and, as a consequence, using an adaptive proposal pdf is recommended, see e.g. [@Haario01]. Moreover, the use $M=20$ improves the results even without employing all the points in the estimators (i.e., in a SG scheme) since, as $M$ increases, we improve the convergence of the internal chain. Moreover, the MH-within-MRG technique provides the smallest MSE values.
We can thus assert that recycling the internal samples provides more efficient estimators, as confirmed by Figure \[FigSIMU2\](a) (represented again in log-scale). Here we fix $T=1000$ and vary $M$. As $M$ increases, the MSE becomes smaller when the MRG technique is employed. When a (SG) sampler is used, the curves show an horizontal asymptote since the internal chains converge after a certain value $M\geq M^*$, and there is not a great benefit from increasing $M$ (recall that in SG we do not recycle the internal samples). Within an MRG scheme, the increase of $M$ yields lower MSE since now we recycle the internal samples. Clearly, the benefit of using MRG w.r.t. SG increases as $M$ grows. Figure \[FigSIMU2\](a) also shows the advantage of using an adaptive MCMC scheme (in this case IA$^2$RMS [@MartinoA2RMS]). The advantage is clearer when the MH and IA$^2$RMS schemes are used within MRG. More specifically note that, as the MH method employed the optimal scale $\sigma^*\approx 3$, Figure \[FigSIMU2\](a) shows the importance of a non-parametric construction of the proposal pdf employed in IA$^2$RMS. Actually, such construction allows adaptation of the entire shape of the proposal, which becomes closer and closer to the target. The performance of IA$^2$RMS and MH within Gibbs becomes more similar as $M$ increases. This is due to the fact that, in this case, with a high enough value of $M$, the MH chain is able to exceed its burn-in period and eventually converges. Finally, note that the adaptation speeds up the convergence of the chain generated by IA$^2$RMS. The advantage of using the adaptation is more evident for intermediate values of $M$, e.g., $10<M<30$, where the difference with the use of a standard MH is higher. As $M$ increases and the chain generated by MH converges, the difference between IA$^2$RMS and MH is reduced.
In Figure \[FigSIMU2\](b), we compare the performance of IA$^2$RMS-within-MRG scheme, setting $M=20$ and varying $T$, with MH-within-a standard Gibbs scheme (i.e., $M=1$) with a longer chain, i.e., a higher value of $T'>T$. In order to provide a comparison as fair as possible, we use the optimal scale parameter $\sigma^*\approx 3$ for the MH method. For each value of $T$ and $T'$, the MSE and computational time (in seconds) is given.[^6] We can observe that, for a fixed time, IA$^2$RMS-within-MRG outperforms in MSE the standard MH-within-Gibbs scheme with a longer chain. These observations confirm the computational and estimation advantages of the proposed MRG approach.
Experiment 3: A third analysis of the efficiency {#ExThirdAn}
------------------------------------------------
In this section, we consider a bi-dimensional target density which presents a strong nonlinear dependence between the variable $x_1$ and $x_2$, i.e., $$\begin{aligned}
\label{FullEx3}
\bar{\pi}(x_1,x_2) & \propto \exp\left(-\frac{(x_1^2+Bx_2^2-A)^2}{4}\right), \end{aligned}$$ with $A=10$ and $B=0.1$. Figure \[FigSIMUdonut\_2\](b) depicts the contour-plot of $ \bar{\pi}(x_1,x_2)$. Note that the difference scale in the first and second axis. We use different Monte Carlo techniques in order to approximate the expected values (groundtruth $\mu_1=0$ and $\mu_2=0$) and the standard deviations of the marginal pdfs (groundtruth $\delta_1\approx \sqrt{5}$ and $\delta_2\approx \sqrt{51}$). We compute the average MSE in the estimation of these $4$ values (also averaged over $2000$ independent runs).
We compare two schemes, MH-within-SG and MH-within-MRG, considering again a random walk proposal pdf $q(x_{d,m}^{(t)}|x_{d,m-1}^{(t)}) \propto \exp\left(-\frac{(x_{d,m}^{(t)}-x_{d,m-1}^{(t)})^2}{2\sigma^2}\right)$, for $d \in \{1,2\}$, $1 \le m \le M$ and $1 \le t \le T$. First of all, we set $T=200$, $\sigma=10$ and vary $M$, as shown in Figure \[FigSIMUdonut\_1\](a). Then, in Figure \[FigSIMUdonut\_1\](b), we set $M=100$, $\sigma=10$ and vary $T$. In Figure \[FigSIMUdonut\_2\](a), we keep fixed $M=100$, $T=200$ and change $\sigma$. The MRG schemes are depicted with solid lines whereas all the SG methods are shown with dashed lines. We can observe that the MH-within-MRG scheme always provides the best results. Figure \[FigSIMUdonut\_2\](a) shows again that performance of the Gibbs schemes depends on the scale parameter of the proposal. In all cases, MRG provides smaller MSE values and the benefit w.r.t. SG is more evident when a good choice of $\sigma$ is employed.
Experiment 4: Learning Hyperparameters in Gaussian Processes {#GPexample}
------------------------------------------------------------
Gaussian processes (GPs) are Bayesian state-of-the-art tools for function approximation and regression [@rasmussen2006gaussian]. As for any kernel method, selecting the covariance function and learning its hyperparameters is the key to attain significant performance. We here evaluate the proposed approach for the estimation of hyperparameters of the Automatic Relevance Determination (ARD) covariance [@Bishop Chapter 6]. Notationally, let us assume observed data pairs $\{y_j,{\bf z}_j\}_{j=1}^{P}$, with $y_j\in \mathbb{R}$ and $${\bf z}_j=[z_{j,1},z_{j,2},\ldots,z_{j,L}]^{\top}\in \mathbb{R}^{L},$$ where $L$ is the dimension of the input features. We also denote the corresponding $P\times 1$ output vector as ${\bf y}=[y_1,\ldots,y_P]^{\top}$ and the $L\times P$ input matrix ${\bf Z}=[{\bf z}_1,\ldots,{\bf z}_P]$. We address the regression problem of inferring the unknown function $f$ which links the variable $y$ and ${\bf z}$. Thus, the assumed model is $$\label{ModelTrue}
y=f({\bf z})+e,$$ where $e\sim N(e;0,\sigma^2)$, and that $f({\bf z})$ is a realization of a Gaussian Process (GP) [@rasmussen2006gaussian]. Hence $f({\bf z}) \sim \mathcal{GP}(\mu({\bf z}),\kappa({\bf z},{\bf r}))$ where $\mu({\bf z})=0$, ${\bf z},{\bf r} \in \mathbb{R}^{L}$, and we consider the ARD kernel function $$\label{EqKernel}
\kappa({\bf z},{\bf r})=\exp\left(-\sum_{\ell=1}^{L}\frac{(z_\ell-r_\ell)^2}{2\delta_\ell^2}\right), \mbox{ } \mbox{ with } \mbox{ } \delta_\ell> 0, \quad \ell=1,\ldots,L.$$ Note that we have a different hyper-parameter $\delta_\ell$ for each input component $z_\ell$, hence we also define ${\bm \delta}=\delta_{1:L}=[\delta_1,\ldots,\delta_L]$. Using ARD allows us to infer the relative importance of different components of inputs: a small value of $\delta_{\ell}$ means that a variation of the $\ell$-component $z_\ell$ impacts the output more, while a high value of $\delta_{\ell}$ shows virtually independence between the $\ell$-component and the output.
Given these assumptions, the vector ${\bf f}=[f({\bf z}_1),\ldots, f({\bf z}_P)]^\top$ is distributed as $$\label{Eq_f}
p({\bf f}|{\bf Z},{\bm \delta}, \kappa)=\mathcal{N}({\bf f};{\bf 0},{\bf K}),$$ where ${\bf 0}$ is a $P\times 1$ null vector, and ${\bf K}_{ij}:=\kappa({\bf z}_i,{\bf z}_j)$, for all $i,j=1,\ldots,P$, is a $P\times P$ matrix. Note that, in Eq. , we have expressed explicitly the dependence on the input matrix ${\bf Z}$, on the vector ${\bm \delta}$ and on the choice of the kernel family $\kappa$. Therefore, the vector containing all the hyper-parameters of the model is $$\begin{aligned}
{\bm \theta}&=&[\theta_{1:L}=\delta_{1:L},\theta_{L+1}=\sigma], \\
{\bm \theta}&=&[{\bm \delta}, \sigma] \in \mathbb{R}^{L+1},\end{aligned}$$ i.e., all the parameters of the kernel function in Eq. and standard deviation $\sigma$ of the observation noise. Considering the filtering scenario and the tuning of the parameters (i.e., inferring the vectors ${\bf f}$ and ${\bm \theta}$), the full Bayesian solution addresses the study of the full posterior pdf involving ${\bf f}$ and ${\bm \theta}$, $$\label{CompletePost}
p({\bf f},{\bm \theta}|{\bf y}, {\bf Z}, \kappa)=\frac{p({\bf y}|{\bf f},{\bf Z},{\bm \theta}, \kappa)p({\bf f}|{\bf z},{\bm \theta},\kappa) p({\bm \theta})}{p({\bf y}|{\bf Z},\kappa)},$$ where $p({\bf y}|{\bf f},{\bf Z},{\bm \theta}, \kappa)=\mathcal{N}({\bf y};{\bf 0},\sigma^2 {\bf I})$ given the observation model in Eq. , $p({\bf f}|{\bf z},{\bm \theta},\kappa)$ is given in Eq. , and $p({\bm \theta})$ is the prior over the hyper-parameters. We assume $p({\bm \theta})=\prod_{\ell=1}^{L+1}\frac{1}{\theta_\ell^{\beta}}\mathbb{I}_{\theta_\ell}$ where $\beta=1.3$ and $\mathbb{I}_{v}=1$ if $v>0$, and $\mathbb{I}_{v}=0$ if $ v\leq 0$. Note that the posterior in Eq. is analytically intractable but, given a fixed vector ${\bm \theta}'$, the marginal posterior of $p({\bf f}|{\bf y}, {\bf Z}, {\bm \theta}',\kappa)=\mathcal{N}({\bf f}; {\bm \mu}_p,{\bm \Sigma}_p)$ is known in closed-form: it is Gaussian with mean $ {\bm \mu}_p={\bf K}({\bf K}+\sigma^2 {\bf I})^{-1} {\bf y}$ and covariance matrix ${\bm \Sigma}_p={\bf K}-{\bf K}({\bf K}+\sigma^2 {\bf I})^{-1} {\bf K}$ [@rasmussen2006gaussian]. For the sake of simplicity, in this experiment we focus on the marginal posterior density of the hyperparameters, $$p({\bm \theta}|{\bf y}, {\bf Z}, \kappa)=\int p({\bf f},{\bm \theta}|{\bf y}, {\bf Z}, \kappa) d{\bf f}\propto p({\bf y}|{\bm \theta}, {\bf Z}, \kappa) p({\bm \theta}),$$ which can be evaluated analytically. Actually, since $p({\bf y}|{\bm \theta}, {\bf Z}, \kappa)=\mathcal{N}({\bf y};{\bf 0},{\bf K}+\sigma^2 {\bf I})$ and $p({\bm \theta}|{\bf y}, {\bf Z}, \kappa) \propto p({\bf y}|{\bm \theta}, {\bf Z}, \kappa)p({\bm \theta})$, we have $$\begin{aligned}
\log \left[p({\bm \theta}|{\bf y}, {\bf Z}, \kappa)\right]
&\propto& -\frac{1}{2} {\bf y}^{\top} ({\bf K}+\sigma^2 {\bf I})^{-1} {\bf y}-\frac{1}{2} \log\left[\mbox{det}\left[{\bf K}+\sigma^2 {\bf I}\right]\right]-\beta \sum_{\ell=1}^{L+1} \log\theta_\ell, \end{aligned}$$ with $\theta_\ell >0$, where clearly ${\bf K}$ depends on $\theta_{1:L}=\delta_{1:L}$ and recall that $\theta_{L+1}=\sigma$ [@rasmussen2006gaussian]. However, the moments of this marginal posterior cannot be computed analytically. Then, in order to compute the Minimum Mean Square Error (MMSE) estimator, i.e., the expected value ${\mathbb E}[{\bm \Theta}]$ with ${\bm \Theta} \sim p({\bm \theta}|{\bf y}, {\bf Z}, \kappa)$, we approximate ${\mathbb E}[{\bm \Theta}]$ via Monte Carlo quadrature. More specifically, we apply a Gibbs-type samplers to draw from $\pi({\bm \theta})\propto p({\bm \theta}|{\bf y}, {\bf Z}, \kappa)$. Note that dimension of the problem is $D=L+1$ since ${\bm \theta}\in \mathbb{R}^{D}$.
We generated the $P=500$ pairs of data, $\{y_j,{\bf z}_j\}_{j=1}^{P}$, drawing ${\bf z}_j\sim\mathcal{U}([0,10]^L)$ and ${\bf y}_j$ according to the model in Eq. , considered $L\in\{1,3\}$ so that $D\in\{2,4\}$, and set $\sigma^*=\frac{1}{2}$ for both cases, $\delta^*=1$ and ${\bm \delta}^*=[1,3,1]$, respectively (recall that ${\bm \theta}^*=[{\bm \delta}^*,\sigma^*]$). Keeping fixed the generated data for each scenario, we then computed the ground-truths using an exhaustive and costly Monte Carlo approximation, in order to be able to compare the different techniques.
We tested the standard MH within SG and MRG [(with a random walk Gaussian proposal $q(x_{d,m}^{(t)}|x_{d,m-1}^{(t)}) \propto \exp\left(-\frac{(x_{d,m}^{(t)}-x_{d,m-1}^{(t)})^2}{2\sigma^2}\right)$, as in the previous examples, with $\sigma=2$)]{}, and also the Single Component Adaptive Metropolis (SCAM) algorithm [@HaarioCW] within SG and MRG. SCAM is a component-wise version of the adaptive MH method [@Haario01] where the covariance matrix of the proposal is automatically adapted. In SCAM, the covariance matrix of the proposal is diagonal and each element is adapted considering only the corresponding component: that is, the variances of the marginal densities of the target pdf are estimated and used as a scale parameter of the proposal pdf in the corresponding component.[^7] We averaged the results using $10^3$ independent runs. Figure \[FigSIMU3\](a) shows the MSE curves (in log-scale) of the different schemes as function of $M\in\{1,10,20,30,40\}$, while keeping fixed $T=100$ (in this case, $D=2$). Figure \[FigSIMU3\](b) depicts the MSE curves ($D=4$) as function of $T$ considering in one case $M=1$ and $M=10$ for the rest. In both figures, we notice that (1) using an $M>1$ is advantageous in any case (SG or MRG), (2) using a procedure to adapt the proposal improves the results, and (3) MRG, i.e., recycling all the generated samples, always outperforms the SG schemes.
Figure \[FigSIMU4\](a) compares the MH-within-SG with the MH-within-MRG, showing the MSE as function of the total number of target evaluations $E=MT$ per full-conditional. We set $M=5$, $T\in\{3, 5, 10, 20, 40, 60, 100\}$ for MH-within-MRG, whereas we have $M=1$ and $T\in\{10, 50,$ $100, 200, 300, 500\}$ for MH-within-SG. Namely, we used a longer Gibbs chain for MH-within-SG. Note that the MH-within-MRG provides always smaller MSEs, considering the same total number of evaluations $E$ of the target density. Figure \[FigSIMU4\](b) depicts the histograms of the samples ${\bm \theta}^{(t)}$ drawn from the posterior $p({\bm \theta}|{\bf y}, {\bf Z}, \kappa)$ in a specific run, with $D=4$, generated using MH-within-MRG with $M=5$ and $T= 2000$. The dashed line represents the mean of the samples (recall that $\delta_1^*=1$, $\delta_2^*=3$, $\delta_3^*=1$ and $(\sigma^*)^2=0.5$). Note that all the samples ${\bm \theta}^{(t)}$ can be employed for approximating the full Bayesian solution of the GP which involves the joint posterior pdf in Eq. .
We recall that the results in Figure \[FigSIMU3\](b) and Figures \[FigSIMU4\] corresponds to $D=4$. We have also tested SG and MRG in higher dimensions as shown in Table \[tabDiffD\]. In order to obtain these results (averaged over $10^3$ runs), we generate $P=500$ pairs of data (as described above; we keep them fixed in all the runs), considering again $\sigma^*=\frac{1}{2}$ and $\delta_\ell=2$ for all $\ell=1,\ldots,L$, with $L\in\{5,7,9\}$. Hence, ${\bm \theta}^*=[{\bm \delta}^*,\sigma^*]$ has dimension $D\in\{6,8,10\}$. We test MH-within-SG and MH-within-MRG with the random walk Gaussian proposal described above ($\sigma=2$), $M=10$ and $T=2000$. Observing Table \[tabDiffD\], we see that the MRG provides the best results for all the dimension $D$ of the inference problem.
\[tabDiffD\]
Experiment 5: Learning Dependencies in Remote Sensing Variables {#Ex3}
---------------------------------------------------------------
We now consider the application of the MRG scheme to study the dependence among different geophysical variables [ (considering real data).]{} Specifically, we consider the case of temperature estimation from thermal infrared (TIR) remotely sensed data. In this scenario, land surface temperature ($T_s$) and emissivity ($\epsilon$) are the two main geo-biophysical variables to be retrieved from TIR data, since most of the energy detected by the sensor in this spectral region is directly emitted by the land surface. The atmosphere status can be considered as a mediating variable in the relations between the satellite measured $T$ and $T_s$, and is here summarized by the integral of the water vapour $W$ through the whole atmospheric column. Both variables $T_s$ and $\epsilon$ are coupled and constitute a typical problem in remote sensing referred as to the “temperature and emissivity separation problem”. On the one hand, models for estimating land temperature $T_s$ typically involve simple parametrizations of at-sensor brightness temperatures $T$, the mean and/or differential emissivities (${\bar \epsilon}$ and $\Delta \epsilon$), and the total atmospheric water vapour content $W$. On the other hand, a plethora of models for estimating $\epsilon$ have been devised [@Jimenez08]. Here we focus on the application of the MRG sampler to infer the statistical dependencies between the observed variables. We aim to obtain the dependence graph between the considered geophysical variables, $X_1=T_s$, $X_2=W$, $X_3=\epsilon$ and $X_4=T$. To assess such relations, we considered synthetic data simulating ASTER sensor conditions [@Sobrino08]. A total of $6588$ data points was available. For simplicity, we focused on band 10 ($\sim 8.3\mu$m) for $T$ and $\epsilon$, and subsampled the dataset to finally work with $220$ data points. The data was subsequently standardized.
### Main procedure
We study the $12$ possible regression models of type $$X_i=f_{j,i}(X_j)+E_{j,i}. \quad \quad i,j\in\{1,2,3,4\}, \quad \mbox{ with } i\neq j,$$ where $E_{j,i}\sim\mathcal{N}(0,\sigma_{i,j}^2)$ and $f_{j,i}(x_j)$ is a realization of a Gaussian Process (GP) [@rasmussen2006gaussian], with zero mean and kernel function defined in Eq. (note that in this case $L=1$). For each regression problem, we trained the corresponding GP model using SCAM-within-MRG with $T=200$ and $M=10$. We analyze the empirical distributions of corresponding hyper-parameters ${\bm \theta}_{i,j}=[\delta_{i,j}, \sigma_{i,j}^2]$ obtained by Monte Carlo. We focus mainly on the distribution of $\delta_{i,j}$, i.e., the hyper-parameter of the kernel in Eq. . Hyper-parameter $\delta_{i,j}$ will tend to be higher and its distribution will exhibit heavier right tail if the Signal-Noise-Ratio (SNR) is low and if $X_i$ and $ X_j$ are close to independence. However, the spread of $\delta_{i,j}$ also depends on the noise power in the system, the unknown mapping (deterministic or stochastic) linking $X_i$ and $X_j$, and the asymmetry of the regression functions, i.e. in general $f_{i,j}\neq f_{j,i}$. Hypothesis testing comes into play to determine the significance of the association between all pairs of variables $X_i$ and $X_j$.
### Hypothesis testing and surrogate data
In order to find significance levels and thresholds about the existence of any possible dependence (strong or weak), we perform an hypothesis test with the null-hypothesis “$\mathcal{H}_0$: independence between $X_i$ and $X_j$”. We build the sampling distribution of $\mathcal{H}_0$ by the surrogate data method in [@Theiler92]. Under the null-hypothesis $\mathcal{H}_0$ of independence, some proper surrogate data can be generated by shuffling the output values (i.e., we permute the outputs keeping fixed the inputs) while keeping fixed the input features. This way we have new data points sharing the same input and output values with the true data, but breaking any structure which links the inputs with the outputs. Clearly, this procedure considers different values for each pair of indices $i$ and $j$ (i.e., each variables $X_i$ and $X_j$).
Given a set of surrogate data, we applied SCAM-within-MRG with $T=200$ and $M=10$ and obtain the empirical distribution of the hyper-parameter $\delta_{i,j}$. We repeated this procedure $500$ times, generating different surrogate data at each run. We computed different empirical moments of the distribution of the hyper-parameter $\delta_{i,j}$, as mean, median and variance from the empirical distributions obtained via Monte Carlo with the true data and the surrogate data. Averaged results over $500$ runs are shown in Table \[tab1\_Ex3\_Results\]. We show mean ${\mathbb E}[\delta]$, median ${\bar \delta}_{0.5}$ and standard deviation $\sqrt{\mbox{var}[\delta]}$, obtained analyzing the true data, and the $p$-values obtained comparing the estimated statistics with the corresponding distributions acquired by the surrogate data method.
Figure \[FigSIMU5\] shows the undirected graphs with significance level set to $\alpha=0.1$. The width of the lines represents the significance of the link according to the estimated $p$-values. If one of the two $p$-values corresponding the two possible regressions between the variable $X_i$ and $X_j$ is greater than $0.1$ the link is depicted in dashed line. The obtained graphs are consistent with a priori physical knowledge about the problem. In particular, it is common sense that the surface temperature $T_s$ and $\epsilon$ depend on $W$ and $T$. After all, remote sensing data processing mainly deals with the estimation of the surface parameters from the acquired satellite brightness temperatures ($T$) and the atmosphere status ($W$)[^8]. In addition, while emissivity $\epsilon$ is generally used for retrieving $T_s$, some simpler methods using only $T$ are successfully used [@Jimenez07]. It is also worth noting that the used data considered only natural surfaces, hence the database was biased towards high values of $\epsilon$, thus explaining why the relationship between $T_s$ and $\epsilon$ was not captured.
[**Link**]{} [**In**]{} [**Out**]{} [**Mean**]{} (${\mathbb E}[\delta]$) [**$p$-value**]{} [**Median**]{} (${\bm {\bar \delta}_{0.5}}$) [**$p$-value**]{} [**Std**]{} ($\sqrt{\mbox{var}[\delta]}$) [**$p$-value**]{}
-------------- ------------ ------------- -------------------------------------- ------------------- ---------------------------------------------- ------------------- ------------------------------------------- -------------------
$T_s$ $W$ 0.68 0.004 0.67 0.002 0.15 0.001
$W$ $T_s$ 0.32 0.001 0.32 0.002 0.15 0.001
$T_s$ $\epsilon$ 11.85 0.22 5.61 0.20 33.16 0.56
$\epsilon$ $T_s$ 11.58 0.23 5.67 0.23 47.36 0.68
$T_s$ $T$ 2.56 0.03 2.53 0.08 0.66 0.006
$T$ $T_s$ 1.83 0.02 1.76 0.03 0.48 0.002
$W$ $T$ 0.33 0.001 0.34 0.001 0.14 0.001
$T$ $W$ 0.40 0.001 0.39 0.004 0.14 0.001
$W$ $\epsilon$ 11.77 0.22 4.04 0.09 32.76 0.54
$\epsilon$ $W$ 11.20 0.21 4.48 0.13 47.97 0.71
$T$ $\epsilon$ 5.22 0.06 3.34 0.09 6.88 0.10
$\epsilon$ $T$ 6.32 0.09 3.95 0.10 10.24 0.14
: [**Exp. 3-**]{} Results for the mean ${\mathbb E}[ \delta]$, the median ${\bar \delta}_{0.5}$ and the std $\sqrt{\mbox{var}[\delta]}$.
\[tab1\_Ex3\_Results\]
Conclusions {#ConclSect}
===========
The Gibbs sampling method is a well-known Markov chain Monte Carlo (MCMC) algorithm, extensively applied in statistics, signal processing and machine learning, in order to obtain samples from complicated a posteriori distributions. A Gibbs sampling approach is particularly useful in high-dimensional inference problems, since the generated samples are constructed component by component. In this sense, it can be considered the MCMC counterpart of the particle filtering methods, for static (i.e., non-sequential inference) and batch (i.e., all the data are processed jointly) frameworks. The key point for the successful application of the SG sampler is the ability to draw efficiently from each the full-conditional densities. However, in real-world applications, drawing from complicated full-conditionals is required, and no direct methods are available in these cases. For solving this issue, several specifically-designed MCMC algorithms has been proposed to be employed within the SG sampler. Most of them require the generation of auxiliary samples that are not included in the resulting estimators. The use of more auxiliary samples accelerates the convergence of the generated Gibbs chain and improves the performance, at the expense of an additional computational effort.
In this work, we have shown that these auxiliary samples can be included within the Gibbs estimators improving their efficiency without any extra computational cost. The consistency of the resulting estimators is ensured since the novel MRG scheme is equivalent to an alternative formulation of the well-known chain rule method. This alternative chain rule procedure has been also described and discussed in this work. Numerical simulations have confirmed the benefits of the novel scheme. First, we have compared the SG and MRG schemes in a toy example, considering the use of several parameter values and the application of different internal MCMC algorithms. MRG yielded clear improvements of the performance w.r.t. the SG approach. Then, we tested the SG and MRG schemes in a hyperparameter estimation problem for GP regression, considering a kernel for automatic relevance determination (ARD). The MRG approach provided the smallest MSEs in estimation of the hyperparameters in all cases. Finally, we studied the application of the proposed MRG sampler to unveil the dependence between different geophysical variables considering remote sensing satellite data.
As future lines, we plan to investigate the use of different number of samples $M_1$,$...$, $M_d$ to be drawn from full-conditionals, and the possible design of an automatic tuning strategy for adapting the number of samples to improve the performance given a specific posterior distribution. This will imply efforts in parallel MRG samplers for scalable learning. We also plan to extend the numerical study with remote sensing data using the MRG scheme in order to infer causal dependences among the geophysical variables, and for that we will consider applying the MRG in (conditional) independence estimation schemes.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank Dr. J. C. Jiménez at IPL for the remote sensing dataset and the fruitful discussions. This work has been supported by the European Research Council (ERC) through the ERC Consolidator Grant SEDAL ERC-2014-CoG 647423.
[^1]: In general these full-conditionals are univariate. Nevertheless, block-wise Gibbs sampling approaches where several random variables are updated simultaneously, have been proposed to speed up the convergence of the Gibbs sampler [@roberts1997updating]. However, unless direct sampling from the multi-variate full-conditionals is feasible, these approaches still result in an increased difficulty of drawing samples and a higher computational cost per iteration. Furthermore, the performance of the overall algorithm can decrease if the blocks are not properly chosen, especially when direct sampling from the multivariate full-conditionals is unfeasible [@Liu04b; @Liang10; @Robert04]. The novel recycling scheme can also be used in the block approach.
[^2]: There also exist intermediate strategies where the same subset of variables are jointly updated, which is often called the Blocked Gibbs Sampler.
[^3]: Sometimes MH-within-Gibbs is also referred as to the Single Component MH algorithm [@HaarioCW] or the Componentwise MH algorithm [@Levine05].
[^4]: [We have also tried the use of a $t$-Student’s density as proposal pdf. Clearly, depending on the proposal parameter employed, the MSE values change, in general. However, the comparison among the different Monte Carlo schemes is not affected by the use of a different proposal density.]{}
[^5]: [ The total number of target evaluations for all the tested algorithms is $E=DMT$ ($D=2$ in this case). Since the factor $D$ (dimension of the problem and number of the full-conditionals) is common for all the samplers, we consider $E=MT$.]{}
[^6]: The computational times are obtained in a Mac processor 2.8 GHz Intel Core i5.
[^7]: More specifically, we have implemented an accelerated version of SCAM which takes more advantage of the MRG scheme, since the variance is also adjusted online during the sampling of the considered full-conditional (for more details, see the code at <http://isp.uv.es/code/RG.zip>).
[^8]: While one could be tempted to infer causal relations, the proposed approach cannot cope with asymmetries in the pdfs of variables through GP hyperparameter estimation.
|
---
abstract: 'Assuming 3-$\nu$ mixing and massive Majorana neutrinos, we analyze the implications of the results of the solar neutrino experiments, including the latest SNO data, which favor the LMA MSW solution of the solar neutrino problem with $\tan^2\theta_{\odot} < 1$, for the predictions of the effective Majorana mass in neutrinoless double beta-decay. Neutrino mass spectra with normal mass hierarchy, with inverted hierarchy and of quasi-degenerate type are considered. For $\cos 2\theta_{\odot} \gtap 0.26$, which follows (at 99.73% C.L.) from the SNO analysis of the solar neutrino data, we find significant lower limits on in the cases of quasi-degenerate and inverted hierarchy neutrino mass spectrum, ${\mbox{$\left| < \! m \! > \right| \ $}}\gtap 0.035$ eV and ${\mbox{$\left| < \! m \! > \right| \ $}}\gtap 8.5\times 10^{-3}$ eV, respectively. If the spectrum is hierarchical the upper limit holds ${\mbox{$\left| < \! m \! > \right| \ $}}\ltap 8.2\times 10^{-3}$ eV. Correspondingly, not only a measured value of ${\mbox{$\left| < \! m \! > \right| \ $}}\neq 0$, but even an experimental upper limit on of the order of $\mbox{few} \times 10^{-2}$ eV can provide information on the type of the neutrino mass spectrum; it can provide also a significant upper limit on the mass of the lightest neutrino $m_1$. A measured value of ${\mbox{$\left| < \! m \! > \right| \ $}}\gtap 0.2$ eV, combined with data on neutrino masses from the $^3$H $\beta-$decay experiment KATRIN, might allow to establish whether the CP-symmetry is violated in the lepton sector.'
---
[**The SNO Solar Neutrino Data, Neutrinoless Double Beta-Decay and Neutrino Mass Spectrum**]{}
S. Pascoli and S. T. Petcov [^1]
[*Scuola Internazionale Superiore di Studi Avanzati, I-34014 Trieste, Italy\
*]{}
Introduction
============
1.0truecm With the publication of the new results of the SNO solar neutrino experiment [@SNO2; @SNO3] (see also [@SNO1]) on i) the measured rates of the charged current (CC) and neutral current (NC) reactions, $\nu_e + D \rightarrow e^{-} + p + p$ and $\nu_l~(\bar{\nu}_l) + D \rightarrow \nu_l~(\bar{\nu}_l) + n + p$, ii) on the day-night (D-N) asymmetries in the CC and NC reaction rates, and iii) on the day and night event energy spectra, further strong evidences for oscillations or transitions of the solar $\nu_e$ into active neutrinos $\nu_{\mu,\tau}$ (and/or antineutrinos $\bar{\nu}_{\mu,\tau}$), taking place when the solar $\nu_e$ travel from the central region of the Sun to the Earth, have been obtained. The evidences for oscillations (or transitions) of the solar $\nu_e$ become even stronger when the SNO data are combined with the data obtained in the other solar neutrino experiments, Homestake, Kamiokande, SAGE, GALLEX/GNO and Super-Kamiokande [@SKsol; @Cl98].
Global analysis of the solar neutrino data [@SNO2; @SNO3; @SNO1; @SKsol; @Cl98], including the latest SNO results, in terms of the hypothesis of oscillations of the solar $\nu_e$ into active neutrinos, $\nu_e \rightarrow \nu_{\mu (\tau)}$, show [@SNO2] that the data favor the large mixing angle (LMA) MSW solution with $\tan^2\theta_{\odot} < 1$, where $\theta_{\odot}$ is the angle which controls the solar neutrino transitions. The LOW solution of the solar neutrino problem with transitions into active neutrinos is only allowed at approximately 99.73% C.L. [@SNO2]; there do not exist other solutions at the indicated confidence level. In the case of the LMA solution, the range of values of the neutrino mass-squared difference $\Delta m^2_{\odot} > 0$, characterizing the solar neutrino transitions, found in [@SNO2] at 99.73% C.L. reads: $${\rm LMA~MSW}:~~~~~~2.2\times 10^{-5}~{\rm eV^2}
\ltap \Delta m^2
\ltap 2.0\times 10^{-4}~{\rm eV^2}~~~~~(99.73\%~{\rm C.L.}).
\label{dmsolLMA}$$ The best fit value of $\Delta m^2_{\odot}$ obtained in [@SNO2] is $(\Delta m^2_{\odot})_{\mathrm{BF}} = 5.0\times 10^{-5}~{\rm eV^2}$. The mixing angle $\theta_{\odot}$ was found in the case of the LMA solution to lie in an interval which at 99.73% C.L. is determined by [@SNO2] $${\rm LMA~~MSW}:~~~~~~~~~~~~~~~~~
0.26 \ltap \cos2\theta_{\odot} \ltap 0.64~~~~~~~(99.73\%~{\rm C.L.}).~~~~~~~~~
\label{thLMA}$$ The best fit value of $\cos2\theta_{\odot}$ in the LMA solution region is given by $(\cos2\theta_{\odot})_{\mathrm{BF}} = 0.50$.
Strong evidences for oscillations of atmospheric neutrinos have been obtained in the Super-Kamiokande experiment [@SKatm00]. As is well known, the atmospheric neutrino data is best described in terms of dominant $\nu_{\mu} \rightarrow \nu_{\tau}$ ($\bar{\nu}_{\mu} \rightarrow \bar{\nu}_{\tau}$) oscillations. The explanation of the solar and atmospheric neutrino data in terms of neutrino oscillations requires the existence of 3-neutrino mixing in the weak charged lepton current (see, e.g., [@BGG99; @P99]).
Assuming 3-$\nu$ mixing and massive Majorana neutrinos, we analyze the implications of the latest results of the SNO experiment for the predictions of the effective Majorana mass in neutrinoless double beta (-) decay (see, e.g., [@BiPet87; @BPP1; @PPW]): $${\mbox{$\left| < \! m \! > \right| \ $}}= \left| m_1 |U_{\mathrm{e} 1}|^2
+ m_2 |U_{\mathrm{e} 2}|^2~e^{i\alpha_{21}}
+ m_3 |U_{\mathrm{e} 3}|^2~e^{i\alpha_{31}} \right|.
\label{effmass2}$$ Here $m_{1,2,3}$ are the masses of 3 Majorana neutrinos with definite mass $\nu_{1,2,3}$, $U_{{\rm e} j}$ are elements of the lepton mixing matrix $U$ - the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) mixing matrix [@BPont57; @MNS62], and $\alpha_{21}$ and $\alpha_{31}$ are two Majorana CP-violating phases [^2] [@BHP80; @Doi81]. If CP-invariance holds, one has [@LW81; @BNP84] $\alpha_{21} = k\pi$, $\alpha_{31} =
k'\pi$, $k,k'=0,1,2,...$, and $$\eta_{21} \equiv e^{i\alpha_{21}} = \pm 1,~~~
\eta_{31} \equiv e^{i\alpha_{31}} = \pm 1 ,
\label{eta2131}$$ represent the relative CP-parities of the neutrinos $\nu_1$ and $\nu_2$, and $\nu_1$ and $\nu_3$, respectively.
The experiments searching for -decay test the underlying symmetries of particle interactions (see, e.g., [@BiPet87]). They can answer the fundamental question about the nature of massive neutrinos, which can be Dirac or Majorana fermions. If the massive neutrinos are Majorana particles, the observation of -decay [^3] can provide unique information on the type of the neutrino mass spectrum and on the lightest neutrino mass [@BPP1; @PPW; @SPAS94; @BGKP96; @BGGKP99; @bbpapers1; @BPP2; @bbpapers2]. Combined with data from the [$\mbox{}^3 {\rm H}$ $\beta$-decay ]{}neutrino mass experiment KATRIN [@KATRIN], it can give also unique information on the CP-violation in the lepton sector induced by the Majorana CP-violating phases, and if CP-invariance holds - on the relative CP-parities of the massive Majorana neutrinos [@BPP1; @PPW; @BGKP96; @bbpapers3].
Rather stringent upper bounds on have been obtained in the $^{76}$Ge experiments by the Heidelberg-Moscow collaboration [@76Ge00], $ {\mbox{$\left| < \! m \! > \right| \ $}}< 0.35~{\rm eV}$ ($90\%$C.L.), and by the IGEX collaboration [@IGEX00], ${\mbox{$\left| < \! m \! > \right| \ $}}< (0.33 \div 1.35)~{\rm eV}$ ($90\%$C.L.). Taking into account a factor of 3 uncertainty in the calculated value of the corresponding nuclear matrix element, we get for the upper limit found in [@76Ge00]: ${\mbox{$\left| < \! m \! > \right| \ $}}< 1.05$ eV. Considerably higher sensitivity to the value of ${\mbox{$\left| < \! m \! > \right| \ $}}$ is planned to be reached in several ${\mbox{$(\beta \beta)_{0 \nu} $}}-$decay experiments of a new generation. The NEMO3 experiment [@NEMO3], which will begin to take data in July of 2002, and the cryogenic detector CUORE [@CUORE], are expected to reach a sensitivity to values of ${\mbox{$\left| < \! m \! > \right| \ $}}\cong 0.1~$eV. An order of magnitude better sensitivity, i.e., to ${\mbox{$\left| < \! m \! > \right| \ $}}\cong 10^{-2}~$eV, is planned to be achieved in the GENIUS experiment [@GENIUS] utilizing one ton of enriched $^{76}$Ge, and in the EXO experiment [@EXO], which will search for ${\mbox{$(\beta \beta)_{0 \nu} $}}-$decay of $^{136}$Xe. Two more detectors, Majorana [@Maj] and MOON [@MOON], are planned to have sensitivity to in the range of $few \times 10^{-2}$ eV.
In what regards the [$\mbox{}^3 {\rm H}$ $\beta$-decay ]{}experiments, the currently existing most stringent upper bounds on the electron (anti-)neutrino mass $m_{\bar{\nu}_e}$ were obtained in the Troitzk [@MoscowH3] and Mainz [@Mainz] experiments and read $m_{\bar{\nu}_e} < 2.2 {\mbox{$ \ \mathrm{eV} \ $}}$. The KATRIN [$\mbox{}^3 {\rm H}$ $\beta$-decay ]{}experiment [@KATRIN] is planned to reach a sensitivity to $m_{\bar{\nu}_e} \sim 0.35$ eV.
The fact that the solar neutrino data implies a relatively large lower limit on the value of $\cos2\theta_{\odot}$, eq. (\[thLMA\]), has important implications for the predictions of the effective Majorana mass parameter in -decay [@BPP1; @PPW] and in the present article we investigate these implications.
The SNO Data and the Predictions for the Effective Majorana Mass
=================================================================
1.0truecm According to the analysis performed in [@SNO2], the solar neutrino data, including the latest SNO results, strongly favor the LMA solution of the solar neutrino problem with $\tan^2 \theta_\odot <1$. We take into account these new development to update the predictions for the effective Majorana mass , derived in [@BPP1], and the analysis of the implications of the measurement of, or obtaining a more stringent upper limit on, performed in [@BPP1; @PPW]. The predicted value of depends in the case of 3-neutrino mixing of interest on (see e.g. [@BPP1; @PPW; @BPP2]): i) the value of the lightest neutrino mass $m_1$, ii) $\Delta m^2_{\odot}$ and $\theta_{\odot}$, iii) the neutrino mass-squared difference which characterizes the atmospheric $\nu_{\mu}$ ($\bar{\nu}_{\mu}$) oscillations, ${\mbox{$\Delta m^2_{\mathrm{atm}} \ $}}$, and iv) the lepton mixing angle $\theta$ which is limited by the CHOOZ and Palo Verde experiments [@CHOOZ; @PaloV]. The ranges of allowed values of $\Delta m^2_{\odot}$ and $\theta_{\odot}$ are determined in [@SNO2], while those of ${\mbox{$\Delta m^2_{\mathrm{atm}} \ $}}$ and $\theta$ are taken from [@Gonza3nu] (we use the best fit values and the 99% C.L. results from [@Gonza3nu]). Given the indicated parameters, the value of depends strongly [@BPP1; @PPW] on the type of the neutrino mass spectrum, as well as on the values of the two Majorana CP-violating phases, $\alpha_{21}$ and $\alpha_{31}$ (see eq. (\[effmass2\])), present in the lepton mixing matrix.
Let us note that if ${\mbox{$\Delta m^2_{\mathrm{atm}} \ $}}$ lies in the interval ${\mbox{$\Delta m^2_{\mathrm{atm}} \ $}}\cong (2.0 - 5.0)\times 10^{-3}~{\rm eV^2}$, as is suggested by the current atmospheric neutrino data [@SKatm00], its value will be determined with a high precision ($\sim 10\%$ uncertainty) by the MINOS experiment [@MINOS]. Similarly, if $\Delta m^2_{\odot} \cong (2.5 - 10.0)\times 10^{-5}~{\rm eV^2}$, which is favored by the solar neutrino data, the KamLAND experiment will be able to measure $\Delta m^2_{\odot}$ with an uncertainty of $\sim (10 - 15)\%$ (99.73% C.L.). Combining the data from the solar neutrino experiments and from KamLAND would permit to determine $\cos2\theta_{\odot}$ with a high precision as well ($\sim 15\%$ uncertainty at 99.73% C.L., see [^4] , e.g., [@Carlos01]). Somewhat better limits on $\sin^2 \theta$ than the existing one can be obtained in the MINOS experiment.
We number the massive neutrinos (without loss of generality) in such a way that $m_1 < m_2 < m_3$. In the analysis which follows we consider neutrino mass spectra with normal mass hierarchy, with inverted hierarchy and of quasi-degenerate type [@BPP1; @PPW; @SPAS94; @BGKP96; @BGGKP99; @bbpapers1; @bbpapers2]. In the case of neutrino mass spectrum with normal mass hierarchy ($m_1 \ll~(<)~ m_2 \ll m_3$) we have ${\mbox{$ \Delta m^2_{\odot} \ $}}\equiv {\mbox{$ \ \Delta m^2_{21} \ $}}$ and $\sin^2\theta \equiv |U_{{\rm e} 3}|^2$, while in the case of spectrum with inverted hierarchy ($m_1 \ll m_2 \cong m_3$) one finds ${\mbox{$ \Delta m^2_{\odot} \ $}}\equiv {\mbox{$ \ \Delta m^2_{32} \ $}}$ and $\sin^2\theta \equiv |U_{{\rm e} 1}|^2$. In both cases one can choose ${\mbox{$\Delta m^2_{\mathrm{atm}} \ $}}\equiv \Delta m^2_{31}$. It should be noted that for $m_1 > 0.2 \ {\rm eV} \gg \sqrt{{\mbox{$\Delta m^2_{\mathrm{atm}} \ $}}}$, the neutrino mass spectrum is of the quasi-degenerate type, $m_1 \cong m_2 \cong m_3$, and the two possibilities, ${\mbox{$ \Delta m^2_{\odot} \ $}}\equiv {\mbox{$ \ \Delta m^2_{21} \ $}}$ and ${\mbox{$ \Delta m^2_{\odot} \ $}}\equiv {\mbox{$ \ \Delta m^2_{32} \ $}}$, lead to the same predictions for .
Normal Mass Hierarchy: ${\mbox{$ \Delta m^2_{\odot} \ $}}\equiv {\mbox{$ \ \Delta m^2_{21} \ $}}$
--------------------------------------------------------------------------------------------------
1.0truecm If ${\mbox{$ \Delta m^2_{\odot} \ $}}= \Delta m^2_{21}$, the effective Majorana mass parameter is given in terms of the oscillation parameters , , $\theta_\odot$ and $|U_{\mathrm{e}3}|^2$ which is constrained by the CHOOZ data, as follows [@BPP1]: $${\mbox{$\left| < \! m \! > \right| \ $}}= \left| \big( m_1 \cos^2 \theta_\odot +
\sqrt{ m_1^2 + {\mbox{$ \Delta m^2_{\odot} \ $}}} \sin^2 \theta_\odot e^{i\alpha_{21}} \big)
(1 - |U_{\mathrm{e}3}|^2)
+
\sqrt{m_1^2 +{\mbox{$\Delta m^2_{\mathrm{atm}} \ $}}} |U_{\mathrm{e}3}|^2 e^{i\alpha_{31} } \right|.
\label{eqmasshierarchy01}$$ The effective Majorana mass can lie anywhere between 0 and the present upper limits, as Fig. 1 (left panels) shows [^5]. This conclusion does not change even under the most favorable conditions for the determination of , namely, even when , , $\theta_{\odot}$ and $\theta$ are known with negligible uncertainty [@PPW]. Our further conclusions for the case of the LMA solution of the solar neutrino problem [@SNO2] are illustrated in Fig. 1 (left panels) and are summarized below.
[**Case A: $m_1 < 0.02 \ {\rm eV}$, $m_1 < m_2 \ll m_3$.**]{}
Taking into account the new constraints on the solar neutrino oscillation parameters following from the SNO data [@SNO2] does not change qualitatively the conclusions reached in ref. [@BPP1; @PPW]. The maximal value of , ${\mbox{$\left| < \! m \! > \right| \ $}}_{ \! \mbox{}_{\rm MAX}}$, for given $m_1$ reads: $$\begin{aligned}
{\mbox{$\left| < \! m \! > \right| \ $}}_{ \! \mbox{}_{\rm MAX}} = &
\Big( m_1 (\cos^2 \theta_\odot)_{ \! \mbox{}_{\rm MIN}} +
\sqrt{ m_1^2 + ({\mbox{$ \Delta m^2_{\odot} \ $}})_{ \! \mbox{}_{\rm MAX}}}
( \sin^2 \theta_\odot)_{ \! \mbox{}_{\rm MAX}} \Big)
(1 - |U_{\mathrm{e}3}|^2_{ \! \mbox{}_{\rm MAX}}) \nonumber \\
&+
\sqrt{m_1^2 + ({\mbox{$\Delta m^2_{\mathrm{atm}} \ $}})_{ \! \mbox{}_{\rm MAX}}}
|U_{\mathrm{e}3}|^2_{ \! \mbox{}_{\rm MAX}} ,
\label{meffmaxhierLMA} \end{aligned}$$ where $(\cos^2 \! \! \theta_\odot)_{\mbox{}_{\rm MIN}}$ and $ (\sin^2 \! \! \theta_\odot)_{\mbox{}_{\rm MAX}}$ are the values corresponding to $(\tan^2 \theta_\odot)_{\mbox{}_{\rm MAX}}$, and ${\mbox{$(\Delta m^2_{\mathrm{atm}})_{ \! \mbox{}_{\mathrm{MAX}}} \ $}}$ is the maximal value of allowed for the ${\mbox{$|U_{\mathrm{e} 3}|^2_{ \! \mbox{}_{\mathrm{MAX}}}$ }}$ [@Gonza3nu].
For the values of and $\tan^2\theta_{\odot}$ from the LMA solution region [@SNO2], eqs. (\[dmsolLMA\]) and (\[thLMA\]), we get for $m_1 \ll 0.02 \ {\rm eV}$: ${\mbox{$\left| < \! m \! > \right| \ $}}_{ \! \mbox{}_{\rm MAX}}
\simeq 8.2 \times 10^{-3}~{\rm eV}$. Using the best fit values of the oscillation parameters found in refs. [@SNO2; @Gonza3nu], one obtains: ${\mbox{$\left| < \! m \! > \right| \ $}}_{ \! \mbox{}_{\rm MAX}}
\simeq 2.0 \times 10^{-3}~{\rm eV}$. The maximal value of corresponds to the case of CP-conservation and $\nu_{1,2,3}$ having identical CP-parities, $\eta_{21} = \eta_{31}=1$.
There is no significant lower bound on because of the possibility of mutual compensations between the terms contributing to and corresponding to the exchange of different virtual massive Majorana neutrinos. Furthermore, the uncertainties in the oscillation parameters do not allow to identify a “just-CP violation” region of values of [@BPP1] (a value of in this region would unambiguously signal the existence of CP-violation in the lepton sector, caused by Majorana CP-violating phases). However, if the neutrinoless double beta-decay will be observed, the measured value of , combined with information on $m_1$ and a better determination of the relevant neutrino oscillation parameters, might allow to determine whether the CP-symmetry is violated due to Majorana CP-violating phases, or to identify which are the allowed patterns of the massive neutrino CP-parities in the case of CP-conservation (for a detailed discussion see ref. [@PPW]).
[**Case B: Neutrino Mass Spectrum with Partial Hierarchy ($0.02 \ {\rm eV} \leq m_1 \leq 0.2 \ {\rm eV}$)**]{}
For $m_1 \geq 0.02 \ {\rm eV}$ there exists a lower bound on the possible values of (Fig. 1, left panels). Using the 99.73% C.L. allowed values of and $\cos2\theta_{\odot}$ from [@SNO2], we find that this lower bound is significant, i.e., ${\mbox{$\left| < \! m \! > \right| \ $}}\gtap 10^{-2}$ eV, for $m_1 \gtap 0.07$ eV. For the best fit values of the oscillation parameters obtained in [@SNO2; @Gonza3nu], one has ${\mbox{$\left| < \! m \! > \right| \ $}}\gtap 10^{-2}$ eV for $m_1 \geq 0.02$ eV.
For a fixed $m_1 \geq 0.02 \ {\rm eV}$, the minimal value of , ${\mbox{$\left| < \! m \! > \right| \ $}}_{ \! \mbox{}_{\rm MIN} }$, is given by $$\label{minmeffLMA}
{\mbox{$\left| < \! m \! > \right| \ $}}_{ \! \mbox{}_{\rm MIN} } \! \cong
m_1 (\cos 2 \theta_\odot )_{\! \mbox{}_{\rm MIN}}
\! (1 - {\mbox{$|U_{\mathrm{e} 3}|^2_{ \! \mbox{}_{\mathrm{MAX}}}$ }}\! \! ) -
\sqrt{m_1^2 + {\mbox{$(\Delta m^2_{\mathrm{atm}})_{ \! \mbox{}_{\mathrm{MAX}}} \ $}}} {\mbox{$|U_{\mathrm{e} 3}|^2_{ \! \mbox{}_{\mathrm{MAX}}}$ }}+
{\cal O} \Big(\frac{{\mbox{$ (\Delta m^2_{\odot})_{ \! \mbox{}_{\mathrm{MAX}}} \ $}}}{4 m_1} \Big) ,$$ where again ${\mbox{$(\Delta m^2_{\mathrm{atm}})_{ \! \mbox{}_{\mathrm{MAX}}} \ $}}$ is the maximal allowed value of for the ${\mbox{$|U_{\mathrm{e} 3}|^2_{ \! \mbox{}_{\mathrm{MAX}}}$ }}$ [@Gonza3nu]. The upper bound on , which corresponds to CP-conservation and $\eta_{21} = \eta_{31} = +1$ ($\nu_{1,2,3}$ possessing identical CP-parities), can be found for given $m_1$ by using eq. (\[meffmaxhierLMA\]). For the allowed values of $m_1$, $ 0.02 \ {\rm eV} \leq m_1 \leq 0.2 \ {\rm eV}$, we have ${\mbox{$\left| < \! m \! > \right| \ $}}\leq 0.2~{\rm eV}$.
Inverted Neutrino Mass Hierarchy: ${\mbox{$ \Delta m^2_{\odot} \ $}}\equiv {\mbox{$ \ \Delta m^2_{32} \ $}}$
-------------------------------------------------------------------------------------------------------------
1.0cm If ${\mbox{$ \Delta m^2_{\odot} \ $}}= \Delta m^2_{32}$, the effective Majorana mass is given in terms of the oscillation parameters , , $\theta_\odot$ and $|U_{\mathrm{e}1}|^2$ which is constrained by the CHOOZ data [@BPP1]: $$\begin{aligned}
{\mbox{$\left| < \! m \! > \right| \ $}}\! = & \Big| m_1 |U_{\mathrm{e}1}|^2
+
\sqrt{m_1^2 + {\mbox{$\Delta m^2_{\mathrm{atm}} \ $}}\! \! - {\mbox{$ \Delta m^2_{\odot} \ $}}}
\cos^2 \theta_\odot (1 - |U_{\mathrm{e}1}|^2) e^{i\alpha_{21}}
\nonumber \\
& +
\sqrt{ m_1^2 + {\mbox{$\Delta m^2_{\mathrm{atm}} \ $}}} \sin^2 \theta_\odot
(1 - |U_{\mathrm{e}1}|^2) e^{i\alpha_{31} }
\Big|.
\label{eqmassinvhierarchy01} \end{aligned}$$ The new predictions for differ substantially from those obtained before the appearance of the latest SNO data due to the existence of a significant lower bound on for every value of $m_1$: even in the case of $m_1 \ll m_2 \cong m_3$ (i.e., even if $m_1 \ll 0.02$ eV), we get $${\mbox{$\left| < \! m \! > \right| \ $}}\gtap 8.5 \times 10^{-3}~{\rm eV}
\label{meffminIMH}$$ (see Fig. 1, right panels). Given the neutrino oscillation parameters, the minimal allowed value of depends on the values of the CP violating phases $\alpha_{21}$ and $\alpha_{31}$.
[**Case A: $m_1 < 0.02 \ {\rm eV}$, $m_1 \ll m_2 \simeq m_3$.** ]{}
1.0cm The effective Majorana mass can be considerably larger than in the case of a hierarchical neutrino mass spectrum [@BPP1; @BGGKP99]. The maximal value of corresponds to CP-conservation and $\eta_{21} = \eta_{31}= + 1$, and for given $m_1$ reads: $$\begin{aligned}
{\mbox{$\left| < \! m \! > \right| \ $}}_{\! \mbox{}_{\rm MAX}}
= & \ m_1 {\mbox{$|U_{\mathrm{e} 1}|^2_{ \! \mbox{}_{\mathrm{MIN}}}$ }}+
\Big( \sqrt{m_1^2 + {\mbox{$(\Delta m^2_{\mathrm{atm}})_{ \! \mbox{}_{\mathrm{MAX}}} \ $}}\! \! - {\mbox{$(\Delta m^2_{\odot})_{ \! \mbox{}_{\mathrm{MIN}}} \ $}}}
(\cos^2 \theta_\odot)_{\! \mbox{}_{\rm MIN}} \nonumber \\
& + \sqrt{m_1^2 + {\mbox{$(\Delta m^2_{\mathrm{atm}})_{ \! \mbox{}_{\mathrm{MAX}}} \ $}}}
(\sin^2 \theta_\odot)_{\! \mbox{}_{\rm MAX}}\Big)
(1 - {\mbox{$|U_{\mathrm{e} 1}|^2_{ \! \mbox{}_{\mathrm{MIN}}}$ }}) ,
\label{meffinvmin} \end{aligned}$$ where $(\cos^2 \! \! \theta_\odot)_{\mbox{}_{\rm MIN}}$ and $ (\sin^2 \! \! \theta_\odot)_{\mbox{}_{\rm MAX}}$ are the values corresponding to $(\tan^2 \theta_\odot)_{\mbox{}_{\rm MAX}}$, and ${\mbox{$|U_{\mathrm{e} 1}|^2_{ \! \mbox{}_{\mathrm{MIN}}}$ }}$ is the minimal allowed value of ${\mbox{$|U_{\mathrm{e} 1}|^2$}}$ for the ${\mbox{$(\Delta m^2_{\mathrm{atm}})_{ \! \mbox{}_{\mathrm{MAX}}} \ $}}$. For the allowed ranges - eqs. (1) and (2) for and $\tan^2\theta_{\odot}$, and the best fit values of the neutrino oscillation parameters, found in [@SNO2; @Gonza3nu], we get $|<m>|_{\mbox{}_{\rm MAX}} \simeq 0.080 \ {\rm eV}$ and $|<m>|_{\mbox{}_{\rm MAX}} \simeq 0.056 \ {\rm eV}$, respectively.
There exists a non-trivial lower bound on in the case of the LMA solution for which $\cos 2 \theta_\odot$ is found to be significantly different from zero. For the 99.73% C.L. allowed values of and $\cos 2 \theta_\odot$ [@SNO2], this lower bound reads: ${\mbox{$\left| < \! m \! > \right| \ $}}\gtap 8.5 \times 10^{-3}~{\rm eV}$. Using the best fit values of the oscillation parameters [@SNO2; @Gonza3nu], we find: $${\mbox{$\left| < \! m \! > \right| \ $}}\gtap 2.8 \times 10^{-2}~{\rm eV}.
\label{bfmeffmin}$$ The lower bound is present even for $\cos 2 \theta_\odot > 0.1$: in this case ${\mbox{$\left| < \! m \! > \right| \ $}}\gtap 4 \times 10^{-3}~{\rm eV}$. The minimal value of , ${\mbox{$\left| < \! m \! > \right| \ $}}_{\! \mbox{}_{\rm MIN}}$, is reached in the case of CP-invariance and $\eta_{21} = - \eta_{31} = - 1$, and is determined by: $$\begin{aligned}
{\mbox{$\left| < \! m \! > \right| \ $}}_{\! \mbox{}_{\rm MIN}}
= & \Big| \ m_1 {\mbox{$|U_{\mathrm{e} 1}|^2_{ \! \mbox{}_{\mathrm{MAX}}}$ }}-
\Big( \sqrt{m_1^2 + {\mbox{$(\Delta m^2_{\mathrm{atm}})_{ \! \mbox{}_{\mathrm{MIN}}} \ $}}\! \! - {\mbox{$ (\Delta m^2_{\odot})_{ \! \mbox{}_{\mathrm{MAX}}} \ $}}}
(\cos^2 \theta_\odot)_{\! \mbox{}_{\rm MIN}} \nonumber \\
& - \sqrt{m_1^2 + {\mbox{$(\Delta m^2_{\mathrm{atm}})_{ \! \mbox{}_{\mathrm{MIN}}} \ $}}} (\sin^2 \theta_\odot)_{\! \mbox{}_{\rm MAX}}
\Big) (1 - {\mbox{$|U_{\mathrm{e} 1}|^2_{ \! \mbox{}_{\mathrm{MAX}}}$ }}) \ \Big|,
\label{meffinvmina} \end{aligned}$$ where $(\cos^2 \! \! \theta_\odot)_{\mbox{}_{\rm MIN}}$ and $ (\sin^2 \! \! \theta_\odot)_{\mbox{}_{\rm MAX}}$ are the values corresponding to $(\tan^2 \theta_\odot)_{\mbox{}_{\rm MAX}}$, and is the maximal allowed value of ${\mbox{$|U_{\mathrm{e} 1}|^2$}}\ $ for the . In the two other CP conserving cases of $\eta_{21} = \eta_{31} = \pm 1$, the lower bound on depends weakly on the allowed values of $\theta_\odot$ and reads ${\mbox{$\left| < \! m \! > \right| \ $}}\gtap 0.03~{\rm eV}$.
If the neutrino mass spectrum is of the inverted hierarchy type, a sufficiently precise determination of , $\theta_\odot$ and (or a better upper limit on ), combined with a measurement of in the current or future -decay experiments, could allow one to get information on the difference of the Majorana CP-violating phases $(\alpha_{31} - \alpha_{21})$ [@BGKP96]. The value of $\sin^2 (\alpha_{31} - \alpha_{21})/2$ is related to the experimentally measurable quantities as follows [@BPP1; @BGKP96]: $$\sin^2 \frac{\alpha_{31} \! - \! \alpha_{21}}{2}
\simeq \Big( 1 - \frac{{\mbox{$\left| < \! m \! > \right| \ $}}^2}{(m_1^2 + {\mbox{$\Delta m^2_{\mathrm{atm}} \ $}}\!)
(1 - {\mbox{$|U_{\mathrm{e} 1}|^2$}})^2} \Big) \frac{1}{\sin^2 2 \theta_\odot}
\simeq \Big( 1 - \frac{{\mbox{$\left| < \! m \! > \right| \ $}}^2}{{\mbox{$\Delta m^2_{\mathrm{atm}} \ $}}\!
(1 - {\mbox{$|U_{\mathrm{e} 1}|^2$}})^2} \Big) \frac{1}{\sin^2 2 \theta_\odot},
\label{alpha2131}$$ ($m_1 < 0.02 \ {\rm eV}$). The constraints on $\sin^2 (\alpha_{31} - \alpha_{21})/2$ one could derive on the basis of eq. (\[alpha2131\]) are illustrated [^6] in Fig. 11 of ref. [@BPP1]. Obtaining an experimental upper limit on of the order of 0.03 eV would permit, in particular, to get a lower bound on the value of $\sin^2 (\alpha_{31} - \alpha_{21})/2$ and possibly exclude the CP conserving case corresponding to $\alpha_{31} - \alpha_{21}= 0$ (i.e., $\eta_{21}= \eta_{31}= \pm 1$).
[**Case B: Spectrum with Partial Inverted Hierarchy ($0.02 \ {\rm eV} \leq m_1 \leq 0.2 \ {\rm eV}$).**]{}
1.0cm The discussion and conclusions in the case of the spectrum with partial inverted hierarchy are identical to those in the same case for the neutrino mass spectrum with normal hierarchy given in sub-section 2.1, Case B, except for the maximal and minimal values of , ${\mbox{$\left| < \! m \! > \right| \ $}}_{ \! \mbox{}_{\rm MAX}}$ and ${\mbox{$\left| < \! m \! > \right| \ $}}_{ \! \mbox{}_{\rm MIN}}$, which for a fixed $m_1$ are determined by: $$\begin{aligned}
\label{maxmeffLMAinva}
{\mbox{$\left| < \! m \! > \right| \ $}}_{ \! \mbox{}_{\rm MAX}} \! \simeq & \
m_1 {\mbox{$|U_{\mathrm{e} 1}|^2_{ \! \mbox{}_{\mathrm{MIN}}}$ }}+
\sqrt{m_1^2 + {\mbox{$(\Delta m^2_{\mathrm{atm}})_{ \! \mbox{}_{\mathrm{MAX}}} \ $}}}(1 - {\mbox{$|U_{\mathrm{e} 1}|^2_{ \! \mbox{}_{\mathrm{MIN}}}$ }}\! ) , \\
\label{maxmeffLMAinvb}
{\mbox{$\left| < \! m \! > \right| \ $}}_{ \! \mbox{}_{\rm MIN} } \! \simeq &
\left| m_1 {\mbox{$|U_{\mathrm{e} 1}|^2_{ \! \mbox{}_{\mathrm{MAX}}}$ }}- \sqrt{m_1^2 + {\mbox{$(\Delta m^2_{\mathrm{atm}})_{ \! \mbox{}_{\mathrm{MIN}}} \ $}}}
(\cos 2 \theta_\odot )_{\! \mbox{}_{\rm MIN}} \! (1 - {\mbox{$|U_{\mathrm{e} 1}|^2_{ \! \mbox{}_{\mathrm{MAX}}}$ }}\! \! )
\right|,\end{aligned}$$ ${\mbox{$|U_{\mathrm{e} 1}|^2_{ \! \mbox{}_{\mathrm{MIN}}}$ }}$ (${\mbox{$|U_{\mathrm{e} 1}|^2_{ \! \mbox{}_{\mathrm{MAX}}}$ }}$) in eq. (\[maxmeffLMAinva\]) (in eq. (\[maxmeffLMAinvb\])) being the minimal (maximal) allowed value of ${\mbox{$|U_{\mathrm{e} 1}|^2$}}$ given the maximal (minimal) value ${\mbox{$(\Delta m^2_{\mathrm{atm}})_{ \! \mbox{}_{\mathrm{MAX}}} \ $}}$ (${\mbox{$(\Delta m^2_{\mathrm{atm}})_{ \! \mbox{}_{\mathrm{MIN}}} \ $}}$).
For any $m_1 \geq 0.02 \ {\rm eV}$, the lower bound on reads: ${\mbox{$\left| < \! m \! > \right| \ $}}\gtap 0.01 \ {\rm eV}$. Using the best fit values of the neutrino oscillation parameters, obtained in [@SNO2; @Gonza3nu], one finds: ${\mbox{$\left| < \! m \! > \right| \ $}}\gtap 0.03 \ {\rm eV}$.
Quasi-Degenerate Mass Spectrum ($m_1 > 0.2 \ {\rm eV}$, $m_1 \simeq m_2 \simeq m_3
\simeq m_{\bar{\nu}_e}$)
-----------------------------------------------------------------------------------
1.0cm The new element in the predictions for in the case of quasi-degenerate neutrino mass spectrum, $m_1 > 0.2 \ {\rm eV}$, is the existence of a lower bound on the possible values of (Fig. 1). The lower limit on is reached in the case of CP-conservation and $\eta_{21}= \eta_{31}= - 1$. One finds a significant lower limit, ${\mbox{$\left| < \! m \! > \right| \ $}}\gtap 0.01 \ {\rm eV}$, if $$(\cos 2 \theta_\odot )_{\! \mbox{}_{\rm MIN}} >
{\rm max} \left (0.05, 1.5
\, {\mbox{$|U_{\mathrm{e} 3}|^2_{ \! \mbox{}_{\mathrm{MAX}}}$ }}/ (1 - {\mbox{$|U_{\mathrm{e} 3}|^2_{ \! \mbox{}_{\mathrm{MAX}}}$ }}\! \! ) \right).$$ More specifically, using the best fit value, and the $90 \%$ C.L. and the $99.73 \%$ C.L. allowed values, of $\cos2\theta_{\odot}$ from [@SNO2], we obtain, respectively: ${\mbox{$\left| < \! m \! > \right| \ $}}\geq 0.10 \ {\rm eV}$, ${\mbox{$\left| < \! m \! > \right| \ $}}\geq 0.06 \ {\rm eV}$ and $${\mbox{$\left| < \! m \! > \right| \ $}}\geq 0.035~{\rm eV}
\label{meffminQDS}$$ (Figs. 1 and 3). These values of $ {\mbox{$\left| < \! m \! > \right| \ $}}$ are in the range of sensitivity of the current and future -decay experiments.
The upper bound on , which corresponds to CP-conservation and $\eta_{21} = \eta_{31} = +1 $ ($\nu_{1,2,3}$ possessing the same CP-parities), can be found for a given $m_1$ by using eq. (\[meffmaxhierLMA\]). For the allowed values of $m_1 > 0.2 \ {\rm eV}$ (which is limited from above by the $^{3}$H $\beta-$decay data [@MoscowH3; @Mainz], $m_{1,2,3} \simeq m_{\bar{\nu}_e}$), ${\mbox{$\left| < \! m \! > \right| \ $}}_{ \! \mbox{}_{\rm MAX}}$ is limited by the upper bounds obtained in the -decay experiments: ${\mbox{$\left| < \! m \! > \right| \ $}}< 0.35~{\rm eV}$ [@76Ge00] and ${\mbox{$\left| < \! m \! > \right| \ $}}< ( 0.33 - 1.35)~{\rm eV}$ [@IGEX00].
In the case of CP conservation and $\eta_{21}= \pm \eta_{31}= + 1$, is constrained to lie in the interval [@BPP1] $m_{\bar{\nu}_\mathrm{e}} ( 1 - 2 {\mbox{$|U_{\mathrm{e} 3}|^2_{ \! \mbox{}_{\mathrm{MAX}}}$ }}) \leq
{\mbox{$\left| < \! m \! > \right| \ $}}\leq m_{\bar{\nu}_\mathrm{e}}$. An upper limit on would lead to an upper limit on $m_{\bar{\nu}_\mathrm{e}}$ which is more stringent than the one obtained in the present [$\mbox{}^3 {\rm H}$ $\beta$-decay ]{}experiments: for ${\mbox{$\left| < \! m \! > \right| \ $}}< 0.35 \ (1.05) \ {\rm eV}$ we have $m_{\bar{\nu}_\mathrm{e}} < 0.41 \ (1.23) \ {\rm eV}$. Furthermore, the upper limit ${\mbox{$\left| < \! m \! > \right| \ $}}< 0.2\ {\rm eV}$ would permit to exclude the CP-parity pattern $\eta_{21}= \pm \eta_{31}= + 1$ for the quasi-degenerate neutrino mass spectrum.
If the CP-symmetry holds and $\eta_{21}= \pm \eta_{31}= - 1$, there are both an upper and a lower limits on , $m_{\bar{\nu}_\mathrm{e}}
((\cos 2 \theta_\odot )_{\! \mbox{}_{\rm MIN}} ( 1 - {\mbox{$|U_{\mathrm{e} 3}|^2_{ \! \mbox{}_{\mathrm{MIN}}} $ }})
+ {\mbox{$|U_{\mathrm{e} 3}|^2_{ \! \mbox{}_{\mathrm{MIN}}} $ }}) \leq {\mbox{$\left| < \! m \! > \right| \ $}}\leq m_{\bar{\nu}_\mathrm{e}}
((\cos 2 \theta_\odot )_{\! \mbox{}_{\rm MAX}} ( 1 - {\mbox{$|U_{\mathrm{e} 3}|^2_{ \! \mbox{}_{\mathrm{MAX}}}$ }})
+ {\mbox{$|U_{\mathrm{e} 3}|^2_{ \! \mbox{}_{\mathrm{MAX}}}$ }})$. Using eq. (\[thLMA\]) and the results on ${\mbox{$|U_{\mathrm{e} 3}|$}}^2$ from ref. [@Gonza3nu], one finds $ 0.26 \ m_{\bar{\nu}_\mathrm{e}} \leq {\mbox{$\left| < \! m \! > \right| \ $}}\leq 0.67 \ m_{\bar{\nu}_\mathrm{e}}$. Given the allowed values of $\cos2\theta_{\odot}$, eq. (\[thLMA\]), the observation of the -decay in the present and/or future -decay experiments, combined with a sufficiently stringent upper bound on $m_{\bar{\nu}_e} \simeq m_{1,2,3}$ from the tritium beta-decay experiments, $m_{\bar{\nu}_e} <
{\mbox{$\left| < \! m \! > \right| \ $}}_{exp} / ((\cos 2 \theta_\odot )_{\! \mbox{}_{\rm MAX}}
(1 - {\mbox{$|U_{\mathrm{e} 3}|^2_{ \! \mbox{}_{\mathrm{MAX}}}$ }}\! \! ) + {\mbox{$|U_{\mathrm{e} 3}|^2_{ \! \mbox{}_{\mathrm{MAX}}}$ }})$, would allow one, in particular, to exclude the case of CP-conservation with $\eta_{21}=\pm \eta_{31}= - 1$ (Fig. 2).
For values of , which are in the range of sensitivity of the future -decay experiments, there exists a “just-CP-violation” region. This is illustrated in Fig. 2, where we show ${\mbox{$\left| < \! m \! > \right| \ $}}/ m_1$ for the case of quasi-degenerate neutrino mass spectrum, $m_1 > 0.2 \ {\rm eV}$, $m_1 \simeq m_2 \simeq m_3 \simeq m_{\bar{\nu}_e}$, as a function of $\cos 2 \theta_\odot$. The “just-CP-violation” interval of values of ${\mbox{$\left| < \! m \! > \right| \ $}}/m_1$ is determined by $$(\cos 2 \theta_\odot )_{\! \mbox{}_{\rm MAX}}
(1 - {\mbox{$|U_{\mathrm{e} 3}|^2_{ \! \mbox{}_{\mathrm{MAX}}}$ }}\! \! ) + {\mbox{$|U_{\mathrm{e} 3}|^2_{ \! \mbox{}_{\mathrm{MAX}}}$ }}<
\frac{{\mbox{$\left| < \! m \! > \right| \ $}}}{m_{\bar{\nu}_e}} < 1 - 2 {\mbox{$|U_{\mathrm{e} 3}|^2_{ \! \mbox{}_{\mathrm{MAX}}}$ }}.
\label{CPviol}$$ Taking into account eq. (\[thLMA\]) and the existing limits on $|U_{{\rm e} 3}|^2$, this gives $0.67 < {\mbox{$\left| < \! m \! > \right| \ $}}/m_{\bar{\nu}_e} < 0.85$. Information about the masses $m_{1,2,3} \cong m_{\bar{\nu}_e}$ can be obtained in the KATRIN experiment [@KATRIN].
A rather precise determination of , $m_1 \cong m_{\bar{\nu}_e}$, $\theta_{\odot}$ and $|U_{\mathrm{e}3}|^2$ would imply an interdependent constraint on the two CP-violating phases $\alpha_{21}$ and $\alpha_{31}$ [@BPP1] (see Fig. 16 in [@BPP1]). For $m_1 \equiv m_{\bar{\nu}_e} > 0.2 $ eV, the CP-violating phase $\alpha_{21}$ could be tightly constrained if ${\mbox{$|U_{\mathrm{e} 3}|$}}^2$ is sufficiently small and the term in containing it can be neglected, as is suggested by the current limits on ${\mbox{$|U_{\mathrm{e} 3}|$}}^2$: $$\sin^2 \frac{\alpha_{21}}{2} \simeq
\Big( 1 - \frac{{\mbox{$\left| < \! m \! > \right| \ $}}^2}{m_{\bar{\nu}_e}^2} \Big)
\frac{1}{\sin^2 2 \theta_\odot}.
\label{CPviol1}$$ The term which depends on the CP-violating phase $\alpha_{31}$ in the expression for , is suppressed by the factor ${\mbox{$|U_{\mathrm{e} 3}|$}}^2$. Therefore the constraint one could possibly obtain on $\cos \alpha_{31}$ is trivial (Fig. 16 in [@BPP1]), unless $ {\mbox{$|U_{\mathrm{e} 3}|$}}^2 \sim {\cal O} ( \sin^2 \theta_\odot)$.
The Effective Majorana Mass and the Determination of the Neutrino Mass Spectrum
===============================================================================
1.0cm The existence of a lower bound on in the cases of inverted mass hierarchy (${\mbox{$ \Delta m^2_{\odot} \ $}}= \Delta m^2_{32}$) and quasi-degenerate neutrino mass spectrum, eqs. (\[meffminIMH\]) and (\[meffminQDS\]), implies that the future -decay experiments might allow to determine the type of the neutrino mass spectrum (under the general assumptions of 3-neutrino mixing and massive Majorana neutrinos, -decay generated only by the (V-A) charged current weak interaction via the exchange of the three Majorana neutrinos, neutrino oscillation explanation of the solar and atmospheric neutrino data). This conclusion is valid not only under the assumption that the -decay will be observed in these experiments and will be measured, but also in the case only a sufficiently stringent upper limit on will be derived.
More specifically, as is illustrated in Fig. 3, the following statements can be made:
1. a measurement of ${\mbox{$\left| < \! m \! > \right| \ $}}= {\mbox{$\left| < \! m \! > \right| \ $}}_{exp} >
0.20 \ {\rm eV}$, would imply that the neutrino mass spectrum is of the quasi-degenerate type ($m_1 > 0.20 \ {\rm eV}$) and that there are both a lower and an upper limit on $m_1$, $(m_1)_{min} \leq m_1 \leq (m_1)_{max}$. The values of $(m_1)_{max}$ and $(m_1)_{min}$ are fixed respectively by the equalities ${\mbox{$\left| < \! m \! > \right| \ $}}_{\mbox{}_{\rm MIN}} = {\mbox{$\left| < \! m \! > \right| \ $}}_{ \! exp}$ and ${\mbox{$\left| < \! m \! > \right| \ $}}_{\mbox{}_{\rm MAX}} = {\mbox{$\left| < \! m \! > \right| \ $}}_{ \! exp}$, where ${\mbox{$\left| < \! m \! > \right| \ $}}_{\mbox{}_{\rm MIN}}$ and ${\mbox{$\left| < \! m \! > \right| \ $}}_{\mbox{}_{\rm MAX}}$ are given by eqs. (\[minmeffLMA\]) and (\[meffmaxhierLMA\]);
2. if is measured and is found to lie in the interval $8.5 \times 10^{-2} \ {\rm eV}\ltap
{\mbox{$\left| < \! m \! > \right| \ $}}_{exp} \ltap 0.20 \ {\rm eV}$, one could conclude that either
i\) ${\mbox{$ \Delta m^2_{\odot} \ $}}\equiv {\mbox{$ \ \Delta m^2_{21} \ $}}$ and the spectrum is of the quasi-degenerate type ($m_1 > 0.20 \ {\rm eV}$) or with partial hierarchy ($0.02 \ {\rm eV} \leq m_1 \leq 0.2 \ {\rm eV}$), with $ 8.4 \times 10^{-2} \ {\rm eV} \ltap m_1
\ltap 1.2\ {\rm eV}$, where the maximal and minimal values of $m_1$ are determined as in the [*Case 1*]{};
or that ii) ${\mbox{$ \Delta m^2_{\odot} \ $}}\equiv {\mbox{$ \ \Delta m^2_{32} \ $}}$ and the spectrum is quasi-degenerate ($m_1 > 0.20 \ {\rm eV}$) or with partial inverted hierarchy ($ 0.02 \ {\rm eV} \leq m_1 \leq 0.2 \ {\rm eV}$), with $(m_1)_{min} = 2.0 \times 10^{-2} \ {\rm eV}$ and $ (m_1)_{max} = 1.2 \ {\rm eV}$, where $(m_1)_{max}$ and $(m_1)_{min}$ are given by the equalities ${\mbox{$\left| < \! m \! > \right| \ $}}_{\mbox{}_{\rm MIN}} = {\mbox{$\left| < \! m \! > \right| \ $}}_{ \! exp}$ and ${\mbox{$\left| < \! m \! > \right| \ $}}_{\mbox{}_{\rm MAX}} = {\mbox{$\left| < \! m \! > \right| \ $}}_{ \! exp}$, and ${\mbox{$\left| < \! m \! > \right| \ $}}_{\mbox{}_{\rm MIN}}$ and ${\mbox{$\left| < \! m \! > \right| \ $}}_{\mbox{}_{\rm MAX}}$ are determined by eqs. (\[maxmeffLMAinvb\]) and (\[maxmeffLMAinva\]);
3. a measured value of satisfying $8.5 \times 10^{-3} \ {\rm eV}\ltap
{\mbox{$\left| < \! m \! > \right| \ $}}_{exp} \ltap 8.0 \times 10^{-2} \ {\rm eV}$, would imply that (see Fig. 3) either
i\) ${\mbox{$ \Delta m^2_{\odot} \ $}}\equiv {\mbox{$ \ \Delta m^2_{21} \ $}}$ and the spectrum is of quasi-degenerate type ($m_1 > 0.20 \ {\rm eV}$), with $(m_1)_{max} \ltap 0.48 \ {\rm eV}$, or with partial hierarchy ($0.02 \ {\rm eV} \leq m_1 \leq 0.2 \ {\rm eV}$),
or that ii) ${\mbox{$ \Delta m^2_{\odot} \ $}}\equiv {\mbox{$ \ \Delta m^2_{32} \ $}}$ and the spectrum is quasi-degenerate ($m_1 > 0.20 \ {\rm eV}$), or with partial inverted hierarchy ($0.02 \ {\rm eV} \leq m_1 \leq 0.2 \ {\rm eV}$), or with inverted hierarchy ($m_1 < 0.02 \ {\rm eV}$), with only a significant upper bound on $m_1$, $ (m_1)_{min} = 0$, $(m_1)_{max} \ltap 0.48 \ {\rm eV}$, where $(m_1)_{max}$ is determined by the equation ${\mbox{$\left| < \! m \! > \right| \ $}}_{\mbox{}_{\rm MIN}} = {\mbox{$\left| < \! m \! > \right| \ $}}_{ \! exp}$, with ${\mbox{$\left| < \! m \! > \right| \ $}}_{\mbox{}_{\rm MIN}}$ given by eq. (\[maxmeffLMAinvb\]);
4. a measurement or an upper limit on , $ {\mbox{$\left| < \! m \! > \right| \ $}}\ltap 8.0 \times 10^{-3} \ {\rm eV}$, would lead to the conclusion that the neutrino mass spectrum is of the normal mass hierarchy type, ${\mbox{$ \Delta m^2_{\odot} \ $}}\equiv {\mbox{$ \ \Delta m^2_{21} \ $}}$, and that $m_1$ is limited from above by $ m_1 \leq (m_1)_{max} \simeq 5.8 \times 10^{-2} \ {\rm eV}$, where $(m_1)_{max}$ is determined by the condition ${\mbox{$\left| < \! m \! > \right| \ $}}_{\mbox{}_{\rm MIN}} = {\mbox{$\left| < \! m \! > \right| \ $}}_{ \! exp}$, with ${\mbox{$\left| < \! m \! > \right| \ $}}_{\mbox{}_{\rm MIN}}$ given by eq. (\[minmeffLMA\]). For the allowed values of the oscillation parameters (at a given confidence level, Fig. 3), an upper bound on , ${\mbox{$\left| < \! m \! > \right| \ $}}< 8 \times 10^{-4} \ {\rm eV}$, would imply an upper limit on $m_1$, $m_1 < 0.01 \ {\rm eV}$ - Fig. 3, middle panel, and $m_1 < 0.025 \ {\rm eV}$ - Fig. 3, lower panel. For the best fit values of , , $\theta_\odot$ and $\theta$, the bound ${\mbox{$\left| < \! m \! > \right| \ $}}\ltap 8 \times 10^{-4} \ {\rm eV}$ would lead to a rather narrow interval of possible values of $m_1$, $ 1 \times 10^{-3} \ {\rm eV}
< m_1 < 4 \times 10^{-3} \ {\rm eV}$ (Fig. 3, upper panel).
Thus, a measured value of (or an upper limit on) the effective Majorana mass ${\mbox{$\left| < \! m \! > \right| \ $}}\ltap 0.03 \ {\rm eV}$ would disfavor (if not rule out) the quasi-degenerate mass spectrum, while a value of ${\mbox{$\left| < \! m \! > \right| \ $}}\ltap 8 \times 10^{-3} \ {\rm eV}$ would rule out the quasi-degenerate mass spectrum, disfavor the spectrum with inverted mass hierarchy and favor the hierarchical neutrino mass spectrum.
Using the best fit values of ${\mbox{$ \Delta m^2_{\odot} \ $}}$, $\cos2\theta_\odot$ from [@SNO2] and of ${\mbox{$\Delta m^2_{\mathrm{atm}} \ $}}$ and $\sin^2\theta$ from [@Gonza3nu], we have found that (Fig. 3, upper panel): i) ${\mbox{$\left| < \! m \! > \right| \ $}}\ltap 2.0\times 10^{-3}~{\rm eV}$ in the case of neutrino mass spectrum with normal hierarchy, ii) $2.8\times 10^{-2}~{\rm eV} \ltap {\mbox{$\left| < \! m \! > \right| \ $}}\ltap 5.6\times 10^{-2}~{\rm eV}$ if the spectrum is with inverted hierarchy, and iii) ${\mbox{$\left| < \! m \! > \right| \ $}}\gtap 0.10~{\rm eV}$ for the quasi-degenerate mass spectrum. Therefore, if ${\mbox{$\Delta m^2_{\mathrm{atm}} \ $}}$, ${\mbox{$ \Delta m^2_{\odot} \ $}}$ and $\cos2\theta_\odot$ will be determined with a high precision ($\sim (10-15) \%$ uncertainty) using the data from the MINOS, KamLAND and the solar neutrino experiments and their best fit values will not change substantially with respect to those used in the present analysis [^7], a measurement of ${\mbox{$\left| < \! m \! > \right| \ $}}\gtap 0.03~{\rm eV}$ would rule out a hierarchical neutrino mass spectrum ($m_1 < m_2 \ll m_3$) even if there exists a factor of $\sim 6$ (or smaller) uncertainty in the value of due to a poor knowledge of the corresponding nuclear matrix element(s). An experimental upper limit of ${\mbox{$\left| < \! m \! > \right| \ $}}< 0.01~{\rm eV}$ suffering from the same factor of $\sim 6$ (or smaller) uncertainty would rule out the quasi-degenerate mass spectrum, while if the uncertainty under discussion is only by a factor which is not bigger than $\sim 3.0$, the spectrum with inverted hierarchy would be strongly disfavored (if not ruled out).
If the minimal value of $\cos2\theta_\odot$ inferred from the solar neutrino data, is somewhat smaller than that in eq. (\[thLMA\]), the upper bound on in the case of neutrino mass spectrum with normal hierarchy (${\mbox{$ \Delta m^2_{\odot} \ $}}\equiv {\mbox{$ \ \Delta m^2_{21} \ $}}$, $m_1 \ll 0.02$ eV) might turn out to be larger than the lower bound on in the case of spectrum with inverted mass hierarchy (${\mbox{$ \Delta m^2_{\odot} \ $}}\equiv {\mbox{$ \ \Delta m^2_{32} \ $}}$, $m_1 \ll 0.02$ eV). Thus, there will be an overlap between the regions of allowed values of in the two cases of neutrino mass spectrum at $m_1 \ll 0.02 \ {\rm eV}$. The minimal value of $\cos 2 \theta_\odot$ for which [*the two regions do not overlap*]{} is determined by the condition: $$(\cos 2 \theta_\odot)_{\mbox{}_{\rm MIN}}
= \frac{\sqrt{{\mbox{$ (\Delta m^2_{\odot})_{ \! \mbox{}_{\mathrm{MAX}}} \ $}}} + 2 \sqrt{{\mbox{$(\Delta m^2_{\mathrm{atm}})_{ \! \mbox{}_{\mathrm{MAX}}} \ $}}}
(\sin^2 \theta)_{\mbox{}_{\rm MAX}}}
{ 2 \sqrt{{\mbox{$(\Delta m^2_{\mathrm{atm}})_{ \! \mbox{}_{\mathrm{MIN}}} \ $}}} + \sqrt{{\mbox{$ (\Delta m^2_{\odot})_{ \! \mbox{}_{\mathrm{MAX}}} \ $}}}}
+ {\cal O} \Big( \frac{{\mbox{$ (\Delta m^2_{\odot})_{ \! \mbox{}_{\mathrm{MAX}}} \ $}}}{4 {\mbox{$(\Delta m^2_{\mathrm{atm}})_{ \! \mbox{}_{\mathrm{MIN}}} \ $}}} \Big),$$ where we have neglected terms of order $(\sin^2 \theta)^2_{\mbox{}_{\rm MAX}}$. For the values of the neutrino oscillation parameters used in the present analysis this “border” value turns out to be $\cos 2 \theta_\odot \cong 0.25$.
Let us note that [@PPW] if the -decay is not observed, a measured value of $m_{\bar{\nu}_e}$ in [$\mbox{}^3 {\rm H}$ $\beta$-decay ]{}experiments, $(m_{\bar{\nu}_e})_{exp} \gtap 0.35$ eV, which is larger than $(m_1)_{max}$, $(m_{\nu_e})_{exp} > (m_1)_{max}$, where $(m_1)_{max}$ is determined as in the [*Case 1*]{} (i.e., from the upper limit on , ${\mbox{$\left| < \! m \! > \right| \ $}}_{\mbox{}_{\rm MIN}} = {\mbox{$\left| < \! m \! > \right| \ $}}_{ \! exp}$, with ${\mbox{$\left| < \! m \! > \right| \ $}}_{\mbox{}_{\rm MIN}}$ given in eq. (\[minmeffLMA\])), might imply that the massive neutrinos are Dirac particles. If the -decay has been observed and measured, the inequality $(m_{\bar{\nu}_e})_{exp} > (m_1)_{max}$, would lead to the conclusion that there exist contribution(s) to the -decay rate other than due to the light Majorana neutrino exchange which partially cancel the contribution due to the Majorana neutrino exchange.
A measured value of , $( {\mbox{$\left| < \! m \! > \right| \ $}})_{exp} > 0.08 \ \mathrm{eV}$, and a measured value of $m_{\bar{\nu}_e}$ or an upper bound on $m_{\bar{\nu}_e}$, such that $m_{\bar{\nu}_e} < (m_1)_{min}$, where $(m_1)_{min}$ is determined by the condition ${\mbox{$\left| < \! m \! > \right| \ $}}_{\mbox{}_{\rm MAX}} = {\mbox{$\left| < \! m \! > \right| \ $}}_{ \! exp}$, with ${\mbox{$\left| < \! m \! > \right| \ $}}_{\mbox{}_{\rm MAX}}$ given by eq. (\[maxmeffLMAinva\]), would imply that [@PPW] there are contributions to the -decay rate in addition to the ones due to the light Majorana neutrino exchange (see, e.g., [@bb0nunmi]), which enhance the -decay rate. This would signal the existence of new $\Delta L =2$ processes beyond those induced by the light Majorana neutrino exchange in the case of left-handed charged current weak interaction.
Conclusions
===========
1.0truecm Assuming 3-$\nu$ mixing and massive Majorana neutrinos, we have analyzed the implications of the results of the solar neutrino experiments, including the latest SNO data, which favor the LMA MSW solution of the solar neutrino problem with $\tan^2\theta_{\odot} < 1$, for the predictions of the effective Majorana mass in -decay. Neutrino mass spectra with normal mass hierarchy, with inverted hierarchy and of quasi-degenerate type are considered. For $\cos 2\theta_{\odot} \geq 0.26$, which follows (at 99.73% C.L.) from the analysis of the solar neutrino data performed in [@SNO2], we find significant lower limits on in the cases of quasi-degenerate and inverted hierarchy neutrino mass spectrum, ${\mbox{$\left| < \! m \! > \right| \ $}}\gtap 0.03$ eV and ${\mbox{$\left| < \! m \! > \right| \ $}}\gtap 8.5\times 10^{-3}$ eV, respectively. If the neutrino mass spectrum is hierarchical (with inverted hierarchy), the upper limit holds ${\mbox{$\left| < \! m \! > \right| \ $}}\ltap 8.2 \times 10^{-3}~(8.0 \times 10^{-2})$ eV. Correspondingly, not only a measured value of ${\mbox{$\left| < \! m \! > \right| \ $}}\neq 0$, but even an experimental upper limit on of the order of ${\rm few} \times 10^{-2}$ eV can provide information on the type of the neutrino mass spectrum; it can provide also a significant upper limit on the mass of the lightest neutrino $m_1$. Further reduction of the LMA solution region due to data, e.g., from the experiments SNO, KamLAND and BOREXINO, leading, in particular, to an increase (a decreasing) of the current lower (upper) bound of $\cos 2\theta_{\odot}$ can strengthen further the above conclusions.
Using the best fit values of ${\mbox{$ \Delta m^2_{\odot} \ $}}$, $\cos2\theta_\odot$ from [@SNO2] and of ${\mbox{$\Delta m^2_{\mathrm{atm}} \ $}}$ and $\sin^2\theta$ from [@Gonza3nu], we have found that (Fig. 3, upper panel): i) ${\mbox{$\left| < \! m \! > \right| \ $}}\ltap 2.0\times 10^{-3}~{\rm eV}$ in the case of neutrino mass spectrum with normal hierarchy, ii) $2.8\times 10^{-2}~{\rm eV} \ltap {\mbox{$\left| < \! m \! > \right| \ $}}\ltap 5.6\times 10^{-2}~{\rm eV}$ if the spectrum is with inverted hierarchy, and iii) ${\mbox{$\left| < \! m \! > \right| \ $}}\gtap 0.10~{\rm eV}$ for the quasi-degenerate neutrino mass spectrum. Therefore, if ${\mbox{$\Delta m^2_{\mathrm{atm}} \ $}}$, ${\mbox{$ \Delta m^2_{\odot} \ $}}$ and $\cos2\theta_\odot$ will be determined with a high precision ($\sim (10-15) \%$ uncertainty) using the data from the MINOS, KamLAND and the solar neutrino experiments and their best fit values will not change substantially with respect to those used in the present analysis, a measurement of ${\mbox{$\left| < \! m \! > \right| \ $}}\gtap 0.03~{\rm eV}$ would rule out a hierarchical neutrino mass spectrum ($m_1 < m_2 \ll m_3$) even if there exists a factor of $\sim 6$ uncertainty in the value of due to a poor knowledge of the corresponding nuclear matrix element(s). An experimental upper limit of ${\mbox{$\left| < \! m \! > \right| \ $}}< 0.01~{\rm eV}$ suffering from the same factor of $\sim 6$ (or smaller) uncertainty would rule out the quasi-degenerate neutrino mass spectrum, while if the uncertainty under discussion is by a factor not bigger than $\sim 3.0$, the spectrum with inverted hierarchy would be strongly disfavored (if not ruled out).
Finally, a measured value of ${\mbox{$\left| < \! m \! > \right| \ $}}\gtap 0.2$ eV, which would imply a quasi-degenerate neutrino mass spectrum, combined with data on neutrino masses from the $^3$H $\beta-$decay experiment KATRIN (an upper limit or a measured value [^8]), might allow to establish whether the CP-symmetry is violated in the lepton sector.
After the work on the present study was essentially completed, few new global analyses of the solar neutrino data have appeared [@BargerSNO2; @StrumiaSNO2; @GoswaSNO2; @ConchaSNO3]. The results obtained in [@BargerSNO2] do not differ substantially from those derived in [@SNO2]; in particular, the (99.73% C.L.) minimal allowed values of $\cos 2\theta_{\odot}$ in the LMA solution region found in [@SNO2] and in [@BargerSNO2] practically coincide. The best fit values of and $\cos 2\theta_{\odot}$ found in [@SNO2; @BargerSNO2; @GoswaSNO2; @ConchaSNO3] also practically coincide, with $\cos 2\theta_{\odot}|_{\mbox{}_\mathrm{BF}}$ lying in the interval (0.41 - 0.50) and ${\mbox{$ \Delta m^2_{\odot} \ $}}|_{\mbox{}_\mathrm{BF}} \simeq 5 \times 10^{-5}~{\rm eV^2}$. The authors of [@StrumiaSNO2] find a similar $\cos 2\theta_{\odot}|_{\mbox{}_\mathrm{BF}}$, but a somewhat larger ${\mbox{$ \Delta m^2_{\odot} \ $}}|_{\mbox{}_\mathrm{BF}} \simeq 7.9\times 10^{-5}~{\rm eV^2}$. According to [@StrumiaSNO2], [@GoswaSNO2] and [@ConchaSNO3], the lower limit $\cos 2\theta_{\odot} > 0.25$ holds approximately at 94% C.L., 90% C.L. and 81% C.L., respectively. Larger maximal allowed values of than that given in eq. (\[dmsolLMA\]) - of the order of $(4 - 5)\times 10^{-4}~{\rm eV^2}$ (99.73% C.L.), have been obtained in the analyses performed in [@StrumiaSNO2; @GoswaSNO2; @ConchaSNO3]. The authors of [@SNO2; @BargerSNO2] used the full SNO data on the day and night event spectra [@SNO2] in their analyses, while the authors of [@StrumiaSNO2; @GoswaSNO2; @ConchaSNO3] did not use at all or used only part of these data.
[99]{}
SNO Collaboration, Q.R. Ahmad et al., nucl-ex/0204009.
SNO Collaboration, Q.R. Ahmad et al., nucl-ex/0204008.
SNO Collaboration, Q.R. Ahmad et al., [*Phys. Rev. Lett.*]{} [**87**]{} (2001) 071301 (nucl-ex/0106015).
Super-Kamiokande Collaboration, Y. Fukuda [*et al.*]{}, [*Phys. Rev. Lett.* ]{} [**86**]{} (2001) 5651 and 5656.
B.T. Cleveland et al., [*Astrophys. J.*]{} [**496**]{} (1998) 505; Y. Fukuda [*et al.*]{}, [*Phys. Rev. Lett. *]{} [**77**]{} (1996) 1683; V. Gavrin, [*Nucl. Phys. Proc. Suppl.*]{} [**91**]{} (2001) 36; W. Hampel [*et al.*]{}, [*Phys. Lett. *]{} [**B447**]{} (1999) 127; M. Altmann [*et al.*]{}, [*Phys. Lett. *]{} [**B490**]{} (2000) 16.
Super-Kamiokande Collaboration, H. Sobel et al., [*Nucl. Phys. Proc. Suppl.*]{} [**91**]{} (2001) 127.
S. M. Bilenky, C. Giunti and W. Grimus, [*Prog. Part. Nucl. Phys.*]{} [**43**]{} (1999) 1.
S.T. Petcov, hep-ph/9907216.
S.M. Bilenky and S.T. Petcov, [*Rev. Mod. Phys.*]{} [**59**]{} (1987) 67.
S.M. Bilenky, S. Pascoli and S.T. Petcov, [*Phys. Rev.*]{} [**D64**]{} (2001) 053010.
S. Pascoli, S.T. Petcov and L. Wolfenstein, [*Phys. Lett.*]{} [**B524**]{} (2002) 319; S. Pascoli and S.T. Petcov, hep-ph/0111203.
B. Pontecorvo, [*Zh. Eksp. Teor. Fiz.*]{} [**33**]{}, 549 (1957), and [**34**]{}, 247 (1958).
Z. Maki, M. Nakagawa and S. Sakata, [*Prog. Theor. Phys.*]{} [**28**]{} (1962) 870.
S. M. Bilenky *et al.*, [*Phys. Lett.*]{} [**B94**]{} (1980) 495.
M. Doi *et al.*, [*Phys. Lett.*]{} **B102** (1981) 323.
L. Wolfenstein, [*Phys. Lett.*]{} [**B107**]{} (1981) 77.
S. M. Bilenky, N. P. Nedelcheva and S. T. Petcov, [*Nucl. Phys.*]{} **B247** (1984) 589; B. Kayser, [*Phys. Rev.*]{} [**D30**]{} (1984) 1023.
H.V. Klapdor-Kleingrothaus *et al.*, [*Mod. Phys. Lett.*]{} [**16**]{} (2001) 2409.
F. Feruglio, A. Strumia and F. Vissani, hep-ph/0201291.
C.E. Aalseth *et al.*, hep-ex/0202018.
S. T. Petcov and A. Yu. Smirnov, [*Phys. Lett.*]{} **B322** (1994) 109.
S.M. Bilenky *et al.*, [*Phys. Rev.*]{} **D54** (1996) 4432.
S.M. Bilenky *et al.*, [*Phys. Lett.*]{} [**B465**]{} (1999) 193.
V. Barger and K. Whisnant, [*Phys. Lett.*]{} **B456** (1999) 194; H. Minakata and O. Yasuda, [*Nucl. Phys.*]{} **B523** (1998) 597; T. Fukuyama *et al.*, [*Phys. Rev.*]{} **D57** (1998) 5844 and hep-ph/0204254; P. Osland and G. Vigdel, [*Phys. Lett.*]{} **B520** (2001) 128; D. Falcone and F. Tramontano, [*Phys. Rev.*]{} **D64** (2001) 077302; T. Hambye, hep-ph/0201307.
S.M. Bilenky, S. Pascoli and S.T. Petcov, [*Phys. Rev.*]{} [**D64**]{} (2001) 113003.
F. Vissani, [*JHEP*]{} **06** (1999) 022; M. Czakon *et al.*, [*Phys. Lett.*]{} **B465** (1999) 211, hep-ph/0003161 and [*Phys. Rev.* ]{} [**D65**]{} (2002) 053008; H. V. Klapdor-Kleingrothaus, H. Pas and A. Yu. Smirnov, [*Phys. Rev.*]{} **D63** (2001) 073005; H. Minakata and H. Sugiyama, [*Phys. Lett.*]{} **B526** (2002) 335; Z. Xing, [*Phys. Rev.*]{} [**D65**]{} (2002) 077302; N. Haba, N. Nakamura and T. Suzuki, hep-ph/0205141.
A. Osipowicz et al. (KATRIN Project), hep-ex/0109033.
W. Rodejohann, [*Nucl. Phys.*]{} [**B597**]{} (2001) 110, and hep-ph/0203214.
H. V. Klapdor-Kleingrothaus *et al.*, [*Nucl. Phys. Proc. Suppl.*]{} [**100**]{} (2001) 309.
C.E. Aalseth, F.T. Avignone III et al., [*Physics of Atomic Nuclei*]{} [**63**]{} (2000) 1225.
X. Sarazin *et al.* (NEMO3 Collaboration), hep-ex/0006031.
E. Fiorini, [*Phys. Rep.*]{} [**307**]{} (1998) 309.
H. V. Klapdor-Kleingrothaus *et al.*, [*J. Phys.*]{} G **24** (1998) 483.
M. Danilov *et al.*, [*Phys. Lett.*]{} [**B480**]{} (2000) 12.
L. De Braeckeleer et al. (Majorana Project), [*Proceedings of the Carolina Conference on Neutrino Physics*]{}, Columbia (SC), USA, March 2000.
H. Ejiri [*et al.*]{}, [*Phys. Rev. Lett.*]{} [**85**]{} (2000) 2917.
V. Lobashev *et al.*, [*Nucl. Phys. Proc. Suppl.*]{} [**91** ]{}(2001) 280.
Ch. Weinheimer *et al.*, talk at the Int. Conf. on Neutrino Physics and Astrophysics “Neutrino’02”, May 25 - 30, 2002, Munich, Germany.
M. Appolonio *et al.*, [*Phys. Lett.* ]{}[**B466**]{} (1999) 415.
F. Boehm et al., [*Phys. Rev. Lett. *]{} [**84**]{} (2000) 3764 and [*Phys. Rev.*]{} [**D62**]{} (2000) 072002.
C. Gonzalez-Garcia *et al.*, [*Phys. Rev.*]{} [**D63**]{} (2001) 033005.
D. Michael (MINOS Collaboration), talk at the Int. Conf. on Neutrino Physics and Astrophysics “Neutrino’02”, May 25 - 30, 2002, Munich, Germany.
J. Shirai (KamLAND Collaboration), talk at the Int. Conf. on Neutrino Physics and Astrophysics “Neutrino’02”, May 25 - 30, 2002, Munich, Germany.
A. de Gouvêa and C. Peña-Garay, hep-ph/0107186.
K.S. Babu and R.N. Mohapatra, [*Phys. Rev. Lett.*]{} [**75**]{} (1995) 2276; H. Pas et al., [*Phys. Lett.* ]{} [**B398**]{} (1997) 311; [*ibid.*]{} [**B459**]{} (1999) 450.
T. J. Weiler, [*Astropart. Phys.* ]{} [**11**]{} (1999) 303; H. Pas and T. J. Weiler, [*Phys. Rev.* ]{} [**D63**]{} (2001) 113015; Z. Fodor [*et al.*]{}, [*Phys. Rev. Lett.* ]{} [**88**]{} (2002) 171101.
V. Barger et al., hep-ph/0204253, version 2 of 29 of April, 2002.
P. Creminelli, G. Signorelli and A. Strumia, hep-ph/0102234, version 3 of 22 of April, 2002.
A. Bandyopadhyay *et al.*, hep-ph/0204286.
J.N. Bachall, M.C. Gonzalez-Garcia and C. Peña-Garay, hep-ph/0204314.
[^1]: Also at: Institute of Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, 1784 Sofia, Bulgaria
[^2]: We assume that the fields of the Majorana neutrinos $\nu_j$ satisfy the Majorana condition: $C(\bar{\nu}_{j})^{T} = \nu_{j},~j=1,2,3$, where $C$ is the charge conjugation matrix.
[^3]: Evidences for -decay taking place with a rate corresponding to $0.11 \ {\rm eV} \leq {\mbox{$\left| < \! m \! > \right| \ $}}\leq 0.56$ eV (95% C.L.) are claimed to have been obtained in [@Klap01]. The results announced in [@Klap01] have been criticized in [@FSViss02; @bb0nu02].
[^4]: We thank C. Peña-Garay for clarifications on this point.
[^5]: This statement is valid as long as $m_1$ and the CP violating phases which enter the effective Majorana mass are not constrained.
[^6]: Note that the CP-violating phase $\alpha_{21}$ is not constrained in the case under discussion. Even if it is found that $\alpha_{31} - \alpha_{21}= 0, \pm \pi$, $\alpha_{21}$ can be a source of CP-violation in $\Delta L =2$ processes other than -decay.
[^7]: The conclusions that follow practically do not depend on $\sin^2 \theta < 0.05 $.
[^8]: Information on the absolute values of neutrino masses in the range of interest might be obtained also from cosmological and astrophysical data, see, e.g., ref. [@Weiler2001].
|
---
abstract: 'In principle, meta-reinforcement learning algorithms leverage experience across many tasks to learn fast reinforcement learning (RL) strategies that transfer to similar tasks. However, current meta-RL approaches rely on manually-defined distributions of training tasks, and hand-crafting these task distributions can be challenging and time-consuming. Can “useful” pre-training tasks be discovered in an unsupervised manner? We develop an unsupervised algorithm for inducing an adaptive meta-training task distribution, i.e. an *automatic curriculum*, by modeling unsupervised interaction in a visual environment. The task distribution is scaffolded by a parametric density model of the meta-learner’s trajectory distribution. We formulate unsupervised meta-RL as information maximization between a latent task variable and the meta-learner’s data distribution, and describe a practical instantiation which alternates between integration of recent experience into the task distribution and meta-learning of the updated tasks. Repeating this procedure leads to iterative reorganization such that the curriculum adapts as the meta-learner’s data distribution shifts. In particular, we show how discriminative clustering for visual representation can support trajectory-level task acquisition and exploration in domains with pixel observations, avoiding pitfalls of alternatives. In experiments on vision-based navigation and manipulation domains, we show that the algorithm allows for unsupervised meta-learning that transfers to downstream tasks specified by hand-crafted reward functions and serves as pre-training for more efficient supervised meta-learning of test task distributions.'
author:
- 'Allan Jabri^[$\alpha$]{}^ Kyle Hsu^[$\beta$]{},$\dagger$^ Benjamin Eysenbach^[$\gamma$]{}^ Abhishek Gupta^[$\alpha$]{}^ Sergey Levine^[$\alpha$]{}^ Chelsea Finn^[$\delta$]{}^'
bibliography:
- 'neurips\_2019.bib'
title: |
Unsupervised Curricula\
for Visual Meta-Reinforcement Learning
---
Introduction
============
The discrepancy between animals and learning machines in their capacity to gracefully adapt and [generalize]{} is a central issue in artificial intelligence research. The simple nematode *C. elegans* is capable of adapting foraging strategies to varying scenarios [@calhoun2014maximally], while many higher animals are driven to acquire reusable behaviors even without extrinsic task-specific rewards [@white1959motivation; @piaget1954]. It is unlikely that we can build machines as adaptive as even the simplest of animals by exhaustively specifying shaped rewards or demonstrations across all possible environments and tasks. This has inspired work in reward-free learning [@hastie2009unsupervised], intrinsic motivation [@singh2005intrinsically], multi-task learning [@rich1997multitask], meta-learning [@schmidhuber1987evolutionary], and continual learning [@thrun].
An important aspect of generalization is the ability to share and transfer ability between related tasks. In reinforcement learning (RL), a common strategy for multi-task learning is conditioning the policy on side-information related to the task. For instance, *contextual* policies [@schaul2015universal] are conditioned on a task description (e.g. a *goal*) that is meant to modulate the strategy enacted by the policy. Meta-learning of reinforcement learning (meta-RL) is yet more general as it places the burden of inferring the task on the learner itself, such that task descriptions can take a wider range of forms, the most general being an MDP. In principle, meta-reinforcement learning (meta-RL) requires an agent to distill previous experience into fast and effective adaptation strategies for new, related tasks. However, the meta-RL framework by itself does not prescribe where this experience should come from; typically, meta-RL algorithms rely on being provided fixed, hand-specified task distributions, which can be tedious to specify for simple behaviors and intractable to design for complex ones [@hadfield2017inverse]. These issues beg the question of whether “useful” task distributions for meta-RL can be generated automatically.
![An illustration of [[CARML]{}]{}, our approach for unsupervised meta-RL. We choose the behavior model $q_\phi$ to be a Gaussian mixture model in a jointly, discriminatively learned embedding space. An automatic curriculum arises from periodically re-organizing past experience via fitting $q_\phi$ and meta-learning an RL algorithm for performance over tasks specified using reward functions from $q_\phi$.[]{data-label="fig:concept"}](figs/fig1_v2.pdf){width="\textwidth"}
In this work, we seek a procedure through which an agent in an environment with visual observations can automatically acquire useful (i.e. utility maximizing) behaviors, as well as how and when to apply them – in effect allowing for *unsupervised* pre-training in visual environments. Two key aspects of this goal are: 1) learning to operationalize strategies so as to adapt to new tasks, i.e. meta-learning, and 2) unsupervised learning and exploration in the absence of explicitly specified tasks, i.e. skill acquisition *without* supervised reward functions. These aspects interact insofar as the former implicitly relies on a task curriculum, while the latter is most effective when compelled by what the learner can and cannot do. Prior work has offered a pipelined approach for unsupervised meta-RL consisting of unsupervised skill discovery followed by meta-learning of discovered skills, experimenting mainly in environments that expose low-dimensional ground truth state [@gupta2018unsupervised]. Yet, the aforementioned relation between skill acquisition and meta-learning suggests that they should not be treated separately.
Here, we argue for closing the loop between skill acquisition and meta-learning in order to induce an *adaptive* task distribution. Such co-adaptation introduces a number of challenges related to the stability of learning and exploration. Most recent unsupervised skill acquisition approaches optimize for the discriminability of induced modes of behavior (i.e. *skills*), typically expressing the discovery problem as a cooperative game between a policy and a learned reward function [@gregor2016variational; @eysenbach2019diversity; @achiam2018variational]. However, relying solely on discriminability becomes problematic in environments with high-dimensional (image-based) observation spaces as it results in an issue akin to mode-collapse in the task space. This problem is further complicated in the setting we propose to study, wherein the policy data distribution is that of a meta-learner rather than a contextual policy. We will see that this can be ameliorated by specifying a hybrid discriminative-generative model for parameterizing the task distribution.
The main contribution of this paper is an approach for inducing a task curriculum for unsupervised meta-RL in a manner that scales to domains with pixel observations. Through the lens of information maximization, we frame our unsupervised meta-RL approach as variational expectation-maximization (EM), in which the E-step corresponds to fitting a task distribution to a meta-learner’s behavior and the M-step to meta-RL on the current task distribution with reinforcement for both skill acquisition and exploration. For the E-step, we show how deep discriminative clustering allows for trajectory-level representations suitable for learning diverse skills from pixel observations. Through experiments in vision-based navigation and robotic control domains, we demonstrate that the approach i) enables an unsupervised meta-learner to discover and meta-learn skills that transfer to downstream tasks specified by human-provided reward functions, and ii) can serve as pre-training for more efficient supervised meta-reinforcement learning of downstream task distributions.
Preliminaries: Meta-Reinforcement Learning {#sec:prelims}
==========================================
*Supervised* meta-RL optimizes an RL algorithm $f_\theta$ for performance on a hand-crafted distribution of tasks $p({\mathcal{T}})$, where $f_\theta$ might take the form of an recurrent neural network (RNN) implementing a learning algorithm [@duan2016rl; @wang2016learning], or a function implementing a gradient-based learning algorithm [@finn2017model]. Tasks are Markov decision processes (MDPs) consisting of state space ${\mathcal{S}}$, action space ${\mathcal{A}}$, reward function , probabilistic transition dynamics $P({\mathbf{s}}_{t+1}|{\mathbf{s}}_t,{\mathbf{a}}_t)$, discount factor $\gamma$, initial state distribution $\rho({\mathbf{s}}_1)$, and finite horizon $T$. Often, and in our setting, tasks are assumed to share ${\mathcal{S}}, {\mathcal{A}}$. For a given ${\mathcal{T}}\sim p({\mathcal{T}})$, $f_\theta$ learns a policy $\pi_\theta({\mathbf{a}}|{\mathbf{s}},{\mathcal{D}_{\mathcal{T}}})$ conditioned on task-specific experience. Thus, a meta-RL algorithm optimizes $f_\theta$ for expected performance of $\pi_\theta({\mathbf{a}}|{\mathbf{s}},{\mathcal{D}_{\mathcal{T}}})$ over $p({\mathcal{T}})$, such that it can generalize to unseen test tasks also sampled from $p({\mathcal{T}})$.
For example, RL$^2$ [@duan2016rl; @wang2016learning] chooses $f_\theta$ to be an RNN with weights $\theta$. For a given task ${\mathcal{T}}$, $f_\theta$ hones $\pi_\theta({\mathbf{a}}| {\mathbf{s}}, {\mathcal{D}_{\mathcal{T}}})$ as it recurrently ingests , the sequence of states, actions, and rewards produced via interaction within the MDP. Crucially, the same task is seen several times, and the hidden state is not reset until the next task. The loss is the negative discounted return obtained by $\pi_\theta$ across episodes of the same task, and $f_\theta$ can be optimized via standard policy gradient methods for RL, backpropagating gradients through time and across episode boundaries.
*Unsupervised* meta-RL aims to break the reliance of the meta-learner on an explicit, upfront specification of $p({\mathcal{T}})$. Following @gupta2018unsupervised, we consider a controlled Markov process (CMP) ${\mathcal{C}}=({\mathcal{S}}, {\mathcal{A}}, P, \gamma, \rho, T)$, which is an MDP without a reward function. We are interested in the problem of learning an RL algorithm $f_\theta$ via unsupervised interaction within the CMP such that once a reward function $r$ is specified at test-time, $f_\theta$ can be readily applied to the resulting MDP to efficiently maximize the expected discounted return.
Prior work [@gupta2018unsupervised] pipelines skill acquisition and meta-learning by pairing an unsupervised RL algorithm DIAYN [@eysenbach2019diversity] and a meta-learning algorithm MAML [@finn2017model]: first, a contextual policy is used to discover skills in the CMP, yielding a finite set of learned reward functions distributed as $p(r)$; then, the CMP is combined with a frozen $p(r)$ to yield $p({\mathcal{T}})$, which is fed to MAML to meta-learn $f_\theta$. In the next section, we describe how we can generalize and improve upon this pipelined approach by jointly performing skill acquisition as the meta-learner learns and explores in the environment.
![ A step for the meta-learner. **(Left) Unsupervised pre-training.** The policy meta-learns self-generated tasks based on the behavior model $q_\phi$. **(Right) Transfer.** Faced with new tasks, the policy transfers acquired meta-learning strategies to maximize unseen reward functions.[]{data-label="fig:train-test"}](figs/arch-diagram-2.pdf){width="\textwidth"}
urricul for Unsupervised Meta-einforceent earning {#sec:method}
=================================================
Meta-learning is intended to prepare an agent to efficiently solve new tasks related to those seen previously. To this end, the meta-RL agent must balance 1) exploring the environment to infer which task it should solve, and 2) visiting states that maximize reward under the inferred task. The duty of unsupervised meta-RL is thus to present the meta-learner with tasks that allow it to practice task inference and execution, without the need for human-specified task distributions. Ideally, the task distribution should exhibit both structure and diversity. That is, the tasks should be distinguishable and not excessively challenging so that a developing meta-learner can infer and execute the right skill, but, for the sake of generalization, they should also encompass a diverse range of associated stimuli and rewards, including some beyond the current scope of the meta-learner. Our aim is to strike this balance by inducing an adaptive task distribution.
With this motivation, we develop an algorithm for unsupervised meta-reinforcement learning in visual environments that constructs a task distribution without supervision. The task distribution is derived from a latent-variable density model of the meta-learner’s cumulative behavior, with exploration based on the density model driving the evolution of the task distribution. As depicted in Figure\[fig:concept\], learning proceeds by alternating between two steps: **organizing experiential data** (i.e., trajectories generated by the meta-learner) by modeling it with a mixture of latent components forming the basis of “skills”, and [meta-reinforcement learning]{} by **treating these skills as a training task distribution**. Learning the task distribution in a data-driven manner ensures that tasks are feasible in the environment. While the induced task distribution is in no way guaranteed to align with test task distributions, it may yet require an implicit understanding of structure in the environment. This can indeed be seen from our visualizations in \[sec:experiments\], which demonstrate that acquired tasks show useful structure, though in some settings this structure is easier to meta-learn than others. In the following, we formalize our approach, [[CARML]{}]{}, through the lens of information maximization and describe a concrete instantiation that scales to the vision-based environments considered in \[sec:experiments\].
An Overview of [[CARML]{}]{} {#sec:overview}
----------------------------
We begin from the principle of information maximization (IM), which has been applied across unsupervised representation learning [@bell_sejnowski_1995; @barber2004im; @oord2018representation] and reinforcement learning [@mohamed15; @gregor2016variational] for organization of data involving latent variables. In what follows, we organize data from our policy by maximizing the mutual information (MI) between state trajectories ${\bm{\tau}}:= ({\mathbf{s}}_1,\dots,{\mathbf{s}}_T)$ and a latent task variable ${\mathbf{z}}$. This objective provides a principled manner of trading-off structure and diversity: from ${I}({\bm{\tau}}; {\mathbf{z}}) := H({\bm{\tau}}) - H({\bm{\tau}}| {\mathbf{z}})$, we see that $H({\bm{\tau}})$ promotes coverage in policy data space (i.e. *diversity*) while $-H({\bm{\tau}}|{\mathbf{z}})$ encourages a lack of diversity under each task (i.e. *structure* that eases task inference).
We approach maximizing ${I}({\bm{\tau}}; {\mathbf{z}})$ exhibited by the meta-learner $f_\theta$ via variational EM [@barber2004im], introducing a variational distribution $q_\phi$ that can intuitively be viewed as a task scaffold for the meta-learner. In the E-step, we fit $q_\phi$ to a reservoir of trajectories produced by $f_\theta$, re-organizing the cumulative experience. In turn, $q_\phi$ gives rise to a task distribution $p({\mathcal{T}})$: each realization of the latent variable ${\mathbf{z}}$ induces a reward function $r_{\mathbf{z}}({\mathbf{s}})$, which we combine with the CMP ${\mathcal{C}}_i$ to produce an MDP ${\mathcal{T}}_i$ (Line \[line:reward\]). In the M-step, $f_\theta$ meta-learns the task distribution $p({\mathcal{T}})$. Repeating these steps forms a curriculum in which the task distribution and meta-learner co-adapt: each M-step adapts the meta-learner $f_\theta$ to the updated task distribution, while each E-step updates the task scaffold $q_\phi$ based on the data collected during meta-training. Pseudocode for our method is presented in Algorithm \[alg:ours\].
${\mathcal{C}}$, an MDP without a reward function Initialize $f_\theta$, an RL algorithm parameterized by $\theta$. Initialize ${\mathcal{D}}$, a reservoir of state trajectories, via a randomly initialized policy.
Fit a task-scaffold $q_\phi$ to ${\mathcal{D}}$, e.g. by using Algorithm \[alg:em\]. **E-step \[sec:e-step\]** \[line:fit\] Sample a latent task variable ${\mathbf{z}}\sim q_\phi({\mathbf{z}})$. Define the reward function $r_{\mathbf{z}}({\mathbf{s}})$, e.g. by Eq. \[eq:gen\_reward\], and a task ${\mathcal{T}}= {\mathcal{C}}\cup r_{\mathbf{z}}({\mathbf{s}})$. \[line:reward\] Apply $f_\theta$ on task ${\mathcal{T}}$ to obtain a policy $\pi_\theta({\mathbf{a}}| {\mathbf{s}}, {\mathcal{D}_{\mathcal{T}}})$ and trajectories $\{{\bm{\tau}}_i\}$. Update $f_\theta$ via a meta-RL algorithm, e.g. RL$^2$ [@duan2016rl]. **M-step \[sec:m-step\]** Add the new trajectories to the reservoir: ${\mathcal{D}}\gets {\mathcal{D}}\cup \{{\bm{\tau}}_i\}$.
a meta-learned RL algorithm $f_\theta$ tailored to ${\mathcal{C}}$
E-Step: Task Acquisition {#sec:e-step}
------------------------
The purpose of the E-step is to update the task distribution by integrating changes in the meta-learner’s data distribution with previous experience, thereby allowing for re-organization of the task scaffold. This data is from the *post-update* policy, meaning that it comes from a policy $\pi_\theta({\mathbf{a}}|{\mathbf{s}},{\mathcal{D}_{\mathcal{T}}})$ conditioned on data collected by the meta-learner for the respective task. In the following, we abuse notation by writing $\pi_\theta({\mathbf{a}}|{\mathbf{s}},{\mathbf{z}})$ – conditioning on the latent task variable ${\mathbf{z}}$ rather than the task experience ${\mathcal{D}_{\mathcal{T}}}$.
The general strategy followed by recent approaches for skill discovery based on IM is to lower bound the objective by introducing a variational posterior $q_\phi({\mathbf{z}}|{\mathbf{s}})$ in the form of a classifier. In these approaches, the E-step amounts to updating the classifier to discriminate between data produced by different skills as much as possible. A potential failure mode of such an approach is an issue akin to mode-collapse in the task distribution, wherein the policy drops modes of behavior to favor easily discriminable trajectories, resulting in a lack of diversity in the task distribution and no incentive for exploration; this is especially problematic when considering high-dimensional observations. Instead, here we derive a generative variant, which allows us to account for explicitly capturing modes of behavior (by optimizing for likelihood), as well as a direct mechanism for exploration.
We introduce a variational distribution $q_\phi$, which could be e.g. a (deep) mixture model with discrete ${\mathbf{z}}$ or a variational autoencoder (VAE) [@kingma2014auto] with continuous ${\mathbf{z}}$, lower-bounding the objective:
$$\begin{aligned}
I({\bm{\tau}};{\mathbf{z}})
&= -\sum_{{\bm{\tau}}} \pi_\theta({\bm{\tau}}) \log \pi_\theta({\bm{\tau}}) + \sum_{{\bm{\tau}},{\mathbf{z}}} \pi_\theta({\bm{\tau}}, {\mathbf{z}}) \log \pi_\theta({\bm{\tau}}|{\mathbf{z}}) \\
&\geq -\sum_{{\bm{\tau}}} \pi_\theta({\bm{\tau}}) \log \pi_{\theta}({\bm{\tau}}) + \sum_{{\bm{\tau}},{\mathbf{z}}} \pi_\theta({\bm{\tau}}|{\mathbf{z}}) q_\phi({\mathbf{z}}) \log q_\phi({\bm{\tau}}|{\mathbf{z}}) \label{eq:gen_variational}
$$
The E-step corresponds to optimizing Eq. \[eq:gen\_variational\] with respect to $\phi$, and thus amounts to fitting $q_\phi$ to a reservoir of trajectories ${\mathcal{D}}{}$ produced by $\pi_\theta$:
$$\max_{\phi} \; {\mathbb{E}}_{{\mathbf{z}}\sim q_\phi({\mathbf{z}}), {\bm{\tau}}\sim {\mathcal{D}}}
\big[
\log q_\phi({\bm{\tau}}|{\mathbf{z}})
\big]$$
What remains is to determine the form of $q_\phi$. We choose the variational distribution to be a state-level mixture density model . Despite using a state-level generative model, we can treat ${\mathbf{z}}$ as a trajectory-level latent by computing the trajectory-level likelihood as the factorized product of state likelihoods (Algorithm \[alg:em\], Line 4). This is useful for obtaining trajectory-level tasks; in the M-step (\[sec:m-step\]), we map samples from $q_\phi({\mathbf{z}})$ to reward functions to define tasks for meta-learning.
a set of trajectories ${\mathcal{D}}=\{({\mathbf{s}}_1,\dots,{\mathbf{s}}_T)\}_{i=1}^N$ Initialize $(\phi_w, \phi_m)$, encoder and mixture parameters. a mixture model $q_\phi({\mathbf{s}},z)$
![ Conditional independence\
assumption for states along a trajectory. []{data-label="fig:graphical_model"}](figs/cond_indep_vizdoom_3.pdf){width="98.00000%"}
**Modeling Trajectories of Pixel Observations.** While models like the variational autoencoder have been used in related settings [@nair2018visual], a basic issue is that optimizing for reconstruction treats all pixels equally. We, rather, will tolerate *lossy* representations as long as they capture *discriminative* features useful for stimulus-reward association. Drawing inspiration from recent work on unsupervised feature learning by clustering [@bojanowski2017unsupervised; @caron2018deep], we propose to fit the trajectory-level mixture model via discriminative clustering, striking a balance between discriminative and generative approaches.
We adopt the optimization scheme of DeepCluster [@caron2018deep], which alternates between i) clustering representations to obtain pseudo-labels and ii) updating the representation by supervised learning of pseudo-labels. In particular, we derive a trajectory-level variant (Algorithm \[alg:em\]) by forcing the responsibilities of all observations in a trajectory to be the same (see Appendix \[app:traj\_derivation\] for a derivation), leading to state-level visual representations optimized with trajectory-level supervision.
The conditional independence assumption in Algorithm \[alg:em\] is a simplification insofar as it discards the order of states in a trajectory. However, if the dynamics exhibit continuity and causality, the visual representation might yet capture temporal structure, since, for example, attaining certain observations might imply certain antecedent subtrajectories. We hypothesize that a state-level model can regulate issues of over-expressive sequence encoders, which have been found to lead to skills with undesirable attention to details in dynamics [@achiam2018variational]. As we will see in \[sec:experiments\], learning representations under this assumption still allows for learning visual features that capture trajectory-level structure.
M-Step: Meta-Learning {#sec:m-step}
---------------------
Using the task scaffold updated via the E-step, we meta-learn $f_\theta$ in the M-step so that $\pi_\theta$ can be quickly adapted to tasks drawn from the task scaffold. To define the task distribution, we must specify a form for the reward functions $r_{\mathbf{z}}({\mathbf{s}})$. To allow for state-conditioned Markovian rewards rather than non-Markovian trajectory-level rewards, we lower-bound the trajectory-level MI objective: $$\begin{aligned}
{I}({\bm{\tau}}; {\mathbf{z}}) &=
\frac{1}{T} \sum_{t=1}^T {H}({\mathbf{z}}) - {H}({\mathbf{z}}|{\mathbf{s}}_{1}, ..., {\mathbf{s}}_{T})
\geq \frac{1}{T} \sum_{t=1}^T {H}({\mathbf{z}}) - {H}({\mathbf{z}}|{\mathbf{s}}_t)
\\
&\geq {\mathbb{E}}_{{\mathbf{z}}\sim q_\phi({\mathbf{z}}), {\mathbf{s}}\sim \pi_\theta({\mathbf{s}}| {\mathbf{z}})}
\big[
\log q_\phi({\mathbf{s}}| {\mathbf{z}}) - \log \pi_\theta({\mathbf{s}})
\big] \label{eq:state_variational_MI}\end{aligned}$$ We would like to optimize the meta-learner under the variational objective in Eq. \[eq:state\_variational\_MI\], but optimizing the second term, the policy’s state entropy, is in general intractable. Thus, we make the simplifying assumption that the fitted variational marginal distribution matches that of the policy: $$\begin{aligned}
&\max_{\theta} \; {\mathbb{E}}_{{\mathbf{z}}\sim q_\phi({\mathbf{z}}), {\mathbf{s}}\sim \pi_\theta({\mathbf{s}}| {\mathbf{z}})} \big[\log q_\phi({\mathbf{s}}| {\mathbf{z}}) - \log q_\phi({\mathbf{s}}) \big] \label{eq:gen_pol} \\
=&\max_{\theta} \; I(\pi_\theta({\mathbf{s}}); q_\phi({\mathbf{z}})) -{D_\text{KL}\infdivx}{\pi_\theta({\mathbf{s}}|{\mathbf{z}})}{q_\phi({\mathbf{s}}|{\mathbf{z}})} + {D_\text{KL}\infdivx}{\pi_\theta({\mathbf{s}})}{q_\phi({\mathbf{s}}))} \label{eq:gen_info}\end{aligned}$$ Optimizing Eq. \[eq:gen\_pol\] amounts to maximizing the reward of . As shown in Eq. \[eq:gen\_info\], this corresponds to information maximization between the policy’s state marginal and the latent task variable, along with terms for matching the task-specific policy data distribution to the corresponding mixture mode and deviating from the mixture’s marginal density. We can trade-off between component-matching and exploration by introducing a weighting term $\lambda \in [0, 1]$ into $r_{\mathbf{z}}({\mathbf{s}})$:
$$\begin{aligned}
r_{\mathbf{z}}({\mathbf{s}}) &= \lambda \log q_\phi({\mathbf{s}}|{\mathbf{z}}) - \log q_\phi({\mathbf{s}}) \label{eq:gen_reward} \\
&= (\lambda - 1) \log q_\phi({\mathbf{s}}|{\mathbf{z}}) + \log q_\phi({\mathbf{z}}|{\mathbf{s}}) + C \label{eq:gen_reward3}\end{aligned}$$ where $C$ is a constant with respect to the optimization of $\theta$. From Eq. \[eq:gen\_reward3\], we can interpret $\lambda$ as trading off between discriminability of skills and task-specific exploration. Figure \[fig:lambda\] shows the effect of tuning $\lambda$ on the structure-diversity trade-off alluded to at the beginning of \[sec:method\].
![Balancing consistency and exploration with $\lambda$ in a simple 2D maze environment. Each row shows a progression of tasks developed over the course of training. Each box presents the mean reconstructions under a VAE $q_\phi$ (Appendix \[app:sec:vae\]) of 2048 trajectories. Varying $\lambda$ of Eq. \[eq:gen\_reward\] across rows, we observe that a small $\lambda$ (top) results in aggressive exploration; a large $\lambda$ (bottom) yields relatively conservative behavior; and a moderate $\lambda$ (middle) produces sufficient exploration and a smooth task distribution.[]{data-label="fig:lambda"}](figs/lambda_smaller_compressed.pdf){width="\textwidth"}
Related Work {#sec:related-work}
============
**Unsupervised Reinforcement Learning**. Unsupervised learning in the context of RL is the problem of enabling an agent to learn about its environment and acquire useful behaviors without human-specified reward functions. A large body of prior work has studied exploration and intrinsic motivation objectives [@schmidhuber2009driven; @salge2014empowerment; @pathak2017curiosity; @fu2017ex2; @burda2019exploration; @bellemare2016unifying; @lehman2011abandoning; @osband2018randomized]. These algorithms do not aim to acquire skills that can be operationalized to solve tasks, but rather try to achieve wide coverage of the state space; our objective (Eq. \[eq:gen\_reward\]) reduces to pure density-based exploration with $\lambda = 0$. Hence, these algorithms still rely on slow RL [@botvinick2019reinforcement] in order to adapt to new tasks posed at test-time. Some prior works consider unsupervised pre-training for efficient RL, but these works typically focus on settings in which exploration is not as much of a challenge [@watter2015embed; @finn2017deep; @ebert2017self], focus on goal-conditioned policies [@pathak2018zero; @nair2018visual], or have not been shown to scale to high-dimensional visual observation spaces [@lopes2012exploration; @shyam2019model]. Perhaps most relevant to our work are unsupervised RL algorithms for learning reward functions via optimizing information-theoretic objectives involving latent skill variables [@gregor2016variational; @achiam2018variational; @eysenbach2019diversity; @warde2019unsupervised]. In particular, with a choice of $\lambda = 1$ in Eq. \[eq:gen\_reward3\] we recover the information maximization objective used in prior work [@achiam2018variational; @eysenbach2019diversity], besides the fact that we simulatenously perform meta-learning. The setting of training a contextual policy with a classifier as $q_\phi$ in our proposed framework (see Appendix \[app:sec:diayn\]) provides an interpretation of DIAYN as implicitly doing trajectory-level clustering. @warde2019unsupervised also considers accumulation of tasks, but with a focus on goal-reaching and by maintaining a goal reservoir via heuristics that promote diversity.
**Meta-Learning**. Our work is distinct from above works in that it formulates a meta-learning approach to explicitly train, without supervision, for the ability to adapt to new downstream RL tasks. Prior work [@hsu2019unsupervised; @khodadadeh2018unsupervised; @antoniou2019assume] has investigated this unsupervised meta-learning setting for image classification; the setting considered herein is complicated by the added challenges of RL-based policy optimization and exploration. @gupta2018unsupervised provides an initial exploration of the unsupervised meta-RL problem, proposing a straightforward combination of unsupervised skill acquisition (via DIAYN) followed by MAML [@finn2017model] with experiments restricted to environments with fully observed, lower-dimensional state. Unlike these works and other meta-RL works [@wang2016learning; @duan2016rl; @mishra2018simple; @rakelly2019efficient; @finn2017model; @houthooft2018evolved; @gupta2018meta; @rothfuss2019promp; @stadie2018some; @sung2017learning], we close the loop to jointly perform task acquisition and meta-learning so as to achieve an automatic curriculum to facilitate joint meta-learning and task-level exploration.
**Automatic Curricula**. The idea of automatic curricula has been widely explored both in supervised learning and RL. In supervised learning, interest in automatic curricula is based on the hypothesis that exposure to data in a specific order (i.e. a non-uniform curriculum) may allow for learning harder tasks more efficiently [@elman1993learning; @schmidhuber2009driven; @graves2017automated]. In RL, an additional challenge is exploration; hence, related work in RL considers the problem of *curriculum generation*, whereby the task distribution is designed to guide exploration towards solving complex tasks [@florensa2017reverse; @matiisen2019teacher; @florensa2017automatic; @schmidhuber2011powerplay] or unsupervised pre-training [@sukhbaatar2018intrinsic; @forestier2017intrinsically]. Our work is driven by similar motivations, though we consider a curriculum in the setting of meta-RL and frame our approach as information maximization.
Experiments {#sec:experiments}
===========
We experiment in visual navigation and visuomotor control domains to study the following questions:
- What kind of tasks are discovered through our task acquisition process (the E-step)?
- Do these tasks allow for meta-training of strategies that transfer to test tasks?
- Does closing the loop to jointly perform task acquisition and meta-learning bring benefits?
- Does pre-training with [[CARML]{}]{} accelerate meta-learning of test task distributions?
Videos are available at the project website <https://sites.google.com/view/carml>.
Experimental Setting
--------------------
The following experimental details are common to the two vision-based environments we consider. Other experimental are explained in more detail in Appendix \[app:sec:implementation\_details\].
**Meta-RL.** [[CARML]{}]{} is agnostic to the meta-RL algorithm used in the M-step. We use the RL$^2$ algorithm [@duan2016rl], which has previously been evaluated on simpler visual meta-RL domains, with a PPO [@schulman2017ppo] optimizer. Unless otherwise stated, we use four episodes per trial (compared to the two episodes per trial used in [@duan2016rl]), since the settings we consider involve more challenging task inference.
**Baselines.** We compare against: 1) PPO from scratch on each evaluation task, 2) pre-training with random network distillation (RND) [@burda2019exploration] for unsupervised exploration, followed by fine-tuning on evaluation tasks, and 3) supervised meta-learning on the test-time task distribution, as an oracle.
**Variants.** We consider variants of our method to ablate the role of design decisions related to task acquisition and joint training: 4) *pipelined* (most similar to [@gupta2018unsupervised]) – task acquisition with a contextual policy, followed by meta-RL with RL$^2$; 5) *online discriminator* – task acquisition with a purely discriminative $q_\phi$ (akin to online DIAYN); and 6) *online pretrained-discriminator* – task acquisition with a discriminative $q_\phi$ initialized with visual features trained via Algorithm \[alg:em\].
Visual Navigation {#sec:viznav}
-----------------
The first domain we consider is first-person visual navigation in ViZDoom [@kempka2016vizdoom], involving a room filled with five different objects (drawn from a set of 50). We consider a setup akin to those featured in [@chaplot2018gated; @xie2018few] (see Figure \[fig:graphical\_model\]). The true state consists of continuous 2D position and continuous orientation, while observations are egocentric images with limited field of view. Three discrete actions allow for turning right or left, and moving forward. We consider two ways of sampling the CMP $\mathcal{C}$. **Fixed**: fix a set of five objects and positions for both unsupervised meta-training and testing. **Random**: sample five objects and randomly place them (thereby randomizing the state space and dynamics).
![Skill maps for visual navigation. We visualize some of the discovered tasks by projecting trajectories of certain mixture components into the true state space. White dots correspond to fixed objects. The legend indicates orientation as color; on its left is an interpretation of the depicted component. Some tasks seem to correspond to exploration of the space (green border), while others are more directed towards specific areas (blue border). Comparing tasks earlier and later in the curriculum (step 1 to step 5), we find an increase in structure. []{data-label="fig:vizdoom_tasks"}](figs/vizdoom_tasks_v2.pdf){width="99.00000%"}
**Visualizing the task distribution**. Modeling pixel observations reveals trajectory-level organization in the underlying true state space (Figure \[fig:vizdoom\_tasks\]). Each map portrays trajectories of a mixture component, with position encoded in 2D space and orientation encoded in the jet color-space; an example of interpreting the maps is shown left of the legend. The components of the mixture model reveal structured groups of trajectories: some components correspond to exploration of the space (marked with green border), while others are more strongly directed towards specific areas (blue border). The skill maps of the fixed and random environments are qualitatively different: tasks in the fixed room tend towards interactions with objects or walls, while many of the tasks in the random setting sweep the space in a particular direction. We can also see the evolution of the task distribution at earlier and later stages of Algorithm \[alg:ours\]. While initial tasks (produced by a randomly initialized policy) tend to be less structured, we later see refinement of certain tasks as well as the emergence of others as the agent collects new data and acquires strategies for performing existing tasks.
**Do acquired skills transfer to test tasks**? We evaluate how well the CARML task distribution prepares the agent for unseen tasks. For both the fixed and randomized CMP experiments, each test task specifies a dense goal-distance reward for reaching a single object in the environment. In the randomized environment setting, the target objects at test-time are held out from meta-training. The PPO and RND-initialized baseline polices, and the finetuned [[CARML]{}]{} meta-policy, are trained for a single target (a specific object in a fixed environment), with 100 episodes per PPO policy update.
[0.32]{} ![[[CARML]{}]{} enables unsupervised meta-learning of skills that transfer to downstream tasks. Direct transfer curves (marker and dotted line) represent a meta-learner deploying for just 200 time steps at test time. Compared to [[CARML]{}]{}, PPO and RND Init sample the test reward function orders of magnitude more times to perform similarly on a single task. Finetuning the [[CARML]{}]{} policy also allows for solving individual tasks with significantly fewer samples. The ablation experiments (c) assess both direct transfer and finetuning for each variant. Compared to variants, the [[CARML]{}]{} task acquisition procedure results in improved transfer due to mitigation of task mode-collapse and adaptation of the task distribution.[]{data-label="fig:transfer"}](figs/vizdoom_plots.pdf "fig:"){width="\textwidth"}
[0.3]{} ![[[CARML]{}]{} enables unsupervised meta-learning of skills that transfer to downstream tasks. Direct transfer curves (marker and dotted line) represent a meta-learner deploying for just 200 time steps at test time. Compared to [[CARML]{}]{}, PPO and RND Init sample the test reward function orders of magnitude more times to perform similarly on a single task. Finetuning the [[CARML]{}]{} policy also allows for solving individual tasks with significantly fewer samples. The ablation experiments (c) assess both direct transfer and finetuning for each variant. Compared to variants, the [[CARML]{}]{} task acquisition procedure results in improved transfer due to mitigation of task mode-collapse and adaptation of the task distribution.[]{data-label="fig:transfer"}](figs/sawyer_plots.pdf "fig:"){width="\textwidth"}
[0.3]{} ![[[CARML]{}]{} enables unsupervised meta-learning of skills that transfer to downstream tasks. Direct transfer curves (marker and dotted line) represent a meta-learner deploying for just 200 time steps at test time. Compared to [[CARML]{}]{}, PPO and RND Init sample the test reward function orders of magnitude more times to perform similarly on a single task. Finetuning the [[CARML]{}]{} policy also allows for solving individual tasks with significantly fewer samples. The ablation experiments (c) assess both direct transfer and finetuning for each variant. Compared to variants, the [[CARML]{}]{} task acquisition procedure results in improved transfer due to mitigation of task mode-collapse and adaptation of the task distribution.[]{data-label="fig:transfer"}](figs/vizdoom_variant_plots.pdf "fig:"){width="\textwidth"}
In Figure \[fig:vizdoom\_fixed\_transfer\], we compare the success rates on test tasks as a function of the number of samples with supervised rewards seen from the environment. Direct transfer performance of meta-learners is shown as points, since in this setting the RL$^2$ agent sees only *four episodes* (200 samples) at test-time, without any parameter updates. We see that direct transfer is significant, achieving up to 71% and 59% success rates on the fixed and randomized settings, respectively. The baselines require over two orders of magnitude more test-time samples to solve a single task at the same level.
While the [[CARML]{}]{} meta-policy does not consistently solve the test tasks, this is not surprising since no information is assumed about target reward functions during unsupervised meta-learning; inevitable discrepancies between the meta-train and test task distributions will mean that meta-learned strategies *will* be suboptimal for the test tasks. For instance, during testing, the agent sometimes ‘stalls’ before the target object (once inferred), in order to exploit the inverse distance reward. Nevertheless, we also see that finetuning the [[CARML]{}]{} meta-policy *trained on random* environments on individual tasks is more sample efficient than learning from scratch. This suggests that deriving reward functions from our mixture model yields useful tasks insofar as they facilitate learning of strategies that transfer.
**Benefit of reorganization**. In Figure \[fig:vizdoom\_fixed\_transfer\], we also compare performance across early and late outer-loop iterations of Algorithm \[alg:ours\], to study the effect of adapting the task distribution (the [[CARML]{}]{} E-step) by reorganizing tasks and incorporating new data. In both cases, number of outer-loop iterations $K=5$. Overall, the refinement of the task distribution, which we saw in Figure \[fig:vizdoom\_tasks\], leads improved to transfer performance. The effect of reorganization is further visualized in the Appendix \[app:sec:task\_distribution\_evolution\].
**Variants**. From Figure \[fig:vizdoom\_variants\], we see that the purely online discriminator variant suffers in direct transfer performance; this is due to the issue of mode-collapse in task distribution, wherein the task distribution lacks diversity. Pretraining the discriminator encoder with Algorithm \[alg:em\] mitigates mode-collapse to an extent, improving task diversity as the features and task decision boundaries are first fit on a corpus of (randomly collected) trajectories. Finally, while the distribution of tasks eventually discovered by the pipelined variant may be diverse and structured, meta-learning the corresponding tasks from scratch is harder. More detailed analysis and visualization is given in Appendix \[app:sec:variants\].
Visual Robotic Manipulation
---------------------------
To experiment in a domain with different challenges, we consider a simulated Sawyer arm interacting with an object in MuJoCo [@todorov2012mujoco], with end-effector continous control in the 2D plane. The observation is a bottom-up view of a surface supporting an object (Figure \[fig:sawyer\_tasks\]); the camera is stationary, but the view is no longer egocentric and part of the observation is proprioceptive. The test tasks involve pushing the object to a goal (drawn from the set of reachable states), where the reward function is the negative distance to the goal state. A subset of the skill maps is provided below.
[1]{} {width="\textwidth"}
**Do acquired skills directly transfer to test tasks**? In Figure \[fig:sawyer\_transfer\], we evaluate the meta-policy on the test task distribution, comparing against baselines as previously. Despite the increased difficulty of control, our approach allows for meta-learning skills that transfer to the goal distance reward task distribution. We find that transfer is weaker compared to the visual navigation (fixed version): one reason may be that the environment is not as visually rich, resulting in a significant gap between the CARML and the object-centric test task distributions.
[r]{}[0.53 ]{}
[0.50]{} {width="\textwidth"}
[0.475]{} {width="\textwidth"}
CARML as Meta-Pretraining
-------------------------
Another compelling form of transfer is pretraining of an initialization for accelerated supervised meta-RL of target task distributions. In Figure \[fig:finetune\], we see that the initialization learned by CARML enables effective supervised meta-RL with significantly fewer samples. To separate the effect of the learning the recurrent meta-policy and the visual representation, we also compare to only initializing the pre-trained encoder. Thus, while direct transfer of the meta-policy may not directly result in optimal behavior on test tasks, accelerated learning of the test task distribution suggests that the acquired meta-learning strategies may be useful for learning related task distributions, effectively acting as pre-training procedure for meta-RL.
Discussion
==========
We proposed a framework for inducing unsupervised, adaptive task distributions for meta-RL that scales to environments with high-dimensional pixel observations. Through experiments in visual navigation and manipulation domains, we showed that this procedure enables unsupervised acquisition of meta-learning strategies that transfer to downstream test task distributions in terms of direct evaluation, more sample-efficient fine-tuning, and more sample-efficient supervised meta-learning. Nevertheless, the following key issues are important to explore in future work.
**Task distribution mismatch**. While our results show that useful structure can be meta-learned in an unsupervised manner, results like the stalling behavior in ViZDoom (see \[sec:viznav\]) suggest that direct transfer of unsupervised meta-learning strategies suffers from a no-free-lunch issue: there will always be a gap between unsupervised and downstream task distributions, and more so with more complex environments. Moreover, the semantics of target tasks may not necessarily align with especially discriminative visual features. This is part of the reason why transfer in the Sawyer domain is less successful. Capturing other forms of structure useful for stimulus-reward association might involve incorporating domain-specific inductive biases into the task-scaffold model. Another way forward is the semi-supervised setting, whereby data-driven bias is incorporated at meta-training time.
**Validation and early stopping**: Since the objective optimized by the proposed method is non-stationary and in no way guaranteed to be correlated with objectives of test tasks, one must provide some mechanism for validation of iterates.
**Form of skill-set**. For the main experiments, we fixed a number of discrete tasks to be learned (without tuning this), but one should consider how the set of skills can be grown or parameterized to have higher capacity (e.g. a multi-label or continuous latent). Otherwise, the task distribution may become overloaded (complicating task inference) or limited in capacity (preventing coverage).
**Accumulation of skill**. We mitigate forgetting with the simple solution of reservoir sampling. Better solutions involve studying an intersection of continual learning and meta-learning.
### Acknowledgments {#acknowledgments .unnumbered}
We thank the BAIR community for helpful discussion, and Michael Janner and Oleh Rybkin in particular for feedback on an earlier draft. AJ thanks Alexei Efros for his steadfastness and advice, and Sasha Sax and Ashish Kumar for discussion. KH thanks his family for their support. AJ is supported by the PD Soros Fellowship. This work was supported in part by the National Science Foundation, IIS-1651843, IIS-1700697, and IIS-1700696, as well as Google.
Derivations
===========
Derivation for Trajectory-Level Responsibilities (Section 3.2.1) {#app:traj_derivation}
----------------------------------------------------------------
Here we show that, assuming independence between states in a trajectory when conditioning on a latent variable, computing the trajectory likelihood as a factorized product of state likelihoods for the E-step in standard EM forces the component responsibilities for all states in the trajectory to be identical. Begin by lower-bounding the log-likelihood of the trajectory dataset with Jensen’s inequality: $$\begin{aligned}
\sum_i \log p({\bm{\tau}}) = &\sum_i \log p({\mathbf{s}}_1^i, {\mathbf{s}}_2^i, ..., {\mathbf{s}}_T^i)\\
= &\sum_i \log \sum_z p({\mathbf{s}}_1^i, {\mathbf{s}}_2^i, ..., {\mathbf{s}}_T^i | z) p(z)\\
\geq &\sum_i \sum_z q_\phi(z | {\mathbf{s}}_1, {\mathbf{s}}_2, ..., {\mathbf{s}}_T) \log \frac{p({\mathbf{s}}_1^i, {\mathbf{s}}_2^i, ..., {\mathbf{s}}_T^i | z) p(z)}{q_\phi(z | {\mathbf{s}}_1, {\mathbf{s}}_2 ...{\mathbf{s}}_T)}\\
=
&\sum_i \mathbb{E}_{z \sim q_\phi(z | {\mathbf{s}}_1, {\mathbf{s}}_2, ..., {\mathbf{s}}_T)} \log \frac{p({\mathbf{s}}_1^i, {\mathbf{s}}_2^i, ..., {\mathbf{s}}_T^i | z) p(z)}{q_\phi(z | {\mathbf{s}}_1, {\mathbf{s}}_2, ..., {\mathbf{s}}_T)}. \label{eq:objective}\end{aligned}$$ We have introduced the variational distribution $q_\phi({\bm{\tau}}, z)$, where $z$ is a categorical variable. Now, to maximize Eq. \[eq:objective\] with respect to $\phi := (\bm{\mu}_1, \bm{\Sigma}_1, \pi_1, ..., \bm{\mu}_N, \bm{\Sigma}_N, \pi_N) $, we alternate between an E-step and an M-step, where the E-step is computing $$\begin{aligned}
q_{ik} &= q_\phi(z = k | {\mathbf{s}}_1^i, {\mathbf{s}}_2^i, ..., {\mathbf{s}}_T^i)\\
&= \frac{q_\phi({\mathbf{s}}_1^i, {\mathbf{s}}_2^i, ..., {\mathbf{s}}_T^i | z=k)q_\phi(z=k)}{\sum_j q_\phi({\mathbf{s}}_1^i, {\mathbf{s}}_2^i, ..., {\mathbf{s}}_T^i | z=j)q_\phi(z=j)}\\
&= \frac{q_\phi({\mathbf{s}}_1^i | z=k)q_\phi({\mathbf{s}}_2^i | z=k) \cdots q_\phi({\mathbf{s}}_T^i | z=k)q_\phi(z=k)}{\sum_j q_\phi({\mathbf{s}}_1^i | z=j)q_\phi({\mathbf{s}}_2^i | z=j) \cdots q_\phi({\mathbf{s}}_T^i | z=j)q_\phi(z=j)}.\end{aligned}$$
We assume that each $q_\phi({\mathbf{s}}| z=k)$ is Gaussian; the M-step amounts to computing the maximum-likelihood estimate of $\phi$, under the mixture responsibilities from the E-step: $$\begin{aligned}
&\bm{\mu}_k = \frac{\sum_{i=1}^N \frac{q_{ik}}{T}\sum_{t=1}^T {\mathbf{s}}_t}{\sum_{i=1}^N q_{ik}}\\
&\bm{\Sigma}_k = \frac{\sum_{i=1}^N \frac{q_{ik}}{T}\sum_{t=1}^T ({\mathbf{s}}_t - \bm{\mu}_k)({\mathbf{s}}_t - \bm{\mu}_k)^\top}{\sum_{i=1}^N q_{ik}}\\
&\pi_k = \frac{1}{N}\sum_{i=1}^N q_{ik}.\end{aligned}$$
In particular, note that the expressions are independent of $t$. Thus, the posterior $q_\phi(z|{\mathbf{s}})$ will be, too.
CARML M-Step
------------
The objective used to optimize the meta-RL algorithm in the CARML M-step can be interpreted as a sum of cross entropies, resulting in the mutual information plus two additional KL terms:
$$\begin{aligned}
&-\mathbb{E}_{{\mathbf{s}}\sim \pi_\theta({\mathbf{s}}|{\mathbf{z}}), {\mathbf{z}}\sim q_\phi({\mathbf{z}})} \big[ \log q_\phi({\mathbf{s}}) - \log q_\phi({\mathbf{s}}|{\mathbf{z}}) \big] \\
=& -\sum_{\mathbf{z}}q_\phi({\mathbf{z}}) \sum_{\mathbf{s}}\pi_\theta({\mathbf{s}}|{\mathbf{z}}) \left(\log q_\phi({\mathbf{s}}) - \log q_\phi({\mathbf{s}}|{\mathbf{z}})\right) \\
=& - \sum_{\mathbf{z}}q_\phi({\mathbf{z}}) \sum_{\mathbf{s}}\pi({\mathbf{s}}|{\mathbf{z}}) \left(\log \frac{q_\phi({\mathbf{s}})}{\pi_\theta({\mathbf{s}})} + \log \pi_\theta({\mathbf{s}}) - \log \frac{q_\phi({\mathbf{s}}|{\mathbf{z}})}{\pi_\theta({\mathbf{s}}|{\mathbf{z}})} - \log \pi_\theta({\mathbf{s}}|{\mathbf{z}})\right) \\
=& H(\pi_\theta({\mathbf{s}})) + {D_\text{KL}\infdivx}{\pi_\theta({\mathbf{s}})}{q_\phi({\mathbf{s}})} - H(\pi_\theta({\mathbf{s}}|{\mathbf{z}})) - {D_\text{KL}\infdivx}{\pi_\theta({\mathbf{s}}|{\mathbf{z}})}{q_\phi({\mathbf{s}}|{\mathbf{z}})} \\
=& I(\pi_\theta({\mathbf{s}}); q_\phi({\mathbf{z}})) + {D_\text{KL}\infdivx}{\pi_\theta({\mathbf{s}})}{q_\phi({\mathbf{s}})} - {D_\text{KL}\infdivx}{\pi_\theta({\mathbf{s}}|{\mathbf{z}})}{q_\phi({\mathbf{s}}|{\mathbf{z}})}.\end{aligned}$$
The first KL term can be interpreted as encouraging exploration with respect to the density of the mixture. The second KL term is the reverse KL term for matching the modes of the mixture.
**Density-based exploration**. In practice, we may want to trade off between exploration and matching the modes of the generative model:
$$\begin{aligned}
r_{\mathbf{z}}({\mathbf{s}}) &= \lambda \log q_\phi({\mathbf{s}}|{\mathbf{z}}) - \log q_\phi({\mathbf{s}}) \label{app:eq:gen_reward} \\
&= (\lambda - 1) \log q_\phi({\mathbf{s}}|{\mathbf{z}}) + \log q_\phi({\mathbf{z}}|{\mathbf{s}}) - \log q_\phi({\mathbf{z}}) \label{app:eq:gen_reward2} \\
&= (\lambda - 1) \log q_\phi({\mathbf{s}}|{\mathbf{z}}) + \log q_\phi({\mathbf{z}}|{\mathbf{s}}) + C \label{app:eq:gen_reward3}\end{aligned}$$
where $C$ is constant with respect to the optimization of $\theta$. Hence, the objective amounts to maximizing discriminability of skills where $\lambda < 1$ yields a bonus for exploring away from the mode of the corresponding skill.
Discriminative [[CARML]{}]{} and DIAYN {#app:sec:diayn}
--------------------------------------
Here, we derive a discriminative instantiation of [[CARML]{}]{}. We begin with the E-step. We leverage the same conditional independence assumption as before, and re-write the trajectory-level MI as the state level MI, assuming that trajectories are all of length $T$: $$I({\bm{\tau}};{\mathbf{z}}) \geq \frac{1}{T} \sum_t I({\mathbf{s}}_t ; {\mathbf{z}}) = I({\mathbf{s}}; {\mathbf{z}})$$ We then decompose MI as the difference between marginal and conditional entropy of the latent, and choose the variational distribution to be the product of a classifier $q_{\phi_c}({\mathbf{z}}| {\mathbf{s}})$ and a density model $q_{\phi_d}({\mathbf{s}})$: $$\begin{aligned}
I({\mathbf{s}};{\mathbf{z}}) &= H({\mathbf{z}}) - H({\mathbf{z}}|{\mathbf{s}}) \\
&= -\sum_{\mathbf{z}}p({\mathbf{z}}) \log p({\mathbf{z}}) + \sum_{{\mathbf{s}},{\mathbf{z}}} \pi_\theta({\mathbf{s}}, {\mathbf{z}}) \log \pi_\theta({\mathbf{z}}|{\mathbf{s}}) \\
& \geq -\sum_{\mathbf{z}}p({\mathbf{z}}) \log p({\mathbf{z}}) + \sum_{{\mathbf{s}},{\mathbf{z}}} \pi_\theta({\mathbf{s}}| {\mathbf{z}}) p({\mathbf{z}}) \log q_{\phi_c}({\mathbf{z}}|{\mathbf{s}}) \label{eq:infomax}\end{aligned}$$ We fix ${\mathbf{z}}$ to be a uniformly-distributed categorical variable. The [[CARML]{}]{} E-step consists of two separate optimizations: supervised learning of $q_{\phi_c}({\mathbf{z}}| {\mathbf{s}})$ with a cross-entropy loss and density estimation of $q_{\phi_d}({\mathbf{s}})$: $$\label{eq:discriminator}
\max_{\phi_c} \; {\mathbb{E}}_{{\mathbf{z}}\sim p({\mathbf{z}}), {\mathbf{s}}\sim \pi_\theta({\mathbf{z}})} \left[\log q_{\phi_c}({\mathbf{z}}| {\mathbf{s}}) \right] \qquad
\max_{\phi_d} \; {\mathbb{E}}_{{\mathbf{z}}\sim p({\mathbf{z}}), {\mathbf{s}}\sim \pi_\theta({\mathbf{z}})} \left[\log q_{\phi_d}({\mathbf{s}}) \right]$$ For the [[CARML]{}]{} M-step, we start from the form of the reward in Eq. \[app:eq:gen\_reward2\] and manipulate via Bayes’: $$\begin{aligned}
r_{\mathbf{z}}({\mathbf{s}})
&= \log q_{\phi_c}({\mathbf{z}}|{\mathbf{s}}) + (\lambda - 1) \log q_{\phi_c}({\mathbf{z}}|{\mathbf{s}}) + (\lambda - 1) \log q_{\phi_d}({\mathbf{s}}) - (\lambda - 1) \log p({\mathbf{z}}) - \log p({\mathbf{z}}) \nonumber \\
&= \lambda \log q_{\phi_c}({\mathbf{z}}|{\mathbf{s}}) + (\lambda - 1) \log q_{\phi_d}({\mathbf{s}}) + C\end{aligned}$$ where $C$ is constant with respect to the optimization of $\theta$ in the M-step $$\max_{\theta} \; {\mathbb{E}}_{{\mathbf{z}}\sim q_\phi({\mathbf{z}}), {\mathbf{s}}\sim \pi_\theta({\mathbf{z}})} \big[
\lambda \log q_{\phi_c}({\mathbf{z}}|{\mathbf{s}}) + (\lambda - 1) \log q_{\phi_d}({\mathbf{s}})
\big] \label{eq:disc_pol}$$ To enable a trajectory-level latent ${\mathbf{z}}$, we want every state in a trajectory to be classified to the same ${\mathbf{z}}$. This is achievable in a straightforward manner: when training the classifier $q_{\phi_c}({\mathbf{z}}| {\mathbf{s}})$ via supervised learning, label each state in a trajectory with the realization of ${\mathbf{z}}$ that the policy $\pi_\theta({\mathbf{a}}| {\mathbf{s}}, {\mathbf{z}})$ was conditioned on when generating that trajectory.
**Connection to DIAYN**. Note that with $\lambda = 1$ in Eq. \[eq:disc\_pol\], we directly obtain the DIAYN [@eysenbach2019diversity] objective without standard policy entropy regularization, and we do away with needing to maintain a density model $\log q_{\phi_d}({\mathbf{s}})$, leaving just the discriminator. If $\pi_\theta({\mathbf{a}}| {\mathbf{s}}, {\mathbf{z}})$ is truly a contextual policy (rather than the policy given by adapting a meta-learner), we have recovered the DIAYN algorithm. This allows us to interpret on DIAYN-style algorithms as implicitly doing trajectory-level clustering with a conditional independence assumption between states in a trajectory given the latent. This arises from the weak trajectory-level supervision specified when training the discriminator: all states in a trajectory are assumed to correspond to the same realization of the latent variable.
Additional Details for Main Experiments {#app:sec:implementation_details}
=======================================
CARML Hyperparameters
---------------------
We train CARML for five iterations, with 500 PPO updates for meta-learning with RL$^2$ in the M-step (i.e. update the mixture model every 500 meta-policy updates). Thus, the CARML unsupervised learning process consumes on the order of 1,000,000 episodes (compared to the \~400,000 episodes needed to train a meta-policy with the true task distribution, as shown in our experiments). We did not heavily tune this number, though we noticed that using too few policy updates (e.g. \~100) before refitting $q_\phi$ resulted in instability insofar as the meta-learner does not adapt to learn the updated task distribution. Each PPO learning update involves sampling 100 tasks with 4 episodes each, for a total of 400 episodes per update. We use 10 PPO epochs per update with a batch size of 100 tasks.
During meta-training, tasks are drawn according to $z \sim q_\phi(z)$, the mixture’s latent prior distribution. Unless otherwise stated, we use $\lambda=0.99$ for all visual meta-RL experiments. For all experiments unless otherwise mentioned, we fix the number of components in our mixture to be $k=16$. We use a reservoir of $1000$ trajectories.
**Temporally Smoothed Reward:** At unsupervised meta-training time, we found it helpful to reward the meta-learner with the average over a small temporal window, i.e. $r^W_z(s_t) = \frac{1}{W} \sum_{i=t-W}^t r_z(s_i)$, choosing $W$ to be $W=10$. This has the effect of smoothing the reward function, thereby regularizing acquired task inference strategies.
**Random Seeds:** The results reported in Figure 6 are averaged across policies (for each treatment) trained with three different random seeds. The performance is averaged across 20 test tasks. The results reported in Figure 7 are based on finetuning CARML policies trained with three different random seeds. We did not observe significant effects of the random seed used in the finetuning procedure of experiments reported for Figure 7.
**Model Selection:** Models used for transfer experiments are selected by performance on a small held-out validation set (ten tasks) for each task, that does not intersect with the test task.
Meta-RL with RL$^2$
-------------------
We adopt the recurrent architecture and hyperparameter settings as specified in the visual maze navigation tasks of @duan2016rl, except we:
- Use PPO for policy optimization ($\textrm{clip}=0.2, \textrm{value\_coef}=0.1$)
- Set the entropy bonus coefficient $\alpha$ in an environment-specific manner. We use $\alpha = 0.001$ for MuJoCo Sawyer and $\alpha = 0.1$ for ViZDoom.
- Enlarge the input observation space to $84 \times 84 \times 3$, adapting the encoder by half the stride in the first convolutional layer.
- Increase the size of the recurrent model (hidden state size 512) and the capacity of the output layer of the RNN (MLP with one hidden layer of dimension 256).
- Allow for four episodes per task (instead of two), since the tasks we consider involve more challenging task inference.
- Use a multi-layer perceptron with one-hidden layer to readout the output for the actor and critic, given the recurrent hidden state.
Reward Normalization
--------------------
A subtle challenge that arises in applying meta-RL across a range of tasks is difference in the statistics of the reward functions encountered, which may affect task inference. Without some form of normalization, the statistics of the rewards of unsupervised meta-training tasks versus those of the downstream tasks may be arbitrarily different, which may interfere with inferring the task. This is especially problematic for RL$^2$ (compared to e.g. MAML [@finn2017model]), which relies on encoding the reward as a feature at each timestep. We address this issue by whitening the reward at each timestep with running mean and variance computed online, separately for each task from the unsupervised task distribution during meta-training. At test-time, we share these statistics across tasks from the same test task distribution.
Learning Visual Representations with DeepCluster
------------------------------------------------
To jointly learn visual representations with the mixture model, we adopt the optimization scheme of DeepCluster [@caron2018deep]. The DeepCluster model is parameterized by the weights of a convolutional neural network encoder as well as a $k$-means model in embedding space. It is trained in an EM-like fashion, where the M-step additionally involves training the encoder weights via supervised learning of the image-cluster mapping.
Our contribution is that we employ a modified E-step, as presented in the main text, such that the cluster responsibilities are ensured to be consensual across states in a trajectory in the training data. As shown in our experiments, this allows the model to learn trajectory-level visual representations. The full [[CARML]{}]{} E-step with DeepCluster is presented below.
a set of trajectories ${\mathcal{D}}=\{({\mathbf{s}}_1,\dots,{\mathbf{s}}_T)\}_{i=1}^N$ Initialize $\phi:=(\phi_w, \phi_m)$, the weights of encoder $g$ and embedding-space mixture model parameters. . a mixture model $q_\phi({\mathbf{s}},z)$
For updating the encoder weights, we use the default hyperparameter settings as described in [@caron2018deep], except 1) we modify the neural network architecture, using a smaller neural network, ResNet-10 [@kaiming2015] with a fixed number of filters (64) for every convolutional layer, and 2) we use number of components $K=16$, which we did not tune. We tried using a more expressive Gaussian mixture model with full covariances instead of $k$-means (when training the visual representation), but found that this resulted in overfitting. Hence, we use $k$-means until the last iteration of EM, wherein a Gaussian mixture model is fitted under the resulting visual representation.
Environments
------------
[0.45]{} ![Example Observation Sequences from the Sawyer (left) and Vizdoom Random (right) environments.[]{data-label="fig:trajs"}](figs/sawyer_first.png "fig:"){width="\linewidth"}
[0.45]{} ![Example Observation Sequences from the Sawyer (left) and Vizdoom Random (right) environments.[]{data-label="fig:trajs"}](figs/vizdoom_first_2.png "fig:"){width="\textwidth"}
### ViZDoom Environment
{width="20.00000%"}
The environment used for visual navigation is a 500x500 room built with ViZDoom [@kempka2016vizdoom]. We consider both fixed and random environments; for randomly placing objects, the only constraint enforced is that objects should not be within a minimal distance of one another. There are 50 train objects and 50 test objects. The agent’s pose is always initialized to be at the top of the room facing forward. We restrict observations from the environment to be $84\times 84$ RGB images. The maximum episode length is set to 50 timesteps. The hand-crafted reward function corresponds to the inverse $l_2$ distance from the specified target object.
The environment considered is relatively simple in layout, but compared to simple mazes, can provide a more complex observation space insofar as objects are constantly viewed from different poses and in various combinations, and are often occluded. The underlying ground-truth state space is the product of continuous 2D position and continuous pose spaces. There are three discrete actions that correspond to turning right, turning left, and moving forward, allowing translation and rotation in the pose space that can vary based on position; the result is that the effective visitable set of poses is not strictly limited to a subset of the pose space, despite discretized actions.
### Sawyer Environment
{width="15.00000%"}
For visual manipulation, we use a MuJoCo [@todorov2012mujoco] environment involving a simulated Sawyer 7-DOF robotic arm in front of a table, on top of which is an object. The Sawyer arm is controlled by 2D continuous control. It is almost identical to the environment used by prior work such as [@nair2018visual], with the exception that our goal space is that of the object position. The robot pose and object are always initialized to the same position at the top of the room facing forward. We restrict observations from the environment to be $84\times 84$ RGB images. The maximum episode length is set to 50 timesteps. The hand-crafted reward function corresponds to the negative $l_2$ distance from the specified target object.
Additional Details for Qualitative Study of $\lambda$ {#app:sec:vae}
=====================================================
Instantiating $q_\phi$ as a VAE
-------------------------------
Three factors motivate the use of a variational auto-encoder (VAE) as a generative model for the 2D toy environment. First, a key inductive bias of DeepCluster, namely that randomly initialized convolutional neural networks work surprisingly well, which @caron2018deep use to motivate its effectiveness in visual domains, does not apply for our 2D state space. Second, components of a standard Gaussian mixture model are inappropriate for modeling trajectories involving turns. Third, using a VAE allows sampling from a continuous latent, potentially affording an unbounded number of skills.
We construct the VAE model in a manner that enables expressive generative densities $p({\mathbf{s}}|z)$ while allowing for computation of the policy reward quantities. We set the VAE latent to be $({\mathbf{z}}, t)$, where $p({\mathbf{z}},t)=p({\mathbf{z}})p(t)=\mathcal{N}(\bm{0}, \bm{I})\frac{1}{T}$. The form of $p(t)$ follows from restricting the policy to sampling trajectories of length $T$. We factorize the posterior as $q_\phi({\mathbf{z}},t|{\mathbf{s}}_{t'})=q({\mathbf{z}}|{\mathbf{s}}_{t'})\delta(t-t')$. Keeping with the idea of having a Markovian reward, we construct the VAE’s recognition network such that it takes as input individual states after training. To incorporate the constraint that all states in a trajectory are mapped to the same posterior, we adopt a particular training scheme: we pass in entire trajectories ${\mathbf{s}}_{1:T}$, and specify the posterior parameters as $\mu_{z} = \frac{1}{T} \sum_t g_{\eta}({\mathbf{s}}_t)$ and $\sigma^2_{z} = \frac{1}{T} \sum_t g_{\eta}({\mathbf{s}}_t)$.
The ELBO for this model is $$\begin{aligned}
&\mathbb{E}_{{\mathbf{z}},t \sim q_\phi({\mathbf{z}},t |{\mathbf{s}}_{t'})}\big[\log q_\phi({\mathbf{s}}_{t'}|{\mathbf{z}}, t) \big] - {D_\text{KL}\infdivx}{q_\phi({\mathbf{z}},t |{\mathbf{s}}_{t'})}{p({\mathbf{z}},t)} \\
=&\mathbb{E}_{{\mathbf{z}}\sim q_\phi({\mathbf{z}}|{\mathbf{s}}_{t'})} \big[ \log q_\phi({\mathbf{s}}_{t'}|{\mathbf{z}}, t') \big] - {D_\text{KL}\infdivx}{q_\phi({\mathbf{z}}|{\mathbf{s}}_{t'})}{p({\mathbf{z}})} - C\label{eq:elbo}\end{aligned}$$ where $C$ is constant with respect to the learnable parameters. The simplification directly follows from the form of the posterior; we have essentially passed $t'$ through the network unchanged. Notice that the computation of the ELBO for a trajectory leverages the conditional independence in our graphical model.
[[CARML]{}]{} Details
---------------------
Since we are not interested in meta-transfer for this experiment, we simplify the learning problem to training a contextual policy $\pi_\theta({\mathbf{a}}| {\mathbf{s}}, z)$. To reward the policy using the VAE $q_\phi$, we compute $$r_z({\mathbf{s}}) = \lambda \log q_\phi({\mathbf{s}}|z) - \log q_\phi({\mathbf{s}})$$ where $$\begin{aligned}
\log q_\phi({\mathbf{s}}|z) = \log \sum_t q_\phi({\mathbf{s}}|z,t)p(t) = \log \frac{1}{T} \sum_t q_\phi({\mathbf{s}}|z,t)\end{aligned}$$ and we approximate $\log q_\phi({\mathbf{s}})$ by its ELBO (Eq. \[eq:elbo\]), substituting the above expression for the reconstruction term.
Sawyer Task Distribution {#app:sec:sawyer}
========================
Visualizing the components of the acquired task distribution for the Sawyer domain reveals structure and diversity related to the position of the object as well as the control path taken to effect movement. Red encodes the true position of the object, and light blue that of the end-effector. We find tasks corresponding to moving the object to various locations in the environment, as well as tasks that correspond to moving the arm in a certain way without object interaction. The tasks provide a scaffold for learning to move the object to various regions of the reachable state space.
Since the Sawyer domain is less visually rich than the VizDoom domain, there may be less visually discriminative states that align with semantics of test task distributions. Moreover, since a large part of the observation is proprioceptive, the discriminative clustering representation used for density modeling captures various proprioceptive features that may not involve object interaction. The consequences are two-fold: 1) the gap in the CARML and the object-centric test task distributions may be large, and 2) the CARML tasks may be too diverse in-so-far as tasks share less structure, and inferring each task involves a different control problem.
{width="\linewidth"}
Mode Collapse in the Task Distribution {#app:sec:variants}
======================================
Here, we present visualizations of the task distributions induced by variants of the presented method, to illustrate the issue of using an entirely discrimination-based task acquisition approach. Using the fixed VizDoom setting, we compare:
1. CARML, the proposed method
2. **online discriminator** – task acquisition with a purely discriminative $q_\phi$ (akin to an online, pixel-observation-based adaptation of [@gupta2018unsupervised]);
3. **online pretrained-discriminator** – task acquisition with a discriminative $q_\phi$ as in **(ii)**, initialized with pre-trained observation encoder.
For all discriminative variants, we found it crucial to use a temperature $\geq 3$ to soften the classifier softmax to prevent immediate task mode-collapse.
[0.3]{} {width="\linewidth"}
[0.3]{} {width="\linewidth"}
[0.3]{} {width="\linewidth"}
\[fig:vizdoom\_shots\]
We find the task acquisition of purely discriminative variants **(ii, iii)** to suffer from an effect akin to mode-collapse; the policy’s data distribution collapses to a smaller subset of the trajectory space (one or two modes), and tasks correspond to minor variations of these modes. Skill acquisition methods such as DIAYN rely purely on discriminability of states/trajectories under skills, which can be more easily satisfied in high-dimensional observation spaces and can thus lead to such mode-collapse. Moreover, they do not a provide a direct mechanism for furthering exploration once skills are discriminable.
On the other hand, the proposed task acquisition approach (Algorithm \[alg:em\], \[sec:e-step\]) fits a generative model over jointly learned discriminative features, and is thus not only less susceptible to mode-collapse (w.r.t the policy data distribution), but also allows for density-based exploration (\[sec:m-step\]). Indeed, we find that **(iii)** seems to mitigate mode-collapse – benefiting from a pretrained encoder from **(i)** – but does not entirely prevent it. As shown in the main text (Figure \[fig:vizdoom\_variants\]), in terms of meta-transfer to hand-crafted test tasks, the online discriminative variants **(ii, iii)** perform worse than CARML **(i)**, due to lesser diversity in the task distribution.
Evolution of Task Distribution {#app:sec:task_distribution_evolution}
==============================
Here we consider the evolution of the task distribution in the Random VizDoom environment. The initial tasks (referred to as CARML It. 1) are produced by fitting our deep mixture model to data from a randomly-initialized meta-policy. CARML Its. 2 and 3 correspond to the task distribution after the first and second CARML E-steps, respectively.
We see that the initial tasks tend to be less structured, in so far as the components appear to be noisier and less distinct. With each E-step we see refinement of certain tasks as well as the emergence of others, as the agent’s data distribution is shifted by 1) learning the learnable tasks in the current data-distribution, and 2) exploration. In particular, tasks that are “refined” tend to correspond to more simple, exploitative behaviors (i.e. directly heading to an object or a region in the environment, trajectories that are more straight), which may not require exploration to discover. On the other hand, the emergent tasks seem to reflect exploration strategies (i.e. sweeping the space in an efficient manner). We also see the benefit of reorganization that comes from refitting the mixture model, as tasks that were once separate can be combined.
![Evolution of the [[CARML]{}]{} task distribution over 3 iterations of fitting $q_\phi$ in the random ViZDoom visual navigation environment. We observe evidence of task refinement and incorporation of new tasks. ](figs/task_evolution.pdf){width="\linewidth"}
|
---
abstract: |
We prove a localization formula in equivariant algebraic $K$-theory for an arbitrary complex algebraic group acting with finite stabilizer on a smooth algebraic space. This extends to non-diagonalizable groups the localization formulas of H.A. Nielsen [@Nie:74] and R. Thomason [@Tho:92]
As an application we give a Riemann-Roch formula for quotients of smooth algebraic spaces by proper group actions. This formula extends previous work of B. Toen [@Toe:99] and the authors [@EdGr:03].
address:
- ' Department of Mathematics, University of Missouri, Columbia, MO 65211 '
- ' Department of Mathematics, University of Georgia, Boyd Graduate Studies Research Center, Athens, GA 30602 '
author:
- Dan Edidin
- William Graham
title: 'Nonabelian localization in equivariant $K$-theory and Riemann-Roch for quotients'
---
[^1]
Introduction
============
Equivariant $K$-theory was developed in the late 1960’s by Atiyah and Segal as a tool for the proof of the index theorem for elliptic operators invariant under the action of a compact Lie group. In the late 1980’s and early 1990’s Thomason constructed an algebraic equivariant $K$-theory modeled on Quillen’s earlier construction of higher $K$-theory for schemes.
In both the topological and algebraic contexts equivariant $K$-theory is studied using its structure as a module for the representation ring of the group $G$. The fundamental theorem of equivariant $K$-theory is the [*localization theorem*]{} for actions of diagonalizable groups. We describe a version of this theorem in the complex algebraic setting. Let $R(G)$ be the representation ring of $G$ tensored with ${{\mathbb C}}$. If $X$ is a $G$-space we let $G(G,X)$ be the equivariant $K$-groups of the category of $G$-equivariant coherent sheaves, also tensored with ${{\mathbb C}}$. If $G$ is diagonalizable and $h \in G$, let $\iota: X^h \rightarrow X$ be the inclusion of the fixed locus of $h$. Let ${{\mathfrak m}}_h \subset R(G)$ denote the maximal ideal of representations whose virtual characters vanish at $h$. The localization theorem states that the natural map $$\iota_*: G(G,X^h)_{{{\mathfrak m}}_h} \rightarrow G(G,X)_{{{\mathfrak m}}_h},$$ is an isomorphism. Much of the power of this theorem comes from the fact that if $X$ is regular, then so is $X^h$, and the localization isomorphism has an explicit inverse, arising from the self-intersection formula for the regular embedding $X^h
\stackrel{\iota} \to X$: If $\alpha \in G(G,X)_{{{\mathfrak m}}_h}$ is an element of localized equivariant $K$-theory then $$\label{eqn.awesome}
\alpha = \iota_*\left( \frac{ \iota^*\alpha}{ \lambda_{-1}(N_\iota^*) } \right).$$ Here $N_\iota^*$ is the conormal bundle to $\iota$, and $\lambda_{-1}(N_\iota^*)$ is defined to be the element in $K$-theory corresponding to the formal sum $\sum_{l = 0}^{\text{rank} N_\iota^*}(-1)^{l} \Lambda^l(N_\iota^*).$
The formula of Equation is extremely useful because it reduces global calculations to those on the fixed locus. It has been applied in a wide range of contexts. For example, the localization theorem on the flag variety $G/B$ can be used to give a proof of the Weyl character formula. In [@EdGr:03] we used the localization theorem to prove a Kawasaki-Riemann-Roch formula for quotients by diagonalizable group actions (similar ideas had been introduced earlier by Atiyah [@Ati:74]).
If we try to generalize Equation to the nonabelian case we immediately run into the problem that $X^h$ is not in general $G$-invariant. However, the locus $X_{\Psi} = \overline{G X^h}$ is $G$-invariant; it is the closure of the union of the fixed point loci of elements in the conjugacy class $\Psi$ of $h$. Let ${{\mathfrak m}}_{\Psi} \subset
R(G)$ denote the maximal ideal of representations whose virtual characters vanish on $\Psi$, and let $i :X_{\Psi} \rightarrow X$ denote the inclusion. Then $i_*\colon G(G,X_{\Psi})_{{{\mathfrak m}}_{\Psi}} \to
G(G,X)_{{{\mathfrak m}}_{\Psi}}$ is an isomorphism (see Theorem \[thm.weakloc\]; this is a variant of a result of Thomason [@Tho:92], adapting a result of Segal [@Seg:68b] from topological $K$-theory). Unfortunately, as Thomason observed [@Tho:88], $X_{\Psi}$ can be singular even when $X$ is smooth, so the self-intersection formula does not apply. To obtain a nonabelian version of Equation new ideas are needed.
Although $X^h$ is not $G$-invariant, it is $Z$-invariant, where $Z =
{\mathcal Z}_G(h)$ is the centralizer in $G$ of $h$. In [@VeVi:02] Vezzosi and Vistoli proved that if $G$ acts on $X$ with finite stabilizers then there is an isomorphism between a localization of $G(Z,X^h)$ and a localization of $G(G,X)$. (Note that their theorem holds in arbitrary characteristic.) In this paper we work with the added hypothesis that the projection $f$ from the global stabilizer $S_X =
\{(g,x) | gx = x\}$ to $X$ is a finite morphism. Our main result states that there is a natural pushforward $\iota_! \colon
G(Z,X^h) \to G(G,X)$, such that when $X$ is smooth, the following formula holds for $\alpha \in G(G,X)_{{{\mathfrak m}}_\Psi}$: $$\label{eqn.irock}
\alpha = \iota_! \left( \frac{ \lambda_{-1}(({\mathfrak g}/{\mathfrak
z})^*)\cap (\iota^!\alpha)_h }{ \lambda_{-1}(N_\iota^*) } \right).$$ Here $\iota^!$ is the composition of the restriction map $G(G,X) \to G(Z,X)$ with the pullback $G(Z,X) \stackrel{\iota^*} \to G(Z,X^h)$; $(\iota^!\alpha)_h$ is the image of $\iota^! \alpha$ in $G(Z,X^h)_{{{\mathfrak m}}_h}$, and $\mathfrak{g}$, $\mathfrak{z}$ are the Lie algebras of $G$ and $Z$ respectively.
To prove this result, we use an equivalent formulation involving the global stabilizer. Let $S_{\Psi} \subset S_X$ be the closed subspace of pairs $(g,x)$ with $g \in \Psi$. The finite map $f:
S_{\Psi} \rightarrow X$ has image $X_{\Psi}$, but unlike $X_{\Psi}$, the space $S_{\Psi}$ is regular (if $X$ is). There is a natural identification of $G(G,S_{\Psi})$ with $G(Z,X^h)$, and the map $\iota_!$ is identified with the pushforward $f_*$ in $G$-equivariant $K$-theory. Moreover, the natural map $f: S_{\Psi} \rightarrow X$ is a local complete intersection morphism, so it has a normal bundle $N_f$. There is a distinguished “central summand” $G(G,S_{\Psi})_{c_{\Psi}}$ of $G(G,S_{\Psi})$; if $\beta \in
G(G,S_{\Psi})$, we let $\beta_{c_{\Psi}}$ denote the component of $\beta$ in the central summand. Equation is equivalent to the statement that if $\alpha \in G(G,X)_{{{\mathfrak m}}_{\Psi}}$, then $$\label{eqn.irock2}
\alpha = f_* \left( \frac{(f^*
\alpha)_{c_{\Psi}}}{\lambda_{-1}(N_f^*)} \right).$$ This formula looks similar to the formula that would hold if $f$ were a regular embedding (the only change would be to replace $(f^*
\alpha)_{c_{\Psi}}$ by $f^*{\alpha}$). However, that formula is not correct, and indeed, the main difficulty in proving is that $f$ is not a regular embedding, so we cannot apply the self-intersection formula. The proof given here is less direct; we first prove the result when $G$ is a product of general linear groups, and then use a change of groups argument to deduce the general case.
The main application of Equations and is to give refined formulas for the Todd classes of sheaves of invariant sections on quotients of smooth algebraic spaces. If $X$ is a smooth, separated, algebraic space and $G$ is an algebraic group acting properly (and thus with finite stabilizer) then the theorem of Keel and Mori [@KeMo:97] implies that there is a (possibly singular) geometric quotient $Y = X/G$. For such quotients there is a map in $K$-theory $\pi_G\colon G(G,X) \to G(Y)$ induced by the exact functor which takes a $G$-equivariant coherent sheaf to its subsheaf of invariant sections (Lemma \[lem.invariants\]). Let $\tau_Y \colon G_0(G,X) \to A_*(Y)$ be the Riemann-Roch map defined by Baum, Fulton, MacPherson [@BFM:75; @Ful:84]. If $\alpha \in
G_0(G,X)$ then we obtain explicit expressions (Theorems \[t.rrtheorem1\] and \[t.rrtheorem2\]) for $\tau_Y(
\pi_G(\alpha_{\Psi}))$ in terms of the restriction of $\alpha$ to a class in $G_0(Z,X^h)$ where $h \in \Psi$ is any element. If we sum over all conjugacy classes $\Psi$ we obtain formulas for $\tau_Y(\pi_G(\alpha))$. When the quotient is quasi-projective our formulas for $\tau_Y(\pi_G(\alpha))$ can be deduced from the Riemann-Roch formula for stacks due to B. Toen [@Toe:99]. Our method of proof is quite different, and makes no essential use of stacks.
The proof of our Riemann-Roch theorem is essentially the same as the proof for diagonalizable $G$ given in [@EdGr:03], with the nonabelian localization theorem of this paper in place of the localization theorem for diagonalizable groups. A key element of the proof in [@EdGr:03] was the fact that, if $G$ is diagonalizable, $X^h$ is $G$-invariant and we may define an $h$-action on $G_0(G,X^h)$ which we called “twisting by $h$”. Intuitively, this twist comes from the $h$-action on the sections of any $G$-equivariant coherent sheaf on $X^h$. In the nonabelian setting one can still twist by a central element, so there is an action of $h$ on $G_0(Z, X^h)$. This can be viewed as a twist of $G(G,S_{\Psi})$, which intuitively comes from the tautological action of the element $g$ on the fiber at $(g,x)$ of any $G$-equivariant vector bundle on $S_{\Psi}$. Versions of this twist and the global stabilizer appear in the Riemann-Roch theorems of Kawasaki and Toen, and motivated our approach to the localization theorem and Riemann-Roch theorem. Interestingly, in the Riemann-Roch formula obtained from , one might expect a term involving $(f^*
\alpha)_{c_{\Psi}}$. However, we prove that the contributions from $(f^* \alpha)_{c_{\Psi}}$ and $f^*
\alpha$ are equal, so our formula does not mention the central summand.
In this paper we work over ${{\mathbb C}}$ and tensor all $K$-groups with ${{\mathbb C}}$. The reason we do this is so that we can identify the representation ring $R(G)$ with class functions on $G$; this idea, which goes back to Atiyah and Segal, allows us to directly relate $G$-equivariant $K$-theory to conjugacy classes in $G$. By working over ${{\mathbb C}}$ we hope that the geometric techniques used to prove our main results are not obscured by technical details.
Nevertheless, we believe that versions of the nonabelian localization and Riemann-Roch theorems should hold over an arbitrary algebraically closed field provided we assume that all stabilizer groups are reduced. In this situation, instead of localizing at maximal ideals ${{\mathfrak m}}_h \in R(G) \otimes {{\mathbb C}}$ where $h \in G$ has finite order, we may localize at the multiplicatively closed set $S_H$ defined on p. 10 of [@VeVi:02], where $H$ is the cyclic group generated by $h$. In a different direction, there should be topological versions of these results for actions of compact Lie groups. This will be pursued elsewhere.
We would like to thank Michel Brion for pointing out an error in an earlier version of the paper. We are also grateful to the referee for several helpful suggestions.
Conventions and notation {#s.conventions}
------------------------
We work entirely over the ground field of complex numbers ${{\mathbb C}}$. All algebraic spaces are assumed to be of finite type over ${{\mathbb C}}$. For a reference on the theory of algebraic spaces, see the book [@Knu:71]. All algebraic groups are assumed to be linear. A basic reference for the theory of algebraic groups is Borel’s book [@Bor:91].
If $G$ is an algebraic group then ${\mathcal Z}(G)$ denotes the center of $G$. If $h \in G$ is any element then ${\mathcal Z}_G(h)$ denotes the centralizer of $h$ in $G$. The conjugacy class of $h$ in $G$ is denoted $C_G(h)$. The map $G \to C_G(h)$, $g \mapsto ghg^{-1}$ identifies $C_G(h)$ with the homogeneous space $G/{\mathcal Z}_G(h)$.
If $G$ is an algebraic group then $R(G)$ denotes the representation ring of $G$ tensored with ${{\mathbb C}}$.
### Group actions {#ss.group}
Let $G$ be an algebraic group acting on an algebraic space $X$. We consider three related conditions on group actions.
\(i) We say that $G$ acts [*properly*]{} if the map $$G\times X \to X \times X, \; (g,x) \mapsto (x,gx)$$ is proper.
Let $G \times X \to X \times X$ be the map we just defined and let $X \to X \times X$ be the diagonal. Then $S_X = G \times X \times_{X \times X} X$ is called the global stabilizer. As a set, $S_X = \{(g,x)| gx =x\}$.
\(ii) We say that $G$ acts with [*finite stabilizer*]{} if the projection $S_X \to X$ is a finite morphism.
\(iii) We say that $G$ acts with [*finite stabilizers*]{} if the projection $S_X \to X$ is quasi-finite; i.e. for every point $x \in X$ the isotropy group $G_x$ is finite.
Since $G$ is affine, the map $G \times X \to X \times X$ is finite if it is proper. It follows that any proper action has finite stabilizer. If $G$ acts properly on $X$, then the diagonal morphism of $X$ factors as the composition of two proper maps $X \stackrel{(e,1_X)} \to G \times X \to X \times X$. This implies that the diagonal is a closed embedding, so $X$ is automatically separated.
Groups, representation rings, and conjugacy classes {#s.repring}
===================================================
This section collects a number of facts about algebraic groups, representation rings, and conjugacy classes which are used in the proof of the localization theorem. We have included proofs of some results that are essentially known but are difficult to find in the literature for groups that are not connected or semisimple.
The representation ring and class functions
-------------------------------------------
Let $G$ be an algebraic group over ${{\mathbb C}}$. The group $G$ is called reductive if the radical of its identity component $G_0$ is a torus. Any reductive algebraic group is the complexification of a maximal compact subgroup [@OnVi:90 Theorem 8, p. 244]. (This result can be deduced from the corresponding result for connected groups, using facts about maximal compact subgroups of Lie groups with finitely many connected components [@Hoc:65 Theorem XV.3.1].) By the unipotent radical of $G$ we mean the unipotent radical of the identity component of $G$; this is a normal subgroup of $G$, and $G$ modulo its unipotent radical is reductive. Any complex algebraic group $G$ has a Levi subgroup $L$. This means $G$ is the semidirect product of $L$ and the unipotent radical of $G$ ([@OnVi:90 Chapter 6, Theorem 4]); $L$ is necessarily reductive.
For a general algebraic group $G$, let $\hat{G}$ denote the set of isomorphism classes of irreducible (finite-dimensional) algebraic representations of $G$. Let ${{\mathbb C}}[G]$ denote the coordinate ring of $G$, and ${{\mathbb C}}[G]^G$ the ring of class functions, i.e., the functions on $G$ which are invariant under conjugation. There is a map $R(G) \rightarrow {{\mathbb C}}[G]^G$ which takes $[V]$ to $\chi_V$, where $V$ is a representation of $G$ and $\chi_V$ its character.
Let $V$ be a representation of $G$, and let $V^*$ denote the dual representation. There is a map $$\label{e.repfunction}
V^* \otimes V \rightarrow {{\mathbb C}}[G]$$ taking $\lambda \otimes v$ to the function $\lambda/v$ defined by $$\lambda/v(x) = \lambda(xv)$$ for $x \in G$. This map is $G \times G$-equivariant, where $G \times G$ acts on $V^* \otimes V$ via the $G$-action on each factor, and $G \times G$ acts on functions by $$( (g_1,g_2) \cdot f)(x) = f(g_1^{-1} x g_2),$$ for $g_1,g_2,x \in G$ and $f \in {{\mathbb C}}[G]$. Functions of the form $\lambda / v$ are called representative functions; we denote the algebra they generate by ${{\mathcal T}}(G)$. If $G$ is a linear algebraic group, then the action of $G \times G$ on ${{\mathbb C}}[G]$ is locally finite, and this can be used to show that ${{\mathcal T}}(G) = {{\mathbb C}}[G]$. If $G$ is the complexification of $K$, then the map ${{\mathcal T}}(G) \rightarrow {{\mathcal T}}(K)$ is an isomorphism (cf. [@BrtD:95]). As a consequence, we obtain the algebraic Peter-Weyl theorem, attributed to Hochschild and Mostow:
If $G$ is a complex reductive algebraic group, then the map $$\label{e.repfunction2}
\oplus_{V \in \hat{G}}V^* \otimes V \rightarrow {{\mathbb C}}[G]$$ is an isomorphism as representations of $G \times G$.
This follows from the usual Peter-Weyl theorem for compact groups ([@Ros:02 p. 201]), in view of the isomorphism ${{\mathbb C}}[G] \rightarrow {{\mathcal T}}(K)$.
In this paper, we frequently use the following result. For connected groups, it can be proved using restriction to a maximal torus.
\[prop.repring\] If $G$ is reductive, the map $R(G) \rightarrow {{\mathbb C}}[G]^G$ taking a representation to its character is an isomorphism. In particular, for any $G$, $R(G)$ is a finitely generated algebra over ${{\mathbb C}}$.
View $G$ as embedded diagonally in $G \times G$. The isomorphism induces an isomorphism of $G$-invariants in the source and target. If $V$ is any representation of $G$, then $(V^*
\otimes V)^G = \mbox{Hom}_G(V,V)$, and if $V$ is irreducible representation, Schur’s lemma implies that this is $1$-dimensional, spanned by the identity map $\mbox{id}_V$. But $\phi(\mbox{id}_V) =
\chi_V$, proving the first statement of the proposition. The second statement follows because the representation rings of $G$ and any Levi factor are isomorphic.
As another application of the relationship between $G$ and $K$, we have the following proposition.
\[prop.finite\] If $H \hookrightarrow G$ is an embedding of groups, then $R(H)$ is a finite $R(G)$-module.
If $H_1$ is a Levi subgroup of $H$ then the restriction map $R(H) \rightarrow R(H_1)$ is an isomorphism, so if necessary replacing $H$ by $H_1$, we may assume $H$ is reductive. Let $G'$ be the quotient of $G$ by its unipotent radical; then the natural map $R(G') \rightarrow R(G)$ is an isomorphism. Moreover, the kernel of $H \rightarrow G'$ is a unipotent normal subgroup of $H$, hence is trivial. Hence the map $H \rightarrow G'$ is injective. Therefore, if necessary replacing $G$ by $G'$, we may assume $G$ is reductive.
Let $L$ be a maximal compact subgroup of $H$; we may assume $L$ is contained in a maximal compact subgroup $K$ of $G$. Then the restriction maps $R(G) \rightarrow R(K)$ and $R(H) \rightarrow R(L)$ are isomorphisms (cf. [@BrtD:95]). Therefore the proposition follows from the analogous result of [@Seg:68a] for compact groups.
Conjugacy classes and representation rings
------------------------------------------
If $H$ is a subgroup of $G$, let $C_H(g)$ denote the $H$-conjugates of $g$, i.e., the image of the map $H \rightarrow G$ taking $h$ to $hgh^{-1}$.
\[prop.conjclosed\] Let $G$ be a reductive algebraic group. A conjugacy class $\Psi$ in $G$ is closed if and only if it is semisimple.
In any algebraic group, the conjugacy class of a semisimple element is closed [@Bor:91 Theorem 9.2]. Conversely, suppose $\Psi$ is a closed conjugacy class. Let $g = su$ be the Jordan decomposition of some $g \in \Psi$. We want to show that $u = 1$, or in other words, that $C_G(s) = C_G(su)$. A general result of Mumford about reductive groups acting on affine schemes ([@MFK:94 Ch. 1, Cor. 1.2]) implies that given two closed disjoint conjugation-invariant subsets of $G$, there is a class function which is $1$ on one subset and $0$ on the other. Therefore, to show that $C_G(s) = C_G(su)$ it suffices to show that any class function takes the same value on $C_G(s)$ as on $C_G(su)$. But this holds because it is true for characters, which span the space of class functions.
Note that if $G$ is not reductive, there may be closed conjugacy classes which are not semisimple, for example if $G = {\mathbb G}_a$.
If $\Psi = C_G(g)$ is a semisimple conjugacy class, let ${{\mathfrak m}}_\Psi
\subset R(G)$ be the ideal of virtual characters which vanish on $\Psi$. Then ${{\mathfrak m}}_\Psi$ is the kernel of the homomorphism of ${{\mathbb C}}$-algebras $R(G) \to {{\mathbb C}}$ defined by the property that $[V] \mapsto
\chi_V(\Psi)$. Since the trivial character does not vanish on $\Psi$, ${{\mathfrak m}}_\Psi$ is a maximal ideal of $R(G)$.
\[prop.specarbitrary\] Let $G$ be an arbitrary algebraic group. The assignment $\Psi \mapsto {{\mathfrak m}}_\Psi$ gives a bijection between the set of semisimple conjugacy classes in $G$ and the maximal ideals in $R(G)$.
First observe that the result holds if $G$ is reductive by Propositions \[prop.repring\] and \[prop.conjclosed\]. For general $G$, let $G = LU$ be a Levi decomposition and let $r \colon R(G) \to R(L)$ be the restriction map. Since $U$ is unipotent, $r$ is an isomorphism.
If ${{\mathfrak m}}\subset R(G)$ is a maximal ideal then, since $L$ is reductive, $r({{\mathfrak m}})= {{\mathfrak m}}_\Phi$ for some semisimple conjugacy class $\Phi = C_L(l)$ in $L$. If $\Psi = C_G(l)$ then $r({{\mathfrak m}}_\Psi) \subset {{\mathfrak m}}_\Phi$. Thus ${{\mathfrak m}}_\Psi \subset {{\mathfrak m}}$. But ${{\mathfrak m}}_\Psi$ is maximal so it equals ${{\mathfrak m}}$.
Suppose $\Psi_1$ and $\Psi_2$ are two semisimple conjugacy classes in $G$ such that ${{\mathfrak m}}_{\Psi_1} = {{\mathfrak m}}_{\Psi_2}$. Since the proposition holds for the reductive subgroup $L$, $r({{\mathfrak m}}_{\Psi_i}) = {{\mathfrak m}}_\Phi$ for some semisimple conjugacy class $\Phi \subset L$. It follows that $\Psi_1 \cap L = \Psi_2 \cap L = \Phi$. This means that $\Psi_1 \cap \Psi_2$ is nonempty, so $\Psi_1 = \Psi_2$.
Given an embedding of groups $G \rightarrow H$, let $r\colon R(H) \to
R(G)$ be the restriction map. Let $\Psi = C_H(h)$ be a semisimple conjugacy class in $H$. By Proposition \[prop.finite\], the ideal ${{\mathfrak m}}_\Psi
R(G)$ is contained in a finite number of maximal ideals of $\operatorname{Spec}R(G)$. If $\Psi'$ is a conjugacy class in $\Psi \cap G$, then the restriction of any virtual character vanishing on $\Psi$ also vanishes on $\Psi'$; that is, ${{\mathfrak m}}_\Psi R(G) \subset {{\mathfrak m}}_{\Psi'}$. Thus, $\Psi \cap G$ decomposes into a finite number of semisimple conjugacy classes $\Psi_1, \ldots , \Psi_l$.
\[prop.conjdecomp\] Let $G \hookrightarrow H$ be an embedding of groups, and let $\Psi'$ and $\Psi$ be semisimple conjugacy classes in $G$ and $H$, respectively. Then ${{\mathfrak m}}_\Psi R(G) \subset {{\mathfrak m}}_{\Psi'}$ if and only if the conjugacy class $\Psi'$ is contained in $(\Psi \cap G)$.
The if direction follows from the discussion immediately preceding the statement of the proposition. Conversely, if $\Psi' $ is a conjugacy class in $G$ not contained in $\Psi \cap G$, then $\Psi'$ is disjoint from the conjugacy classes $\Psi_1, \ldots , \Psi_l$ in $\Psi \cap G$. By Proposition \[prop.specarbitrary\], ${{\mathfrak m}}_{{\Psi}'}$ is distinct from each of the ${{\mathfrak m}}_{\Psi_i}$. Therefore there is a virtual character $f \in {{\mathfrak m}}_{\Psi'}$ such $f$ is not in some ${{\mathfrak m}}_{\Psi_i}$. Thus, the restriction of $f$ to $\Psi \cap G$ is not zero, so $f \notin {{\mathfrak m}}_\Psi R(G)$.
\[rem.pickbill\] Proposition \[prop.conjdecomp\] implies that if $\Psi \subset H$ is a semisimple conjugacy class then $R(G)_{{{\mathfrak m}}_{\Psi}}$ is a semilocal ring with maximal ideals ${{\mathfrak m}}_{\Psi_{1}}R(G)_{{{\mathfrak m}}_{\Psi}},
\ldots {{\mathfrak m}}_{\Psi_{l}}R(G)_{{{\mathfrak m}}_{\Psi}}$.
As noted above, if $G$ is a subgroup of $H$ and $g \in
G$, the intersection of $C_H(g)$ with $G$ may consist of more than one $G$-conjugacy class in $G$. The following result shows that it is possible to find embeddings where this does not occur.
\[prop.goodembedding\] Suppose $G$ is an algebraic group and $\Psi = C_G(g)$ is a semisimple conjugacy class in $G$. There is an embedding $G \rightarrow H$, where $H = \prod_i \operatorname{GL}_{n_i}$, such that $C_H(g) \cap G = C_G(g)$.
Since $R(G) = R(L)$, where $L$ is a Levi factor of $G$, $R(G)$ is Noetherian. Therefore we can find a finite set $f_1,\ldots f_l$ of elements which generate the maximal ideal ${{\mathfrak m}}_\Psi \subset R(G)$. Each function $f_i$ can be written as a finite sum $f_i = \sum_j
a_{ij}\chi_{ij}$ where $\chi_{ij}$ is the character of a $G$-module $V_{ij}$. Let $V$ be a faithful representation of $G$ and set $H = \operatorname{GL}(V) \times \prod_{i,j} \operatorname{GL}(V_{ij})$. Then $G$ embeds as a subgroup of $H$ since it embeds as a subgroup of the first factor, $\operatorname{GL}(V)$.
Let $g'$ be an element of $G$ such that $g'$ is conjugate to $g$ in $H$. Since $H$ is a direct product, the image of $g$ and $g'$ in each $\operatorname{GL}(V_{ij})$ must have the same trace. Thus, $\chi_{ij}(g) =
\chi_{ij}(g')$ for all $i,j$. Hence $f_{i}(g)= f_i(g') = 0$ for each generator $f_i$ of ${{\mathfrak m}}_\Psi$ (since $g$ is, by definition, an element of the conjugacy class $\Psi$). Therefore, by Proposition \[prop.specarbitrary\], $g'$ is in the conjugacy class of $g$ in $G$.
Equivariant $K$-theory {#s.ekt}
======================
This section contains some $K$-theoretic results needed for the proof of our main result, the nonabelian localization theorem.
Basic facts and notation
------------------------
Let $G$ be an algebraic group acting on an algebraic space $X$. We use the notation ${\tt coh}^G_X$ to denote the abelian category of $G$-equivariant coherent ${\mathcal O}_X$-modules. Let $$G(G,X) = \oplus_{i = 0}^\infty G_i(G,X) \otimes {{\mathbb C}}$$ where $G_i(G,X)$ is the $i$-th Quillen $K$-group of the category ${\tt
coh}^G_X$. Since all our coefficients are taken to be complex, we will simply write $G_0(G,X)$ (rather than $G_0(G,X) \otimes {{\mathbb C}}$) for the Grothendieck group of $G$-equivariant coherent sheaves, tensored with ${{\mathbb C}}$. Likewise, we write $K_0(G,X)$ for the Grothendieck ring of $G$-equivariant bundles, also tensored with ${{\mathbb C}}$. When $X$ is a smooth scheme, Thomason’s equivariant resolution theorem implies that $K_0(G,X) = G_0(G,X)$ (this is true even without tensoring with ${{\mathbb C}}$). We use analogous notation in the non-equivariant setting, writing $\tt{coh}_X$ for the category of coherent ${\mathcal O}_X$-modules, and write $G(X)$, $G_0(X)$, and $K_0(X)$ for the non-equivariant versions of $K$-theory.
If ${\mathcal E}$ is a $G$-equivariant locally free sheaf then the assignment ${\mathcal F} \mapsto {\mathcal E \otimes F}$ defines an exact functor ${\tt coh}_X^G \to {\tt coh}_X^G$. This implies that there is an action of $K_0(G,X)$ on $G(G,X)$.
If $X$ and $Y$ are $G$-spaces and $p \colon X \to Y$ is a $G$-map, then there is a pullback $p^* \colon K_0(G,X) \to K_0(G,X)$. When $Y =
\operatorname{Spec}{{\mathbb C}}$, this pullback makes $G(G,X)$ an $R(G) = K_0(G,\operatorname{Spec}{{\mathbb C}})$ module.
If $p\colon X \to Y$ is a $G$-equivariant proper morphism then there is a pushforward $p_* \colon G(G,X) \to G(G,Y)$ which is an $R(G)$-module homomorphism [@Tho:86i 1.11-12]. The pushforward is defined on the level of Grothendieck groups by the formula $p_*[{\mathcal F}] =
\sum (-1)^{i}[R^i p_* {\mathcal F}].$
If $p \colon X \to Y$ is flat and $G$-equivariant then there is a pullback $p^*\colon G(G,X) \to G(G,Y)$ which is also an $R(G)$-module homomorphism. It is induced by the exact functor which takes a coherent sheaf ${\mathcal F}$ to its pullback $p^*{\mathcal F}$.
More generally, if $X$ is a regular algebraic space then Vezzosi and Vistoli proved [@VeVi:02 Theorem A4] that $G(G,X)$ is isomorphic to the Waldhausen $K$-theory of the category ${\mathcal W}_{3,X}$ of complexes of flat quasi-coherent $G$-equivariant ${\mathcal O}_X$ modules with bounded coherent cohomology. It follows that if $p \colon X \to Y$ is a map of regular algebraic spaces then there is a pullback $p^* \colon G(G,X) \to G(G,Y)$ which is an $R(G)$-module homomorphism. If $p$ is a regular embedding of smooth algebraic spaces then there is a self-intersection formula for $ \alpha \in G(G,X)$: $$p^* p_* \alpha = \lambda_{-1}(N_p^*) \cap \alpha,$$ where $N_p^*$ is the conormal bundle to map $p$ and $\lambda_{-1}(N_p^*) \in K_0(G,X)$ is the formal sum $\sum_{i = 0}^{rk\; N_p^*} (-1)^i
[\Lambda^i(N_p^*)]$. This fact is proved in the course of the proof of Theorem 3.7 of [@VeVi:02].
Morita equivalence {#s.morita}
------------------
In this paper we make extensive use of a particular instance of Morita equivalence, which we briefly describe. If $Z \subset G$ is a closed subgroup and $X$ is a $Z$-space then we may consider the $G \times Z$-space $G \times X$ where $(k,z) \cdot (g,x) = (kgz^{-1},zx)$. We write $G \times_Z X$ for the quotient of $G \times X$ by the free action of the subgroup $1 \times Z$. The space $G \times_Z X$ will often be referred to as a [*mixed space*]{}. If the $Z$ action on $X$ is the restriction of a $G$ action, then the automorphism of $G \times X$ given by $(g,x) \mapsto (g,gx)$ induces an isomorphism of quotients $G\times_Z X \to G/Z \times X$.
Since the actions of $G$ and $Z$ on $G \times X$ commute, the action of $G \times 1$ on $G \times X$ descends to an action on the quotient $G \times_Z X$. The Morita equivalence we use is the equivalence of categories between the category of $Z$-equivariant coherent sheaves on $X$ and the category $G$-equivariant coherent sheaves on $G \times_Z X$. The equivalence is given by pulling a $Z$-module on $X$ back to $G \times
X$ to obtain a $G \times Z$-module on $G \times X$ and then taking the subsheaf of $1 \times Z$-invariant sections to obtain a $G$-module on $G
\times_Z X$.
\[rem.tantomixedbund\] When $X$ is a point, Morita equivalence between the categories of $G$-equivariant coherent sheaves on $G/Z$ and $Z$-modules is obtained by taking the fiber of a sheaf at the identity coset $Z$. Under this equivalence the tangent bundle of $G/Z$ corresponds to the $Z$-module ${\mathfrak g}/{\mathfrak z}$, where ${\mathfrak g}$ and ${\mathfrak
z}$ denote the Lie algebras of $G$ and $Z$ respectively [@Bor:91 Proposition 6.7]. If $X$ is a $G$-space then the identification $G \times_Z X = G/Z \times X$ identifies the relative tangent bundle of the projection $p \colon G \times_Z X
\to X$ with the bundle $G \times_Z (X \times {\mathfrak g}/{\mathfrak
z})$.
\[rem.moritaequiv\] The Morita equivalence of categories above induces an $R(G)$-module isomorphism in $K$-theory $G(Z,X) \to G(G, G \times_Z X)$. Here $R(G)$ acts on $G(Z,X)$ via the restriction map $R(G) \to R(Z)$. This observation will be used repeatedly in the sequel.
Localization in equivariant $K$-theory {#sec.loc}
--------------------------------------
In this section we extend the localization theorem of [@Tho:92 Theorem 2.2] to arbitrary algebraic groups. The proof is similar to Thomason’s. However, because we work with ideals in $R(G) \otimes {{\mathbb C}}$ which may have zero intersection with the integral representation ring, we cannot directly quote his results.
\[thm.weakloc\] Let $G$ be an algebraic group acting on an algebraic space $X$. Let $\Psi = C_G(h)$ be a semisimple conjugacy class and let $X_\Psi$ be the closure of $GX^h$ in $X$.
\(a) The proper pushforward $$i_*\colon G(G,X_\Psi) \to G(G,X)$$ is an isomorphism of $R(G)$-modules after localizing at ${{\mathfrak m}}_\Psi$.
\(b) If $X$ is smooth and $h \in {\mathcal Z}(G)$ (so $\Psi = h$ and $X^h$ is $G$-invariant) then the map of $R(G)$-modules $$\cap\;\lambda_{-1}(N_i^*) \colon G(G,X^h) \to G(G,X^h)$$ is invertible after localizing at ${{\mathfrak m}}_h$, and if $\alpha \in G(G,X)_{{{\mathfrak m}}_h}$, then $$\label{eqn.prettygood}
\alpha = i_*\left(\lambda_{-1}(N_i^*)^{-1} \cap i^*\alpha\right).$$ Here the notation $\lambda_{-1}(N_i^*)^{-1} \cap i^*\alpha$ means the image of $i^*\alpha$ under the inverse of the isomorphism $\cap \; \lambda_{-1}(N_i^*)$.
If $X$ is a smooth scheme then $X^h$ is as well. By Thomason’s equivariant resolution theorem we may identify $K_0(G,X^h)$ with $G_0(G,X^h)$ [@Tho:87 Theorem 5.7] and view $(\lambda_{-1}(N_i^*))^{-1}$ as an element in $G(T,X^h)_{{{\mathfrak m}}_h}$.
[*Step 1: $G = T$ is a torus.*]{} If $T$ is a torus then $C_G(h) = h$. Using Noetherian induction and the localization long exact sequence [@Tho:87 Theorem 2.7] it suffices to prove that $G(T,X)_{{{\mathfrak m}}_h} = 0$ whenever $X^h$ is empty. Since $T$ is diagonalizable, we can, by Proposition \[prop.repring\], identify $R(T)$ with the coordinate ring of $T$, and ${{\mathfrak m}}_h$ with the maximal ideal of $h \in T$. If $T' \subset T$ is a closed subgroup then $R(T')_{{{\mathfrak m}}_h} = 0$ unless $h \in T'$. By Thomason’s generic slice theorem [@Tho:86d Theorem 4.10] there is a $T$-invariant open set $U$, a closed subgroup $T'$ acting trivially on $U$ and a $T$-equivariant isomorphism $U \simeq T/T' \times U/T$. Since $T'$ acts trivially on $U$, and $X^h$ is empty, $h \notin T'$. By Morita equivalence, $G(T,U) = G(T',U/T)$. But $T'$ acts trivially on $U/T$, so $G(T',U/T) = R(T') \otimes G(U/T)$. Thus $G(T',U/T)_{{{\mathfrak m}}_h} = 0$. Applying Noetherian induction and using the localization long exact sequence we conclude that $G(T,X)_{{{\mathfrak m}}_h} =0$. This proves part (a).
Next we must show that when $X$ is smooth, the multiplication map $$\cap\;\lambda_{-1}(N_i^*) \colon G(T,X^h)_{{{\mathfrak m}}_h} \to G(T,X^h)_{{{\mathfrak m}}_h}$$ is an isomorphism. This may be done using Noetherian induction on $X^h$. Again we apply Thomason’s generic slice theorem. Thus we may assume that $X^h = T/T'
\times X^h/T$. As above $G(T,X^h)_{{{\mathfrak m}}_h} = R(T')_{{{\mathfrak m}}_h} \otimes G(X^h/T)$.
Let ${\mathcal N}_{x}$ be the fiber of $N_i^*$ over a point $x\in
X^h$. Since $T'$ acts trivially on $X^h$, ${\mathcal N}_x$ is a $T'$-module. As in the proof of [@Tho:92 Lemma 3.2], the identification $G(T,X^h)_{{{\mathfrak m}}_h} = R(T')_{{{\mathfrak m}}_h} \otimes G(X^h/T)$ implies that the action of $\lambda_{-1}(N_i^*)$ on $G(T,X^h)_{{{\mathfrak m}}_h}$ is invertible if and only if $\lambda_{-1}({\mathcal N}_x)$ is invertible in $R(T')_{{{\mathfrak m}}_h}$ for some $x$ in each connected component of $X^h$; i.e. $\lambda_{-1}({\mathcal N}_x) \notin {{\mathfrak m}}_h$. Now ${\mathcal N}_x$ decomposes into $1$-dimensional eigenspaces for the action of $T'$ so we can write $\lambda_{-1}({\mathcal N}_x)=
\prod_{l=1}^s (1 - \chi_l)$, where $\chi_1, \ldots , \chi_s$ are (not necessarily distinct) characters of $T'$.
Let $H$ be the closure of the cyclic subgroup of $T$ generated by $h$. Then $X^H = X^h$. Since $X^h = T/T' \times X^h/T$, $T'$ is the biggest subgroup of $T$ acting trivially on $X^h = X^H$. Thus, we see that $H \subset T'$ and the characters in the $T'$-module decomposition restrict to characters of $H$. None of these characters can be trivial on $H$ since the normal space to $X^H$ at $x$ is the quotient of the tangent space $T_{x,X}$ by the invariant subspace $T_{x,X}^H$. Since the cyclic subgroup generated by $h$ is dense in $H$, we see that, for each $\chi_l$, we have $\chi_l(h^n) =
\chi_l(h)^n \neq 1$ for some exponent $n$. Thus $(1 - \chi_l) \notin
{{\mathfrak m}}_h$. Therefore, the action of $\lambda_{-1}(N_i^*)$ on $G(T,X^h)_{{{\mathfrak m}}_h}$ is invertible.
Finally, the formula in Equation can be deduced as follows. Since $i_*$ is surjective, $\alpha = i_* \beta$ for some $\beta \in G(T,X^h)_{{{\mathfrak m}}_h}$. Thus, $$\begin{aligned}
i^*\alpha & = & i^* i_* \beta\\
& = & \lambda_{-1}(N_i^*)\cap\beta\end{aligned}$$ where the second equality follows from the self-intersection formula in equivariant $K$-theory. Since the action of $\lambda_{-1}(N_i^*)$ is invertible after localizing at ${{\mathfrak m}}_h$, the formula follows.\
[*Step 2. $G$ is connected and reductive.*]{} We use a standard reduction to a maximal torus argument. Let $T$ be any maximal torus in $G$ and let $B$ be a Borel subgroup of $G$ containing $T$. Since $B/T$ is isomorphic to affine space, the restriction $G(B,X) \to G(T,X)$ is an isomorphism [@Tho:88 Proof of Theorem 1.13]. The same proof implies that the projection $p \colon G \times_B X \to X$ induces a pullback $p^!\colon
G(G,X) \to G(G,G \times_B X)= G(T,X)$ and a pushforward $p_!\colon
G(T,X) \to G(G,X)$, with the properties that $p_! 1 =1$, and if $\beta \in K_0(G,X)$, then $p_! (p^!\beta \cap \alpha) = \beta \cap
p_!\alpha$. In particular, $p^!$ is a split monomorphism. Moreover, standard functorial properties of equivariant $K$-theory imply that $p^!$ and $p_!$ are functorial for $G$-equivariant morphisms.
To prove (a), as in the torus case it suffices to prove that if $X^h$ is empty then $G(G,X)_{{{\mathfrak m}}_{\Psi}} = 0$. Since $G(G,X)$ embeds in $G(T,X)$ it actually suffices to prove that $G(T,X)_{{{\mathfrak m}}_{\Psi}} = 0$. Let $h_1, h_2, \ldots h_n$ be the conjugates of $h$ contained in the maximal torus $T$. By Remark \[rem.pickbill\], $R(T)_{{{\mathfrak m}}_{\Psi}}$ is a semilocal ring whose maximal ideals are the ${{\mathfrak m}}_{h_i} R(T)_{{{\mathfrak m}}_{\Psi}}$. Hence, if $M$ is any $R(T)$-module, to show that $M_{{{\mathfrak m}}_{\Psi}} = 0$ it suffices to show that $M_{{{\mathfrak m}}_{h_i}} = 0$ for all $i$.
Since each $h_i$ is $G$-conjugate to $h$, each $X^{h_i}$ is empty. By the torus case, $G(T,X)_{{{\mathfrak m}}_{h_i}} =
0$ for each $h_i$. This implies that $G(G,X)_{{{\mathfrak m}}_{\Psi}} = 0$, proving (a).
Now suppose that $h \in {\mathcal Z}(G)$. To avoid confusion, we write ${{\mathfrak m}}_h^G$ for the maximal ideal in $R(G)$ corresponding to the one-element conjugacy class $h \in G$ and ${{\mathfrak m}}_h^T$ the maximal ideal in $R(T)$ corresponding to the one-element conjugacy class $h \in T$. By Remark \[rem.pickbill\], $R(T)_{{{\mathfrak m}}_h^G} = R(T)_{{{\mathfrak m}}_h^T}.$ If $X$ is smooth then, by Step 1, the action of $\lambda_{-1}(N_i^*)\in K_0(G,X^h)$ on $G(T,X^h)_{{{\mathfrak m}}_h^G} = G(T,X^h)_{{{\mathfrak m}}_h^T}$ is invertible. Thus the action of $\lambda_{-1}(N_i^*)$ on $G(G,X^h)_{{{\mathfrak m}}_h^G} = p_!(G(T,X^h)_{{{\mathfrak m}}_h^G})$ is as well.
Once we know that the action of $\lambda_{-1}(N_i^*)$ is invertible after localizing at ${{\mathfrak m}}_h^G$, the formula of equation follows from the self-intersection formula.\
[*Step 3: $G$ is arbitrary.*]{} As in the previous steps, to prove part (a) it suffices to show that if $\Psi = C_G(h)$ and $X^h$ is empty, then $G(G,X)_{{{\mathfrak m}}_\Psi} = 0$. By Proposition \[prop.goodembedding\], there is an embedding of $G$ into a product of general linear groups $Q$ such that if $\Psi_1 = C_Q(h)$, then $\Psi_1 \cap Q = \Psi$. This implies that if $X^h$ is empty then so is $(Q \times_G X)^h$. Thus, by Remark \[rem.moritaequiv\] and Step 2, we conclude that $G(G,X)_{{{\mathfrak m}}_{\Psi_1}} = 0$. Since ${{\mathfrak m}}_{\Psi_1} R(G) \subset {{\mathfrak m}}_\Psi$ it follows that $G(G,X)_{{{\mathfrak m}}_\Psi}= 0$ as well.
Now suppose that $h$ is central in $G$; then $h$ is central in $Q$ as well. Let $i \colon X^h \to X$ be the inclusion of the fixed locus and let $\iota_h \colon Q \times_G X^h \to Q \times_G X^h$ be the corresponding inclusion of mixed spaces. By Morita equivalence and Remark \[rem.pickbill\], it suffices to show that the action of $\lambda_{-1}(N_{\iota}^*)$ on $G(Q, Q \times_G X^h)$ is invertible after localizing at ${{\mathfrak m}}_h^Q \subset R(Q)$. This follows from Step 2 and the following lemma.
\[lem.changefixed\] Let $G$ be a closed subgroup of an algebraic group $Q$ and let $h \in G \cap {\mathcal Z}(Q)$. If $X$ is a smooth $G$-space then $(Q \times_G X^h) = (Q \times_G X)^h$ as closed subspaces of $Q \times_G X$.
It is clear that $Q \times_G X^h \subset (Q \times_G X)^h$, so we need only show the reverse inclusion. Since $Q \times_G X^h$ and $(Q \times_G X)^h$ are closed smooth subspaces of the algebraic space $Q \times_G X$ it suffices to show that they have the same closed points (since we work over the algebraically closed field ${{\mathbb C}}$).
A point corresponding to the $G$-orbit of $(q,x)\in Q \times X$ is fixed by $h$ if and only if $h(q,x) = (hq,x)$ is in the same $G$ orbit as $(q,x)$. This means that there is an element $g \in G$ such that $(hq,x) = (q g^{-1},g x)$. Thus $g$ fixes $x$ and $g^{-1} = q^{-1}h q$. Since $h \in {\mathcal Z}(Q)$ this means $g= h^{-1}$. Since $h$ and $h^{-1}$ have the same fixed locus we conclude that $x \in X^{h}$; i.e. $(q,x) \in Q \times X^h$.
This concludes the proof of Theorem \[thm.weakloc\].
Decomposition of equivariant $K$-theory {#sec.decomp}
---------------------------------------
Let $G$ be an algebraic group acting on an algebraic space $X$. Assume that $G$ acts with finite stabilizers. In this case, there is a decomposition of $G(G,X)$ into a direct sum of pieces, which we now describe.
Since $X$ is assumed to be Noetherian there is a finite set of conjugacy classes $\Phi_1, \ldots \Phi_m$ of elements of finite order such that $X^g$ is nonempty if and only if $g \in \Phi_i$ for some $i$ [@VeVi:02 Theorem 5.4].
\[prop.decomp\] With the assumptions above, the localization maps $G(G,X) \to G(G,X)_{{{\mathfrak m}}_{\Phi_i}}$ induce a direct sum decomposition $$G(G,X) = \oplus_i G(G,X)_{{{\mathfrak m}}_{\Phi_i}}.$$
By [@EdGr:00 Remark 5.1], there is a ideal $J \subset R(G)$ that annihilates $G(G,X)$ and such that $R(G)/J$ is supported at a finite number of points[^2] of $\operatorname{Spec}R(G)$. This implies that $G(G,X) = \oplus_i G(G,X)_{{{\mathfrak m}}_{\Phi_i}}$ for some set of semi-simple conjugacy classes $\{\Phi_1, \Phi_2, \ldots \Phi_m\}$. If $X^h$ is empty for $h \in \Phi$, then by Theorem \[thm.weakloc\], $G(G,X)_{{{\mathfrak m}}_\Psi} = 0$.
Suppose that $G$ acts on $X$ with finite stabilizers. If $\alpha \in G(G,X)$, then we denote the component of $\alpha$ in the summand $G(G,X)_{{{\mathfrak m}}_\Psi}$ by $\alpha_\Psi$. Note that if $\beta \in K_0(G,X)$ then $$(\beta \cap \alpha)_\Psi = \beta \cap (\alpha_\Psi).$$ Also, suppose that $f$ is a $G$-equivariant morphism of algebraic spaces such that $G$ acts with finite stabilizers on the source and target. If $f$ is proper morphism, then $f_*(\alpha_\Psi) = (f_* \alpha)_\Psi$. Likewise, if $f$ is flat or a map of regular algebraic spaces then $f^*\alpha_\Psi = (f^* \alpha)_\Psi$. These basic facts follow immediately from the fact that $f_*$ and $f^*$ are $R(G)$-module homomorphisms. They will be used repeatedly in the proof of Theorem \[thm.localization\].
Let $X$ be a $Z$-space where $Z$ acts with finite stabilizers and let $Z
\subset G$ be an embedding of $Z$ into another algebraic group $G$. Morita equivalence (Section \[s.morita\]) identifies $G(G, G \times_Z X)$ with $G(Z,X)$ giving an $R(Z)$-module structure on $G(G,G \times_Z X)$. As a result we may obtain a more refined decomposition of $G(G, G \times_Z X)$.
\[prop.changedecomp\] Let $Z$ act on $X$ with finite stabilizers. If $\Psi$ is a semisimple conjugacy class in $G$ and $(\Psi \cap Z)$ decomposes into the union of conjugacy classes $\Psi_1, \ldots , \Psi_l$, then $$G(G,G\times_Z X)_{{{\mathfrak m}}_\Psi} = \oplus_{i=1}^l G(G,G \times_Z X)_{{{\mathfrak m}}_{\Psi_i}}.$$
By Remark \[rem.pickbill\], $R(Z)_{{{\mathfrak m}}_\Psi}$ is a semi-local ring with maximal ideals ${{\mathfrak m}}_{\Psi_1},\ldots , {{\mathfrak m}}_{\Psi_l}$ where $\Psi_1, \ldots \Psi_l$ are the conjugacy classes in $\Psi \cap Z$. The proposition follows.
Projection formulas for flag bundles
------------------------------------
In this section we prove some projection formulas for maps of flag bundles which will be needed in the proof of the nonabelian localization theorem. We begin with a lemma which is certainly known in greater generality, but for which we lack a suitable reference.
\[lem.projection\] Let $p \colon P_1 \to P_2$ be a proper, flat, $G$-equivariant map of quasi-projective schemes such that action of $G$ is linearized with respect to an ample line bundle on each $P_i$. Let $X$ be an algebraic space with a $G$-action. Let $p_1 \colon P_1 \times X \to P_1$ and $p_2 \colon P_2 \times X \to P_2$ be the projections. Set $\phi = (p \times 1) \colon P_1 \times X \to P_2 \times X$. If $A \in K_0(G,P_1)$ and $\alpha \in G(G,P_2 \times X)$ then $$\label{eqn.projection}
\phi_*(p_1^*A \cap \phi^*\alpha) = p_2^*p_*A \cap \alpha$$
We use the ideas in the proof of the projection formula given in [@Qui:73 Proposition 7.2].
By assumption the action of $G$ on $P_1$ is linearized with respect to an ample line bundle. Hence $K_0(G,P_1)$ is generated by classes of equivariant vector bundles ${\mathcal E}$ with $R^ip_*{\mathcal E} = 0$ for $i > 0$ [@Qui:73 Section 7.2]. Thus we may assume $A = [{\mathcal E}]$ and $R^ip_*{\mathcal E} = 0$ for $i>0$. Since $p$ is flat it follows that $p_*{\mathcal E}$ is a locally free $G$-equivariant sheaf on $P_2$. Thus, if ${\mathcal F}$ is any coherent sheaf on $P_2 \times X$ then $R^i\phi_*(p_1^*{\mathcal E} \otimes \phi^*{\mathcal F}) = 0$. (This can be checked locally in the étale topology so we may assume $X$ is an affine scheme. The proof in that case is given on p. 59 of Srinivas’s book [@Sri:96].) Thus the functor $${\tt coh}^G_{(P_2 \times X)} \to {\tt coh}^G_{(P_2 \times
X)},\; \; {\mathcal F} \mapsto \phi_*(p_1^*{\mathcal E} \otimes
\phi^*{\mathcal F})$$ is exact. This functor induces the endomorphism of $G(G,X)$ given by $\alpha \mapsto \phi_*(p_1^*A \cap \phi^*\alpha)$.
On the other hand, the endomorphism of $G(G,P_2 \times X)$ given by $\alpha \mapsto p_2^*p_*A \cap \alpha$ is induced by the exact functor $${\tt coh}^G_{(P_2 \times X)} \to {\tt coh}^G_{(P_2 \times
X)},\; \; {\mathcal F} \mapsto p_2^*p_*{\mathcal E} \otimes {\mathcal F}.$$
Srinivas also proves that there is a natural (and hence $G$-equivariant) isomorphism $\phi_*(p_1^*{\mathcal E} \otimes \phi^*{\mathcal F}) \simeq
\phi_*p_1^*{\mathcal E} \otimes {\mathcal F}$. Since $\phi_* p_1^*{\mathcal E}$ is naturally (and thus $G$-equivariantly) isomorphic to $p_2^*p_*{\mathcal E}$ the two exact functors we have defined are isomorphic. Therefore, the formula of Equation holds.
Let $G$ be a connected group and let $P \subset G$ be a parabolic subgroup containing a Borel subgroup $B$ and having Levi factor $Z$. Choose a maximal torus $T
\subset Z$. Since $P$ is parabolic, $T$ is a maximal torus of $G$ as well. Let $W(G,T) = N_G(T)/{\mathcal Z}_G(T)$ and $W(Z,T)= N_Z(T)/{\mathcal Z}_Z(T)$ be the Weyl groups of $G$ and $Z$ respectively. If $X$ is a $G$-space then we have projections $$G \times_B X\stackrel{p}{\rightarrow} G \times_P X\stackrel{q}{\rightarrow} X.$$ Set $\pi = q \circ p$. The flag bundle projection formulas are given by the next result.
\[prop.tanpush\] If $\alpha \in G(G,X)$, $\beta \in G(G,G \times_P X)$ then the following identities hold:\
(i) $\pi_* (\lambda_{-1}(T^*_{\pi}) \cap \pi^*\alpha) = |W(G,T)|\alpha$.\
(ii) $q_* (\lambda_{-1}(T^*_q) \cap q^*\alpha) = \frac{|W(G,T)|}
{|W(Z,T)|} \alpha$.\
(iii) $p_* (\lambda_{-1}(T^*_{\pi}) \cap p^*\beta) = |W(Z,T)|
(\lambda_{-1}(T^*_q)\cap \beta)$.\
Since $G$ acts on $X$, the mixed spaces $G \times_B X$ and $G \times_P X$ are isomorphic to $G/B \times X$ and $G/P \times X$ respectively. The maps $p,q, \pi$ are all smooth and projective and the bundles $T_{\pi}$, $T_{q}$ and $T_p$ are all obtained by pullback from the smooth projective schemes $G/B$ or $G/P$. Therefore, by Lemma \[lem.projection\] we may assume that $X = \operatorname{Spec}{{\mathbb C}}$ and $\alpha = 1$.
Observe that since $B$ is a parabolic with Levi factor $T$, (i) is a special case of (ii). To prove (ii), it suffices to show that if $P \subset G$ is a parabolic with Levi factor $Z$ and $q \colon G/P \to \operatorname{Spec}{{\mathbb C}}$ is the projection, then $$q_*(\lambda_{-1}(T_{G/P}^*)) = |W(G,T)|/|W(Z,T)| \in R(G).$$ Since $R(G) \subset R(T)$ we may check this identity in $R(T)$. The torus $T$ acts on $G/P$ with a finite number of fixed points $p_1, \ldots , p_n$ where $n = |W(G,T)|/|W(Z,T)|$. Let $i_l \colon p_l \to G/P$ be the inclusion. The fixed points are isolated and $N_{i_l} = i_l^*T_{G/P}$. We claim that $$\label{e.tanpush}
\lambda_{-1}(T_q^*) = \sum_{l=1}^n i_{l*} 1.$$ Indeed, since $G/P$ has a $T$-equivariant cell decomposition, [@ChGi:97 Lemma 5.5.1] implies that $G_0(T,G/P) = K_0(T,G/P)$ is a free $R(T)$-module. In addition $R(T)$ is an integral domain, we may check after localizing at any prime ideal in $R(T)$. By Theorem \[thm.weakloc\], the equation holds after localizing at ${{\mathfrak m}}_a \in R(T)$ where $a\in T$ is any element with $(G/P)^a = (G/P)^T$. This proves . Pushing forward to $G(T,\operatorname{Spec}k) = R(T)$ completes the proof of (ii).
If $G$ is connected then $G$-equivariant $K$-theory is a summand in $T$-equivariant $K$-theory [@Tho:88 Theorem 1.13]. Thus, to prove (iii) we may again work in $T$-equivariant $K$-theory. The $T$ action on $G/B$ has $|W(G,T)|$ fixed points while the $T$ action on $G/P$ has $|W(G,T)/W(Z,T)|$. For each fixed point $P \in G/P$ the fiber $p^{-1}(P)$ contains exactly $|W(Z,T)|$ of the $T$-fixed points in $G/B$. Applying to $G/B$ and $G/P$, we see that $p_*\lambda_{-1}(T^*_{G/B}) =
|W(Z,T)| \lambda_{-1}(T_{G/P}^*)$.
The global stabilizer and its equivariant $K$-theory {#s.globalstabilizer}
====================================================
General facts about the global stabilizer
-----------------------------------------
If $X$ is a $G$-space then $G$ acts on $G \times X$ by conjugation on the first factor and by the original action on the second factor. The global stabilizer $S_X \subset G \times X$ is a $G$-invariant subspace and the projection $f \colon S_X \to X$ is $G$-equivariant.
Let $\Psi$ be a semisimple conjugacy class in $G$. Define $S_{\Psi} \subset S_X$ to be the inverse image of $\Psi$ under the projection $S_X \to G$. Set theoretically, $S_\Psi = \{(g,x) | g \in \Psi
\text{ and } gx = x\}$.
\[rem.finite\] Since semisimple conjugacy classes are closed, $S_\Psi$ is closed in $S_X$. Thus, if $G$ acts with finite stabilizer then the projection $f\colon S_\Psi \to X$ is also finite.
If $\Psi$ and $\Psi'$ are disjoint conjugacy classes then $S_\Psi \cap S_{\Psi^{'}} = \emptyset$. When $G$ acts with finite stabilizers then $S_X$ is the disjoint sum of the closed subspaces $S_{\Psi_{1}} \coprod S_{\Psi_{2}} \ldots \coprod S_{\Psi_{l}}$ where $\{\Psi_1, \ldots , \Psi_l\}$ is the set of conjugacy classes whose elements have non-trivial stabilizer (these conjugacy classes are necessarily semisimple because in characteristic 0 any element of finite order is semisimple). Hence $S_\Psi$ is also open in $S_X$ in this case.
Let $h$ be an element of $\Psi$ and let $Z = {\mathcal Z}_G(h)$ be the centralizer of $h$. The map $G \times X^h \stackrel{\Phi_h} \to S_\Psi$ given by $(g,x) \mapsto (ghg^{-1},gx)$ is invariant under the free action of $Z$ on $G \times X^h$ given by $z(g,x) = (gz^{-1}, zx)$.
\[lem.whatisspsi\] $G \times X^h \stackrel{\Phi_h} \to S_\Psi$ is a $Z$-torsor. Hence $S_\Psi$ is identified with the quotient $G \times_Z X^h$.
The map $G \to \Psi$, $g \mapsto ghg^{-1}$ identifies $\Psi$ with $G/Z$. Thus, by base change, the map $\Phi_h \colon G \times X \to \Psi \times X$ given by $(g,x) \mapsto (ghg^{-1}, gx)$ is also a torsor. Since $\Phi_h^{-1}(S_\Psi) = G\times X^h$ the lemma follows.
\[rem.spsismooth\] If $X$ is smooth then the identification of $S_\Psi$ with $G \times_Z X^h$ implies that $S_\Psi$ is smooth and the projection $f\colon S_\Psi \to X$ factors as the composition of the regular embedding $S_\Psi \stackrel{i} \to G \times_Z X$ with the smooth projection morphism $G \times_Z X \stackrel{\eta}\to X$.
If $X$ is smooth, the map $S_\Psi \stackrel{i} \to G \times_Z X$ is a regular embedding but the projection $G \times_Z X \rightarrow Z$ is not proper. This makes it difficult to use $G \times_Z X$ in the proof of our main result. However, suppose that $G$ is connected and $Z$ is a Levi factor of a parabolic subgroup $P$. Let $\rho: G \times_Z X \rightarrow G \times_P
X$ be the projection, and let $j = \rho \circ i: S_\Psi \to G \times_P
X$. The following result holds.
\[lemma.zpsitoxp\] Let $G$ be connected and let $\Psi = C_G(h)$, where $h$ is semisimple. Assume that $Z = {\mathcal Z}_G(h)$ is a Levi factor of a parabolic subgroup $P$. If $G$ acts on $X$ with finite stabilizer then $j \colon S_{\Psi} \to G \times_P X$ is a regular embedding.
Since the morphisms of algebraic spaces $i$ and $\rho$ are representable[^3] the composite $j = \rho \circ i$ is as well. By [@EGA4 Proposition 19.1.1] any closed immersion of regular schemes is a regular embedding. The property of being a regular embedding is local for the étale topology of the target so we may apply this proposition to representable morphisms of algebraic spaces. Since $S_\Psi$ and $G \times_P X$ are smooth we are reduced to showing that $j$ is a closed immersion.
The finite map $f\colon S_{\Psi} \to X$ factors as $p \circ j$ where $p\colon G \times_P X \to X$ is the projection, so $j$ is finite and thus closed. To prove that it is an immersion we will show that is unramified and injective on geometric points. These two conditions suffice because by [@EGA4 Cor. 18.4.7] an unramified morphism (Zariski) locally factors as the composition of an étale morphism with a closed immersion. If, in addition, the map is injective on geometric points then the étale morphism is an open immersion by [@EGA4 Theorem 17.9.1]. Hence these conditions will imply that our morphism is locally the composition of an open immersion with a closed immersion.
To show that $j$ is unramified it suffices to show that the map $f = j \circ q \colon S_\Psi \to X$ is unramified. Since $S_\Psi$ is open in $S_X$ we need only show that the projection $S_X \to X$ is unramified. This last fact follows from the fact the fiber of $S_X \to X$ over a geometric point $x \to X$ is a group scheme over $x$. Since we work in characteristic 0 this must be reduced.
Next we show that $j$ is injective on geometric points. Consider the morphism $G \times X^h \to G \times_Z X^h \to G \times_P X$. Suppose that $(g_1,x_1)$ and $(g_2,x_2)$ have the same image in $G \times_P
X$. By definition this means that there is an element $p \in P$ such that $x_2 = px_1$ and $g_2 = g_1 p^{-1}$. Since $P$ is a semidirect product of $Z$ and $U$ with $U$ unipotent and normal, we can write $p
= uz$ with $u \in U$ and $z \in Z$. Thus $x_2 = u z x_1$; we want to show that $u = 1$. Because $z x_1$ and $x_2$ are both in $X^h$, it suffices to show that if $u$ is an element of $U$ with $x$ and $ux$ both in $X^h$, then $u = 1$. This last assertion follows because the assumptions imply that $h u x = h u h^{-1} x = u x$, or $u^{-1}
(huh^{-1})$ fixes $x$. Now, $u^{-1} (huh^{-1})$ is unipotent since it is in $U$. However, this element has nontrivial fixed locus, so by Remark \[rem.finite\] it is semisimple. Hence $u^{-1} (huh^{-1}) =
1$, so $u \in Z$. Since $Z \cap U = \{ 1 \}$ we obtain $u = 1$, as desired.
Localization theorems related to the global stabilizer
------------------------------------------------------
Let $\Psi = C_G(h)$ be a semisimple conjugacy class, and let $Z =
{\mathcal Z}_G(h)$. Recall that by Lemma \[lem.whatisspsi\], $\Phi_h: G \times X^h \rightarrow S_{\Psi}$ is a $Z$-torsor.
By Morita equivalence, $G(G,S_\Psi)$ is isomorphic to $G(Z,X^h)$, so $G(G,S_\Psi)$ is an $R(Z)$-module. The restriction homomorphism $R(G) \rightarrow
R(Z)$ is compatible with the $R(G)$ and $R(Z)$-module structures on $G(G,S_\Psi)$ (cf. Section \[s.morita\]). Since $h$ is central in $Z$ it is a one-element conjugacy class with corresponding maximal ideal ${{\mathfrak m}}_h \subset R(Z)$.
The next lemma shows that the action of $R(Z)$ on $G(G,S_\Psi)$ is independent of the choice of $h \in \Psi$. This fact is crucial for the proof of the nonabelian localization theorem.
\[lem.hindep\] Let $\Psi$ be a semisimple conjugacy class and let $h_1$ and $h_2= kh_1k^{-1}$ be in $\Psi$. Set $Z_1 = {\mathcal Z}_G(h_1)$ and $Z_2 = {\mathcal Z}_G(h_2)$. Let $C_k: Z_1 \rightarrow Z_2$ denote conjugation by $k$, and let $C_k^*: R(Z_2) \rightarrow R(Z_1)$ denote the pullback. Then $C_k^*({{\mathfrak m}}_{h_2}) = {{\mathfrak m}}_{h_1}$ and the pullback $C_k^*$ is compatible with the actions of $R(Z_1)$ and $R(Z_2)$ on $G(G,S_\Psi)$ as defined above.
Since $C_k(h_1) = h_2$, we have $C_k^*({{\mathfrak m}}_{h_2}) = {{\mathfrak m}}_{h_1}$. To verify the compatibility of the $R(Z_i)$-actions, we must unwind the definitions. The maps $a_i: G/Z_i \rightarrow \Psi$ defined by $a_i(g Z_i) = g h_i g^{-1}$ give $G$-equivariant identifications of $G/Z_i$ with $\Psi$. These in turn give change of group identifications of $R(Z_i)$ with $K_0(G, \Psi)$, and the action of $R(Z_i)$ on $G(G, S_{\Psi})$ is obtained composing these identifications with the pullback $K_0 (G, \Psi) \rightarrow K_0(G, S_{\Psi})$. Therefore, it suffices to prove that $C_k^*$ is compatible with the identifications of $R(Z_i)$ with $K_0(G, \Psi)$.
The identifications of $G/Z_i$ with $\Psi$ are compatible with the isomorphism $\kappa: G /Z_1 \rightarrow G/Z_2$ taking $g Z_1$ to $g k^{-1} Z_2$. If $V_2$ is a $Z_2$-module, write $C_k^* V_2$ for the same underlying vector space, but with $Z_1$-module structure obtained by pullback via the map $C_k$. The identification of $R(Z_i)$ with $K_0(G, \Psi)$ takes the class of a $Z_i$-module $V_i$ to the class of the vector bundle $G \times_{Z_i} V_i$ on $G/Z_i$. The pullback via $\kappa$ of the vector bundle $G \times_{Z_2} V_2$ is $G$-equivariantly isomorphic to the vector bundle $G \times_{Z_1} C_k^* V_2$, which proves the result.
\[prop.nfinvertible\] Let $G$ be an algebraic group acting on a smooth algebraic space $X$. Let $\Psi = C_G(h)$ be a semi-simple conjugacy class, let $Z = {\mathcal Z}_G(h)$, and let $f \colon S_\Psi \to X$ be the projection. Let $N_f^* = [T_{S_{\Psi}}^*] - [f^*(T_X^*)] \in K_0(G,S_\Psi)$ be the class of the relative cotangent bundle. The map $\cap\; \lambda_{-1}(N_f^*)\colon G(G,S_\Psi) \to
G(G,S_\Psi)$ restricts to an isomorphism after localizing at the maximal ideal ${{\mathfrak m}}_h \in R(Z)$.
\[rem.nfinvertible\] Recall that $S_\Psi$ is smooth when $X$ is smooth (Remark \[rem.spsismooth\]) so that $f \colon S_\Psi \to X$ is a morphism of smooth algebraic spaces.
Let $\iota\colon X^h \to X$ be the inclusion of the fixed locus of $h$. By Theorem \[thm.weakloc\] the map $$\cap \; \lambda_{-1}(N^*_\iota) \colon G(Z,X^h)_{{{\mathfrak m}}_{h}}
\to G(Z,X^h)_{{{\mathfrak m}}_{h}}$$ is an isomorphism. Let $i \colon S_\Psi \to G \times_Z X$ be the corresponding inclusion of mixed spaces. By Morita equivalence the map $$\cap \; \lambda_{-1}(N_i^*) \colon G(G,S_\Psi)_{{{\mathfrak m}}_h} \to
G(G,S_\Psi)_{{{\mathfrak m}}_h}$$ is an isomorphism.
Let $\eta \colon G \times_Z X \to X$ be the projection. Then $f = \eta \circ i$, so $[T_f] = [N_i] - [i^*T_\eta] \in K_0(G,S_\Psi)$ Thus $\lambda_{-1}(N_i^*) = \lambda_{-1}(N_f^*) \lambda_{-1}(i^*T_\eta^*)$. Since $\lambda_{-1}(N_i^*)$ acts by automorphisms on $G(G,S_\Psi)_{{{\mathfrak m}}_h}$ it follows that $\lambda_{-1}(N_f^*)$ must as well.
Now assume that $G$ is connected and that $Z = {\mathcal Z}_G(h)$ is a Levi factor of a parabolic subgroup $P \subset G$. Since $P/Z$ is isomorphic to affine space we may identify identify $G(G, G\times_P X)$ with $G(Z,X)$.
\[prop.plocal\] Let $G$ be a connected algebraic group acting on a smooth algebraic space $X$. Let $\Psi = C_G(h)$; assume that $Z = {\mathcal Z}_G(h)$ is a Levi factor of a parabolic subgroup $P$. Let $i$ and $j$ be the maps of $S_{\Psi}$ into $G \times_Z X$ and $G \times_P X$ defined above.
\(a) The map $$\cap \; \lambda_{-1}(N_{j}^*) \colon G(G,S_\Psi) \to G(G,S_\Psi)$$ is an isomorphism after localizing at the maximal ideal ${{\mathfrak m}}_h \subset R(Z)$.
\(b) If in addition $G$ acts on $X$ with finite stabilizer and if $\beta \in G(G,G \times_P X)_{{{\mathfrak m}}_h}$, then $$\label{eqn.parabolic} \beta =
j_* \left(\lambda_{-1}(N_j^*)^{-1} \cap j^* \beta\right).$$ where $\lambda_{-1}(N_j^*)^{-1} \cap j^*\beta$ denotes the image of $j^*\beta$ under the inverse of the map $\cap\; \lambda_{-1}(N_j^*)$.
\(a) The map $j$ factors as $\rho \circ i$ where $\rho \colon G \times_P X
to G \times_Z X$ is the projection. Thus $[N_j] = [N_i] - [i^*T_\rho]$. Hence $\lambda_{-1}(N_i^*) = \lambda_{-1}(N_j^*)\lambda_{-1}(T_\rho^*)$. By Morita equivalence and Theorem \[thm.weakloc\], $\lambda_{-1}(N_i^*)$ acts by automorphisms on $G(G,S_\Psi)_{{{\mathfrak m}}_h}$. Therefore, $\lambda_{-1}(N_j^*)$ must as well.
\(b) By Morita equivalence and Theorem \[thm.weakloc\] (for the group $Z$) the pushforward $i_*\colon G(G,S_\Psi)_{c_\Psi} \to G(G,G \times_Z X)_{{{\mathfrak m}}_h}$ is an isomorphism. Since the action of $\lambda_{-1}(N_i^*)$ on $G(G,S_\Psi)_{c_\Psi}$ is invertible, it follows from the self intersection formula that $i^*\colon G(G,G \times_Z X)_h \to G(G,S_\Psi)_{c_\Psi}$ is also an isomorphism. Hence $$j^* = (\rho \circ i)^* \colon G(G,G \times_P X)_{{{\mathfrak m}}_h}
\to G(G,S_\Psi)_{c_\Psi}$$ is as well. Thus, it suffices to prove that Equation holds after applying $j^*$ to both sides. By the self-intersection formula for the regular embedding $j$, we have $$j^*j_*\left(\lambda_{-1}(N_j^*)^{-1} \cap j^*\beta \right)
= \lambda_{-1}(N_j^*) \cap \left( \lambda_{-1}(N_j^*)^{-1} \cap j^*\beta\right)
= j^*\beta$$
The central summand
-------------------
Suppose that $G$ acts with finite stabilizers on $X$. Let $\Psi = C_G(h)$, and keep the notation above. Since $h$ is a one-element conjugacy class contained in $\Psi \cap Z$, Proposition \[prop.decomp\] implies that $G(G,S_\Psi)_{{{\mathfrak m}}_h}$ is a summand in $G(G,S_\Psi)_{{{\mathfrak m}}_\Psi}$. By Lemma \[lem.hindep\], this summand is independent of choice of $h \in \Psi$.
\[def.centralpiece\] Let $G$ act with finite stabilizers on $X$, and let $\Psi = C_G(h)$. With notation as above, the summand $G(G,S_{\Psi})_{{{\mathfrak m}}_h} \subset G(G,S_{\Psi})_{{{\mathfrak m}}_\Psi}$ (which is independent of $h \in \Psi$) will be called the central summand and denoted $G(G,S_\Psi)_{c_\Psi}$. The component of $\beta \in G(G,S_\Psi)$ in this summand will be denoted $\beta_{c_\Psi}$.
The nonabelian localization theorem {#s.nonabelian}
===================================
The main theorem of our paper is the following nonabelian localization theorem.
\[thm.localization\] Let $G$ be an algebraic group acting with finite stabilizer on a smooth algebraic space $X$. Let $\Psi = C_G(h)$ be a semisimple conjugacy class and let $f \colon S_\Psi \to X$ be the projection. If $\alpha \in G(G,X)$ let $\alpha_\Psi$ be the component of $\alpha$ supported at the maximal ideal ${{\mathfrak m}}_\Psi \subset R(G)$. Then $$\label{eqn.werock}
\alpha_\Psi =
f_* \left( \lambda_{-1}(N_f^*)^{-1} \cap (f^* \alpha)_{c_\Psi}\right)$$ where $\lambda_{-1}(N_f^*)^{-1} \cap (f^*\alpha)_{c_\Psi}$ is the image of $(f^*\alpha)_{c_\Psi}$ under the inverse of the automorphism $\cap \; \lambda_{-1}(N_f^*)$ of $G(G,S_\Psi)_{c_\Psi}$.
The theorem can be restated in way that is sometimes more useful for calculations. As usual, let $Z = {\mathcal Z}_G(h)$. Let $\iota^! \colon G(G,X) \to G(Z,X^h)$ be the composition of the restriction functor $G(G,X) \to G(Z,X)$ with the pullback $G(Z,X) \stackrel{\iota^*} \to
G(Z,X^h)$. Let $\beta_h$ denote the component of $\beta \in G(Z,X^h)$ in the summand $G(Z,X^h)_{{{\mathfrak m}}_h}$. Let ${\mathfrak g}$ (resp. ${\mathfrak z}$) be the adjoint representation of $G$ (resp. $Z$). The restriction of the adjoint representation to the subgroup $Z$ makes ${\mathfrak g}$ a $Z$-module, so ${\mathfrak g}/{\mathfrak z}$ is a $Z$-module. Let $\eta \colon G \times_Z X \to X$ be the projection. By Remark \[rem.tantomixedbund\], $T_\eta = G \times_Z (X \times
{\mathfrak g}/{\mathfrak z})$. We therefore obtain the following corollary.
\[cor.localization\] With assumptions as in Theorem \[thm.localization\], let $h$ be an element of $\Psi$. Let $Z = {\mathcal Z}_G(h)$, and let $\iota_! \colon G(Z,X^h) \to G(G,X)$ be the map obtained by composing $f_*$ with the Morita equivalence isomorphism $G(Z,X^h) \to
G(G, S_\Psi)$. If $\alpha \in G(G,X)_{{{\mathfrak m}}_\Psi}$, then $$\alpha = \iota_! \left( \lambda_{-1}(N_\iota^*)^{-1} \cap
\lambda_{-1}(({\mathfrak g}/{\mathfrak z})^*)\cap
(\iota^!\alpha)_h\right).$$
Proof of Theorem \[thm.localization\] if $G$ is connected and $Z$ is a Levi factor of a parabolic subgroup {#sect.localizationproof1}
----------------------------------------------------------------------------------------------------------
In this section we prove Theorem \[thm.localization\] under the assumptions that $G$ is connected and that $Z=
{\mathcal Z}_G(h)$ is a Levi factor in a parabolic subgroup $P \subset G$. (In fact, it would suffice for our purposes to take $G$ equal to a product of general linear groups, and then $Z$ is automatically such a Levi factor.) Choose a maximal torus $T$ and and Borel subgroup $B$ such that $h \in T \subset B \subset P$. We have maps of mixed spaces $$G \times_B X\stackrel{p}{\rightarrow} G \times_P X\stackrel{q}{\rightarrow} X;$$ and we write $\pi = q \circ p$. Let $j: S_{\Psi} \rightarrow G
\times_P X$ denote the regular embedding of Lemma \[lemma.zpsitoxp\].
Since $G \times_T X \rightarrow G \times_B X$ is a bundle with fibers isomorphic to $B/T$, pullback gives an isomorphism of $G(G, G \times_B X)$ with $G(G, G \times^T X)$. Now, $\Psi \cap T$ consists of a finite number of elements $h= h_1, \ldots , h_w$, where $w = |W(G,T)|/|W(Z,T)|$. By Proposition \[prop.changedecomp\], if $\alpha \in G(G,X)_{{{\mathfrak m}}_{\Psi}}$, then there is a decomposition $$\pi^*\alpha = \sum_{l = 1}^w (\pi^*\alpha)_{h_l},$$ where $(\pi^*\alpha)_{h_l}$ refers to the component supported at ${{\mathfrak m}}_{h_l} \subset R(T)$. Because $h$ is central in $Z$, there is a also component $(q^* \alpha)_h$ of $q^* \alpha$ supported at the maximal ideal ${{\mathfrak m}}_h \subset R(P)= R(Z)$.
The key step is given by the following proposition.
\[prop.decomposition\] Keep the assumptions of Theorem \[thm.localization\], and in addition assume that $G$ is connected and $Z$ is a Levi factor of a parabolic subgroup $P$. Then the following identity holds in $G(G,X)_{{{\mathfrak m}}_\Psi}$: $$f_* \left(\lambda_{-1}(N_f^*)^{-1} \cap (f^*\alpha)_{c_{\Psi}} \right) =
\frac{1}{|W(Z,T)|} \pi_*
\left(\lambda_{-1}(T_\pi^*) \cap (\pi^*\alpha)_h\right).$$
We keep the notation introduced before the statement of the proposition. Since $f = q \circ j$ and $j^*$ is an $R(Z)$-module homomorphism $j^*((q^*\alpha)_h) =
(f^*\alpha)_{h}$. But $(f^*\alpha)_h$ is independent of $h$ and equals $(f^*\alpha)_{c_\Psi}$ by definition of the central summand. Now, $[N_f^*] = [N_j^*] - [j^*T_q^*]$, so $$\label{eqn.firstlambda}
\begin{array}{rcl}
\lambda_{-1}(N_f^*)^{-1} \cap (f^*\alpha)_{c_{\Psi}} & = & \lambda_{-1}(N_j^*)^{-1}
\cap (\lambda_{-1}(j^*T_q^*) \cap (f^*\alpha)_{c_{\Psi}})\\
& = &
\lambda_{-1}(N_j^*)^{-1} \cap j^*\left(\lambda_{-1}(T_q^*) \cap (q^*\alpha)_h
\right).
\end{array}$$ Applying $f_* = q_* \circ j_*$ to both sides of Equation , we obtain $$\label{eqn.secondlambda}
\begin{array}{rcl}
f_*\left(\lambda_{-1}(N_f^*)^{-1} \cap (f^*\alpha)_{c_\Psi}\right)
& = & q_* j_* \left(\lambda_{1}(N_j^*)^{-1} \cap j^*(\lambda_{-1}(T_q^*)
\cap (q^*\alpha)_h)\right)\\
& = & q_*\left(\lambda_{-1}(T_q^*) \cap (q^*\alpha)_h\right)
\end{array}$$ where the second equality follows from Proposition \[prop.plocal\]. By compatibility of pullback with support, $(\pi^*\alpha)_h = p^*((q^*\alpha)_h)$. By Proposition \[prop.tanpush\], $$\lambda_{-1}(T_q^*) \cap (q^*\alpha)_h = \frac{1}{|W(Z,T)|}
p_*\left(\lambda_{-1}(T_\pi^*) \cap (\pi^*\alpha)_h\right)$$ so $$q_*\left(\lambda_{-1}(T_q^*) \cap (q^*\alpha)_h\right)
= \frac{1}{|W(Z,T)|}\pi_*\left(\lambda_{-1}(T_\pi^*) \cap
(\pi^*\alpha)_h\right).$$
The proof of the theorem for $G$ connected and $Z$ a Levi factor of a parabolic subgroup is an easy consequence of the preceding proposition. By Proposition \[prop.tanpush\], $$\alpha = \frac{1}{|W(G,T)|} \pi_*\left(
\lambda_{-1}(T_\pi^*) \cap \pi^* \alpha\right).$$ Therefore, $$\begin{aligned}
\alpha & = & \frac{1}{|W(G,T)|}\sum_{l = 1}^w \pi_*\left(
\lambda_{-1}(T_\pi^*) \cap (\pi^* \alpha)_{h_l}\right)\\
& = &
\frac{|W(Z,T)|}{|W(G,T)|}\sum_{l = 1}^w
f_*\left( \lambda_{-1}(N_f^*)^{-1} \cap (f^*\alpha)_{c_\Psi}\right)\\
& = & f_*\left( \lambda_{-1}(N_f^*) \cap (f^*\alpha)_{c_\Psi}\right),\end{aligned}$$ as desired.
As a corollary of the proof in this case we obtain the following induction formula for $\alpha \in G(G,X)_{{{\mathfrak m}}_\Psi}$. $$\alpha = \frac{|W(G,T)|}{|W(Z,T)|}\pi_*\left(\lambda_{-1}(T_\pi^*) \cap (\pi^*\alpha)_h\right)$$
Proof of Theorem \[thm.localization\] for arbitrary $G$ {#sect.localizationproof2}
-------------------------------------------------------
By Proposition \[prop.goodembedding\], we can embed $G$ as a closed subgroup of a product of general linear groups $Q$ such that, writing $\Psi = C_G(h)$ and $\Psi_Q = C_Q(h)$, we have $\Psi_Q \cap G = \Psi$. Write $Z = {\mathcal Z}_G(h)$ and $Z_Q = {\mathcal Z}_Q(h)$. Let $Y = Q
\times_G X$; then $Q$ acts with finite stabilizer on $Y$. The group $Q$ is connected, and direct calculation shows that $Z_Q$ is a a product of general linear groups which is a Levi factor of a parabolic. Therefore, the nonabelian localization theorem applies to the $Q$-action on $Y$.
By Proposition \[prop.changedecomp\] and the fact that $\Psi_Q \cap
G = \Psi$, we have $G(Q,Y)_{{{\mathfrak m}}_{\Psi_Q}} = G(Q,Y)_{{{\mathfrak m}}_\Psi}$. As usual, $S_{\Psi_Q}$ denotes the part of the global stabilizer $S_Y$ corresponding to $\Psi_Q$. For simplicity of notation we will write $S_Q$ for $S_{\Psi_Q}$ and $S$ for $S_{\Psi}$. We denote by $f_Q$ the morphism $S_Q \rightarrow
Y$. If $\beta \in G(Q,S_Q)$ then by definition, $\beta_{c_{\Psi_Q}}$ is the component of $\beta$ supported at the maximal ideal ${{\mathfrak m}}_h^Q \subset R(Z_Q)$. Likewise, if $\gamma \in G(Q, Q \times_G S)=
G(G,S)$ then $\gamma_{c_\Psi}$ is the component supported at ${{\mathfrak m}}_h \subset
R(Z)$. Since $h$ is central in $Z$ and $Z_Q$, Remark \[rem.pickbill\] implies that if $M$ is any $R(Z)$-module then $M_{{{\mathfrak m}}_h} = M_{{{\mathfrak m}}_h^Q}$, where the $R(Z_Q)$ action on $M$ is given by the restriction homomorphism $R(Z_Q) \to R(Z)$. Thus $\gamma_{c_\Psi}$ may also be identified with the component of $\gamma$ supported at ${{\mathfrak m}}_h^Q$.
Note that $R(Z_Q)$ acts on $G(G,S_Q)$ and $R(Z)$ acts on $G(Q, Q \times_G S) = G(G,S)$. By Morita equivalence, the localization theorem for $G$ acting on $X$ is a consequence of the following lemma.
Keep the assumptions and notation of this subsection. Then there is a $Q$-equivariant isomorphism $\Phi: Q \times_G S \to S_Q$ such that:
i\) $\Phi^* \colon G(Q,S_Q) \to G(Q, Q \times_G S)$ is an $R(Z_Q)$-module homomorphism where the action of $R(Z_Q)$ on $G(Q,Q \times_G S)$ is given by the restriction homomorphism $R(Z_Q) \to R(Z)$.
ii\) $f_Q \circ \Phi = (1 \times_G f)$.
Consider the map $$T \colon Q \times S \to \Psi_Q \times Q \times X, \; \; (q,g,x) \mapsto
(qgq^{-1},q,x).$$ This map induces a map of quotient spaces $\tilde{\Phi} \colon Q \times_G S \to \Psi_Q \times Y$ such that the following diagram commutes: $$\label{diag.qhom}
\begin{array}{ccc} Q \times_G S & \to & \Psi_Q \times Y\\
\downarrow & & \downarrow \\
Q \times_G \Psi & \stackrel{\phi}{\to} & \Psi_Q
\end{array}$$ Here the vertical arrows are the obvious projections and the bottom horizontal arrow is given by $\phi(\overline{(q,k)} ) = qkq^{-1}$. Note that $S_Q$ is a closed subspace of $\Psi_Q \times Y$. We have a commutative diagram $$\begin{array}{ccccc}
R(Z_Q) & \stackrel{=}\to & K_0(Q,\Psi_Q) & \to & K_0(Q,\Psi_Q \times Y) \\
\downarrow & & \downarrow \phi^* & & \downarrow \tilde{\Phi}^* \\
R(Z) & \stackrel{=}\to & K_0(Q, Q \times_G \Psi ) & \to & K_0(Q,Q \times_G S).
\end{array}$$ Here the second arrow in each row is a pullback map. The first arrow in the top row takes $[V] \in R(Z_Q)$ to the class of the vector bundle $Q \times_{Z_Q} V$ on $Q/Z_Q = \Psi_Q$. The first arrow in the bottom row takes $[W] \in R(Z)$ to the class of the vector bundle $Q
\times_Z W$ on $Q/Z = Q \times_G \Psi$. The action of $R(Z_Q)$ on $G(Q,S_Q)$ (resp. of $R(Z)$ on $G(Q,Q \times_G S)$) is defined using the composition along the top (resp. bottom) row. The commutativity of this diagram implies that $\tilde{\Phi}^*$ is a $R(Z_Q)$-module homomorphism.
Next we show that $\tilde{\Phi}$ induces an isomorphism of $Q \times_G S$ onto the subspace $S_Q \subset \Psi_Q \times Y$. Let $W$ be the inverse image of $S_Q$ in $\Psi_Q \times Q \times X$. Then $W$ is the closed subspace consisting of triples $(k,q, x)$ such that $g = q^{-1}kq$ is in $G$ and $g x = x$. If $(g,x) \in S$ and $q \in Q$ then clearly $(qgq^{-1},q,x) \in W$. Thus $\tilde{\Phi}$ factors through a morphism $\Phi \colon Q \times_G S \to S_Q$. Since $\tilde{\Phi}^*$ is an $R(Z_Q)$-module homomorphism, so is $\Phi^*$. Next we show that $\Phi$ is an isomorphism. Since we work over ${{\mathbb C}}$ and $\Phi$ is a representable morphism of smooth (hence normal) algebraic spaces, it suffices, by Zariski’s main theorem (proved for algebraic spaces in [@Knu:71 Theorem V4.2]), to prove that $\Phi$ is bijective on geometric points.
First we show that $\Phi$ is injective. Let $(q_1,g_1,x_1)$ and $(q_2,g_2,x_2)$ be two points of $Q \times S$ such that $T(q_1,g_1,x_1)= (q_1g_1q_1^{-1},q_1,x_1)$ and $T(q_2,g_2,x_2) = (q_2g_2q_2^{-1},q_2,x_2) $ have the same image in $\Psi_Q \times Y$. Then $q_2g_2q_2^{-1} = q_1g_1q_1^{-1}$ and there is an element $g \in G$ such that $x_2 = gx_1$ and $q_2 = q_1 g^{-1}$. Thus $g_2 = gg_1g^{-1}$ and hence $(q_1,g_1,x_1)$ and $(q_2, g_2, x_2)$ are in the same $G$-orbit in $Q \times S$. Therefore, $\Phi$ is injective on geometric points.
Conversely, suppose that $(k,q,x) \in W$. Then $g =q^{-1}kq \in \Psi_Q \cap G = \Psi$. Thus $(k,q,x) = T(q,g,x)$ so $\Phi$ is surjective on geometric points.
Finally, the fact that $f_Q \circ \Phi = (1 \times_G f)$ is clear from the definition of $\Phi$. This completes the proof of the lemma and with it Theorem \[thm.localization\].
Riemann-Roch for quotients {#s.err}
==========================
As an application of the nonabelian localization theorem, we can give an explicit formula for the Riemann-Roch map for quotients of smooth algebraic spaces by proper actions of algebraic groups. Recall, (Section \[ss.group\]) that any algebraic space with a proper group action is automatically separated.
Before stating this, we need some preliminary results about invariants.
Invariants and equivariant $K$-theory
-------------------------------------
Let $G$ be an algebraic group acting properly on an algebraic space $X$. The theorem of Keel and Mori [@KeMo:97] states that the quotient stack $[X/G]$ has a moduli space $Y = X/G$ in the category of algebraic spaces. Translated in terms of group actions, this means that the map of algebraic spaces $X \stackrel{\pi} \to Y$ is a categorical geometric quotient in the category of algebraic spaces.
Let $X$ be an algebraic space with a proper $G$ action and let $X \stackrel{\pi} \to Y$ be the geometric quotient. If ${\mathcal F}$ is a $G$-equivariant quasi-coherent ${\mathcal O}_X$-module, define ${\mathcal F}^G = (\pi_*{\mathcal F})^G$. We call this the functor of taking $G$-invariants.
To obtain a map from equivariant $K$-theory to the $K$-theory of the quotient, we need the following lemma.
\[lem.invariants\] Let $X$ be an algebraic space with a proper $G$-action (see Section \[ss.group\]) and let $X \stackrel{\pi} \to Y$ be the geometric quotient. The assignment ${\mathcal F} \mapsto {\mathcal F}^G$ defines an exact functor ${\tt coh}^G_X \to {\tt coh}_Y$.
If $X$ and $Y$ are both schemes, and $G$ is reductive, then this lemma is a consequence of the facts that the quotient map $X
\rightarrow Y$ is affine [@MFK:94 Proposition 0.7], and that taking invariants by a locally finite action of a reductive group is an exact functor (in characteristic $0$).
The theorem of Keel and Mori is proved using an étale local description of the moduli space $Y = X/G$. In particular they prove that $[X/G]$ has a representable étale cover by quotient stacks $\{ [U_i/H_i]\}$ with $U_i$ affine, $H_i$ finite and such that the following diagram of stacks and moduli spaces is Cartesian: $$\begin{array}{ccc}
[U_i/H_i] & \to & [X/G] \\
\downarrow & & \downarrow\\
U_i/H_i & \to & X/G
\end{array}$$ Let $V_i = U_i \times_{[X/G]} X$. Then $V_i$ is affine and has commuting and free actions of $G$ and $H_i$. Let $X_i = Y_i/H_i$. Since $H_i$ acts freely, $X_i \to X$ is étale. If we let $Y_i = X_i/G_i$, then the map of quotients $Y_i \to Y$ is also étale, and the following diagram of spaces and quotients is Cartesian: $$\label{diag.structure}
\begin{array}{ccc}
X_i & \to & X\\
\downarrow & & \downarrow\\
Y_i & \to & Y
\end{array}$$ The actions of $G$ and $H_i$ on $Y_i$ commute so $X_i/G = (Y_i/G_i)/H_i$. The local structure of geometric quotients given by Diagram implies that the quotient map $X \to Y$ is an affine morphism in the category of algebraic spaces.
The question is local in the étale topology on the quotient $Y$. Thus we may assume that there is an affine scheme $V$ and a finite group $H$ such that $V$ has commuting free actions by $H$ and $G$ and $X = V/G$. It follows that the $G \times H$-quotient map $V \stackrel{p} \to Y$ factors as $ p = q\circ \pi_V = \pi \circ q_V$ where $\pi_V \colon V \to V/G$ is a $G$-torsor, $q_V \colon V \to X$ is an $H$-torsor and $q\colon V/G \to Y$ is a quotient by $H$.
Let ${\mathcal G}$ be a $(G \times H)$-equivariant coherent sheaf on $V$. Then $$\begin{aligned}
{\mathcal G}^{G \times H} & = & ({\mathcal G}^G)^H \\
& = & ({\mathcal G}^H)^G\end{aligned}$$ The group $H$ is finite and $Y = (V/G)/H$ so the assignment ${\mathcal H} \mapsto {\mathcal H}^H$ is an exact functor ${\tt coh}_{V/G}^H \to {\tt coh}_Y$. Since $G$ and $H$ act freely on $V$ the assignments ${\mathcal G} \mapsto {\mathcal G}^G$ and ${\mathcal G} \mapsto
{\mathcal G}^H$ define equivalences ${\tt coh}^{G \times H}_V \to {\tt
coh}^H_{V/G}$ and ${\tt coh}^{G \times H}_V \to {\tt coh}^G_X$. Thus, the assignment ${\mathcal G} \mapsto {\mathcal G}^{G \times H}$ is an exact functor ${\tt coh}^{G \times H}_V \to
{\tt coh}_X$. Since the assignment ${\mathcal G} \mapsto {\mathcal G}^{H}$ is an equivalence ${\tt coh}^{G \times H}_V \to {\tt coh}^G_X$ it follows that the assignment ${\mathcal F} \to {\mathcal F}^G$ is an exact functor ${\tt coh}^G_X \to {\tt coh}_Y$.
If $X$ is an algebraic space with a proper $G$-action and geometric quotient $X \stackrel{\pi} \to Y$ let $\pi_G \colon G(G,X) \to G(Y)$ be the map on $K$-theory induced by the exact functor ${\tt coh}_X^G \to {\tt coh}_Y$ given by ${\mathcal F} \mapsto {\mathcal F}^G$. We will usually denote $\pi_G(\alpha)$ by $\alpha^G$.
If $G$ acts properly on $X$ and $q \colon X' \to X$ is a finite $G$-equivariant map then $G$ acts properly on $X'$ [@EdGr:03 Prop. 2.1]. Let $\pi \colon X \to Y$, $\pi'\colon X' \to Y'$ be the geometric quotients. Since the composite map $\pi \circ q \colon X' \to Y$ is $G$-invariant there is a map of quotient $q'\colon Y' \to Y$ such that the diagram commutes. $$\label{diag.invcommute}
\begin{array}{ccc}
X' & \stackrel{q} \to & X \\
\pi^{'} \downarrow & & \pi \downarrow\\
Y^{'} & \stackrel{q^{'}} \to & Y
\end{array}$$
\[lem.invcommute\] The map $q'$ is finite and $\pi_G \circ q_* = q'_* \circ \pi'_G $ as maps $G(G,X') \to G(Y)$.
Working locally in the étale topology we may assume that $Y$ is affine. It follows (since $q$ is finite) that all of the other spaces in Diagram are affine. Let $X = \operatorname{Spec}A$, $X'
= \operatorname{Spec}B$. Then $Y' = \operatorname{Spec}B^G$ and $Y = \operatorname{Spec}A^G$. If $M$ is a $G$-equivariant $B$-module then $_{A^G} M^G = (_A M)^G$ (c.f. [@EdGr:03 Proposition 2.3]). Translated to sheaves this means that ${\mathcal F}$ is a $G$-equivariant quasi-coherent sheaf on $X'$ then $(q_* {\mathcal F})^G = q'_*({\mathcal F}^G)$ as quasi-coherent sheaves on $Y$.
To complete the proof of the Lemma we need to show that $q'$ is finite. Since $q'$ is affine we need to show that $q'_* {\mathcal
O}_{Y'}$ is coherent. Now $\pi'$ is a geometric quotient so ${\mathcal
O}_{Y'} = {\mathcal O_{X'}^G}$. Thus $q_*{\mathcal O}_{Y'} = (q_*
{\mathcal O}_{X'})^G$ is coherent because $q$ is finite.
Twisting equivariant $K$-theory by a central subgroup
-----------------------------------------------------
Let $Z$ be an algebraic group and let $H$ be a subgroup (not necessarily closed) of the center of $Z$ consisting of semisimple elements. If $V$ is any representation of $Z$ then, since any commuting family of semisimple endomorphisms is simultaneously diagonalizable, $V$ can be written as a direct sum of $H$-eigenspaces $V
= \oplus V_{\chi}$ where the sum is over all characters $\chi: H
\rightarrow {{\mathbb C}}^*$ and $H$ acts on $V_{\chi}$ by $h\cdot v = \chi(h)v$. Since elements of $H$ are central in $Z$, each eigenspace $V_{\chi}$ is $Z$-stable, and therefore we have $[V] = \sum [V_{\chi}]$ in $R(Z)$. We define an action of $H$ on the representation ring $R(Z)$, by $$h \cdot [V] = \sum \chi (h)^{-1} [V_{\chi}],$$ for $h \in H$ and $[V] = \sum [V_{\chi}] \in R(Z)$. Because $H$ is a central subgroup, it acts on ${{\mathbb C}}[Z]^Z$ by the rule $(h \cdot f)(z)
=f(h^{-1}z)$, for $h \in H$, $z \in Z$, $f \in {{\mathbb C}}[Z]^Z$. With these actions, the character map from $R(Z)$ to ${{\mathbb C}}[Z]^Z$ is $H$-equivariant. Recall that maximal ideals of $R(Z)$ are of the form ${{\mathfrak m}}_{\Psi}$, where $\Psi$ is a semisimple conjugacy class in $Z$. The $H$-equivariance of the character map implies that if $h \in H$, then $$h \cdot {{\mathfrak m}}_{\Psi} = {{\mathfrak m}}_{h \Psi}.$$ In particular, $$h^{-1} {{\mathfrak m}}_h = {{\mathfrak m}}_1,$$ the augmentation ideal of $R(Z)$.
More generally, let $X$ be an algebraic space with a $Z$-action, such that $H$ acts trivially on $X$. In this case, if ${{\mathcal E}}$ is any $Z$-equivariant coherent sheaf on $X$, and $\chi$ a character of $H$, let ${{\mathcal E}}_{\chi}$ be the subsheaf of ${{\mathcal E}}$ whose sections (on any étale open set $U$) are given by $${{\mathcal E}}_{\chi} (U) = \{ s \in {{\mathcal E}}(U) \ | \ h \cdot s = \chi (h) s \}.$$ Then ${{\mathcal E}}= \oplus_{\chi} {{\mathcal E}}_{\chi}$ is a decomposition of ${{\mathcal E}}$ into a direct sum of $H$-eigensheaves for ${{\mathcal E}}$. We define an action of $H$ on $G_0(Z,X)$ by $$h \cdot [{{\mathcal E}}] = \sum \chi(h)^{-1} [{{\mathcal E}}_{\chi}].$$ If $X$ is a point, this reduces to the previous definition of the action of $H$ on $R(Z)$. Thus, the actions of $H$ on $R(Z)$ and on $G_0(Z,X)$ are compatible: if $r \in R(Z)$, $\alpha \in G_0(Z,X)$, then $h
\cdot (r \alpha) = (h \cdot r) (h \cdot \alpha)$. If $Z$ acts on $X$ with finite stabilizers, then we can identify $G_0(Z,X) = \oplus
G_0(Z,X)_{{{\mathfrak m}}_{\Psi}}$, where the sum is over finitely many conjugacy classes $\Psi$. The compatibility of the actions implies that $$\label{eqn.shift}
h^{-1} G_0(Z,X)_{{{\mathfrak m}}_h} = G_0(Z,X)_{{{\mathfrak m}}_1}.$$ We refer to the action of $H$ on $G_0(Z,X)$ as twisting. If $\alpha \in G_0(Z,X)$, we will often write $\alpha(h)$ for $h^{-1} \cdot \alpha$ and refer to this map as twisting by $h$.
Twisting and the global stabilizer
----------------------------------
We now define a central twist. Let $\Psi = C_G(h)$ be a semisimple conjugacy class and let $Z = {\mathcal Z}_G(h)$ and let $H$ be the cyclic subgroup of $Z$ generated by $h$. Since $h$ is central in $Z$, there is an action of $H$ on $G_0(Z,X^h)$. The map $\Phi_h \colon G
\times X^h \to S_\Psi$, $(g,x)\mapsto (ghg^{-1},gx)$ identifies $S_\Psi$ with $G \times_Z X$. Thus by Morita equivalence there is an action of $H$ on $G_0(G,S_\Psi)$.
We define a twisting map $$\begin{array}{c}
G_0(G,S_{\Psi}) \rightarrow G_0(G,S_{\Psi}) \\
\alpha \rightarrow \alpha(c_{\Psi})
\end{array}$$ to be the map induced by Morita equivalence corresponding to the endomorphism of $G_0(Z,X^h)$ given by twisting by $h$. We will call this map the central twist.
\[lemma.independenttwist\] With notation as above, the map $\alpha \mapsto \alpha(c_{\Psi})$ is independent of the choice of $h \in \Psi$. Moreover, if $\alpha \in G_0(G, S_{\Psi})_{c_{\Psi}}$, then $\alpha(c_{\Psi})
\in G_0(G,S_{\Psi})_{{{\mathfrak m}}_1}$. For arbitrary $\beta \in G_0(G,S_{\Psi})$, the component of $\beta(c_{\Psi})$ in $G_0(G,S_{\Psi})_{{{\mathfrak m}}_1}$ equals $\beta_{c_{\Psi}}(c_{\Psi})$.
Suppose that $h_1$ and $h_2 = k h_1 k^{-1}$ are two elements of $\Psi$; let $Z_i = {\mathcal Z}_G(h_i)$. Then $k X^{h_1} = X^{h_2}$ and $k Z_1
k^{-1} = Z_2$. We have inclusions $f_i: X^{h_i} \rightarrow S_{\Psi}$ given by $f_i(x) = (h_i, x)$ for $x \in X^{h_i}$. The equivalence of categories between ${\tt coh}^G_{S_{\Psi}}$ and ${\tt coh}^{Z_i}_{X^{h_i}}$ takes a $G$-equivariant coherent sheaf ${{\mathcal F}}$ on $S_{\Psi}$ to the sheaf $f_i^* {{\mathcal F}}$ on $X^{h_i}$ (since $f_i$ is $Z_i$-equivariant, the pullback $f_i^* {{\mathcal F}}$ has a natural structure of $Z_i$-equivariant sheaf). Now, $k^*$ induces an equivalence of categories between ${\tt coh}^{Z_2}_{X^{h_2}}$ and ${\tt coh}^{Z_1}_{X^{h_1}}$, and moreover $k^* f_2^* = f_1^*$. Therefore, to show the independence of the twisting map, it suffices to show that a $Z_2$-equivariant sheaf ${{\mathcal E}}$ on $X^{h_2}$ is an $h_2$-eigensheaf with eigenvalue $\chi$ if and only if the pullback sheaf $k^* {{\mathcal E}}$ on $X^{h_1}$ is an $h_1$-eigensheaf with eigenvalue $\chi$. This holds because if $U_2 \to X_2$ is any étale open set, and $U_1 = k^* U_1$, then $(k^*{{\mathcal E}})(U_1) = {{\mathcal E}}(U_2)$, and the automorphism of $(k^*{{\mathcal E}})(U_1)$ coming from $h_1$ coincides with the automorphism of ${{\mathcal E}}(U_2)$ induced by $h_2$.
Next, if $\alpha \in G_0(G, S_{\Psi})_{c_{\Psi}}$, then under the Morita equivalence isomorphism, $\alpha$ corresponds to an element in $G_0(Z,X^h)_{{{\mathfrak m}}_h}$. By definition of the central twist and , $\alpha(c_{\Psi})$ corresponds to an element of $G_0(Z,X^h)_{{{\mathfrak m}}_1}$. By Proposition \[prop.changedecomp\], the Morita equivalence isomorphism identifies $G_0(Z,X^h)_{{{\mathfrak m}}_1}$ with $G(G_0(G,S_{\Psi})_{{{\mathfrak m}}_1}$. Hence $\alpha(c_{\Psi}) \in G_0(G,S_{\Psi})_{{{\mathfrak m}}_1}$.
Finally, if $\beta$ is an arbitrary element of $G_0(G, S_{\Psi})$, then under the Morita equivalence isomorphism the elements $\beta(c_{\Psi})$ and $\beta_{c_{\Psi}}(c_{\Psi})$ correspond to elements of $G_0(Z,X^h)$ whose components at ${{\mathfrak m}}_1$ are equal, so the result follows.
Now assume that $G$ acts properly on $X$. Then it also acts properly on $S_\Psi$, since the map $f\colon S_\Psi \to X$ is finite. By Lemma \[lem.invariants\] there is a map in $K$-theory (the map of taking $G$-invariants) $G_0(G,S_{\Psi}) \rightarrow G_0(S_{\Psi}/G)$, $\alpha \mapsto \alpha^G$.
\[l.ctwistinv\] If $\alpha \in G_0(G,S_{\Psi})$, then $\alpha^G = (\alpha(c_{\Psi}))^G$.
Let $h \in \Psi$ and $Z = {\mathcal Z}_G(h)$. Since the map $\Phi_h: G \times
X^h \rightarrow S_{\Psi}$ is a $Z$-torsor, the quotients $X^h/Z$ and $S_{\Psi}/G$ are both identified with the quotient $M = (G \times X^h)/(G
\times Z)$. We have a map $G(Z, X^h)
\rightarrow G(M)$ (the map of taking $Z$-invariants) and a map $G(G,S_{\Psi}) \rightarrow G(M)$ (the map of taking $G$-invariants). By Morita equivalence, both $G(Z,X^h)$ and $G(G,S_{\Psi})$ are isomorphic to $G(G \times Z, G \times X^h)$ and under these isomorphisms, both maps coincide with the map $G(G \times Z, G \times
X^h) \rightarrow G(M)$ (the map of taking $G \times Z$-invariants).
In view of the definition of the central twist, it suffices to prove that if $\alpha \in G_0(Z,X^h)$ then $\alpha^Z = (\alpha(h))^Z$. For this we may assume that $\alpha = [{{\mathcal E}}]$, where ${{\mathcal E}}$ is a $Z$-equivariant coherent sheaf on $X^h$. As above, write ${{\mathcal E}}= \oplus
{{\mathcal E}}_{\chi}$. Note that $[{{\mathcal E}}_{\chi}]^Z = 0$ unless $\chi$ is the trivial character; so, denoting the trivial character by $1$, we have $\alpha^Z = \sum [{{\mathcal E}}_{\chi}]^Z = [{{\mathcal E}}_1]^Z$ and $(\alpha(h))^Z = \sum
\chi(h)^{-1} [{{\mathcal E}}_{\chi}]^Z = [{{\mathcal E}}_1]^Z$, completing the proof.
Riemann-Roch for quotients: statement and proof
-----------------------------------------------
If $X$ is a $G$-space let $CH^*_G(X) = \oplus_i CH^i_G(X) \otimes {{\mathbb C}}$ where $CH^i_G(X)$ denotes the “codimension” $i$ equivariant Chow groups of $X$ as in [@EdGr:00 p. 569]. Recall [@EdGr:98 Theorem 3] that if $G$ acts properly on $X$, with quotient $X/G$, then there is an isomorphism $\phi^G_X: CH^*_G(X) \rightarrow CH^*(X/G)$, where $CH^*(X/G) = \oplus_i CH^i(X/G) \otimes {{\mathbb C}}$. The map is defined as follows: Because $G$ acts with finite stabilizers $CH^*_G(X)$ is generated by fundamental classes of $G$-invariant cycles. If $V$ is a closed $G$-invariant subspace let $[V/G]$ be the image of $V$ under the quotient map. Then $\phi^G_X([V]) =\frac{1}{e_V}[V/G]$ where $e_V$ is the order of the stabilizer of a general point of $V$.
If $G$ and $X$ are understood, we may write simply $\phi$ or $\phi_X$ for $\phi^G_X$.
If $Y$ is an algebraic space let $\tau_Y \colon G_0(Y) \to CH^*(Y)$ be the Todd isomorphism of [@Ful:84 Theorem 18.3] (as extended to algebraic spaces). Likewise if $X$ is a $G$-space let $\tau_X^G \colon
G_0(G,X) \to \Pi_{i=0}^\infty CH^i_G(X)$ be the equivariant Todd map of [@EdGr:00]. When $G$ acts with finite stabilizers, $CH^i_G(X) =
0$ for $i > \dim X$ so the target of the equivariant Todd map is $CH^*_G(X)$ (cf. [@EdGr:00 Cor 5.1]). Note that $\tau^G_X$ is an isomorphism only when $G$ acts freely. However, it factors through an isomorphism $\widehat{G_0(G,X)} \to \Pi_{i = 0}^\infty CH^i_G(X)$ where $\widehat{G_0(G,X)}$ is the completion of $G_0(G,X)$ at the augmentation ideal of $R(G)$ [@EdGr:00 Theorem 4.1].
\[t.rrtheorem1\] Let $G$ be an algebraic group acting properly on a smooth algebraic space $X$, and let $Y = X/G$ be the quotient. Fix a conjugacy class $\Psi$ in $G$ and let $S_{\Psi}$ be the corresponding part of the global stabilizer, so we have a $G$-equivariant map $f:
S_{\Psi} \rightarrow X$ and an induced map $g: S_{\Psi}/G \rightarrow
Y$ on the quotients. Let $\alpha \in G_0(G,X)$ and let $\alpha_{\Psi}$ denote the part of $\alpha$ in $G_0(G,X)_{{{\mathfrak m}}_{\Psi}}$. Then $$\label{e.rrtheorem1}
\begin{array}{ccl}
\tau_Y((\alpha_{\Psi})^G) &= & \phi_G^X\circ \tau^G_X \circ f_*
\left((\lambda_{-1}(N_f^*)^{-1} \cap f^*\alpha)(c_\Psi)\right) \\\\
& = & g_* \circ \phi^G_{S_{\Psi}} \circ \tau^G_{S_{\Psi}}
\left( (\lambda_{-1}(N_f^*)^{-1} \cap f^* \alpha)(c_\Psi) \right).
\end{array}$$
This theorem can be stated in the following equivalent form, which can be more convenient in applications.
\[t.rrtheorem2\] Keep the notation and hypotheses of the previous theorem, and in addition, fix $h \in \Psi$, let $Z = {\mathcal Z}_G(h)$, and let $\iota:
X^h \rightarrow X$ denote the closed embedding. Identify $X^h/Z$ with $S_{\Psi}/G$. Let $\alpha |_{Z}$ denote the image of $\alpha$ under the natural map of $G$-equivariant $K$-theory to $Z$-equivariant $K$-theory. Then $$\label{e.rrtheorem2}
\tau_Y((\alpha_{\Psi})^G) = g_* \circ \phi^Z_{X^h} \circ \tau^Z_{X^h}
\left( \lambda_{-1}(N^*_{\iota})^{-1} \cap \lambda_{-1}({{\mathfrak g}}^*/ {{\mathfrak z}}^*)
\cap \iota^*(\alpha|_Z)(h) \right) .$$
Note that we do not need to compute $\alpha_{\Psi}$ to apply the formulas of Theorems \[t.rrtheorem1\] and \[t.rrtheorem2\]. However, the answer would be the same if on the right side of these formulas we replaced $\alpha$ with $\alpha_{\Psi}$. The reason is that the part of the formula corresponding to $\alpha_{\Psi'}$ for any $\Psi' \neq \Psi$ vanishes. Also, since the invariant map is linear, $\tau_Y(\alpha^G) = \sum_{\Psi} \tau_Y((\alpha_{\Psi}^G)$ can be computed using the formulas of Theorems \[t.rrtheorem1\] and \[t.rrtheorem2\].
If $X$ (and hence also $X^h$) is a smooth scheme, then $K_0(G,X) = G_0(G,X)$ and the equivariant $\tau$ maps can be calculated using the equivariant Chern character map, and the equivariant Todd class of the tangent bundle. If if $\beta \in K_0(Z,X^h)$ then using the formulas of Theorem 3.3 [@EdGr:00] $$\label{eqn.rrr}
\tau^Z_{X^h}(\beta) = \operatorname{ch}^{Z}(\beta)\tau^Z([{\mathcal O_{X^h}}])$$
If in addition, $Z$ is connected or $X^h$ has a $Z$-linearized ample line bundle then $$\tau^Z(\mathcal O_{X^h}) = \frac{\operatorname{Td}^Z(T_{X^h})}{\operatorname{Td}^Z({{\mathfrak z}})}$$ where $\operatorname{Td}^Z$ is the equivariant Todd class of [@EdGr:00 Definition 3.1]. This follows from [@EdGr:00 Theorem 3.1(d)] and the following observation about the definition of the equivariant Riemann-Roch map of [@EdGr:00]: If a group $G$ acts freely on a smooth space $X$ with quotient $X \stackrel{\pi} \to X/G$ then, identifying $CH^*_G(X)$ with $CH^*(X/G)$, we have $\tau^G({\mathcal O}_X) = \tau({\mathcal O}_{X/G})$ When $X/G$ (and thus $X$) is a smooth scheme then $\tau({\mathcal O}_{X}) = \operatorname{Td}(T_{X/G})$. In this case $\tau^G({\mathcal O}_X) = \operatorname{Td}^G(\pi^*T_{X/G})$. By Lemma \[lem.tantorsor\], $T_\pi = X \times {\mathfrak g}$. Therefore, $\operatorname{Td}^G(\pi^*T_{X/G}) = \operatorname{Td}^G(T_X)/\operatorname{Td}^G({\mathfrak g})$.
It suffices to prove Theorem \[t.rrtheorem1\], since this implies Theorem \[t.rrtheorem2\] using the Morita equivalence isomorphism $G_0(G, S_{\Psi}) \simeq G_0(Z, X^h)$. We also need only prove the second formula in Equation ; this implies the first formula because the maps $\tau$ and $\phi$ are covariant for finite morphisms.
The proof of Theorem \[t.rrtheorem1\] is almost the same as the proof in the case where $G$ is diagonalizable, given in [@EdGr:03 Theorem 3.1]. The nonabelian localization theorem replaces the localization theorem for actions of diagonalizable groups used in [@EdGr:03]. We give the general proof here, referring to [@EdGr:03] for some omitted details. Throughout the proof we write $\phi_M$ for $\phi^G_M$, where $G$ acts properly on $M$. We write $S$ for $S_{\Psi}$.
The proof proceeds in three steps. First, we observe that the theorem is true if the action of $G$ on $X$ is free (and that in this case, it holds without the assumption that $X$ is smooth). In this case, if $\Psi \neq \{ 1 \}$, then $S$ is empty and $\alpha_{\Psi} = 0$, so both sides of vanish. If $\Psi = \{1 \}$, then $f: S
\rightarrow X$ is an isomorphism, and the theorem amounts to the assertion that $$\tau_Y(\alpha^G) = \phi_X \circ \tau^G_X(\alpha),$$ which follows from [@EdGr:00 Theorem 3.1(e)].
Second, we prove the theorem for $\Psi = \{ 1 \}$. In this case $f\colon S \rightarrow X$ is an isomorphism, and the theorem amounts to the assertion that $$\tau_Y((\alpha_1)^G) = \phi_X \circ \tau^G_X(\alpha).$$ Since $\tau^G$ maps components of $K$-theory supported at maximal ideals other than ${{\mathfrak m}}_1$ to zero (this is proved as in [@EdGr:03 Proposition 2.6]), on the right side of this equation we can replace $\alpha$ by $\alpha_1$. By [@EdGr:98 Proposition 10] there exists a finite surjective morphism $p: X' \rightarrow X$ of $G$-spaces such that $G$ acts freely on $X'$ and then induced map of quotients $q\colon X' \to X$ is also finite and surjective. (Note that $X'$ need not be smooth.) By [@EdGr:03 Lemma 3.5] (which holds without the diagonalizability assumption on $G$), the map $p_*: G_0(G,X')_{{{\mathfrak m}}_1} \rightarrow
G_0(G,X)_{{{\mathfrak m}}_1}$ is surjective. Therefore there exists $\beta_1 \in
G_0(G,X')_{{{\mathfrak m}}_1}$ such that $p_* (\beta_1) = \alpha_1$. There is a geometric quotient $X' \rightarrow Y'$ such that the induced map $q:
Y' \rightarrow Y$ is also finite. By Lemma \[lem.invcommute\], taking invariants commutes with pushforward by a finite morphism, so $q_*((\beta_1)^G) = (\alpha_1)^G$. Therefore, $$\tau_Y((\alpha_1)^G) = \tau_Y(q_*((\beta_1)^G) = q_* \tau_{Y'}((\beta_1)^G).$$ By the first step of the proof, this equals $q_* \circ \phi_{X'} \circ
\tau^G_{X'}(\beta_1)$. Since $q_* \circ \phi_{X'} = \phi_X \circ
p_*$, we have $$q_* \circ \phi_{X'} \circ \tau^G_{X'}(\beta_1) = \phi_X \circ p_*
\circ \tau^G_{X'}(\beta_1) = \phi_X \circ \tau^G_X \circ p_* (\beta_1)
= \phi_X \circ \tau^G_X (\alpha_1),$$ as desired.
Third, we prove the theorem for general $\Psi$. Let $\beta = \lambda_{-1}(N_f^*)^{-1} \cap f^* \alpha$. We need to prove that $$\tau_Y((\alpha_{\Psi}^G) = g_* \circ \phi_S \circ \tau^G_S (\beta (c_{\Psi})).$$ By Lemma \[lemma.independenttwist\], the components of $\beta (c_{\Psi})$ and $\beta_{c_{\Psi}} (c_{\Psi})$ supported at ${{\mathfrak m}}_1 \in R(G)$ are equal, so arguing as in Step 2, we see that $\tau^G_S (\beta (c_{\Psi})) = \tau^G_S (\beta_{c_{\Psi}}(c_{\Psi}))$. But $\beta_{c_{\Psi}} (c_{\Psi})$ is in $G_0(S,G)_{{{\mathfrak m}}_1}$. Therefore we can apply the second step of the proof to $\beta_{c_{\Psi}}(c_{\Psi})$. Thus, $$\begin{array}{ccll}
\tau_Y(\alpha_{\Psi}^G) & = &
\tau_Y \circ ((f_* (\beta_{c_{\Psi}}))^G)
& \mbox{ localization} \\\\
& = & g_* \circ \tau_{S/G} ( \beta_{c_{\Psi}}^G)
& \mbox{ finite-pushforward commutes with invariants} \\\\
& = & g_* \circ \tau_{S/G} ( (\beta_{c_{\Psi}}(c_{\Psi}))^G)
& \mbox{ Lemma \ref{l.ctwistinv}} \\\\
& = & g_*\circ \phi_S \circ \tau^G_S (\beta_{c_{\Psi}}(c_{\Psi}))
& \mbox { Step 2} \\\\
& = & g_* \circ \phi_S \circ \tau^G_S (\beta(c_{\Psi})). &
\end{array}$$ This completes the proof.
Appendix
========
This appendix contains a result about the tangent bundle to a torsor which is difficult to find rigorously proved in the literature.
\[lem.tantorsor\] Let $X \stackrel{f} \to Y$ be a (left) $G$-torsor. Then $T_f$ is canonically isomorphic to the $G$-bundle $X \times \mathfrak{g}$ where the $G$-action on the Lie algebra $\mathfrak{g}$ is the adjoint action.
By definition $T_f$ is the normal bundle to the diagonal morphism $X \stackrel{\Delta_{f}} \to X \times_Y X$. Since $f$ is a $G$-torsor the diagram $$\label{eq.diagtorsor}
\begin{array}{rrr}
G \times X & \stackrel{\sigma} \to & X\\
\pi \downarrow & & f \downarrow\\
X & \stackrel{f} \to & Y
\end{array}$$ is a cartesian, where $\sigma \colon G \times X\to X$ is the action map and $\pi\colon G \times X \to X$ is projection. If $G$ acts on $G
\times X$ by conjugation on the first factor and the usual action on the second factor then all morphisms in (\[eq.diagtorsor\]) are $G$-invariant. Thus there is a canonical identification of $G$-spaces $X \times_Y X \to G \times X$. Under this identification the diagonal corresponds to the section $X \stackrel{(e_G, 1_X)} \to G \times
X$. This map is obtained by base change from the $G$-equivariant inclusion $e_G \to G$ (where $G$ acts on itself by conjugation) whose normal bundle is $\mathfrak{g}$. Therefore, $T_f = X \times \mathfrak{g}$.
\[rem.tantorsor\] Suppose $N \subset G$ is a closed normal subgroup with Lie algebra $\mathfrak{n} \subset \mathfrak{g}$. Normality of $N$ implies that if $X \stackrel{f} \to Y$ is an $N$-torsor then there is a natural left $G$-action on $Y$ such that $f$ is $G$-equivariant. Essentially the same argument as above implies that $T_f$ is naturally isomorphic to ${\mathfrak n} \otimes {\mathcal O}_X$ where the $G$-action on ${\mathfrak n}$ is the restriction of the adjoint action to the $ad$-invariant subalgebra ${\mathfrak n}\subset
{\mathfrak g}$.
[MFK]{}
Michael Francis Atiyah, [*Elliptic operators and compact groups*]{}, Springer-Verlag, Berlin, 1974, Lecture Notes in Mathematics, Vol. 401.
Paul Baum, William Fulton, and Robert MacPherson, [*Riemann-[R]{}och for singular varieties*]{}, Inst. Hautes Études Sci. Publ. Math. (1975), no. 45, 101–145.
Armand Borel, [*Linear algebraic groups*]{}, second ed., Springer-Verlag, New York, 1991.
Theodor Br[ö]{}cker and Tammo tom Dieck, [*Representations of compact [L]{}ie groups*]{}, Graduate Texts in Mathematics, vol. 98, Springer-Verlag, New York, 1995, Translated from the German manuscript, Corrected reprint of the 1985 translation.
Neil Chriss and Victor Ginzburg, [*Representation theory and complex geometry*]{}, Birkhäuser Boston Inc., Boston, MA, 1997.
Dan Edidin and William Graham, [*Equivariant intersection theory*]{}, Invent. Math. **131** (1998), no. 3, 595–634.
, [*Riemann-[R]{}och for equivariant [C]{}how groups*]{}, Duke Math. J. **102** (2000), no. 3, 567–594.
, [*Riemann-[R]{}och for quotients and [T]{}odd classes of simplicial toric varieties*]{}, Comm. in Alg. **31** (2003), 3735–3752.
John Fogarty, [*Invariant theory*]{}, W. A. Benjamin, Inc., New York-Amsterdam, 1969.
William Fulton, [*Intersection theory*]{}, Springer-Verlag, Berlin, 1984.
A. Grothendieck and J. Dieudonné, [*Élements de [G]{}éométrie [A]{}lgébrique [I]{}[V]{}. Étude locale des schemas et des morphismes de schémas*]{}, Inst. Hautes Études Sci. Publ. Math. No. **20, 24, 28, 32** (1964, 1965, 1966, 1967).
G. Hochschild, [*The structure of [L]{}ie groups*]{}, Holden-Day Inc., San Francisco, 1965.
James E. Humphreys, [*Conjugacy classes in semisimple algebraic groups*]{}, Mathematical Surveys and Monographs, vol. 43, American Mathematical Society, Providence, RI, 1995.
Se[á]{}n Keel and Shigefumi Mori, [*Quotients by groupoids*]{}, Ann. of Math. (2) **145** (1997), no. 1, 193–213.
Donald Knutson, [*Algebraic spaces*]{}, Springer-Verlag, Berlin, 1971, Lecture Notes in Mathematics, Vol. 203.
D. Mumford, J. Fogarty, and F. Kirwan, [*Geometric invariant theory*]{}, third ed., Springer-Verlag, Berlin, 1994.
H. Andreas Nielsen, [*Diagonalizably linearized coherent sheaves*]{}, Bull. Soc. Math. France **102** (1974), 85–97.
A. L. Onishchik and [È]{}. B. Vinberg, [*Lie groups and algebraic groups*]{}, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1990, Translated from the Russian and with a preface by D. A. Leites.
Daniel Quillen, [*Higher algebraic [$K$]{}-theory. [I]{}*]{}, Algebraic $K$-theory, I: Higher $K$-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), Springer, Berlin, 1973, pp. 85–147. Lecture Notes in Math., Vol. 341.
Wulf Rossmann, [*Lie groups*]{}, Oxford Graduate Texts in Mathematics, vol. 5, Oxford University Press, Oxford, 2002, An introduction through linear groups.
, [*The representation ring of a compact [L]{}ie group*]{}, Inst. Hautes Études Sci. Publ. Math. (1968), no. 34, 113–128.
Graeme Segal, [*Equivariant [$K$]{}-theory*]{}, Inst. Hautes Études Sci. Publ. Math. (1968), no. 34, 129–151.
V. Srinivas, [*Algebraic [$K$]{}-theory*]{}, second ed., Progress in Mathematics, vol. 90, Birkhäuser Boston Inc., Boston, MA, 1996.
R. W. Thomason, [*Comparison of equivariant algebraic and topological [$K$]{}-theory*]{}, Duke Math. J. **53** (1986), no. 3, 795–825.
, [*Lefschetz-[R]{}iemann-[R]{}och theorem and coherent trace formula*]{}, Invent. Math. **85** (1986), no. 3, 515–543.
, [*Algebraic [$K$]{}-theory of group scheme actions*]{}, Algebraic topology and algebraic $K$-theory (Princeton, N.J., 1983), Ann. of Math. Stud., vol. 113, Princeton Univ. Press, Princeton, NJ, 1987, pp. 539–563.
, [*Equivariant algebraic vs. topological [$K$]{}-homology [A]{}tiyah-[S]{}egal-style*]{}, Duke Math. J. **56** (1988), no. 3, 589–636.
, [*Une formule de [L]{}efschetz en [$K$]{}-théorie équivariante algébrique*]{}, Duke Math. J. **68** (1992), no. 3, 447–462.
B. Toen, [*Théorèmes de [R]{}iemann-[R]{}och pour les champs de [D]{}eligne-[M]{}umford*]{}, $K$-Theory **18** (1999), no. 1, 33–76.
Gabriele Vezzosi and Angelo Vistoli, [*Higher algebraic [$K$]{}-theory of group actions with finite stabilizers*]{}, Duke Math. J. **113** (2002), no. 1, 1–55.
[^1]: The first author was supported by N.S.A. and the second by N.S.F
[^2]: If $G(G,X)$ is a finitely generated $R(G)$-module this also follows from Theorem \[thm.weakloc\] and [@VeVi:02 Theorem 5.4].
[^3]: We say that a morphism of algebraic space $f\colon X \to Y$ is representable if for any scheme $Y'$ and map $Y' \to Y$, the fiber product $X \times_Y Y'$ is a scheme.
|
---
abstract: 'The interface between a topological insulator and a normal insulator hosts localized states that appear due to the change in band structure topology. While these states are topologically protected, states at separate interfaces can hybridize and gap out. An important question is whether there are other factors, such as local fields and/or disorder, that may also impact these states. In this paper, we use a combination of experiment and theory to study heterostructures of GeTe (normal insulator) and Sb$_2$Te$_3$ (topological insulator) in which, due to the strong chemical affinity between the materials, there is significant intermixing at the interface where the topological state is expected. To characterize the interface, we evaluate the band offset between GeTe and Sb$_2$Te$_3$ using X-ray photoemission spectroscopy and use atom probe tomography to chart the elemental composition along the stacking direction. We then use first-principles calculations to independently calculate the band offset value and also to parametrize the band structure within a four-band continuum framework. Strikingly, the continuum model reveals that the interfacial topological modes can couple over significantly longer distances when the normal insulator medium is intermixed GeTe and Sb$_2$Te$_3$ rather than simply GeTe. We confirm this finding using first-principles calculation for thin heterostructure, which indicate that a disordered interface is sufficient to induce long-range coupling between embedded topological modes. Our study provides insights into how to use bulk-like structures to manipulate topological modes. Our results also provide a microscopic basis for recent experimental findings \[Nguyen *et al.*, Sci. Rep. **6**, 27716 (2016)\] where topological interface states were seen to couple over relatively large distances.'
address:
- '$^1$ CD-FMat, National Institute of Advanced Industrial Science and Technology (AIST), 1-1-1 Umezono, Tsukuba Central 2, Tsukuba, Japan'
- '$^2$ Department of Materials Science and Metallurgy, University of Cambridge, 27 Charles Babbage Road, Cambridge CB3 0FS, United Kingdom'
- '$^3$ Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Centre for Mathematical Sciences, Cambridge CB3 0WA, United Kingdom'
- '$^4$ TCM Group, Cavendish Laboratory, University of Cambridge, Cambridge CB3 0HE, United Kingdom'
- '$^5$ RIKEN Center for Computational Science, 7-1-26 Minatojima-minami, Cyuo-ku, Kobe, Hyogo 650-0047, Japan'
- '$^6$ Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands'
- '$^7$ Peter Grünberg Institute (PGI-9), Forschungszentrum Jülich, 52425 Jülich, Germany'
- '$^8$ Department of Physics, University of Cambridge, J. J. Thomson Avenue, Cambridge CB3 0HE'
author:
- 'Hisao Nakamura$^{1,2}$\*, Johannes Hofmann$^{3,4}$\*, Nobuki Inoue$^5$, Sebastian Koelling$^6$, Paul M. Koenraad$^6$, Gregor Mussler$^7$, Detlev Grützmacher$^7$, and Vijay Narayan$^8$\*'
bibliography:
- 'bib.bib'
title: 'Robustness of embedded topological modes in bulk-like GeTe-Sb$_2$Te$_3$ heterostructures'
---
INTRODUCTION
============
The Ge-Sb-Te (GST) system has been widely studied due to its multi-functional character including phase-change properties [@raoux14; @kato06; @yamada87; @simpson11; @chen86; @inoue19], ferroelectric characteristics [@tominaga14; @tominaga15], and potential thermoelectric properties [@ibarra18]. The GST system may also have a non-trivial band topology as found in, for example, superlattices of alternating Sb$_2$Te$_3$ and GeTe layers (GST-SL) [@sa12]. Here, Sb$_2$Te$_3$ is a topological insulator (TI) [@hiseh09; @zhang09] and GeTe is a normal insulator (NI) [@tsu68], which becomes superconducting below $1 K$ [@hein64; @narayan16; @narayan19]. The superlattice system consisting of $L$ GST units, labeled as \[(Sb$_2$Te$_3$)$_M$(GeTe)$_N$\]$_L$, is overall a topological or non-topological insulator depending on the relative layer thickness denoted by the indices $M$ and $N$ [@tominaga14; @halasz12].
There are two mechanisms that contribute to the character of the band structure: (i) the physical intermixing of adjacent layers, and (ii) the coupling of topological modes [@burkov11; @halasz12] that appear at the interface of Sb$_2$Te$_3$ and GeTe (SG interface) in the superlattice. These two-dimensional modes are localized on the edge of (Sb$_2$Te$_3$)$_M$ layer and have a linear Dirac dispersion with a helical spin structure. More strictly, the overall phase is governed by the competing hybridization of topological modes of the top and bottom edge of the (Sb$_2$Te$_3$)$_M$ layer (intralayer coupling) and those of the neighboring layers through (GeTe)$_N$ layer (interlayer coupling). To date, much of the investigation on GST as TI/NI multilayers, theoretical and experimental, focuses on monolayer/few monolayer superlattice units, e.g., ($M$,$N$) = (1,2), or (2,2) [@tominaga14; @tominaga15; @sa12; @nakamura17; @bang16; @qian16]. GST structures in which the individual GeTe and Sb$_2$Te$_3$ are more bulk-like have received significantly less attention. In this context, a recent finding by some of the authors showed, unexpectedly, that in heterostructures of bulk-like Sb$_2$Te$_3$ and GeTe, the net number of topological modes is controlled by the GeTe layer even in a range where $M$, $N \gg 10$ [@nguyen16]. In Ref. [@nguyen16], the low-temperature magneto-transport of a Sb$_2$Te$_3$-GeTe-Sb$_2$Te$_3$ tri-layer structure was measured and the number of two-dimensional (2D) modes was inferred from the weak anti-localization characteristic [@hikami80; @backes17; @backes19]. Whereas one might expect that each TI/NI interface should host a topological mode, it was observed that for samples with a $15$ nm-thick GeTe layer, the number of 2D modes was two fewer than expected. It was suggested that in a Sb$_2$Te$_3$-GeTe-Sb$_2$Te$_3$ (SGS) tri-layer, the topological modes flanking the GeTe layer hybridized even when separated by up to 15 nm and developed a gap of approximately $30$ K. In order to understand this SGS tri-layer, i.e., TI/NI/TI tri-layer system, the role of band offsets at the SG interface, atomistic structure, and electronic structure of the junction have to be clarified.
In this paper, we find that the physical structure at the Sb$_2$Te$_3$-GeTe interface plays a crucial role in mediating interlayer interactions within the heterostructure. In particular, we find that under conditions of a sharp, well-defined interface, interactions between topological modes separated by more than a few nm is relatively weak. However, under more realistic conditions where the interface has some degree of intermixing, topological modes separated by as much as $10$ nm couple and develop a gap of several meV. This strongly supports the findings of Nguyen *et al.* [@nguyen16].
This paper is structured as follows: In Sec. \[sec:2\], we present experimental measurements of the band offset between Sb$_2$Te$_3$ and GeTe on molecular-beam-epitaxy (MBE)-grown samples using X-ray photoemission spectroscopy (XPS). We then use first-principles calculations to obtain an independent estimate of the band offset at the SG interface using a range of experimentally relevant microstructures. Our estimates compare favorably with the XPS data, thereby validating the first-principles result. Moreover, the our experimental XPS indicates a significant intermixing between. Next, in Sec. \[sec:3\], we develop a continuum model of an SGS tri-layer within a four-band framework, the parameters of which are obtained from our first-principles calculations. The continuum model can be used to study large systems that would be computationally too expensive to study directly using first-principles methods. Finally, in Sec. \[sec:IV\], we analyze the robustness of topological modes to chemical interactions and examine how the microscopic structure of the interfacial region may impact the interlayer coupling of topological modes.
BAND OFFSET {#sec:2}
===========
In this section, we discuss the band offset between the GeTe and Sb$_2$Te$_3$ layers in the GST heterostructure. First, in Sec. \[sec:experiment\], we present experimental measurements of the band offset at the interface of GeTe and Sb$_2$Te$_3$ using XPS depth profiles. In Sec. \[sec:II2\], we then use first-principles calculations to evaluate the bulk crystal structure of GeTe and Sb$_2$Te$_3$, from which we obtain an independent evaluation of the band offset, which agrees with the experimental results. The agreement between experiment and theory validates the theoretical calculation, and provides a basis for the continuum model developed in Section \[continuum\] to describe larger systems.
EXPERIMENTAL EVALUATION {#sec:experiment}
-----------------------
We use XPS measurements to determine the band offset between bulk GeTe and bulk Sb$_2$Te$_3$. Following Refs. [@kim07; @kraut80; @fang11], this is done by evaluating the difference of the core electron energy levels in bulk samples, and comparing this to the difference in the core energy levels in a heterostructure. Conventionally, this would require XPS spectra of three separate samples: a bulk GeTe film, a bulk Sb$_2$Te$_3$ film, and a heterostructure of the two in which the top layer is sufficiently thin ($\sim 5$ nm) that the X-rays can penetrate it fully and sample both materials. However, such a procedure will have unknown systematic errors when considering Sb$_2$Te$_3$-GeTe heterostructures as the two compounds have a strong chemical affinity and will undergo significant intermixing, especially in the vicinity of the interface. Here, we obtain instead a “depth profile” of a single GeTe-Sb$_2$Te$_3$ heterostructure where XPS spectra are taken between successive Ar-ion etches of the sample, which successively remove the top layers of the heterostructure. The measurements are continued for the entire depth of the sample, i.e., until the Ar-ion etch fully depletes the material. This approach eliminates variations due to the different growth conditions for separate samples. The samples considered here are MBE-grown heterostructures in which the GeTe (top) layer has a thickness of 11 nm and the Sb$_2$Te$_3$ layer is 25 nm thick. The samples are grown on a Si(111) substrate as described in Ref. [@nguyen16].
Figure \[fig:1\] shows XPS depth profiles for different etching times $t=25, 65, 75, 90, 120, 130, 140$, and $170$s in which the Ge(2p3) transition \[Fig. \[fig:1a\]\], the Sb(3d) transition \[Fig. \[fig:1b\]\], and the Te(3d) transition \[Fig. \[fig:1c\]\] is monitored. As expected, initially there is a pronounced Ge peak which begins to diminish at the same time the Sb peak appears, indicating that the top GeTe layer is completely eroded after $130$s of etching. The Te(3d) transition \[Fig. \[fig:1c\]\] shows little depth dependence, which is expected. The traces taken between $120$s and $140$s show features corresponding to both Ge and Sb, suggesting that the X-rays probe both layers in this range, which points to an intermixing between the layers. These traces also reflect the diffuse nature of interface between the two materials. The band offset $\Delta E_v$ is obtained as described in Refs. [@kim07; @kraut80; @fang11]: $$\label{XPS_Eq}
\Delta E_v = (E_{\rm CL}^{Ge} - E_v^{Ge}) - (E_{\rm CL}^{Sb} - E_v^{Sb}) + \Delta E_{\rm CL} .$$ Here, the first two terms on the right-hand side represent the difference in energy between the core level (CL) and valence band edge ($E_v$) of Ge and Sb, respectively. These are obtained from the bulk spectra as shown in Figs. \[fig:2a\] and \[fig:2b\]. The third term is the difference in energy between the core levels of Ge and Sb obtained from the combined spectrum shown in Fig. \[fig:2c\]. The result for the band offset is $\Delta E_v = 0.4 \pm 0.1$ eV.
Further evidence for intermixing region is obtained in Fig. \[APT\] where we show atom probe tomography (APT) of a Sb$_2$Te$_3$/GeTe/Sb$_2$Te$_3$ sample. APT is based on the evaporation of atoms in the form of ions from a single tip-shaped sample by means of an electric field. During the analyses, ions are projected from the apex of the tip onto a position-sensitive single ion detector [@blavett93] by the electric field. On the basis of the measured positions and the time-of-flight between the tip apex and the detector surface a 3D reconstruction of the analyzed volume is created [@bas95]. Further details of the ATP can be found in the Supplementary Material [@SOM].
FIRST-PRINCIPLES CALCULATIONS {#sec:II2}
-----------------------------
[@lllll]{} & Sb$_2$Te$_3$ (Rh) & Sb$_2$Te$_3$ (Rk) & GeTe (dRk) & GeTe (Rk)\
Fermi level & 0.00 & -0.08 & -0.13 & 0.22\
VBM & -0.02 & -0.11 & -0.36 & 0.13\
CBM & 0.08 & -0.06 & 0.11 & 0.22\
Band Gap & 0.10 & 0.05 & 0.47 & 0.09\
In this section, we present results for the band offset obtained from electronic structure calculations of bulk Sb$_2$Te$_3$ and GeTe via density functional theory (DFT). In our calculations, we use the SIESTA program package [@soler02] and adopt the double zeta plus polarization function (DZP) level basis set. Our results are consistent with the experimental findings presented in the previous section. To evaluate the band offset and identify the SG interfacial electronic states, we make use of the non-equilibrium Green’s function technique [@datta97] combined with DFT (NEGF-DFT). Since NEGF-DFT satisfies semi-infinite boundary condition, our results are free from artificial size effects, which are often problematic in slab model calculations. NEGF-DFT calculations were carried out using the Smeagol program package [@rocha06; @rungger08]. We adopt an exchange correlation (XC) functional of the van der Waals (vW) correction, DF2 [@klime09], for total energy calculations and use the local density approximation (LDA) to determine the band structure in NEGF-DFT calculations, where spin-orbit (SO) interaction are included.
At room temperature, bulk GeTe crystallizes in a disordered rock salt (dRk) structure and Sb$_2$Te$_3$ crystallizes in a rhombohedral layered (Rh) structure [@dasilva08]. Hence, the conventional hexagonal cell can be taken as the unit cell for both, where the (111) direction of the rock salt corresponds to the (0001) in the conventional hexagonal cell. We set the c-axis to the (0001) direction and define the $z$-coordinate along the c-axis. The SG interface plane is then perpendicular to the (0001) direction. In the conventional hexagonal cell, both GeTe (dRk) and Sb$_2$Te$_3$ (Rh) have a CBA/CBA/CBA/... stacking, where A, B, and C represent monolayers of Sb$_2$Te$_3$ or GeTe. In other words, the bulk unit cell of Sb$_2$Te$_3$ and GeTe consists of three monolayers, which may be denoted as (Sb$_2$Te$_3$)$_3$ and (GeTe)$_3$, respectively [@dasilva08; @nonaka00]. The experimental values of the lattice constants are $a_0=b_0=4.26~\AA$ and $c_0=30.75~\AA$ for Sb$_2$Te$_3$ (Rh) and $a_0=b_0=4.17~\AA$ and $c_0=10.90~\AA$ for GeTe (dRk) [@nonaka00]. Thus, there is a very small lattice mismatch at the SG interface. For simplicity, we fix the lattice constant as $a_0=b_0=4.25~\AA$ for both of Sb$_2$Te$_3$ and GeTe (as well as the SGS tri-layer) in our computational models and then allow all atoms in the cell to relax to their atomic positions. The atomic structures are shown in Figs. \[fig:3a\] and \[fig:3b\], respectively.
The band offset is calculated as follows: first, the Fermi level $E_F$ is obtained from DFT calculations of bulk systems that include spin-orbit interactions. Then, we define an extended cell C by taking $1 \times 1 \times 3$ unit cells and apply the self-consistent NEGF-DFT. The left and right sides of C are connected to the bulk semi-infinitely by the self-energy terms, such that our calculations give the Green’s function projected on C. Using the resulting Green’s functions, we analyze the spectral density and evaluate the conduction band minima (CBM) and valence band maxima (VBM). Next, we carried out NEGF-DFT for the same C while the right side of the cell C is now terminated by vacuum. Practically, we took a vaccuum region of $z_{\rm vac} = 15.0~\AA$ in the $z$-direction. Now, we can introduce the unique definition of the Fermi level $E_F^0$ using the vacuum level, i.e., $$E_F^0 = E_F - V_H(z=z_{\rm vac}) ,$$ where $V_H$ is the Hartree potential averaged over the $xy$ plane. As the last step, we corrected the values of VBM and CBM by Eq. (2), which are denoted as $E_v^0$ and $E_c^0$, respectively. We applied the above procedures to Sb$_2$Te$_3$ (Rh) and GeTe (dRk), and as a reference, also to the rock salt (Rk) structures of Sb$_2$Te$_3$ and GeTe which are possible crystal phases representing vacancy states or at high temperature [@sun06; @ohyanagi14; @disante13]. The value of $E_F^0$ of Sb$_2$Te$_3$ (Rh) is $-4.61$ eV.
In Table 1, we set $E_F^0$ of Sb$_2$Te$_3$ (Rh) to zero and list the values of the VBM, the CMB and the Fermi level of different structures relative to this. The band offset between Sb$_2$Te$_3$ (Rh) and GeTe (dRk), given by the difference between the respective VBM is $\Delta E_v \approx 0.36$eV, which agrees well with our experimental value reported in Sec. 2. The validity of the calculations is also confirmed by noting that the calculated band gap of Sb$_2$Te$_3$ is $0.1$eV, which is consistent with its narrow gap, $p$-type semiconductor character. Likewise, the calculated band gap of GeTe (dRk)$=0.47$eV is close to the experimental value of $0.6$eV [@tsu68]. We note here that the band gap of GeTe (Rk) is underestimated by our theoretical calculation, but disorder is known to increase the experimental band gap, which is consistent with our results [@dasilva08].
The present results suggest the validity of our computational model in order to analyze the topological modes of an SGS tri-layer quantitatively. In the next section, in order to treat large heterostructures beyond the range of numerical DFT simulations, we construct an effective four-band model with parameters derived from our first-principles calculations.
FOUR-BAND CONTINUUM MODEL {#sec:3}
=========================
\[continuum\]
In the previous section, we have both experimentally and theoretically elucidated the structure of the SG interface. Experimental measurements of the band gap obtained using XPS were shown to be consistent with theoretical results from [*ab-initio*]{} DFT calculations of a semi-infinite slab structure, which indicates that our computational DFT model is predictive for these systems. The aim of this section is to extend the theoretical model to tri-layer structures and to address the recent experiments by Nguyen *et al.* [@nguyen16] by considering the qualitative effect of a thick, bulk-like GeTe intermediate layer on the embedded interface states.
While the numerical DFT method is in principle exact, i.e., it will accurately describe the inter- and intralayer coupling as well as the chemical intermixing at the interfaces, modeling very thick bulk-like heterostructures comes with a prohibitive numerical cost. In practice, we are restricted to very thin structures of typically less than ten layers. In order to make contact with the experiments on bulk-like structures of Ref. [@nguyen16], in this section, we introduce an effective four-band model using parameter values derived from the bulk calculations presented in the previous section. This model allows us to describe tri-layer structures of arbitrary thickness. Indeed, as a main result of this paper, our findings indicate a significant interlayer-coupling of surface states across the GeTe layer, which is consistent with the experiment [@nguyen16].
An effective four-band model is predictive for inter- and intralayer coupling effects, but it neglects the physical intermixing of the GeTe and Sb$_2$Te$_3$ phases at the interface, i.e., a reconfiguration of atomic positions. In order to take this into account, we consider additionally a model in which the GeTe film is replaced by a Ge$_2$Sb$_2$Te$_5$ (GST225) crystal phase, which is one of the most standard compositions of the GST alloy. We adopt the Kooi structure of GST225 [@kooi02], the crystal structure of which is shown in Fig. \[fig:3c\]. To further support our effective model, in Sec. \[sec:IV\], we present ab-initio results for thin heterostructures that are consistent with the results obtained by the four-band model.
[@lllllllllll]{} & $A_0$ & $A_2$ & $B_0$ & $B_2$ & $C_0$ & $C_1$ & $C_2$ & $M_0$ & $M_1$ & $M_2$\
Sb$_2$Te$_3$ & 3.40 & 0.00 & 0.84 & 0.00 & 0.00 & -12.39 & -10.78 & -0.22 & 19.64 & 48.51\
GeTe & 5.51 & 0.00 & 2.75 & 0.00 & -0.13 & -3.56 & 12.41 & 0.79 & -25.17 & -120.40\
GST225 & 0.02 & 0.00 & 0.00 & 0.00 & 0.04 & -6.99 & 2.21 & 0.14 & 16.74 & 14.82\
\[tab:2\]
CONTINUUM MODEL OF HETEROSTRUCTURE
----------------------------------
Topological insulators in the Bi$_2$Se$_3$ and Sb$_2$Te$_3$ family are often described by a four-band model Hamiltonian that captures the band structure near the $\Gamma$ point. The model was first derived by Zhang [*et al.*]{} [@zhang09] and by Liu [*et al.*]{} [@liu10]. The effective four-band Hamiltonian is $$H = \pmatrix{
\varepsilon({\bf k}) + M({\bf k}) & B(k_z) k_z & 0 & A(k_\parallel) k_- \cr
B(k_z) k_z & \varepsilon({\bf k}) - M({\bf k})& A(k_\parallel) k_- & 0\cr
0 & A(k_\parallel) k_+ & \varepsilon({\bf k}) + M({\bf k})& - B(k_z) k_z \cr
A(k_\parallel) k_+ & 0 & - B(k_z) k_z & \varepsilon({\bf k}) - M({\bf k})
} , \label{eq:hamiltonian}$$ where, adopting the notation of Ref. [@liu10], $\varepsilon({\bf k}) = C_0 + C_1 k_z^2 + C_2 k_\parallel^2$, $M({\bf k}) = M_0 + M_1 k_z^2 + M_2 k_\parallel^2$, $A(k_\parallel) = A_0 + A_2 k_\parallel^2$, $B(k_z) = B_0 + B_2 k_z^2$, $k_\parallel^2 = k_x^2 + k_y^2$, and $k_\pm = k_x \pm i k_y$. Here, the interface is aligned in the $xy$-plane, and the $z$-direction is perpendicular to the interface.
The parameters of Sb$_2$Te$_3$ were derived in Ref. [@zhang09] and are summarized in Tab. \[tab:2\], where we define the zero of the energy scale at the Fermi level of Sb$_2$Te$_3$. Note that the parameters $A_2$ and $B_2$, which relate to higher-order terms in $k$, are set to zero since we only consider the region close to $\Gamma$ point. For calculations of multilayer structures, we also require parameter values for an effective model of the GeTe phase. Due to band inversion, the band gap of GeTe (dRk) increases at the $L$ point in the rock salt structure [@sa12; @liebmann16]. The four-band model (\[eq:hamiltonian\]) is still applicable, where the $L$ point of the rock salt cell relates to the $\Gamma$ point in the conventional hexagonal cell. We construct an effective Hamiltonian of GeTe (dRk) by fitting the first-principles data of the bulk unit cell presented in Sec. \[sec:II2\] near the $\Gamma$ point to our effective model. The parameter set is given in Tab. \[tab:2\]. Related parameters for a Hamiltonian that describes the GST225 (Kooi) phase are also given in Tab. \[tab:2\].
For a slab geometry with an interface in the $xy$-plane, we loose translational invariance in the $z$ direction. The Hamiltonian of a heterostructure takes the form (\[eq:hamiltonian\]) with $z$-dependent coefficients $A_1(z), A_2(z),\ldots$ etc: $$\hat{H}(k_x,k_y,k_z) \to \hat{H}(k_x, k_y, - i\partial_z) . \label{eq:hamiltoniantrilayer}$$ Note that in the heterostructure, the second $z$-derivative must be written in a manifestly symmetric form to ensure a hermitian Hamiltonian, i.e., we replace [@bastard86] $$- B_2 k_z^2 \to \frac{d}{dz} B_2(z) \frac{d}{dz} .$$ Eigenstates that solve $\hat{H} \Psi(z) = E \Psi(z)$ depend on the parallel momentum $k_\parallel$ as well as the $z$ coordinate. We consider solutions within a layer with the ansatz [@bastard86] $$\Psi_{\alpha\beta}(z,E) = \Psi_{\alpha\beta}(E) e^{\beta \lambda_{\alpha}(E) z} ,$$ where $\lambda_\alpha$ is a complex-valued function of $E$ and $\alpha,\beta=\pm$. The eigenstate equation for $\Psi(z,E)$ reads $$\eqalign{
\pmatrix{
-\lambda_\alpha^2 D_- + L_1 & - i (\beta \lambda_\alpha) B_0 & 0 & A_0 k_- \cr
- i (\beta \lambda_\alpha) B_0 & - \lambda_\alpha^2 D_+ + L_2 & A_0 k_- & 0 \cr\
0 & A_0 k_+ & - \lambda_\alpha^2 D_- + L_1 & i (\beta \lambda_\alpha) B_0 \cr\
A_0 k_+ & 0 & i (\beta \lambda_\alpha) B_0 & - \lambda_\alpha^2 D_+ + L_2
} \Psi_{\alpha\beta}(E) \cr
\qquad = E \Psi_{\alpha\beta}(E)
}$$ with $$\eqalign{
D_\pm = C_1 \mp M_1 \cr
L_1 = C_0 + M_0 + (C_2 + M_2) k_\parallel^2 \cr
L_2 = C_0 - M_0 + (C_2 - M_2) k_\parallel^2 .
}$$ The value of $\lambda_\alpha$ is be obtained in closed analytical form by solving Eq. (5), which gives [@zhou08; @shan10] $$\lambda_\alpha^2(E) = - \frac{F}{2 D_+ D_-} + \alpha \frac{\sqrt{R}}{2 D_+ D_-} ,$$ where $$\eqalign{
F = B_0^2 + D_+ (E - L_1) + D_- (E - L_2) \cr
R = F^2 - 4 D_+ D_- [(E-L_1) (E-L_2) - A_0^2 k_+ k_-] .
}$$ For each value of $\beta \lambda_\alpha(E)$, there are two linearly independent eigenvectors: $$\Psi_{\alpha\beta1}(E) =
\pmatrix{
E + \lambda_\alpha^2 D_+ - L_2 \cr - i (\beta \lambda_\alpha) B_0 \cr 0 \cr A_0 k_+
}
, \quad
\Psi_{\alpha\beta2}(E) =
\pmatrix{
A_0 k_- \cr 0 \cr i (\beta \lambda_\alpha) B_0 \cr E + \lambda_\alpha^2 D_- - L_1
} .
\label{eq:eigenvectors}$$ The full solution in each layer is composed of these eight eigenvectors. Eight boundary conditions then determine the relative magnitude of the eigenvectors (with the overall magnitude fixed by the normalization) and the energy $E$. The condition on $E$ follows from the determinant of the boundary conditions.
It remains to determine boundary conditions. First, we require $\Psi = 0$ at the two boundaries of the slab (eight boundary conditions). Second, at each interface, we impose the continuity of the wavefunction and the current (which gives four boundary conditions per interface). This can be written as $$\eqalign{
\Psi(-\varepsilon) = \Psi(+ \varepsilon) \\
{\cal M} \Psi(-\varepsilon) = {\cal M} \Psi(+ \varepsilon) ,
}$$ where the flux operator ${\cal M}$ is obtained by integrating the Hamiltonian across the interface $${\cal M} = \pmatrix{
(C_1+M_1) \frac{\partial}{\partial z} & i B_0 & 0 & 0 \cr
i B_0 & (C_1-M_1) \frac{\partial}{\partial z} & 0 & 0 \cr
0 & 0 & (C_1+M_1) \frac{\partial}{\partial z} & - i B_0 \cr
0 & 0 & - i B_0 & (C_1-M_1) \frac{\partial}{\partial z}
} .$$
FOUR-BAND PARAMETERS AND APPLICATION TO SGS TRI-LAYER
-----------------------------------------------------
Before presenting results for the SGS tri-layer structure, we consider a single Sb$_2$Te$_3$ (Rh) film to estimate the strength of the interlayer coupling. The single slab geometry has been solved in Refs. [@zhou08; @shan10; @linder09]. Results of this calculation using Sb$_2$Te$_3$ parameters are shown in Fig. \[fig:4\]. Figure \[fig:4a\] shows the position of the valence band maximum and the conduction band minimum at the $\Gamma$ point as a function of the slab thickness $L_s$ (orange lines). These states are surface states that are localized at the edge of the slab and gap out due to the intralayer coupling between the surface modes. The plot also includes the first bulk bands as blue lines. As for Bi$_2$Se$_3$, there is an oscillating exponential decay of the gap with increasing slab thickness $L_s$. The inset illustrated the band structure near the $\Gamma$ point for two values of the thickness $L_s = 5$ nm and $10$ nm, where the massive gapped surface Dirac states is indicated by the orange lines and bulk bands by blue lines. Figure \[fig:4b\] shows the gap as a function of slab thickness $L_s$. As is apparent from the figure, surface states on the left and right edges are sufficiently decoupled for $L_s = 15$ nm, at which point the energy gap caused by intralayer coupling between the surface states is negligible (less than 0.01 eV). Furthermore, as is apparent from Fig. \[fig:4a\], the upper and lower energies of the surface states at $\Gamma$ point are converged to $0.14$ eV below the Fermi level of Sb$_2$Te$_3$. The Sb$_2$Te$_3$ film is“bulk-like” when its thickness is larger than $15$ nm.
We now discuss the band structure of an SGS tri-layer structure that consist of two outer Sb$_2$Te$_3$ layers and an embedded GeTe middle layer. Figure \[fig:5a\] shows results for the band structure near the $\Gamma$ point computed as discussed in the previous section, where we choose an outer bulk-like Sb$_2$Te$_3$ layer of thickness $L_s=15$ nm, and an inner GeTe layer of thickness $L_g=5$ nm. While the surface states at the outer edges of the tri-layer are still localized at the slab energy of $-0.14$ eV, the inner Dirac states are shifted in energy, but no band gap opens at the $\Gamma$ point. Note that the numerical result is essentially unchanged for different values of the thickness $L_g$ of the inner GeTe layer, and a gap opening at the $\Gamma$ point is only observed for very small values $L_g < 2$ nm. This result suggests that (i) the interlayer coupling is completely suppressed even for moderate GeTe layers and (ii) that the perturbation of the GeTe wave function is sufficiently weak that the band structure of SG interfacial state is unchanged. This is a very reasonable result considering the large offset of the GeTe valence band maximum and conduction band minimum compared to the Sb$_2$Te$_3$ layer: The VBM and CBM of Sb$_2$Te$_3$ are located within the band gap of GeTe and there are no GeTe states in the energy range of surface states. Interlayer coupling of topological mode is suppressed.
As an alternative model, we examine an SGS structure in which the GeTe layer is replaced by GST225 (Kooi), the thickness of which $L_k$ is also fixed at $5$ nm. The results are summarized in Fig. \[fig:5b\]. Interestingly, and in contrast to the previous case, we find a band gap opening at the $\Gamma$ point. The band gap appears to close at points away from the $\Gamma$ point, but this closed gap should be considered as an accidental degeneracy of the wave functions originating from the band structure of Sb$_2$Te$_3$ and GST225(Kooi), rather than any topological modes.
FIRST-PRINCIPLES CALCULATIONS OF INTERFACIAL STATES IN SGS TRI-LAYER {#sec:IV}
====================================================================
The continuum model assumes that the Hamiltonian derived from the (homogeneous) bulk structure is applicable to the heterostructure junction. Although we mimicked the effect of physical intermixing of chemical species by introducing the GST225 layer, effects due to local electronic states and/or vacancies, which we refer to as “chemical interactions”, are not represented. While charge transfer or charge accumulation by impurities at the SG interface is expected to be sufficiently small (as the electron affinity of both Sb$_2$Te$_3$ and GeTe is strong), the robustness of any topological mode against local fields arising from chemical interactions effect is not clear. In order to address this, in the following, we extract interfacial states of various thin SGS tri-layers directly by using NEGF-DFT.
In our computational model of the tri-layer, the left- and right-hand side of Sb$_2$Te$_3$ is represented explicitly by $1\times1\times3$ unit cells, where the outermost cells are connected to the bulk by self-energy terms as discussed in Sec. \[sec:II2\]. Hence, different from the previous section, we can only consider the embedded interfacial state on the SG interface sides to analyze the topological mode, i.e., the intralayer-coupling is automatically eliminated. We consider three separate models with three separate structures for the NI part:\
(Model A): \[(Sb$_2$Te$_3$)$_9$\] /(GeTe)$_{3n}$/\[(Sb$_2$Te$_3$)$_9$\]
(Model B): \[(Sb$_2$Te$_3$)$_9$\]/(Ge$_2$Te$_2$)(GeTe)$_{3(n-2)}$(Ge$_2$Te$_2$) /\[(Sb$_2$Te$_3$)$_9$\]
(Model C): \[(Sb$_2$Te$_3$)$_9$\] /(GST225)$_m$(GeTe)$_{3(n-2)}$(GST225)$_m$\[(Sb$_2$Te$_3$)$_9$\]\
In our notation, \[(Sb$_2$Te$_3$)$_3$\]$_2$, for example, denotes a staking of the two (conventional hexagonal) unit cells of the Sb$_2$Te$_3$ crystal, i.e., it is a stacking of six Sb$_2$Te$_3$ quintuple monolayers (QLs). The intermediate layer in model A is a GeTe (dRk) layer. Here, the stacked numbers of (GeTe)$_3$ units, $n$, is taken as $n = 6$ (recall that a stacking of three GeTe monolayers is also the unit cell of the GeTe (dRK) bulk crystal in the conventional hexagonal cell). A change in the interface structure is taken into account in model (B), which contains a vacancy layer at the boundary of a GeTe (dRk) phase. Here, the label (Ge$_2$Te$_2$) represents a vacancy layer that consists of a single (GeTe)$_3$ block. Finally, in model (C) we introduce two unit cells of GST225 on either side of GeTe as an intermixing region, i.e., $m=2$. The outermost regions are connected to bulk Sb$_2$Te$_3$ (Rh). The 2D band dispersion of the interface is extracted by projecting the density of states (DOS) on the interface, and is exactly calculated from the Green’s function as a function of energy $E$ and wave vector $k_{||}$. We take $k$ to point along M-$\Gamma$-K line and the DOS was projected on each QL in the junction. We present the projected band structure at three separate positions: (a) the Sb$_2$Te$_3$-QL closest to the SG interface plane, (b) the secondary neighboring QL, and (c) the third QL in order to analyze the localization of topological mode. We labeled the above the three QLs as QL$^{(1)}$, QL$^{(2)}$, and QL$^{(3)}$, respectively. In our notation, \[(Sb$_2$Te$_3$)$_3$\]$_2$, for example, denotes a staking of the two (conventional hexagonal) unit cells of the Sb$_2$Te$_3$ crystal, i.e., it is a stacking of six Sb$_2$Te$_3$ quintuple monolayers (QLs). The intermediate layer in model A is a GeTe (dRk) layer. Here, the stacked numbers of (GeTe)$_3$ units, $n$, is taken as $n = 6$ (recall that a stacking of three GeTe monolayers is also the unit cell of the GeTe (dRK) bulk crystal in the conventional hexagonal cell). A change in the interface structure is taken into account in model (B), which contains a vacancy layer at the boundary of a GeTe (dRk) phase. Here, the label (Ge$_2$Te$_2$) represents a vacancy layer that consists of a single (GeTe)$_3$ block. Finally, in model (C) we introduce two unit cells of GST225 on either side of GeTe as an intermixing region, i.e., $m=2$. The outermost regions are connected to bulk Sb$_2$Te$_3$ (Rh). The 2D band dispersion of the interface is extracted by projecting the density of states (DOS) on the interface, and is exactly calculated from the Green’s function as a function of energy $E$ and wave vector $k_{||}$. We take $k$ to point along M-$\Gamma$-K line and the DOS was projected on each QL in the junction. We present the projected band structure at three separate positions: (a) the Sb$_2$Te$_3$-QL closest to the SG interface plane, (b) the secondary neighboring QL, and (c) the third QL in order to analyze the localization of topological mode. We labeled the above the three QLs as QL$^{(1)}$, QL$^{(2)}$, and QL$^{(3)}$, respectively.
In models (A) and (B), the extracted 2D band structure on QL$^{(3)}$ is very similar to that of bulk Sb$_2$Te$_3$ (Rh) as given in Fig. \[fig:5a\]. Although a very weak spectral density coming from the DOS of QL$^{(2)}$ is found, the electronic state in QL$^{(3)}$ is essentially a bulk state. In contrast, the projected band dispersion of QL$^{(1)}$ is more complicated and shows strong hybridization with the states of GeTe, i.e., the Sb$_2$Te$_3$ layer immediately adjacent to GeTe is strongly perturbed by chemical interactions. In the QL$^{(2)}$ of model (B), we find a clear Dirac cone similar to the clean Sb$_2$Te$_3$ surface state. However, for model (A), we observe a Rashba-type split band rather than topological mode. These results lead to the following conclusions: first, the interfacial state characterized as the “surface” state of Sb$_2$Te$_3$ can be localized narrowly in the secondary neighboring Sb$_2$Te$_3$ monolayer in the junction; and second, the topological mode is not robust to chemical interaction even when the band offset is sufficiently large. By comparing models (A) and (B), the existence of a vacancy layer at the SG interface induces a significant chemical interaction effect. The local electric field due to a (GeTe)$_3$ block in the SG interface works as a built-in asymmetric external field and gives rise to a Rashba-type interfacial state even though the topological mode is protected by GeTe block containing a vacancy layer, i.e., Ge$_2$Te$_2$. Interestingly, in contrast to model (A) and model (B), we found that 2D band on QL$^{(3)}$ of model (C) does not converge to that of bulk. The presence of GST225 opens large band gap and the band dispersion is more NI-like near the $\Gamma$ point. This result is also found in the continuum model, and thus the interlayer coupling of the wave function across GST225 is more long-ranged than in the cases with GeTe or vacancy layer + GeTe.
Model (C) represents a somewhat more realistic intermixing at the SG interface than the continuum model, i.e, GST225 is only narrow sublayer of NI as (GST225)$_m$\[(GeTe)3\]$_{(n-2)}$(GST225)$_m$, where $(m,n) = (2,6)$. According to the results of model (A) and (B), the above GeTe block is sufficiently thick to suppress interlayer coupling of topological modes from the two Sb$_2$Te$_3$ slabs. In other words, the large band gap at $\Gamma$ is a result of local interaction (and hybridization) of each topological mode and electronic state narrowly localized on GST225. However, if the intermixing region of GST225 is sufficiently thick in the normal insulator part, the topological modes can be made to couple by tuning “bulk-like” thickness of the NI in the SGS tri-layer. This offers a very plausible explanation for the results observed by Nguyen *et al.* [@nguyen16] since, experimentally, we find clear evidence of an intermixing region even in bulk-like SGS heterostructures (Figs. 1 and \[APT\]).
\
\
CONCLUSIONS
===========
In this paper, we have studied the band structure of bulk-like heterostructures of Sb$_2$Te$_2$ and GeTe as a prototypical TI-NI system. We have focused on the interfacial region between these materials and shed light on the conditions under which the topological mode may or may not be present. We have laid particular emphasis on understanding realistic structures in which the interface is not perfect, but rather contains an intermixed phase of the parent compounds Sb$_2$Te$_3$ and GeTe. Our principal finding is that the presence of this intermediate phase serves to enhance the length over which topological modes may interact with each other. Importantly, we confirm experimentally that SGS heterostructures show a degree of intermixing, thereby underlining the relevance of our findings to the existing experimental literature on SGS systems. Of particular importance in this context is the experimental work by Nguyen *et al.* [@nguyen16] which observes unexpectedly long-ranged interactions between topological modes in an SGS heterostructure. Our findings provide a natural explanation for these results and lay the foundation for future work wherein superlattices of bulk-like Sb$_2$Te$_3$ and GeTe layers can be used to deterministically produce different topological phases [@burkov11].
This work was supported by EPSRC, UK, CREST, JST (Grant No. JPMJCR14F1) and Peterhouse, Cambridge.
References {#references .unnumbered}
==========
|
---
abstract: |
A new class of stochastic variables, governed by a specifice set of rules, is introduced. These rules force them to loose some properties usually assumed for this kind of variables. We demonstrate that stochastic processes driven by these random sources must be described using an probability amplitude formalism in a close resemblance to Quantum Theory. This fact shows, for the first time, that probability amplitudes are a general concept and is not exclusive to the formalism of Quantum Theory. Application of these rules to a noisy, one-dimensional motion, leds to a probability structure homomorphic to Quantum Mechanics.
Stochastic Models, Game Theory, Quantum Theory
author:
- |
J. M. A. Figueiredo\
Universidade Federal de Minas Gerais - Dept. de Física\
Caixa Postal 702 - Belo Horizonte - 30.123-970\
Brazil\
email: josef@fisica.ufmg.br
title: Games with Quantum Analogues
---
The understanding that noise drives a wide variety of phenomena in Nature brought to many branches of Science the conviction that randomness is a fundamental property that must be considered in any real modeling of the physical world. In fact, all theories yet developed that include noise as a element driving an otherwise deterministic process still remain making use of this concept even when specific changes on its mathematical modeling may eventually be refined. The only exception is Quantum Theory, which makes use of probabilistic tools without specific reference to any noise source. In its early developments Einstein, Hopf and Stern [@miloni] tried to explain some electrodynamic phenomena using a noisy electromagnetic field, in addition to the usual determinist one, in the dynamical equations describing particle motion. However, later developments of the Quantum Theory led to a noiseless model, that acts as a noise source but not depending, on its fundamental epistemological components, of a noise source. Despite this very important exception we can assert, without doubt, that stochastic physical theories are epistemologically robust and confident.
It is clear, from the above arguments, that two general classes of probability theories exist. The classical ones makes use of probabilities or probability densities and is used to describe stochastic processes. On the other hand quantum probabilities must be determined from a probability amplitude and do not describe a stochastic process. In fact quantum probability is considered a intrinsic property of Nature; this way it is strongly believed that probability amplitudes are exclusive to quantum phenomena which drives any other macroscopic manifestation of physical phenomena. No other manifestation of statistical phenomena, in any field, may have, in principle, the characteristics of the quantum probability. That is the point we wanto to discuss in this work. We present a new model for the random variable itself thus affecting all existing stochastic theories whenever the novelties introduced here are applicable. Stochastic processes driven by this kind of variable make use of a probability amplitude formalism even thought no reference to quantum phenomena is made.
Usual stochastic processes present two main components: a determinist law plus an additional stochastic term depending on the value of a random variable. This variable assumes its values in a unpredictable way but once its instantaneous fluctuating value is defined the system responds to it in a predicable way. No doubt concerning the ”reading” process is introduced. A causal relation exists between the actual value assumed by the random variable and the one in fact used by the system in its deterministic response to noise. This reading process then has a property, borrowed from similar concept, usual in Quantum Theory, we call ”realism” because the actual value of the random variable may, at least in principle, be known.
We want to deny the necessity of realism in a stochastic theory by introducing the concept of *incomplete* random variable. A system responds to this kind of variable in a two steps process. In the first one the variable fluctuates in the usual way but its value is not readily available to the system that must read it. In this second step some errors may occur. Two types of errors will be considered here. They are as defined by the Q-rules set postulated below:
- rule one: the fluctuating value is not read; the fluctuation is lost. System’s state is frozen until the next round. A clear breakdown of causality is introduced at this point.
- rule two: some readings are wrong. Another value for the fluctuating variable, not the real one, is used by the system to drive its dynamics. The value is lost but not the reading.
Before a explicitly mathematical formalization of these rules it is worth to stress that all them are algorithmic even though they break causality and realism. In fact, we can’t assert to an incomplete random variable any criteria of realism because it is impossible to know its exact value; at the same time the relation between the system and the noise source is not causal because nobody can guarantee system’s dynamics will be dictated by a given fluctuation. In some sense, however, reality cannot be completely discarded because fluctuations itself are real. Notwithstanding, realism, as defined above, definitively does not exist. More formally, let us consider a discrete random variable $Y$ that can have $M$ values, and its associated probability distribution $P\left( y\right) $. For definiteness assume that $P\left(
y_{j}\right) >0$ for all $j\in\left\{ 1,2,...M\right\} $. If $Y$ is a incomplete random variable then its value is not directly accessible to the system. The actual histogram for $Y$ is not any more given by $P\left(
y\right) $. Instead, a modified one that takes into account the Q-rules must be used. This modified probability distribution, the non-classical one, will be written as $\textsl{p}_{j}=P_{j}+C_{j}$ where $P_{j}\equiv P\left(
y_{j}\right) $ and $C_{j}$ is the non-classical term. This term has contributions from both Q-rules. From the first one we have the probability of lost readings given by $\gamma_{j}P_{j}$. Thus, $P_{j}\left( 1-\gamma
_{j}\right) $ is the probability of successful readings. Two kinds of contributions arise from rule two. The probability of reading state $j$ may be enhanced by a wrong reading of another state; on the other hand this probability may be decreased by a wrong reading that converts state $j$ to another one. If $\Gamma_{j,l}$ $\left( j\neq l\right) $ is the probability that a realization giving the value $y_{l}$ be read as $y_{j}$ then we write rule two as $\sum_{l=1,l\neq j}^{M}\left( \Gamma_{j,l}P_{l}-\Gamma_{l,j}P_{j}\right) $. Defining $\tilde{C}_{j,l}\equiv\Gamma_{j,l}P_{l}-\left(
\Gamma_{l,j}+\frac{\gamma_{j}}{M-1}\right) P_{j}$ $\left( j\neq l\right) $ we get the histogram effectively read for variable $Y $$$\textsl{p}_{j}=P_{j}+\sum_{l=1,l\neq j}^{M}\tilde{C}_{j,l} \label{Q-histog}$$ Observe that the non-classical terms $\tilde{C}_{j,l}$ may have any sign; consequently they cannot be seen as a truly probability term. Only the non classical distribution $\textsl{p}$, that possesses the desired positiveness, is accessible to the system. That means we cannot write $p$ as the sum of two independent process, one driven by $P$ and the other by the non-classical terms. At this point we explicitly break realism because the observed histogram corresponds to $p$, not to $P$. So we cannot directly measure or observe the noise source. This way we see that incomplete random variables present a simple algorithm in their realization, although their conception are more complicated than the usual ones. For instance, consider a roulette with $M$ stops and an imperfect camera that reads pointer’s position at each round. The roulette drives a game but pointer’s position is only accessible to players through the camera that may generate wrong readings (or no readings at all), according to the Q-rules. The roulette is real but this reality is only conceptual, since the actual value used in the game comes from camera readings.
We are now in position to study a stochastic process driven by a incomplete random variable. Let us initially consider a discrete time process (a game) [@note], of the following general form $$f\left( x_{n+1},x_{n}\right) =g\left( x_{n}\right) w_{n} \label{game}$$ where $W$ is the (uncorrelated) stochastic variable and $\varphi\left(
w\right) $ its associated probability distribution. For $g\left( x\right)
=0$ the game evolves according a deterministic rule, defining player’s strategy. A noise source includes a chance ingredient, that makes best strategy choices non-trivial. We regard eqn$\left( \ref{game}\right) $ as defining the value the stochastic variable $x_{n+1}$ will take at round $n+1$, considering that it takes the value $x_{n}$ at round $n$. The joint probability density $\Pi\left( x_{N},x_{N-1},....,x_{1}\right) $ completely describe this process and for a given sampling sequence of $W$ it is given by [@kampen]: $$\Pi\left( x_{N},x_{N-1},....,x_{1}\right) =\int\prod_{n=1}^{N}\delta\left(
f\left( x_{n+1},x_{n}\right) -g\left( x_{n}\right) w_{n}\right)
\varphi\left( w_{n}\right) dw_{n}$$ which in the discretized model assumes the form $$\Pi\left( x_{N},x_{N-1},....,x_{1}\right) =\sum_{j1=1}...\sum_{jN=1}\prod_{n=1}^{N}\delta\left( f\left( x_{n+1},x_{n}\right) -g\left(
x_{n}\right) w_{jn}\right) \varphi\left( w_{jn}\right) \label{paths}$$ The product $\prod_{n=1}^{N}\varphi\left( w_{jn}\right) $ defines a ordered sampling sequence $\left\{ w_{j1}...w_{jN}\right\} $ of the bare random variable $W$ which we will call a ”path”; it is easy to see that in the above summation there are $M^{N}$ paths. Path concept in stochastic processes was used initially by Onsager and Machlup [@onsager] in order to describe the effect of fluctuations on non-equilibrium systems. Later Graham [@graham] developed a more general formalism to include sophisticated dynamics satisfying the Liouville equation which successfully allowed works in systems subject to colored noise [@colored]. We shall adopt eqn$\left(
\ref{paths}\right) $ as a strong support to later developments in the present work.
Now it is natural to introduce the effects of the Q-rules on stochastic processes by imposing that $W$ be an incomplete random variable $Y$ and studying how incompleteness affects eqn$\left( \ref{paths}\right) $. Histogram of $Y$ is given by eqn$\left( \ref{Q-histog}\right) $ and at round $n$ it assumes the value $y_{jn}$. Then to any path there is an associated product of the form $$\prod_{n=1}^{N}\textsl{p}\left( y_{jn}\right) =\prod_{n=1}^{N}\left(
P_{jn}+\sum_{l=1,l\neq j}^{M}\tilde{C}_{jn,l}\right) \label{qpaths}$$ which, for a incomplete variable, branches in a sum having $M^{N}$ terms, as seen from the left side of the above equation. There are $M^{N}$ paths so the sum over all paths, involved in eqn$\left( \ref{paths}\right) $, has $M^{2N}$ terms. The set of all paths, $\Xi$, may be represented as the union of two disjoint subsets: $\Xi=\Omega\bigcup\Lambda$ where $\Omega$ is the set of all paths containing only $P$-terms and $\Lambda$ contains elements with at least one non-classical contribution. Thus $\Omega$ consists in the set of all ”classical” paths, those valid for a complete random variable, and has $M^{N}$ elements.
In sequence we make a simplification in the model, assuming that $$\tilde{C}_{j,l}=\tilde{C}_{l,j}\Longrightarrow\tilde{C}_{j,l}=-\frac
{\gamma_{j}P_{j}+\gamma_{l}P_{l}}{2\left( M-1\right) }\leq0 \label{symm}$$ This means that vacuum losses $\left( \gamma\right) $ completely dominate the chance of non-classical effects during the reading process. The number of elements in $\Lambda$ is reduced due to this symmetry as shown below. Writing $M^{2N}=\left( M^{2}-M+M\right) ^{N}$ we get $$M^{2N}=\sum_{l=0}^{N}\binom{N}{l}M^{\left( N-l\right) }\left(
M^{2}-M\right) ^{l}$$ showing that there are $\binom{N}{l}M^{\left( N-l\right) }\left(
M^{2}-M\right) ^{l}$paths with $l$ non-classical contributions. The symmetry expressed by eqn$\left( \ref{symm}\right) $ demands this number be reduced to $\binom{N}{l}M^{\left( N-l\right) }\left( \frac{M^{2}-M}{2}\right)
^{l}$ terms, generating a total number of independent non-classical paths equal to $$\sum_{l=1}^{N}\binom{N}{l}M^{\left( N-l\right) }\left( \frac{M^{2}-M}{2}\right) ^{l}=\left( \frac{M^{2}+M}{2}\right) ^{N}-M^{N}$$ caused by the existence of twin paths, those having labels in the non-classical contributions exchanged. This result reduces the number of elements in $\Xi$ to $\left( \frac{M^{2}+M}{2}\right) ^{N}$.
We will now show that classical paths may index all non-classical ones as well, in a very specific way. To this end we rewrite $$\tilde{C}_{j,l}=-\frac{\sqrt{P_{l}P_{j}}}{2\left( M-1\right) }\left(
\gamma_{l}\sqrt{\frac{P_{l}}{P_{j}}}+\gamma_{j}\sqrt{\frac{P_{j}}{P_{l}}}\right) \equiv\sqrt{P_{j}}\sqrt{P_{l}}d_{j,l} \label{symm1}$$ and $P_{j}=\sqrt{P_{j}}\sqrt{P_{j}}$, in such a way that each non-classical path has the structure of a product of $2N$ factors of the type $\sqrt{P}$, followed by a product of $d^{\prime}s$, in the same number of the existing non-classical terms in the considered path. Note that classical paths can also be rewritten as a sequence of type $\sqrt{P}$ having, obviously, no $d$-terms. Since all $d\acute{}s$ are negative the value of a non-classical path may have any sign. In considering this type of sequences, another marked difference between elements in $\Omega$ and $\Lambda$ is that each non-classical contribution has necessarily cross terms like $\sqrt{P_{j}}\sqrt{P_{l}}$, with $l\neq j$. However, note that non-classical paths may have classical-like segments with the same structure of elements in $\Omega$. We call *radix* $\left(
R\right) $ of a sequence its non-$d$ terms part. That is, the radix is a sequence of terms like $\sqrt{P_{j}}\sqrt{P_{l}}$ (for all $l$ and $j$). Consequently all elements in $\Lambda$ have the structure $R\prod d$. The product runs over the number $L_{path}$ of non-classical terms in the considered sequence. Thus we define an auxiliary set $\Delta$ composed by all possible radices. It’s easy to see that $\Omega$ is a subset of $\Delta$ which have the same number of elements of $\Xi$. In what follows we shall use these facts in order to construct a unified structure for that set.
Let us consider the set $\tilde{H}$ of all classical-like paths of type $\sqrt{P}$; that is, paths in $\tilde{H}$ are just those sequences in $\Omega$ where $P^{\prime}s$ are substituted by the corresponding $\sqrt{P^{\prime}s}$. Thus each element $R_{\tilde{H}}$ in $\tilde{H}$ may be written as $R_{\tilde{H}}=\sqrt{R_{\Omega}}$, where $R_{\Omega}$ is the corresponding path in $\Omega$. On the other hand since $\Delta$ consists of all combinations of ordered sequences of $\sqrt{P}$-type we may write each element there as a cross-product of some elements in $\tilde{H}$. That is, we always can write each element $R_{\Delta}$ in $\Delta$ as $R_{\Delta}=R_{\tilde{H}}R_{\tilde{H}}^{\prime}$ for two carefully chosen elements in $\tilde{H}$. As a result we write the important result that $\Delta=\tilde{H}\otimes\tilde{H}$. Now we define an extended set $H$ by assigning to each element in $\tilde{H}$ a “phase”, a complex number $\exp(i\varphi_{\Omega})$, in such a way that the $M^{N}$ elements in $H$ have the general form $S_{H}=R_{\tilde
{H}}\exp(i\varphi_{\Omega})=\sqrt{R_{\Omega}}\exp(i\varphi_{\Omega})$. Consequently, since $S_{H}\left( S_{H}\right) ^{\ast}=R_{\Omega}$, we see that $\Omega\supset H\otimes H^{\ast}$. On the other hand the radix of an element in $H$ is just the corresponding element in $\tilde{H}$. This way we see that elements of $H\otimes H^{\ast}$ have the general form $R_{\Delta}\exp(i\left( \varphi_{\Omega}-\varphi_{\Omega^{\acute{}}}\right) )$ or $R_{\Delta}\exp(-i\left( \varphi_{\Omega}-\varphi_{\Omega^{\acute{}}}\right) )$. The sum $\tilde{\Xi}=$ $\sum H\otimes H^{\ast}$ has $M^{N}+M^{N}\left( M^{N}-1\right) /2$ terms (all them real numbers); out of these there are $M^{N}$ terms having no phase contribution. They correspond to the classical paths belonging to $\Omega$; the remaining ones can be collected into $\left( \frac{M^{2}+M}{2}\right) ^{N}-M^{N}$ independent radices $R_{\Delta}$ times a linar combination of phase terms, each one expressed as a $2\cos\left( \varphi_{\Omega}-\varphi_{\Omega^{\acute{}}}\right) $. This last term is symmetric under exchange of argument indices, in as much as eqn$\left( \ref{symm1}\right) $is. Hence the same argument used to count the twin paths is applicable. This reduces the number of terms in $\tilde{\Xi}$ to $\left( \frac{M^{2}+M}{2}\right) ^{N}$ which is the same number as the sum over all paths in $\Xi$. The above results allow us to write the main result of this work: $$\sum_{\Xi}=\left| \sum_{H}\right| ^{2} \label{summ}$$ valid if we make the association $$\sum_{L_{path}}\prod^{L_{ph}}d_{ji}\equiv\sum_{L_{path}}\prod^{L_{ph}}\cos\left( \varphi_{i}-\varphi_{j}\right) \label{coss}$$ where $i$ and $j$ are classical paths used to index the corresponding common radix $R_{\Delta}$ of elements in $\tilde{H}$. That is the sum over all paths in $\Xi$ may be written as a squared sum over all paths in $H$. In analogy to quantum theory we call amplitudes the elements in $H$. Thus, in order to sum up the right result in $\Xi$ sample paths in $H$ (which consists of classical paths for the amplitudes) must ”interfere”; that is, in this space, paths, not single realizations of the random variable, are the basic objects. Notice that the phase of a path is a non-local object in the sense we cannot assign to it a single specific process, once it is defined for combinations of $d$-terms. In fact, even when $Y$ is uncorrelated, the phase depends on the whole sequence in a path. In this space realism breaks down since amplitudes cannot be observed in any particular realization of $Y$. In the same way it will be shown below that causality is not present as well. Therefore, in the context of this work probability amplitudes describe in a unified way the whole effects of incomplete random variables in such way that interference of these amplitudes means that cross-effects on probabilities due to the Q-rules are relevant.
In order to obtain values for the $M^{N}$ phases we expand them in a (truncated) path-dependent power series, each one having a number of terms equal to the integer part of $\left[ \left( M+1\right) /2\right] ^{N}$, as follows $$\varphi_{l}=\varphi_{N}+\sum_{nl=1}^{N}A_{nl}y_{nl}+\sum_{nl=1}^{N}\sum
_{ml=1}^{N}\tilde{B}_{nl}^{ml}y_{nl}y_{ml}+... \label{series}$$ where $l\in\left[ 1,M^{N}\right] $ and $\varphi_{N}$ is zero if $M^{N}$ is odd. The sequence of values the random variable takes in a given path may be rescaled by incorporation of the linear coefficients $\left\{ A_{nl}\right\}
$ once a similar renormalization in the high order coefficients is consistently performed. This way we get a new set of (path-dependent) values for $Y$. This new set is obtained by linearization of eqn$\left(
\ref{coss}\right) $ and does not represent any restriction on the formulation of the problem since, as seen from eqn$\left( \ref{paths}\right) $, their elements are dummy variables. As a result it is possible to rewrite eqn$\left( \ref{series}\right) $ as $$\varphi_{l}=\varphi_{N}+\sum_{nl=1}^{N}\tilde{y}_{nl}+\sum_{nl=1}^{N}\sum_{ml=1}^{N}B_{nl}^{ml}\tilde{y}_{nl}\tilde{y}_{ml}+.. \label{series1}$$ where now it is implicit that the set of sampling variables $\tilde{y}$ is path-dependent. For each path we have chosen the truncation schema carefully by collecting exactly $\left[ \left( M+1\right) /2\right] ^{N}$ coefficients in the above series. This sums up to a set of $M^{N}\left[
\left( M+1\right) /2\right] ^{N}$ elements in the whole set $H$. Then just make use of the same number of equations displayed in eqn$\left(
\ref{coss}\right) $ to solve for these coefficients, completing this way the construction of $H$. Now we have at hand a space consisting of classical paths, homomorphic to $\Omega$, but whose elements are probability amplitudes. These amplitudes are the mathematical objects used to treat incomplete random variables.
To the phase of a path it is not possible to assign any single realization of the stochastic variable. However it is possible to assign a phase to a segment of a path although single-event association still remains invalid. To define the phase of a segment we rewrite eqn$\left( \ref{series1}\right) $ as a sum of $N$ terms: $$\varphi_{l}=\sum_{n=1}^{N}\left[ \frac{\varphi_{Nl}}{N}+y_{nl}\left(
1+\sum_{m=1}^{N}B_{nl}^{ml}y_{ml}+...\right) \right] \equiv\sum_{n=1}^{N}\phi_{nl} \label{phase}$$ allowing us to write an element in $H$ as $S_{H}=\prod_{n=1}^{N}\sqrt{P_{jn}}e^{i\phi_{nl}}$. That is, incomplete random variables must be described by a probability amplitude $$U\left( j,n,N\right) \equiv\sqrt{P_{j}}e^{i\varphi\left( j,n,N\right) }
\label{amplit}$$ in such a way that $\left| U\left( j,n,N\right) \right| ^{2}=P_{j}$. The phase of this probability amplitude is process-dependent and cannot be observed in any single realization of the variable, which always results on its effective histogram $p$. In this case we say that the bare distribution $P$ is *hidden*. Furthermore, the phase value, for a given path, depends on all realizations of $y$ that closes the considered path, including those chosen at future rounds (in the considered path). This property breaks down causality in the phase definition for individual realizations in the sense that future rounds define the present. However this fact does not configure a violation of causality for the whole process once the paths, which form the basic blocks in constructing the transition amplitudes, are causal.
Returning back to the continuum, we must then describe an incomplete random variable $y$ by a probability amplitude $U\left( y,n,N\right) =\sqrt
{P\left( y\right) }e^{i\varphi\left( y,n,N\right) }$ that must be considered in any stochastic process it drives. We cannot measure the phase of a probability amplitude in any single event measurement; thus it is not possible to assign to it an element of reality. This is the main difference from usual (classic) stochastic variables that are completely described by their bare probability distribution $P\left( y\right) $. Now we are in position of analyze how the Q-rules affect a specific game.
To do this let us consider the process defined in eqn$\left( \ref{game}\right) $. For any given initial state $E\left( x_{0}\right) $, the probability distribution after $N$ rounds is [@gardner] $$E\left( x_{N}\mid x_{0}\right) =\int E\left( x_{0}\right) \prod_{n=0}^{N}\delta\left( f\left( x_{n+1},x_{n}\right) -y_{n}\right) P\left(
y_{n}\right) dy_{n}dx_{n}$$ but as written this equation describes the classical version of $Y$. If the Q-rules apply up, the phase of the probability distribution should be considered. So, as it stands this equation cannot be used unless we do make the substitution $P\rightarrow p$; we also perform an integration over the random variable $y$ that results in $$E\left( x_{N}\right) =\int E\left( x_{0}\right) \prod_{n=0}^{N-1}p\left(
f\left( x_{n+1},x_{n}\right) \right) dx_{n}$$ which is recognized as the Chapman-Kolmogorov equation for a Markov process having transition probabilities given by $p\left( f\left( x_{n+1},x_{n}\right) \right) $. After discretization this equation seems to have the same path topology of eqns$\left( \ref{paths}\right) $ and $\left(
\ref{qpaths}\right) $; however, an additional sum over the state at $n=0$ is present. It can naturally be inserted in the paths if we define $E\left(
x_{0}\right) \equiv\left| \psi\left( x_{0}\right) \right| ^{2}$ and use eqn$\left( \ref{summ}\right) $ to get an analogous expression for the probability distribution at time $N$ given by $E\left( x_{N}\right) =\left|
\psi\left( x_{N}\right) \right| ^{2}$, where $$\psi\left( x_{N}\right) =\sum_{H}\sqrt{P\left( f\left( x_{n+1},x_{n}\right) \right) }e^{i\varphi\left( path\right) }\psi\left(
x_{0}\right) \label{feymman}$$ which, after recovering the continuum, may be written as $$\psi\left( x_{N}\right) =\int\psi\left( x_{0}\right) \prod_{n=0}^{N-1}\sqrt{P\left( f\left( x_{n+1},x_{n}\right) \right) }e^{i\varphi
\left( path\right) }dx_{n} \label{feymman1}$$ Notice that now the bare probability $P$ is used in place of $p$ as prescribed by eqn$\left( \ref{amplit}\right) $. We arrived at a stochastic version of the Feynman-Kac formula, generalized to any kind of (one-player) game. Note that it is possible to control intensities of each kind of bare process altering both the radix $\left( \sqrt{P}\right) $ and the phase $\left(
d^{\prime}s\right) $ of a given path in the above cited equation.
We can use this equation in any Markovian process satisfying eqn$\left(
\ref{game}\right) $ driven by incomplete random variables. Since the set of rules defining this kind of variables are algorithmic, a new generation of games may be defined and constructed, impinging new challenges to Game Theory. These new games are not the same as the incomplete games [@game], those where players don’t know about the decisions the others have taken. The use of incomplete variables in a game makes the reading process imperfect but this limitation does not deny the knowledge of the readings be shared by all players. However, it may occur that players have their own reading apparatus. In this case we have a truly incomplete non-classical game. In this case decision theory and best strategy modeling must take into account the incompleteness of the random source. The possibilities opened by this approach should improve our comprehension about algorithmic probability amplitude effects through modeling and implementation of Q-rules in any specific game. Because we make use of probability amplitude formalism in these games it is reasonable to call them Quantum Games. In this work we do not develop further any general discussion about the consequences of the concepts presented here on Game Theory. Our main interest here was solely to show the existence of this class of games and how the probability amplitude formalism may arise in a truly stochastic theory. However, it would be interesting to analyze some real physical game since a similarity of eqn$\left( \ref{feymman1}\right) $ to path integrals in Quantum Theory is unavoidable. To this end we present below an important sample-game describing the motion of a classical particle subject to a noisy environment, which is very similar to those processes considered in ref [@colored]. Then we will show that a probability amplitude for this game can be constructed in a way that resembles the properties of a quantum particle. A remarkable point is the evidence shown here that probability amplitude effects are not exclusive of Quantum Theory neither a mysterious fundamental working mechanism in the Universe. In fact this formalism is now trivial once infinity types of stochastic processes may use it where some basic, algorithmic rules are the fundamental assumptions, not the use of the probability amplitude formalism itself. This fact changes naturally our attention in direction to Quantum Mechanics in order to look for a more fundamental phenomena that justify its basics axioms. We shall now prove that in a restricted sense this possibility may be real.
Let us consider the one-dimensional motion modeled by a particle of mass $m$ subject to a conservative force field plus the random effects of a vacuum field. At the present stage of our reasoning what is matter is not the physical origin of these random effects, but the way they change the particle motion. The immediate impact falls over particle’s energy whose fluctuations are driven by this vacuum field. We describe this effect as $$H\left( x\left( t\right) ,\dot{x}\left( t\right) \right) =E_{0}+\mathcal{E}y\left( t\right)$$ where $\mathcal{E}$ is the noise source intensity and $y\left( t\right) $ is the realization of a (dimensionless, zero mean) incomplete random variable $Y$. Time discretization of this equation, with $\Delta t\equiv\varepsilon$, leads to $$\frac{m\left( x_{n+1}-x_{n}\right) ^{2}}{2\varepsilon^{2}}+V\left(
x_{n}\right) -E_{0}=\mathcal{E}y_{n} \label{hamiltonian}$$ We are interested in a high noise intensity limit given by $\mathcal{E}\rightarrow\infty$, but subject to the condition that $\mathcal{E}\left\langle
y^{2}\right\rangle $ is finite. More specifically, we write $\mathcal{E}\equiv\alpha/\epsilon$ in such a way that $\left\langle H-E_{0}\right\rangle
=\mathcal{E}\left\langle y\right\rangle =0$ and $$\sqrt{\left\langle \left( H-E_{0}\right) ^{2}\right\rangle }=\alpha
\frac{\sqrt{\left\langle y^{2}\right\rangle }}{\varepsilon}\equiv\frac{\alpha
}{\tau} \label{time-energy}$$ leading to uncertainty in the energy given by $\left( \Delta E\right)
\tau\sim\alpha$. The number $\tau$ measures the characteristic time fluctuations taking place during system evolution. Therefore, it must be process-dependent. Its finiteness demands the limit of small fluctuations for the variable $Y$. In order to use eqn$\left( \ref{feymman1}\right) $ for the probability amplitude associated to this process we use normalized volume integrals in this equation; thus, for fixed $\varepsilon$ , we do make the substitution $dx\rightarrow\sqrt{\frac{m}{\varepsilon\alpha}}dx$ resulting in the following expression for probability amplitudes $$\psi\left( x_{N}\right) =\left( \frac{m}{\varepsilon\alpha}\right)
^{N/2}\int\psi\left( x_{0}\right) \prod_{n=0}^{N-1}\sqrt{P\left(
\varepsilon\frac{\frac{m\left( x_{n+1}-x_{n}\right) ^{2}}{2\varepsilon^{2}}+V-E_{0}}{\alpha}\right) }e^{i\varphi\left( path\right) }dx_{n}
\label{amplitude}$$ Considering the very small values of the random variable $Y$ we may limit ourselves to first order terms in the expansion for the phase, so we have $\varphi_{l}\left( y_{0},....y_{N}\right) =\varphi_{Nl}+\sum_{path}y+\mathcal{O}\left( y^{2}\right) $. We also scale energy reference level choosing $E_{0}$ in such a way that $\sum\varphi_{Nl}=E_{0}/\mathcal{E}$, so eqn$\left( \ref{amplitude}\right) $ leads to $$\psi\left( x_{N}\right) =\left( \frac{m}{\varepsilon\alpha}\right)
^{N/2}\int\psi\left( x_{0}\right) \exp\left( \frac{i\epsilon\sum_{path}H}{\alpha}+\mathcal{O}\left( \epsilon^{2}\right) \right) \prod_{n=0}^{N-1}\sqrt{P\left( \varepsilon\frac{\left( H-E_{0}\right) }{\alpha
}\right) }dx_{n}$$ This linearization procedure hiddes the particular choice of the non-classical therms present in the power series expansion for the phase. A more convenient form for this equation comes out in the phase space representation, obtained by the use of the following expression $$\exp\left( i\frac{m\left( x_{n+1}-x_{n}\right) ^{2}}{2\varepsilon\alpha
}\right) =\sqrt{\frac{i\varepsilon}{2\pi m\alpha}}\int\exp\left(
\frac{i\varepsilon}{\alpha}\left( \frac{p_{n}^{2}}{2m}-\frac{\left(
x_{n+1}-x_{n}\right) }{\varepsilon}p_{n}\right) \right) dp_{n}
\label{Heisenb}$$ which after insertion on the equation for $\psi\left( x_{N}\right) $ and discarding second order terms in $\epsilon$ results in the following naive expression $$\psi\left( x_{N}\right) =\left( \frac{i}{2\pi}\right) ^{N/2}\int
\psi\left( x_{0}\right) \exp\left( \frac{-i}{\alpha}\sum_{path}\epsilon\mathcal{L}\right) \prod_{n=0}^{N-1}\sqrt{P\left( \varepsilon
\frac{\left( H-E_{0}\right) }{\alpha}\right) }\frac{dx_{n}dp_{n}}{\alpha}
\label{FKac}$$ where $-\mathcal{L\equiv}\frac{p_{n}^{2}}{2m}+V\left( x_{n}\right)
-\frac{\left( x_{n+1}-x_{n}\right) }{\varepsilon}p_{n}=H-p\dot{x}$ has to be interpreted as the (”phase space”) particle’s Lagrangean. Phase space Lagrangean has a dual interpretation: the kinetic energy term is partially determined by the velocity $\frac{\left( x_{n+1}-x_{n}\right) }{\varepsilon
}$ and partially by the independent momentum variable $p$. Application of Euler-Lagrange equation to this Lagrangean selects the classical path and gives directly the set of dynamical equations for the system $$\begin{aligned}
p\left( t\right) & =m\dot{x}\left( t\right) \label{motion1}\\
\dot{p}\left( t\right) & =-\frac{dV}{dx}\nonumber\end{aligned}$$ In our stochastic model noise decouples momentum and velocity so all orbits in phase space are now permitted. No deterministic relation between momentum and velocity, like that shown in eqn$\left( \ref{motion1}\right) $, exists any more although the velocity still is real and given by the time derivative of the position. The same effect is verified when the motion is subject to a Wiener noise and a Fokker-Planck equation, using the Ito approach, is constructed. In this case it is possible to show that the obtained Fokker-Planck equation admits a probability amplitude formalism which results from a perturbation series on a variable conjugate to the momentum. The lowest order terms are fully compatible with a quantum dynamics for the particle [@fp].
The above developments shows that space and momentum variables are related by the unitary transformation of the eqn$\left( \ref{Heisenb}\right) $; consequently they satisfy an equal time ”uncertainty principle” since from known properties of Fourier transform we must have $$\frac{\varepsilon}{\alpha}\left( \Delta\left[ \frac{\left( x_{n+1}-x_{n}\right) }{\varepsilon}\right] \Delta\left[ p_{n}\right] \right)
=\frac{1}{\alpha}\left( \Delta x_{n}\right) \left( \Delta p_{n}\right)
\geq2\pi$$ valid for fixed $x_{n+1}$.
Probability amplitudes are then given by eqn$\left( \ref{FKac}\right) $ which is a phase-space Feynman-Kac formula in a most striking resemblance to Quantum Mechanics. The difference is concentrated on the $\sqrt{P}$ term associated to the still undefined bare probability $P$ and in the value of the constant $\alpha$. In the limit of continuous time $\left( \epsilon
\rightarrow0\right) $ the condition of finite rms for the noise source, as displayed in eqn$\left( \ref{time-energy}\right) $, demands that only vanishingly small values of the random variable are relevant at infinitely large noise intensity $\mathcal{E}$. This fact allows the substitution of values for $P$ by its value at $y=0$ (equal to $P_{0}$), which is then incorporated as a normalization constant in the probability amplitude. As a result a finite probability distribution is obtained, since the lost apodization induced by large fluctuations is compensated by this normalization procedure. All these considerations complete a theory for amplitudes calculated as $$\psi\left( x_{N}\right) =\left( \frac{iP_{0}}{2\pi}\right) ^{N/2}\int
\psi\left( x_{0}\right) \exp\left( \frac{-i\mathcal{A}_{path}}{\alpha
}\right) \prod_{n=0}^{N-1}\frac{dx_{n}dp_{n}}{\alpha} \label{Feynm}$$ where $\mathcal{A}_{path}\mathcal{\equiv}\epsilon\sum_{path}\mathcal{L}$ is the discrete time phase-space classical action and the value of $P_{0}$ being incorporated in a proper Hilbert space normalization for $\psi\left(
x_{N}\right) $. Notice that the bare probability structure is hidden in this approximation which leads to a hidden process (or variable?) theory. An important fact is that eqn$\left( \ref{Feynm}\right) $ is fully algorithmic (e.g. using Monte Carlo) by the use of the prescriptions given here for path space construction making trivial the mystery of how an equation like the above one may naturally appear in theoretical models. The result we have obtained is a Feynman-like path integral for a quantum particle; apart from a normalization factor it seems that non-classical stochastic process, as described here, is sufficiently rich to explain the probability amplitude structure of one-dimensional Quantum Mechanics. The classical action being the Onsager-Machlup functional [@onsager] for the underlying stochastic process which acts not on probabilities but on amplitudes as required by the non-classical nature of this random process. However the universal character of the Planck’s constant demands that the kind of noise we considered must also be universal in order to set $\alpha=\hbar$ in the above reasoning and in particular in the eqn$\left( \ref{FKac}\right) $, the generalized Feynman-Kac formula. Saying differently, must exist a kind of noise field capable to couple to any elementary particle wherever its physical character. This is not a trivial task and go further in this direction now is premature considering the primitive informations we have at hand besides the possibility of incomplete nature for an existing fundamental stochastic vacuum. The converse sense is also true. Feynman and Hibbs [@hibbs] showed that the important paths for a quantum mechanical particle are those that are not differentiable. In particular they have shown that $\left\langle \left(
x_{n+1}-x_{n}\right) ^{2}/\epsilon^{2}\right\rangle \sim\epsilon^{-1}$ a result compatible with fractal trajectories activated by a noise source. Thus should exist, following the prescription given here, a non-classical random variable associated to quantum processes, not available inside the limits of the Quantum Theory because it should be the linearization of some hidden incomplete stochastic process. What we have showed is how the hidden mechanism (the HV mechanism) happens, masking this subjacent stochastic process and leaving out only quantal effects, which shall depend on the linearized form of the nonclassical terms through the $d$-terms set. Consequently it turns out quite impossible distinguish Quantum Mechanics, at least for the simple case treated here, from a incomplete random variable process.
As strange may appear the use of the phase space Lagrangean presents no additional difficulties as well. In fact our model clearly defines the role of each dynamical concept in quantum processes, specifying its character of reality and their formal intrinsic relationship. We can advance further in this reasoning if we look for a differential equation that solves the amplitude problem. The first step is to find $\delta\psi=\psi\left(
x,t+\varepsilon\right) -\psi\left( x,t\right) $ and owing to eqn$\left(
\ref{Feynm}\right) $ this configures a moving boundary variational problem since the end point of the action integral is changed. We have $$\delta\psi=\frac{-1}{i\alpha C}\int\psi\left( x_{0}\right) \left(
\delta\mathcal{A}\right) \exp\left( \frac{-i\mathcal{A}\left( y,\dot
{y}\right) }{\alpha}\right) \mathcal{D}y\mathcal{D}p$$ and to calculate $\delta\mathcal{A}$ the effect of changing the end point due to particle’s velocity must be considered. The result is [@elsgolc] $$\delta\mathcal{A=}\left( \mathcal{L}-\dot{y}\left( t\right) \frac
{\partial\mathcal{L}}{\partial\dot{y}\left( t\right) }\right) _{y\left(
t\right) =x}dt+\frac{\partial\mathcal{L}}{\partial\dot{y}\left( t\right)
}_{_{y\left( t\right) =x}}dx+\int^{t}\left( \frac{\partial\mathcal{L}}{\partial y}-\frac{d}{dt^{\acute{}}}\frac{\partial\mathcal{L}}{\partial\dot{y}}\right) \delta y\left( t^{\acute{}}\right) dt^{\acute{}}$$ where the second term in the coefficient of $dt$ corrects the time derivative of the action due to particle’s motion. From this equation we get the partial derivatives of the amplitude: $$\begin{aligned}
\frac{\partial\psi}{\partial x} & =\frac{-1}{i\alpha C}\int\psi\left(
x_{0}\right) p\exp\left( \frac{-i\mathcal{A}}{\alpha}\right) \mathcal{D}y\mathcal{D}p\\
\frac{\partial\psi}{\partial t} & =\frac{1}{i\alpha C}\int\psi\left(
x_{0}\right) \left( \frac{p^{2}}{2m}+V\left( x\right) \right) \exp\left(
\frac{-i\mathcal{A}}{\alpha}\right) \mathcal{D}y\mathcal{D}p\end{aligned}$$ Notice that information about particle’s velocity is lost whilst the surviving associated variable, the momentum, has no reality content; at most a statistical interpretation of its value can be given. This view becomes more evident if we note that $$\frac{\partial^{2}\psi}{\partial x^{2}}=\frac{-1}{\alpha^{2}C}\int\psi\left(
x_{0}\right) p^{2}\exp\left( \frac{-i\mathcal{A}}{\alpha}\right)
\mathcal{D}y\mathcal{D}p$$ so a closed differential equation is obtained for the amplitude $$i\alpha\frac{\partial\psi}{\partial t}=-\frac{\alpha^{2}}{2m}\frac
{\partial^{2}\psi}{\partial x^{2}}+V\left( x\right) \psi$$ defining its Hilbert space operator structure. Observe that the probability amplitude depends only on spatial variables and time. No information concerning the velocity is present although a resemblance to the (classical) momentum through spatial derivatives are permitted with some care allowing the usual association of the operator $-\frac{\alpha^{2}}{2m}\frac{\partial^{2}}{\partial x^{2}}+V\left( x\right) $ to the classical Hamiltonian. Our formalism show how this exactly happen and to what extend this association is valid. However remember that this equation is valid for a specific noise type and its universal validity, which permits the substitution $\alpha=\hbar$, demands a non-trivial vacuum physics.
The present theory has a quite general range and for the specific mechanical model we are considering the formalism of Quantum Mechanics, if applicable, is just the simplest description, that using the lowest order in the phase and noise probability distribution expansion. It seems reasonable that the weakest nonlinear contribution may generate corrections to eqn$\left( \ref{FKac}\right) $ which lies outside Quantum Theory itself while still maintaining the nonclassical character of the formalism. Expressing differently we able to predict hipper-quantum phenomena, those explicitly depending of the Q-rules (in the present mechanical model means $d$-dependent terms) but not explained by the use of the simple Feynman formula. This new class of quantum phenomena would be described by considering the second order correction of eqn$\left(
\ref{phase}\right) $ given by the set of $B$ coefficients. Thus, at least in principle, we know how generalizations to Quantum Theory may appear and how to test if Quantum Theory is fundamental. Nobody makes no doubt about the capabilities of Quantum Mechanics in explain non-classical word. However once it is considered a fundamental theory all possible predictions of nonclassical phenomena must lie within the range of its formalism. The nonlinear correction to the phase enable us to test wether or not in fact it is fundamental as well as hopefully predict new phenomena never yet considered. We can say more. In the continuum $\left( \epsilon\rightarrow0\right) $ only vanishingly small values of the fluctuating variable $Y$ will survive and in this case the linear approximation is rigorously true if the kernel matrix $B_{nl}^{ml}$ as well as all higher order phase coefficients are topologically dense to survive in the continuum. This means that Quantum Mechanics may be, in fact, a truly hidden-variable theory of first type [@belinfante] and any tentative of find or detect its stochastic nature be definitively unfruitful. In this case an evidence of hipper-quantum phenomena means that space-time continuity is broken at some scale thus generating information about the underground vacuum physics necessarily hidden in usual quantum processes.
On the contrary, if the weakest nonlinear term survives in the continuum the generalized Feynman-Kac formula, eqn$\left( \ref{Feynm}\right) $, changes to $$\begin{aligned}
\psi\left( x,t\right) & =\frac{1}{C}\int\psi\left( x_{0}\right)
\exp\left( \frac{-i\mathcal{A}_{path}}{\alpha}\right) .\\
& .\exp\left( \frac{i}{\alpha^{2}}\int^{t}B\left( u\left( \tau\right)
,\dot{u}\left( \tau\right) ,u\left( \varsigma\right) ,\dot{u}\left(
\tau\right) \right) H\left( u\left( \tau\right) ,\dot{u}\left(
\tau\right) \right) H\left( u\left( \varsigma\right) ,\dot{u}\left(
\varsigma\right) \right) d\tau d\varsigma\right) \mathcal{D}u\mathcal{D}p\end{aligned}$$ so in order to derive a differential equation for the wave function in a similar way developed above, a generalization of the moving boundary variational calculus must be done. Keeping terms linear in $B$ this may be done yet with some involved calculations. The result presents a correction to Schröedinger equation (linear in $B$) where the important fact is that vacuum terms are present. The HV mechanism is broken. This involved questions transcends, by its nature, the limits of the discussions we intend to present in this work so we deserve for the future additional tracks on this line.
Concluding, observe that the rules we introduced here for a incomplete random variable may, at first glance, be so strange as the axioms of Quantum Theory are. But once those rules are algoritmic, we are able to test them from a heuristic point of view since they must belong to the vacuum phenomenology which, at least in principle, is accessible to experiments. Breakdown of causality and determinism, as introduced here, is not a big problem too because no physical principle demands these reasonable assumptions be valid outside our common sense perception. The main result of this work still is the fact that a probability amplitude formalism is possible in describing the class of stochastic process we introduced here. This is a general result which may include physical processes as well. In this case a probability amplitude description of the motion of a particle is obtained, with properties very similar to those Quantum Theory makes use of. The price paid is the need of an universal vacuum field in order to explain the generality of Quantum Mechanics. At the present stage of our formulation we are not able to predict all properties this vacuum field must possess. The apparent advantage over existing hidden-variable theories is the algorithmic procedure and the explicit demonstration of the HV mechanism. However it is important to cite that we are not worried about justifications to Quantum Theory. If Nature truly admits incomplete random variables in its basic realm Quantum Theory shouldn’t be considered as fundamental because in this case it cannot capture the basic processes physical world possesses. Its success would be consequence of fortuit epistemological and practical rules the HV mechanism enables. This way we believe the present work opens tips to looking for new phenomenology in this field. It should be indispensable, within the present context, a serious investigation on vacuum properties, powered by an incomplete stochastic random field, as a proper source for quantum behavior in Nature.
[99]{} A. Einstein and L. Hopf, Ann. d. Phys. **33**, 1105 (1910), A. Einstein and O. Stern, Ann. d. Phys. **40**, 551 (1913), P. W. Milonni, *The Quantum Vacuum* (Academic Press, San Diego, 1994).
In fact a game demands more than one (competiting) player so what we have defined is properly a lottery. A game concerns a multivariate process. We do not treat this case in the present work because the main ideas we intend show here are not affected by the number of players and can be easily implemented for a true game. Nevertheless we still keep game’s terminology.
N. G. van Kampen, *Stochastic Processes in Physics and Chemistry* (North-Holland, Amsterdan, 1981).
L. Onsager and S. Machlup, Phys. Rev. **91**, 1505 (1953)
R. Graham, Z. Physik B **26**, 281 (1977)
See for example Pesquera et all, Phys. Lett. **94A** 287 (1983), Wio et all, Phys. Rev. A **40**, 7312 (1989), Lehmann et all, Phys. Rev E **62**, 6282 (2000)
C. W. Gardiner, *Handbook of Stochastic Methods* (Springer-Verlag, Berlin, 1990)
Drew Fudenberg and Jean Tirole, *Game Theory* (Cambridge, Mass., 1991)
M. S. Torres Jr. and J. M. A. Figueiredo, quant-ph/0204123 (2002)
R. P. Feynman and A. R. Hibbs, *Quantum Mechanics and Path Integrals* (MacGraw-Hill, 1965)
F. J. Belinfante, *A survey on Hidden-Variables Theory*, (Pergamon Press, 1973)
L. E. Elsgolc, *Calculus of Variations* (Pergamon Press, Mass., 1962)
|
Over the past 20 years it has been shown that the low energy physics of a variety of models of one-dimensional (1D) correlated electrons can be described by the Luttinger liquid (LL) phenomenology[@haldane; @johannes]. Because of the progress in the experimental realization of quasi 1D systems and speculations about possible LL behavior in the normal state of the high-temperature superconductors LL phenomenology has lately attracted considerable attention. In Fermi liquids the low energy physics and low temperature thermodynamics is dominated by the excitation of quasi-particles which are in a one-to-one correspondence to the particle-hole excitations of the non-interacting system. The elementary excitations in LL’s are of collective bosonic nature. In LL’s the discontinuity of the momentum distribution function $n(k)$ at the Fermi wave vector $k_F$ vanishes and the density of states of the one-particle Green’s function at the Fermi energy is zero. Close to the Fermi surface both functions are dominated by power law behavior with exponents which are given by the so called LL parameters $K_{\rho}$ and $K_{\sigma}$. According to LL phenomenology thermodynamic quantities (e. g. the compressibility and spin susceptibility) and the non-analytic behavior of correlation functions can be expressed in terms of these two parameters and the velocities $v_{\rho}$ and $v_{\sigma}$ of charge and spin excitations. By calculating the one-particle Green’s function of the Tomonaga-Luttinger (TL) model[@tomonaga; @luttinger] we will show that the asymptotic behavior of the Green’s function of a general LL at large space-time distances is nonetheless less universal than is widely believed. The exponents of the algebraic decay in the directions within the $x$-$v_F t$ plane which are determined by the velocities $ v_{\rho}$ and $v_{\sigma}$ are [*not*]{} given by the LL parameters of the model alone. Here $v_F$ denotes the Fermi velocity. We will furthermore discuss the implications for the spectral function. The TL model is a continuum model of interacting 1D electrons. The linearization of the electron dispersion around the two Fermi points and neglecting the backscattering processes between electrons makes it feasible to determine the spectrum of the Hamiltonian and calculate all correlation functions. Following Luttinger[@luttinger] we introduce right- ($\alpha=+$) and left-moving ($\alpha=-$) Fermions with spin $s$, creation operators $a^{\dag}_{k,\alpha,s}$, dispersion $\xi_{\alpha}(k)= \alpha v_F (k- \alpha k_F)$, density operators ($q \neq 0$) $\rho_{\alpha,s}(q)= \sum_k
a_{k,\alpha,s}^{\dagger} a_{k+q,\alpha,s}^{}$, and particle number operators $n_{k,\alpha,s}=a_{k,\alpha,s}^{\dagger}
a_{k,\alpha,s}^{}$. To simplify the mathematical treatment Luttinger added an infinite filled Fermi sea to the ground state[@luttinger]. The Hamiltonian for a system of length $L$ is given by $$\begin{aligned}
&& H = \sum_k \sum_{\alpha,s} \xi_{\alpha}(k)
\left[ n_{k,\alpha,s} - \left< n_{k,\alpha,s} \right>_0 \right] \nonumber \\
& & + \frac{1}{2 L} \sum_{{q \neq 0} \atop {\alpha, s, s'}}
\left[g_{4,\parallel}(q) \delta_{s,s'} +
g_{4,\perp}(q) \delta_{s,-s'} \right]
\rho_{\alpha,s}(q) \rho^{\dag}_{\alpha,s'}(q) \nonumber \\
&& + \frac{1}{L} \sum_{{q \neq 0} \atop {s, s'}}
\left[g_{2,\parallel}(q) \delta_{s,s'} +
g_{2,\perp}(q) \delta_{s,-s'} \right]
\rho_{+,s}(q) \rho^{\dag}_{-,s'}(q) .
\label{hamiltonian}\end{aligned}$$ Here we use “g-ology” notation[@solyom] and $\left< \ldots \right>_0$ denotes the (non-interacting) ground state expectation value. Contrary to many authors we keep the explicit $q$ dependence of the coupling functions. We assume that the Fourier transforms $g_{i,\kappa}(q)$ ($i=2,4$; $\kappa=\parallel , \perp$) of the two-particle interaction have only contributions for $q {< \atop \sim} q_c \ll k_F$ with an interaction cut-off $q_c$. At [*no*]{} stage of the discussion it will be necessary to introduce any further cut-offs “by hand” despite the infinite (filled) Fermi sea at negative energies. The model only belongs to the LL universality class if $g_{2,\kappa}(q=0)$ is finite for $\kappa = \parallel$ and $\perp$ and at least one of the two coupling constants is non-zero. Thus we restrict ourselves to these kind of interactions.
Bosonization of the Hamiltonian and a canonical transformation leads to[@johannes] $$\begin{aligned}
H = \sum_{q \neq 0} \sum_{\nu = \rho,\sigma} \varepsilon_{\nu}(q)
\beta_{\nu}^{\dagger}(q) \beta_{\nu}^{}(q) ,
\label{hamiltoniandiag} \end{aligned}$$ with bosonic operators $\beta_{\nu}(q)^{\dagger}$ describing charge ($\nu=\rho$) and spin ($\nu=\sigma$) excitations (spin-charge separation). The energies $\varepsilon_{q,\nu}$ are given by $$\begin{aligned}
\frac{\varepsilon_{q,\nu}}{|q|} = v_F \sqrt{\left( 1+
\frac{g_{4,\nu}(q)}{\pi v_F} \right)^2 -
\left( \frac{g_{2,\nu}(q)}{\pi v_F}\right)^2 }
\equiv v_{\nu}(q) ,
\label{energies} \end{aligned}$$ where we have introduced the renormalized charge and spin density velocities $v_{\nu}(q)$ and interactions $g_{i,\rho/\sigma}(q) \equiv \left[g_{i,\parallel}(q)
\pm g_{i,\perp}(q) \right]/2$. The one-particle Green’s function $i G_{\alpha,s}^{<} (x,t) =
\left< \psi_{\alpha,s}^{\dagger}(0,0)
\psi_{\alpha,s}^{}(x,t)\right>$, which after a double Fourier transformation leads to the spectral function $\rho_{\alpha,s}^{<}(k,\omega)$ relevant for photoemission experiments, can be calculated using the bosonization of the fermion fields $\psi_{\alpha,s}^{\dagger}(x)= \frac{1}{\sqrt{L}} \sum_{k} e^{-ikx}
a_{k,\alpha,s}^{\dagger}$[@haldane]. In the thermodynamic limit and at zero temperature we obtain $ G_{+}^{<}(x,t) = [G_{+}^{<}]^0(x,t) \exp{\{F(x,t)\}}$ with $$\begin{aligned}
F(x,t) & = & \frac{1}{2}\sum_{\nu=\rho, \sigma}
\int_{0}^{\infty} \frac{dq}{q} \left\{
e^{- i q \left( x - v_{\nu}(q) t \right) }
- e^{- i q \left( x - v_F t \right) } \right. \nonumber \\* && +
\left. 2 \gamma_{\nu}(q) \left[ \cos{(qx)} e^{i q v_{\nu}(q) t} -1 \right]
\right\}
\label{fdef} \end{aligned}$$ and the non-interacting Green’s function $$\begin{aligned}
[G_{+}^{<}]^0(x,t)
= - \frac{1}{2\pi} \frac{e^{i k_F x} }{ x-v_F t -i0 }.
\label{g0t0}\end{aligned}$$ Because the Green’s function is the same for both spin directions we have suppressed the spin index $s$. By leaving out irrelevant particle number contributions in the Hamiltonian we effectively shifted the energy scale so that the chemical potential is zero. $\gamma_{\nu}(q)$ in Eq. (\[fdef\]) is given by the eigenvectors of the canonical transformation and can be written as $\gamma_{\nu}(q)=\left[ K_{\nu}(q)+ 1/K_{\nu}(q) -2 \right]/4$, with $$\begin{aligned}
K_{\nu} (q)= \sqrt{\frac{1+g_{4,\nu}(q)/(\pi v_F)-g_{2,\nu}(q)/(\pi
v_F)}{1+g_{4,\nu}(q)/(\pi v_F)+g_{2,\nu}(q)/(\pi
v_F)}} .
\label{kdef}\end{aligned}$$ Due to the interaction cut-off the momentum integral in Eq. (\[fdef\]) is regular in the ultraviolet limit. The $q \to 0$ limits of the four functions $v_{\nu}(q)$ and $K_{\nu} (q)$ define the two velocities and two LL parameters ($v_{\rho}, v_{\sigma}, K_{\rho}, K_{\sigma}$). Roughly speaking the non-analyticities in spectral functions at low energies and small momenta are given by the behavior of $G_{+}^{<}(x,t)$ at large arguments. To determine the non-analytic behavior of the momentum distribution $n_+(k) = \int_{-\infty}^{\infty} dx e^{-ikx} i G_+^<(x,0)$ and the momentum [*integrated*]{} spectral function $\rho_+^<(\omega)= \int_{-\infty}^{\infty} dt e^{i \omega t}
i G_+^<(0,t)/(2 \pi)$ it is sufficient to discuss the behavior of $G_{+}^{<}(x,t)$ along the $x$ and $v_F t$ axis. To emphasize the difference to the general case of non-vanishing $x$ [*and*]{} $v_F t$ we first focus on these two cases. With a little less mathematical rigor they have already been discussed several times[@approxg]. Using integration by parts[@regular] in the first derivative of $F(x,0)$ and $F(0,t)$ we obtain the [*leading*]{} behavior $F(x,0) \sim - \alpha \ln{ \left| x \right|}$ and $F(0,t) \sim - \alpha \ln{ \left| v_F t \right|}$, with $\alpha =
\gamma_{\rho}(0) + \gamma_{\sigma}(0)$. In $G_{+}^{<}(x,t)$ this leads to power law behavior along the $x$ and $v_F t$ axis with an exponent which [*only*]{} depends on the strength of the interaction at vanishing momentum and thus the LL parameter $K_{\rho}$ and $K_{\sigma}$ of the model at hand. This is in accordance with the LL phenomenology. From general theorems about Fourier transforms the by now well known LL behavior of the momentum distribution function follows for $|k-k_F|/q_c \to 0$ $$\begin{aligned}
\frac{1}{2} - n_+(k) \sim \left\{
\begin{array}{r@{\quad:\quad}l}
|k-k_F|^{\alpha} \mbox{sign} (k-k_F) & \mbox{for} \,\, 0 <\alpha <1 \\
(k-k_F) \ln{|k-k_F|} & \mbox{for} \,\, \alpha =1 \\
(k-k_F) & \mbox{for} \,\, \alpha > 1 .
\end{array} \right. \end{aligned}$$ $$\begin{aligned}
\label{nk}\end{aligned}$$ For $\alpha > 1$ the leading behavior of $n_+(k)$ is dominated by a linear term but a higher derivative still diverges at $k_F$. Due to the different analytic properties of $F(0,t)$ (analytic in the upper half of the complex $t$ plane) and $F(x,0)$ ([*not*]{} analytic in either the upper or the lower half of the complex $x$ plane) $\rho_+^<(\omega)$ is dominated by a non-analytic power law behavior even for $\alpha > 1$[@vmdoc]. Thus anomalous dimensions $\alpha >1$ should in principle be observable in momentum integrated photoemission spectra[@experiments]. In the literature the mathematical reason for this important difference in the behavior of $n_+(k)$ and $\rho_+^<(\omega)$ has thus far not been properly pointed out. For $\omega/(v_F q_c) \to 0$ we obtain $$\begin{aligned}
\rho_+^<(\omega) \sim \left\{
\begin{array}{r@{\quad:\quad}l}
\Theta(-\omega) (-\omega)^{\alpha} & \mbox{for} \,\, \alpha \not\in
{\rm I \! N} \\
\Theta(-\omega) (-\omega)^{\alpha} \ln{|\omega|} & \mbox{for}
\,\, \alpha \in {\rm I \! N} .
\end{array} \right.
\label{rhoomega}\end{aligned}$$ The prefactors and the range over which the leading behavior given in Eqs. (\[nk\]) and (\[rhoomega\]) can be observed depend on the interaction at [*all*]{} $q$. As a consequence numerically calculated curves for different interaction potentials $g_{i,\kappa}(q)$ but the same anomalous dimension might appear quite different[@vmdoc; @ksvm].
To find a more explicit form of the Green’s function the $q$ integral in Eq. (\[fdef\]) has often been evaluated after replacing $v_{\nu}(q) \to v_{\nu}(0)$, $\gamma_{\nu}(q) \to
\gamma_{\nu}(0)$ and multiplying the integrand by a factor $\exp{\left(-q\Lambda\right)}$[@approxg; @vmks]. The $q$ integral can then be performed and one obtains $$\begin{aligned}
&& [G_{+}^{<}]_A (x,t) = \frac{-e^{ik_F x}/(2 \pi)}
{x- v_F t -i0} \prod_{\nu =\rho,\sigma}
\left[ \frac{x- v_F t - i \Lambda}
{x- v_{\nu} t - i \Lambda } \right]^{1/2} \nonumber \\*
&& \times \left[ \frac{\Lambda^2}
{\left( x- v_{\nu} t - i \Lambda \right)
\left( x+ v_{\nu} t + i \Lambda \right) }
\right]^{\gamma_{\nu}/2} .
\label{gapprox}\end{aligned}$$ For the Hamiltonian Eq. (\[hamiltonian\]) there exists [*no*]{} special interaction potential so that this [*approximation*]{} becomes exact. As in the discussion of the general case below we will transform onto new variables $s=x-ct$ and $s'=x+ct$ with an arbitrary velocity $c$ and discuss the behavior of $ [G_{+}^{<}]_A \left(
x[s,s'],t[s,s']\right)$ for large $s$ with a fixed $s'$ and vice versa and for different values of $c$. For all velocities $c$ but $v_{\rho}$ and $v_{\sigma}$, $ [G_{+}^{<}]_A \left( x[s,s'],t[s,s']\right)$ falls off like $s^{-(1+\alpha)}$ and $s'^{-(1+\alpha)}$. This is the behavior we already found for $v_F t =0$ or $x=0$. For $c=v_{\rho}$ the Green’s function falls off like $$\begin{aligned}
[G_{+}^{<}]_A \left( x[s,s'],t[s,s'] \right)
& \sim & s^{-(1 + \gamma_{\sigma} + \gamma_{\rho}/2)} ,
\label{gabfallvfs1} \\
\lbrack G_{+}^{<} \rbrack_A \left( x[s,s'],t[s,s'] \right)
& \sim & s'^{-(1/2+ \gamma_{\sigma} +\gamma_{\rho}/2)} .
\label{gabfallvfss1} \end{aligned}$$ The behavior of the Green’s function for $c=v_{\sigma}$ follows from Eqs. (\[gabfallvfs1\]) and (\[gabfallvfss1\]) by interchanging $\gamma_{\rho}$ and $\gamma_{\sigma}$. Within the above approximation the exponents of the asymptotic behavior for all $c$ and thus for all directions within the $x$-$v_F t$ plane can be expressed in terms of $K_{\rho}$ and $K_{\sigma}$. The resulting spectral function displays power law singularities (bounded non-analyticities in case the $\gamma_{\nu}$ are too large) at $\omega = \pm v_{\nu} (k-k_F)$ for all $\Lambda (-k+k_F)>0$[@vmks].
Without using any approximations it has been shown that Eq. (\[gapprox\]) gives the correct asymptotic behavior of $G_{+}^{<} \left( x,t\right)$ for a [*box potential*]{} $g_{i,\kappa}(q) = g_{i,\kappa} \Theta(q_c - |q|)$[@ksvm]. In this case the momentum range over which $\rho_{+}^{<}(k,\omega)$ is dominated by two power law singularities at the charge and spin excitation energies is limited to $0<(-k+k_F)/q_c < 1$. For other momenta further non-analyticities occur[@ksvm].
Next we will show that $[G_{+}^{<}]_A (x,t)$ does [*not*]{} give the correct asymptotic behavior for an arbitrary shape of the interaction. After transforming onto $s$ and $s'$ the Green’s function is given by $$\begin{aligned}
G_{+}^{<} \left( x[s,s'],t[s,s'] \right) =
[G_{+}^{<}]^0 \left( x[s,s'],t[s,s'] \right) e^{\tilde{F}(s,s')} ,
\label{gt0sss}\end{aligned}$$ where $\tilde{F}(s,s')$ follows from Eq. (\[fdef\]) and $(x,t) \to (s,s')$. As an example we will discuss the behavior of $\tilde{F}(s,s')$ for large $s'$ and fixed $s$ in more detail and only present the results for the other case. We first take the derivative with respect to $s'$. This gives $$\begin{aligned}
&& \frac{d\tilde{F}(s,s')}{ds'} = \frac{1}{2}\sum_{\nu=\rho,\sigma}
\int_{0}^{\infty}
dq \left\{ - i \frac{1}{2} \left( 1- \frac{ v_{\nu}(q)}{c}\right)
\nonumber \right. \\
&& \times e^{- i q \frac{1}{2} \left( 1+ \frac{ v_{\nu}(q)}{c}\right) s }
e^{- i q \frac{1}{2} \left( 1- \frac{ v_{\nu}(q)}{c}\right) s' }
\nonumber \\
&& + i \frac{1}{2} \left( 1- \frac{v_F}{c}\right)
e^{- i q \frac{1}{2} \left( 1+ \frac{ v_F}{c}\right) s }
e^{- i q \frac{1}{2} \left( 1- \frac{ v_F}{c}\right) s' } \nonumber
\\ && + \gamma_{\nu}(q) \left[
i \frac{1}{2} \left( 1+ \frac{ v_{\nu}(q)}{c}\right)
e^{ i q \frac{1}{2} \left( 1- \frac{ v_{\nu}(q)}{c}\right) s }
e^{ i q \frac{1}{2} \left( 1+ \frac{ v_{\nu}(q)}{c}\right) s' }
\nonumber \right. \\
&& \left. \left. - i \frac{1}{2}
\left( 1- \frac{ v_{\nu}(q)}{c}\right)
e^{- i q \frac{1}{2} \left( 1+ \frac{ v_{\nu}(q)}{c}\right) s }
e^{- i q \frac{1}{2} \left( 1- \frac{ v_{\nu}(q)}{c}\right) s' }
\right] \right\} .\end{aligned}$$ $$\begin{aligned}
\label{ftildeabgel}\end{aligned}$$ To simplify the discussion we assume that $ v_{\nu}(q)$ are monotonic functions. Using integration by parts and the method of stationary phase in the asymptotic expansion of the integral we find for all velocities $c$ but $v_{\rho}(0)$, $v_{\sigma}(0)$, and $v_F$ that $\tilde{F}(s,s')$ goes like $- \alpha \ln{(s')}$ and thus $$\begin{aligned}
\tilde{G}_{+}^{<} \left( s,s' \right) \equiv
G_{+}^{<} \left( x[s,s'],t[s,s'] \right) \sim s'^{
-(1+\alpha) } .
\label{gallg} \end{aligned}$$ For large $s$ and fixed $s'$ we obtain the same behavior.
If $c=v_{\rho}(0)$ the phase $q \left[ 1-v_{\rho}(q)/v_{\rho}(0)
\right]$ in the first and fourth term of Eq. (\[ftildeabgel\]) becomes stationary at $q=0$. At the stationary point also the prefactor $\left[ 1-v_{\rho}(q)/v_{\rho}(0) \right]$ vanishes and thus we have to [*generalize*]{} the method of stationary phase. The details of this generalization will be given elsewhere and here we will only present the results. The leading contribution of the fourth term of Eq. (\[ftildeabgel\]) to the large $s'$ behavior is $- \frac{\gamma_{\rho}}{2} \frac{1}{p_{\rho} +1} \frac{1}{s'}$, where $p_{\rho}$ is the smallest number $n \in {\rm I \! N} \cup \{ \infty \}$ with $\left[v_{\rho} \right]^{(n)}(0) \neq 0$, where $\left[v_{\rho} \right]^{(n)}(q)$ denotes the $n$-th derivative. According to the definition of $v_{\rho} (q)$ Eq. (\[energies\]) $p_{\rho}$ is a measure of the “smoothness” of the interaction at vanishing momentum. Evaluating the first term in Eq.(\[ftildeabgel\]) in the same way and using integration by parts in the other terms leads to $$\begin{aligned}
\tilde{G}_{+}^{<} \left( s,s' \right)
\sim s'^{-(1/2+ \gamma_{\sigma} +\gamma_{\rho}/2
+ 1/[2p_{\rho}+2]+ \gamma_{\rho}/ [2 p_{\rho}+2])}.
\label{gabfallvfss1allg} \end{aligned}$$ This result is different from Eq. (\[gabfallvfss1\]) and more importantly it shows that the asymptotic behavior [*cannot*]{} be obtained from the LL parameters $K_{\rho}$ and $K_{\sigma}$ of the TL model alone. This is in [*contrast*]{} to the wide spread belief that Eq. (\[gapprox\]) displays the asymptotic behavior of [*all*]{} models which belong to the LL universality class provided the exponents are expressed in terms of the LL parameters of the specific model. The behavior Eq.(\[gabfallvfss1\]) is only recovered if $p_{\rho}=\infty$, i. e.if all derivatives of $v_{\rho} (q)$ and thus all derivatives of the interaction potential vanish at $q=0$. For this reason Eq. (\[gapprox\]) gives the correct leading behavior in case of a box potential. For the large $s$ behavior and $c=v_{\rho}(0)$ we obtain $$\begin{aligned}
\tilde{G}_{+}^{<} \left( s,s' \right)
\sim s^{-(1+ \gamma_{\sigma} +\gamma_{\rho}/2
+ \gamma_{\rho}/ [2 p_{\rho}+2])}.
\label{gabfallvfs1allg} \end{aligned}$$ For $c=v_{\sigma}(0)$ we have to analyze the Green’s function following the same route and obtain $$\begin{aligned}
\tilde{G}_{+}^{<} \left( s,s' \right)
& \sim & s^{-(1 + \gamma_{\rho} + \gamma_{\sigma}/2
+ \gamma_{\sigma}/ [2 p_{\sigma}+2])}
,
\label{gabfallvfs2allg} \\
\tilde{G}_{+}^{<} \left( s,s' \right)
& \sim & s'^{-(1/2+ \gamma_{\rho} +\gamma_{\sigma}/2
+ 1/[2p_{\sigma}+2]+ \gamma_{\sigma}/ [2 p_{\sigma}+2])} ,
\label{gabfallvfss2allg} \end{aligned}$$ where $p_{\sigma}$ is defined in analogy to $p_{\rho}$.
The case $c=v_F$ has so far been excluded. If $v_F \neq v_{\rho}(0)$ and $v_F \neq v_{\sigma}(0)$ the behavior of the Green’s function is given by Eq. (\[gallg\]). In the relevant case of a spin independent interaction with $v_{\sigma}(q) \equiv v_F$ and $\gamma_{\sigma}
=0$ the Green’s function is given by Eqs. (\[gabfallvfs2allg\]) and (\[gabfallvfss2allg\]) with $p_{\sigma} = \infty$.
To gain more insight into the asymptotic behavior of the Green’s function we have evaluated the momentum integral in $\tilde{F}(s,s')$ numerically. As an illustration of the new predictions Eqs.(\[gabfallvfss1allg\])-(\[gabfallvfss2allg\]) we present $\mbox{Re} \, \{
\tilde{F}(s,s') \}$ for $s=0$ as a function of $s'$ and $c=v_{\rho}(0)$ on a log-linear scale in Fig. \[fig1\]. The curves have been calculated for a spin independent interaction with $g_{4,\parallel}(q) \equiv g_{4,\perp}(q) \equiv
g_{2,\parallel}(q) \equiv g_{2,\perp}(q) \equiv g \exp{ \{
( -q/q_c)^{p_{\rho}} \} }$, $2 g/(\pi v_F) = 3$, i. e. $v_{\rho}(0)/v_F =2$, $\gamma_{\rho} =1/8$ and different $p_{\rho}$. From Eqs. (\[gt0sss\]) and (\[gabfallvfss1allg\]) we expect $$\begin{aligned}
\tilde{F}(s,s') \sim \left\{ \frac{1}{2} -
\left( \frac{\gamma_{\rho}}{2} + \frac{1}{2p_{\rho}+2}
+ \frac{\gamma_{\rho}}{2 p_{\rho}+2} \right)
\right\} \ln{\left(s' \right)}
\label{expect}\end{aligned}$$ Fits of the data for $s' > 10^3$ reproduce the expected prefactors of the $\ln{(s')}$ within a relative error of less than 0.02%.
For increasing $s$ we have to go to larger $s'$ to find the asymptotic behavior Eq. (\[expect\]). In an intermediate regime of $s'$, which increases with increasing $s$, $\mbox{Re} \, \{ \tilde{F}(s,s') \}$ displays the behavior of the $p_{\rho} = \infty$ case. This is illustrated in Fig. \[fig2\]. It shows $\mbox{Re} \, \{ \tilde{F}(s,s') \}$ as a function of $s'$ on a log-linear scale for the above form of coupling functions, $c=v_{\rho}(0)$ and different $s$. The parameters are $2 g/(\pi v_F) = 5$, i. e. $v_{\rho}(0)/v_F =\sqrt{6}$, $\gamma_{\rho} =0.2144$ and $p_{\rho} =2$. A fit for $s q_c=0$ again reproduces the expected behavior Eq. (\[expect\]) with a very high accuracy. The data for $s q_c=10$ only show this behavior for $s'q_c > 10^5$. For $s q_c=1000$ a fit for $10^6<s' q_c< 10^7$ gives $0.3934 \ln{(s')}$ which is very close to $0.3929 \ln{(s')}$, the expected behavior for $p_{\rho} = \infty$. Similar to the $s q_c=100$ curve for arguments between $10^6$ and $10^7$, the curve for $s q_c=1000$ shows a cross over at very large $s' q_c$ and the prefactor of the logarithmic term is again given by the $p_{\rho} = 2$ value $0.1904$.
For the approximated Green’s function Eq. (\[gapprox\]) and the box potential the two-dimensional Fourier transformation which leads to the momentum resolved spectral function can be performed analytically[@vmks; @ksvm]. Thus far we have not succeeded in calculating $\rho_{\alpha,s}^{<}(k,\omega)$ for an interaction with $p_{\nu} <
\infty$. From the Fourier transformation of the approximated Green’s function it is known, that the exponents of the algebraic decay along the special directions ($c=v_{\nu}$) determine the exponents of the non-analyticities at $\omega = \pm v_{\nu} (k-k_F)$. As we have shown above the exponents of the algebraic decay of $G_{+}^{<} ( x,t)$ along the special directions are different from the exponents of $[G_{+}^{<}]_A (x,t) $, thus we have [*no*]{} reason to believe that $\rho_{+}^{<}(k,\omega)$ shows power law singularities with the same exponents as $[\rho_{+}^{<}]_A(k,\omega)$. [*It is not even obvious that the exact spectral function shows power law singularities at all.*]{} Certainly we expect the two peak structure of spin and charge excitations, but it is not clear if, for any non-vanishing $k-k_F$, these peaks are given by algebraic singularities. The extended region in which $G_{+}^{<} ( x,t)$ displays the asymptotic behavior of $[G_{+}^{<}]_A (x,t) $ on the other hand indicates that the resulting spectral functions might look very similar at least for very small $|k-k_F|$. For $|k-k_F| \to 0$ we expect to find growing regions in which the exact spectral function resembles the power law behavior of the approximation (with the same exponents as the approximation) up to energies very close to $\pm v_{\nu} (k-k_F)$ but [*not*]{} exactly at these energies. A comparison of broadened spectral functions for a finite system, i. e. of spectral functions with no “real” algebraic singularities, for a box and Gaussian potential is presented in Ref. [@vmdoc] and indeed shows a prominent similarity between both spectra. A more detailed comparison will be given elsewhere.
The results presented here are important for the interpretation of numerically calculated Green’s functions and spectra of microscopic models. They show that it is impossible to determine the LL parameters of the considered model from the asymptotic behavior of the Green’s function within the special directions. The numerical evaluation of the asymptotic behavior of the Green’s function gives on the other hand a possibility to confirm our predictions. Furthermore it should be possible to confirm the predictions by analyzing the finite size scaling of the weight of single peaks in the spectral function of a microscopic model of finite length. For energies close to but different from $\pm v_{\nu} (k-k_F)$ we expect to find power law behavior with exponents given by the LL paramters as in the approximation. Not so for the scaling of the peaks exactly at $\pm v_{\nu} (k-k_F)$. Our results have also consequences for the comparison of angle resolved and angle integrated spectral functions which have been measured by high resolution photoemission[@experiments].
The author would like to thank K. Schönhammer, W. Metzner, and N.Shannon for very helpful discussions.
F. D. M. Haldane, J. Phys. C [**14**]{}, 2585 (1981). For a review see J. Voit, Rep. Prog. Phys. [**58**]{}, 977 (1995). S. Tomonaga, Prog. Theo. Phys. [**5**]{}, 544 (1950). J. M. Luttinger, J. Math. Phys. [**4**]{}, 1154 (1963). J. Sólyom, Adv. Phys. [**28**]{}, 201 (1979). A. Theumann, J. Math. Phys. [**8**]{}, 2460 (1967); I. E. Dzyaloshinskiǐ and A. I. Larkin, Zh. Eksp. Teor. Fiz. [**65**]{}, 411 (1973) \[Sov. Phys. JETP [**38**]{}, 202 (1974)\]; A. Luther and I. Peschel, Phys. Rev. B [**9**]{}, 2911 (1974); H. Gutfreund and M. Schick, Phys. Rev. [**168**]{}, 418 (1968). We assume that the interaction is sufficiently regular. V. Meden, Ph.D. thesis, Universität Göttingen, 1996. B. Dardel [*et al.,*]{} Europhys. Lett. [**19**]{}, 525 (1992); G.-H. Gweon [*et al.,*]{} J. Phys.: Condens. Matter [**8**]{}, 9923 (1996). K. Schönhammer and V. Meden, Phys. Rev. B [**47**]{}, 16205 (1993). V. Meden and K. Schönhammer, Phys. Rev. B [**46**]{}, 15753 (1992); J. Voit, Phys. Rev. B [**47**]{}, 6740 (1993).
|
{#section .unnumbered}
**Supplementary Information**
This Supplementary Information is organized as follows: Supplementary Note \[sec:implementation-ions\] contains a detailed discussion of the implementation of the QND measurement scheme in a trapped-ion quantum simulator, including an analysis of the experimental feasibility of the scheme with different species of ions and using transverse or axial phonon modes. We provide additional information on applications of the QND scheme in Supplementary Note \[sec:numer-study-therm\] and \[sec:eigenst-therm-hypoth\]. In particular, we give details on the numerical analysis of the ferromagnetic transition in the transverse-field Ising, and discuss prospects of an experimental test of the ETH.
Implementation with trapped ions {#sec:implementation-ions}
================================
In the main text we outline the implementation of our QND measurement scheme in a trapped-ion quantum simulator. In this section we elaborate on the detailed derivations behind the short presentation in the main text, and discuss the experimental feasibility of the proposed scheme. The section is structured as follows.
In Supplementary Note \[subsec:Double-Molmer-Sorensen-interaction\] we introduce and provide an analytical study of the double Mølmer-Sørensen (MS) laser configuration (see Fig. 2 of the main text). We show that the low-frequency dynamics is governed by the effective system-meter coupling Hamiltonian $\hat{H}_{{\cal SM}}$, defined in Eq. (2) of the main text (we set $\hbar=1$ hereafter) $$\hat{H}_{{\cal SM}}=\hat{H}^{\prime}\otimes\mathbb{I}+\vartheta\hat{H}\otimes\hat{P},\label{eq:H_QND}$$ where $\hat{P}\equiv i(\hat{a}_{0}^{\dagger}-\hat{a}_{0})/\sqrt{2}$ is the quadrature operator of the center-of-mass (COM) phonon mode, with $\hat{a}_{0}(\hat{a}_{0}^{\dag})$ the corresponding annihilation(creation) operator. Both $\hat{H}$ and $\hat{H}^{\prime}$ are many-body spin Hamiltonians of the Ising type, $$\begin{aligned}
\hat{H} & =-\sum_{i<j}^{N}J_{ij}\hat{\sigma}_{i}^{x}\hat{\sigma}_{j}^{x}-h\sum_{j=1}^{N}\hat{\sigma}_{j}^{z},\label{eq:H_Ising_coupling_to_meter}\\
\hat{H}' & =-\text{\ensuremath{\sum_{i<j}^{N}J_{ij}\hat{\sigma}_{i}^{x}\hat{\sigma}_{j}^{x}}}-(B-h)\sum_{j=1}^{N}\hat{\sigma}_{j}^{z}.\label{eq:H_Ising_intro}\end{aligned}$$ By adjusting the transverse field strength $B$, we are able to tune the measurement from QND $(B=2h)$ to imperfect QND $(B\simeq2h)$ which supports the observation of quantum jumps.
In Supplementary Note \[sec: Numerical\] we provide a numerical study of the double MS scheme in different parameter regimes which supports the validity of the system-meter coupling Hamiltonian Eq. .
In Supplementary Note \[sec:continuous\_readout\] we describe the continuous readout of the spin Hamiltonian $\hat{H}$, achieved by sideband laser cooling of the motion of an ancilla ion at the edge of the ion chain and homodyne detection of its fluorescence, as schematically shown in Fig. 2(a) of the main text. We derive the resulting dynamics of the spin system as described by the stochastic master equation (SME) $$\begin{aligned}
d\hat{\rho}_{c}(t) &=&-i[\hat{{H}}',\hat{\rho}_{c}(t)]dt+\gamma\mathcal{D}[\hat{H}/J]\hat{\rho}_{c}(t)dt \nonumber
\\
&&+\sqrt{\epsilon\gamma}\mathcal{H}[\hat{H}/J]\hat{\rho}_{c}(t) dW(t).\label{eq:SME_measureH}\end{aligned}$$ Here $\hat{\rho}_c(t)$ is the density matrix of the spin system conditioned on the homodyne detection signal, $J$ is the characteristic energy scale of the spin Hamiltonian, $\gamma$ is an effective measurement rate, $\epsilon$ is an overall detection efficiency and $dW(t)$ a white noise Wiener increment. For the detailed expressions of $J$ and $\gamma$, cf. the main text or Supplementary Note \[sec:continuous\_readout\] below. The corresponding homodyne current reads $$I(t)=2\sqrt{\epsilon\gamma}\langle\hat{H}/J\rangle_{c}+\xi(t),\label{eq:Ih_measureH}$$ with $\xi(t)$ white (shot) noise $dW(t)\equiv \xi(t)dt$. We conclude Supplementary Note \[sec:continuous\_readout\] with a brief discussion on the filtering of the homodyne current.
In Supplementary Note \[sec:experiment\] we discuss some experimental considerations on the proposed trapped-ion implementation, including the analysis of its scalability, and the discussion of its robustness against major experimental imperfections. Supplementary Note \[sec:experiment\] also provides typical numbers for a proof-of-principle experiment.
We remark that our QND measurement scheme can be implemented with both transverse ($x$ direction) and axial ($z$ direction) phonon modes of the 1D ion string. While transverse phonon modes give rise to the power-law spin interactions $J_{ij}\propto |i-j|^{-\alpha}$ with $1<\alpha<3$ as is considered in the main text, axial phonon modes provide rich opportunities for engineering exotic spin couplings [@Porras2004]. The derivation of our scheme for both cases is essentially the same. For notational concreteness, in the following Supplementary Notes \[subsec:Double-Molmer-Sorensen-interaction\] and \[sec:continuous\_readout\], we derive the equations by assuming transverse phonon modes. With the simple replacement $x\to z$ for the ionic motional operators, the same derivation applies to the axial case. In Supplementary Note \[sec:experiment\], we discuss the features and experimental requirements of the transverse and the axial implementation separately.
Analytical study of the double Mølmer-Sørensen configuration {#subsec:Double-Molmer-Sorensen-interaction}
------------------------------------------------------------
In this section we analyze in detail the laser configuration which generates the desired system-meter coupling $\hat{H}_{{\cal SM}}$ as an effective Hamiltonian derived in perturbation theory for the laser assisted spin-mode couplings. While the model Hamiltonian in the main text refers to the lowest order terms in this expansion, we also derive the higher-order corrections to $\hat{H}_{{\cal SM}}$ and argue that they are indeed negligible under typical experimental conditions.
### Light-ion coupling\[subsec:Light-ion-coupling\]
We consider $N$ ions trapped in a linear Paul trap. The internal structure of each ion is assumed to be a two level system (TLS), consisting of two qubit states $\left|\downarrow\right\rangle $ and $\left|\uparrow\right\rangle $. The transition $\left|\downarrow\right\rangle \rightarrow\left|\uparrow\right\rangle $ is driven by two pairs of laser beams, such that each pair realizes a Mølmer-Sørensen (MS) configuration, as shown schematically in Fig. 2 of the main text. The lasers which form the first pair, shown as the amber beams, are detuned by $\pm\Delta$ from the qubit transition frequency $\omega_{\uparrow\downarrow}\equiv E_{\uparrow}-E_{\downarrow}$ respectively, and have wave vector projections $\pm k$ along the $x$ direction; The lasers corresponding to the second pair, shown as the blue beams, are detuned by $\pm\Delta'$ from $\omega_{\uparrow\downarrow}$, and have wave vector projections $\mp k$ along the $x$ direction. In the frame rotating at $\omega_{\uparrow\downarrow}$, the full Hamiltonian of the internal and motional degrees of freedom (DOFs) of the ion chain reads $$\hat{H}_{\text{full}}=\hat{H}_{0}+\hat{V}.\label{eq:H_t}$$ Here $\hat{H}_{0}$ is the Hamiltonian of the external motion of the ions (along the $x$ direction), and can be expressed in terms of the collective phonon modes $$\label{eq: phonon_free_H}
\hat{H}_{0}=\sum_{q}\omega_{q}\hat{a}_{q}^{\dagger}\hat{a}_{q}.$$ where $\omega_{q}$ and $\hat{a}_{q}$ respectively denote the frequency and the annihilation operator of the mode $q$. Here, the modes are ordered according to their energy. For transverse phonon modes, the COM mode $q = 0$ has the highest frequency, and therefore $\omega_{q} > \omega_{q+1}$. The order of modes is reversed for axial phonon modes. The interaction between the ions and the lasers is described by the Hamiltonian $$\begin{aligned}
\hat{V}= & \frac{1}{2}\sum_{j=1}^{N}\hat{\sigma}_{j}^{+}\Big(\Omega_{1}e^{-i\Delta t+ik\hat{X}_{j}+i\zeta_{j}^{1}}+\Omega_{2}e^{i\Delta t-ik\hat{X}_{j}+i\zeta_{j}^{2}}\nonumber \\
& +\Omega_{3}e^{-i\Delta't-ik\hat{X}_{j}+i\zeta_{j}^{3}}+\Omega_{4}e^{i\Delta't+ik\hat{X}_{j}+i\zeta_{j}^{4}}\Big)+{\rm {H.c}.}\label{eq:Hint_schro_pic}\end{aligned}$$ Here $\Omega_{m}$ with $m = 1, \dotsc, 4$ denotes the Rabi frequency of the laser beams, which is assumed to be real and positive for concreteness. The lasers with indices $m = 1,2$ correspond to the first MS pair and are shown as amber beams in Fig. 2 of the main text. The blue beams, which correspond to the lasers with indices $m = 3,4$, form the second MS pair. For the $j$-th ion, $\hat{\sigma}_{j}^{+}\equiv\left|\uparrow\right\rangle _{j}\left\langle
\downarrow\right|$ is its internal raising operator, and $\zeta_{j}^{m}$ is the phase of laser $m$ at its equilibrium position. The operator $\hat{X}_{j}$ describes the small-amplitude displacement from the equilibrium position along the $x$ direction, and can be expressed in terms of the phonon operators as $k\hat{X}_{j}=\sum_{q}\eta_{q}M_{jq}\left(\hat{a}_{q}+\hat{a}_{q}^{\dagger}\right)$, where $M_{jq}$ is the distribution matrix element of mode $q$, and the Lamb-Dicke (LD) parameters are defined as $\eta_{q}=\eta\sqrt{\omega_{0}/\omega_{q}}$, with $\eta=k/\sqrt{2m\omega_{0}}$.
Hereafter we consider the Rabi frequencies of the four laser beams being approximately equal up to a small offset, $$\begin{aligned}
\Omega_{1} & =\Omega_{3}=\Omega,\nonumber \\
\Omega_{2} & =\Omega_{4}=\Omega+\delta\Omega.\label{eq:Rabi_frequency_choice}\end{aligned}$$ According to Supplementary Note \[subsec:Expansion-with-respect\], the small Rabi frequency mismatch $\delta\Omega$ creates the desired transverse field term of the Ising Hamiltonians and , with the transverse field strength $h\propto\delta\Omega$. A pair of Mølmer-Sørensen laser beams is known to create the Ising spin Hamiltonian Eq. in the off-resonant regime $\Delta^{(\prime)}\gg\Omega$, $|\Delta^{(\prime)}-\omega_{q}|\gg\eta_{q}\Omega$ (see Refs. [@Kim2009] and the discussion below). In our double MS configuration, however, an additional term describing the QND coupling between the Ising spin Hamiltonian and the COM phonon mode is generated [\[]{}see the second term of Eq. . This is achieved by tuning $\Delta^{\prime}=\Delta+\omega_{0}$, i.e., by choosing the beating between the two pairs of MS lasers to match the COM phonon excitation frequency. It leads to a resonant crosstalk between the two MS configurations, which results in the desired QND coupling term.
In the following, we derive Eq. via a Magnus expansion of the time evolution of the ion chain in the interaction picture. We shall first introduce our method, which is a combined Magnus expansion and Lamb-Dicke expansion, in Supplementary Notes \[subsec:methodology\_magnus\] and \[subsec:Expansion-with-respect\], respectively, and then work out the detailed expression of Eq. order by order. We summarize the results in Supplementary Note \[subsec:summary\_and\_tuning\].
### Magnus expansion: effective Hamiltonian {#subsec:methodology_magnus}
Performing the gauge transformation $\hat{\sigma}_{j}^{+}\to\hat{\sigma}_{j}^{+}\exp\left[\left.-i\left(\zeta_{j}^{1}+\zeta_{j}^{2}\right)\right/2\right]$ and moving into the interaction picture with respect to $\hat{H}_{0}$, Eq. becomes $$\begin{aligned}
\hat{V}_{I}= & \frac{\Omega}{2}\sum_{j=1}^{N}\hat{\sigma}_{j}^{+}\Big[ e^{-i\Delta t+ik\hat{X}_{j}\left(t\right)+i\varphi_{j}}\nonumber \\
& +{\left( 1+\frac{\delta\Omega}{\Omega} \right)} e^{i\Delta t-ik\hat{X}_{j}\left(t\right)-i\varphi_{j}}\nonumber \\
& +e^{-i\Delta't-ik\hat{X}_{j}\left(t\right)+i(\theta+\varphi_{j}^{\prime})}\nonumber \\
& + \left( 1+\frac{\delta\Omega}{\Omega} \right) e^{i\Delta't+ik\hat{X}_{j}(t)+i(\theta-\varphi_{j}^{\prime})}\Big]+{\rm {H.c}.}\label{eq:H_int}\end{aligned}$$ where the time-dependent position operator can be expressed in terms of the phonon modes as $k\hat{X}_{j}\left(t\right)=\sum_{q}\eta_{q}M_{jq}\hat{x}_{q}(t)$ with $\hat{x}_{q}(t)\equiv\hat{a}_{q}{\rm exp}(-i\omega_{q}t)+{\rm H.c.}$, and the relative laser phases are denoted as $\theta=\left(\zeta_{j}^{3}+\zeta_{j}^{4}-\zeta_{j}^{1}-\zeta_{j}^{2}\right)/2,$ $\varphi_{j}=\left(\zeta_{j}^{1}-\zeta_{j}^{2}\right)/2,$ and $\varphi_{j}'=\left(\zeta_{j}^{3}-\zeta_{j}^{4}\right)/2$. We note that the phase $\theta$ is independent of the ion index $j$. For the implementation with transverse phonons considered here, this is simply because the laser phase $\zeta_j^m$ is independent of $j$. For the axial implementation, this is also true, as the position dependencies of the phases of two counter-propagating lasers cancel each other.
We consider the regime where the MS lasers drive the qubit transition and the phonon sidebands off-resonantly, $\Delta^{(\prime)}\gg\Omega$, $|\Delta^{(\prime)}-\omega_{q}|\gg\eta_{q}\Omega$. The evolution operator corresponding to Eq. can be formally written as a Magnus series, $$\begin{aligned}
\hat{U}(t) & \equiv\exp\left[-i\hat{G}(t)\right]=\text{\ensuremath{\mathcal{T}}exp}\left[-i\int_{0}^{t}dt_{1}\hat{V}_{I}\left(t_{1}\right)\right],\nonumber \\
\hat{G}\left(t\right) & =\sum_{l=1}^{\infty}\hat{G}_{l}(t).\label{eq:G}\end{aligned}$$ Correspondingly, it allows us to define an effective Hamiltonian $\hat{H}_{\text{eff}}\equiv\lim_{t\rightarrow\infty}\hat{G}\left(t\right)/t$ which describes the slow dynamics of the ion chain on a time scale much longer than the phononic oscillation period $\sim 1/\omega_{0}$ [@Kim2009]. The lowest-order terms of the Magnus series are given by $$\begin{aligned}
\hat{G}_{1}(t) =&\int_{0}^{t}dt_{1}\hat{V}_{I}\left(t_{1}\right)dt_{1},\label{eq:G_1}\\
\hat{G}_{2}(t) =&-\frac{i}{2}\int_{0}^{t}dt_{1}\int_{0}^{t_{1}}dt_{2}\left[\hat{V}_{I}\left(t_{1}\right),\hat{V}_{I}\left(t_{2}\right)\right]\label{eq:G_2},\\
\hat{G}_{3}(t) =&-\frac{1}{6}\int_{0}^{t}dt_{1}\int_{0}^{t_{1}}dt_{2}\int_0^{t_2}dt_3\nonumber\\
&\Big\{\left[\hat{V}_{I}\left(t_{1}\right),\left[\hat{V}_{I}\left(t_{2}\right),\hat{V}_I\left(t_3\right)\right]\right]\nonumber\\
&+
\left[\left[\hat{V}_{I}\left(t_{1}\right),\hat{V}_{I}\left(t_{2}\right)\right],\hat{V}_I\left(t_3\right)\right]
\Big\}.
\label{eq:G_3}\end{aligned}$$ In the next section we derive $\hat{H}_{{\rm eff}}$ via explicit calculation of $\hat{G}(t)$ in the long-time limit. In this calculation we further perturbatively expand $\hat{V}_{I}$ in terms of the small Lamb-Dicke parameter $\eta\ll1$. This allows us to construct $\hat{H}_{{\rm eff}}$ order by order as a systematic expansion in $\eta$.
### Expansion with respect to the Lamb-Dicke parameter $\eta$ {#subsec:Expansion-with-respect}
We now construct the effective Hamiltonian $\hat{H}_{{\rm eff}}$ as an expansion with respect to the Lamb-Dicke parameter, $\hat{H}_{\text{eff}}=\sum_{\ell=0}^{\infty}\hat{H}_{\text{eff}}^{(\ell)}$ with $\hat{H}_{\text{eff}}^{(\ell)}\propto\eta^{\ell}$. In order to do that, we first expand the interaction Hamiltonian Eq. as $\hat{V}_{I}=\sum_{\ell=0}^{\infty}\hat{V}_{I}^{(\ell)}$, with $$\begin{aligned}
\hat{V}_{I}^{(\ell)}\left(t\right)= & \frac{1}{2\times \ell !}\sum_{j=1}^{N}\hat{\sigma}_{j}^{+}\bigg[i\sum_{q}\eta_{q}M_{jq}\hat{x}_{q}\left(t\right)\bigg]^{\ell}\nonumber \\
& \times\bigg[{\Omega}e^{-i\Delta t+i\varphi_{j}}+\left(-1\right)^{\ell}(\Omega+\delta\Omega)e^{i\Delta t-i\varphi_{j}}\nonumber \\
& +\left(-1\right)^{\ell}\Omega e^{-i\Delta't+i(\theta+\varphi_{j}^{\prime})}\nonumber \\
& +(\Omega+\delta\Omega)e^{i\Delta't+i(\theta-\varphi_{j}^{\prime})}\bigg]+{\rm {H.c}.}\label{eq:V_I_expansion}\end{aligned}$$ To simplify the analysis, hereafter we consider $\theta=0$. Moreover, we choose $\varphi_{j+1}-\varphi_{j}=2\pi s,$ $\varphi_{j+1}^{\prime}-\varphi_{j}^{\prime}=-2\pi s$ with $s\in\mathbb{Z}$. For the implementation using transverse phonon modes, this condition is automatically satisfied, with $s=0$. For the implementation using axial phonon modes, this can be achieved by using the central part of an ion chain in a standard Paul trap with nearly uniform spacing $d$ (or alternatively by using ions in equal-distance ion traps [@Schulz_2008; @Harlander2011; @Mehta2014; @Wilson2014]) and by choosing an appropriate wavevector $k$ of the MS beams such that $kd=2\pi s$.
### Contributions from $\hat{G}_1(t)$
Substituting the expression Eq. into Eq. and taking into account the conditions $|\Delta|\gg\Omega$ and $|\Delta-\omega_{q}|\gg\eta_{q}\Omega$, we immediately see that $\hat{G}_{1}(t)$ does not contribute to $\hat{H}_{\text{eff}}$. Indeed, $\hat{G}_{1}(t)$ describes small-amplitude fast oscillations at frequency $\sim\Delta$, which average to zero in the long-time regime.
### Contributions from $\hat{G}_2(t)$ up to $\eta^3$: $\hat{H}_{\cal SM}$
Below, we derive the contribution from $\hat{G}_{2}\left(t\right)$ as an expansion in the Lamb-Dicke parameter. In this derivation, we implicitly assume that the small offset of the Rabi frequency [\[]{}see Eq. satisfies $\delta\Omega/\Omega\sim O(\eta_{q}^{2})$.
#### Transverse field terms.
The zeroth-order expansion of $\hat{H}_{{\rm eff}}$ is readily constructed by plugging $\hat{V}_{I}^{(0)}$ into Eq. , $$\begin{aligned}
\hat{H}_{\text{eff}}^{\left(0\right)} & =-\frac{i}{2t}\int_{0}^{t}dt_{1}\int_{0}^{t_{1}}dt_{2}\left[\hat{V}_{I}^{\left(0\right)}\left(t_{1}\right),\hat{V}_{I}^{\left(0\right)}\left(t_{2}\right)\right]\nonumber \\
& =h\sum_{j=1}^{N}\hat{\sigma}_{j}^{z}.\label{eq:H^0}\end{aligned}$$ We note that here and below, we implicitly assume the long-time limit $t \to \infty$. The zeroth-order contribution provides the transverse field term of the quantum Ising Hamiltonian, with the transverse field strength given by $$h=\frac{\Omega\delta\Omega}{2}\left(\frac{1}{\Delta}+\frac{1}{\Delta^{\prime}}\right).\label{eq:transverse_field}$$ Similarly, the first-order expansion $\hat{H}_{{\rm eff}}^{(1)}$ is given by $$\begin{aligned}
\hat{H}_{\text{eff}}^{\left(1\right)} & =-\frac{i}{2t}\sum_{\ell+m=1}\int_{0}^{t}dt_{1}\int_{0}^{t_{1}}dt_{2}\left[\hat{V}_{I}^{(\ell)}\left(t_{1}\right),\hat{V}_{I}^{(m)}\left(t_{2}\right)\right]\nonumber \\
& =-\vartheta h\sum_{j=1}^{N}\hat{\sigma}_{j}^{z}\otimes\hat{P},\label{eq:H^1}\end{aligned}$$ where $\hat{P}\equiv i(\hat{a}_{0}e^{i\varphi}-\hat{a}_{0}^{\dagger}e^{-i\varphi})/\sqrt{2}$ is a quadrature operator of the COM phonon mode, with $\varphi\equiv\varphi_{j}-\varphi_{j}'$ an angle dependent on the laser phases, and $\vartheta=-\eta_{0}\sqrt{2}M_{i0}\simeq-\eta_{0}\sqrt{2/N}$ the dimensionless coupling strength. In the following, we absorb the phase $\varphi$ into the definition of $\hat{a}_{0}$, $\hat{a}_{0}e^{i\varphi}\to-\hat{a}_{0}$, thus $\hat{P}=i(\hat{a}_{0}^{\dag}-\hat{a}_{0})/\sqrt{2}$.
#### Ising terms.
The second order expansion of $\hat{H}_{{\rm eff}}$ can be constructed analogously, $$\begin{aligned}
\hat{H}_{\text{eff}}^{\left(2\right)} & =-\frac{i}{2t}\sum_{\ell+m=2}\int_{0}^{t}dt_{1}\int_{0}^{t_{1}}dt_{2}\left[\hat{V}_{I}^{(\ell)}\left(t_{1}\right),\hat{V}_{I}^{(m)}\left(t_{2}\right)\right]\nonumber \\
& =-\sum_{i<j}J_{ij}\hat{\sigma}_{i}^{x}\hat{\sigma}_{j}^{x}.\label{eq:H^2}\end{aligned}$$ In this derivation we dropped terms $\sim\eta^{2}\delta\Omega$ under our assumption $\delta\Omega/\Omega\propto\eta^{2}$. We discuss the effect of these higher-order terms in Supplementary Note \[sec:higher-order\_correction\]. Equation describes an Ising spin-spin coupling with coupling strength $$\begin{aligned}
J_{ij}= & -\Omega^{2}\sum_{q}\eta_{q}^{2}\omega_{q}M_{iq}M_{jq}\nonumber \\
& \times\left[\frac{1}{\Delta^{2}-\left(\omega_{q}\right)^{2}}+\frac{1}{(\Delta')^{2}-\left(\omega_{q}\right)^{2}}\right],\label{eq:J_mn}\end{aligned}$$ which includes two independent contributions from the two MS laser configuration.
Finally, the third order expansion of $\hat{H}_{{\rm eff}}$ can be calculated in an analogous (though lengthy) way $$\begin{aligned}
\hat{H}_{\text{eff}}^{\left(3\right)} & =-\frac{i}{2t}\sum_{\ell+m=3}\int_{0}^{t}dt_{1}\int_{0}^{t_{1}}dt_{2}\left[\hat{V}_{I}^{(\ell)}\left(t_{1}\right),\hat{V}_{I}^{(m)}\left(t_{2}\right)\right]\nonumber \\
& =-\vartheta\Bigg(\sum_{i<j}J_{ij}\hat{\sigma}_{i}^{x}\hat{\sigma}_{j}^{x}+\mathcal{E}\Bigg)\otimes\hat{P}.\label{eq:H^3}\end{aligned}$$ Here, the spin-spin interaction strength $J_{ij}$ is defined in Eq. , and $\vartheta$ and $\hat{P}$ are defined below Eq. . $\mathcal{E}\equiv\sum_{j}J_{jj}/2$ is a constant driving field for the COM quadrature, which we neglect in the following as it just leads to a constant component in the measured signal. Equation results from a resonant cross-talk between the two MS laser configuration under the condition $\Delta'=\Delta+\omega_{z}$, and describes the QND coupling between the spin Hamiltonian and the quadrature of the COM phonon mode.
In deriving Eq. , an important assumption we made is that no other phonon mode is resonantly excited except the COM mode up to third order in the Lamb-Dicke parameter. This “single sideband addressibility” is guaranteed by the condition $$\sum_q \frac{\eta^2_q\Omega^2\omega_q\eta_p}{[\Delta^{(\prime)}]^2-\omega_q^2}\ll |\omega_p-\omega_0|,\quad \quad\forall p\neq 0.
\label{eq:validity}$$ Equation can be interpreted physically as follows: In our scheme, the excitation of sideband phonons is achieved by simultaneously flipping *two* spins. The strength for simutaneously flipping spin $i$ and $j$ is given by $J_{ij}$ in Eq. . As a result, the sideband addressing strength is $\sim \eta \Omega^2\sum_q \eta^2_q\omega_q/\{[\Delta^{(\prime)}]^2-\omega_q^2\}$ (we set $M_{iq}=1$ for a worst-scenario analysis), and should be much smaller than the spectral gap between the COM mode and other modes.
In a transverse-phonon implementation, the phonon spectrum gets denser for increasing number of ions. Thus the validity of condition sets a limit on the scalability of our QND scheme. This is analyzed in detail later in Supplementary Note \[sec:experiment\]. Here, we only note that Eq. is much less stringent than the requirement for sideband addressing via laser flipping *individual* spins, $\eta_q \Omega \ll |\omega_q-\omega_0|,\quad\forall q\neq 0$, as is required, e.g., by the Cirac-Zoller gate or the near-resonant Mølmer-Sørensen gate. This leads to nice scalability of our QND measurement scheme for a given laser power. Combining Eqs. , , and , the effective Hamiltonian of the ion chain $\hat{H}_{{\rm eff}}$ can be written in the form of $\hat{H}_{\cal SM}$ in Eq. , with the identification $$\begin{aligned}
\hat{H} & =-\sum_{i<j}J_{ij}\hat{\sigma}_{i}^{x}\hat{\sigma}_{j}^{x}-h\sum_{j=1}^{N}\hat{\sigma}_{j}^{z},\nonumber \\
\hat{H}' & =-\text{\ensuremath{\sum_{i<j}J_{ij}\hat{\sigma}_{i}^{x}\hat{\sigma}_{j}^{x}}}+h\sum_{j=1}^{N}\hat{\sigma}_{j}^{z},\label{eq:spinHamiltonian_details}\end{aligned}$$ $\vartheta\simeq-\eta_{0}\sqrt{2/N}$ and $\hat{P}=i(\hat{a}_{0}^{\dag}-\hat{a}_{0})/\sqrt{2}$.
### Tuning of $\hat{H}_{{\cal SM}}$ {#subsec:summary_and_tuning}
Here we describe a method to further tune the transverse field in $\hat{H}$ and $\hat{H}'$ [\[]{}cf. Eq. independently, thus allowing to reach the QND sweetspot $\hat{H}=\hat{H}'$. To this end, we consider the same laser configuration as in Supplementary Note \[subsec:Light-ion-coupling\], nevertheless the detunings of the MS lasers are now respectively modified to $B\pm\Delta,$ $B\pm\Delta^{\prime}$, with $B\sim J\ll\Delta,\Delta^{\prime}$. In the frame rotating at frequency $\omega_{\uparrow\downarrow}+B$, we get an additional term $B\sum_{j=1}^{N}\hat{\sigma}_{j}^{z}$ in the Hamiltonian of the laser-driven ion chain $\hat{H}_{\text{full}}$ [\[]{}cf. Eq. . Repeating the same derivation as described in Supplementary Notes \[subsec:Light-ion-coupling\] and \[subsec:Expansion-with-respect\], we recover exactly the same $\hat{H}$ that is coupled to the meter DOFs, while $H_{}^{\prime}$ is modified as $$H^{\prime}=\text{\ensuremath{\sum_{i<j}J_{ij}\hat{\sigma}_{i}^{x}\hat{\sigma}_{j}^{x}}}-\left(h-B\right)\sum_{j}\hat{\sigma}_{j}^{z}.$$ By choosing $B=2h$ we realize the QND condition $\hat{H}^{\prime}=\hat{H}$, while offsetting $B$ slightly from $2h$ allows us to observe quantum jumps between different eigenstates of $\hat{H}$.
### Contributions from $\hat{G}_2(t)$ beyond $\eta^3$: higher-order corrections {#sec:higher-order_correction}
In this section we derive the corrections to the QND Hamiltonian Eq. resulting from higher-order terms in the Lamb-Dicke expansion of $\hat{G}_2(t)$. We show that these terms do not change the QND character of the proposed measurement scheme.
By straight forward calculation, we find that the fourth order expansion of the effective Hamiltonian can be written as $$\begin{aligned}
\hat{H}_{\text{eff}}^{\left(4\right)} & =-\frac{i}{2t}\sum_{\ell+m=4}\int_{0}^{t}dt_{1}\int_{0}^{t_{1}}dt_{2}\left[\hat{V}_{I}^{(\ell)}\left(t_{1}\right),\hat{V}_{I}^{(m)}\left(t_{2}\right)\right]\nonumber \\
& =-\sum_{i<j}J_{ij}^{(4)}\hat{\sigma}_{i}^{x}\hat{\sigma}_{j}^{x},\label{eq:H^4}\end{aligned}$$ with the spin-spin coupling $$\begin{aligned}
J_{ij}^{(4)}= & -\frac{\Omega^{2}}{2}\sum_{qp}\eta_{q}^{2}\eta_{p}^{2}M_{iq}M_{jq}M_{ip}M_{jp}(\omega_{q}+\omega_{p})\nonumber\\
& \times\left[\frac{1}{\Delta^{2}-(\omega_{q}+\omega_{p})^{2}}+\frac{1}{(\Delta')^{2}-(\omega_{q}+\omega_{p})^{2}}\right]\nonumber\\
& +\frac{\Omega^{2}}{2}\eta_{0}^{2}\omega_{z}\sum_{p}\eta_{p}^{2}(M_{ip}^{2}+M_{jp}^{2})\nonumber\\
& \times\sum_{q}M_{iq}M_{jq}\left[\frac{1}{\Delta^{2}-\omega_{q}^{2}}+\frac{1}{(\Delta')^{2}-\omega_{q}^{2}}\right].\label{eq:Jij4}\end{aligned}$$ In the derivation of Eq. , we dropped terms proportional to the phonon-occupation under the assumption $\langle\hat{a}_{q}^{\dag}\hat{a}_{q}\rangle\ll1$.
Besides $\hat{H}_{\text{eff}}^{\left(4\right)}$, another correction to the QND Hamiltonian that is fourth order in $\eta$ comes from the term $\sim\eta^{2}\delta\Omega\propto\eta^{4}$ which we dropped in Eq. . Via straightforward calculation we find this term can be written as a transverse field Ising Hamiltonian with site-dependent transverse field, $$-\sum_{i<j}t_{ij}\hat{\sigma}_{i}^{x}\hat{\sigma}_{j}^{x}-\sum_{j}\lambda_{j}\hat{\sigma}_{j}^{z},\label{eq:Hcorrection}$$ with the coefficients $$\begin{aligned}
t_{ij}= & -\Omega\delta\Omega\sum_{q}\omega_{q}\eta_{q}^{2}M_{iq}M_{jq}\nonumber \\
& \times\left[\frac{1}{\Delta^{2}-\omega_{q}^{2}}+\frac{1}{(\Delta')^{2}-\omega_{q}^{2}}\right],\nonumber \\
\lambda_{j}= & -2\Omega\delta\Omega\sum_{q}\eta_{q}^{2}M_{jq}^{2}\left(\frac{1}{\Delta}+\frac{1}{\Delta'}\right).\end{aligned}$$
Importantly, the corrections to the QND Hamiltonian up to fourth order in the Lamb-Dicke parameter, Eq. and , only involve spin DOFs and do not involve phonon DOFs. Thus, they only slightly renormalize the coefficients of $\hat{H}'$ [\[]{}cf. Eq. , introducing tiny mismatch between $\hat{H}'$ and $\hat{H}$. As described in the main text, these mismatch only introduce rare quantum jumps between energy eigenstates [\[]{}cf. Fig. 1(3) of the main text[\]]{}, whereas the QND character of the measurement is maintained.
### Effects of higher order Magnus series
In this section, we show that the contributions to the effective Hamiltonian from higher-order terms in the Magnus series, $\hat{G}_{n}(t),n\geq 3$, are of higher order than $\eta^4$. Thus they are much smaller than the system-meter coupling $\hat{H}_{\cal SM}$ and do not change the QND character of the proposed measurement scheme. The following analysis is based on power counting and physical arguments. Let us first consider the contributions from $\hat{G}_3(t)$, cf. Eq. . For simplicity, we temporarily assume balanced MS configurations, i.e., $\delta\Omega=0$. In this case, the spin operators $\hat{\sigma}_{j}^\pm$ in Eq. can be combined to $\hat{\sigma}_j^x$. Therefore, spin operators always commute with each other at different time. As a result, the double commutator in the integrand of Eq. is nonzero only if both the inner and outer commutators contain at least a pair of phonon annihilation and creation operators. Such an integrand, contains at least four phonon operators, and its contribution is $O(\eta^4)$ after the integration. Moreover, it is straightforward to see such a contribution contains a fast oscillating phase $\sim \Delta^{(\prime)} t$ and do not contribute to the effective Hamiltonian. Physically, it corresponds to processes which involve two virtual phonon excitations, which are nevertheless off-resonant. Thus, $\hat{G}_3(t)$ contributes $O(\eta^5)$ to the effective Hamiltonian when $\delta\Omega=0$.
We now reintroduce the imbalance $\delta\Omega$. Keeping in mind that $\delta\Omega/\Omega\sim O(\eta^2)$, in the integrand of Eq. we can restrict ourselves to ‘relevant’ terms that contain $\delta\Omega$ and at most two phonon operators, since the other terms contribute at $O(\eta^5)$ after multiplication with $\delta\Omega$. Under the conditions $\Delta^{(\prime)}\gg\Omega$ and $|\Delta^{(\prime)}-\omega_{q}|\gg\eta_{q}\Omega$, which are met in our off-resonant double MS configuration, it is straightforward to verify that the ‘relevant’ terms involving zero or one phonon operator contain fast oscillating phase $\sim \omega_z t$ and average to zero in the long time limit. The analysis of the terms consisting of $\delta\Omega$ and two phonon operators is more involved. First, we note that these terms are of the order $O(\eta^4)$, i.e., much smaller than the QND coupling Hamiltonian $\hat{H}_{\cal SM}$. Secondly, we find that they are off-resonant and average to zero if we avoid ‘accidental’ resonances, by requiring $|\omega_q-\omega_p|\neq |\Delta-\omega_0|,\, \forall p,q$, which can be achieved by choosing appropriate $\Delta$ in experiments. To summarize, the contribution of $\hat{G}_3(t)$ can be made as small as $O(\eta^5)$ and is thus negligible.
Next, we consider the contributions from $\hat{G}_4(t)$. Adopting similar arguments as above, we find that if $\delta\Omega=0$ the contribution from $\hat{G}_4(t)$ is on the order of $O(\eta^6)$, and corresponds physically to processes involving three virtual phonon excitations. Reintroducing $\delta\Omega$ and looking at ‘relevant’ terms that contain at most two phonon operators, we find only one term that does not oscillate and remains finite in the long-time limit, under the assumption that accidental resonances are avoided and the phonon occupations are small, $\langle\hat{a}_{q}^{\dag}\hat{a}_{q}\rangle\ll1$. This term is proportional to $\delta \Omega\times\Omega^3/\Delta^3\sum_j \hat{\sigma}_j^z$ and describes the AC-Stark shift from fourth-order perturbation theory. Comparing this term to the QND coupling terms and , we find it has a relative strength $\Omega^2/(\eta\Delta^2)$. To ensure that this term has negligible effect on the performance of our QND scheme, we thus require $\Omega^2/(\eta\Delta^2)\ll 1$. This is typically true, as $\eta$ and $\Omega/\Delta$ are small parameters which have similar magnitudes.
Along the same lines, we can show that the contributions from $\hat{G}_n,n> 4$ are all negligible.
Numerical study of the double Mølmer-Sørensen configuration {#sec: Numerical}
-----------------------------------------------------------
We complement the analytical investigation of the double Mølmer-Sørensen laser configuration scheme above by a numerical study of the evolution of a system of $N=3$ ions in a truncated phonon basis, and we identify parameter regimes in which a description of the system in terms of the effective Hamiltonian Eq. is valid. We consider two scenarios: First, we study the evolution with $\delta \Omega = 0$, which corresponds to the Ising model with no transverse field $h=0$, and we assume that the QND scheme is implemented with transverse phonon modes. Under these conditions, we show that in a wide range of detunings $\Delta$ the exact dynamics of the joint system consisting of spin and phonon degrees of freedom modes is well reproduced by the effective Hamiltonian Eq. . Second, for a fixed value of the detuning $\Delta$, we show that the effective Hamiltonian remains valid for a range of values of the transverse field $h$. This analysis assumes the use of axial phonon modes and is based on Floquet theory.
### Dynamics of phonon modes
The starting point of our consideration is the full Hamiltonian of the system Eq. . We focus on the transverse phonon modes and analyse the tunability of the detuning $\Delta$ which defines the spin-spin interaction parameter $\alpha$ as discussed. We assume all Rabi frequencies to be equal ($ \delta \Omega=0$), which corresponds to the Ising model with no transverse field Eq. . In this case the full Hamiltonian simplifies to: $$\hat{V}_I=\sum_{j=1}^{N} \sigma^{x}_{j} \Big[ \cos \Big(\Delta t-k \hat{X}_{j} (t) \Big) + \cos \Big(\Delta' t+k \hat{X}_{j}(t) \Big) \Big]. \label{eq:ReducedH}$$ This implies that for an initial product state in the $\sigma^x$ basis there is no dynamics of spin variables, and only the phonon modes evolve in time. If they are initially prepared in their vacuum states then, according to the effective Hamiltonian given in Eqs. and , at some final time $t_f$ we expect $\langle\hat{x}_q\rangle \propto \sqrt{\langle a_q^{\dag} a_q \rangle} \propto \delta_{q,0}(\sum J_{ij} s^x_i s^x_j )\cdot t_f $, where $\{s^x_i\}$ denote the initial spin configuration.
To check the validity of our model we compute the mean phonon number $ \langle a_q^{\dag} a_q \rangle$ using the full Hamiltonian Eq. for the corresponding initial spin states as a function of the detuning $\Delta$, while keeping $\Delta'=\Delta+\omega_0$. The result is shown in Supplementary Fig. \[Fig:verification0\](a). We observe the appearance of resonances at certain values of $\Delta$, where the evolution does not correspond to the effective Hamiltonian Eq. . Partially this can be explained by the poles of the fourth order expansion term of the $J_{ij}$ matrix Eq. . The rest of resonances have lower amplitude and higher frequency and we attribute them to the many-phonon resonant processes which correspond to the higher-than-fourth order $\eta$-expansion terms.
Remarkably, there exist wide regions of detuning $\Delta$ free of resonances provided that there is space between the higher order harmonics of the phononic eigenfrequencies $\omega_0\gg |\omega_0-\omega_{q}|,\,\forall q$. As shown in Supplementary Fig. \[Fig:verification0\](b) the system dynamics in these regions is well reproduced by the effective model Eqs. , such that the COM mode population reflects the eigenenergies of the spin system.
In order to show that the scheme scales well with the number of ions we present in Supplementary Fig. \[Fig:verification0\_6\](a,b) results for $N=6$ ions interacting via 6 phonon modes. One can see that effective QND dynamics represents the exact results as expected.
The setup which is described in this section is also of interest as a first proof-of-principle experimental test of the proposed QND coupling of the spin model to the COM phonons $\vartheta\hat{H}\otimes\hat{P}$. Importantly, it does not require implementation of the full QND scheme with the continuous readout as the mean phonon-number measurement can be done in a multi-shot fashion.
![(a) Transverse phonon mode occupation at the final time $t_f=2\pi \times 600/\omega_0$ for different initial states of $N=3$ spins indicated with blue, green, and orange colors. Solid, dot-dashed and dashed curves stand for the $q=0,1,2$ modes, respectively. Vertical lines represent frequencies of the higher-order harmonics of the phonon modes. (b) Zoomed-in part \[red rectangle in (a)\] including the first resonances. Dashed curves show the analytical results obtained from Eqs. , . The numerical parameters: $\eta=0.1$, $\Omega/\omega_0=1/6, \omega_0/\omega_z=3$, the phonon modes are described by 5 Fock states. []{data-label="Fig:verification0"}](Fig_delta_scan.pdf "fig:"){width="1\columnwidth"} ![(a) Transverse phonon mode occupation at the final time $t_f=2\pi \times 600/\omega_0$ for different initial states of $N=3$ spins indicated with blue, green, and orange colors. Solid, dot-dashed and dashed curves stand for the $q=0,1,2$ modes, respectively. Vertical lines represent frequencies of the higher-order harmonics of the phonon modes. (b) Zoomed-in part \[red rectangle in (a)\] including the first resonances. Dashed curves show the analytical results obtained from Eqs. , . The numerical parameters: $\eta=0.1$, $\Omega/\omega_0=1/6, \omega_0/\omega_z=3$, the phonon modes are described by 5 Fock states. []{data-label="Fig:verification0"}](Fig_delta_scan_inset.pdf "fig:"){width="1\columnwidth"}
![(a) Transverse phonon mode occupation at the final time $t_f=2\pi \times 400/\omega_0$ for different initial states of $N=6$ spins indicated with various colors. Solid curves stand for the COM mode ($q=0$) population, dashed curves represent the rest of the modes $q\neq0$. Vertical lines represent frequencies of the higher-order harmonics of the phonon modes. (b) Comparison with the analytical results. Dashed curves show the analytical results obtained from Eqs. , . Parameters: $\eta=0.1$, $\Omega/\omega_0=1/8, \omega_0/\omega_z=4$. The phonon modes are described by 3 Fock states. []{data-label="Fig:verification0_6"}](Fig_delta_scan6.pdf "fig:"){width="1\columnwidth"} ![(a) Transverse phonon mode occupation at the final time $t_f=2\pi \times 400/\omega_0$ for different initial states of $N=6$ spins indicated with various colors. Solid curves stand for the COM mode ($q=0$) population, dashed curves represent the rest of the modes $q\neq0$. Vertical lines represent frequencies of the higher-order harmonics of the phonon modes. (b) Comparison with the analytical results. Dashed curves show the analytical results obtained from Eqs. , . Parameters: $\eta=0.1$, $\Omega/\omega_0=1/8, \omega_0/\omega_z=4$. The phonon modes are described by 3 Fock states. []{data-label="Fig:verification0_6"}](Fig_delta_scan6_inset.pdf "fig:"){width="1\columnwidth"}
### Floquet spectrum analysis
Here we verify the effective QND dynamics for various transverse field values. In particular, we perform a numerical simulation of the periodically driven system of $N=3$ ions interacting via 3 axial phonon modes according to the full Hamiltonian $\hat{H}_{{\rm full}}(t)$ given by Eq. . We choose commensurate detunings $\Delta=-7\omega_{0}$, $\Delta'=-6\omega_{0}$, such that the overall dynamics is periodic with frequency $\omega_{0}$. Next, the operator of unitary evolution $\hat{U}(t)=\mathcal{T}\exp\left[-i\int_{0}^{t}dt_{1}\hat{H}_{{\rm
full}}(t_{1})\right]$ is numerically evaluated for one period of the oscillation. The logarithm of eigenvalues of $\hat{U}(2\pi/\omega_{0})$ provides $E_{\ell}^{{\rm Floquet}}$ the quasi energies of the effective Hamiltonian.
In Supplementary Fig. \[Fig:verification\](a) we compare the Floquet quasi energies $E_{\ell}^{{\rm Floquet}}$ (blue dotted lines) with the spectrum $E_{\ell}^{{\rm Ising}}$ of the effective Ising Hamiltonian with adjusted transverse field $B=2h$ (dashed lines) for various values of the Rabi frequency mismatch $\delta\Omega$ expressed as a transverse field $h$ via Eq. . The figure clearly shows that the exact eigenvalues $E_{\ell}^{{\rm Floquet}}$ are well represented by the effective Ising model.
Next, we study the coupling of the Ising Hamiltonian to the COM phonon mode. Here we consider the non-hermitian Hamiltonian of the full system (ions+phonons) $\hat{H}_{{\rm full}}(t)-i\frac{\gamma_{s}}{2}a_{0}^{\dagger}a_{0}$ with the non-hermitian term describing the decay of the COM mode due to the read-out. The Floquet eigenstates with the quasi energies around 0 and small imaginary parts represent the steady states of the open system. The COM mode displacements $\braket{a_{0}+a_{0}^{\dagger}}$ averaged over these Floquet states are shown in Supplementary Fig. \[Fig:verification\](b) with red lines. The displacement is proportional to the corresponding eigenenergy of the Ising Hamiltonian shown with dashed lines. The resulting read-out photocurrent is sensitive to the amplitude of the COM mode oscillations and, therefore, reveals the eigenenergies of the desired Ising model.
![Numerical verification of the double Mølmer-Sørensen configuration via Floquet analysis. (a) Test of the free evolution term $\hat{H}\otimes\mathbb{{I}}$ of the effective system-meter Hamiltonian . Blue lines show the exact Floquet spectrum depending on the effective transverse field $h$ (see text), dashed lines represent eigenenergies of the effective Ising model. (b) The system-meter coupling $\vartheta\hat{H}\otimes\hat{P}$ test. Red lines show the COM mode displacement $\braket{a_{0}+a_{0}^{\dagger}}$ averaged over the exact Floquet eigenstates (see text), dashed lines represent the corresponding eigenenergies of the effective Ising model. The following parameters are used: $\eta=0.3$, $\Delta=-7\omega_{0}$, $\Delta'=-6\omega_{0}$, $\Omega=\omega_{0}$, $0<\frac{\delta\Omega}{\Omega}<2\times10^{-3}$, $\gamma_{s}=3\frac{\eta
J}{\sqrt{N}}\left(\frac{\eta\Omega}{\Delta}\right)^{2}$. The COM mode is described by 6 Fock states, the other modes use 3 Fock states.[]{data-label="Fig:verification"}](Floquet_test_h_scan.pdf){width="1\columnwidth"}
Continuous readout of the spin Hamiltonian {#sec:continuous_readout}
------------------------------------------
With the implementation of the system-meter coupling Hamiltonian Eq. at hand, in this section we present the detailed discussion on the readout of the transverse Ising Hamiltonian via continuous monitoring the center-of-mass phonon quadrature $\hat{X}$, extending the short description presented in the Method section of the main text.
The experimental setup we have in mind is shown schematically in Fig. 2 of the main text. Here, aside from the ions $j\in\{1, \dotsc, N\}$ which generates the QND Hamiltonian Eq. , an ancilla ion $j=0$ is trapped at the edge of the ion chain and is subjected to sideband resolved laser cooling. The fluorescence emitted by the ancilla ion is collected by a lens setup and is continuously detected by a homodyne apparatus. We assume the MS lasers doesn’t interact with the ancilla ion, nor does the cooling laser impact the ions $j\in\{1, \dotsc, N\}$. As such, the ancilla ion participates in the collective vibrations of the ion chain and serves as a ‘transducer’ to couple light and phonons, thus allowing for monitoring the latter.
In the following, we introduce the quantum optical model for our considered setup in Supplementary Note \[sec:QSSE\], using the language of a quantum stochastic Schrödinger equation (QSSE) (see, e.g., Chap. 9 in Ref. [@gardiner2015quantum] for an introduction). Based on it, in Supplementary Note \[sec:adiabatic elimination\] we derive a QSSE describing the coupling between the phonons and light by adiabatically eliminating the internal DOFs of the ancilla ion. Finally, in Supplementary Note \[sec:SME\] we derive the stochastic master equation for continuous homodyne detection of the spontaneously emitted light and arrive at Eqs. and .
### Quantum stochastic Schrödinger equation {#sec:QSSE}
To be specific, we consider a standing-wave cooling configuration, i.e., the ancilla ion locates at the node of the standing wave [@cirac1992]. In the interaction picture with respect to $\hat{H}_0$ \[c.f. Eq. \], and in the frame rotating with the frequency of the cooling laser $\omega_{L}$, the internal dynamics of the auxiliary ion is described by $$\hat{H}_{\text{TLS}}=-\Delta_{e}|e\rangle\langle e|+\frac{\Omega_{0}}{2}(|e\rangle\langle g|+{\rm H.c.})\sin[k_{0}\hat{X}_{0}(t)].$$ Here, $|g\rangle(|e\rangle)$ is the ground(excited) level of the cooling transition respectively and $\Delta_{e}=\omega_{L}-\omega_{eg}$ is the frequency detuning between the cooling laser and the $|g\rangle\to|e\rangle$ transition. We assume the cooling laser is along the $x$ axis, with wavevector $k_{0}$ and Rabi frequency $\Omega_{0}$. The operator $\hat{X}_{0}(t)$ describes the (small-amplitude) displacement of the ancilla ion around its equilibrium position, and is related to the collective phonon modes of the ion chain by $\hat{X}_{0}(t)=\sum_{q}M_{0q}[\hat{a}_{q}(t)+\hat{a}_{q}^\dag(t)]/\sqrt{2m_{0}\omega_{q}}$ with $m_{0}$ the mass of the ancilla ion, and $\hat{a}_{q}(t)=\hat{a}_q{\rm exp}(-i\omega_q t)$.
Besides the internal structure of the ancilla ion, the rest DOFs of our model includes the internal pseudo-spins of ion $j\in\{1, \dotsc, N\}$ and the $N+1$ axial phonon modes. In the interaction picture with respect to $\hat{H}_0$, the time evolution of the total system is described by the (Itô) QSSE [@gardiner2015quantum] for the ions and the external electromagnetic field (bath DOFs), $$\begin{aligned}
d|\Psi\rangle= & -i\left(\hat{H}_{\cal{SM}}+\hat{H}_{\text{TLS}}-\frac{i}{2}\Gamma_{e}|e\rangle\langle e|\right)|\Psi\rangle dt\nonumber \\
& +\int du\sqrt{\Gamma_{e}N(u)}|g\rangle\langle e|e^{-ik_{0}u\hat{X}_{0}(t)}d\hat{B}^{\dagger}(u,t)|\Psi\rangle.\label{eq:QSSE}\end{aligned}$$ In Eq. , the first line includes the spin-phonon Hamiltonian $\hat{H}_{{\rm sys}}$, the internal Hamiltonian of the ancilla ion $\hat{H}_{{\rm TLS}}$, and the spontaneous decay of the ancilla ion at a rate $\Gamma_{e}$. The second line describes spontaneous emission of the ancilla ion into the 3D electromagnetic modes. Here, the function $N(u)$ reflects the dipole emission pattern of the cooling transition, which, for the 1D ionic motion considered here, depends on a single variable $u\equiv\cos\nu\in[-1,1]$ with $\nu$ the angle between the wavevector of the emitted photon and the $x$ axis. The spontaneous emission is accompanied by the momentum recoil described by the operator $e^{-ik_{0}u\hat{X}_{0}}$, with $k_{0}$ the wavevector of the emitted photon (approximately the same as the wavevector of the cooling laser). To account for the relevant electromagnetic modes in the emission direction $u$, quantum optics introduces the corresponding bosonic noise operators $\hat{b}_{u}(t)$ and $\hat{b}_{u}^{\dag}(t)$, satisfying the white-noise commutation relations $[\hat{b}_{u}(t),\hat{b}_{u}^{\dag}(t')]=\delta(u-u')\delta({t-t'})$ [@gardiner2015quantum]. In the Itô QSSE these noise operators are transcribed as Wiener operator noise increments, $\hat{b}_{u}(t)dt\to d\hat{B}(u,t)$. Assuming the 3D bath is initially in the vacuum state, they obey the Itô table [@gardiner2015quantum], $$\begin{split}d\hat{B}(u,t)d\hat{B}^{\dag}(u',t) & =dt\delta(u-u'),\\
d\hat{B}^{\dag}(u,t)d\hat{B}(u',t) & =0,\\
d\hat{B}(u,t)d\hat{B}(u',t) & =d\hat{B}^{\dag}(u,t)d\hat{B}^{\dag}(u',t)=0.
\end{split}
\label{eq:vacuumItoLaw2}$$ We note, apart from the explicit ion-bath coupling in the second line of Eq. , the inclusion of the 3D electromagnetic field bath also introduces a decay term $-i\Gamma_{e}|e\rangle\langle e|/2$ in the first line of Eq. . Mathematically, this non-Hermitian term appears as an “Itô correction" when applying the Itô stochastic calculus to describe physical systems [@gardiner2015quantum].
Based on Eq. , in the next section we derive a QSSE describing the coupling between the phonon modes and the electromagnetic field bath by adiabatically eliminating the internal dynamics of the ancilla ion.
### Adiabatic elimination of the internal dynamics of the ancilla ion {#sec:adiabatic elimination}
We consider the following parameter regime. (i) The ancilla ion is weakly excited by the cooling laser, $\eta_{q}^{0}\Omega_{0}\ll\Gamma_{e}$, where $\eta_{q}^{0}\equiv k_{0}/\sqrt{2m_{0}\omega_{q}}$ is the Lamb-Dicke parameter corresponding to the cooling laser. (ii) The QND interaction is much weaker than the spontaneous emission strength of the ancilla ion, $|\hat{H}_{{\cal SM}}|\ll\Gamma_{e}$. (iii) The sideband resolved regime $\omega_{q}\gg\Gamma_{e}$. Condition (i) and (ii) guarantees that the internal dynamics of the ancilla ion is much faster than the dynamics of the rest of the system, allowing us to adiabatically eliminate the internal dynamics of the ancilla ion. Condition (iii) enables us to selectively enhance the center-of-mass phonon contribution in the detected photon current (see detailed discussion in Supplementary Note \[sec:SME\]).
To perform the adiabatic elimination, we formally decompose the state of the total system [\[]{}see Eq. into two components, $|\Psi\rangle=|\psi_{e}\rangle|e\rangle+|\psi_{g}\rangle|g\rangle$, with $|\psi_{e(g)}\rangle\equiv\langle e(g)|\Psi\rangle$. By the expansion up to second order in the small Lamb-Dicke parameter $\eta_{q}^{0}$, Eq. becomes two coupled equations for $|\psi_{e(g)}\rangle$, $$\begin{aligned}
d|\psi_{e}\rangle= & -i\left[\hat{H}_{{\cal SM}}-\left(\Delta_{e}+\frac{i}{2}\Gamma_{e}\right)\right]|\psi_{e}\rangle dt\nonumber \\
& -i\frac{\Omega_{0}}{2}\sum_{q}\eta_{q}^{0}M_{0q}\left[\hat{a}_{q}^{\dagger}(t)+\hat{a}_{q}(t)\right]|\psi_{g}\rangle dt,\label{eq:elimination_psi_e}\\
d|\psi_{g}\rangle= & -i\hat{H}_{\cal SM}|\psi_{g}\rangle dt-i\frac{\Omega_{0}}{2}\sum_{q}\eta_{q}^{0}M_{0q}\left[\hat{a}_{q}^{\dagger}(t)+\hat{a}_{q}(t)\right]|\psi_{e}\rangle dt\nonumber \\
& +\int du\sqrt{\Gamma_{e}N(u)}\bigg\{1-i\sum_{q}\eta_{q}^{0}M_{0q}\left[\hat{a}_{q}^{\dagger}(t)+\hat{a}_{q}(t)\right]\nonumber \\
& -\frac{1}{2}\Big[\sum_{q}\eta_{q}^{0}M_{0q}\left[\hat{a}_{q}^{\dagger}(t)+\hat{a}_{q}(t)\right]\Big]^{2}\bigg\}d\hat{B}^{\dagger}(u,t)|\psi_{e}\rangle.\label{eq:elimination_psi_g}\end{aligned}$$ From Eq. it is easy to see $|{\psi}_{e}\rangle\sim O(\eta_{q}^{0})$. To keep $|\psi_{g}\rangle$ accurate to $O[(\eta_{q}^{0})^{2}]$, we can neglect the second order Taylor expansion in the last term of Eq. .
Under conditions (i) and (ii) introduced in the beginning of this section, Eq. can be solved adiabatically $$\begin{aligned}
|{\psi}_{e}\rangle=\frac{\Omega_{0}}{2}\sum_{q}\eta_{q}^{0}M_{0q}\bigg[\frac{\hat{a}_{q}^{\dagger}(t)}{\Delta_{e}\!-\!\omega_{q}\!+\!\frac{i}{2}\Gamma_{\!e}}\!+\!\frac{\hat{a}_{q}(t)}{\Delta_{e}\!+\!\omega_{q}\!+\!\frac{i}{2}\Gamma_{\!e}}\bigg]|{\psi}_{g}\rangle.\end{aligned}$$ Plugging the solution into Eq. , we arrive at a QSSE which describes the slow dynamics of the system assuming the ancilla ion staying in its internal stationary (ground) state, $$\begin{aligned}
d|\psi_{g}\rangle= & -i\left(\hat{{H}}_{{\cal SM}}+\sum_{q}\delta{\omega}_{q}\hat{a}_{q}^{\dagger}\hat{a}_{q}\right) |\psi_{g}\rangle dt\nonumber \\
&-\frac{1}{2}\left(A_{q}^{+}\hat{a}_{q}\hat{a}_{q}^{\dagger}+A_{q}^{-}\hat{a}_{q}^{\dagger}\hat{a}_{q}\right)|\psi_{g}\rangle dt\nonumber \\
& +\int du\sqrt{\Gamma_{e}N(u)}\hat{{\cal J}}d\hat{B}^{\dagger}(u,t)|\psi_{g}\rangle,\label{eq:SSE_phonon}\end{aligned}$$ where $dt\gg1/\Gamma_{e}$ is the coarse-grained time increment and $d\hat{B}^{\dag}(u,t)$ is the corresponding coarse-grained quantum noise increment. $\delta\omega_{q}$ is a (tiny) frequency renormalization of the $q$-th phonon mode, $$\begin{aligned}
\delta\omega_{q}=(\eta_{q}^{0}M_{0q}\Omega_{0})^{2}\bigg[\frac{\Delta_{e}+\omega_{q}}{4(\Delta_{e}\!+\!\omega_{q})^{2}\!+\!\Gamma_{e}^{2}}+\frac{\Delta_{e}-\omega_{q}}{4(\Delta_{e}\!-\!\omega_{q})^{2}\!+\!\Gamma_{e}^{2}}\bigg].\end{aligned}$$ In the following we neglect such a tiny frequency shift. The damping rates $A_{q}^{\pm}$ for the $q$-th phonon mode are defined as $$A_{q}^{\pm}=\frac{(\eta_{q}^{0}M_{0q}\Omega_{0})^{2}}{4(\Delta_{e}\mp\omega_{q})^{2}+\Gamma_{e}^{2}}\Gamma_{e}.$$ The operator $\hat{{\cal J}}$ is a collective quantum jump operator including all phonon modes, $$\begin{aligned}
\hat{{\cal J}}=\frac{\Omega_{0}}{2}\sum_{q}\eta_{q}^{0}M_{0q}\bigg(\frac{\hat{a}_{q}^{\dagger}(t)}{\Delta_{e}-\omega_{q}+\frac{i}{2}\Gamma_{e}}+\frac{\hat{a}_{q}(t)}{\Delta_{e}+\omega_{q}+\frac{i}{2}\Gamma_{e}}\bigg).\label{eq:collapse_OP_full}\end{aligned}$$
The QSSE describes the coupling between the phonon DOFs and the external electromagnetic field bath. This allows us to read out the COM quadrature $\hat{X}$ via homodyne detection of the external bath, as detailed in the next section.
### Homodyne detection of the fluorescence {#sec:SME}
We consider continuous homodyne detection of the laser cooling fluorescence, as shown schematically in Fig. 1 of the main text. In such a measurement, the fluorescence photons are collected by linear optical elements, e.g., by a lens setup, and are then mixed with a reference laser at a beam splitter. Photon counting of the mixed beam then allows for the measurement of the phase information of the fluorescence photons.
We assume the lens system covers a solid angle $\Omega$, and define $$\epsilon=\int_{\Omega}duN(u)$$ as the fraction of photons collected by the lens setup. The corresponding quantum noise increment is $$d\hat{B}(t)=\frac{1}{\sqrt{\epsilon}}\int_{\Omega}du\sqrt{N(u)}d\hat{B}(u,t).$$
The homodyne measurement corresponds to making a measurement of the following quadrature operator [@wiseman2009quantum; @gardiner2015quantum] $$d\hat{Q}(t)=d\hat{B}(t)e^{-{i\phi}}+d\hat{B}^{\dagger}(t)e^{{i\phi}},$$ with $\phi=(\omega_{\rm LO}-\omega_L)t+\phi_0$ and $\omega_{\rm LO}$ and $\phi_0$ being the frequency and phase of the local oscillator. The measurement projects the state of the bath onto an eigenstate of $d\hat{Q}(t)$ corresponding to the eigenvalue $dq(t)$, which defines the homodyne current via $dq(t)\equiv
I(t)dt$. It can be shown [@wiseman2009quantum; @gardiner2015quantum] that the measurement outcome $dq(t)$ obeys a normal distribution centered at the mean value of the quantum jump operator $\hat{{\cal J}}$, i.e., $$dq(t)\equiv I(t)dt=\sqrt{\epsilon\Gamma_{e}}\langle\hat{{\cal J}}e^{-i\phi}+\hat{{\cal J}}^{\dag}e^{i\phi}\rangle_{c}+dW(t),\label{eq:Ih_full}$$ where $dW(t)$ is a random Wiener increment, which is related to the shot noise by $dW(t)=\xi(t)dt$. The expectation value $\langle\dots\rangle_{c}={\rm Tr}(\dots\mu_{c})$ is taken with a conditional density matrix $\mu_{c}$ of the spin-phonon system. The evolution of $\mu_{c}$ is given by a SME derived from Eq. by projecting out the bath DOFs following the standard procedure [@wiseman2009quantum; @gardiner2015quantum], $$\begin{aligned}
d\mu_{c}= & -i[\hat{{H}}_{{\cal SM}},\mu_{c}]dt\nonumber +\sum_{q}\left(A_{q}^{+}\mathcal{D}[\hat{a}_{q}^{\dagger}]+A_{q}^{-}\mathcal{D}[\hat{a}_{q}]\right)\mu_{c}dt\nonumber \\
& +\sqrt{\epsilon\Gamma_{e}}\mathcal{H}[\hat{{\cal J}}e^{-i\phi}]\mu_{c}dW(t),\label{eq:SME_app}\end{aligned}$$ with $\mathcal{D}[\hat{O}]\rho\equiv\hat{O}\rho\hat{O}^{\dagger}-\frac{1}{2}\hat{O}^{\dagger}\hat{O}\rho-\frac{1}{2}\rho\hat{O}^{\dagger}\hat{O}$ being the Lindblad superoperator, and ${\cal H}[\hat{O}]\rho\equiv\hat{O}\rho-{\rm Tr}(\hat{O}\rho)\rho+\text{H.c.}$ a superoperator corresponding to homodyne measurement. The first two lines of Eq. is akin to the laser cooling master equation of trapped particles [@cirac1992; @gardiner2015quantum], while the third line describes the measurement backaction of a continuous homodyne detection.
Under the condition of resolved sideband $\omega_{q}\gg\Gamma_{e}$, we can enhance the component corresponding to the COM phonon mode in the homodyne signal Eq. , by tuning the cooling laser in resonance with the red sideband of the COM mode, $\Delta_{e}=-\omega_{0}$. Under this condition, we have $\hat{{\cal J}}\simeq-i\Omega_{0}\eta_{0}^{0}M_{00}\hat{a}_{0}{\rm exp}(-i\omega_0 t)/\Gamma_{e}$ [\[]{}see Eq. , and $A_{0}^{+}\simeq A_{q}^{\pm}\simeq0$ for $q\neq0$. Defining $\hat{\rho}_{c}^{{\cal SM}}={\rm Tr}_{{\rm ph},q\neq0}({\mu}_{c})$ by trancing out the phonon modes except for the COM mode, we have $$\begin{aligned}
I(t)= & \sqrt{2\epsilon\gamma_{s}}\langle\hat{X}\rangle_{c}+\xi(t),\nonumber \\
d\rho_{c}^{{\cal SM}}= & -i[\hat{{H}}_{{\cal SM}},\rho_{c}^{{\cal SM}}]dt+\gamma_{s}\mathcal{D}\left[\hat{a}_{0}\right]\rho_{c}^{{\cal SM}}dt\nonumber \\
& +\sqrt{\epsilon\gamma_{s}}\mathcal{H}\left[\hat{a}_{0}\right]\rho_{c}^{{\cal SM}}dW(t).\label{eq:SME_eff}\end{aligned}$$ where $\hat{X}=(\hat{a}_{0}+\hat{a}_{0}^{\dag})/\sqrt{2}$ is the $x$-quadrature of the COM phonon mode, $\gamma_{s}=(\Omega_{0}\eta_{0}^{0}M_{00})^{2}/\Gamma_{e}$ is an effective measurement rate, with $M_{00}\simeq1/\sqrt{N}$, and we choose $\omega_{\rm LO}-\omega_L=\omega_0$ and $\phi_0=-\pi/2$ for the local oscillator to maximize the homodyne current.
Equation already describes continuous QND readout of the transverse field Ising Hamiltonian. To simplify the analysis, we can further adiabatically eliminating the COM phonon mode in Eq. under the condition $\gamma_{s}\gg\vartheta J$, and arrive at Eqs. and with the identification $\gamma\equiv2J^{2}\vartheta^{2}/\gamma_{s}=2\Gamma_{e}(\vartheta J/\Omega_{0}\eta_{0}^{0}M_{00})^{2}$.
### Filtering of the homodyne current
The homodyne current Eq. is noisy, as it contains the (white) shot noise $\xi(t)$ inherited from the vacuum fluctuation of the electromagnetic field environment. To suppress the noise, we filter the homodyne current with a suitable linear lowpass filter $$\mathcal{I}_{\tau}(t)=\int dt'h_{\tau}(t-t')I(t'),\label{eq:filteredIh}$$ where $h_{\tau}(t)$ is the filter function with a frequency bandwidth $\sim1/\tau$, and $\mathcal{I}(t)$ is the *filtered homodyne current*. The filter attenuates the component of the shot noise with frequency higher than $1/\tau$ thus allowing us to extract out the signal we are interested in.
We adopt two filters in the main text. The first one is a simple *cumulative time-average*, $\overline{\mathcal{I}}(\tau)=(2N\sqrt{\gamma\epsilon\tau})^{-1}\int_{0}^{\tau}dtI(t)$. This allows us to attenuate the shot noise as much as possible, and is especially suitable for QND measurement (cf. Fig. 1e of the main text). In contrast, for imperfect QND measurement we are interested in resolving the quantum jumps between different energy eigenstates as a competition between coherent evolution and measurement backaction. To achieve this, we filter the homodyne current via $\mathcal{I}_{\tau}(t)=(2N\sqrt{\gamma\epsilon}\tau)^{-1}\int_{0}^{\infty}dt'e^{-t'/\tau}I(t-t')$ and call $\mathcal{I}_{\tau}(t)$ the *window-filtered homodyne current*. The time window $\tau$ is chosen to ensure $1/\gamma\ll\tau\ll T_{{\rm dwell}}$ with $\gamma$ the measurement rate and $T_{{\rm dwell}}$ the typical time that the system dwells in particular eigenstates. This allows us to attenuate the shot noise as much as possible while still being able to resolve the quantum jumps.
Experimental feasibility {#sec:experiment}
------------------------
Having discussed our QND measurement scheme for the transverse-field Ising Hamiltonian in trapped-ion setups, in this section, we show that state-of-the-art trapped-ion experiments provide all ingredients for the implementation of the QND scheme. First, in Supplementary Note \[sec:exp\_general\], we summarize the experimental requirements of our scheme and discuss experimental imperfections including multiple sources of decoherence. We then discuss some practical points. These include the implementation of our scheme with axial and transverse phonon modes, analyzed in Supplementary Note \[sec:axial\_phonon\] and Supplementary Note \[sec:transverse\_phonon\] respectively, as well as the implementation with different ion species, discussed in Supplementary Note \[sec:Ion\_species\]. Finally, in Supplementary Note \[sec:numbers\], we present experimental parameters for proof-of-principle realizations of our scheme.
### Experimental requirements and practical imperfections {#sec:exp_general}
The performance of our QND measurement scheme depends on the collection efficiency $\epsilon$ of the photons scattered by the ancilla ion. A collection efficiency of $15 \, \%$ is experimentally feasible for a single trapped ion [@PhysRevLett.96.043003], and we expect that a similar collection efficiency can be reached in our proposed setup. Even larger photon collection rates can be achieved by coupling the ancilla ion to optical cavities [@Stute2012], or by simultaneous detection of the fluorescence of several ancilla ions.
In the implementation of homodyne detection of the spin system, we assume that the MS lasers do not interact with the ancilla ion, and that the cooling laser does not impact the ions $j\in\{1, \dotsc, N\}$. These requirements can be met by individual addressing of each ion in realizing the MS configuration. Alternatively, this can be achieved by using global MS lasers and by choosing the ancilla ion from a different ion species [@Tan:2015aa; @Negnevitsky:2018aa], so that the ancilla is decoupled from the MS lasers due to its different internal electronic structure. We note, however, that an ancilla ion with a different mass changes the structure of the COM mode. This has to be rectified in order to perform our QND measurement scheme, e.g., via local adjustments of the trapping potential near the ancilla ion using optical potentials [@Schneider:2010aa].
Realistic trapped-ion systems have multiple sources of decoherence. The coherence time of current trapped-ion quantum simulators is limited by dephasing of the internal spins due to fluctuations of the global magnetic field which defines the quantization axis. Encoding the spins in ionic internal states which are first-order insensitive to magnetic field fluctuations greatly suppresses dephasing and extends the single-spin coherence time. This has been implemented, e.g., for $^9{\rm Be}^+$ ions (with a [single-spin coherence time $t_{\rm coh}$]{} $\sim 1.5 \, {\rm s}$ [@PhysRevLett.95.060502]) and for $^{171}{\rm Yb}^+$ ions (with [$t_{\rm coh}\sim 2.5 \, {\rm s}$]{} [@PhysRevA.76.052314]). Without this type of encoding, the coherence time is typically one order of magnitude shorter. For example, [for $^{40}{\rm Ca}^+$ ions $t_{\rm coh}\sim 95 \, \mathrm{ms}$]{} [@PhysRevLett.106.130506].
Another important source of decoherence is phonon heating due to electromagnetic field noise. In standard linear Paul traps, the phonon heating rate is typically below $1/$s and is thus negligible [@PhysRevLett.83.4713]. Nevertheless, in surface ion traps, phonon heating is much more significant due to the short distance between the ions and the trap electrodes. Operating at cryogenic temperature can reduce phonon heating significantly. For example, the phonon heating rate of axial phonons is reduced to values as low as $70/$s for ion spacings of $d\sim30 \, \mu \mathrm{m}$ in the cryogenic surface traps which are used by the NIST group [@Brown2011]. Even lower phonon heating rates are being actively pursued [@McConnell2015].
In Supplementary Notes \[sec:Ion\_species\] and \[sec:numbers\] further below, we show that the proposed QND measurement requires a time much shorter than the coherence time of trapped ions which is limited by the factors outlined above. Thus, current trapped-ion technology allows for robust implementation of our QND measurement scheme.
### Implementation with axial phonon modes {#sec:axial_phonon}
In this section, we present some considerations on implementation of our QND measurement scheme with axial phonon modes.
The spectrum of axial phonon modes of an ion string in a linear Paul trap is extensive, i.e., it broadens with increasing number of ions $N$. To implement the long range Ising model with dipolar interaction $J_{ij}\propto 1/|i-j|^3$, the detunings $\Delta(\Delta')$ of the double MS configuration should also increase with the number of ions. Thus, to keep the spin-spin coupling $J\propto(\Omega/\Delta)^{2}(k^{2}/2m)$ [\[]{}see Eq. finite, the power of the MS laser beams also goes up with increasing $N$. The achievable laser power in the laboratory thus puts a practical limitation on the scalability of the implementation wit axial phonon modes. On the other hand, the implementation with axial phonon modes benefits a relative large system-meter coupling $\hat{H}_{{\cal SM}}$, thanks to the large Lamb-Dicke parameter $\eta$ associated with axial phonon modes (we note that in $\hat{H}_{{\cal SM}}$, $\vartheta\propto\eta$). In view of these, the implementation with axial phonon modes best serves as a small-scale proof-of-principle experiment, which demonstrates our proposed QND measurement and its applications. We provide the typical experimental parameters for such an implementation in Supplementary Note \[sec:numbers\].
Moreover, we comment that the extensive feature of the axial phonon spectrum allows for engineering exotic spin coupling that goes beyond the power-law coupling, e.g., frustrated spin models, by carefully adjusting the laser detunings $\Delta^{(\prime)}$ with respect to the phonon spectrum [@Porras2004]. The associated QND measurement could possibly enable rich opportunities for the study of these models and the preparation of their eigenstates.
### Implementation with transverse phonon modes {#sec:transverse_phonon}
Here we present some considerations concerning the implementation of our QND measurement scheme with transverse phonon modes.
In contrast to axial phonon modes, the transverse phonon modes in a linear Paul trap have a dense spectrum of width $\propto\omega_{z}^{2}/\omega_{x}$, which is almost independent of the number of ions $N$; Here, $\omega_{z(x)}$ is the trapping frequency along the axial (transverse) direction, respectively. As a result, the long range-Ising model and the associated QND measurement can be implemented by a double MS configuration for which the detunings $\Delta^{(\prime)}$ and the Rabi frequency $\Omega$ can be held fixed upon increasing the number of ions. This leads to better scalability regarding the laser power as compared to the implementation with axial phonon modes.
The scalability of an implementation with transverse phonon modes is limited by the condition Eq. , since longer ion chain leads to denser phonon spectrum which eventually violates Eq. . To estimate an upper limit of the ion number $N$, we note that the LHS of Eq. is much smaller than $\sum_q\eta_q^2\Omega^2\eta/|\Delta^{(\prime)}-\omega_0|\ll \sum_q\eta_q^2\Omega$, the latter “$\ll$" coming from our off-resonance condition $|\Delta^{(\prime)}-\omega_0|\gg\eta\Omega$. Thus, Eq. is well satisfied as long as $\sum_q\eta_q^2\Omega\leq |\omega_1-\omega_0|$. We can estimate $\sum_q\eta_q^2\Omega$ as $N\eta^2\Omega$ and $|\omega_1-\omega_0|$ as $\omega_z^2/\omega_x$, thus the above condition becomes $N\eta^2\Omega\leq\omega_z^2/\omega_x$. Moreover, to prevent zig-zag transition of a linear ion chain we require $\omega_{x}/\omega_{z}\geq0.73N^{0.86}$ [@Wineland1998]. Combining these two conditions we find $N\leq[\omega_{z}/(\eta^{2}\Omega)]^{0.54}$. The latter quantity is typically around $100$ in experiments. Thus, according to these estimates, the implementation with transverse phonon modes allows for scaling up to hundreds of ions.
### Implementation with different ion species {#sec:Ion_species}
![Energy resolution of a single run of the QND measurement for different ions species. The (dimensionless) energy resolution $\Delta E/J$ over the mean level spacing $N/2^N$ is plotted against the ion number $N$ and the dimensionless parameter $\epsilon \eta E_r\tau$, where $\epsilon$ is the detection efficency, $\eta$ the Lamb-Dicke paramter $E_r$ the recoil energy and $\tau$ the measurement time. [The three dashed lines corresponds to estimations for three different ion species, for which we fix $\epsilon=0.15, \eta=0.1$ and choose $\tau=t_{\rm coh}/N$ with $t_{\rm coh}$ the single-spin coherence time. ]{}[]{data-label="Fig:IonSpecies"}](Fig3_IonSpecies_New.pdf){width="1\columnwidth"}
We now discuss and compare the implementation with different ion species. As already mentioned in Supplementary Note \[sec:exp\_general\], different ion species have different coupling strengths to the laser fields, and thus different energy scales of the targeted spin models and the associated QND measurements. Further, different ion species possess different coherence time. These parameters impact the performance of an implementation of our QND measurement scheme. As examples, we consider three ion species that are commonly used in current trapped-ion experiments: $^{171}{\rm Yb}^+$, $^{40}{\rm Ca}^+$ and $^{9}{\rm Be}^+$.
A key parameter to quantify the performance of the proposed QND measurement is the signal-to-noise ratio (SNR) of a single measurement run, see the discussion in Methods of the main text. This can be equivalently expressed in terms of the (dimensionless) energy resolution $\Delta E/J\sim 1/\sqrt{2\gamma\epsilon\tau}$, for given measurement strength $\gamma$, photon collection efficiency $\epsilon$ and filtering time $\tau$ (which is the same as the measurement time). The energy resolution represents our ability to distinguish two adjacent energy levels from the documented data of a single measurement run, and should be compared to the (dimensionless) mean many-body level spacing, which can be estimated as $N/2^N$.
[The achievable measurement time $\tau$ can be estimated using the typical many-body dephasing time as $\tau= t_{\rm coh}/N$, with $t_{\rm coh}$ the single-spin coherence time and $N$ the number of ions.]{} The measurement strength $\gamma$ of our QND scheme is controlled by the parameter $\vartheta J\sim (\eta/\sqrt{N})E_r(\Omega/\Delta)^2$, see the discussion in Supplementary Note \[subsec:Double-Molmer-Sorensen-interaction\]. Here, $E_r$ is the recoil energy of the qubit transition, whereas $\Omega/\Delta\ll 1$ is a small quantity independent of the ion species. As a result, we have the energy resolution [$\Delta E/J\sim (\Delta/\Omega)N^{3/4}/\sqrt{2\eta \epsilon E_r t_{\rm coh}}$]{}. Taking realistic experimental parameters, we find that the quantity $\eta \epsilon E_r t_{\rm coh}$ differs for different ion species and spans a range from $10^2$ for $^{40}{\rm Ca}^{+}$ to $10^4$ for $^9{\rm Be}^{+}$, see the vertical lines in Supplementary Fig. \[Fig:IonSpecies\]. Among the three ion species, $^9{\rm Be}^{+}$ holds the promise of achieving the best energy resolution due to its light mass and long coherence time.
In Supplementary Fig. \[Fig:IonSpecies\], we further plot the ratio between the energy resolution $\Delta E/J$ and the mean level spacing $N/2^N$, for increasing ion number $N$. Under current experimental conditions, the maximum system size $N$ for which a single run of the QND measurement is able to resolve the eigenenergies can be estimated as [$N\simeq 4$ for $^{40}{\rm Ca}^{+}$, $N\simeq 6$ for $^{171}{\rm Yb}^{+}$ and $N\simeq 8$ for $^{9}{\rm Be}^{+}$]{}. As a result, all three ion species are good candidates for building an intermediate-size interacting spin system for testing quantum fluctuation relations. The favorable scalability of $^{9}{\rm Be}^{+}$ facilitates testing the eigenstate thermalization hypothesis. Finally, we note that these estimations concern a single measurement run and better energy resolution could be achieved by repeated measurements.
### Parameters for proof-of-principle experiments {#sec:numbers}
We proceed to present experimental parameters for a proof-of-principle implementation of our QND measurement scheme.
First, let us consider an implementation with $^{9}$Be$^{+}$ ions and with axial phonon modes. The experimental system we have in mind is similar to the one reported in Ref. [@PhysRevLett.117.060505]. To be concrete, we consider $N=5$ $^{9}$Be$^{+}$ ions in a linear Paul trap. The internal spin of a $^{9}$Be$^{+}$ ion consists of two hyperfine states driven by a Raman transition involving two $313 \, \mathrm{nm}$ single-photon transitions with recoil energy $E_{r}=2\pi\times226.5 \, \mathrm{kHZ}$. We choose the axial trapping frequency $\omega_{z}=2\pi\times3 \, \mathrm{MHz}$, leading to a moderate Lamb-Dicke parameter of $\eta\simeq\sqrt{E_{r}/\omega_{z}}\simeq0.27$. We further choose $\Delta=3\omega_{q=4}$ and $\Omega=0.15\Delta$ to stay in the off-resonant regime. The resulting spin-spin coupling strength is $J\simeq2\pi\times5 \, \mathrm{kHz}$, and the system-meter coupling is $\vartheta\simeq-0.17$. We choose the laser cooling rate of the ancilla ion $\gamma_{s}=2\pi\times5 \, \mathrm{kHz}$. Consequently, the effective measurement rate is $\gamma\simeq2\pi\times290 \, \mathrm{Hz}$. Assuming a photon collection efficiency $\epsilon=0.15$ as discussed above, our QND measurement has a resulting characteristic time scale $1/\epsilon\gamma\simeq3.7 \, \mathrm{ms}$, which is much shorter than the typical single qubit dephasing time $\sim1 \, \mathrm{s}$ [@PhysRevLett.117.060505]. Specifically, an averaging time $\tau=10/\epsilon\gamma$ leads to an energy resolution (see Methods) $\Delta E/J\sim0.22$, smaller than the minimal energy gap in this five-spin Ising model. This enables the preparation of single energy eigenstates via QND measurement, which suffices, e.g., for testing quantum fluctuation relations. This also allows for the observation of quantum jumps between different eigenstates in the imperfect QND regime as discussed in the main text.
Next, we provide experimental parameters for a transverse-phonon implementation realizing the power-law decaying spin-spin interactions, and discuss the associated energy resolution. These are relevant to the discussion in Supplementary Note \[sec:eigenst-therm-hypoth\] below on testing the eigenstate thermalization hypothesis. We consider $N=6$ $^{9}$Be$^{+}$ ions in a linear Paul trap with axial trapping frequency $\omega_{z}=2\pi\times 2 \, \mathrm{MHz}$ and transverse trapping frequency $\omega_{x}=2\pi\times 8 \, \mathrm{MHz}$. We choose $\Omega=2\pi\times 1.76 \, \mathrm{MHz}$, $\Delta=2\pi\times 8.8 \, \mathrm{MHz}$, $\Delta^\prime=\Delta+\omega_x$, and the Lamb-Dicke parameter along the transverse direction $\eta=0.09<\sqrt{E_r/\omega_x}$, which can be realized by properly choosing the direction of the double MS beams with respect to the ion string. The resulting spin-spin coupling strength obeys an approximate power law decay $J_{ij}\sim J/|i-j|^\alpha$ with $\alpha=1.5$ and $J\simeq2\pi\times 2.6 \, \mathrm{kHz}$. Further, this generate a QND coupling with strength $|\vartheta J|\simeq 2\pi\times 0.1 \, \mathrm{kHz}$. We choose the laser cooling rate of the ancilla ion $\gamma_{s}=2\pi\times 0.5 \, \mathrm{kHz}$. Consequently, the effective measurement rate is $\gamma\simeq2\pi\times 40 \, \mathrm{Hz}$. Assuming a photon collection efficiency $\epsilon=0.15$, and a measurement time $\tau=50 \, \mathrm{ms}$, the achieved energy resolution is $\Delta E/J\simeq 0.35$. This corresponds to a resolution of the energy density $\Delta\varepsilon=\Delta E/(JN)=0.06$, which is indicated as horizontal error bars in Supplementary Fig. \[FIG\_eth\].
Numerical study of thermal properties of energy eigenstates {#sec:numer-study-therm}
===========================================================
In this section we provide additional details on the numerical simulations used in our study of thermal properties of the energy eigenstates in the main text.
In Supplementary Note \[sec: MC\] we describe the canonical-ensemble quantum Monte-Carlo simulations of the transverse-field Ising model used in Fig. 3 of the main text. We discuss the phase transition in the case of of long- and short-range interactions.
In Supplementary Note \[sec: Realistic\] we discuss the phase diagram of the transverse field Ising model for realistic spin-spin interaction $J_{ij}$ and show that it qualitatively agree with the one obtained using an approximate power-law.
![Phase transition of the Ising model in the canonical ensemble. (a), (c) Binder cumulant as a function of temperature for different system sizes. (b), (d) the corresponding energy of the system.[]{data-label="Fig1"}](fig_Binder.pdf "fig:"){width="0.49\columnwidth"}![Phase transition of the Ising model in the canonical ensemble. (a), (c) Binder cumulant as a function of temperature for different system sizes. (b), (d) the corresponding energy of the system.[]{data-label="Fig1"}](fig_Binder_2.pdf "fig:"){width="0.49\columnwidth"}\
![Phase transition of the Ising model in the canonical ensemble. (a), (c) Binder cumulant as a function of temperature for different system sizes. (b), (d) the corresponding energy of the system.[]{data-label="Fig1"}](fig_energy_density.pdf "fig:"){width="0.49\columnwidth"} ![Phase transition of the Ising model in the canonical ensemble. (a), (c) Binder cumulant as a function of temperature for different system sizes. (b), (d) the corresponding energy of the system.[]{data-label="Fig1"}](fig_energy_density_2.pdf "fig:"){width="0.49\columnwidth"}
Monte-Carlo simulations {#sec: MC}
-----------------------
Here we provide details on the numerical simulations of the phase transition of the Ising model in canonical ensemble $\hat{\rho}_{\text{th}}\left(T\right)\equiv
e^{-\hat{H}/T}/\text{Tr}\left[e^{-\hat{H}/T}\right]$ using the quantum Monte-Carlo technique. It allows us to calculate the critical energy $\varepsilon$ using the finite-size scaling analysis of the Binder cumulant, defined as $$U_{4}\equiv1-\frac{\left\langle \hat{m}_{x}^{4}\right\rangle }{3\left\langle \hat{m}_{x}^{2}\right\rangle ^{2}}$$ By its construction [@Binder81], this cumulant distinguishes the ordered phase with $U_{4}\approx2/3$, from the disordered phase with $U_{4}\approx0$. As a result, when crossing the phase transition, the Binder cumulant has a sharp jump between these two values at the critical temperature $T_{c}$. This allows us to determine $T_{c}$ for the Ising model. The results of the Monte-Carlo simulation for $\alpha=1.5,$ $h/J=1$ and different system sizes $N$ are shown in Supplementary Fig. \[Fig1\](a). For a sufficiently large number of spins the curves of $U_{4}$ cross approximately at the same temperature, which provides a good estimate of $T_{c}$. The corresponding critical energy density $\varepsilon\equiv\text{Tr}\left[\hat{H}\hat{\rho}_{\text{th}}\left(T_{c}\right)\right]/NJ$ can be easily determined from the energy-temperature conversion curve shown in Supplementary Fig. \[Fig1\](b).
![Excited-state phase transition in the Ising model with $\alpha=3$. (a) Ferro-paramagnet crossover in the Ising model of $N=14$ spins prepared by the energy measurements in microcanonical ensembles of width $\Delta E/(JN)=0.1$. The transition between magnetically ordered phase $\braket{\hat{m}_{x}^{2}}_{\text{mc}}\approx1$ (dark blue) to disordered phase $\braket{\hat{m}_{x}^{2}}_{\text{mc}}\approx0$ (light blue) is shown as function of the mean energy density $\varepsilon=\braket{\hat{H}}_{{\rm mc}}/(JN)$ and the transverse field $h$. Test of ETH for the symmetry sector $\{+1,+1\}$ is shown in the inset: only the ground state has a bimodal distribution $P\left(m_{x}\right)$.[]{data-label="Fig2"}](Fig2SM.pdf){width="0.8\columnwidth"}
We also study the Ising model with $\alpha=3$ shown in Supplementary Fig. \[Fig1\](c) and (d). The Binder cumulant curves show no crossing at finite temperature, which indicates the absence of thermal phase transitions as it should be in case of short-range interactions $\alpha>2$ [@Dutta_01]. Below, we study if the same thermodynamic properties are exhibited by the individual eigenstates as can be expected if the ETH holds in this regime.
Realistic spin-spin interaction {#sec: Realistic}
-------------------------------
For a finite ion chain, the spin-spin coupling $J_{ij}$ Eq. only approximately satisfy the power law. Here we study the phase diagram of the Ising model using these realistic spin-spin interaction coefficients $J_{ij}$, and show that it agrees well with the phase diagram of the power-law interaction model studied in the main text. We consider $14$ ions in a linear Paul trap and use the transverse phonons to implement our QND measurement. We choose $\omega_x/\omega_z=10$, $\Delta/\omega_z=10.22$ and $\Delta^\prime/\omega_z=20.22$. By numerically calculate the phonon modes, we find that the spin-spin couplings in Eq. (8) of the main text satisfy approximately power-law decay, with the exponent $\alpha\simeq 1.5$ from a least-square fit. Based on these realistic spin-spin couplings, we calculate the phase diagram of the system as is shown in Supplementary Fig. \[Fig\_realistic\]. As can be seen all the features of the phase diagram are in good agreement with those shown in the main text, where we approximated $J_{ij}\propto1/|i-j|^{1.5}$.
Case of short-range interactions
--------------------------------
We now study the phase transition for the case $\alpha=3$ in the microcanonical ensemble and on the level of individual eigenstates. The phase diagram in the microcanonical ensemble is shown in Supplementary Fig. \[Fig2\]. It is clearly visible that contrary to the long-range Ising model, the ordering remains only in the vicinity of $\epsilon\approx\epsilon_{0}$. This is also reflected by the order parameter probability distribution $P\left(m_{x}\right)$ for the individual eigenstates shown in inset of Supplementary Fig. \[Fig2\], which shows bimodal behavior only for the ground state. We note that this observation is compatible with the eigenstate thermalization hypothesis.
![Phase diagram of the transverse field Ising model $N=14$ with the interaction coefficients $J_{ij}$ computed according to Eq. . All the qualitative features are in good agreement with those shown in Fig. 3 (a) of the main text.[]{data-label="Fig_realistic"}](Fig_realistic.pdf){width="\columnwidth"}
Eigenstate thermalization hypothesis {#sec:eigenst-therm-hypoth}
====================================
As explained in the main text, the ETH asserts a specific form of matrix elements of few-body observables in the energy eigenbasis of an ergodic many-body Hamiltonian [@Srednicki1999]: $$\label{eq:ETH}
\braket{\ell'|\hat{O}|\ell}=O(\bar{E})\delta_{\ell'\ell}+e^{-S(\bar{E})/2}f_{\hat{O}}(\bar{E},\omega)R_{\ell'\ell}.$$ In particular, diagonal and off-diagonal matrix elements are determined by functions $O(\bar{E})$ and $f_{\hat{O}}(\bar{E},\omega)$, respectively, which depend smoothly on their arguments $\bar{E}=(E_{\ell}+E_{\ell'})/2$ and $\omega=E_{\ell'}-E_{\ell}$. $S(\bar{E})$ is the thermodynamic entropy at the mean energy $\bar{E}$, and $R_{\ell'\ell}$ is a random number with zero mean and unit variance. From the above form of matrix elements it follows that single energy eigenstates $\ket{\ell}$ encode thermodynamic properties such as phases and phase transitions which we typically associate with a microcanonical or canonical ensemble describing systems in thermodynamic equilibrium. In the main text, we describe how qualitative aspects of eigenstate thermalization can be probed across the ferromagnetic phase transition of the long-range transverse Ising model at finite energy density. Here, we elaborate on direct and more quantitative experimental tests for diagonal and off-diagonal matrix elements.
Diagonal matrix elements {#sec:diag-matr-elem}
------------------------
![Structure factor for single energy eigenstates for $N=6,10,14$ (black, blue, red dots). The narrowing of fluctuations of the eigenstate expectation values with increasing system size is a clear indication of the occurrence of eigenstate thermalization. The Ising model parameters are $\alpha=1.5$ and $h/J=0.75$. The horizontal error bars indicate the estimated energy resolution for 6 spins. The vertical error bars provide an estimate of the result dispersion for 100 simulated experimental runs per eigenvalue.[]{data-label="FIG_eth"}](Fig_ETH_alt){width="0.9\columnwidth"}
Assessing ETH quantitatively requires to show that fluctuations of single-eigenstate expectation values $\braket{\ell|\hat{O}|\ell}$ around the microcanonical average $\tr(\hat{O}\hat{\rho}_{E_{\ell}}^{\mathrm{mc}})$ are suppressed with increasing system size [@DAlessio2016]. Suitable expectation values for this purpose are fluctuations of the magnetization, $\langle\ell|\hat{m}_{x}^{2}|\ell\rangle$ where $\hat{m}_{x}=N^{-1}\sum_{j}\hat{\sigma}_{j}^{x}$, and the structure factor, $S_{\ell}\equiv N\langle\ell|\hat{m}_{x}^{2}|\ell\rangle$, which remain finite in the thermodynamic limit in the ordered and disordered phase, respectively. Using these quantities, numerical tests of ETH have been performed for the two-dimensional transverse Ising model with nearest-neighbor interactions [@Mondaini2016], and in the one-dimensional model [@Fratus2016]. In experiments with the trapped ion toolbox, the system sizes for which single eigenstates can be prepared are limited by the increasing measurement time which is required to resolve many-body energy level splittings. Since for intermediate system sizes the number of states in the disordered phase exceeds the one in the ordered phase (see Fig. 3 in the main text), the most promising prospect to test ETH quantitatively in experiments is to consider the structure factor $S_{\ell}$ in the disordered phase. We demonstrate the narrowing with system size of state-to-state fluctuations of the structure factor for the one-dimensional transverse Ising model numerically in Supplementary Fig. \[FIG\_eth\]. Even for $6$ spins, for which we expect experiments with current technology to be able to resolve individual energy eigenstates as discussed in Supplementary Note \[sec:numbers\] above, the structure factor exhibits relatively small state-to-state fluctuations, consistent with ETH. In the figure, we also indicate the estimated energy resolution for $6$ spins, and the expected error in determining the structure factor from 1000 measurements. A further suppression of state-to-state fluctuations is clearly visible for 10 and 14 spin systems, which, however, require improved experimental coherence times and detection efficiencies.
Off-diagonal matrix elements {#sec:off-diag-matr-elem}
----------------------------
![Measuring off-diagonal matrix elements of a local observable $\hat{V}=\tilde{h}\,\hat{\sigma}_{j}^{z}$ in the energy eigenbasis via the work probability distribution $P(W)$. (a) Protocol to measure the work distribution $P(W)$ as described in the text. (i) The system of $N=5$ spins is prepared in an energy eigenstate $\ket{\ell=6}$ with energy $E_{\ell=6}$. (ii) By applying the perturbation $\hat{V}$ at the middle spin ($j=3$ and $\tilde{h}=J$) during a time $J\Delta t=0.5$, the system is driven into a superposition of eigenstates $\ket{\ell}$ with probabilities $P_{\ell}(t)$ (blue shading). (iii) A second measurement of energy collapses the state of the system continuously to a final state $\ket{\ell'}$. An exemplary trajectory is shown in dark red. Repeating steps (i), (ii), and (iii) produces a sample of trajectories (light red), which gives access to the full distribution $P(W)$ of work $W=E_{\ell'}-E_{\ell}$. (b) Normalized work distribution $\mathcal{P}(W)=P(W)/\sum_{W}P(W)$ (blue columns). For weak perturbations, the work distribution is determined by off-diagonal matrix elements $\braket{\ell'|\hat{V}|\ell}$ of the perturbation. The corresponding approximation to $\mathcal{P}(W)$ is indicated by black dots.[]{data-label="fig:13"}](Fig4SM){width="1\columnwidth"}
Off-diagonal matrix elements are encoded in the dynamics, e.g. in transition probabilities between energy eigenstates in response to a weak perturbation. Using the trapped-ion QND toolbox, these transition probabilities are accessible through the protocol, which is illustrated in Supplementary Fig. \[fig:13\](a): The preparation (i) of an eigenstate of $\hat{H}$ with energy $E_{\ell}$ at a given value $h$ of the transverse field is followed by a period (ii) of length $\Delta t$ of free evolution [\[]{}$\vartheta=0$ in Eq. according to a perturbed Hamiltonian $\hat{H}'=\hat{H}+\hat{V}$; this is followed by another measurement (iii) of $\hat{H}$ which yields a value $E_{\ell'}$. The measurement outcomes determine the work $W=E_{\ell'}-E_{\ell}$ performed on the system by the perturbation $\hat{V}$. To the lowest order in the perturbation, the work distribution is determined by off-diagonal matrix elements $\braket{\ell'|\hat{V}|\ell}$ in the energy eigenbasis, $P(W)=\delta_{\ell'\ell}+\left(\Delta t\right)^{2}\lvert\braket{\ell'|\hat{V}|\ell}\rvert^{2}$. For $\lvert\braket{\ell'|\hat{V}|\ell}\rvert,W\ll(\Delta t)^{-1}$, we find good agreement between the exact work distribution and the lowest-order approximation as illustrated in Supplementary Fig. \[fig:13\](b).
|
---
abstract: 'We present the results of Karl G. Jansky Very Large Array (VLA) observations to study the properties of FR 0 radio galaxies, the compact radio sources associated with early-type galaxies which represent the bulk of the local radio-loud AGN population. We obtained A-array observations at 1.5, 4.5, and 7.5 GHz for 18 FR 0s from the FR0[*[CAT]{}*]{} sample: these are sources at $z<0.05$, unresolved in the FIRST images and spectroscopically classified as low excitation galaxies (LEG). Although we reach an angular resolution of $\sim$0.3 arcsec, the majority of the 18 FR 0s is still unresolved. Only four objects show extended emission. Six have steep radio spectra, 11 are flat cores, while one shows an inverted spectrum. We find that 1) the ratio between core and total emission in FR 0s is $\sim$30 times higher than in FR I and 2) FR 0s share the same properties with FR Is from the nuclear and host point of view. FR 0s differ from FR I only for the paucity of extended radio emission. Different scenarios were investigated: 1) the possibility that all FR 0s are young sources eventually evolving into extended sources is ruled out by the distribution of radio sizes; 2) similarly, a time-dependent scenario, where a variation of accretion or jet launching prevents the formation of large-scales radio structures, appears to be rather implausible due to the large abundance of sub-kpc objects 3) a scenario in which FR 0s are produced by mildly relativistic jets is consistent with the data but requires observations of a larger sample to be properly tested.'
author:
- |
Ranieri D. Baldi$^{1}$[^1], Alessandro Capetti$^{2}$, Gabriele Giovannini$^{3,4}$\
$^{1}$ School of Physics and Astronomy, University of Southampton, Southampton, SO17 1BJ, UK\
$^{2}$ INAF - Osservatorio Astrofisico di Torino, Strada Osservatorio 20, I-10025 Pino Torinese, Italy\
$^{3}$ Dipartimento di Fisica e Astronomia, Università di Bologna, via Gobetti 93/2, 40129 Bologna, Italy\
$^{4}$ INAF - Istituto di Radio Astronomia, via P. Gobetti 101, I-40129, Bologna, Italy
bibliography:
- 'my.bib'
title: |
High-resolution VLA observations of FR0 radio galaxies:\
properties and nature of compact radio sources.
---
\[firstpage\]
galaxies: active $-$ galaxies: elliptical and lenticular, cD $-$ galaxies: nuclei - galaxies: jets $-$ radio continuum: galaxies
Introduction {#intro}
============
Among the variety of observed morphologies of radio-emitting Active Galactic Nuclei (AGN) in the local Universe, the most common one is the presence of a single compact emitting region [@baldi10a]. This conclusion could be derived only after the advent of deep large area radio surveys in opposition to high-flux limited sample studies (such as the 3C, 2Jy, and B2 catalogues @bennett62a [@wall85; @colla75]) which typically selected radio sources extending on scale of many kpc and belonging to the Fanaroff & Riley classes I and II. The cross-match of optical and radio surveys (SDSS, NVSS, and FIRST dataset, SDSS/NVSS sample, @best05a [@best12]) showed that compact radio sources, at 5$\arcsec$ resolution, represent the vast majority of the local radio AGN population [@baldi09; @baldi10a; @sadler14; @banfield15; @whittam16; @whittam17; @miraghaei17; @lukic18]. Earlier radio studies [@rogstad69; @heeschen70; @ekers74; @wrobel91b; @slee94; @giroletti05] already pointed out that most of the radio sources in the local universe are flat-spectrum compact sources, but the attention of the community has been mainly focused on the study of extended radio sources (FR I/FR II).
Compact radio sources can potentially be associated with different classes of AGN, including radio-quiet AGN, compact steep-spectrum sources (CSS) and blazars. By using multiwavelength data provided by SDSS, @baldi10a selected from the SDSS/NVSS sample [@best05a; @best12] a large population of massive red early-type galaxies (ETGs) associated with compact radio sources. These objects have been named [*FR0s*]{} by @ghisellini11 as a convenient way of linking the compact radio sources seen in nearby galaxies into the canonical Fanaroff-Riley classification scheme [@sadler14]. Adopting these selection criteria, FR 0s form a rather homogeneous population of low-luminosity radio galaxies [@baldi18a]. FR 0s appear indistinguishable from low-power FR I/FR II LEG radio sources [@capetti17a; @capetti17b], sharing similar range of AGN bolometric luminosities, host galaxy properties and BH masses: apparently, the only feature which characterise FR 0s from the other FR classes is their lack of substantial extended radio emission.
Since this vast population is still virtually unexplored, @baldi15 carried out a pilot study of Karl G. Jansky Very Large Array (VLA) radio imaging of a small sample of FR 0s. The main result is that most of the sources still appears compact at higher resolution, with $\sim$80 per cent of the total radio emission unresolved in the core (i.e., they are highly core-dominated). The few extended sources show a symmetric radio structure. Their radio spectra generally are flat or steep, but with an emerging flat core at higher radio frequencies. In addition, these sources show radio core energetics, line and X-ray luminosities [@torresi18], similar to FR Is. @baldi15 conclude that FR 0s are able to launch a jet whose unresolved radio core base appears indistinguishable from those of FR Is, but not emitting prominently at large scales. This radio behaviour is similar to what is observed in nearby giant ETGs which harbour low-power RL AGN (10$^{36-38}$ erg s$^{-1}$; @baldi09 [@baldi16]), named Core Galaxies (CoreG, @balmaverde06core) or very low power (10$^{33-38}$ erg s$^{-1}$) LINERs recently studied with eMERLIN [@baldi18b] and VLA [@nyland16]. These sources exhibit compact radio emission at the VLA resolution, only occasionally associated with diffuse extended emission [@filho00; @filho02; @falcke00; @nagar00; @nagar02], and consistent with a FR 0 classification.
The multiband properties of FR 0s indicate that their compactness and high core dominance are genuine and not due to a geometric effect [@baldi15; @torresi18]. All these characteristics point out the uniqueness of the radio properties of the FR 0s as a stand-alone class, different from the other FR classes, blazars, CSS and radio-quiet AGN.
We proposed various explanations to account for the radio properties of FR 0s. Their small size might indicate the youth of their radio activity, but @baldi18a showed that a scenario in which all FR 0s eventually evolve into extended radio sources, cannot account for the space numbers of different FR classes. However, other open possibilities are: i) FR0s could be short-lived and/or recurrent episodes of AGN activity, not long enough for radio jets to develop at large scales [@sadler14], ii) FR0s produce slow jets, possibly due to small BH spin, experiencing instabilities and entrainment in the dense interstellar medium of the host galaxy corona that causes their premature disruption [@baldi09; @bodo13], and iii) the differences between FR 0 and extended radio galaxies are due to a distinct environment.
In order to explore the FR 0 with a more systematic approach we selected a homogeneous and well defined sample of such sources, named FR0[*[CAT]{}*]{} [@baldi18a]. The FR0[*[CAT]{}*]{} includes 104 compact radio sources selected by combining observations from the NVSS, FIRST, and SDSS surveys (see Appendix A for an update of the catalogue). We included in the catalogue the sources with redshift $\leq 0.05$ and with an optical spectrum characteristic of low excitation galaxies. We imposed a limit on their deconvolved angular sizes of 4$\arcsec$, corresponding to a linear size $\lesssim$ 5 kpc, based on the FIRST images. Their FIRST radio luminosities at 1.4 GHz are mostly in the range $10^{38} \lesssim \nu L_{1.4} \lesssim 10^{40} {\>{\rm erg}\,{\rm s}^{-1}}$. The FR0[*[CAT]{}*]{} hosts are mostly (86 %) luminous ($-21 \gtrsim M_r \gtrsim -23$) red early type galaxies with black hole (BH) masses $10^8 \lesssim M_{\rm BH} \lesssim
10^9 M_\odot$ with a small tail down to $10^{7.5} M_\odot$.
Compactness is not a well defined property as it depends on the resolution, sensitivity and the frequency of the available observations. Furthermore, we lack of any information on their radio spectral shape. For these reasons we started a comprehensive study of FR0[*[CAT]{}*]{} sources with VLA observations at higher frequency and resolution with respect to FIRST, in order to explore their morphology at different scales and wavelengths. Following the pilot study [@baldi15], here we present VLA A-array radio observations for 18 sources at 1.5, 4.5 and 7.5 GHz reaching a resolution of $\sim$0.3 arcsec to: i) study the extended emission, morphology and asymmetry ; ii) resolve the radio cores; iii) derive the radio spectral distributions.
The paper is organised as follows. In Sect. \[sample\] we define the sample and present the new VLA observations for 18 sources. In Sect \[results\] we analyse the radio and spectro-photometric properties of the sample which are discussed in Sect. \[discussion\]. The summary and conclusions to our findings are given in Sect. \[conclusion\]. We also present a revision of the FR0[*[CAT]{}*]{} sample in Appendix A.
![Total radio luminosity at 1.4 GHz from NVSS vs \[O III\] line luminosity (erg s$^{-1}$). The large red dots represent the sub-sample of 18 galaxies presented in this paper, the medium blue points represent the 104 sources forming FR0[*[CAT]{}*]{}. The small black dots are the FR I radio galaxies of the FRI[*[CAT]{}*]{} [@capetti17a] while the green crosses are the FRI in the 3C sample. The solid line represents the correlation between line and radio-luminosity derived for the 3CR/FR I sample [@buttiglione10].[]{data-label="lrlo3"}](lrlo3-jvla.epsi)
The sample and the VLA observations {#sample}
===================================
Eighteen objects, whose main properties are listed in Table \[tab1\], were randomly extracted from the FR0[*[CAT]{}*]{} sample and observed with the VLA. More specifically, we formed groups of three or four FR 0s located at small angular separations, as to reduce the telescope overheads and the time needed for observations of the calibrators. All groups were included in the schedule and five of them were actually executed. Because the observing strategy was only based on the location in the sky, this source selection does not introduce specific biases. The observed FR 0s reside in the redshift range 0.019–0.050 and are well representative of the whole FR0[*[CAT]{}*]{} sample in terms of radio and AGN power, see Figure \[lrlo3\]. This figure also shows the large deficit of radio emission of FR 0s when compared to the FR Is part of the 3C sample by a factor ranging from $\sim$30 to $\sim$1000 in the same range of bolometric AGN luminosity (represented by the \[O III\] line luminosity).
We obtained 8.67 hours of observations with the VLA in its A-array configuration between December 27, 2016 and January 23, 2017. We observed the 18 objects in 5 separate scans, ranging from 1 to 2 hours, including 3 or 4 sources. Similar to the observation strategy for the VLA pilot study of 7 FR 0s used in @baldi15, the targets have been observed in L and C bands splitting the exposure times in two. While the L band configuration corresponds to the default 1 GHz-wide band centred at 1.5 GHz, the C band was modified based on our purposes. We divided the available 2-GHz bandwidth into two sub-bands of 1 GHz centred at 4.5 and 7.5 GHz (hereafter C1 and C2 bands, respectively). This strategy allows to obtain images in 3 different radio frequencies in two integration scan. Each of the three bands was configured in 7 sub-bands of 64 channels of 1 MHz. Each source was observed for $\sim$10 minutes in both the L and C band, spaced out by the pointing to the phase calibrators for 2-4 minutes. The flux calibrator was 3C 286 observed for $\sim$6-7 minutes.
The data were calibrated by the CASA 5.0.0 pipeline v1.4.0, adding further manual flagging to remove low-level radio-frequency interferences and noisy scans to increase the general quality of the data. The final imaging process was performed with [*AIPS*]{} (Astronomical Image Processing System) package according to standard procedures. The images were then produced in the L, C1, and C2 bands from the calibrated data using the task [*IMAGR*]{}. The angular resolutions reached in the three bands are, respectively, $\sim$ 1.7, 0.5, 0.3 arcsec. We self-calibrated the maps of the sources with flux density $\gtrsim$5 mJy. The typical rms of the final images is $\sim$0.02 mJy, measured in background regions near the target. We measured the flux densities of the unresolved core components with the task [*JMFIT*]{} and the total radio emission from the extended sources with the task [*TVSTAT*]{}. In Tab. \[tab2\] we give the main parameters of the resulting images. Fig. \[maps\] presents the maps of the extended radio sources with radio contour level listed in Tab. \[tab3\].
Results
=======
The VLA observations show radio emission with flux densities in the three bands ranging between 1 and 281 mJy (with typical errors of 0.04 mJy in L band and 0.02 mJy in the two C bands). Most of the sources (14/18) appears unresolved down to a resolution of $\sim$03 which corresponds to 100–300 pc.[^2] Four sources show instead radio emission extended on a few arcseconds, on a scale of 2–14 kpc (see Table \[tab4\] and Fig. \[maps\]). In J0907+32 two symmetric jets reach a distance of $\sim$7$\arcsec$ ($\sim$14 kpc in largest linear size, LLS). In J1213+50 the jets are best seen in the 4.5 GHz image where they extend out to $\sim1\farcs5$ (LLS $\sim$2 kpc) from the nucleus. Only one jet, $\sim3\arcsec$ long (LLS $\sim$3 kpc), is detected in J1559+25. Finally, two diffuse radio structures are found in J1703+24, with an angular size of $\sim$14$\arcsec$ (LLS $\sim$9 kpc). In the Table \[tab4\] we also provide the total flux densities on the entire radio source. For the four extended sources, we estimated the jet counter-jet ratio measuring the brightness ratio in two symmetric regions as near as possible to the nuclear emission by avoiding the core and considering the surface brightness at similar distance from the core. The brightness jet ratio range between $\sim$1 and 2, while one source is fully one-sided ($>$8).
Our high resolution VLA observations, obtained with a relatively short exposure time, might miss faint extended emission. In order to explore this possibility we compared the observed total VLA 1.5 GHz flux densities with those measured by FIRST and NVSS at larger resolutions ($\sim$5 and $\sim$45 arcsec, respectively, see Table \[tab4\]). The flux densities measured in the L band from our maps are generally consistent with those from the FIRST catalogue (derived from observations between 1994 and 1999 for our sources), an indication that we recovered most of the radio emission. Indeed, all but four sources show differences of less than 20% over a timescale of $\sim$20 years. One of them (J1136+51) decreased in flux from 7.8 to 5.3 mJy. Three sources (namely J0943+36, 1025+10, and J1530+27) instead increased their flux densities by a factor between 1.5 to 3.2.
All the 18 FR0s show a ratio between NVSS and FIRST flux densities very close to unity ($0.89 < F_{\rm NVSS}/F_{\rm FIRST} < 1.11$) with only two exceptions. One is J1559+25 with a ratio of $\sim$5: this is due to the presence of a second compact source with a FIRST flux density of 117 mJy located 28$\arcsec$ to the West, blended with our target in the NVSS image. The second is J1703+24 with a ratio of 1.43: this is one of the FR0s with extended emission, suggesting that some extended emission is lost in the FIRST images.
We also obtained matched-beam radio maps[^3] to derive the radio spectra at the three frequencies for all sources. The resolution-matched flux densities of the central components are reported in Table \[tab4\] and plotted in the radio spectra in Fig. \[SED\]. The typical flux errors for the central components are smaller than 0.1 mJy. Only for the brightest sources, exceeding $\sim100$ mJy, they can be as high as 0.5 mJy. We do not report them in the Table for sake of clarity.
The spectral indices $\alpha$ between 1.5 and 4.5 GHz ($F_{\nu} \propto
\nu^{\alpha}$) cover a broad range, from $\sim$1 to -0.2: most spectra are flat ($-0.2< \alpha < 0.4$ is measured for 11 objects), six are steep ($0.49 <
\alpha < 1.03$). In one case (J0943+36) the spectrum is strongly inverted, with $\alpha$ = -0.6. If we consider the spectral indices measured between the two adjacent bands, we typically observe a spectral steepening at 7.5 GHz, with a median difference between the two indices of $\Delta\alpha \sim$ 0.16. We also note that four flat-spectrum sources (1025+10, 1530+27 1628+25 1658+25) have slightly convex radio spectra: the flux density at 4.5 GHz is typically $\sim$20 per cent higher than the one at 1.5 GHz and 15 per cent higher than at 7.5 GHz.
{width="19cm" height="22cm"}
![Distribution of core dominance $R$, i.e., the ratio between the flux density of the central component at 7.5 GHz and the NVSS flux density at 1.4 GHz. The hatched histogram corresponds to upper limits on $R$ for the sources with no radio core detection.[]{data-label="coredom"}](coredom.epsi)
![Core radio power vs. \[O III\] line luminosity (erg s$^{-1}$) for CoreG (pink triangles), 3CR/FR I radio-galaxies (green asterisks), and FR0s (red dots) from this study and the 7 FR 0s (red squares) from the VLA pilot study [@baldi15]. The line indicates the best linear correlation found for 3CR/FR Is.[]{data-label="coreo3"}](lcore_lo3_allFR0_all.epsi)
Interestingly, results of VLBI observations for four of FR 0s studied here have been recently presented by @cheng18. J0943+36, the inverted-spectrum source, shows an unresolved VLBI core with an inverted spectrum and significant variability, with a flux density increasing from 0.17 to 0.25 Jy over four years. J1213+50, a twin-sided jet FR 0, show a similar morphology but on mas-scale and perpendicular to the VLA jets. J1230+47, a flat-spectrum FR 0, shows twin jets on mas scale with the VLBI. J1559+25, the one-sided FR 0, appears similarly one-sided on mas scale with the VLBI with no sign of variability ($<20$%).
One of the aims of this study is the measurement of the radio core. In the four extended sources a compact central component is always clearly visible. However, the spectrum of the central source of, e.g., J0907+32 is very steep ($\alpha\sim1$) indicating a substantial contribution from optically thin, extended emission. We must then rely on the radio spectra to isolate the core emission. We considered as core-dominated the 11 sources with central component characterised by a flat spectrum ($\alpha\lesssim0.4$). For the remaining objects the 7.5 GHz flux density is adopted as upper limit to the core. In @baldi15 we noticed several sources with a flattening of the radio spectrum between 4.5 and 7.5 GHz with respect to that measured between 1.5 and 4.5 GHz and interpreted this behaviour as the emerging of a flat core at the higher frequencies. In the observations we are presenting here, there is not a clear evidence of a flattening of the radio spectra at 7.5 GHz: the extrapolation of the 1.5 $-$ 4.5 GHz agrees or slightly over-predicts the actual measurement at 7.5 GHz by $\sim$20 per cent.
The distribution of core dominance of our sources, $R$, defined as the ratio between the nuclear emission at 7.5 GHz and the total NVSS flux density at 1.4 GHz[^4] is presented in Fig. \[coredom\]. The 11 sources with a flat spectrum have $-0.4 \lesssim
{\rm log} R \lesssim 0.5$. The upper limits on $ {\rm log} R$ for those with a steep spectrum are in the range -0.9 to -0.4. This distribution is significantly different from from that of 3CR/FRIs (with a $>$99.9% probability, according to a Kolmogorov Smirnoff test) while it is not distinguishable from the $R$ distribution of CoreG and the other FR 0s studied in the previous VLA project [@baldi10a; @baldi15].[^5]
We can now include our sources in the $L_{core} (= L_{\rm 7.5 GHz})$ vs. [L$_{\rm{\tiny{ [O~III]}}}$]{} plane similarly to what done in @baldi15 for the VLA FR 0 pilot study. This diagnostic plot compares the radio core energetics with the line luminosity, adopted as a proxy for the AGN luminosity [@heckman04]. The FR 0s lie in the same region populated by the lower luminosity 3CR/FR Is and follow the same core-line relation found for FR Is and CoreG. This relation further strengthens the similarity between the nuclei of FR 0s and FR Is. However, we remind that for seven of them we only derived upper limits to their radio core flux densities for the lack of a clear flat-spectrum unresolved core. Although the point scatter of FR 0s is consistent with that of FR Is, the presence of upper-limits on the core measurements in the correlation challenges the FR 0–FR I similarity scenario.
Discussion
==========
Despite the fact that FR 0s share the same properties with FR I radio galaxies from the nuclear and host point of view, FR 0s differ from the other FR classes for their remarkable lack of significant extended radio emission, and for being dominated by a sub-kpc scale flat-spectrum component. In @baldi15 and @baldi18a we discussed various scenarios to interpret this unique feature of FR 0 and we now review them in the light of the results obtained with the new VLA observations.
@baldi18a conclude that a scenario in which all FR 0s are young radio sources that will eventually evolve into extended radio source does not reconcile with the relative number densities of these classes. Thanks to the new high-resolution observations, we can further test this possibility, by exploring the distribution of radio sizes of FR 0s. In Fig. \[size\] we show the size distribution of all 182 SDSS/NVSS LEG radio galaxies with $z<$0.05, 104 of which are included in FR0[*[CAT]{}*]{}. The fraction of FR0s unresolved in the VLA observations is $\sim78$%, for which we set a conservative size limit of 1 kpc (represented by the arrow in the Fig. \[size\] left panel), and they represent $\sim44$% of the whole population of radio emitting AGN within this volume. The remaining sources extend up to $\sim$100 kpc and about one third of them are FR Is, part of the FRI[*[CAT]{}*]{} and sFRI[*[CAT]{}*]{} samples. In the right panel of this Figure we present the number of objects in each bin, divided by the bin size in kpc. As already discussed in @baldi18a, in case of constant expansion speed all bins should be equally populated. Conversely, the number density of sources with size $\lesssim$1 kpc exceed by $\sim$2 orders of magnitude that of the extended sources indicating that FR 0s do not generally grow to become large radio galaxies.
The distribution of sizes also sets strong constraints of the interpretation that FR 0 are short-lived recurrent sources. As we mentioned in @baldi15, the results obtained by @shin12 indicate that the more massive galaxies spend a larger fraction of their time in active states than satellite galaxies. The possibility that this is the origin of the strong peak in the size distribution appears to rather contrived considering the relatively small differences in host galaxy masses between FR 0s and FR Is, on average only a factor 1.6 and with substantial overlap between the two classes [@baldi18a; @miraghaei17]. Nevertheless, it is correct to mention that a low/moderate amplitude radio flux density variation has been detected for a few FR 0s over time scale of years [@cheng18]. This variability is not necessary associated with a nuclear recurrence, but a phenomenon expected within an evolutionary scenario of the source [@morganti17].
Another scenario for the origin of FR 0s is related to the jet launching region. The similar nuclear luminosities of FR 0s and FR Is and the FR 0 radio morphologies we observed suggest that within the radio core, less than 1 kpc, the jets of FR 0s should start relativistic [@cheng18]. However, at larger scales ($>$ 1 kpc), we envisaged that the jet $\Gamma$ factors of FR 0s are lower than in FR Is by affecting their ability to the penetrate the host’s interstellar medium and transforming the relativistic jets to turbulent flow not far outside the optical core of the galaxy as seen in nearby FR I radio galaxies (e.g. @killeen86 [@venturi93; @bicknell95]). Nevertheless, no evidence of deceleration sites along the jet on different scales has been observed in our FR 0 sources. We can test the possibility of a jet deceleration and the bulk jet speed by measuring the sidedness of jets in FR 0s and to compare it with that of FR Is. By considering together this new dataset and the results of the pilot program there are now five FR 0s with rather symmetrical extended or slightly sided structures (jet brightness ratio 1–2, J0907+32, J1213+50, J1703+24 in this work and ID 547 and ID 590 from @baldi15). We also found one highly asymmetric source but the minimum jet Lorentz factor to obtain the observed flux ratio between the jet and counter-jet of $\gtrsim 8$ is only $\Gamma \gtrsim 1.1$, not necessarily indicative of a highly relativistic jet. From the comparison between our VLA and VLBI images from @cheng18 for 5 FR 0s, we learnt that, when observed, sub-kpc scale jets are typically present also at mas scale. However, for our source (J1213+504) the mas-scale jets appear perpendicular to those observed with VLA, possibly due to jet procession, while in one compact FR 0 (J1230+47), two pc-scale twin jets emerge, pointing to a sub-relativistic jet speed on sizes smaller than what probed by the VLA. An inferred mildly relativistic jet bulk speed on sub-kpc scale is in agreement with the Doppler boosting factors estimated by @cheng18 from VLBI observations. The modelling of the multiband spectrum of the first FR 0 detected in $\gamma$-ray with Fermi [@grandi16] with standard beamed and misaligned jet models [@maraschi03; @ghisellini05] produce bulk jet $\Gamma$ factor 2 – 10 [@tavecchio18]. Clearly, the statistic on the jets asymmetries in FR 0s is still insufficient to draw a firm conclusion on their speed, but an evidence of mildly-relativistic jet speed is increasing.
Another tool that can be used to measure the jets speed is to consider the dispersion of the core powers with respect to a quantity independent of orientation, such as the emission line luminosity, see Fig. \[coreo3\]. Because there might be a substantial level of intrinsic scatter, this approach can provide us with an [*upper limit*]{} on the jet $\Gamma$. However, due to the presence of a significant fraction of non detections of flat cores, we can only derive a [*lower limit*]{} to the cores dispersion. Higher frequency observations are needed to isolate the core emission also in the sources with steep spectra and fully exploit this method.
We must also mention that a different matter content of the FR 0 jets from the other FR classes might account for the reduced jet extension observed in FR 0s. Lighter jets, possibly composed mainly of electron-positron pairs [@ghisellini12] would correspond a much lower jet power, again, hampering the formation of large scale radio structures, with respect to what is predicted by a leptonic model [@maraschi03].
While the majority of FR 0s conform with the idea that they are compact flat spectrum sources, some of them show steep spectra. Interestingly, the fraction of flat and steep sources in our sample (with a two to one ratio) is the same found by @sadler14 in their sample of FR 0 LEG, despite the rather different selection criteria. FR 0s certainly are a mixed population low-power radio sources, one expected contaminant being a small population of genuinely young FR Is, as also suggested by @cheng18.
We must mention that four sources show a slightly convex radio spectra and one source (J0943+36) has an inverted spectrum, similar to what seen in GHz peaked sources (GPS, @peacock82). Nevertheless, VLBI observations [@cheng18] did not resolve a mas-scale double-lobe structure as expected by this class of radio sources. Therefore, its particular radio behaviour requires more attention to investigate whether its compactness and spectrum reconciles with being a young radio galaxy, which expands at a very low rate due to its much lower power with respect to GPS [@odea98] or, instead, is consistent with what expected from young FR Is.
Summary and Conclusions {#conclusion}
=======================
We presented new VLA observations at array A at three frequencies 1.5, 4.5 and 7.5 GHz for a sample of 18 FR 0 radio galaxies selected from the FR0[*[CAT]{}*]{} sample. At the highest angular resolution ($\sim$0.3 arcsec), most of the sources are still unresolved with with $\sim$80 per cent of the total radio emission enclosed in the core (i.e., they are more core-dominated than FR Is by a factor $\sim$30). Only four objects show extended emission up to 14 kpc, with various morphologies: one- and two-sided jets, and double lobes. Six have steep radio spectra, one shows an inverted spectrum, while 11 are flat cores. Four of the flat-spectrum cores in FR 0s are actually slightly convex, with a small steepening between 4.5 and 7.5 GHz with respect to the slope measured at lower frequencies.
For 11 the sources, where a radio core is detected, the core and emission line luminosities correlate following the relation valid for FR I and CoreG radio galaxies, respectively at high and low luminosities. However, for 7 sources due to their their steep spectra and morphology, we can give only an upper limit to the radio core luminosity, which is still in agreement with the correlation. More high-resolution data are necessary to isolate their core components. Adding the 7 FR 0s studied in a previous VLA exploratory pilot @baldi15, we confirm the similarity between the radio cores of FR 0s and FR Is and the absence of strong beaming effect in the FR 0 cores.
The size distribution of nearby $z<0.05$ RL AGN shows, thanks to the new observations, a strong peak for sources smaller than $\sim
1$kpc. This rules out the possibility that FR 0s are young sources that will all evolve into more extended radio galaxies and it also disfavours the interpretation that they are recurrent sources, with very short periods of activity. The most likely possibility is that FR 0s are associated with jets that are mildly relativistic at sub-kpc scale.
The combination of high-resolution, large collecting area, and wide frequency range of the new generation of radio telescopes (e.g., SKA, ngVLA, LOFAR) will enable to satisfy the increasing interest in low-power radio sources [@nyland18], and, consequently, in FR 0s [@whittam17]. The advent of larger radio surveys will give the opportunity to study the pc-scale emission of FR 0s and test whether their jets have smaller bulk jet speed than the other FR classes on firmer statistical grounds, by expanding the size of the observed sample with high quality radio observations. Higher frequency observations are also needed to isolate the core emission in sources in which the radio spectra is steep and detect their jets. On the opposite side, lower frequency observations will be very useful to establish whether among them there are other inverted-spectrum sources, which will provide indications on the nature of the FR0 population: slow jets or young radio sources?
Acknowledgments {#acknowledgments .unnumbered}
===============
The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. RDB acknowledge the support of STFC under grant \[ST/M001326/1\]. We thank Francesca Panessa, Graziano Chiaro and Matt Malkan for a helpful discussion on the nature of compact radio sources. We also thank the reviewer for the useful comments, which helped us improve the quality of the manuscript.
{width="19cm" height="5cm"}
{width="8.8cm" height="7.8cm"}   
The case of IC 711 and an update of the FR0[*[CAT]{}*]{} {#notes}
=========================================================
Most of the 108 sources in the FR0[*[CAT]{}*]{} catalogue show a ratio between the FIRST and NVSS flux densities between 0.8-1.2, indicating that, in general, in FR 0s there is no significant amount of extended low-brightness radio emission lost due to the missing short baselines of the FIRST observations. However, individual exceptions might exist. In Fig. \[ic711\] we present the images obtained as part of our program of VLA observations of SDSS J113446.55+485721.9 which shows a pair of bent asymmetric jets extending for $\sim$90 from the radio core. This source, also known as IC 711, is actually the longest head-tail radio source known [@vallee76], extending over $\sim$ 500 kpc [@vallee87]. Its extended nature is clearly seen also in the NVSS image (see Fig. \[NVSS\]).
The NVSS images were not used for the FR 0s selection because of their low spatial resolution (45$\arcsec$) and the resulting high level of confusion. Nonetheless, the case of IC 711 indicates that they can still be used to improve the sources morphological classification. We then retrieved the NVSS images of all 108 FR 0s in FR0[*[CAT]{}*]{} looking for extended emission. Elongations in the iso-contours are rather common, usually extending just in one direction. As expected, in most cases this is the result of confusion, due to the presence of a nearby radio source, well visible in the FIRST images. Nonetheless, there are four sources in which the radio emission is clearly extended . Their NVSS images are shown in Fig. \[NVSS\]. These sources (namely SDSS J112039.95+504938.2 SDSS J113446.55+485721.9 aka IC 711, SDSS J160616.02+181459.8, and SDSS J162549.96+402919.4) are not genuine compact FR 0s and should be then removed from the FR0[*[CAT]{}*]{}catalogue, which now counts 104 sources.
Regarding IC 711, the core flux densities measured from the VLA observations are 25.92 $\pm$ 0.03 at 1.5 GHz, 24.30 $\pm$ 0.02 at 4.5 GHz, and 23.23 $\pm$ 0.01 at 7.5 GHz, respectively, showing a flat spectrum with $\alpha$ = - 0.07. In this source the jets are highly asymmetric in the innermost regions, but their brightness ratio decreases at larger distances, possibly due to a deceleration from relativistic to subsonic [@bicknell95].
\[lastpage\]
[^1]: E-mail: r.baldi@soton.ac.uk
[^2]: In Appendix \[notes\], we also show the radio maps at the three frequencies of IC 711, originally included in the FR0[*[CAT]{}*]{} sample (and observed in this VLA project) but subsequently discarded based on the presence of extended emission in its NVSS image.
[^3]: We matched the C-band radio images at the resolution obtained in L band by using the parameter [*uvtaper*]{} in the [*IMAGR*]{} task in AIPS.
[^4]: Except for J1559+25 for which we used the FIRST data because of the contamination to the NVSS flux density from the nearby source.
[^5]: We estimated $R$ as the ratio between the 7.5 GHz core emission and the total 1.4 GHz flux density, while for the 3CR/FRIs sources we used the 5 GHz core flux density against the 1.4 total flux density. However, since the radio core emission has generally a flat spectrum this quantity is only weakly dependent on the frequency used for the core measurement and this comparison is robust.
|
---
abstract: 'The influence of graphene islands on the electronic structure of the Ir(111) surface is investigated. Scanning tunneling spectroscopy (STS) indicates the presence of a two-dimensional electron gas with a binding energy of $-160{\mathrm {\,meV}}$ and an effective mass of $-0.18{\mathrm {\,m_e}}$ underneath single-layer graphene on the Ir(111) surface. Density functional calculations reveal that the STS features are predominantly due to a holelike surface resonance of the Ir(111) substrate. Nanometer-sized graphene islands act as local gates, which shift and confine the surface resonance.'
author:
- 'S. J. Altenburg'
- 'J. Kröger'
- 'T. O. Wehling'
- 'B. Sachs'
- 'A. I. Lichtenstein'
- 'R. Berndt'
bibliography:
- 'cites2.bib'
title: 'Local Gating of an Ir(111) Surface Resonance by Graphene Islands'
---
On a number of metal surfaces, single layers of graphene may be grown. Owing to their different graphene–metal interactions [@Wintterlin20091841], these substrates may modify the electronic structure of the graphene layer. For example, the $\pi$ and $\pi^*$ bands of graphene, which give rise to a Dirac cone dispersion in the pristine material, are significantly altered on Ni(111) and Ru(0001) [@PhysRevB.50.17487; @Himpsel1982L159], where a gap in the Dirac bands opens. In contrast, from Ir(111), an almost unchanged band structure of graphene has been reported [@art:grapheneMinigapsIr]. The modification of the electronic structure of the substrate upon graphene adsorption has hardly been investigated. From calculations, quenching of the Ni(111) surface state upon graphene adsorption was predicted [@dze_11]. The only experiment related to this issue addressed a spatial variation of the Ru(0001) $d$ states, which arises from a graphene-induced moiré pattern [@mgy_11]. This state of affairs is surprising as electronic states at surfaces are sensitive probes of the interaction with adsorbates.
Here, we combine scanning tunneling spectroscopy and density functional theory (DFT) calculations of Ir(111) covered with a single layer of graphene. Spectra of the differential conductance ($\text{d}I/\text{d}V$; $I$: current; $V$: sample voltage) of pristine and graphene-covered Ir(111) reveal a holelike surface resonance. Intriguingly, the most prominent features in the spectra are due to this resonance at the Ir–graphene interface rather than to any states of the graphene layer. This effect is attributed to the selectivity of the tunneling current for states with small parallel momentum. Although the electronic structure of graphene is only weakly perturbed by the Ir(111) substrate, graphene shifts the Ir surface resonance by $\approx\,190{\mathrm {\,meV}}$ towards the Fermi level. As a result, graphene islands act as local gates which confine the surface resonance and induce characteristic standing wave patterns. The resonance shift and an effective mass $m^*= -0.18{\mathrm {\,m_e}}$ ($\text{m}_{\text{e}}$: free electron mass) determined from these patterns are consistent with DFT results.
![Main figure: Spectra of $\text{d}I/\text{d}V/(I/V)$ recorded along a line from Ir onto a graphene island as indicated in Inset (b). A clear shift of the surface resonance onset occurs. Contours of constant LDOS calculated using a scattering model (see text) are indicated by black and light gray lines for Ir and graphene, respectively. A horizontal blue (gray) line indicates the position of the graphene edge. Inset (a): Spectra of $\text{d}I/\text{d}V$ recorded near the beginning (blue(black)) and end (red(dark gray)) of the line in Inset (b). The beginning of the line is defined by the cross at the dotted line in Inset (b). Inset (b): Constant-current STM image of a graphene patch on Ir(111) ($-220{\mathrm {\,mV}}$, $100{\mathrm {\,pA}}$). White dots mark positions where $\text{d}I/\text{d}V$ spectra were recorded. A white cross denotes zero distance. The false colors used in the main figure and in Inset (b) are defined in the upper-right-hand corner.[]{data-label="fig:step"}](fig1.eps){width="85mm"}
Experiments were performed with a home-built scanning tunneling microscope (STM), operated at $5.2{\mathrm {\,K}}$ in an ultrahigh vacuum. Ir(111) surfaces were cleaned by cycles of Ar$^+$ bombardment and annealing. Graphene films were grown by exposing the sample to $\approx 6\times 10^{-4}{\mathrm {\,Pa}}{\mathrm {\,s}}$ of $\text{C}_2\text{H}_4$ at room temperature and subsequent annealing at $\approx 1400{\mathrm {\,K}}$. This procedure leads to a partial coverage of Ir(111) by highly ordered graphene [@art:IrClusters]. Au tips were prepared by [*ex-situ*]{} cutting, [*in-vacuo*]{} heating and Ar$^+$ bombardment. Spectra of $\text{d}I/\text{d}V$ were acquired by a standard lock-in technique (modulation frequency: 9.1[kHz]{}, modulation amplitude: 10[mV$_{\text{rms}}$]{}) and subsequently normalized by $I/V$ to compensate for the voltage-dependent transmission of the tunneling barrier [@art:FeenstraNorm].
Figure \[fig:step\] shows normalized $\text{d}I/\text{d}V$ data, which were obtained along a slightly curved line \[Inset (b) to Fig.\[fig:step\]\] crossing a step from bare Ir onto a graphene island. Above the bare Ir(111) surface, a steplike decrease in the $\text{d}I/\text{d}V$ signal at $V \approx\,-350{\mathrm {\,mV}}$ \[Inset (a) to Fig.\[fig:step\], blue (black) line\] occurs, which is indicative of a holelike surface state or resonance[^1]. Such a resonance was previously observed with photoelectron spectroscopy [@veen_80; @ARPESIr]. Above the graphene layer, the steplike feature is shifted to $V \approx\,-150{\mathrm {\,mV}}$ \[Inset (a) to Fig.\[fig:step\], red (dark gray) line\]. When approaching the edge of the graphene island from either side, the $\text{d}I/\text{d}V$ step moves towards lower voltages and disappears on top of the graphene edge. This spatial variation of the shift can be explained by scattering at the graphene edge. The rather strong interaction between graphene edges and the Ir(111) substrate bends the graphene edges towards the metal and leads to the formation of chemical bonds between Ir and C atoms at the island edge [@PhysRevLett.103.166101]. The Ir(111) surface resonance is considered as a free electron gas with a binding energy $E_0$ and effective mass ${m^*}$ scattered from a hard-wall potential provided by the graphene edges. The spatial variation of the local density of states (LDOS), $\rho_{\text{s}}$, can then be described as $$\rho_{\text{s}}(E,x)=1-J_0(2{k_{\|}}x),$$ where $J_0$ is the zeroth-order Bessel function, ${k_{\|}}=\sqrt{2{m^*}(E-E_0)}/\hbar$ is the parallel momentum, $E$ is the energy, and $x=0$ is the position of the hard-wall potential [@avouris:1447; @jacklevic; @Eigler]. Lines in Fig.\[fig:step\] show contours of constant LDOS calculated for ${m^*}=-0.18{\mathrm {\,m_e}}$ (see confinement analysis below), $E_{0,\,\text{Ir}}=-350{\mathrm {\,meV}}$ (curved black line) and $E_{0,\,\text{gr}}=-160{\mathrm {\,meV}}$ (curved white line), for Ir(111) and graphene-covered Ir(111), respectively. To match experimental data, a Gaussian broadening of $40{\mathrm {\,meV}}$ was applied. Further, the energies $E_{0,\,\text{gr}}$ and $E_{0,\,\text{Ir}}$ were chosen to yield a good fit between the calculated first maximum of the oscillation (embraced by contour lines) and the experimental data. The simple hard-wall model reproduces the curvature and position of the LDOS maxima quite well and yields an energetic shift of the surface resonance between bare and graphene-covered Ir of $\Delta E\approx\,190{\mathrm {\,meV}}$. The energies fit well to the energies extracted from the single spectra; $E_{0,\,\text{gr}}$ also matches the energy obtained by confinement analysis (see below). It is important to note that the data do not reveal any particle-hole symmetric counterpart of these confinement features above the Fermi level. At energies below $\approx\,-650{\mathrm {\,meV}}$ (not shown in Fig.\[fig:step\]) variations in the LDOS with the periodicity of the moiré pattern predominate, probably due to weak periodic potential modulations.
![ Calculated band structures of (a) pristine and (b) graphene-covered Ir(111). Light gray lines in (a) show the dispersion of all states of the supercell used. The contribution of each state to the tunneling current is indicated by the widths of blue (black) and red (dark gray) lines. On Ir, the current is essentially due to a surface resonance at the center of the surface Brillouin zone ([$\overline{\Gamma}$]{}). On the graphene-covered surface, the resonance is shifted upwards. It still carries most of the current, while the Dirac cone around [[$\overline{\text{K}}$]{}]{} is less conducting. Green (gray) parabolas show a fit with an effective mass ${m^*}\approx-0.17{\mathrm {\,m_e}}$.[]{data-label="fig:DFT"}](fig2.eps){width="85mm"}
To further support the above model, DFT calculations of the pristine and graphene-covered Ir(111) surface were performed [^2]. We used the projector-augmented plane wave method [@PAWbloechl; @PAWkresse], as implemented in the Vienna *ab initio* simulation package [@kresse_vasp]. Light gray lines in Fig.\[fig:DFT\](a) show the calculated band structure of an Ir(111) surface. Near the Fermi energy of the Ir(111) system, various bands originate from bulk bands with $p$ or $d$ character and there are several surface states around the bulk band gap near [[$\overline{\text{K}}$]{}]{} and a surface resonance near [$\overline{\Gamma}$]{}. This resonance is mainly derived from Ir $p_z$ orbitals of the first few atomic layers near the surface. In the energy range $-1{\mathrm {\,eV}}<E<0{\mathrm {\,eV}}$ probed in the STM experiments, the calculated dispersion of the surface resonance is approximately parabolic, with an effective mass of ${m^*}\approx-0.17{\mathrm {\,m_e}}$ \[Fig.\[fig:DFT\](a), green (gray) line\]. At the graphene-covered surface \[Fig.\[fig:DFT\](b)\], two additional bands derived from the C $p_z$ orbitals occur and form a Dirac cone near [[$\overline{\text{K}}$]{}]{} in agreement with previous photoemission [@ARPESIr] and DFT [@art:grapheneMinigapsIr] studies. On the graphene-covered surface, the resonance is shifted upwards \[Fig.\[fig:DFT\](b), green (gray) line\] by an amount which depends on the distance between the graphene sheet and the topmost Ir layer. For typical spacings between $0.327$ and $0.362{\mathrm {\,nm}}$ [@michelyvdW], the calculated shift is between $100$ and $200{\mathrm {\,meV}}$, which is consistent with our experimental value of $\approx\,190{\mathrm {\,meV}}$.
To trace back the origin of the resonance shift upon graphene adsorption, calculations were performed in which the graphene C atoms were replaced by chemically fully inert Ne atoms. As a result, a Ne layer shifts the Ir surface resonance upwards by virtually the same amount as the graphene layer. As Ne provides no states at the Fermi level which could donate or accept charge from the Ir surface, the upward shift of the resonance is most likely due to a significant Pauli repulsion. Nevertheless, Coulomb potential effects, e.g., via charge redistribution [@michelyvdW], occur and cannot be disregarded in modeling the full electronic structure of graphene/Ir(111). The calculations further show a downward shift of the Dirac cone when the graphene sheet is pushed towards the Ir substrate. This shift cannot be explained by Pauli repulsion and demonstrates that Coulomb potential effects are predominant for the energy of the graphene Dirac point.
To determine the contributions of the various states to the tunneling current, the approach of Tersoff and Hamann [@art:Tersoff; @*art:Tersoff2] was used and the tip was modeled as an $s$ orbital ${\ensuremath{|L\rangle }}$. Based on an estimate of the experimental tip–sample distance[@fnote1] the orbital is placed $0.48{\mathrm {\,nm}}$ above the surface. The overlap $|{\ensuremath{\langle \Psi_{n,k}|L\rangle}}|^2$, where ${\ensuremath{|\Psi_{n,k}\rangle }}$ is the wave function of a band $n$ at wave vector $k$, is indicated by the width of the colored bands in Fig.\[fig:DFT\]. On both surfaces, clean and graphene-covered Ir(111), the main contribution to the current is due to the aforementioned surface resonance. The current due to the Dirac bands of graphene is significantly smaller. This may be understood from the parallel momenta ${k_{\|}}$ of these states, which affect their decay into vacuum. The surface resonance is located around [[$\overline{\Gamma}$]{}]{} and thus decays less rapidly than the Dirac cone states near [[$\overline{\text{K}}$]{}]{} [^3]. Therefore, the steps in the $\text{d}I/\text{d}V$ spectra may safely be attributed to the (shifted) Ir(111) surface resonance. This result is also in agreement with the absence of electron–hole symmetry from the experimental spectra. In recent publications [@PhysRevLett.107.236803; @doi:10.1021/nn2028105], scanning tunneling spectroscopy data from graphene on Ir(111) are attributed to tunneling from graphene states. However, the analyses of Refs. neglect the substrate electronic states at the Brillouin zone center. In contrast, the present results show the importance of substrate states at [[$\overline{\Gamma}$]{}]{} which give the dominant contribution to the current in our STM experiments.
![(a) Constant-current STM image of a graphene island on Ir(111) ($-220{\mathrm {\,mV}}$, $1{\mathrm {\,nA}}$). A blue (light gray) circle indicates the effective island diameter of $8.3{\mathrm {\,nm}}$ used for further analysis. (b)–(d) Normalized $\text{d}I/\text{d}V$ maps of the graphene island in (a), recorded at the indicated voltages. The observed LDOS oscillations evolve as expected for the confinement of an electron gas in a quantum dot. (e) Energies of confined states with significant LDOS at the island center ($n=0$) measured from various graphene islands. States with $l=1$ (open blue circles) and $l=2$ (filled red circles) are resolved. Lines show a fit according to Eq.(\[eqn:Enl\]) with $E_0=-160{\mathrm {\,meV}}$ and ${m^*}=-0.18{\mathrm {\,m_e}}$, which are in good agreement with values calculated within DFT.[]{data-label="fig:maps"}](fig3.eps){width="85mm"}
In addition to scattering at their edges, graphene islands lead to confinement of the hole states. Figure\[fig:maps\] shows an STM image of a graphene island along with normalized $\text{d}I/\text{d}V$ maps recorded at constant current. At increasingly negative sample bias \[Figs.\[fig:maps\](b)–(d)\] the pattern inside the island evolves from a central maximum over a ring to a ring with a central maximum, as expected for confined states with zero, one, and two nodes, respectively. For a more detailed analysis, we model the island by a circular quantum dot with hard walls. The eigenenergies $E_{n,l}$ of a confined electron gas are [@platt:1448] $$E_{n,l} = E_0 + \frac{2\hbar^2u_{n,l}^2}{{m^*}d^2},
\label{eqn:Enl}$$ where $u_{n,l}$ is the $l$th root of the $n$th-order Bessel function $J_n$ and $d$ is the island diameter. Figure \[fig:maps\](e) displays the energies of the first two resonances which exhibit an LDOS maximum at the island center ($E_{0,1}$ and $E_{0,2}$) evaluated from spatially resolved $\text{d}I/\text{d}V$ spectra of 8 nearly circular islands with diameters between $5$ and $12{\mathrm {\,nm}}$. The effective island diameters $d$ were determined from an inscribed circle, which touches the island boundaries at the midpoint of the step edge \[Fig.\[fig:maps\](a), blue (light gray) line\]. Energies calculated according to Eq.(\[eqn:Enl\]) with $E_0=-160{\mathrm {\,meV}}$ and ${m^*}=-0.18{\mathrm {\,m_e}}$ \[Fig.\[fig:maps\](e), lines\] match the experimental data very well.
![Spatially resolved normalized $\text{d}I/\text{d}V$ spectra recorded along a line through a graphene island (inset). The experimental distance scale was recalibrated to account for the small curvature of the measurement path. Light gray lines mark contours of constant LDOS calculated for the circular island which is indicated in the inset. Blue (gray) lines indicate the boundaries of the model island. Inset: Constant-current STM image of the graphene island ($-220{\mathrm {\,mV}}$, $100{\mathrm {\,pA}}$). White dots indicate positions where spectra of $\text{d}I/\text{d}V$ were recorded. A white ring denotes zero distance. An inscribed blue (gray) circle shows the effective island diameter of $11.8{\mathrm {\,nm}}$ used in modeling.[]{data-label="fig:largeIsland"}](fig4.eps){width="85mm"}
As a final test of the model spatially resolved $\text{d}I/\text{d}V$ spectra from a roundish island are compared with the calculated LDOS. The LDOS, $\rho(E,x)$, of the surface resonance confined to a disk is $$\rho(E,x) = \sum_{n,l}
|\Psi_{n,l}|^2
\exp\left[
-\frac{1}{2}\left(
\frac{E_{n,l}-E}{\delta E}
\right)^2
\right].
\label{eqn:rhoIsland}$$ $\Psi_{n,l}$ are solutions to the Schrödinger equation as described by Platt [[*et al. *]{}]{}[@platt:1448], $$\Psi_{n,l}(r,\varphi) = J_n\left(
u_{n,l}\frac{2r}{d}
\right)
\exp(\text{i}n\varphi).
\label{eqn:PsiIsland}$$ $\delta E$ is a Gaussian broadening reflecting a finite lifetime of the states. While the broadening may depend on energy [@PhysRevB.71.155417], the constant broadening $\delta E=40{\mathrm {\,meV}}$ assumed here is sufficient to match the experimental observations. Figure \[fig:largeIsland\] shows a series of 29 normalized $\text{d}I/\text{d}V$ spectra measured along a line across a graphene island (inset). Light gray contour lines show the calculated LDOS using the measured diameter of $11.8{\mathrm {\,nm}}$ (blue (gray) horizontal lines). The qualitative agreement of the theoretical and experimental data is further evidence that graphene islands confine the Ir(111) resonance.
In conclusion, van der Waals-bonded graphene on Ir(111) induces a pronounced shift in the Ir(111) surface resonance. The disappearance of the resonance at graphene edges indicates the covalent carbon-metal interaction, which acts as a hard-wall potential for scattering of resonance electrons. Nanometer-sized graphene flakes can therefore confine quasi-two-dimensional electron gases to artificial quantum dots.
Funding by the Deutsche Forschungsgemeinschaft via SPP 1459 and SFB 668, and the Schleswig-Holstein-Fonds, as well as computer time at HLRN, are acknowledged.
Note added in proof: Results concerning confined electronic states in graphene islands on Ir(111) [@PhysRevLett.108.046801] have been published after submission of this manuscript. Ref. attributes the confined states to either graphene states or the scattered Ir(111) surface resonance, depending on the graphene island size. This is in contrast to our interpretation that the surface resonance predominates the STM data for any island size.
During the refereeing process an experimental observation of a graphene-induced shift of the Ir(111) surface resonance has been reported [@PhysRevLett.108.066804].
[^1]: The large width of the onset is consistent with the fact that the resonance is degenerate with bulk states. It may also be affected by residual adsorbates on Ir(111), which attenuate the signal of the surface resonance in their vicinity. Hence, spectra were obtained as far away from such defects as possible.
[^2]: The Ir surface was modeled using a slab of 18 layers of Ir atoms and a vacuum gap of $5.5{\mathrm {\,nm}}$. A $k$-mesh of $15\times 15\times 1$ points and a kinetic energy cut-off of $400{\mathrm {\,eV}}$ were used. To study graphene on Ir(111), the slab was covered with graphene on one side with one of the two C atoms of the graphene unit cell on top of an Ir atom and the graphene lattice constant adjusted to match the Ir lattice. Spin-orbit coupling was taken into account to correctly model splitting of surface states [@ARPESIr].
[^3]: The periodic potential associated with the moiré superlattice of graphene on Ir(111), which is not included in the calculations, may in principle scatter states from [[$\overline{\text{K}}$]{}]{} to the center of the Brillouin zone. However, owing to the large lattice constant of the superstructure, the associated momentum is small and multiple scattering steps are required, which makes this mechanism unlikely.
|
---
abstract: 'In this article, we present a discrete time modeling framework, in which the shape and dynamics of a Limit Order Book (LOB) arise endogenously from an equilibrium between multiple market participants (agents). We use the proposed modeling framework to analyze the effects of trading frequency on market liquidity in a very general setting. In particular, we demonstrate the dual effect of high trading frequency. On the one hand, the higher frequency increases market efficiency, if the agents choose to provide liquidity in equilibrium. On the other hand, it also makes markets more fragile, in the sense that the agents choose to provide liquidity in equilibrium only if they are market-neutral (i.e., their beliefs satisfy certain martingale property). Even a very small deviation from market-neutrality may cause the agents to stop providing liquidity, if the trading frequency is sufficiently high, which represents an endogenous liquidity crisis (aka flash crash) in the market. This framework enables us to provide more insight into how such a liquidity crisis unfolds, connecting it to the so-called adverse selection effect.'
author:
- |
Roman Gayduk and Sergey Nadtochiy[^1] [^2] [^3]\
$\,\,\,\,$\
*University of Michigan*
bibliography:
- 'MFGLOB\_refs.bib'
title: 'Liquidity Effects of Trading Frequency [^4]'
---
[**Key words**]{}: liquidity, trading frequency, Limit Order Book, continuum-player games, conditional tails of Itô processes.
Introduction
============
This paper is concerned with liquidity effects of trading frequency on an auction-style exchange, in which the participating agents can post limit or market orders. On the one hand, higher trading frequency provides more opportunities for the market participants to trade, hence, improving the liquidity of the market and increasing its efficiency. On the other hand, higher trading frequency also provides more opportunities for some participants to manipulate the price and disrupt the market liquidity. Such a manipulation creates a new type of risk, which reveals itself in unusually high price deviations, which cannot be explained by changes in the fundamental value of the asset. The most famous example of this phenomenon is the “flash crash” of $2010$. This example motivates the need for a comprehensive study of the tradeoff between the liquidity providing role of strategic players and the liquidity risk they generate, and its relation to trading frequency. The collective liquidity of the market is captured by the *Limit Order Book (LOB)*, which contains all the limit buy and sell orders.
The goal of the present paper is two-fold. First, we develop a new framework for modeling market microstructure, in which the shape of the LOB, and its dynamics, arise *endogenously* from the interactions between the agents. Among the many advantages of such an approach is the possibility of modeling the market reaction to changes in the rules of the exchange: e.g., limited trading frequency, transaction tax, etc. The second, and most important, goal of the present work is to investigate the *liquidity effects* of *trading frequency*, using the proposed modeling framework. In particular, the main results of this paper (cf. the discussion in Section \[se:examples\], as well as Theorems \[le:main.zeroTermSpread\], \[thm:main.necessary\] and Corollary \[prop:main.smallspread\], in Section \[se:main\]) describe the dual effect of high trading frequency. On the one hand, if the agents choose to provide liquidity in equilibrium, higher trading frequency decreases the bid-ask spread and makes the expected profits of all market participants converge to the same (fundamental) value, thus, improving the market efficiency. On the other hand, higher trading frequency also makes the LOB more sensitive to the deviations of the agents’ attitudes from market-neutrality. It is, of course, clear that a strong bullish or bearish signal induces market participants to trade at a higher or lower price. However, the novelty of our observation is in the role that the trading frequency plays in amplifying this effect. Namely, we show that, if the trading frequency is high, even if agents have plenty of inventory, a very small deviation from market-neutrality may cause them to stop providing liquidity, by either withdrawing from the market completely, or by posting limit orders far away from the fundamental price. Such actions cause disproportional deviations in the LOB, which cannot be explained by any fundamental reasons: they are much higher than the trading signal (i.e., the expected change in the fundamental price), and they occur without any shortage of supply or demand for the asset. We refer to such a deviation as an *endogenous* liquidity crisis, because it is due to the trading mechanism (i.e., the rules by which the market participants interact), rather than any fundamental reasons (note the similarity with the flash crash). Our framework provides insights into how such a liquidity crisis unfolds, connecting it to the so-called *adverse selection* effect. In particular, Section \[se:examples\] constructs an equilibrium in which an endogenous liquidity crisis does not occur because of an abnormally large market order, wiping out the liquidity on one side of the LOB, but because the optimal strategies of the agents require them to stop providing liquidity on one side of the LOB. On the mathematical side, our analysis uses the properties of conditional tails of the increments of a general It[ô]{} process. The main result in that regard, in Lemma \[le:necessary.marginal.maximum\], provides a uniform exponential bound on the conditional tails of the increments of a general It[ô]{} process. We believe that this result is useful in its own right, and, to the best of our knowledge, it is not available in the existing literature.
In recent years, we observed an explosion in the amount of literature devoted to the study of market microstructure. In addition to various empirical studies, a large part of the existing theoretical work focuses on the problem of optimal execution: see, among others, [@MMS1], [@MMS2], [@MMS6], [@MMS7], [@MMS11], [@MMS13], [@MMS15], [@MMS16], [@MMS20], [@MMS22], [@MMS23], [@MMS26], [@MMS25], and references therein. In these articles, the dynamics and shape of the LOB are modeled exogenously, or, equivalently, the arrival processes of the limit and market orders are specified exogenously. In particular, none of these articles attempts to explain the shape and dynamics of the LOB, arising directly from the interaction between the market participants. A different approach to the analysis of market microstructure has its roots in the economic literature. For example, [@MMS.g1], [@MMS.g2], [@MMS.g3], [@MMS.g4], [@MMS.g5], [@MMS.g6], [@DA.DuZhu], [@Bressan1], [@Bressan2], [@Bressan4] consider equilibrium models of market microstructure, and they are more closely related to the present work. However, the models proposed in the aforementioned papers do not aim to represent the mechanics of an auction-style exchange with sufficient precision, and, in particular, they are not well suited for analyzing the liquidity effects of trading frequency, which is the main focus of the present paper. A somewhat related strand of literature focuses on the endogenous formation of LOB in markets with a designated market maker: see e.g., [@MMS.gmm1], [@MMS.gmm2], [@MMS.gmm3], [@MMS.gmm4], [@MMS.gmm5]. In these papers, the LOB is not an outcome of a multi-agent equilibrium: instead, it is controlled by a single agent, the market maker. In the present paper, we model the entire LOB as an output of an equilibrium between a large number of agents, each of whom is allowed to both consume and provide liquidity (in particular, we have no designated market maker). Our setting is related to the literature on *double auctions* (cf. [@DA.Vayanos], [@DA.DuZhu]), with the crucial difference that the participants of each auction are allowed to choose two “asymmetric" types of strategies: market or limit orders. In addition, the present framework assumes that, ex ante, all agents have access to the same information, and, in this sense, it is similar to [@MMS.g1], [@MMS.g3], [@MMS.g6]. In particular, the *adverse selection* effect, herein, does not arise from any a priori information asymmetry between agents, instead, it is caused by the *mechanics* of the exchange. We formulate the problem as a *continuum-player game* – this abstraction allows us to obtain computationally tractable results (cf. [@Aumann], [@Schmeidler], [@GCarmona] for the concept of a continuum-player game, and [@MFG1], [@MFG2], [@MFG3], [@MFG4] for the subclass of mean field games).
The paper is organized as follows. Subsection \[se:setup\] describes the probabilistic setting, along with the execution rules of the exchange and the resulting state processes of the agents. Subsection \[se:equil.def\] defines the equilibrium and introduces the notion of *degeneracy* of the market (which represents an endogenous liquidity crisis). In Section \[se:examples\], we construct an equilibrium in a simple model, illustrating how an endogenous liquidity crisis unfolds, and how it is connected to the adverse selection effect. Theorems \[le:main.zeroTermSpread\], \[thm:main.necessary\], and Corollary \[prop:main.smallspread\], in Section \[se:main\], are the main results of the paper: they formalize and generalize the conclusions of Section \[se:examples\]. In Section \[se:tails\], we prove the key technical results on the (conditional) tails of marginal distributions of Itô processes. Sections \[se:pf.1\], \[se:pf.2\] contain the proofs of the main results. We conclude in Section \[se:conclusion\].
Modeling framework for a finite-frequency auction-style exchange {#se:setup}
================================================================
Mechanics of the exchange {#se:setup}
-------------------------
We consider an exchange in which trading can only occur at discrete times $n=0,1,\ldots,N$. We assume that the market participants are split into two groups: the *external investors*, who are “impatient", in the sense that they only submit market orders, and the *strategic players*, who can submit both market and limit orders, and who are willing to optimize their actions over a given (short) time horizon, in order to get a better execution price.[^5] In our study, we focus on the strategic players, who are referred to as *agents*, and we model the behavior of the external investors exogenously, via an *exogenous demand*. The interpretation of the external investors is clear: these are the investors who either have a longer-term view on the market, or who simply need to buy or sell the asset for reasons other than short-term profits. The strategic players (i.e., agents), on the contrary, are short-term traders, who attempt to maximize their objective at a shorter time horizon $N$. During every time period $[n,n+1)$, all the orders coming to the exchange are split into *limit* and *market* orders. The limit orders are collected in the so-called *Limit Order Book (LOB)*, and the market orders form the *demand curve*. At time $n+1$, the market orders in the demand curve are executed against the limit orders in the LOB. Then, this process is repeated in the next time interval. In particular, during a time period $[n,n+1)$ (for simplicity, we say “at time $n$"), an agent is allowed to submit a market order, post a limit buy or sell order, or wait (i.e., do nothing). If a limit order is not executed in a given time period, it costs nothing to cancel or re-position it for the next time period. Notice that our framework does not model the time-priority of limit orders. However, introducing a time-priority would not change the agents’ maximum objective value, as the “tick size" is assumed to be zero (i.e., the set of possible price levels is ${\mathbb R}$), and, hence, an agent can always achieve a priority by posting her order “infinitesimally" above or below a given competing order. Further details on modeling the formation of an LOB and the execution rules are presented below.
The demand curves are modeled exogenously by a random field $D=\left(D_n(p)\right)_{p\in{\mathbb R},n=1,\ldots,N}$ on a filtered probability space $\left(\Omega,\mathbb{F}=\left(\mathcal{F}_n\right)_{n=0}^N,{\mathbb P}\right)$, such that $\mathcal{F}_0$ is a trivial sigma-algebra, completed w.r.t. ${\mathbb P}$. The random variable $D^+_n(p) = \max(D_n(p),0)$ denotes the number of shares of the asset that the external investors and the agents submitting market orders are willing to purchase at or below the price $p$, accumulated over the time period $[n-1,n)$, and $D^-_n(p) = -\min(D_n(p),0)$ denotes the number of shares of the asset that the external investors and the agents submitting market orders are willing to sell at or above the price $p$, in the same time period. We assume that $D_n(\cdot)$ is a.s. nonincreasing and measurable w.r.t. $\mathcal{F}_n\otimes\mathcal{B}({\mathbb R})$. We denote by $\mathbb{A}$ a Borel space of *beliefs*, and, for each $\alpha\in\mathbb{A}$, there exists a *subjective probability measure* ${\mathbb P}^{\alpha}$ on $\left(\Omega,\mathcal{F}_N\right)$, which is absolutely continuous with resect to ${\mathbb P}$. We assume that, for any $n=0,\ldots,N$ and any $\alpha\in\mathbb{A}$, there exists a regular version of the conditional probability ${\mathbb P}^{\alpha}$ given $\mathcal{F}_n$, denoted ${\mathbb P}^{\alpha}_n$.[^6] We denote the associated conditional expectations by ${\mathbb E}^{\alpha}_n$. We also need to assume that, for any $\alpha\in\mathbb{A}$, there exists a modification of the family $\left\{{\mathbb P}^{\alpha}_n\right\}_{n=0}^N$, which satisfies the *tower property with respect to ${\mathbb P}$*, in the following sense: for any $n\leq m$ and any r.v. $\xi$, such that ${\mathbb E}^{\alpha} \xi^+ < \infty$, we have $${\mathbb E}^{\alpha}_n {\mathbb E}^{\alpha}_m \xi = {\mathbb E}^{\alpha}_n \xi,\,\,\,\,\,\,\,\,\,{\mathbb P}\text{-a.s.}$$ There exists such a modification, for example, if ${\mathbb P}^{\alpha}\sim{\mathbb P}$. In any market model, for every $\alpha$, we fix such a modification of conditional probabilities (up to a set of ${\mathbb P}$-measure zero) and assume that all conditional expectations $\left\{{\mathbb E}^{\alpha}_n\right\}$ are taken under this family of measures. The *Limit Order Book (LOB)* is given by a pair of adapted processes $\nu=(\nu^+_n,\nu^-_n)_{n=0}^{N}$, such that every $\nu^+_n$ and $\nu^-_n$ is a finite sigma-additive random measure on ${\mathbb R}$ (w.r.t. $\mathcal{F}_n\otimes\mathcal{B}({\mathbb R})$). Herein, $\nu^+_n$ corresponds to the cumulative limit sell orders, and $\nu^-_n$ corresponds to the cumulative limit buy orders, posted at time $n$.The bid and ask prices at any time $n=0,\ldots,N$ are given by the random variables $$p^b_n = \sup \text{supp}(\nu^-_n),
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,p^a_n = \inf \text{supp}(\nu^+_n),$$ respectively. Notice that these extended random variables are always well defined but may take infinite values.
We define the *state space* of an agent as $\mathbb{S}={\mathbb R}\times\mathbb{A}$, where the first component denotes the *inventory* of an agent, and the second component denotes her *beliefs*. Every agent in state $(s,\alpha)$ models the future outcomes using the subjective probability measure ${\mathbb P}^{\alpha}$. There are infinitely many agents, and their distribution over the state space is given by the *empirical distribution* process $\mu=(\mu_n)_{n=0}^{N}$, such that every $\mu$ is a finite sigma-additive random measure on $\mathbb{S}$ (w.r.t. $\mathcal{F}_n\otimes\mathcal{B}(\mathbb{S})$). In particular, the total mass of agents in the set $S\subset\mathbb{S}$ at time $n$ is given by $\mu_n(S)$. The inventory level $s$ represents the *number of shares per agent*, held in state $(s,\alpha)$. In particular, the total number of shares held by all agents in the set $S\subset\mathbb{S}$ is given by $\int_{S} s\mu_n(ds,d\alpha)$. The interpretation of this definition in a finite-player game is discussed in Remark \[rem:finplayer\] below. We refer the reader to [@GCarmona] for the general concept of a continuum-player game.
\[rem:finplayer\] The continuum-player game defined in this section can be related to a finite-player game as follows. Denote by $\mu_0$ the empirical distribution of the agents’ states at a given time. Recall that $\mu_0$ is a measure on $\mathbb{S}={\mathbb R}\times\mathbb{A}$, and assume that it is a finite linear combination of Dirac measures: $\mu_0 = \frac{1}{M} \sum_{i=1}^M \delta_{(s^{i},\alpha^{i})}$. In this case, we interpret $s^i$ as the [**number of shares per agent**]{} held by the agents in the $i$th group. Let us explain how this notion is related to the actual inventory levels (i.e., the actual numbers of shares held by the agents) in the associated finite-player game. To this end, consider a collection of $M$ agents, whose states are given by their (actual) inventories and beliefs, $(s,\alpha)$, with the current states being $\{(\tilde{s}^i = s^i/M,\alpha^i)\}$. Define the “unit mass" of agents to be $M$. In this finite-player collection, the mass of agents (measured relative to the unit mass, $M$) at any state $(Ms,\alpha)$ is precisely $\mu_0(\{(s,\alpha)\})$, and their total inventory is $Ms\mu_0(\{(s,\alpha)\})$. The number of shares per agent is, then, defined as the total inventory held by these agents divided by their mass, and it is equal to $Ms$. Choosing $s=\tilde{s}^i$, we conclude that, in the finite-player collection, the number of shares per agent held by the agents at state $(\tilde{s}^i,\alpha^i)$ is given by $M\tilde{s}^i = s^i$, which coincides with our interpretation of $s^i$ in the continuum-player game. It is also easy to show that an equilibrium in the proposed continuum-player game (defined in the next subsection) produces an approximate equilibrium in the associated finite-player game, when the inventory levels $\{\tilde{s}^i\}$ are small (cf. Subsection 2.3 in the extended version of this paper, [@GaydukNadtochiy1])
As the parameter $\alpha$ does not change over time, the state process of an agent, denoted $(S_n)$, is an adapted ${\mathbb R}$-valued process, representing her inventory.[^7] The control of every agent is given by a triplet of adapted processes $(p,q,r) = (p_n,q_n,r_n)_{n=0}^{N-1}$ on $\left(\Omega,\mathbb{F}\right)$, with values in ${\mathbb R}^2\times\left\{0,1\right\}$. The first coordinate, $p_n$, indicates the location of a limit order placed at time $n$, and $q_n$ indicates the size of the order (measured in shares per agent, and with negative values corresponding to buy orders).[^8] The last coordinate $r_n$ shows whether the agent submits a market order (if $r_n = 1$) or a limit order (if $r_n=0$). Assume that an agent posts a limit sell order at a price level $p_n$. If the demand to buy the asset at this price level, $D^+_{n+1}(p_n)$, exceeds the amount of all limit sell orders posted below $p_n$ at time $n$, then (and only then) the limit sell order of the agent is executed. Market orders of the agents are always executed at the bid or ask prices available at the time when the order is submitted. We interpret an internal market order (i.e., the one submitted by an agent) as the decision of an agent to join the external investors, in the given time period. Summing up the above, we obtain the following dynamics for the state process of an agent, starting with initial inventory $s\in{\mathbb R}$ at time $m=0,\ldots,N-1$: $$S^{(p,q,r)}_m(m,s,\nu) = s,\,\,\,\,\,\, \Delta S^{(p,q,r)}_{n+1}(m,s,\nu) = S^{(p,q,r)}_{n+1}(m,s,\nu) - S^{(p,q,r)}_n(m,s,\nu)
= -q_n \bone_{\left\{ r_n=1 \right\}}$$ $$\label{eq.stateProc.def}
- \bone_{\left\{ r_n=0 \right\}}
\left( q^+_n \bone_{\left\{D^+_{n+1}(p_n) > \nu^+_n((-\infty,p_n))\right\}}
- q^-_n \bone_{\left\{D^-_{n+1}(p_n) > \nu^-_n((p_n,\infty))\right\}}\right),
\,\,\,\,n=m,\ldots,N-1.$$ The above dynamics represent an optimistic view on the execution by the agents. In particular, they imply that all limit orders at the same price level are executed in full, once the demand reaches them: i.e., each agent believes that her limit order will be executed first among all orders at a given price level. In addition, all agents’ market orders are executed at the bid and ask prices: i.e., each agent believes that her market order will be executed first, when the demand curve is cleared against the LOB, at the end of a given time period. These assumptions can be partially justified by the fact that the agents’ orders are infinitesimal: $q_n$ is measured in shares per agent, and an individual agent has zero mass. However, if a non-zero mass of agents submit limit orders at the same price level, or execute market orders, at the same time, then, the above state dynamics may violate the market clearing condition: the total size of executed market orders (both in shares and in dollars) may not coincide with the total size of executed limit orders (at least, as viewed by the agents). Nevertheless, this issue is resolved if, at any time, the mass of agents posting limit orders at the same price level or posting market orders is zero. In other words, $(\nu,p,q,r)$ satisfy, ${\mathbb P}$-a.s.: $\nu_n$ is continuous, as a measure on ${\mathbb R}$ (i.e., it has no atoms), and $r_n=0$. Such an equilibrium is constructed in Section 8 of the extended version of this paper, [@GaydukNadtochiy1]. The general definition of a continuum-player game and its connection to a finite-player game can be found, e.g., in [@GCarmona] and in the references therein (see also Subsection 2.3 in the extended version of this paper, [@GaydukNadtochiy1]).
The modeling framework proposed herein has a close connection to the models of *double auctions*, in the economic literature (cf. [@DA.DuZhu], [@DA.Vayanos]). The main difference is in the non-standard design of the auction. Namely, in the proposed setting, the auction participants may choose different styles of trading, i.e., market or limit orders, which generates an ex-post information asymmetry between participants: the limit orders have to be submitted before the demand curve is observed, while the market orders are submitted using complete information about the LOB. This difference is not coincidental – it is, in fact, crucial for a realistic representation of the risks associated with each order type, and it is at the core of the results established herein. A more detailed discussion of the information structure is provided in the next subsection.
Equilibrium {#se:equil.def}
-----------
The objective function of an agent, starting at the initial state $(s,\alpha)\in\mathbb{S}$, at any time $m=0,\ldots,N$, and using the control $(p,q,r)$, is given by the $\mathcal{F}_m$-measurable random variable $$\label{eq.intro.Jm.def}
J^{(p,q,r)}(m,s,\alpha,\nu) =
{\mathbb E}^{\alpha}_m \left[ \left(S^{(p,q,r)}_N(m,s,\nu)\right)^+ p^b_N - \left(S^{(p,q,r)}_N(m,s,\nu)\right)^- p^a_N
\right.$$ $$\left.
- \sum_{n=m}^{N-1} \left(p_n\bone_{\left\{ r_n = 0\right\}} + p^a_n\bone_{\left\{ r_n = 1, q_n <0\right\}} + p^b_n\bone_{\left\{ r_n = 1, q_n >0\right\}} \right) \Delta S^{(p,q,r)}_{n+1}(m,s,\nu) \right],$$ where we assume that $0\cdot\infty = 0$. In the above expression, we assume that, at the final time $n=N$, each agent is forced to liquidate her position at the bid or ask prices available at that time. Alternatively, one can think of it as *marking to market* the residual inventory, right after the last external market order is executed.
\[def:admis\] For a given LOB $\nu$, integer $m=0,\ldots,N-1$, and state $(s,\alpha)\in\mathbb{S}$, the triplet of adapted processes $(p,q,r)$ is an [**admissible control**]{} if the positive part of the expression inside the expectation in (\[eq.intro.Jm.def\]) has a finite expectation under ${\mathbb P}^{\alpha}$.
For a given LOB $\nu$, an initial condition $(m,s,\alpha)$, and a triplet of $\mathbb{F}\times\mathcal{B}(\mathbb{S})$-adapted random fields $(p,q,r)$, we identify the latter (whenever it causes no confusion) with stochastic processes $(p,q,r)$ via: $$p_n=p_n\left(S^{(p,q,r)}_n(m,s,\nu),\alpha\right),\,\,\,
q_n=q_n\left(S^{(p,q,r)}_n(m,s,\nu),\alpha\right),\,\,\,
r_n=r_n\left(S^{(p,q,r)}_n(m,s,\nu),\alpha\right),$$ and the state dynamics (\[eq.stateProc.def\]), for $n=m,\ldots,N$. This system determines $(p,q,r)$ and $S^{(p,q,r)}$ recursively.
\[def:optControl\] For a given LOB $\nu$, we call the triplet of progressively measurable random fields $(p,q,r)$ an [**optimal control**]{} if, for any $m=0,\ldots,N$ and any $(s,\alpha)\in\mathbb{S}$, we have:
- $(p,q,r)$ is admissible,
- $J^{(p,q,r)}(m,s,\alpha,\nu) \geq J^{(p',q',r')}(m,s,\alpha,\nu)$, ${\mathbb P}$-a.s., for any admissible control $(p',q',r')$.
In the above, we make the standard simplifying assumptions of continuum-player games: each agent is too small to affect the empirical distribution of cumulative controls (reflected in $\nu$) when she changes her control (cf. [@GCarmona]). Note also that our definition of the optimal control implies that it is time consistent: re-evaluation of the optimality at any future step, using the same terminal criteria, must lead to the same optimal strategy. Next, we discuss the notion of equilibrium in the proposed game. First, we notice that, if $p^b_N$ or $p^a_N$ becomes infinite, the agents with positive or negative inventory may face the objective value of “$-\infty$", for any control they use. In such a case, their optimal controls may be chosen in an arbitrary way, resulting in unrealistic equilibria. To avoid this, we impose the additional regularity condition on $\nu$.
\[def:admis.LOB\] A given LOB $\nu$ is admissible if, for any $m=0,\ldots,N-1$ and any $\alpha\in\mathbb{A}$, we have, ${\mathbb P}$-a.s.: $${\mathbb E}^{\alpha}_m |p^a_N|\vee|p^b_N| < \infty.$$
Let us consider the (stochastic) value function of an agent for a fixed $(m,s,\alpha,\nu)$: $$\label{eq.gen.Val.randField}
V^{\nu}_m(s,\alpha) = \text{esssup}_{p,q,r} J^{(p,q,r)}\left(m,s,\alpha,\nu\right),$$ where the essential supremum is taken under ${\mathbb P}$, over all admissible controls $(p,q,r)$, and $J^{(p,q,r)}$ is given by (\[eq.intro.Jm.def\]). Appendix A shows that, for any admissible $\nu$, $V^{\nu}_m(\cdot,\alpha)$ has a continuous modification under ${\mathbb P}$, which we refer to as the value function of an agent with beliefs $\alpha$. Using the Dynamic Programming Principle, Appendix A provides an explicit system of recursive equations that characterize optimal strategies and the value function. In particular, the results of Appendix A (cf. Corollary \[cor:piecewiseLin\]) yield the following proposition.
\[cor:piecewiseLin.new\] Assume that, for an admissible LOB $\nu$, there exists an optimal control $(\hat{p},\hat{q},\hat{r})$. Then, for any $(s,\alpha)\in\mathbb{S}$, the following holds ${\mathbb P}$-a.s., for all $n=0,\ldots,N-1$: $$V^{\nu}_n(s,\alpha) = s^+ \lambda^a_n(\alpha) - s^- \lambda^b_n(\alpha),$$ with some adapted processes $\lambda^a(\alpha)$ and $\lambda^b(\alpha)$, such that $\lambda^a_N(\alpha) = p^b_N$ and $\lambda^b_N(\alpha) = p^a_N$.
The values of $\lambda^a(\alpha)$ and $\lambda^b(\alpha)$ can be interpreted as the *expected execution prices* of the agents with beliefs $\alpha$, who are long and short the asset, respectively.
\[def:equil.def\] Consider an empirical distribution process $\mu=(\mu_n)_{n=0}^N$ and a market model, as described in Subsection \[se:setup\]. We say that a given LOB process $\nu$ and a control $(p,q,r)$ form an [**equilibrium**]{}, if there exists a Borel set $\tilde{\mathbb{A}}\subset \mathbb{A}$, called the [**support**]{} of the equilibrium, such that:
1. $\mu_n\left(\mathbb{{\mathbb R}}\times\left(\mathbb{A}\setminus \tilde{\mathbb{A}}\right)\right)=0$, ${\mathbb P}$-a.s., for all $n$,
2. $\nu$ is admissible, and $(p,q,r)$ is an optimal control for $\nu$, on the state space $\tilde{\mathbb{S}}={\mathbb R}\times\tilde{\mathbb{A}}$,
3. and, for any $n=0,\ldots,N-1$, we have, ${\mathbb P}$-a.s., $$\label{eq.nuplus.fixedpoint.def}
\nu^+_n((-\infty,x]) = \int_{\tilde{\mathbb{S}}} \bone_{\left\{p_n(s,\alpha)\leq x, r_n(s,\alpha)=0\right\}}\, q^+_n(s,\alpha) \mu_n(ds,d\alpha),
\,\,\,\,\,\,\forall\, x\in{\mathbb R},$$ $$\label{eq.numinus.fixedpoint.def}
\nu^-_n((-\infty,x]) = \int_{\tilde{\mathbb{S}}} \bone_{\left\{p_n(s,\alpha)\leq x, r_n(s,\alpha)=0\right\}}\, q^-_n(s,\alpha) \mu_n(ds,d\alpha),
\,\,\,\,\,\,\forall\, x\in{\mathbb R}.$$
It follows from Proposition \[cor:piecewiseLin.new\] that, in equilibrium, it is optimal for an agent with zero initial inventory to do nothing. Hence, in equilibrium, roundtrip strategies are impossible. To allow for roundtrip strategies in equilibrium, one can, e.g., introduce an upper bound on $|q|$ or on the total inventory of an agent (as it is done, e.g., in [@MMS.gliq1]). However, we do not believe that such a modification would change the qualitative behavior of market liquidity as a function of trading frequency, which is the main focus of the present paper.
Notice that, because the optimal controls are required to be time consistent under ${\mathbb P}$, the above definition, in fact, defines a *sub-game perfect equilibrium*. It is also worth mentioning that Definition \[def:equil.def\] defines a *partial equilibrium*, as the empirical distribution process $\mu$ is given exogenously. A more traditional version of Nash equilibrium would require $\mu$ to be determined by the initial distribution and the values of the state processes: $$\label{eq.endog.mu}
\mu_n = \mu_0 \circ \left( (s,\alpha)\mapsto \left(S_n^{(p,q,r)}(0,s,\nu),\alpha\right) \right)^{-1},$$ which must hold ${\mathbb P}$-a.s., for all $n=0,\ldots,N$, with $S_n^{(p,q,r)}(0,s,\nu)$ defined via (\[eq.stateProc.def\]), in addition to the other conditions in Definition \[def:equil.def\]. Nevertheless, we choose not to enforce the condition (\[eq.endog.mu\]) in the definition of equilibrium, in order to allow new agents to enter the game, which, in effect, amounts to modeling $\mu$ exogenously. If one assumes that no new agent arrives to the market, then, the fixed-point condition (\[eq.endog.mu\]) has to be enforced. Note also that our interpretation of the demand curve $D_n(\cdot)$ implies that it consists of both the external (i.e., due external investors) and internal (i.e., due to the agents) market orders. Therefore, it may be reasonable to consider an additional consistency condition for an equilibrium. A part of this condition is to ensure that a non-zero mass of agents submit market buy orders only if the fundamental price rises above the ask price (i.e., only if a market buy order is actually executed), and, similarly, a non-zero mass of agents submit market sell orders only if the fundamental price falls below the bid price. We assume that the agents’ market orders enter into the demand curve with the highest level of priority: e.g., their market buy orders enter the demand curve at the price level infinitesimally close to, but below, the fundamental price, in order to guarantee that they are the first ones to be executed. Thus, another part of the aforementioned consistency condition is to ensure that the absolute value of the demand curve to the left or to the right of the fundamental price is sufficiently large to account for all internal market orders. Mathematically, such consistency condition can be formulated as follows: $$\label{eq.endog.D.1}
d^b_n:=\mu_n\left( \left\{(s,\alpha)\,:\,q_n(s,\alpha)<0,\,r_n(s,\alpha)=1\right\}\right)>0\,\,
\Rightarrow\,\,p^0_{n+1}>p^a_{n},\,\,\lim_{p\uparrow p^0_{n+1}}D^+_{n+1}(p)\geq d^b_n,$$ $$\label{eq.endog.D.2}
d^a_n:=\mu_n\left( \left\{(s,\alpha)\,:\,q_n(s,\alpha)>0,\,r_n(s,\alpha)=1\right\}\right)>0\,\,
\Rightarrow\,\,p^0_{n+1}<p^b_{n},\,\,\lim_{p\downarrow p^0_{n+1}}D^-_{n+1}(p)\geq d^a_n.$$ The above conditions become redundant if the agents never submit market orders in equilibrium. Section 8 of the extended version of this paper, [@GaydukNadtochiy1], shows how to construct an equilibrium which satisfies condition (\[eq.endog.mu\]), and in which the agents never submit market orders (hence, (\[eq.endog.D.1\]) and (\[eq.endog.D.2\]) are also satisfied). However, it is important to emphasize that the main results of the present work (cf. Section \[se:main\]) provide necessary conditions for [**all**]{} equilibria: for those satisfying the conditions (\[eq.endog.mu\]), (\[eq.endog.D.1\]), (\[eq.endog.D.2\]) and for the ones that do not.
Let us comment on the information structure of the game. In the present setting, all agents observe the same information, given by the filtration ${\mathbb F}$. We consider an open-loop Nash equilibrium, in which the agent’s strategy is viewed as an adapted stochastic process (rather than a function of the states and controls of other players), and the definition of optimality is chosen accordingly. In addition, as $\mu$ is adapted to ${\mathbb F}$, each agent has complete information about the present and past states of other agents, and their beliefs. However, as the agents use different (subjective) measures $\{{\mathbb P}^{\alpha}\}$, their views on the future values of $\mu$ may be different. Of course, it would be more realistic to assume that the agents do not have complete information about each other’s current states, but this would make the problem significantly more complicated. In the present setting, the agents also have complete information about the current location of the fundamental price. In our follow-up paper, [@GaydukNadtochiy2], we relax this assumption, which allows us to develop a more realistic model for the “local" behavior of an individual agent. However, such a relaxation does not seem necessary for the questions analyzed herein.
As all agents use the same information, the present article belongs to the strand of literature that attempts to explain microstructure phenomena without information asymmetry (cf. [@MMS.g3], [@MMS.g6], [@MMS.g1], [@MMS.g2]). Nevertheless, it is important to mention that information asymmetry arises ex-post, between the market participants submitting market and limit orders. This asymmetry is not due to superior information a priori available to any of the agents. Instead, it stems from the very nature of limit orders, which are “passive" by design (cf. the discussion on the last paragraph of Subsection \[se:setup\]). Similar observation is made in [@MMS.g3].
Next, we need to add another condition to the notion of equilibrium. Notice that equations (\[eq.nuplus.fixedpoint.def\])–(\[eq.numinus.fixedpoint.def\]) should serve as the fixed-point constraints that enable one to obtain the optimal controls $(p,q,r)$, along with the LOB $\nu$. However, these equations only hold for $n=0,\ldots,N-1$: indeed, the agents do not need to choose their controls at time $n=N$, as the game is over and their residual inventory is marked to the bid and ask prices. However, the terminal bid and ask prices are determined by the LOB $\nu_N$, which, in turn, can be chosen arbitrarily. To avoid such ambiguity, we impose an additional constraint on the equilibria studied herein. First, we introduce the notion of a *fundamental price*.
\[def:het.p0\] Assume that ${\mathbb P}$-a.s., for any $n=1,\ldots,N$, there exists a unique $p^0_n$ satisfying $D_n\left(p^0_n\right)=0$. Then, the adapted process $(p^0_n)_{n=1}^N$ is called the [**fundamental price process**]{}.
Whenever the notion of a fundamental price is invoked, we assume that it is well defined. The intuition behind $p^0$ is clear: it is a price level at which the immediate demand is balanced. However, it is important to stress that we do not assume that the asset can be traded at the fundamental price level. Rather, $p^0$ is a feature of the abstract current demand curve, whereas all actual trading happens on the exchange, against the current LOB. This aspect of our setting differs from many other approaches in the literature.
\[def:het.\] Assume that the fundamental price is well defined and denote $\xi_N = p^0_N - p^0_{N-1}$. Then, an equilibrium with LOB $\nu$ is [**linear at terminal crossing (LTC)**]{} if $$\label{eq.LTC.def}
\nu_N = \nu_{N-1}\circ (x\mapsto x+\xi_N)^{-1},\,\,\,\,\,\,\,\,{\mathbb P}\text{-a.s.}$$
The above definition assumes that the terminal LOB $\nu_N$ is obtained from $\nu_{N-1}$ by a simple shift, with the size of the shift equal to the increment in the fundamental price. This definition connects the LOB at the terminal time with the demand process, ruling out many unnatural equilibria. In particular, the question of existence of an equilibrium becomes non-trivial. However, the mere existence of an equilibrium is not the main focus of the present work: the existence results, established herein, are limited to Section \[se:examples\], which constructs an LTC equilibrium in a specific Gaussian random walk model (a slightly more general existence result is given in Section 8 of the extended version of this paper, [@GaydukNadtochiy1]). What is central to the present investigation is the observation that the agents may reach an equilibrium in which one side of the LOB becomes empty (as demonstrated by the example of Section \[se:examples\]). We call such LOB, and the associated equilibrium, *degenerate*.
We say that an equilibrium with LOB $\nu$ is [**non-degenerate**]{} if $\nu^{+}_n({\mathbb R})>0$ and $\nu^{-}_n({\mathbb R})>0$, for all $n=0,\ldots,N-1$, ${\mathbb P}$-a.s..
Intuitively, the degeneracy of the LOB refers to a situation where, with positive probability, one side of the LOB disappears from the market: i.e., $\nu^+_n({\mathbb R})$ or $\nu^-_n({\mathbb R})$ becomes zero. Clearly, this happens when the agents who are supposed to provide liquidity choose to post market orders (i.e. consume liquidity) or wait (neither provide nor consume liquidity). Such a degeneracy can be interpreted as the *endogenous liquidity crisis* – the one that arises purely from the interaction between the agents, and cannot be justified by any fundamental economic reasons (e.g., the external demand for the asset may still be high, on both sides). Taking an optimistic point of view, we assume that the agents choose a non-degenerate equilibrium, whenever one is available. However, if a non-degenerate equilibrium does not exist, an endogenous liquidity crisis may occur with positive probability. One of the main goals of this paper is to provide insights into the occurrence of an endogenous liquidity crisis and its relation to trading frequency.
Example: a Gaussian random walk model {#se:examples}
=====================================
In this section, we consider a specific market model for the external demand $D$, to construct a non-degenerate LTC equilibrium. More importantly, using this model, we illustrate the liquidity effects of trading frequency. The present example, albeit very simplistic, enables us to identify the important changes in the optimal strategies of the agents (and, hence, to the LOB) as the trading frequency increases. In particular, we demonstrate how the *adverse selection* effect may be amplified disproportionally by the high trading frequency and may cause a liquidity crisis. Note that the adverse selection phenomenon, in the present setting, is not a consequence of any ex-ante information asymmetry but is due to the mechanics of the exchange (i.e., the nature of limit orders), which is similar to the phenomena documented in [@MMS.g3], [@MMS.g2]. In the rest of the paper, we show that the conclusions of this section are not due to the particular choice of a model made in the present section and, in fact, persist in a much more general setting.
On a complete stochastic basis $(\Omega,\tilde{{\mathbb F}}=(\tilde{\mathcal{F}}_t)_{t\in[0,T]},{\mathbb P})$, we consider a continuous time process $\tilde{p}_0$: $$\label{eq.p0.BM}
\tilde{p}^0_t = p^0_0 + \alpha t + \sigma W_t,\quad p^0_0 \in {\mathbb R}, \quad t\in[0,T],$$ where $\alpha\in{\mathbb R}$ and $\sigma>0$ are constants, and $W$ is a Brownian motion. We also consider an arbitrary progressively measurable random field $(\tilde{D}_t(p))$, s.t., ${\mathbb P}$-a.s., the function $\tilde{D}_t(\cdot)-\tilde{D}_s(\cdot)$ is strictly decreasing and vanishing at zero, for any $0\leq s < t\leq T$. Finally, we introduce the empirical distribution process $(\tilde{\mu}_t)$, with values in the space of finite sigma-additive measures on $\mathbb{S}$. We partition the time interval $[0,T]$ into $N$ subintervals of size $\Delta t=T/N$. A discrete time model is obtained by discretizing the continuous time one[^9] $$\mathcal{F}_n = \tilde{\mathcal{F}}_{n\Delta t},\quad p^0_n = \tilde{p}^0_{n\Delta t},
\quad D_n(p) = (\tilde{D}_{n\Delta t}-\tilde{D}_{(n-1)\Delta t})(p-p^0_n),\quad \mu_n = \tilde{\mu}_{n\Delta t}.$$ In this section, for simplicity, we assume that the set of agents’ beliefs is a singleton: $\mathbb{A}=\left\{\alpha\right\}$ and ${\mathbb P}^{\alpha}={\mathbb P}$. We also assume that (at least, from the agents’ point of view) there are always some long and short agents present in the market: $\mu_n\left((0,\infty)\times\mathbb{A}\right),\mu_n\left((-\infty,0)\times\mathbb{A}\right)>0$, ${\mathbb P}$-a.s., for all $n$. Clearly, $N$ represents the trading frequency, and the continuous time model represents the “limiting model," which the agents use as a benchmark, in order to make consistent predictions in the markets with different trading frequencies. We assume that the benchmark model is fixed, and $N$ is allowed to vary. In the remainder of this section, we propose a method for constructing a non-degenerate LTC equilibrium in the above discrete time model. We show that the method succeeds for any $(N,\sigma)$ if $\alpha=0$. However, for $\alpha\neq 0$, we demonstrate numerically that the method fails as $N$ becomes large enough. We show why, precisely, the proposed construction fails, providing an economic interpretation of this phenomenon. Moreover, we analyze the market close to the moment when a non-degenerate equilibrium fails to exist and demonstrate that the agents’ behavior at this time follows the pattern typical for an endogenous liquidity crisis.
In view of Proposition \[cor:piecewiseLin.new\], in order to construct a non-degenerate LTC equilibrium, we need to find a control $(\hat{p},\hat{q},\hat{r})$, and the expected execution prices $(\hat{\lambda}^a,\hat{\lambda}^b)$, s.t. the value function of an agent with inventory $s$ is given by $V_n(s) = s^+\hat{\lambda}^a_n - s^-\hat{\lambda}^b_n$, and it is attained by the strategy $(\hat{p},\hat{q},\hat{r})$. In addition, we need to find a non-degenerate LOB $\nu$, s.t. (\[eq.nuplus.fixedpoint.def\]), (\[eq.numinus.fixedpoint.def\]) and (\[eq.LTC.def\]) hold. Our ansatz is as follows $$\nu_n = \left(h^a_n \delta_{p^a_n}, h^b_n \delta_{p^b_n}\right),
\quad p^a_n = \hat{p}^a_n + p^0_n,
\quad p^b_n = \hat{p}^b_n + p^0_n,
\quad -\infty < \hat{p}^b_n,\, \hat{p}^a_n<\infty,$$ $$\hat{p}_n(s) = p^a_n\bone_{\{s>0\}} + p^b_n\bone_{\{s<0\}},
\quad \hat{q}_n(s) = s,
\quad \hat{r}_n(s) = 0,
\quad \lambda^a_n = \hat{\lambda}^a_n + p^0_n,
\quad \lambda^b_n = \hat{\lambda}^b_n + p^0_n,$$ where $\delta$ is the Dirac measure, $(\hat{p}^a,\hat{p}^b,\hat{\lambda}^a,\hat{\lambda}^b)$ are deterministic processes, and $h^a_n=\int_0^{\infty}s\mu_n(ds)>0$, $h^b_n=\int_{-\infty}^0|s|\mu_n(ds)>0$. With such an ansatz, the conditions (\[eq.nuplus.fixedpoint.def\]), (\[eq.numinus.fixedpoint.def\]) are satisfied automatically. Thus, we only need to choose finite deterministic processes $(\hat{p}^a,\hat{p}^b,\hat{\lambda}^a,\hat{\lambda}^b)$ s.t.: $\hat{p}^a_N = \hat{p}^a_{N-1}$, $\hat{p}^b_N = \hat{p}^b_{N-1}$ (so that the equilibrium is LTC) and the associated $(\hat{p},\hat{q},0)$ form an optimal control, producing the value function $V_n(s) = s^+\lambda^a_n - s^-\lambda^b_n$. Appendix A contains necessary and sufficient conditions for characterizing such families $(p^a,p^b,\lambda^a,\lambda^b)$. In particular, we deduce from Corollaries \[cor:piecewiseLin\] and \[cor:piecewiseLin.verif\] that $(\hat{p}^a_{N-1},\hat{p}^b_{N-1},\hat{\lambda}^a_{N-1},\hat{\lambda}^b_{N-1})$ form a suitable family in a single-period case, $[N-1,N]$, if they solve the following system: $$\label{eq.ex.singleStep.1}
\left\{
\begin{array}{l}
{\hat{p}^a_{N-1} \in \text{arg}\max_{p\in{\mathbb R}} {\mathbb E}\left((p - \hat{p}^b_{N-1} - \xi) \bone_{\left\{ \xi > p \right\}}\right),
\quad \hat{p}^b_{N-1}<0,\phantom{\frac{\frac{1}{2}}{2}}}\\
{\hat{p}^b_{N-1} \in \text{arg}\max_{p\in{\mathbb R}} {\mathbb E}\left((\hat{p}^a_{N-1} - p + \xi) \bone_{\left\{ \xi < p \right\}}\right),
\quad \hat{p}^a_{N-1}>0,\phantom{\frac{\frac{1}{2}}{2}}}\\
{\hat{\lambda}^a_{N-1} = \hat{p}^b_{N-1} + \alpha\Delta t + {\mathbb E}\left((\hat{p}^a_{N-1} - \hat{p}^b_{N-1} - \xi) \bone_{\left\{ \xi > \hat{p}^a_{N-1} \right\}}\right), \phantom{\frac{\frac{\frac{1}{2}}{2}}{2}}}\\
{\hat{\lambda}^b_{N-1} = \hat{p}^a_{N-1} + \alpha\Delta t - {\mathbb E}\left((\hat{p}^a_{N-1} - \hat{p}^b_{N-1} + \xi) \bone_{\left\{ \xi < \hat{p}^b_{N-1} \right\}}\right), \phantom{\frac{\frac{\frac{1}{2}}{2}}{2}}}\\
{\hat{p}^b_{N-1}\leq \hat{\lambda}^a_{N-1},
\quad \hat{\lambda}^b_{N-1} \leq \hat{p}^a_{N-1},
\quad \hat{p}^a_{N-1} \geq \hat{p}^b_{N-1} + |\alpha|\Delta t, \phantom{\frac{\frac{1}{2}}{2}}}
\end{array}
\right.$$ where $\xi=\Delta p^0_N\sim \mathcal{N}(\alpha\Delta t, \sigma^2 \Delta t)$. Let us comment on the economic meaning of the equations in (\[eq.ex.singleStep.1\]). The expectations in the first two lines represent the *relative expected profit* from executing a limit order at time $N$, at the chosen price level $p+p^0_{N-1}$, versus marking the inventory to market at time $N$, at the best price available on the other side of the book: i.e., $p^b_N = \hat{p}^b_{N-1} + \xi + p^0_{N-1}$ or $p^a_N = \hat{p}^a_{N-1} + \xi + p^0_{N-1}$. Notice that a limit order is executed if and only if the fundamental price at time $N$ is above or below the chosen limit order: i.e., if $p^0_{N-1} + \xi > p + p^0_{N-1}$ or $p^0_{N-1} + \xi < p + p^0_{N-1}$.[^10] Clearly, it is only optimal for an agent to post a limit order if the relative expected profit is nonnegative, which is the case if and only if $\hat{p}^b_{N-1}<0<\hat{p}^a_{N-1}$. The third and fourth lines in (\[eq.ex.singleStep.1\]) represent the expected execution prices of the agents at time $N-1$, assuming they use the controls given by $(\hat{p}^a_{N-1},\hat{p}^b_{N-1})$. Each of the right hand sides is a sum of two components: the relative expected profit from posting a limit order and the expected value of marking to market at time $N$, measured relative to $p^0_{N-1}$. Let us analyze the inequalities in the last line of (\[eq.ex.singleStep.1\]). If the bid price at time $N-1$ exceeds the expected execution price of a long agent, i.e., $\hat{p}^b_{N-1} + p^0_{N-1}> \hat{\lambda}^a_{N-1}+ p^0_{N-1}$, then every agent with positive inventory prefers to submit a market order, rather than a limit order, at time $N-1$, which causes the ask side of the LOB to degenerate. Similarly, we establish $\hat{\lambda}^b_{N-1} \leq \hat{p}^a_{N-1}$. Finally, if $\alpha>0$ and $\hat{p}^a_{N-1} < \hat{p}^b_{N-1} + \alpha\Delta t$, an agent may buy the asset using a market order at time $N-1$, at the price $\hat{p}^a_{N-1}+p^0_{N-1}$, and sell it at time $N$, at the expected price $\hat{p}^b_{N-1} + p^0_{N-1} + \alpha \Delta t > \hat{p}^a_{N-1}+p^0_{N-1}$ (a reverse strategy works for $\alpha<0$). This strategy can be scaled to generate infinite expected profit and, hence, is excluded by the last inequality in the last line of (\[eq.ex.singleStep.1\]).
We construct a solution to (\[eq.ex.singleStep.1\]) by solving a fixed-point problem given by the first two lines of (\[eq.ex.singleStep.1\]) and verifying that the desired inequalities hold.[^11] We implement this computation in MatLab, and the results can be seen as the right-most points on the graphs in Figure \[fig:2\]. From the numerical solution, we see that, whenever $\Delta t$ is small enough, the conditions $\hat{p}^b_{N-1}\leq \hat{\lambda}^a_{N-1}$ and $\hat{\lambda}^b_{N-1} \leq \hat{p}^a_{N-1}$ are satisfied (cf. the right part of Figure \[fig:2\]).[^12] In addition, for $\alpha\geq0$, we have $$0 < {\mathbb E}\left(\hat{p}^a_{N-1} - \hat{p}^b_{N-1} - \xi\,\vert\,\xi > \hat{p}^a_{N-1} \right)
= \hat{p}^a_{N-1} - \hat{p}^b_{N-1} - {\mathbb E}\left(\xi\,\vert\,\xi > \hat{p}^a_{N-1} \right)
\leq \hat{p}^a_{N-1} - \hat{p}^b_{N-1} - \alpha\Delta t,$$ which yields the last inequality in (\[eq.ex.singleStep.1\]). The case of $\alpha<0$ is treated similarly. Notice that $\hat{\lambda}^a_N = \hat{p}^b_N = \hat{p}^b_{N-1}$ and $\hat{p}^a_{N-1} = \hat{p}^a_N = \hat{\lambda}^b_N$. Thus, the single-period equilibrium we have constructed satisfies: $$\label{eq.example.1period.ineq.1}
\hat{p}^b_{n}\leq \hat{\lambda}^a_{n},\quad \hat{\lambda}^b_{n} \leq \hat{p}^a_{n},
\quad \hat{\lambda}^a_{n+1} < 0,\quad \hat{\lambda}^b_{n+1}>0,$$ for $n=N-1$. If one of the first two inequalities in (\[eq.example.1period.ineq.1\]) fails, the agents choose to submit market orders, as opposed to limit orders, which leads to *degeneracy* of the LOB – one side of it disappears. If one of the last two inequalities fails, the execution of a limit order, at any price level, yields a negative relative expected profit for the agents on one side of the book (given by the expectation in the first or second line of (\[eq.ex.singleStep.1\])). As a result, it becomes optimal for all such agents to stop posting any limit orders, and the LOB degenerates. The latter is interpreted as the *adverse selection* effect. For example, if the third inequality in (\[eq.example.1period.ineq.1\]) fails, then, every long agent believes that, no matter the price at which her limit order is posted, if it is executed in the next time period, her expected execution price at the next time step will be higher than the price at which the limit order is executed. Hence, it suboptimal to post a limit order at all. In a single period $[N-1,N]$, by choosing small enough $\Delta t$, we can ensure that the inequalities in (\[eq.example.1period.ineq.1\]) are satisfied. However, it turns out that, as we progress recursively, constructing an equilibrium, we may encounter a time step at which one of the inequalities in (\[eq.example.1period.ineq.1\]) fails, implying that a non-degenerate LTC equilibrium cannot be constructed for the given time period (at least, using the proposed method). To see this, consider the recursive equations for $(\hat{p}^a,\hat{\lambda}^a)$ (which are chosen to satisfy the conditions of Corollary \[cor:piecewiseLin\], in Appendix A, given our ansatz): $$\label{eq.RW.pan}
\left\{
\begin{array}{l}
{\hat{p}^a_n \in \text{arg}\max_{p\in{\mathbb R}} {\mathbb E}\left(\left(p-\hat{\lambda}^a_{n+1} - \xi\right) \bone_{\left\{ \xi> p \right\}}\right),
\phantom{\frac{\frac{1}{2}}{2}}}\\
{\hat{\lambda}^a_n = \hat{\lambda}^a_{n+1} + \alpha\Delta t + {\mathbb E}\left( \left(\hat{p}^a_n - \hat{\lambda}^a_{n+1} - \xi\right) \bone_{\left\{\xi> \hat{p}^a_n ) \right\}} \right) <0,\phantom{\frac{\frac{1}{2}}{2}}}
\end{array}
\right.$$ and similarly for $(\hat{p}^b,\hat{\lambda}^b)$. Using the properties of the Gaussian distribution, it is easy to see that, if $\hat{\lambda}^a_{n+1}<0$, we have $\hat{p}^a_n>0$. Similar conclusion holds for $(\hat{\lambda}^b,\hat{p}^b)$. Thus, if $\hat{\lambda}^a_k< 0 < \hat{\lambda}^b_k$, for $k=n+1,\ldots,N$, our method allows us to construct a non-degenerate LTC equilibrium on the time interval $[n,N]$, with $\hat{p}^b<0<\hat{p}^a$. Such a construction always succeeds if the agents are market-neutral: i.e., $\alpha=0$. Indeed, in this case, assuming $\hat{\lambda}^a_{n+1} < 0 < \hat{\lambda}^b_{n+1}$, we have $\hat{p}^b_n < 0 < \hat{p}^a_n$ and $$\hat{\lambda}^a_{n+1} + \left({\mathbb E}\left( \left(\hat{p}^a_n - \hat{\lambda}^a_{n+1} - \xi\right) \bone_{\left\{\xi> \hat{p}^a_n ) \right\}} \right)\right)^+
= {\mathbb E}\left( \hat{\lambda}^a_{n+1} \bone_{\left\{\xi> \hat{p}^a_n ) \right\}} \right)
+ {\mathbb E}\left( \left(\hat{p}^a_n - \xi\right) \bone_{\left\{\xi> \hat{p}^a_n ) \right\}} \right)
< 0.$$ Hence, $\hat{\lambda}^a_n < 0$, and, similarly, we deduce that $\hat{\lambda}^b_n>0$. By induction, we obtain a non-degenerate LTC equilibrium on $[0,N]$, for any $(N,\sigma)$, as long as $\alpha=0$. Corollary \[prop:main.smallspread\] shows that, as $N\rightarrow\infty$, the processes $(\hat{\lambda}^a,\hat{\lambda}^b)$ converge to zero, which means that the expected execution prices converge to the fundamental price. The latter is interpreted as *market efficiency* in the high-frequency trading regime: any market participant expects to buy or sell the asset at the fundamental price. The left hand side of Figure \[fig:3\] shows that the bid and ask prices also converge to the fundamental price if $\alpha=0$. This can be interpreted as a *positive liquidity effect* of increasing the trading frequency.
However, the situation is quite different if $\alpha\neq 0$. Assume, for example, that $\alpha>0$. Then, the second line of (\[eq.RW.pan\]) implies that $\hat{\lambda}^a$ increases by, at least, $\alpha\Delta t$ at each step of the (backward) recursion. Recall that the number of steps is $N=T/\Delta t$, hence, $\hat{\lambda}^a_0 \geq \hat{\lambda}^a_N + \alpha T$. If $|\hat{\lambda}^a_N|$ is small (which is typically the case if $N$ is large), then, we may obtain $\hat{\lambda}^a_{n+1}\geq 0$, at some time $n$, which violates the third inequality in (\[eq.example.1period.ineq.1\]), or, equivalently, implies that the objective in the first line of (\[eq.RW.pan\]) is strictly negative for all $p$. The latter implies that it is suboptimal for the agents with positive inventory to post limit orders, and the proposed method fails to produce a non-degenerate LTC equilibrium in the interval $[n,N]$. Figure \[fig:2\] shows that this does, indeed, occur. Figures \[fig:2\] and \[fig:3\] also show that, for a given (finite) frequency $N$, if $|\alpha|$ is small enough, a non-degenerate equilibrium may still be constructed. Nevertheless, for any $|\alpha|\neq0$, however small it is, there exists a large enough $N$, s.t. the non-degenerate LTC equilibrium fails to exist (at least, within the class defined by the proposed method). This is illustrated in Figure \[fig:3\].
It is important to provide an economic interpretation of why such degeneracy occurs. A careful examination of Figure \[fig:2\] reveals that, around the time when $\hat{\lambda}^a$ becomes nonnegative, the ask price $\hat{p}^a$ explodes. This means that the agents who want to sell the asset are only willing to sell it at a very high price. Notice also that this price is several magnitudes larger than the expected change in the fundamental price (represented by the black dashed line in the left hand side of Figure \[fig:2\]). Hence, such a behavior cannot be justified by the behavior of the fundamental. Indeed, this is precisely what is called an *endogenous liquidity crisis*. So, what causes such a liquidity crisis? Recall that there are two potential reasons for the market to degenerate: agents may choose to submit market orders (if $\hat{p}^b_n>\hat{\lambda}^a_n$ or $\hat{p}^a_n<\hat{\lambda}^b_n$), or they may choose to wait and do nothing (if $\hat{\lambda}^a_{n+1}\geq 0$ or $\hat{\lambda}^b_{n+1}\leq 0$). The right hand side of Figure \[fig:2\] shows that the degeneracy is caused by the second scenario. This means that the naive explanation of an endogenous liquidity crisis, based on the claim that, in a bullish market, those who need to buy the asset will submit market orders wiping out liquidity on the sell side of the book, is wrong. Instead, if the agents on the sell side of the book have the same beliefs, they will increase the ask price so that it is no longer profitable for the agents who want to buy the asset to submit market buy orders. In fact, the ask price may increase disproportionally to the expected change in the fundamental price (i.e., the signal), and this is what causes an endogenous liquidity crisis. The size of the resulting change in the bid or ask price depends not only on the signal, but also on the trading frequency, which demonstrates the *negative liquidity effect* of increasing the trading frequency: it fragilizes the market with respect to deviations of the agents from market-neutrality. The latter, in turn, is explained by the fact that higher trading frequency exacerbates the *adverse selection* effect. To see this, consider, e.g., an agent who is trying to sell one share of the asset. Increasing the trading frequency increases the expected execution value of this agent, bringing it closer to the fundamental price: this corresponds to $\hat{\lambda}^a$ approaching zero (from below). Assume that the agent posts a limit sell order at a price level $p$. If this order is executed in the next period, then, the agent receives $p$, but, for this to happen, the fundamental price value at the next time step, $p^0_{n+1}$, has to be above $p$. On the other hand, the expected execution price of the agent at the next time step is $p^0_{n+1} + \hat{\lambda}^a_{n+1}$. Thus, the expected relative profit, given the execution of her limit order, is ${\mathbb E}_n (p - p^0_{n+1} - \hat{\lambda}^a_{n+1} \, |\,p^0_{n+1}>p )$. The latter expression cannot be positive, unless $\hat{\lambda}^a_{n+1}<0$ and $|\hat{\lambda}^a_{n+1}|$ is sufficiently large. Therefore, if $|\hat{\lambda}^a_{n+1}|$ is small relative to ${\mathbb E}_n (p^0_{n+1} - p\, |\,p^0_{n+1}>p)$, the agent is reluctant to post a limit order at the price level $p$. Hence, $p$ needs to be sufficiently large, to ensure that ${\mathbb E}_n (p^0_{n+1} - p\, |\,p^0_{n+1}>p )$ is smaller than $|\hat{\lambda}^a_{n+1}|$ (in the Gaussian model of this section, the latter expectation vanishes as $p\rightarrow\infty$) – and the extent to which $p$ needs to increase determines the effect of adverse selection. It turns out that, if the agents are market-neutral (i.e. $\alpha=0$), as the frequency $N$ increases, the quantity ${\mathbb E}_n (p^0_{n+1} - p\, |\,p^0_{n+1}>p )$, for any fixed $p$, converges to zero at the same rate as $|\hat{\lambda}^a_{n+1}|$, hence, the above adverse selection effect does not get amplified. On the contrary, if the agents are not market-neutral, $\hat{\lambda}^a_{n+1}$ reaches zero at some high enough (but finite) frequency, while ${\mathbb E}_n (p^0_{n+1} - p\, |\,p^0_{n+1}>p )$ remains strictly positive, for any finite $p$, which produces an “infinite" adverse selection effect and causes the market to degenerate. Of course, so far, these conclusions are based on a very specific example and on a particular method of constructing an equilibrium. The next section shows that they remain valid in any model (with, possibly, heterogeneous beliefs) in which the fundamental price is given by an It[ô]{} process.
It is worth mentioning that a similar adverse selection effect arises in [@MMS.g3], and it is referred to as the “winner’s curse" in [@MMS.g2]. However, the latter papers do not investigate the nature of this phenomenon and focus on other questions instead. In the literature on double auctions (cf. [@DA.DuZhu], [@DA.Vayanos]), a similar effect arises when the auction participants choose to decrease their trading activity in a given auction, because they expect many more opportunities to trade in the future. The latter is similar to the agents choosing to forgo limit orders and wait, in the present example.
Main results {#se:main}
============
In this section, we generalize the previous conclusions, so that they hold in a general model and for any equilibrium. As before, we begin with the “limiting" continuous time model. Consider a terminal time horizon $T>0$ and a complete stochastic basis $(\Omega,\tilde{{\mathbb F}}=(\tilde{\mathcal{F}}_t)_{t\in[0,T]},{\mathbb P})$, with a Brownian motion $W$ on it.[^13] We define the adapted process $\tilde{p}^0$ as a continuous modification of $$\label{eq.p0.cont}
\tilde{p}^0_t = p^0_0 + \int_0^t \sigma_s dW_s,\,\,\,\,\,\,\,\,\,\,\,p^0_0 \in {\mathbb R},$$ where $\sigma$ is a progressively measurable locally square integrable process.
\[ass:sigma\] There exists a constant $C>1$, such that, $1/C\leq \sigma_t\leq C$, for all $t\in[0,T]$, ${\mathbb P}$-a.s..
Consider a Borel set of beliefs $\mathbb{A}$ and the associated family of measures $\left\{{\mathbb P}^{\alpha}\right\}_{\alpha\in\mathbb{A}}$ on $(\Omega,\tilde{\mathcal{F}}_T)$, absolutely continuous with respect to ${\mathbb P}$. Then, for any $\alpha\in\mathbb{A}$, we have $$\label{eq.p0.cont.a}
\tilde{p}^0_t = p^0_0 + A^{\alpha}_t + \int_0^t \sigma_s dW^{\alpha}_s,\quad p^0_0 \in {\mathbb R},
\quad {\mathbb P}^{\alpha}\text{-a.s.},\,\,\forall t\in[0,T],$$ where $W^{\alpha}$ is a Brownian motion under ${\mathbb P}^{\alpha}$, and $A^{\alpha}$ is a process of finite variation. We assume that $A^{\alpha}$ is absolutely continuous: i.e., for any $\alpha\in\mathbb{A}$, there exists a locally integrable process $\mu^{\alpha}$, such that $$A^{\alpha}_t = \int_0^t \mu^{\alpha}_s ds,\quad {\mathbb P}^{\alpha}\text{-a.s.},\,\,\forall t\in[0,T].$$
\[ass:A.alpha\] For any $\alpha\in\mathbb{A}$, the process $\mu^{\alpha}$ is ${\mathbb P}$-a.s. right-continuous, and there exists a constant $C>0$, such that $|\mu^{\alpha}_t| \leq C$, for all $t\in[0,T]$, ${\mathbb P}$-a.s..
Thus, we can rewrite the dynamics of $\tilde{p}^0$, under each ${\mathbb P}^{\alpha}$, as follows: ${\mathbb P}^{\alpha}$-a.s., the following holds for all $t\in[0,T]$ $$\label{eq.p0.cont.a.alpha}
\tilde{p}^0_t = p^0_0 + \int_0^t \mu^{\alpha}_s ds + \int_0^t \sigma_s dW^{\alpha}_s,\quad p^0_0 \in {\mathbb R}.$$ In addition, we modify the above stochastic integral on a set of ${\mathbb P}^{\alpha}$-measure zero, so that (\[eq.p0.cont.a.alpha\]) holds for *all* $(t,\omega)$. In what follows, we often need to analyze the future dynamics of $\tilde{p}^0$ under ${\mathbb P}^{\alpha}$, conditional on $\tilde{\mathcal{F}}_t$, for various $(t,\alpha)$ simultaneously. This is why we need the following joint regularity assumption.
\[ass:joint.cond.reg\] There exists a modification of regular conditional probabilities $$\left\{\tilde{{\mathbb P}}^{\alpha}_t={\mathbb P}^{\alpha}\left(\cdot\,|\,\tilde{\mathcal{F}}_t\right)
\right\}_{t\in[0,T],\,\alpha\in\mathbb{A},}$$ such that it satisfies the tower property with respect to ${\mathbb P}$ (as described in Section \[se:setup\]).
Assumption \[ass:joint.cond.reg\] is satisfied, for example, if ${\mathbb P}^{\alpha}\sim {\mathbb P}$, for all $\alpha\in\mathbb{A}$, or if the set $\mathbb{A}$ is countable. Throughout the rest of the paper, $\tilde{{\mathbb P}}^{\alpha}_t$ refers to a member of the family appearing in Assumption \[ass:joint.cond.reg\]. All conditional expectations $\tilde{{\mathbb E}}^{\alpha}_t$ are taken under such $\tilde{{\mathbb P}}^{\alpha}_t$.
The main results of this section require additional continuity assumptions on $\sigma$ and $\mu^{\alpha}$. The following assumption can be viewed as a stronger version of $\mathbb{L}^2$-continuity of $\sigma$.
\[ass:main.L2.strong\] There exists a function $\varepsilon(\cdot)\geq0$, such that $\varepsilon(\Delta t)\rightarrow0$, as $\Delta t\rightarrow0$, and, ${\mathbb P}$-a.s., $$\tilde{{\mathbb P}}^{\alpha}_{t} \left({\mathbb E}^{\alpha}\left(\left(\sigma_{s\vee\tau} - \sigma_{\tau} \right)^2 \,|\,\mathcal{F}_{\tau} \right) \leq \varepsilon(\Delta t) \right) = 1$$ holds for all $t\in[0,T-\Delta t]$, all $s\in[t,t+\Delta t]$, all stopping times $t\leq\tau\leq s$, and all $\alpha\in\mathbb{A}$.
The above assumption is satisfied, for example, if $\sigma$ is an Itô process with bounded drift and diffusion coefficients. Next, we state a continuity assumption on the drift, which can be interpreted as a uniform right-continuity in probability of the martingale $\tilde{{\mathbb E}}^{\alpha}_{t} \mu^{\alpha}_s$.
\[ass:main.mu.cont.strong\] For any $\alpha\in\mathbb{A}$ and any $t\in[0,T)$, there exists a deterministic function $\varepsilon(\cdot)\geq0$, such that $\varepsilon(\Delta t)\rightarrow0$, as $\Delta t\rightarrow0$, and, ${\mathbb P}^{\alpha}$-a.s., $$\tilde{{\mathbb P}}^{\alpha}_{t'} \left( \left| \int_{t}^T\left(\tilde{{\mathbb E}}^{\alpha}_{t''} \mu^{\alpha}_s - \tilde{{\mathbb E}}^{\alpha}_{t'} \mu^{\alpha}_s\right) ds\right| \geq \varepsilon(\Delta t)\right) \leq \varepsilon(\Delta t)$$ holds for all $t\leq t' \leq t'' \leq t+\Delta t\leq T$.
Notice that Assumptions \[ass:joint.cond.reg\], \[ass:main.L2.strong\], and \[ass:main.mu.cont.strong\] are not quite standard. Therefore, below, we describe a more specific (although, still, rather general) diffusion-based framework, in which the Assumptions \[ass:sigma\]–\[ass:main.mu.cont.strong\] reduce to standard regularity conditions on the diffusion coefficients, and are easily verified. To this end, consider a model in which $\mu^{\alpha}_t = \bar{\mu}^{\alpha}(t,Y_t)$, $\sigma_t = \bar{\sigma}(t,Y_t)$, and, under ${\mathbb P}$, the process $Y$ is a diffusion taking values in ${\mathbb R}^d$ $$dY_t = \Gamma(t,Y_t)dt + \Sigma(t,Y_t) d\bar{B}_t,$$ where $\Gamma:[0,T]\times{\mathbb R}^d\rightarrow{\mathbb R}^d$, $\Sigma=(\Sigma^{i,j})$ is a mapping on $[0,T]\times{\mathbb R}^d$ with values in the space of $d\times m$ matrices, and $\bar{B}$ is $m$-dimensional Brownian motion under ${\mathbb P}$ (on the original stochastic basis). We assume that $\Gamma$ and $\Sigma$ possess enough regularity to conclude that $Y$ is a strongly Markov process. Notice that Assumptions \[ass:sigma\] and \[ass:A.alpha\] reduce to the upper and lower bounds on the functions $\bar{\mu}^{\alpha}$ and $\bar{\sigma}$. Assumption \[ass:joint.cond.reg\] is satisfied if we assume that ${\mathbb P}^{\alpha}\sim{\mathbb P}$, for all $\alpha\in\mathbb{A}$. Let us further assume that the Radon-Nikodym derivative of each measure is in Girsanov form: $$\frac{d{\mathbb P}^{\alpha}}{d{\mathbb P}} = \exp\left(-\frac{1}{2} \int_0^t \|\gamma^{\alpha}(s,Y_s)\|^2 ds + \int_0^t \gamma^{\alpha}(s,Y_s) d\bar{B}_s \right),$$ with an ${\mathbb R}^d$-valued function $\gamma^{\alpha}$, for each $\alpha\in\mathbb{A}$. Let us assume that all entries of $\Gamma$, $\gamma^{\alpha}$ and $\Sigma$ are absolutely bounded by a constant (uniformly over $\alpha\in\mathbb{A}$). Assuming, in addition, that $\bar{\sigma}$ is globally Lipschitz, we easily verify Assumption \[ass:main.L2.strong\]. In order to verify Assumption \[ass:main.mu.cont.strong\], we assume that the quadratic form generated by $A(t,y):=\Sigma(t,y) \Sigma^T(t,y)$ is bounded away from zero, uniformly over all $(t,y)$, and that the entries of $\Gamma$, $\gamma^{\alpha}$ and $\Sigma$ are continuously differentiable with absolutely bounded derivatives (uniformly over $\alpha\in\mathbb{A}$). Then, the Feynman-Kac formula implies that, for any $t\leq s$, $$\tilde{{\mathbb E}}^{\alpha}_t \mu^{\alpha}_s = u^{s,\alpha}(t,Y_t),$$ where $u^{s,\alpha}$ is the unique solution to the associated partial differential equation (PDE) $$\partial_t u^{s,\alpha} + \sum_{i=1}^d \Gamma^{\alpha,i} \partial_{y_i} u^{s,\alpha} + \frac{1}{2}\sum_{i,j=1}^d A^{i,j} \partial^2_{y_i y_j} u^{s,\alpha} = 0,\,\,\,\,(t,y)\in (0,s)\times{\mathbb R}^d,
\quad u^{s,\alpha}(s,y)=\bar{\mu}^{\alpha}(s,y),$$ and $\Gamma^{\alpha}=\Gamma + \Sigma \gamma^{\alpha}$. Assume that, for each $s\in[0,T]$, the function $\bar{\mu}^{\alpha}(s,\cdot)$ is continuously differentiable with absolutely bounded derivatives, uniformly over all $(s,\alpha)$. Then, the standard Gaussian estimates for derivatives of the fundamental solution to the above PDE (cf. Theorem 9.4.2 in [@Friedman.book]) imply that every $\partial_{y_i}u^{s,\alpha}$ is absolutely bounded, uniformly over all $(s,\alpha)$. Then, Itô’s formula and Itô’s isometry yield, for all $t'\leq t''$ and $s\geq t'$: $$\tilde{{\mathbb E}}^{\alpha}_{t'}\left(\tilde{{\mathbb E}}^{\alpha}_{t''} \mu^{\alpha}_s - \tilde{{\mathbb E}}^{\alpha}_{t'} \mu^{\alpha}_s\right)^2
= \sum_{j=1}^m \int_{t'}^{t''\wedge s} \tilde{{\mathbb E}}^{\alpha}_{t'}\left(\sum_{i=1}^d\partial_{y_i} u^{s,\alpha}(v,Y_v)\Sigma^{i,j}(v,Y_v) \right)^2 dv \leq C_1 (t''\wedge s\,-\,t'),$$ with some constant $C_1>0$. The above estimate and Jensen’s inequality imply the statement of Assumption \[ass:main.mu.cont.strong\] and complete the description of the diffusion-based setting. As in Section \[se:examples\], we also consider a progressively measurable random field $\tilde{D}$, s.t. ${\mathbb P}$-a.s. the function $\tilde{D}_t(\cdot)-\tilde{D}_s(\cdot)$ is strictly decreasing and vanishing at zero, for any $0\leq s < t \leq T$. We assume that the demand curve, $\tilde{D}_t(\cdot)-\tilde{D}_s(\cdot)$, cannot be “too flat".
\[ass:main.demandInv.unif\] There exists $\varepsilon>0$, s.t., for any $0\leq t - \varepsilon \leq s < t \leq T$, there exists a $\tilde{\mathcal{F}}_{s}\otimes \mathcal{B}({\mathbb R})$-measurable random function $\kappa_s(\cdot)$, s.t., ${\mathbb P}$-a.s., $\kappa_{s}(\cdot)$ is strictly decreasing and $\left|\tilde{D}_t(p)-\tilde{D}_s(p)\right| \geq \left| \kappa_{s}(p) \right|$, for all $p\in{\mathbb R}$.
Finally, we introduce the empirical distribution process $(\tilde{\mu}_t)$, with values in the space of finite sigma-additive measures on $\mathbb{S}$. The next assumption states that every $\tilde{\mu}_t$ is dominated by a deterministic measure.
\[ass:dom.mu\] For any $t\in[0,T]$, there exists a finite sigma-additive measure $\mu^{0}_t$ on $\left(\mathbb{S}, \mathcal{B}\left(\mathbb{S} \right)\right)$, s.t., ${\mathbb P}$-a.s., $\tilde{\mu}_t$ is absolutely continuous w.r.t. $\mu^0_t$.
We partition the time interval $[0,T]$ into $N$ subintervals of size $\Delta t=T/N$. A discrete time model is obtained by discretizing the continuous time one $$\mathcal{F}_n = \tilde{\mathcal{F}}_{n\Delta t},\quad p^0_n = \tilde{p}^0_{n\Delta t},\quad D_n(p) = (\tilde{D}_{n\Delta t}-\tilde{D}_{(n-1)\Delta t})(p-p^0_n),\quad \mu_n = \tilde{\mu}_{n\Delta t}.
$$ Before we present the main results, let us comment on the above assumptions. These assumptions are important from a technical point of view, however, some of them have economic interpretation that may provide (partial) intuitive explanations of the results that follow. In particular, Assumption \[ass:sigma\] ensures that the fundamental price remains “noisy," which implies that an agent can execute a limit order very quickly by posting it close to the present value of $p^0$, if there are no other orders posted there. In combination with Assumption \[ass:main.demandInv.unif\], the latter implies that, when the frequency, $N$, is high, an agent has a lot of opportunities to execute her limit order at a price close to the fundamental price (at least, if no other orders are posted too close to the fundamental price). Intuitively, this means that the agent’s execution value should improve as the frequency increases. Assumption \[ass:main.mu.cont.strong\] ensures that, if an agent has a signal about the direction of the fundamental price, this signal is persistent – i.e., it is continuous in the appropriate sense. When the trading frequency $N$ is large, such persistency means that an agent has a large number of opportunities to exploit the signal, implying that she is in no rush to have her order executed immediately. The main results of this work, presented below, along with their proofs, confirm that these heuristic conclusions are, indeed, correct.
As mentioned in the preceding sections, our main goal is to analyze the liquidity effects of increasing the trading frequency. Therefore, we fix a limiting continuous time model, and consider a sequence of discrete time models, obtained from the limiting one as described above, for $N\rightarrow\infty$. This can be interpreted as observing the same population of agents, each of whom has a fixed continuous time model for future demand, in various exchanges that allow for different trading frequencies. We begin with the following theorem, which shows that, if every market model in a given sequence admits a non-degenerate equilibrium, then, the terminal bid and ask prices converge to the fundamental price, as the trading frequency goes to infinity.
\[le:main.zeroTermSpread\] Let Assumptions \[ass:sigma\], \[ass:A.alpha\], \[ass:joint.cond.reg\], \[ass:main.L2.strong\], \[ass:main.demandInv.unif\], \[ass:dom.mu\] hold. Consider a family of uniform partitions of a given time interval $[0,T]$, with diameters $\left\{\Delta t=T/N>0\right\}$ and with the associated family of discrete time models, and denote the associated fundamental price process by $p^{0,\Delta t}$. Assume that every such model admits a non-degenerate LTC equilibrium, and denote the associated bid and ask prices by $p^{b,\Delta t}$ and $p^{a,\Delta t}$ respectively. Then, there exists a deterministic function $\varepsilon(\cdot)$, s.t. $\varepsilon(\Delta t)\rightarrow0$, as $\Delta t\rightarrow0$, and, for all small enough $\Delta t>0$, the following holds ${\mathbb P}$-a.s.: $$\left|p^{a,\Delta t}_{N} - p^{0,\Delta t}_{N}\right| + \left|p^{b,\Delta t}_{N} - p^{0,\Delta t}_{N}\right| \leq \varepsilon(\Delta t)$$
The above theorem has a useful corollary, which can be interpreted as follows: *if the market does not degenerate as the frequency increases, then, such an increase improves market efficiency*. Here, we understand the “improving efficiency" in the sense that the expected execution price (i.e., the price per share that an agent expects to receive or pay by the end of the game) of every agent converges to the fundamental price.
\[prop:main.smallspread\] Under the assumptions of Theorem \[le:main.zeroTermSpread\], denote the support of every equilibrium by $\tilde{\mathbb{A}}^{\Delta t}$ and the associated expected execution prices by $\lambda^{a,\Delta t}$ and $\lambda^{b,\Delta t}$. Then, there exists a deterministic function $\varepsilon(\cdot)$, such that $\varepsilon(\Delta t)\rightarrow0$, as $\Delta t\rightarrow0$, and, ${\mathbb P}$-a.s., $$\sup_{n=0,\ldots,N,\,\alpha\in\tilde{\mathbb{A}}^{\Delta t}}\left(\left|\lambda^{a,\Delta t}_n(\alpha) - p^{0,\Delta t}_n\right| + \left|\lambda^{b,\Delta t}_n(\alpha) - p^{0,\Delta t}_n\right|\right) \leq \varepsilon(\Delta t),$$ for all small enough $\Delta t>0$.
Denote ${\mathbb E}^{\alpha}_n = \tilde{{\mathbb E}}^{\alpha}_{n\Delta t}$. It follows from Corollary \[cor:piecewiseLin\], in Appendix A, and the definition of LTC equilibrium that $\lambda^{a,\Delta t}_{N}(\alpha) = p^{b,\Delta t}_N$ and $\lambda^{b,\Delta t}_{N}(\alpha)=p^{a,\Delta t}_N$. It also follows from Corollary \[cor:piecewiseLin\] (or, more generally, from the definition of a value function) that $\lambda^{a,\Delta t}(\alpha)$ is a supermartingale, and $\lambda^{b,\Delta t}(\alpha)$ is a submartingale, under ${\mathbb P}^{\alpha}$. Thus, we have: $\lambda^{a,\Delta t}_{n}(\alpha) \geq {\mathbb E}^{\alpha}_{n} p^{b,\Delta t}_{N}$ and $\lambda^{b,\Delta t}_{n}(\alpha) \leq {\mathbb E}^{\alpha}_{n} p^{a,\Delta t}_{N}$. On the other hand, notice that we must have: $\lambda^{a,\Delta t}_{n}(\alpha) \leq {\mathbb E}^{\alpha}_{n} p^{a,\Delta t}_{N}$ and $\lambda^{b,\Delta t}_{n}(\alpha) \geq {\mathbb E}^{\alpha}_{n} p^{b,\Delta t}_{N}$. Assume, for example, that $\lambda^{a,\Delta t}_{n}(\alpha) > {\mathbb E}^{\alpha}_{n} p^{a,\Delta t}_{N}$ on the event $\Omega'$ of positive ${\mathbb P}^{\alpha}$-probability. Consider an agent at state $(0,\alpha)$, who follows the optimal strategy of an agent at state $(1,\alpha)$, starting from time $n$ and onward, on the event $\Omega'$ (otherwise, she does not do anything). It is easy to see that the objective value of this strategy is $${\mathbb E}^{\alpha}\left( \bone_{\Omega'} \left( \lambda^{a,\Delta t}_{n}(\alpha) - {\mathbb E}^{\alpha}_{n} p^{a,\Delta t}_{N} \right)\right) > 0,$$ which contradicts Corollary \[cor:piecewiseLin\]. The second inequality is shown similarly. Thus, we conclude that, for any $n=0,\ldots,N-1$, both $\lambda^{a,\Delta}_{n}(\alpha)$ and $\lambda^{b,\Delta}_{n}(\alpha)$ belong to the interval $$\left[{\mathbb E}^{\alpha}_{n} p^{b,\Delta t}_{N},\,{\mathbb E}^{\alpha}_{n} p^{a,\Delta t}_{N}\right],$$ which, in turn, converges to zero, as $\Delta t\rightarrow0$, due to the deterministic bounds obtained in the proof of Proposition \[le:main.zeroTermSpread\].
The results of Theorem \[le:main.zeroTermSpread\] and Corollary \[prop:main.smallspread\] can be viewed as a specific case of a more general observation: markets become more efficient as the frictions become smaller. In the present setting, the limited trading frequency is viewed as friction, and the market efficiency is measured by the difference between the bid and ask prices, or between the expected execution prices. Many more instances of analogous results can be found in the literature, depending on the choice of a friction type. For example, the markets become efficient in [@MMS.gmm1] and [@MMS.gmm2] as the number of insiders vanishes. Similarly, the markets become efficient in [@DA.DuZhu] as the trading frequency increases and the size of private signals vanishes. It is also mentioned in [@MMS.gliq1] that the market would become efficient if there was no restriction on the size of agents’ inventories therein.
The above results demonstrate the positive role of high trading frequency. However, they are based on the assumption that the market does not degenerate as frequency increases. In the context of Section \[se:examples\], we saw that the markets do not degenerate only if the agents are market-neutral (i.e. $\alpha=0$). If this condition is violated and the frequency $N$ is sufficiently high, the market does not admit any non-degenerate equilibrium (i.e., there exists no safe regime, in which the liquidity crisis would never occur). It turns out that this conclusion still holds in the general setting considered herein.
\[thm:main.necessary\] Let Assumptions \[ass:sigma\], \[ass:A.alpha\], \[ass:joint.cond.reg\], \[ass:main.L2.strong\], \[ass:main.mu.cont.strong\], \[ass:main.demandInv.unif\], \[ass:dom.mu\] hold. Consider a family of uniform partitions of a given time interval $[0,T]$, with diameters $\left\{\Delta t=T/N>0\right\}$, containing arbitrarily small $\Delta t$, and with the associated family of discrete time models. Assume that every such model admits a non-degenerate LTC equilibrium, with the same support $\tilde{\mathbb{A}}$. Then, for all $\alpha\in\tilde{\mathbb{A}}$, we have: $\tilde{p}^{0}$ is a [**martingale**]{} under ${\mathbb P}^{\alpha}$.
The above theorem shows that the market degenerates even if the signal $\mu^{\alpha}$ is very small (but non-zero), provided the trading frequency $N$ is large enough. Therefore, as discussed at the end of Section \[se:examples\], such degeneracy cannot be attributed to any fundamental reasons, and we refer to it as the *endogenous liquidity crisis*. Let us provide an intuitive (heuristic) argument for why the statement of Theorem \[thm:main.necessary\] holds. Assume, first, that all long agents (i.e., those having positive inventory) are bullish about the asset (i.e., have a positive drift $\mu^{\alpha}$). Then, similar to Section \[se:examples\], the higher trading frequency amplifies the *adverse selection effect*, forcing the long agents to withdraw liquidity from the market (i.e., they prefer to do nothing and wait for a higher fundamental price level). Note that, in the present setting, the agents may have different beliefs, the LOB may have a complicated shape and dynamics, and the expected execution prices are no longer deterministic. All this makes it difficult to provide a simple description of how the high frequency amplifies the adverse selection. Nevertheless, the general analysis of this case is still based on the idea discussed at the end of Section \[se:examples\]: it has to do with how fast $\tilde{{\mathbb E}}^{\alpha}_{n\Delta t} (p^0_{n+1} - p\, |\,p^0_{n+1}>p )$ vanishes (as the frequency increases), relative to the rate at which the expected execution prices approach the fundamental price. Thus, to avoid market degeneracy, there must be a non-zero mass of long agents who are market-neutral or bearish. As the trading frequency grows, these agents will post their limit orders at lower levels. Next, assume that there exists a bullish agent (long, short, or with zero inventory). Then, at a sufficiently high trading frequency, the agent’s expected value of a long position in a single share of the asset will exceed the ask prices posted by the market-neutral and bearish long agents. In this case, the bullish agent prefers to buy more shares at the posted ask price, in order to sell them later. As the agents are small and their objectives are linear, the bullish agent can scale up her strategy to generate infinite expected profits. This contradicts the definition of optimality and implies that an equilibrium fails to exist. Thus, all agents have to be either market-neutral or bearish. Applying a symmetric argument, we conclude that all agents must be market-neutral.[^14] A rigorous formulation of the above arguments, which constitutes the proof of Theorem \[thm:main.necessary\], is given in Section \[se:pf.2\]. It is worth mentioning that the possible degeneracy of the LOB is also documented in [@MMS.gmm1], and is referred to as a “market shut down". The setting used in the latter paper is very different: it analyzes a quote-driven exchange (i.e., the one with a designated market maker) and assumes the existence of insiders with superior information. Nevertheless, it is possible to draw a parallel with the LOB degeneracy in the present setting. Namely, the degeneracy in [@MMS.gmm1] occurs when the number of insiders increases, which implies that the signal, generated by the insiders’ trading, becomes sufficiently large. The latter is similar to the deviation from martingality of the fundamental price in the present setting. However, an increase in the number of insiders in [@MMS.gmm1] also implies an increase in frictions (since the insiders can be interpreted as friction in [@MMS.gmm1]). Theorem \[thm:main.necessary\], on the other hand, states that a market degeneracy will occur when the frictions are sufficiently small. Perhaps, this dual role of the number of insiders did not allow for a detailed analysis of market shut downs in [@MMS.gmm1]. Many other models of market microstructure (cf. [@MMS.g3], [@MMS.g6], [@MMS.g1], [@MMS.g2], [@DA.DuZhu]) are not well suited for the analysis of market degeneracy, either because the agents in these models pursue “one-shot" strategies (i.e., they cannot choose to wait and post a limit order later) or because the fundamental price (or its analogue) is restricted to be a martingale.
Conditional tails of the marginal distributions of Itô processes {#se:tails}
================================================================
As follows from the discussion in the preceding sections, in order to prove the main results of the paper, we need to investigate the properties of marginal distributions of the fundamental price $\tilde{p}^0$ (more precisely, the distributions of its increments). In order to prove Theorem \[le:main.zeroTermSpread\], we need to show that the difference between the fundamental price and the bid or ask prices converges to zero, as the frequency $N$ increases to infinity. It turns out that, for this purpose, it suffices to show that the distribution of a normalized increment of $\tilde{p}^0$ converges to the standard normal distribution. The following lemma summarizes these results. It is rather simple, but technical, hence, its proof is postponed to Appendix B. In order to formulate the result (and to facilitate the derivations in subsequent sections), we introduce addiitonal notation. For notational convenience, we drop the superscript $\Delta t$ for some variables (we only emphasize this dependence when it is important). For any market model on the time interval $[0,T]$, associated with a uniform partition with diameter $\Delta t=T/N>0$, and having a fundamental price process $p^0$, we define $$\label{eq.xi.not}
\xi_n = p^{0}_n - p^{0}_{n-1} = \tilde{p}^0_{t_n} - \tilde{p}^0_{t_{n-1}},
\quad {\mathbb E}^{\alpha}_n = \tilde{{\mathbb E}}^{\alpha}_{t_n},
\quad {\mathbb P}^{\alpha}_n = \tilde{{\mathbb P}}^{\alpha}_{t_n},
\quad t_n = n \Delta t,
\quad n=1,\ldots,NT/\Delta t.$$ We denote by $\eta_0$ a standard normal random variable (on a, possibly, extended probability space), which is independent of $\mathcal{F}_N$ under every ${\mathbb P}^{\alpha}$.
\[gapproxapplied\] Let Assumptions \[ass:sigma\], \[ass:A.alpha\], \[ass:joint.cond.reg\], \[ass:main.L2.strong\] hold. Then, there exists a function $\varepsilon(\cdot)\ge0$, s.t. $\varepsilon(\Delta t)\to0$, as $\Delta t\to0$, and the following holds ${\mathbb P}$-a.s., for all $p\in{\mathbb R}$, all $\alpha\in\mathbb{A}$, and all $n=1,\ldots,N$,
- $(|p|\vee 1)\left|{\mathbb P}^{\alpha}_{n-1}\left(\frac{\xi_n}{\sqrt{\Delta t}} >p\right)
- {\mathbb P}^{\alpha}_{n-1}\left(\sigma_{t_{n-1}}\eta_0>p\right)\right|
\le\varepsilon(\Delta t)$,
- $\left|{\mathbb E}^{\alpha}_{n-1}\left( \frac{\xi_n}{\sqrt{\Delta t}}\bone_{\left\{\xi_n/\sqrt{\Delta t}>p\right\}}\right)
- {\mathbb E}^{\alpha}_{n-1}\left(\sigma_{t_{n-1}}\eta_0 \bone_{\left\{\sigma_{t_{n-1}}\eta_0>p\right\}}\right) \right|\le\varepsilon(\Delta t)$.
In addition, the above estimates hold if we replace $(\xi_n,\eta_0,p)$ by $(-\xi_n,-\eta_0,-p)$.
In order to prove Theorem \[thm:main.necessary\] we need to compare the rates at which the conditional expectations ${\mathbb E}^{\alpha}_n (p^0_{n+1} - p\, |\,p^0_{n+1}>p )$ vanish (as the frequency $N$ goes to infinity) to the rate at which the expected execution prices converge to the fundamental price. This requires a more delicate analysis – in particular, the mere proximity of the distribution of a (normalized) fundamental price increment to the Gaussian distribution is no longer sufficient. In fact, what we need is a precise uniform estimate of the conditional tail of the distribution of a fundamental price increment. The desired property is formulated in the following lemma, which, we believe, is valuable in its own right. This result enables us to estimate the tails of the conditional marginal distribution of an It[ô]{} process $X$ uniformly by an exponential. To the best of our knowledge, this result is new. The main difficulties in establishing the desired estimates are: (a) the fact that we estimate the *conditional*, as opposed to the regular, tail, and (b) the fact that the estimates need to be uniform over the values of the argument. Note that, even in the case of a diffusion process $X$, the classical Gaussian-type bounds for the tails of the marginal distributions of $X$ are not sufficient to establish the desired estimates. The reason is that, in general, the Gaussian estimates of the regular tails from above and from below have different orders of decay, for the large values of the argument, which makes them useless for estimating the conditional tail (which is a ratio of two regular tails).
\[le:necessary.marginal.maximum\] Consider the following continuous semimartingale on a stochastic basis $(\hat{\Omega},(\hat{\mathcal{F}}_t)_{t\in[0,1]},\hat{{\mathbb P}})$: $$X_t = \int_0^t \hat{\mu}_u du + \int_{0}^t \hat{\sigma}_u dB_u,\,\,\,\,\,\,\,\,\,\,\,\,t\in[0,1],$$ where $B$ is a Brownian motion (with respect to the given stochastic basis), $\hat{\mu}$ and $\hat{\sigma}$ are progressively measurable processes, such that the above integrals are well defined. Assume that, for any stopping time $\tau$ with values in $[0,1]$, $c\leq |\hat{\sigma}_{\tau}| \leq C$ holds a.s. with some constants $c,C>0$. Then, there exists $\varepsilon>0$, depending only on $(c,C)$, s.t., if $$\hat{\mu}^2_{\tau}\leq \varepsilon,\quad \hat{{\mathbb E}}\left( (\hat{\sigma}_{s\vee\tau} - \hat{\sigma}_{\tau})^2 \,|\,\hat{\mathcal{F}}_{\tau} \right) \leq \varepsilon\,\,\,\, \text{a.s.},$$ for all $s\in[0,1]$ and all stopping time $\tau$, with values in $[0,1]$, then, for any $c_1>0$, there exists $C_1>0$, depending only on $(c,C,\varepsilon,c_1)$, s.t. the following holds: $$\hat{{\mathbb P}}(X_1 > x+z\,\vert\, X_1>x) \leq C_1 e^{-c_1 z},\quad\forall x,z\geq0.$$
In the course of this proof, we will use the shorthand notation, $\hat{{\mathbb E}}_{\tau}$ and $\hat{{\mathbb P}}_{\tau}$, to denote the conditional expectation and the conditional probability w.r.t $\hat{\mathcal{F}}_{\tau}$. We also denote $$A_t = \int_0^t \hat{\mu}_u du,
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,G_t = \int_{0}^t \hat{\sigma}_u dB_u.$$ For any $x\geq0$, let us introduce $\tau_x = 1\wedge\inf\left\{t\in[0,1]\,:\, X_t = x \right\}$. Then $$\hat{{\mathbb P}}(X_1 > x+z)
\leq \hat{{\mathbb P}}(\sup_{t\in[0,1]} X_t > x+z)
= \hat{{\mathbb E}} \left( \bone_{\left\{ \tau_x < 1 \right\}} \hat{{\mathbb P}}_{\tau_x} \left(\sup_{s\in[\tau_x,1]} (X_s - x) > z \right) \right)$$ Notice that, on $\left\{\tau_x\leq s \right\}$, we have: $X_s - x = A_{s\vee\tau_x} - A_{\tau_x} + G_{s\vee\tau_x} - G_{\tau_x}$. In addition, the process $(Y)_{s\in[0,1]}$, with $Y_s = A_{s\vee\tau_x} - A_{\tau_x}$, is adapted to the filtration $(\hat{\mathcal{F}}_{\tau_x\vee s})$, while the process $(Z)_{s\in[0,1]}$, with $Z_s = G_{s\vee\tau_x} - G_{\tau_x}$, is a martingale with respect to it. Next, on $\left\{\tau_x < 1 \right\}$, we have: $$\hat{{\mathbb P}}_{\tau_x} \left(\sup_{s\in[\tau_x,1]} (X_s - x) > z \right)
= \hat{{\mathbb P}}_{\tau_x} \left(\sup_{s\in[0,1]} (Y_s + Z_s) > z\right)$$ $$\leq \hat{{\mathbb P}}_{\tau_x} \left(\sup_{s\in[0,1]} \exp\left(c_1Z_s - \frac{1}{2}c_1^2\langle Z\rangle_s \right) > \exp\left( c_1z - c_1\sqrt{\varepsilon} - \frac{1}{2} c_1^2C^2\right)\right),$$ where we used the fact that $\langle Z\rangle_s \leq \langle X\rangle_1 \leq C^2$, for all $s\in[0,1]$. Using the Novikov’s condition, it is easy to check that $$M_s = \exp\left(c_1Z_s - \frac{1}{2}c_1^2\langle Z\rangle_s \right),\,\,\,\,\,\,\,\,\,s\in[0,1],$$ is a true martingale, and, hence, we can apply the Doob’s martingale inequality to obtain, on $\left\{\tau_x < 1 \right\}$: $$\hat{{\mathbb P}}_{\tau_x} \left(\sup_{s\in[0,1]} \exp\left(c_1Z_s - \frac{1}{2}c_1^2\langle Z\rangle_s \right) > \exp\left( c_1z - c_1\sqrt{\varepsilon} - \frac{1}{2}c_1^2 C^2\right)\right)
\leq \exp\left( -c_1z + c_1\sqrt{\varepsilon} + \frac{1}{2} c_1^2 C^2\right).$$ Collecting the above inequalities, we obtain $$\label{eq.necessary.biglemma.step1.res}
\hat{{\mathbb P}}(X_1 > x+z) \leq \hat{{\mathbb P}}(\sup_{t\in[0,1]} X_t > x+z) \leq C_2(\varepsilon) e^{-c_1z} \hat{{\mathbb P}}(\tau_x < 1) = C_2(\varepsilon) e^{-c_1z} \hat{{\mathbb P}}(\sup_{t\in[0,1]} X_t > x).$$ The next step is to estimate the distribution tails of a running maximum via the tails of the distribution of $X_1$. To do this, we proceed as before: $$\label{eq.necessary.biglemma.step2.1}
\hat{{\mathbb P}}(X_1 > x)
= \hat{{\mathbb E}} \left( \bone_{\left\{ \tau_x < 1 \right\}} \hat{{\mathbb P}}_{\tau_x}\left(Y_1 + Z_1 > 0\right) \right),$$ with $Y$ and $Z$ defined above. Notice that, on $\left\{\tau_x < 1 \right\}$, $$\hat{{\mathbb P}}_{\tau_x}\left(Y_1 + Z_1 > 0\right)
= \hat{{\mathbb P}}_{\tau_x}\left(\hat{\sigma}_{\tau_x} \frac{B_1-B_{\tau_x}}{\sqrt{1-\tau_x}}
+ \frac{1}{\sqrt{1-\tau_x}} \int_{\tau_x}^{1} \hat{\mu}_{u} du
+ \frac{1}{\sqrt{1-\tau_x}} \int_{0}^{1} (\hat{\sigma}_{u\vee\tau_x} - \hat{\sigma}_{\tau_x}) dB^x_u > 0\right),$$ where $B^x_s = B_{s\vee\tau_x}$ is a continuous square-integrable martingale with respect to $(\hat{\mathcal{F}}_{s\vee\tau_x})$. Denote $$R_s = \int_{0}^{s} (\hat{\sigma}_{u\vee\tau_x} - \hat{\sigma}_{\tau_x}) dB^x_u,
\quad s\in[0,1],$$ and notice that it is a square-integrable martingale with respect to $(\hat{\mathcal{F}}_{s\vee\tau_x})$. Then, on $\left\{\tau_x<1\right\}$ (possibly, without a set of measure zero), we have: $$\hat{{\mathbb E}}_{\tau_x} \left(\frac{1}{\sqrt{1-\tau_x}} R_1 \right)^2
=\frac{1}{1-\tau_x} \hat{{\mathbb E}}_{\tau_x} R^2_1
\leq \frac{1}{1-\tau_x}\int_{\tau_x}^{1} \hat{{\mathbb E}}_{\tau_x}(\hat{\sigma}_{u\vee\tau_x} - \hat{\sigma}_{\tau_x})^2 du
\leq \varepsilon.$$ In addition, $$\hat{{\mathbb E}}_{\tau_x} \left( \frac{1}{\sqrt{1-\tau_x}} \int_{\tau_x}^{1} \hat{\mu}_{u} du \right)^2
\leq \varepsilon.$$ Collecting the above and using Chebyshev’s inequality, we obtain, on $\left\{\tau_x < 1 \right\}$: $$\left|\hat{{\mathbb P}}_{\tau_x}\left(Y_1 + Z_1 > 0\right)
- \hat{{\mathbb P}}_{\tau_x}\left(\hat{\sigma}_{\tau_x} \frac{B_1-B_{\tau_x}}{\sqrt{1-\tau_x}} \leq -\varepsilon^{1/3} \right)\right|
\leq 2\varepsilon^{1/6}.$$ On the other hand, due to the strong Markov property of Brownian motion, on $\left\{\tau_x<1\right\}$, we have, a.s.: $$\hat{{\mathbb P}}_{\tau_x}\left(\hat{\sigma}_{\tau_x} \frac{B_1-B_{\tau_x}}{\sqrt{1-\tau_x}} \leq -\varepsilon^{1/3} \right)
= \left.\hat{{\mathbb P}} \left(\xi \leq -\frac{\varepsilon^{1/3}}{\sigma} \right)\right|_{\sigma=\hat{\sigma}_{\tau_x}},$$ where $\xi$ is a standard normal. As $\hat{\sigma}_{\tau_x}\in[c,C]$, we conclude that the right hand side of the above converges to $1/2$, as $\varepsilon\rightarrow0$, uniformly over almost all random outcomes in $\left\{\tau_x<1\right\}$. In particular, for all small enough $\varepsilon>0$, we have: $$\bone_{\left\{\tau_x<1 \right\}} \left|\hat{{\mathbb P}}_{\tau_x}\left(Y_1 + Z_1 \leq 0\right) - \hat{{\mathbb P}}_{\tau_x}\left(Y_1 + Z_1 > 0\right) \right| \leq \bone_{\left\{\tau_x<1 \right\}} \delta(\varepsilon)<1,$$ and, in view of (\[eq.necessary.biglemma.step2.1\]), $$\hat{{\mathbb P}}(X_1>x) \geq \hat{{\mathbb E}} \left( \bone_{\left\{ \tau_x < 1 \right\}} \hat{{\mathbb P}}_{\tau_x}\left(Y_1 + Z_1 \leq 0\right) \right) - \delta(\varepsilon) \hat{{\mathbb P}}(\tau_x<1)$$ Summing up the above inequality and (\[eq.necessary.biglemma.step2.1\]), we obtain $$2\hat{{\mathbb P}}(X_1>x) \geq (1-\delta(\varepsilon))\hat{{\mathbb P}}(\tau_x<1) = (1-\delta(\varepsilon))\hat{{\mathbb P}}(\sup_{t\in[0,1]}X_t > x),$$ which, along with (\[eq.necessary.biglemma.step1.res\]), yields the statement of the lemma.
Proof of Theorem \[le:main.zeroTermSpread\] {#se:pf.1}
===========================================
Within the scope of this proof, we adopt the notation introduced in (\[eq.xi.not\]) and use the following convention.
\[not:shift\] The LOB, the bid and ask prices, the expected execution prices, and the demand, are all measured relative to $p^0$. Namely, we use $\nu_n$ to denote $\nu_n\circ (x\mapsto x+p^0_n)^{-1}$, $p^a_n$ to denote $p^a_n-p^0_n$, $p^b_n$ to denote $p^b_n-p^0_n$, $\lambda^a_n$ to denote $\lambda^a_n-p^0_n$, $\lambda^b_n$ to denote $\lambda^b_n-p^0_n$, and $D_n(p)$ to denote $D_n(p^0_n+p)$.
Herein, we are only concerned with what happens in the last trading period – at time $(N-1)$, where $N=T/\Delta t$. Hence, we omit the subscript $N-1$ whenever it is clear from the context. In particular, we write $p^a$ and $p^b$ for $p^a_{N-1}$ and $p^b_{N-1}$, $\nu$ for $\nu_{N-1}$, and $\xi$ for $\xi_N$. Note also that, in an LTC equilibrium, we have: $p^a=p^a_N=p^a_{N-1}$, with similar equalities for $p^b$ and $\nu$. For convenience, we also drop the superscript $\Delta t$ in the LOB and the associated bid and ask prices. Finally, we denote by $\tilde{\mathbb{A}}$ the support of a given equilibrium. As the roles of $p^a$ and $p^b$ in our model are symmetric, we will only prove the statement of the proposition for $p^b$. We are going to show that, under the assumptions of the theorem, there exists a constant $C_0>0$, depending only on the constant $C$ in Assumptions \[ass:sigma\] and \[ass:A.alpha\], such that, for all small enough $\Delta t$, we have, ${\mathbb P}$-a.s.: $$\label{eq.prop1.target}
-C_0\leq p^b/\sqrt{\Delta t} < 0$$ First, we introduce $\hat{A}^\alpha(p;x)$, which we refer to as the simplified objective: $$\label{eq.simp.obj.def}
\hat{A}^\alpha(p;x)={\mathbb E}^\alpha_{N-1}\left((p-x-\xi)\bone_{\{\xi>p\}}\right).$$ Recall that the expected relative profit from posting a limit sell order at price level $p$, in the last time period,[^15] is given by $A^\alpha(p;p^b_{N})$, where $$\label{eq.true.obj.def}
A^\alpha(p;x)={\mathbb E}^\alpha_{N-1}\left((p-x-\xi)\bone_{\{D^+_N(p-\xi)>\nu^+((-\infty,p))\}}\right).$$ The simplified objective is similar to $A^\alpha$, but it assumes that there are no orders posted at better prices than the one posted by the agent. In particular, $\hat{A}^\alpha(p;x)=A^\alpha(p;x)$ for $p\le p^a$. Corollary \[cor:piecewiseLin\], in Appendix A, states that, in equilibrium, ${\mathbb P}$-a.s., if the agents in the state $(s,\alpha)$ post limit sell orders, then they post them at a price level $p$ that maximizes the true objective $A^\alpha(p;p^b)$. The following lemma shows that the value of the modified objective becomes close to the value of the true objective, for the agents posting limit sell orders close to the ask price.
\[le:simp.to.true.val\] ${\mathbb P}$-a.s., either $\nu^+(\{p^a\})>0$ or we have: $$\left\vert A^{\alpha}(p;p^b) - \hat{A}^{\alpha}(p^a;p^b)\right\vert\to0,$$ as $p\downarrow p^a$, uniformly over all $\alpha\in\tilde{\mathbb{A}}$.
If $\nu^+(\{p^a\})=0$, then $\nu^+$ is continuous at $p^a$, and $\nu^+((-\infty,p])\rightarrow0$, as $p\downarrow p^a$. Then, we have $$\left| A^{\alpha}(p;p^b) - \hat{A}^{\alpha}(p^a;p^b)\right|$$ $$=\left| {\mathbb E}^{\alpha}_{N-1}\left((p - p^b-\xi)\bone_{\{D^+_N(p-\xi)>\nu^+((-\infty,p))\}}\right)
- {\mathbb E}^{\alpha}_{N-1}\left((p^a-p^b-\xi)\bone_{\{\xi>p^a\}}\right)\right|$$ $$\le|p-p^a|+\left\Vert p^a-p^b-\xi\right\Vert_{\mathbb{L}^2\left({\mathbb P}^{\alpha}_{N-1}\right)}{\mathbb P}^{\alpha}_{N-1}\left(\xi>p^a,\,D^+_N(p-\xi)\le\nu^+((-\infty,p))\right)$$ Thus, it suffices to show that: (i) $\left\Vert p^a-p^b-\xi \right\Vert_{\mathbb{L}^2({\mathbb P}^{\alpha}_{N-1})}$ is bounded by a finite random variable independent of $\alpha$, and (ii) $${\mathbb P}^{\alpha}_{N-1}\left(\xi_N>p^a,\, D^+_N(p-\xi)\le\nu^+((-\infty, p))\right) \to0,
\quad {\mathbb P}\text{-a.s.},$$ as $p\downarrow p^a$, uniformly over $\alpha$. For (i), we have: $$\left\Vert p^a-p^b-\xi \right\Vert_{\mathbb{L}^2({\mathbb P}^{\alpha}_{N-1})}\le |p^a-p^b|+\left\Vert\xi\right\Vert_{\mathbb{L}^2({\mathbb P}^{\alpha}_{N-1})} \leq |p^a-p^b| + 2C \sqrt{\Delta t},$$ where the constant $C$ appears in Assumptions \[ass:sigma\] and \[ass:A.alpha\]. For (ii), we note that $$\{\xi_N>p^a,\, D^+_N(p-\xi) \le \nu^+((-\infty,p))\}
= \{\xi_N>p^a,\, \xi \le p - D^{-1}_N\left(\nu^+((-\infty,p))\right)\},$$ as $D_N(\cdot)$ is strictly decreasing, with $D_N(0)=0$. Assumption \[ass:main.demandInv.unif\] implies that $$\kappa^{-1}(\nu^+((-\infty,p)))\leq D^{-1}_N(\nu^+((-\infty,p))) < 0,$$ where $\kappa$ is known at time $N-1$. Therefore, $${\mathbb P}^{\alpha}_{N-1}\left(\xi>p^a,\, D^+_N(p-\xi)\le\nu^+((-\infty, p))\right)
\leq {\mathbb P}^{\alpha}_{N-1} \left( \xi \in \left(p^a, p - \kappa^{-1}(\nu^+((-\infty,p))) \right] \right).$$ It remains to show that, ${\mathbb P}$-a.s., the right hand side of the above converges to zero, uniformly over all $\alpha$. Assume that it does not hold. Then, with positive probability ${\mathbb P}$, there exists $\varepsilon>0$ and a sequence of $(p_k,\alpha_k)$, such that $p_k\downarrow p^a$ and $${\mathbb P}^{\alpha_k}_{N-1} \left( \xi \in (p^a, p_k - \kappa^{-1}(\nu^+((-\infty,p_k))) ] \right) \geq \varepsilon.$$ Notice that, ${\mathbb P}$-a.s., the family of measures $\left\{ \hat{\mu}_k = {\mathbb P}^{\alpha_k}_{N-1}\circ \xi^{-1} \right\}_k$ is tight. The latter follows, for example, from the fact that, ${\mathbb P}$-a.s., the conditional second moments of $\xi$ are bounded uniformly over all $\alpha$ (which, in turn, is a standard exercise in stochastic calculus). Prokhorov’s theorem, then, implies that there is a subsequence of these measures that converges weakly to some measure $\hat{\mu}$ on ${\mathbb R}$. Next, notice that, for any fixed $k$ in the chosen subsequence, there exists a large enough $k'$, such that $$\left|\hat{\mu} \left( \left(p^a, p_k - \kappa^{-1}(\nu^+((-\infty,p_k))) \right] \right) - \mu_{k'}\left( \left(p^a, p_k - \kappa^{-1}(\nu^+((-\infty,p_k))) \right] \right)\right| \leq \varepsilon/2.$$ Thus, for any $k$ in the subsequence, we have $$\hat{\mu} \left( \left(p^a, p_k - \kappa^{-1}(\nu^+((-\infty,p_k))) \right] \right) \geq \varepsilon/2.$$ The above is a contradiction, as the intersection of the corresponding intervals, $(p^a, p_k - \kappa^{-1}(\nu^+((-\infty,p_k))) ]$, over all $k$ is empty.
Now we are ready to prove the upper bound in (\[eq.prop1.target\]).
\[bidasksigns\] In any non-degenerate LTC equilibrium, $p^b<0<p^a$, ${\mathbb P}$-a.s..
We only show that $p^b<0$ holds, the other inequality being very similar. Assume that $p^b\ge0$ on some positive ${\mathbb P}$-probability set $\Omega'\in\mathcal{F}_{N-1}$. We are going to show that this results in a contradiction. First, Corollary \[cor:piecewiseLin\], in Appendix A, implies that, ${\mathbb P}$-a.s., if the agents in state $(s,\alpha)$ post a limit sell order, then we must have: $\sup\limits_{p\in{\mathbb R}} A^{\alpha}(p;p^b) \geq0$. In addition, on $\Omega'$, we have: $\hat{A}^{\alpha}(p^a;p^b)<0$ for all $\alpha\in\tilde{\mathbb{A}}$, as $\xi$ has full support in ${\mathbb R}$ under every ${\mathbb P}^{\alpha}_{N-1}$ (which, in turn, follows from the fact that $\sigma$ is bounded uniformly away from zero). Then, Lemma \[le:simp.to.true.val\] implies that there exists a $\mathcal{F}_{N-1}$-measurable $\bar{p}\geq p^a$, such that, on $\Omega'$, the following holds a.s.: if $\nu^+(\{p^a\})=0$ then $\bar{p}>p^a$, and, in all cases, $$\label{eq.bidasksigns.ubopt}
A^{\alpha}(p;p^b) < 0,\,\,\,\,\,\,\,\,\,\forall p\in[p^a, \bar{p}], \,\,\,\,\forall \alpha\in\tilde{\mathbb{A}}$$ Clearly, it is suboptimal for an agent to post a limit sell order below $\bar{p}$. However, an agent’s strategy only needs to be optimal up to a set of ${\mathbb P}$-measure zero, and these sets can be different for different $(s,\alpha)$. Therefore, a little more work is required to obtain the desired contradiction. Consider the set $B\subset \Omega'\times\mathbb{{\mathbb R}}\times\tilde{\mathbb{A}}$: $$B = \left\{(\omega,s,\alpha)\,|\, \hat{q}(s,\alpha)>0,\,\,\hat{p}(s,\alpha)\leq \bar{p} \right\}.$$ This set is measurable with respect to $\mathcal{F}_{N-1}\otimes \mathcal{B}\left(\mathbb{{\mathbb R}}\times\tilde{\mathbb{A}}\right)$, due to the measurability properties of $\hat{q}$ and $\hat{p}$. Notice that, due to the above discussion and the optimality of agents’ actions (cf. Corollary \[cor:piecewiseLin\], in Appendix A), for any $(s,\alpha)\in\mathbb{{\mathbb R}}\times\tilde{\mathbb{A}}$, we have: $${\mathbb P}(\left\{\omega\,|\, (\omega,s,\alpha)\in B \right\}) = 0,$$ and hence $${\mathbb E}_{N-1} \int_{\mathbb{{\mathbb R}}\times\tilde{\mathbb{A}}} \bone_{B}(\omega,s,\alpha) \mu_{N-1}(ds,d\alpha)
= \int_{\mathbb{{\mathbb R}}\times\tilde{\mathbb{A}}} {\mathbb E}_{N-1}\left(\bone_{B}(\omega,s,\alpha) \rho_{N-1}(\omega,s,\alpha)\right) \mu^0_{N-1}(ds,d\alpha)
= 0,$$ where $\rho_{N-1}$ is the Radon-Nikodym density of $\mu_{N-1}$ w.r.t. to the deterministic measure $\mu^0_{N-1}$ (cf. Assumption \[ass:dom.mu\]). The above implies that, ${\mathbb P}_{N-1}$-a.s., $\bone_{B}(\omega,s,\alpha)\rho_{N-1}(\omega,s,\alpha)=0$, for $\mu^0_{N-1}$-a.e. $(s,\alpha)$. Notice also that, for all $(\omega,s,\alpha)\in \Omega'\times\mathbb{{\mathbb R}}\times\tilde{\mathbb{A}}$, $$\bone_{\left\{\hat{p}(s,\alpha)\leq \bar{p} \right\}} \hat{q}^+(s,\alpha) \bone_{B^c} = 0.$$ From the above observations and the condition (\[eq.nuplus.fixedpoint.def\]) in the definition of equilibrium (cf. Definition \[def:equil.def\]), we conclude that, on $\Omega'$, the following holds a.s.: $$\nu^+([p^a,\bar{p}])=0,$$ where $\bar{p}\geq p^a$, and, if $\nu^+(\{p^a\})=0$, then $\bar{p}> p^a$. This contradicts the definition of $p^a$ (recall that $p^a$ is ${\mathbb P}$-a.s. finite, due to non-degeneracy of the LOB).
It only remains to prove the lower bound on $p^b$ in (\[eq.prop1.target\]). Assume that it does not hold. That is, assume that there exists a family of equilibria, with arbitrary small $\Delta t$, and positive ${\mathbb P}$-probability $\mathcal{F}_{N-1}$-measurable sets $\Omega^{\Delta t}$, such that $p^b<-C_0\sqrt{\Delta t}$ on $\Omega^{\Delta t}$. We are going to show that this leads to a contradiction with $p^a>0$. To this end, assume that the agents maximize the simplified objective function, $\hat{A}^{\alpha}$, instead of the true one, $A^{\alpha}$. Then, if $p^b$ is negative enough, the optimal price levels become negative for all $\alpha$. The precise formulation of this is given by the following lemma.
\[gap\] There exists a constant $C_0>0$, s.t., for any small enough $\Delta t$, there exist constants $\epsilon,\delta>0$, s.t., ${\mathbb P}$-a.s., we have $$\hat{A}^{\alpha}(-\delta;x)\ge\epsilon+\sup\limits_{y\ge0}\hat{A}^\alpha(y;x),$$ for all $\alpha\in\tilde{\mathbb{A}}$ and all $x\le-C_0\sqrt{\Delta t}$.
Denote $\bar{\xi}=\xi/\sqrt{\Delta t}$ and consider the random function $$\bar{A}^\alpha(p;x)={{\mathbb E}^\alpha_{N-1}}\left((p-x-\bar{\xi})\bone_{\{\bar{\xi}>p\}}\right).$$ Notice that $$\hat{A}^\alpha(p;x)=\sqrt{\Delta t}\bar{A}^\alpha\left(p/\sqrt{\Delta t}; x/\sqrt{\Delta t}\right),$$ and, hence, we can reformulate the statement of the lemma as follows: there exists a constant $C_0>0$, s.t., for any small enough $\Delta t$, there exist constants $\epsilon,\delta>0$, s.t., ${\mathbb P}$-a.s., we have $$\bar{A}^{\alpha}(-\delta;x)\ge\epsilon+\sup\limits_{y\ge0} \bar{A}^\alpha(y;x),$$ for all $\alpha\in\tilde{\mathbb{A}}$ and all $x\le-C_0$. Notice that $$\begin{gathered}
\bar{A}^\alpha(-\delta;x)-\bar{A}^\alpha(y;x)
= -x{{\mathbb E}^\alpha_{N-1}}\left(\bone_{\{-\delta<\bar{\xi}\le y\}}\right) - {{\mathbb E}^\alpha_{N-1}}\left(\xi\bone_{\{-\delta<\bar{\xi}\le y\}}\right)
- \delta{{\mathbb E}^\alpha_{N-1}}\left(\bone_{\{\bar{\xi}>-\delta\}}\right) - y{{\mathbb E}^\alpha_{N-1}}\left(\bone_{\{\bar{\xi}>y\}}\right)\end{gathered}$$ is non-increasing in $x$, and, hence, such is $\bar{A}^{\alpha}(-\delta;x)-\sup\limits_{y\ge0}\bar{A}^\alpha(y;x)$. Hence, it suffices to prove the above statement for $x=-C_0$. Next, consider the deterministic function $A_\sigma(p;x)$, defined via $$\label{eq.Asigma.def}
A_\sigma(p;x)=\hat{{\mathbb E}}\left((p-x-\sigma\eta_0)\bone_{\{\sigma\eta_0>p\}}\right),$$ where $\eta_0$ is a standard normal random variable on some auxiliary probability space $(\hat{\Omega},\hat{{\mathbb P}})$. It follows from Lemma \[gapproxapplied\] that there exists a function $\varepsilon_2(\cdot)\ge0$, s.t. $\varepsilon_2(\Delta t)\to0$, as $\Delta t\to0$, and, ${\mathbb P}$-a.s., we have $$\left|\bar{A}^\alpha(p;-C_0)-A_{\sigma_{t_{N-1}}}(p;-C_0)\right|\le\varepsilon_2(\Delta t),$$ for all $\alpha\in\tilde{\mathbb{A}}$ and all $p\in{\mathbb R}$. Then, as we can always choose $\Delta t$ small enough, so that $\varepsilon_2(\Delta t)<\epsilon$, the statements of the lemma would follow if we can show that there exist constants $\epsilon,\delta,C_0>0$, s.t., ${\mathbb P}$-a.s., $$A_{\sigma_{t_{N-1}}}(-\delta;-C_0)\ge3\epsilon+\sup\limits_{y\ge0}A_{\sigma_{t_{N-1}}}(y;-C_0)$$ As $\sigma_{t_{N-1}}(\omega)\in[1/C,C]$, ${\mathbb P}$-a.s., it suffices to find $\epsilon,\delta,C_0>0$, s.t. $$A_\sigma(-\delta;-C_0)\ge3\epsilon+\sup\limits_{y\ge0}A_\sigma(y;-C_0), \quad\forall\,\sigma\in[1/C,C].$$ Note that the above inequality does not involve $\omega$ or $\xi$, and it is simply a property of a deterministic function. Notice also that $A_\sigma(p;x)=\sigma A_1\left(p/\sigma;x/\sigma\right)$, with $A_1$ given in (\[eq.Asigma.def\]). Then, if we denote by $F(x)$ and $f(x)$, respectively, the cdf and pdf of a standard normal, we obtain $$A_1(p;x)=(p-x)(1-F(p))-\int_p^{\infty} t f(t)\text{d}t.$$ A straightforward calculation gives us the following useful properties of $A_1$ and $A_\sigma$
\(i) For any $\sigma>0$ and any $x<0$, the function $p\mapsto A_\sigma(p;x)$ has a unique maximizer $p_\sigma(x)$, in particular, it is increasing in $p\le p_\sigma(x)$ and decreasing in $p\ge p_\sigma(x)$.
\(ii) The function $$x\mapsto p_\sigma(x)=\sigma p_1(x/\sigma)=\sigma\left((1-F)/f\right)^{-1}(-x/\sigma)$$ is increasing in $x<0$ and converges to $-\infty$, as $x\to-\infty$.
Then, choosing $C_0$ large enough, so that $p_1(-C_0/C)<0$, ensures $p_\sigma(-C_0)<0$, for all $\sigma\in[1/C,C]$. Setting $\delta=-p_1(-C_0/C)/C$ guarantees that $p_\sigma(-C_0)\le-\delta$, for all $\sigma\in[1/C,C]$. Then, by property (i) above, we have, for all $\sigma\in[1/C,C]$ $$A_\sigma(-\delta;-C_0)>A_\sigma(0;-C_0)=\sup\limits_{y\ge0}A_\sigma(y;-C_0).$$ Finally, as $A_\sigma(-\delta;-C_0)-A_\sigma(0;-C_0)$ is a continuous function of $\sigma\in[1/C,C]$, we can find $\epsilon$, such that $$A_\sigma(-\delta;-C_0)\ge3\epsilon+\sup\limits_{y\ge0}A_\sigma(y;-C_0), \quad\forall\,\sigma\in[1/C,C].$$
Recall that our assumption is that $p^b<-C_0\sqrt{\Delta t}$ holds on a set $\Omega^{\Delta t}$ of positive ${\mathbb P}$-measure. Recall also that $p^a>0$, ${\mathbb P}$-a.s., due to Lemma \[bidasksigns\]. Then, Lemmas \[le:simp.to.true.val\] and \[gap\] imply that there exists $\mathcal{F}_{N-1}$-measurable $\bar{p}\geq p^a$, s.t., on $\Omega^{\Delta t}$, we have a.s.: if $\nu^+(\{p^a\})=0$ then $\bar{p}>p^a$, and, in all cases, $$A^{\alpha}(p;p^b) < \sup_{p'\in{\mathbb R}} A^{\alpha}(p';p^b),\,\,\,\,\,\,\,\,\forall p\in[p^a,\bar{p}],\,\,\,\,\forall \alpha\in\tilde{\mathbb{A}}.$$ It is intuitively clear that posting limit sell orders at the above price levels $p$ must be suboptimal for the agents. However, the above inequality, on its own, does not yield a contradiction, as the agents’ strategies are only optimal up to a set of ${\mathbb P}$-probability zero, and these sets may be different for different states $(s,\alpha)$. To obtain a contradiction with the definition of $p^a$, we simply repeat the last part of the proof of Lemma \[bidasksigns\] (following equation (\[eq.bidasksigns.ubopt\])). This ensures that (\[eq.prop1.target\]) holds and completes the proof of the theorem.
Proof of Theorem \[thm:main.necessary\] {#se:pf.2}
=======================================
Within the scope of this proof, we adopt the notation introduced in (\[eq.xi.not\]) and use Notational Convention \[not:shift\] (i.e. we measure the LOB, the expected execution prices, and the demand, relative to $p^0$, but keep the same variables’ names). Assume that the statement of the theorem does not hold: i.e., there exists $\alpha_0\in\tilde{\mathbb{A}}$, such that $\tilde{p}^0$ is not a martingale under ${\mathbb P}^{\alpha_0}$. Then, there exists $s\in[0,T)$, s.t., with positive probability ${\mathbb P}^{\alpha_0}$, we have $$\tilde{{\mathbb E}}^{\alpha_0}_{s}\tilde{p}^0_T \neq \tilde{p}^0_{s}.$$ Without loss of generality, we assume that there exists a constant $\delta>0$ and a set $\Omega'\in\mathcal{F}_{s}$, having positive probability ${\mathbb P}^{\alpha_0}$ (and hence ${\mathbb P}$), s.t., for all random outcomes in $\Omega'$, we have $$\label{eq.thm1.delta.def}
\tilde{{\mathbb E}}^{\alpha_0}_{s}(\tilde{p}^0_T - \tilde{p}^0_{s})\geq \delta$$ (the case of negative values is analogous). Next, we fix an arbitrary $\Delta t$ from a given family and consider the associated non-degenerate LTC equilibrium.
\[le:necessary.1\] There exists a deterministic function $\varepsilon(\cdot)\geq0$, s.t. $\varepsilon(\Delta t)\rightarrow0$, as $\Delta t\rightarrow0$, and, for any small enough $\Delta t>0$, there exists $n=0,\ldots,N - 3$ and $\Omega''\in\mathcal{F}_n$, s.t. ${\mathbb P}^{\alpha_0}_n(\Omega'')>0$ and the following holds on $\Omega''$ $${\mathbb P}^{\alpha_0}_{n+2} \left( {\mathbb E}^{\alpha_0}_{n+3} \left(p^0_N - p^0_{n+3} \right) \leq \delta/2\right) \leq \varepsilon(\Delta t).$$
The proof follows from Assumption \[ass:main.mu.cont.strong\]. Consider $t=t'=s$ and $t'' = t_{n+2}$. Then, Assumption \[ass:main.mu.cont.strong\] implies $$\tilde{{\mathbb P}}^{\alpha_0}_{s} \left( \left|\tilde{{\mathbb E}}^{\alpha_0}_{t_{n+2}} \int_{s}^T \mu^{\alpha_0}_u du - \tilde{{\mathbb E}}^{\alpha_0}_{s} \int_{s}^T \mu^{\alpha_0}_u du\right| \geq \varepsilon(\Delta t)\right) \leq \varepsilon(\Delta t)$$ on $\Omega'$, a.s.. Notice also that $$\tilde{{\mathbb E}}^{\alpha_0}_{s} (\tilde{p}^0_T - \tilde{p}^0_{s})
= \tilde{{\mathbb E}}^{\alpha_0}_{s} \int \limits_{s}^T \mu^{\alpha_0}_u \text{d}u.$$ Then, assuming that $\varepsilon(\Delta t)$ is small enough and recalling (\[eq.thm1.delta.def\]), we obtain $$\tilde{{\mathbb P}}^{\alpha_0}_{s} \left( \tilde{{\mathbb E}}^{\alpha_0}_{t_{n+2}} \int_{s}^T \mu^{\alpha_0}_u du \leq 3\delta/4 \right) \leq \varepsilon(\Delta t),$$ on $\Omega'$. Therefore, there exists a set $\Omega''\in\mathcal{F}_{s}\subset\mathcal{F}_{t_{n}}$, s.t. $\tilde{{\mathbb P}}^{\alpha_0}_{t_{n}}(\Omega'')>0$ and $$\tilde{{\mathbb E}}^{\alpha_0}_{t_{n+2}} \int_{s}^T \mu^{\alpha_0}_u du \geq 3\delta/4,$$ on $\Omega''$. Next, we choose $t=s$, $t'=t_{n+2}$, $t'' = t_{n+3}$, and use Assumption \[ass:main.mu.cont.strong\], to obtain $$\tilde{{\mathbb P}}^{\alpha_0}_{t_{n+2}} \left( \left|\tilde{{\mathbb E}}^{\alpha_0}_{t_{n+3}} \int_{s}^T \mu^{\alpha_0}_u du - \tilde{{\mathbb E}}^{\alpha_0}_{t_{n+2}} \int_{s}^T \mu^{\alpha_0}_u du\right| \geq \varepsilon(\Delta t)\right) \leq \varepsilon(\Delta t),$$ on $\Omega''$, a.s.. Assuming that $\varepsilon(\Delta t)$ is small enough and using the last two inequalities, we obtain $$\tilde{{\mathbb P}}^{\alpha_0}_{t_{n+2}} \left( \tilde{{\mathbb E}}^{\alpha_0}_{t_{n+3}} \int_{s}^T \mu^{\alpha_0}_u du \leq \delta/2 \right) \leq \varepsilon(\Delta t).$$ Finally, due to Assumption \[ass:A.alpha\], and as $\Delta t$ is small, we can replace $\int_{s}^T \mu^{\alpha_0}_u du$ by $\int_{t_{n+3}}^T \mu^{\alpha_0}_u du$, and $\delta/2$ by $\delta/4$, in the above equation. This completes the proof of the lemma.
Using the strategy at which the agent in state $(1,\alpha_0)$ waits until the last moment $n=N$, we conclude that the process $(\lambda^a_n(\alpha_0) + p^0_n)$ must be a supermartingale under ${\mathbb P}^{\alpha_0}$. More precisely, due to the definition of an optimal strategy, we have, ${\mathbb P}$-a.s. $$\lambda^a_{n+2}(\alpha_0) \geq {\mathbb E}^{\alpha_0}_{n+2} \lambda^a_N(\alpha_0) + {\mathbb E}^{\alpha_0}_{n+2}\left( {\mathbb E}^{\alpha_0}_{n+3}(p^0_N - p^0_{n+3}) + \xi_{n+3} \right).$$ Recall that $\lambda^a_N(\alpha_0) = p^b_N$ and, due to Theorem \[le:main.zeroTermSpread\] (more precisely, it follows from the proof of the theorem), there exists a constant $C_0>0$, s.t., for all small enough $\Delta t>0$, the following holds ${\mathbb P}$-a.s. $$-C_0\sqrt{\Delta t}\le p^{b}_N <0<p^{a}_N\le C_0\sqrt{\Delta t}.$$ Thus, we have, ${\mathbb P}$-a.s. $$\label{eq.necessary.lambdab.est.1}
\lambda^a_{n+2}(\alpha_0) \geq -C_0\sqrt{\Delta t}
+ {\mathbb E}^{\alpha_0}_{n+2}\left( {\mathbb E}^{\alpha_0}_{n+3}(p^0_N - p^0_{n+3}) \right)
+ {\mathbb E}^{\alpha_0}_{n+2} \xi_{n+3}.$$ Due to Assumption \[ass:A.alpha\], we have, ${\mathbb P}$-a.s. $${\mathbb E}^{\alpha_0}_{n+2} \xi_{n+3} \leq C\Delta t,
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
\left|{\mathbb E}^{\alpha_0}_{n+3}(p^0_N - p^0_{n+3})\right|
\leq C T,$$ and, hence, $$\lambda^a_{n+2}(\alpha_0) \geq -C_0\sqrt{\Delta t} + CT + C\Delta t.$$ In addition, making use of Lemma \[le:necessary.1\], we conclude that, for any small enough $\Delta t$, there exist $n=0,\ldots,N - 2$ and $\Omega''\in\mathcal{F}_n$, s.t. ${\mathbb P}^{\alpha_0}_n(\Omega'')>0$ and $${\mathbb P}^{\alpha}_{n+2} \left( {\mathbb E}^{\alpha}_{n+3} \left(p^0_N - p^0_{n+3} \right) \leq \delta/2\right) \leq \varepsilon(\Delta t),\,\,\,\,\,\text{on}\,\,\Omega''.$$ Using (\[eq.necessary.lambdab.est.1\]) and assuming that $\Delta t$ is small enough, we obtain $$\lambda^a_{n+2}(\alpha_0) \geq \delta/4,\,\,\,\,\,\,\,\text{on}\,\,\Omega''.$$ Next, Corollary \[cor:piecewiseLin\], in Appendix A, implies that, ${\mathbb P}$-a.s., $$p^b_{n+1}\geq {\mathbb E}^{\alpha_0}_{n+1}\left(\lambda^a_{n+2}(\alpha_0)+\xi_{n+2}\big\vert \xi_{n+2}<p^b_{n+1}\right).$$ Thus, on $\Omega''$, we obtain $$\label{eq.necessary.pa.est.1}
p^b_{n+1} - {\mathbb E}^{\alpha_0}_{n+1}\left(\xi_{n+2}\big\vert\xi_{n+2}<p^b_{n+1}\right) \geq \delta /4.$$
The following lemma shows that, for any number $p$, the conditional expectation of the fundamental price increment, ${\mathbb E}^{\alpha_0}_{n+1}(\xi_{n+2}|\xi_{n+2}<p)$, approaches $p$ as the size of the time interval vanishes. This result follows from Lemma \[le:necessary.marginal.maximum\].
\[le:necessary.2\] There exists a constant $C_3>0$, s.t., for all small enough $\Delta t>0$, and for any $t\in[0,T-\Delta t]$, the following holds ${\mathbb P}$-a.s. $$\sup\limits_{p\le0}\left| p - \tilde{{\mathbb E}}^{\alpha_0}_t\left(\tilde{p}^0_{t+\Delta t} - \tilde{p}^0_t\,\big\vert\, \tilde{p}^0_{t+\Delta t}-\tilde{p}^0_t<p\right) \right| \leq C_3 \sqrt{\Delta t}.$$
Fix $t$ and $\Delta t>0$ and consider the evolution of $\tilde{p}^0_s$, for $s\in[t,t+\Delta t]$, under ${\mathbb P}^{\alpha_0}_t$ $$\tilde{p}^0_{s} - \tilde{p}^0_t = \int_t^s \mu^{\alpha_0}_u du + \int_t^s \sigma_u dW^{\alpha_0}_u,$$ where $W^{\alpha_0}$ is a Brownian motion under ${\mathbb P}^{\alpha_0}$. Rescaling by $\sqrt{\Delta t}$, we obtain $$(\tilde{p}^0_{s} - \tilde{p}^0_t)/\sqrt{\Delta t} = X_{(s-t)/\Delta t},
\quad X_s = \int_0^s \hat{\mu}_u du + \int_0^s \hat{\sigma}_u d\hat{W}_u,
\quad s\in[0,1],$$ with $$\hat{\mu}_s = \sqrt{\Delta t} \, \mu^{\alpha_0}_{t+s\Delta t},
\quad \hat{\sigma}_s = \sigma_{t+s\Delta t},
\quad \hat{W}_s = \frac{1}{\sqrt{\Delta t}} \left(W^{\alpha_0}_{t+s\Delta t} - W^{\alpha_0}_t\right),
\quad s\in[0,1].$$ Notice that the above processes are adapted to the new filtration $\hat{\mathbb{F}}$, with $\hat{\mathcal{F}}_s = \tilde{\mathcal{F}}_{t+s\Delta t}$, and, ${\mathbb P}$-a.s., under $\tilde{{\mathbb P}}^{\alpha_0}_t$, $\hat{W}$ is a Brownian motion with respect to $\hat{\mathbb{F}}$. Next, due to Assumptions \[ass:sigma\] and \[ass:main.L2.strong\], for any small enough $\Delta t>0$, ${\mathbb P}$-a.s., the dynamics of $(-X_s)$, under $\tilde{{\mathbb P}}^{\alpha_0}_t$, satisfy all the assumptions of Lemma \[le:necessary.marginal.maximum\]. As a result, we obtain $$\tilde{{\mathbb P}}^{\alpha_0}_t(X_1 < -x-z)
\leq C_1 e^{-z} \tilde{{\mathbb P}}^{\alpha_0}_t(X_1 < -x),
\,\,\,\,\,\,\,\,\,\,\,\,\,\forall x,z\geq0.$$ Finally, we notice that $$\sup\limits_{p\le0}\left| p - \tilde{{\mathbb E}}^{\alpha_0}_t\left(\tilde{p}^0_{t+\Delta t} - \tilde{p}^0_t\big\vert \tilde{p}^0_{t+\Delta t}-\tilde{p}^0_t<p\right)\right|
= \sqrt{\Delta t} \sup\limits_{p\le0}\left| p - \tilde{{\mathbb E}}^{\alpha_0}_t\left(X_1\big\vert X_1<p\right)\right|$$ $$= \sqrt{\Delta t} \sup\limits_{p\le0}\left| p - \frac{\int_{-p}^{\infty} x \,d\,\tilde{{\mathbb P}}^{\alpha_0}_t(X_1 < -x) }{\tilde{{\mathbb P}}^{\alpha_0}_t(X_1 < p)} \right|
= \sqrt{\Delta t} \sup\limits_{p\le0}\left| \frac{\int_{0}^{\infty} \tilde{{\mathbb P}}^{\alpha_0}_t(X_1 < p - z) dz}{\tilde{{\mathbb P}}^{\alpha_0}_t(X_1 < p)} \right|
\leq C_1 \sqrt{\Delta t},$$ which completes the proof of the lemma.
Using (\[eq.necessary.pa.est.1\]) and Lemma \[le:necessary.2\], we conclude that, for all small enough $\Delta t$, we have: $p^b_{n+1} > 0$ on $\Omega''$, ${\mathbb P}$-a.s.. In addition, Corollary \[cor:piecewiseLin\], in Appendix A, implies that, for any $\alpha\in\tilde{\mathbb{A}}$, the following holds ${\mathbb P}$-a.s. $$\lambda^a_{n+1}(\alpha) \geq p^b_{n+1}.$$ Next, with a slight abuse of notation (similar notation was introduced in the proof of Proposition \[le:main.zeroTermSpread\]), we consider the simplified objective of an agent who posts a limit sell order at the ask price $p^a_n$ $$\hat{A}^{\alpha}(p^a_n;\lambda^a_{n+1}) = {\mathbb E}^{\alpha}_n\left( p^a_n - \lambda^a_{n+1} - \xi_{n+1} \,|\, \xi_{n+1} > p^a_n \right).$$ The above estimates imply that, on $\Omega''$, we have, ${\mathbb P}$-a.s. $$\label{eq.necessary.simpObj.neg}
\hat{A}^{\alpha}(p^a_n;\lambda^a_{n+1}) \leq
{\mathbb E}^{\alpha}_n\left( p^a_n - \xi_{n+1} \,|\, \xi_{n+1} > p^a_n \right)
- {\mathbb E}^{\alpha}_n\left( p^b_{n+1}\bone_{\Omega''} \,|\, \xi_{n+1} > p^a_n \right)
< 0,\,\,\,\,\,\,\,\,\,\,\,\forall \alpha\in\tilde{\mathbb{A}}.$$ To obtain the last inequality in the above, we recall that $\Omega''\in\mathcal{F}_n$ and, ${\mathbb P}$-a.s., $\bone_{\Omega''}{\mathbb P}_n(\Omega\setminus\Omega'')=0$, $p^b_{n+1} > 0$ on $\Omega''$, and ${\mathbb P}^{\alpha}_n(\xi_{n+1} > p^a_n)>0$, for all $\alpha\in\tilde{\mathbb{A}}$. Next, repeating the proof of Lemma \[le:simp.to.true.val\] (and using the fact that $\lambda^a_{n+1}$ is absolutely bounded, as shown in Corollary \[prop:main.smallspread\]), we conclude that, ${\mathbb P}$-a.s., either $\nu^+_n(\{p^a_n\})>0$, or we have $$\left\vert A^{\alpha}(p;\lambda^a_{n+1}) - \hat{A}^{\alpha}(p^a_n;\lambda^a_{n+1})\right\vert\to0,$$ as $p\downarrow p^a$, uniformly over all $\alpha\in\tilde{\mathbb{A}}$, where we introduce the true objective, $$A^\alpha(p;\lambda^a_{n+1})={\mathbb E}^\alpha_{n}\left(\left(p-\lambda^a_{n+1}-\xi_{n+1}\right)\bone_{\{D^+_{n+1}(p-\xi_{n+1})>\nu^+_n((-\infty,p))\}}\right).$$ This convergence, along with (\[eq.necessary.simpObj.neg\]), implies that there exists a $\mathcal{F}_{n}$-measurable $\bar{p}\geq p^a_n$, such that, on $\Omega''$, the following holds ${\mathbb P}$-a.s.: if $\nu^+_n(\{p^a_n\})=0$ then $\bar{p}>p^a_n$, and, in all cases, $$A^{\alpha}(p;\lambda^a_{n+1}) < 0,\,\,\,\,\,\,\,\,\,\forall p\in[p^a_n, \bar{p}], \,\,\,\,\forall \alpha\in\tilde{\mathbb{A}}.$$ Finally, we repeat the last part of the proof of Lemma \[bidasksigns\] (following equation (\[eq.bidasksigns.ubopt\])), to obtain a contradiction with the definition of $p^a_n$, and complete the proof of the theorem. The last argument also shows that, when $\Delta t$ is small enough, it becomes suboptimal for the agents to post limit sell orders, as the expected relative profit from this action becomes negative, causing the market to degenerate.
Summary and future work {#se:conclusion}
=======================
In this paper, we present a new framework for modeling market microstructure, which does not require the existence of a designate market maker, and in which the LOB arises endogenously, as a result of equilibrium between multiple strategic players (aka agents). This framework is based on a continuum-player game. It closely approximates the mechanics of an auction-style exchange, so that, in particular, it can be used to analyze the liquidity effects of changes in the rules of the exchange. We use the proposed framework to study the liquidity effects of high trading frequency. In particular, we demonstrate the dual nature of high trading frequency. On the one hand, in the absence of a bullish or bearish signal about the asset, the higher trading frequency improves the efficiency of the market. On the other hand, at a sufficiently high trading frequency, even a very small trading signal may amplify the adverse selection effect, creating a disproportionally large change in the LOB, which is interpreted as an endogenous liquidity crisis.
The present article raises many questions for further research. Notice that our main results are of a qualitative nature: they demonstrate the general behavior of the LOB, as a function of trading frequency, but do not immediately allow for computations. It would also be interesting to establish quantitative results. In particular, we would like to construct an equilibrium in a more realistic, and more concrete, model than the one used in Section \[se:examples\]. Such a model would allow for heterogeneous beliefs, and it would prescribe the specific sources of information (i.e., relevant market factors) used by the agents to form their beliefs. A model of this type could be calibrated to market data and used to study the effects of changes in relevant market parameters on the LOB. Finally, it would be interesting to develop a continuous time version of the proposed framework, in order to better capture the present state of the markets, where the trading frequency is not restricted. All these questions are the subject of our follow-up paper [@GaydukNadtochiy2].
Appendix A
==========
This section contains several useful technical results on the representation of the value function of an agent in the proposed game. Notice that (\[eq.stateProc.def\]) and (\[eq.intro.Jm.def\]) imply that, if $\nu$ is admissible, then, for any $(\alpha,m,p,q,r)$, we have, ${\mathbb P}$-a.s. $$\left|J^{(p,q,r)}\left(m,s,\alpha,\nu\right) - J^{(p,q,r)}\left(m,s',\alpha,\nu\right)\right|
\leq |s-s'|\, {\mathbb E}^{\alpha}_m |p^a_N| \vee |p^b_N|,
\,\,\,\,\,\,\,\,\,\,\forall s,s'\in\mathbb{{\mathbb R}}$$ This implies that every $J^{(p,q,r)}\left(m,\cdot,\alpha,\nu\right)$ and $V^{\nu}_m(\cdot,\alpha)$ has a continuous modification under ${\mathbb P}$. Thus, whenever $\nu$ is admissible, we define the value function of an agent as the aforementioned continuous modification of the left hand side of (\[eq.gen.Val.randField\]).
\[le:DPP\] Assume that an optimal control exists for an admissible LOB $\nu$. Assume also that, for any $\alpha\in\mathbb{A}$, the associated value function $V^{\nu}_n(\cdot,\alpha)$, defined in (\[eq.gen.Val.randField\]), is measurable with respect to $\mathcal{F}_n\otimes\mathcal{B}({\mathbb R})$. Then, it satisfies the following Dynamic Programming Principle.
- For $n=N$ and all $(s,\alpha)\in\mathbb{S}$, we have, ${\mathbb P}$-a.s. $$\label{eq.het.VN}
V^{\nu}_N(s,\alpha) = s^+ p^b_N - s^- p^a_N$$
- For all $n=N-1,\ldots,0$ and all $(s,\alpha)\in\mathbb{S}$, we have $$V^{\nu}_n(s,\alpha) = \text{esssup}_{p,q,r}\left\{\bone_{\left\{r_n=0\right\}}{\mathbb E}_n^{\alpha} \left( V^{\nu}_{n+1}\left(s,\alpha\right)
+ \left(q_n p_n + V^{\nu}_{n+1}\left(s-q_n,\alpha\right) - V^{\nu}_{n+1}\left(s,\alpha\right)\right)\cdot
\right.\right.$$ $$\label{eq.het.Vn}
\left.\left.
\cdot\left( \bone_{\left\{q_n\geq0,\,D^+_{n+1}(p_n) > \nu^+_n((-\infty,p_n)) \right\}} + \bone_{\left\{q_n<0,\,D^-_{n+1}(p_n) > \nu^-_n((p_n,\infty)) \right\}} \right) \right)\right.$$ $$\left.
+ \bone_{\left\{r_n=1\right\}} \left( q^+_n p^b_n - q^-_n p^a_n + {\mathbb E}_n^{\alpha} V^{\nu}_{n+1}\left(s-q_n,\alpha\right) \right)
\right\},$$ where the essential supremum is taken under ${\mathbb P}$, over all admissible controls $(p,q,r)$.
The most important step is to show that, for all $n=0,\ldots N-1$ and $(s,\alpha)\in\mathbb{S}$, $$\label{eq.DPP.aux.1}
V^{\nu}_n(s,\alpha) = \text{esssup}_{p,q,r}
{\mathbb E}^{\alpha}_n \left( V^{\nu}_{n+1}\left(S^{n,s,(p,q,r)}_{n+1},\alpha\right)
- g^{\nu}_{n}\left(p_n,q_n,r_n,D_{n+1}\right) \right),$$ where the essential supremum is taken under ${\mathbb P}$, over all admissible controls $(p,q,r)$, and $$g^{\nu}_{n}\left(p_n,q_n,r_n,D_{n+1}\right) = \left(p_n\bone_{\left\{ r_n = 0\right\}} + p^a_n\bone_{\left\{ r_n = 1, q_n <0\right\}} + p^b_n\bone_{\left\{ r_n = 1, q_n >0\right\}} \right) \Delta S^{n,s,(p,q,r)}_{n+1}$$ does not depend on $s$. Assume that $J^{(p,q,r)}\left(n,\cdot,\alpha,\nu\right)$ is a continuous modification of the objective function. Notice that, for all $m\leq k \leq n$, we have, ${\mathbb P}$-a.s. $${\mathbb E}^{\alpha}_k J^{(p,q,r)}\left(n,S_n^{m,s,(p,q,r)},\alpha,\nu\right)
= J^{(p,q,r)}\left(k,S_k^{m,s,(p,q,r)},\alpha,\nu\right)
+ {\mathbb E}^{\alpha}_k \sum_{j=k}^{n-1} g^{\nu}_{j}\left(p_j,q_j,r_j,D_{j+1}\right)$$ Notice also that, for any $(p,q,r)$ we have, ${\mathbb P}$-a.s.: $J^{(p,q,r)}\left(m,s,\alpha,\nu\right) \leq V_m^\nu(s,\alpha)$, for all $s\in\mathbb{S}$. Let us show that the left hand side of (\[eq.DPP.aux.1\]) is less than its right hand side $$V_m^\nu(s,\alpha)
=\text{essup}_{p,q,r} J^{(p,q,r)}\left(m,S_m^{m,s,(p,q,r)},\alpha,\nu\right)$$ $$=\text{essup}_{p,q,r} {\mathbb E}^{\alpha}_m \left( J^{(p,q,r)}\left(m+1,S_{m+1}^{m,s,(p,q,r)},\alpha,\nu\right)
- g^{\nu}_{m}\left(p_m,q_m,r_m,D_{m+1}\right) \right)$$ $$\leq \text{essup}_{p,q,r} {\mathbb E}^{\alpha}_m \left( V^{\nu}_{m+1}\left(S_{m+1}^{m,s,(p,q,r)},\alpha\right)
- g^{\nu}_{m}\left(p_m,q_m,r_m,D_{m+1}\right) \right)$$ Next, we show that the right hand side of (\[eq.DPP.aux.1\]) is less than its left hand side. For any $(p,q,r)$, we have, ${\mathbb P}$-a.s. $${\mathbb E}^\alpha_m \left(V^\nu_{m+1}\left(S_{m+1}^{m,s,(p,q,r)},\alpha\right)
- g^{\nu}_{m}\left(p_m,q_m,r_m,D_{m+1}\right)\right)$$ $$={\mathbb E}^\alpha_m \left(J^{(\hat{p},\hat{q},\hat{r})}\left(m+1,S_{m+1}^{m,s,(p,q,r)},\alpha,\nu\right)
- g^{\nu}_{m}\left(p_m,q_m,r_m,D_{m+1}\right)\right)
= J^{(\tilde{p},\tilde{q},\tilde{r})}\left(m,s,\alpha,\nu\right)
\leq V^{\nu}_m(s,\alpha),$$ where $(\tilde{p}_n,\tilde{q}_n,\tilde{r}_n)$ coincide with $(\hat{p}_n,\hat{q}_n,\hat{r}_n)$, for $n\geq m+1$, while they are equal to $(p_m,q_m,r_m)$, for $n=m$. The proof is completed easily by plugging the dynamics of the state process, (\[eq.stateProc.def\]), into (\[eq.DPP.aux.1\]).
The following corollary provides a more explicit recursive formula for the value function and optimal control. In particular, it states that the value function of an agent at any time remains linear in $s$, in both positive and negative half lines (with possibly different slopes).
\[cor:piecewiseLin\] Assume that an admissible LOB $\nu$ has an optimal control $(\hat{p},\hat{q},\hat{r})$. Then, for any $(s,\alpha)\in\mathbb{S}$, the following holds ${\mathbb P}$-a.s., for all $n=0,\ldots,N-1$
1. $V^{\nu}_n(s,\alpha) = s^+ \lambda^a_n(\alpha) - s^- \lambda^b_n(\alpha)$, with some adapted processes $\lambda^a(\alpha)$ and $\lambda^b(\alpha)$, such that $\lambda^a_N(\alpha) = p^b_N$ and $\lambda^b_N(\alpha) = p^a_N$;
2. $p^a_n \geq {\mathbb E}_{n}^{\alpha} \left( \lambda^a_{n+1}(\alpha) \right)$ and $p^b_n \leq {\mathbb E}_{n}^{\alpha} \left( \lambda^b_{n+1}(\alpha) \right)$;
3. if, for some $p\in{\mathbb R}$, ${\mathbb P}^{\alpha}_n\left(D^+_{n+1}(p) > \nu^+_{n}((-\infty,p))\right)>0$, then $$\label{eq.het.NoPred.1}
p \leq {\mathbb E}_{n}^{\alpha} \left( \lambda^b_{n+1}(\alpha) \,|\, D^+_{n+1}(p) > \nu^+_{n}((-\infty,p)) \right);$$
4. if, for some $p\in{\mathbb R}$, ${\mathbb P}^{\alpha}_n\left(D^-_{n+1}(p) > \nu^-_{n}((p,\infty))\right)>0$, then $$\label{eq.het.NoPred.2}
p \geq {\mathbb E}_{n}^{\alpha} \left( \lambda^a_{n+1}(\alpha) \,|\, D^-_{n+1}(p) > \nu^-_{n}((p,\infty)) \right);$$
5. for all $s>0$,
- $\lambda^a_n(\alpha) = \max\left\{ p^b_n,
{\mathbb E}^{\alpha}_n \lambda^a_{n+1}(\alpha) + \left(\sup_{p\in{\mathbb R}} {\mathbb E}^{\alpha}_n \left( \left( p - \lambda^a_{n+1}(\alpha) \right) \bone_{\left\{ D^+_{n+1}(p) > \nu^+_{n}((-\infty,p)) \right\}} \right)\right)^+ \right\}$,
- if $\hat{q}_n(s,\alpha)\neq 0$ and $\hat{r}_n(s,\alpha)=0$, then $$\lambda^a_n(\alpha) =
{\mathbb E}^{\alpha}_n \lambda^a_{n+1}(\alpha) + \sup_{p\in{\mathbb R}} {\mathbb E}^{\alpha}_n \left( \left( p - \lambda^a_{n+1}(\alpha) \right) \bone_{\left\{ D^+_{n+1}(p) > \nu^+_{n}((-\infty,p)) \right\}} \right),$$ and $p=\hat{p}_n(s,\alpha)$ attains the above supremum,
- if $\hat{q}_n(s,\alpha) = 0$ and $\hat{r}_n(s,\alpha)=0$, then $\lambda^a_n(\alpha) = {\mathbb E}^{\alpha}_n \lambda^a_{n+1}(\alpha)$,
- if $\hat{r}_n(s,\alpha)=1$, then $\lambda^a_n(\alpha) = p^b_n$;
6. for all $s<0$,
- $\lambda^b_n(\alpha) = \min\left\{ p^a_n,
{\mathbb E}^{\alpha}_n \lambda^b_{n+1}(\alpha) - \left(\sup_{p\in{\mathbb R}} {\mathbb E}^{\alpha}_n \left( \left( \lambda^b_{n+1}(\alpha) - p \right) \bone_{\left\{ D^-_{n}(p) > \nu^-_{n-1}((p,\infty)) \right\}} \right)\right)^+ \right\}$,
- if $\hat{q}_n(s,\alpha)\neq 0$ and $\hat{r}_n(s,\alpha)=0$, then $$\lambda^b_n(\alpha) =
{\mathbb E}^{\alpha}_n \lambda^b_{n+1}(\alpha) - \sup_{p\in{\mathbb R}} {\mathbb E}^{\alpha}_n \left( \left( \lambda^b_{n+1}(\alpha) - p \right) \bone_{\left\{ D^-_{n}(p) > \nu^-_{n-1}((p,\infty)) \right\}} \right),$$ and $p=\hat{p}_n(s,\alpha)$ attains the above supremum,
- if $\hat{q}_n(s,\alpha) = 0$ and $\hat{r}_n(s,\alpha)=0$, then $\lambda^b_n(\alpha) = {\mathbb E}^{\alpha}_n \lambda^b_{n+1}(\alpha)$,
- if $\hat{r}_n(s,\alpha)=1$, then $\lambda^b_n(\alpha) = p^a_n$.
Let us plug the piecewise-linear form of the value function into (\[eq.het.Vn\]) $$V^{\nu}_n(s,\alpha) = \text{esssup}_{p,q,r}\left\{\bone_{\left\{r_n=0\right\}}
\left( s^+ {\mathbb E}_n^{\alpha} \lambda^a_{n+1}(\alpha) - s^- {\mathbb E}_n^{\alpha} \lambda^b_{n+1}(\alpha)
\right.\right.$$ $$\left.\left.
+ {\mathbb E}_n^{\alpha} \left( \left(q_n p_n + (s-q_n)^+ \lambda^a_{n+1}(\alpha) - (s-q_n)^- \lambda^b_{n+1}(\alpha) - s^+ \lambda^a_{n+1}(\alpha) + s^- \lambda^b_{n+1}(\alpha) \right)\cdot
\right.\right.\right.$$ $$\left.\left.\left.
\left( \bone_{\left\{q_n\geq0,\,D^+_{n+1}(p_n) > \nu^+_n((-\infty,p_n)) \right\}} + \bone_{\left\{q_n<0,\,D^-_{n+1}(p_n) > \nu^-_n((p_n,\infty)) \right\}} \right)\right) \right)\right.$$ $$\left.
+ \bone_{\left\{r=1\right\}} \left( q^+_n p^b_n - q^-_n p^a_n + (s-q_n)^+ {\mathbb E}_n^{\alpha} \lambda^a_{n+1}(\alpha) - (s-q_n)^- {\mathbb E}_n^{\alpha} \lambda^b_{n+1}(\alpha) \right)
\right\}$$ First, notice that it suffices to consider the essential supremum over all random variables $(p_n,q_n,r_n)$.[^16] Moreover, the essential supremum can be replaced by the supremum over all deterministic $(p_n,q_n,r_n)\in{\mathbb R}^2\times\{0,1\}$. To see the latter, it suffices to assume that the supremum is not attained by the optimal strategy (with positive probability), and construct a superior strategy via the standard measurable selection argument (cf. Corollary 18.27 and Theorem 18.26 in [@Aliprantis]), which results in a contradiction. It is easy to see that, for any fixed $(p_n,s,r_n)$, the above function is piece-wise linear in $q_n$, with the slope changing at $q_n=0$ and $q_n=s$. Hence, for a finite maximum to exists, the slope of this function must be nonnegative, at $q_n\rightarrow-\infty$, and non-positive, at $q_n\rightarrow \infty$. This must hold for any $(p_n,r_n,s)$, to ensure that the value function of an agent is finite: otherwise, an agent can scale up her position to increase the value function arbitrarily. Considering $r_n=1$, we obtain condition 2 of the corollary. The case $r_n=0$ yields conditions 3 and 4. Notice also that the maximum of the aforementioned function is always attained at $q_n=0$ or $q_n=s$. Considering all possible cases: $r_n=0,1$, $q_n=0,s$, $s=0$, $s>0$ and $s<0$ – we obtain the recursive formulas for $\lambda^a_n$ and $\lambda^b_n$ (i.e., conditions 5 and 6 of the corollary). In addition, as the optimal $q_n$ takes values $0$ and $s$, it is easy to see that the piece-wise linear structure of the value function in $s$ is propagated backwards, and, hence, condition 1 of the corollary holds.
It is also useful to have a converse statement.
\[cor:piecewiseLin.verif\] Consider an admissible LOB $\nu$ and admissible control $(\hat{p},\hat{q},\hat{r})$, such that $\hat{q}_n(s,\alpha)\in\left\{ 0,s\right\}$. Assume that, for any $\alpha\in\mathbb{A}$ and any $n=0,\ldots,N$, there exists a progressively measurable random function $V^{\nu}_{\cdot}(\cdot,\alpha)$, such that, for any $s\in\mathbb{{\mathbb R}}$, ${\mathbb P}$-a.s., $(\hat{p},\hat{q},\hat{r},V^{\nu})$ satisfy the conditions 1–6 of Corollary \[cor:piecewiseLin\]. Then, $(\hat{p},\hat{q},\hat{r})$ is an optimal control for the LOB $\nu$.
It suffices to revert the arguments in the proof of Corollary \[cor:piecewiseLin\], and recall that $\hat{q}$ can always be chosen to be equal to $0$ or $s$, without compromising the optimality.
Appendix B
==========
*Proof of Lemma \[gapproxapplied\]*. The following lemma shows that the normalized price increments are close to Gaussian in the conditional $\mathbb{L}^2$ norm.
\[l2conv\] Let Assumptions \[ass:sigma\], \[ass:A.alpha\], \[ass:joint.cond.reg\], \[ass:main.L2.strong\] hold. Then, there exists a deterministic function $\epsilon(\cdot)\ge0$, such that $\epsilon(\Delta t)\to0$, as $\Delta t\to0$, and, ${\mathbb P}$-a.s., for all $\alpha\in\mathbb{A}$ and all $n=1,\ldots,N$, we have $${\mathbb E}^\alpha_{n-1}\left(\left(\xi_n/\sqrt{\Delta t} - \sigma_{t_{n-1}}(W^\alpha_{t_n}-W^\alpha_{t_{n-1}})/\sqrt{\Delta t}\right)^2\right) \le \epsilon(\Delta t).$$
Notice: $\xi_n/\sqrt{\Delta t} - \sigma_{t_{n-1}}(W^\alpha_{t_n}-W^\alpha_{t_{n-1}})/\sqrt{\Delta t}
=\frac{1}{\sqrt{\Delta t}} \int\limits_{t_{n-1}}^{t_n}\mu^\alpha_s \text{d}s
+ \frac{1}{\sqrt{\Delta t}} \int\limits_{t_{n-1}}^{t_n}(\sigma_s-\sigma_{t_{n-1}}) \text{d}W^\alpha_s$. Then, using Assumptions \[ass:A.alpha\], \[ass:main.L2.strong\], and Itô’s isometry, we obtain the statement of the lemma.
The next lemma connects the proximity in terms of $\mathbb{L}^2$ norm and the proximity of expectations of certain functions of random variables. This result would follow trivially from the classical theory, but, in the present case, we require additional uniformity – hence, a separate lemma is needed (whose proof is, nevertheless, quite simple).
\[gapprox\] For any constant $C>1$, there exists a deterministic function $\gamma(\cdot)\ge0$, s.t. $\gamma(\varepsilon)\to0$, as $\varepsilon\to0$, and, for any $\varepsilon>0$, $\sigma\in[1/C,C]$, and any random variables $\eta\sim\mathcal{N}(0,\sigma^2)$ and $\xi$ (the latter is not necessarily Gaussian), satisfying ${\mathbb E}(\xi-\eta)^2\le\varepsilon$, the following holds for all $p\in{\mathbb R}$
- $(|p|\vee 1)\left| {\mathbb P}(\xi>p) - {\mathbb P}(\eta>p) \right|\le\gamma(\varepsilon)$,
- $\left| {\mathbb E}(\xi\bone_{\{\xi>p\}}) - {\mathbb E}(\eta\bone_{\{\eta>p\}}) \right|\le\gamma(\varepsilon)$.
\(ii) Note that $$\left| {\mathbb E}(\xi\bone_{\{\xi>p\}}) - {\mathbb E}(\eta\bone_{\{\eta>p\}}) \right| \le
\left| {\mathbb E}\left((\xi-\eta)\bone_{\{\xi>p\}}\right) \right| + \left|{\mathbb E}\left(\eta(\bone_{\{\xi>p\}}-\bone_{\{\eta>p\}})\right)\right|$$ $$\leq \sqrt{\varepsilon} + \left\Vert\eta\right\Vert_2 \sqrt{{\mathbb P}(\xi>p,\eta\le p) + {\mathbb P}(\xi\le p,\eta>p)},$$ and $${\mathbb P}(\xi>p,\eta\le p) \le {\mathbb P}(p\ge\eta\ge p-\sqrt[3]{\varepsilon}) + {\mathbb P}(|\xi-\eta|>\sqrt[3]{\varepsilon}) \le M\sqrt[3]{\varepsilon}+\frac{{\mathbb E}(\xi-\eta)^2}{(\sqrt[3]{\varepsilon})^2}\le (M+1)\sqrt[3]{\varepsilon},$$ where we used the fact that $\eta$ has a density bounded by a fixed constant $M$. We can similarly show that ${\mathbb P}[\xi\le p,\eta>p]\le(M+1)\sqrt[3]{\varepsilon}$. The resulting estimates yield the statement of the lemma.
Taking $\varepsilon(\Delta t)=\gamma(\epsilon(\Delta t))$ and applying the above lemmas, we get the statement of Lemma \[gapproxapplied\], with $(W^\alpha_{t_n}-W^\alpha_{t_{n-1}})/\sqrt{\Delta t}$ in place of $\eta_0$. Finally, we note that the laws of the two random variables coincide under ${\mathbb P}^{\alpha}_{n-1}$, and the statement depends only on these laws. The last statement of Lemma \[gapproxapplied\] follows from the fact that Lemma \[gapprox\] is stable under analogous substitution.
[cc]{} [ ]{} & [ ]{}\
[cc]{} [ ]{} & [ ]{}\
[^1]: Partial support from the NSF grant DMS-1411824 is acknowledged by both authors.
[^2]: We thank the anonymous referees and the Associate Editor for constructive comments that helped us improve the paper significantly.
[^3]: Address the correspondence to Sergey Nadtochiy, Mathematics Department, University of Michigan, 530 Church Street, Ann Arbor, MI 48109, USA; e-mail: sergeyn@umich.edu.
[^4]: This version: February 13, 2017. First version August 28, 2015.
[^5]: We do not distinguish the “aggressive" limit orders, which are posted at the price level of an opposite limit order, and treat them as market orders. This causes no loss of generality, as the market participants in our setting have a perfect observation of the LOB.
[^6]: This assumption holds, for example, if $\mathcal{F}_N$ is generated by a random element with values in a standard Borel space.
[^7]: Note that, although ${\mathbb P}^{\alpha}$ does not change over time, the conditional distribution of the future demand, as perceived by the agent, changes dynamically, according to the new information received.
[^8]: Note each agent is only allowed to place her limit order at a single price level, at any given time. However, this entails no loss of optimality. Indeed, using the Dynamic Programming Principle derived in Appendix A, one can show, by induction, that, in equilibrium, an agent does not benefit from posting multiple limit orders at the same time. As shown in [@Schmeidler], this is typical for a continuum-player game.
[^9]: In order to ensure the existence of regular conditional probabilities for the discrete time model, we can, for example, assume that $\tilde{\mathcal{F}}_T$ is generated by a random element with values in a standard Borel space.
[^10]: The execution of limit orders simplifies in the chosen ansatz, because the agents on each side of the book (i.e., long or short) post orders at the same prices.
[^11]: In fact, it is not difficult to prove rigorously that, for any $(\alpha,\sigma)$, there exists a unique solution to such a system, provided $\Delta t$ is small enough. We omit this result for the sake of brevity.
[^12]: This is easy to explain intuitively, as the optimal objective values in the first two lines of (\[eq.ex.singleStep.1\]) are of the form $C\sqrt{\Delta t} + \alpha \underline{\underline{O}}(\Delta t)$.
[^13]: In order to ensure the existence of regular conditional probabilities for the discrete time model, we can, for example, assume that $\tilde{\mathcal{F}}_T$ is generated by a random element with values in a standard Borel space.
[^14]: This argument, along with the fact that Definition \[def:optControl\] requires an optimal control to be optimal for *all* $\alpha$, explains why the statement of Theorem \[thm:main.necessary\] holds for *all*, as opposed to $\mu_n$-a.e., $\alpha\in\tilde{\mathbb{A}}$.
[^15]: Recall that everything is measured relative to the fundamental price, according to the Notational Convention \[not:shift\]
[^16]: The admissibility constraint does not cause any difficulties here, as, in the case where $(p_n,q_n,r_n)$ do not attain the supremum, they can be improved, so that $(p_n,q_n)$ increase by no more than a fixed constant.
|
---
bibliography:
- 'pp.bib'
---
[**Measurement of neutrino flux from the primary protonproton fusion process in the Sun with Borexino detector. [^1]**]{} O.Yu.Smirnov$^{\mbox{a}}$ on behalf of the Borexino collaboration: M. Agostini$^{\mbox{b}}$, S. Appel$^{\mbox{b}}$, G. Bellini$^{\mbox{c}}$, J. Benziger$^{\mbox{d}}$, D. Bick$^{\mbox{e}}$, G. Bonfini$^{\mbox{f}}$, D. Bravo$^{\mbox{g}}$, B. Caccianiga$^{\mbox{c}}$, F. Calaprice$^{\mbox{h}}$, A. Caminata$^{\mbox{i}}$, P. Cavalcante$^{\mbox{f}}$, A. Chepurnov$^{\mbox{j}}$, K. Choi$^{\mbox{k}}$, D. D’Angelo$^{\mbox{c}}$, S. Davini$^{\mbox{l}}$, A. Derbin$^{\mbox{m}}$, L. Di Noto$^{\mbox{i}}$, I. Drachnev$^{\mbox{l}}$, A. Empl$^{\mbox{n}}$, A. Etenko$^{\mbox{o}}$, K. Fomenko$^{\mbox{a}}$, D. Franco$^{\mbox{p}}$, F. Gabriele$^{\mbox{f}}$, C. Galbiati$^{\mbox{h}}$, C. Ghiano$^{\mbox{i}}$, M. Giammarchi$^{\mbox{c}}$, M. Goeger-Neff$^{\mbox{b}}$, A. Goretti$^{\mbox{h}}$, M. Gromov$^{\mbox{j}}$, C. Hagner$^{\mbox{e}}$, E. Hungerford$^{\mbox{n}}$, Aldo Ianni$^{\mbox{f}}$, Andrea Ianni$^{\mbox{h}}$, K. Jedrzejczak$^{\mbox{r}}$, M. Kaiser$^{\mbox{e}}$, V. Kobychev$^{\mbox{s}}$, D. Korablev$^{\mbox{a}}$, G. Korga$^{\mbox{f}}$, D. Kryn$^{\mbox{p}}$, M. Laubenstein$^{\mbox{f}}$, B. Lehnert$^{\mbox{t}}$, E. Litvinovich$^{\mbox{o}}$$^{\mbox{u}}$, F. Lombardi$^{\mbox{f}}$, P. Lombardi$^{\mbox{c}}$, L. Ludhova$^{\mbox{c}}$, G. Lukyanchenko$^{\mbox{o}}$$^{\mbox{u}}$, I. Machulin$^{\mbox{o}}$$^{\mbox{u}}$, S. Manecki$^{\mbox{g}}$, W. Maneschg$^{\mbox{v}}$, S. Marcocci$^{\mbox{l}}$, E. Meroni$^{\mbox{c}}$, M. Meyer$^{\mbox{e}}$, L. Miramonti$^{\mbox{c}}$, M. Misiaszek$^{\mbox{r}}$$^{\mbox{f}}$, P. Mosteiro$^{\mbox{h}}$, V. Muratova$^{\mbox{m}}$, B. Neumair$^{\mbox{b}}$, L. Oberauer$^{\mbox{b}}$, M. Obolensky$^{\mbox{p}}$, F. Ortica$^{\mbox{w}}$, K. Otis$^{\mbox{x}}$, L. Pagani$^{\mbox{i}}$, M. Pallavicini$^{\mbox{i}}$, L. Papp$^{\mbox{b}}$, L. Perasso$^{\mbox{i}}$, A. Pocar$^{\mbox{x}}$, G. Ranucci$^{\mbox{c}}$, A. Razeto$^{\mbox{f}}$, A. Re$^{\mbox{c}}$, A. Romani$^{\mbox{w}}$, R. Roncin$^{\mbox{f}}$$^{\mbox{p}}$, N. Rossi$^{\mbox{f}}$, S. Schönert$^{\mbox{b}}$, D. Semenov$^{\mbox{m}}$, H. Simgen$^{\mbox{v}}$, M. Skorokhvatov$^{\mbox{o}}$$^{\mbox{u}}$, A. Sotnikov$^{\mbox{a}}$, S. Sukhotin$^{\mbox{o}}$, Y. Suvorov$^{\mbox{y}}$$^{\mbox{o}}$, R. Tartaglia$^{\mbox{f}}$, G. Testera$^{\mbox{i}}$, J. Thurn$^{\mbox{t}}$, M. Toropova$^{\mbox{o}}$, E. Unzhakov$^{\mbox{m}}$, R.B. Vogelaar$^{\mbox{g}}$, F. von Feilitzsch$^{\mbox{b}}$, H. Wang$^{\mbox{y}}$, S. Weinz$^{\mbox{z}}$, J. Winter$^{\mbox{z}}$, M. Wojcik$^{\mbox{r}}$, M. Wurm$^{\mbox{z}}$, Z. Yokley$^{\mbox{g}}$, O. Zaimidoroga$^{\mbox{a}}$, S. Zavatarelli$^{\mbox{i}}$, K. Zuber$^{\mbox{t}}$, G. Zuzel$^{\mbox{r}}$.
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
**Abstract**
Neutrino produced in a chain of nuclear reactions in the Sun starting from the fusion of two protons, for the first time has been detected in a real-time detector in spectrometric mode. The unique properties of the Borexino detector provided an oppurtunity to disentangle pp-neutrino spectrum from the background components. A comparison of the total neutrino flux from the Sun with Solar luminosity in photons provides a test of the stability of the Sun on the 10$^{5}$ years time scale, and sets a strong limit on the power production in the unknown energy sources in the Sun of no more than 4% of the total energy production at 90% C.L.
Introduction
============
The solar photon luminosity (a total power radiated in form of photons into space) is determined by measuring the total solar irradiance by spacecrafts over the wide subrange of the electromagnetic spectrum, from x-rays to radio wavelengths; it has been accurately monitored for decades. The luminosity $L_{\odot}=3.846\times10^{26}$ W is measured for a precision of 0.4% with the largest uncertainty of about 0.3% due to disagreements between the measurements of different satellite detectors [@Chapman; @SunTotal]. The energy lost by neutrinos adds $L_{\nu}=0.023\cdot L_{\odot}$ to this value [@Bahcall1989]. The solar luminosity constraint on the solar neutrino fluxes can be written as [@LumConstr]:
$$\frac{L_{\odot}}{4\pi(1a.u.)^{2}}=\sum\alpha_{i}\Phi_{i}\label{Lum}$$
where 1 a.u. is the average earth-sun distance, the coefficient $\alpha_{i}$ is the amount of energy provided to the star by nuclear fusion reactions associated with each of the important solar neutrino fluxes, $\Phi_{i}$. The numerical values of the $\alpha$s are determined to an accuracy of $10^{-4}$ and better.
The estimated uncertainty in the luminosity of the Sun corresponds to less than 3% uncertainty in total solar neutrino flux.
The Sun is a weakly variable star, its luminosity has short term fluctuations [@SunTotal; @SolarVars]. The major fluctuation occurs during the eleven-year solar cycle with amplitude of about 0.1%. Long-term solar variability (such as the Maunder minimum in the 16th and 17th century) is commonly beleived to do not exceed the short term variations.
Because of the relation (\[Lum\]) between the solar photon and neutrino luminosity, the measurement of the total neutrino luminosity will provide a test of the stability of the Sun at the time scale of 40000 years [@FR99], the time needed for the radiation born at the center of the Sun to arrive to its surface. Finding a disagreement between $L_{\odot}$ and $L_{\nu}$ would have significant long term enviromental implications, and in the case of an agreement of two measurements it would be possible to limit the unknown sources of the solar energy, different from the known thermonuclear fusion of light elements in the pp-chain and CNO-cycle.
The main neutrino sources in the Sun are the pp- and $^{7}$Be reactions, providing roughly 91 and 7% of the total neutrino flux respectively. Borexino already measured $^{7}$Be neutrino flux with $5\%$ precision [@Be11], but till recent time the pp-neutrino flux was derived in a differential measurement using the data of solar detectors.
Solar pp neutrinos measurement is a critical test of stellar evolution theory, discussion of the physics potential of the pp solar neutrino flux measurement can be found in [@pp_potential; @WhyPP; @Raghavan] (at the moment of the discussion the parameters space for MSW solution was not established yet, thus the authors were giving priority to this part of the physical potential).
A number of projects aiming to perform pp-neutrino detection have been put forward in past two decades, but with all the time passed since the proposals, none of them started the operation facing the technical problems with realization. The principal characteristics of the proposals are presented in table$\:$ \[Tab:Experiments\]. The radiochemical experiments sensitive to the solar pp- neutrinos (SAGE [@SAGE09] and GALLEX [@Gallex]) are not cited in the table, the combined best fit of the radiochemical and other solar experiments gives solar pp-neutrino flux of $(6.0\pm0.8)\times10^{10}$ cm$^{-2}$s$^{-1}$ [@SAGE09] in a good agreement with expected value of $6.0\times(1.000\pm0.006)\times10^{10}$ cm$^{-2}$s$^{-1}$.
A possibility to use ultrapure liquid organic scintillator as a low energy solar neutrino detector for a first time was discussed in [@ppCTF; @ppCTF-2]. The authors come to the conclusion that a liquid scintillator detector with an active volume of 10 tons is a feasible tool to register the solar pp-neutrino if operated at the target level of radiopurity for Borexino and good energy resolution (5% at 200 keV) is achieved.
![\[Borexino\]Schematic view of the Borexino detector.](Figures/BX){width="50.00000%"}
Borexino detector
=================
The Borexino detector consists of a dome-like structure (see Fig.\[Borexino\]), 16 meters in diameter, filled with a mass of 2,400 tons of highly purified water which acts as the shield against the external radioactive emissions of the rocks and the environment that surround the facility. The water buffer acts also as an effective detector of the residual cosmic rays. Within the volume of water a steel sphere is mounted which hosts 2,200 looking inward photomultiplier tubes providing 34% geometrical coverage. On the outer side of the stainless sphere 200 PMTs of the outer muon veto detector are mounted, these PMTs detects the Cherenkov light caused by muons passing through.
The sphere contains one thousand tonnes of pseudocumene. Finally, the innermost core of the facility contains roughly 280 tons of the scintillating liquid bounded within a 100 $\mu$m thick nylon transparent bag with $\sim$4.2 m radius. The water and the pseudocumene buffers, as well as the scintillator itself, have a record-low level of radioactive purity. The energy of each event is measured using light response of the scintillator, and the position of the interaction is determined using timing information from the PMTs. The latter is important for the selection of the innermost cleanest part of the detector within 3 meters radius, as only the internal 100 tonnes of scintillator have the radioactive background low enough to allow the solar neutrino detection, the scintillator layer close to the nylon serves as an active shield against the $\gamma$ originating from the nylon trace radioactive contamination. The threshold of the detector is set as low as possible to exclude triggering from the random dark count of PMTs. The Borexino has excellent energy resolution for its size, this is the result of the high light yield of $\sim$500 p.e./MeV/2000 PMTs. The energy resolution is as low as 5% at 1 MeV.
Data processing
===============
The low-energy range, namely 165-590 keV, of the Borexino experimental spectrum has been recently carefully analyzed with a purpose of the pp-neutrino flux measurement [@PP14]. The data were acquired from January 2012 to May 2013 and correspond to 408 live days of the data taking. These are the data were collected at the beginning of the second phase of Borexino which had started after the additional purification of the liquid scintillator following the calibration campaign of 2010-2011 [@Calib12]. The main backgrounds for the solar neutrino studies were significantly reduced in the Phase 2, the content of $^{85}$Kr is compatible with zero, and background from $^{210}$Bi reduced by a factor 3 to 4 compared to the values observed at the end of the Phase 1 just before the purification.
The experimental spectrum is presented in Fig.1 The main features of the experimental spectrum can be seen in the figure: the main contribution comes from the $^{14}$C decays at low energies (below 200 keV), the monoenergetic peak corresponds to 5.3 MeV $\alpha-$particles from $^{210}$Po decay. The statistics in the first bins used in the analysis is very high, of the order of $5\times10^{5}$events, demanding development of the very precise model for the studies - the allowed systematic precision at low energy part should be comparable to the statistical fluctuations of 0.14%.
Data analysis
-------------
The Borexino spectrum in the low energy range is composed mainly of the events from $\beta$- decays of $^{14}$C present in liquid organic oscillator in trace quantities, its measured abundance with respect to the $^{12}$C is $(2.7\pm0.1)\times10^{-18}$ g/g. The $\beta$ -decay of $^{14}$ C is an allowed ground-state to ground-state ($0^{+}\rightarrow1^{+}$ ) Gamow-Teller transition with an endpoint energy of $E_{0}=156.476\pm0.004$ $\;$ keV.
In Borexino the amount of the active PMTs is high ($\sim$2000), demanding setting of the high acquisition threshold in order to exclude detector triggering from random coincidence of dark count in PMTs: hardware trigger was set at the level of 25 PMTs in coincidence within 30 ns window, providing negligible random events count. The acquisition efficiency corresponding to 25 triggered PMTs is roughly 50% and corresponds to the energy release of $\sim$50 keV. In present analysis, the same as in the “pp”-analysis, the threshold was set at the lowest possible value at $\sim$60 triggered PMTs ($\sim$160 keV). In independent measurement with laser the trigger inefficiency was found to be below $10^{-5}$ for energies above 120 keV [@Brx14].
Energy resolution
-----------------
The most sensitive part of the analysis is the behaviour of the energy resolution with energy. The variance of the signal is smeared by the dark noise of the detector (composed of the dark noise from individual PMTs). In order to account for the dark noise the data were sampled every two seconds forcing randomly fired triggers. Some additional smearing of the signal is expected because of the continuously decreasing number of the PMTs in operation. The amount of live PMTs is followed in real time and we know precisely its distribution, so in principle this additional smearing can be precisely accounted for. It was found that the following approximation works well in the energy region of interest:
$\sigma_{N}^{2}=N(p_{0}-p_{1}v_{1})+N^{2}(v_{T}(N)+v_{f})$,
where $N=N_{0}<f(t)>_{T}$ is average number of working PMTs during the period of the data taking, $f(t)$ is a function describing the amount of working PMTs in time with $f(0)=N_{0}$. The last parameter here is $v_{f}(N)=<f^{2}(t)>_{T}-<f(t)>_{T}^{2}$, it is the variance of the $f(t)$ function over the time period of the data taking.
An additional contribution to the variance of the signal was identified, it is the intrinsic width of the scintillation response. From the simple consideration this contribution reflects the additional variations due to the fluctuations of the delta- electrons production and the energy scale non-linearity. It should scale inversely proportional to the energy loss. Because of the limited range of the sensitivity to this contribution, basically restricted to the very tail of the $^{14}$C spectrum, the precise energy dependence could be neglected and we used a constant additional term in the resolution. Taking it all together, the variance of the energy resolution (in terms of the used energy estimator) is:
$\sigma_{N}^{2}=N(p_{0}-p_{1}v_{1})+N^{2}(v_{T}(N)+v_{f})+\sigma_{d}^{2}+\sigma_{int}^{2}$,
where $\sigma_{d}$ is contribution of the dark noise (fixed) and $\sigma_{int}$ is contribution from the intrinsic line shape smearing. The probability $p_{1}$is linked to the energy estimator with relation $n=Np_{1}$.
Scintillation line shape
------------------------
The shape of the scintillation line (i.e. the response of the detector to the monoenergetic source uniformly distributed over the detector’s volume) is another sensitive component of the analysis. A common approximation with a normal distribution is failing to describe the tails of the MC-generated monoenergetic response already at the statistics of the order of $10^{3}$ events. This was notified already in the first phase of Borexino and the approximation of the scintillation line shape with generalized gamma- function [@SM07] have been used to fit monoenergetic $^{210}$Po peak in the solar $^{7}$Be neutrino analysis [@Be707; @Be708; @Be11]. The generalized gamma- function (GGC) was developed for the energy estimator based on the total collected charge, but it provided a reasonable approximation for the energy estimator based on the number of triggered PMTs given the moderate statistics corresponding to the total amount of the events in $^{210}$Po peak. The fit quality of the $^{210}$Po peak is rather insensitive to the residual deviations in the tails. This is not the case for the precise $^{14}$C spectrum modeling, as all the events in the fraction of the $^{14}$C spectrum above the energy threshold originate from the spectral smearing. An ideal detector’s response to the point-like monoenergetic source at the center is a perfect binomial distribution and it would be well approximated by a Poisson distribution. When dealing with real response one should adjust the “base distribution” width to take into account at least the additional smearing of the signal due to the various factors. The problem with binomial “base function” (or with its Poisson approximation) is that its width is defined by the mean value. In case of Poisson the variance of the signal coincide with mean $\mu$. A better approximation of the response function was tested with MC model, namely the scaled Poisson (SP) distribution:
$$f(x)=\frac{\mu^{xs}}{(xs)!}e^{-\mu},\label{ScaledPoisson}$$
featuring two parameters, that could be evaluated using expected mean and variance of the response:
$$s=\frac{\sigma_{n}^{2}}{n}\text{ and }\mu=\frac{n^{2}}{\sigma_{n}^{2}}.\label{Pars}$$
The agreement of the approximation and the detector response function was tested with Borexino MC model and it was found that at low energies (\[ScaledPoisson\]) better reproduces the scintillation line shape compared to the generalized gamma function up to the statistics of $10^{8}$, while at energies just above the $^{14}$C tail both distribution gives comparable approximation. The quality of the fit was estimated using $\chi^{2}$ criterion, for example with $10^{7}$ total statistics (these events are uniformly distributed in the detector and then the FV is selected) for $n=50$ (approximately 140 keV) we found $\chi^{2}/n.d.f.$=88.0/61 for the GGC compared to $\chi^{2}/n.d.f.$=59.3/61 for the SP distribution.
As proven by MC tests, the SP distribution as a base function works well in the energy region of interest despite of the additional smearing due to the factors enlisted in the previous paragraph. This is a result of the “absorption” of the relatively narrow non-statistical distributions by the much wider base function, as follows from MC such an absorption results in the smearing of the total distribution without changing its shape.
As it was noted above, the fit was performed in $n$ scale. All the theoretical spectra involved in the fit were first translated into the $n$ scale and then smeared using resolution function (\[ScaledPoisson\]) with $\mu$ and scale factor $s$ calculated using (\[Pars\]). As it is clear from the discussion above, the detector’s response has the shape described by (\[ScaledPoisson\]) only in the “natural” $n$ scale. If the measured values of $n$ would been converted into the energy, the shape will be deformed because of the non-linearity of the energy estimator scale with respect to the energy, complicating the construction of the precise energy response.
Standard fit
-------------
The “standard” options of the spectral fit are: number of triggered PMTs in a fixed time window of 230 ns (npmts) used as energy estimator; 62–220 npmts fit range; 75.5$\pm$1.5 tonnes fiducial volume (defined by the condition R$<$3.02 m and $|Z|<$1.67 m).
The rate of the solar neutrinos is constrained either at the value found by Borexino in the different energy range ($\mathbf{R({}^{7}Be)=}$ $48\pm2.3$ cpd [@Be11]), or fixed at the prediction of the SSM in the SMW/LMA oscillation scenario **(**R(pep)=2.80 cpd**,** R(CNO)=5.36 cpd). All counts here and below are quoted for 100 tonnes of LS.
The $^{14}$C rate was constrained at the value found in independent measurement with the second cluster $\mathbf{R(^{14}C})=40\pm1$ Bq (or $R(^{14}\text{C})=(3.456\pm0.0864)\times10^{6}$ cpd). The synthetic pile-up rate was constrained at the values found with the algorithm. The normalization factors for other background components were mainly left free ($\mathbf{^{85}Kr}$,$\mathbf{^{210}Bi}$ and $\mathbf{^{210}Po}$) and the fixed rate of $\mathbf{^{214}Pb}$ ($R(\mathbf{^{214}Pb)=0.06}$ cpd) was calculated on the base of the amount of identified radon events. The light yield and two energy resolution parameters ($v_{T}$ and $\sigma_{int}$) are left free. The position of the $\mathbf{^{210}Po}$ is also left free in the analysis, decoupling it from the energy scale.
Systematics study
-----------------
An evident source of systematics is uncertainty of the FV. The FV mass is defined using position reconstruction code, residual bias in the reconstructed position is possible. The systematic error of the position reconstruction code was defined during the calibration campaign, comparing the reconstructed source position with the nominal one [@PP14; @Calib12]. At the energies of interest the systematic error on the FV mass is 2%.
The stability and robustness of the measured pp neutrino interaction rate was verified by performing fits varying initial conditions, including fit energy range, method of pile-up construction, and energy estimator. The distribution of the central values for pp-neutrino interaction rates obtained for all these fit conditions was then used as an estimate of the maximal systematic error (partial correlations between different factors are not excluded).
The remaining external background in the fiducial volume at energies relevant for the pp neutrino study is negligible. In the particular case of the very low-energy part of the spectrum, the fit was repeated in five smaller fiducial volumes (with smaller radial and/or z-cut), which yields very similar results, indicating the absence of the influence of the external backgrounds at low energies.
Results and Implications
========================
![\[Pee-fig\]Survival probabilities for electron neutrino (Borexino only data from [@PP14; @Be11; @BrxB8; @BrxPEP])](Figures/PeeBrx){width="80.00000%"}
The solar pp neutrino interaction rate measured by Borexino is $pp=144\pm13(stat)\pm10(syst)$ cpd/100 t, compatible with the expected rate of $pp_{theor}=131\pm2$ cpd/100 t. The corresponding total solar pp-neutrino flux is $\phi_{pp}(Borex)=(6.6\pm0.7)\times10^{10}$ cm$^{-2}$s$^{-1}$, in a good agreement with the combined best fit value of the radiochemical and other solar experiments $\phi_{pp}(other)=(6.0\pm0.8)\times10^{10}$ cm$^{-2}$s$^{-1}$ [@SAGE09]. Both are in agreement with the expected value of $6.0\times(1.000\pm0.006)\times10^{10}$ cm$^{-2}$s$^{-1}$.
The survival probability for electron neutrino from pp-reaction is $P_{ee}(Borex)=0.64\pm0.12$. This is the fourth energy range explored by Borexino, all the Borexino results on the electron neutrino survival probability are presented graphically in Fig.\[Pee-fig\].
Taking into account that Borexino and other experiments measurements are independent, the results can be combined:
$$\phi_{pp}=(6.37\pm0.46)\times10^{10}cm^{-2}s^{-1}.$$
The electron neutrino survival probability measured in all solar but Borexino experiment is $P_{ee}(other)=0.56\pm0.08$, combining it with Borexino one we obtain:
$$P_{ee}=0.60\pm0.07,$$ well compatible with theoretical prediction of the MSW/LMA model $0.561_{-0.042}^{+0.030}$.
All available measurements of the solar neutrino fluxes are shown in Tab.\[SSMvsData\]. The total energy production in the solar reactions observed till now (by detecting corresponding neutrino fluxes) is $4.04\pm0.28$ W$\cdot$s$^{-1}$ in a good agreement with a total measured $L_{\odot}=3.846\times10^{26}$ W$\cdot$s$^{-1}$. There is not much space left for the unknown energy sources, the 90% C.L. lower limit for the total energy production (conservatively assuming zero contribution from the not-observed reactions) is $L_{tot}=3.68\times10^{26}$ W$\cdot$s$^{-1}$. If one assumes that such an unknown source exists, its total power with 90% probability can’t exceed $0.15\times10^{26}$ W$\cdot$s$^{-1}$. In other words no more than 4% of the total energy production in the Sun is left for the unknown energy sources, confirming that the Sun shines due to the thermonuclear fusion reactions.
Acknowledgments
===============
The Borexino program is made possible by funding from INFN (Italy), NSF (USA), BMBF, DFG, and MPG (Germany), RFBR: Grants 14-22-03031 and 13-02-12140, RFBR-ASPERA-13-02-92440 (Russia), and NCN Poland (UMO-2012/06/M/ST2/00426). We acknowledge the generous support and hospitality of the Laboratori Nazionali del Gran Sasso (LNGS).
[^1]: Talk at the International Workshop on Prospects of Particle Physics: “Neutrino Physics and Astrophysics”, JINR, INR, 1 February - 8 February 2015, Valday, Russia.
|
---
abstract: 'By making the second quantization for the Cini Model of quantum measurement without wave function collapse \[M. Cini, Nuovo Cimento, B73 27(1983)\], the second order quantum decoherence (SOQD) is studied with a two mode boson system interacting with an idealized apparatus composed by two quantum oscillators. In the classical limit that the apparatus is prepared in a Fock state with a very large quantum number, or in a coherent state with average quantum numbers large enough, the SOQD phenomenon appears similar to the first order case of quantum decoherence.'
address: |
Institute of Theoretical Physics, Academia Sinica,\
P. O. Box 2735, Beijing 100080, China
author:
- 'D. L. Zhou, G. R. Jin and C. P. Sun$^{a,b}$'
title: Second Quantization of Cini Model for High Order Quantum Decoherence in Quantum Measurement
---
Introduction
============
In usual the quantum coherence is reflected by the spatial interference of two or more “paths” in terms of single particle wave function. Correspondingly the decoherence phenomenon losing coherence can be understood in term of an “which-path” detection implied by the quantum entanglement of the considered system with the environment or the measuring apparatus \[1-3\]. Most recently, we have shown \[4,5\] that this more simple, but most profound observation can be also implemented in the many particle picture to account for the losing of the high order quantum coherence (HOQC) described by the high order correlation function \[6,7\]. In this letter we will give a detailed study of this novel context specified for the quantum measurement problem.
To this end we first make the second quantization for the Cini Model of quantum measurement without wave function collapse \[8\] to obtain a modelled system — a two mode boson system interacting with an idealized measuring apparatus composed also by two quantum oscillators. Then, the second order quantum decoherence (SOQD) is studied with this model in the classical limit that the apparatus is prepared in a Fock state with a very large quantum number, or in a coherent state with average quantum numbers large enough.
The crucial point that we understand the higher order quantum decoherence problem in the “which-path” picture is to introduce the concept of the multi-particle wave amplitude (MPWA), whose norm square is just the high order correlation function \[4,5\]. Before the measurement, as an effective wave function, this multi-time amplitude can be shown to be a supposition of several components. When the an apparatus entangles with them to make an effective measurement, the high order quantum coherence loses dynamically. This decoherence process can be explained as a generalized “which-path” measurement for the defined multi-particle paths in the MPWA .
Second Quantization of Cini Model
==================================
The original Cini model for quantum measurement emphasizes the production of quantum entanglement between the states of measured system S (a two-level system) and the measuring apparatus D — many indistinguishable particles with two possible modes $\omega
_1$ and $\omega _2$. The two states $u_g$ and $u_e $ of S have different interaction strengths $d_g$ and $d_e$ with D. Then, the large number N of “ionized” particles in the “ionized” mode $%
\omega _2$ transiting from the “un-ionized” model $\omega _2$ shows this quantum entanglement. In the following we wish to make a second quantization for the system components to built a novel model for SOQD with Hamiltonian
$$\begin{aligned}
\hat{H}_0 &=&\omega _e\hat{b}_e^{\dagger }\hat{b}_e, \\
\hat{V} &=&\omega _1\hat{a}_1^{\dagger }\hat{a}_1+\omega _2\hat{a}%
_2^{\dagger }\hat{a}_2+(d_e\hat{b}_e^{\dagger }\hat{b}_e+d_g\hat{b}%
_g^{\dagger }\hat{b}_g)(\hat{a}_1^{\dagger }\hat{a}_2+\hat{a}_2^{\dagger }%
\hat{a}_1),\end{aligned}$$
where $\hat{H}_0$ is the free Hamiltonian of the system, $\hat{V}$ the free Hamiltonian of the apparatus D plus the interaction between S and D; and $%
\hat{b}_e^{\dagger }(\hat{b}_e), \hat{b}_g^{\dagger }(\hat{b}_g)$ the creation (annihilation) operators of two modes labelled by index $e$ and $g$. Their frequencies are $\omega _e$ and $\omega
_g=0$ respectively. The operators $\hat{a}_j^{\dagger
}(\hat{a}_j)$ are creation (annihilation) operators of the modes which labelled by index $j$ for the mode frequency $\omega _j$, ${j=1,2}$. The frequency-dependent constant $%
d_e$ ($d_g$) measures the coupling constant between the $e$ ($g$) mode of the system and the apparatus. The most important feature of the model is that $[H_0,V]=0$, i.e. the system does not dissipate energy to the apparatus. Notice this model is equivalent to the generalized Cini model with many-levels given by us \[9\].
Starting with this concrete model we first consider the meaning of the “path” for the high order quantum correlation in the free particle case. The typical example of the higher order quantum coherence is that the single-component state $|1_e,1_g\rangle $ of the two independent particles shows its quantum coherence in its second order quantum correlation function $G^{(2)}(t_1,t_2)$, which can just be written as the norm square $G^{(2)}=|\psi |^2$ \[10\] of the equivalent “two-time wave function” $\Psi (t_1,t_2)$ (it was also called the bipparticle wavepacket \[11\] for photons), namely, $$\begin{aligned}
G^{(2)} &=&\langle 1_e1_g|\hat{\phi}^{\dagger }(t_1)\hat{\phi}%
^{\dagger }(t_2)\hat{\phi}(t_2)\hat{\phi}(t_1)|1_e,1_g\rangle \nonumber \\
\mbox{} &=&|\langle 0,0|\hat{\phi}(t_2)\hat{\phi}(t_1)|1_e,1_g\rangle
|^2\equiv |\Psi (t_1,t_2)|^2\end{aligned}$$ Here, we define a “measuring” field operator of two modes $g$ and $e$ $$\hat{\phi}=c_g\hat{b}_ge^{-i\omega _gt}+c_e\hat{b}_ee^{-i\omega
_et}\equiv c_g(t)\hat{b}_e+c_e(t)\hat{b}_e.$$ The two time wave function $\Psi (t_1,t_2)$ can be understood in terms of the two “paths” picture from the initial state $|1_g,1_e\rangle $ to the finial one $|0,0\rangle $ \[5\]:
---------------------------------------------------------------------------
$
\begin{array}{ccccc}
|1_g,1_e\rangle & \stackrel{c_e(t_1)}{\longrightarrow } &
|1_g,0_e\rangle &
\stackrel{c_{_g}(t_2)}{\longrightarrow } & |0,0\rangle \\
& \searrow \stackrel{c_g(t_1)}{} & & \stackrel{c_e(t_2)}{}\nearrow & \\
& & |1_e,0_g\rangle & &
\end{array}
$
---------------------------------------------------------------------------
---------------------------------------------------------------------------
Obviously, they are just associated with the two amplitudes forming a coherent superposition
$$\begin{aligned}
\Psi (t_1,t_2) &=&\langle 0,0|\hat{\phi}(t_2)\hat{\phi}(t_1)|1_e,1_g\rangle
\\
&=&c_ec_ge^{-i\omega _et_2-i\omega _gt_1}+c_gc_ee^{-i\omega _{_g}t_2-i\omega
_{_e}t_1}\end{aligned}$$
Correspondingly, the second order correlation function $$G^{(2)}=2|c_ec_g|^2[1+\cos ([\omega _g-\omega _e][t_2-t_1])]$$ shows the HOQC in the time domain. The above observation for the second order quantum coherence can also be discovered in the higher order case. It is noticed that our present arguments will be based on the equivalent field operator $\hat{\Phi}=\sum
c_n\hat{b}_n$ is specified for a quantum measurement to a superposition state $|\phi \rangle =\sum c_n|n\rangle$. In reference \[5\], we have point out its observability in an idealized cavity QED experiment.
Many-Particle “Which-Way” Detection
===================================
In the case with interaction, we consider the generalized second order correlation functions $$G[t,t^{\prime },\hat{\rho}(0)]=Tr(\hat{\rho}(0)\hat{B}^{\dagger }(t)\hat{B}%
^{\dagger }(t^{\prime })\hat{B}(t^{\prime })\hat{B}(t)),$$ with respect to “measuring” field operator \[4\]. It is defined as a functional of the density operator $\hat{\rho}(0)$ of the whole system for a given time $0$. Here, the bosonic field operator $$\begin{aligned}
\hat{B}(t) &=&\exp (i\hat{H}t)[c_1\hat{b}_g+c_2\hat{b}_e]\exp (-i\hat{H}t) \\
&=&c_1\exp (i\hat{V}t)[c_1\hat{b}_g+\hat{b}_ec_2\exp (-i\omega _et)]\exp (-i%
\hat{V}t)\end{aligned}$$ describes a specific quantum measurement with respect to the superpositions $%
|+\rangle =$ $c_1|e\rangle $ $+c_2|g\rangle $ and $|-\rangle =$ $%
c_2|e\rangle $ $-c_1|g\rangle $ where $c_1$ and $c_2$ satisfy the normalization relation $|c_1|^2+|c_2|^2=1$. Without loss of the generality, we take $c_1=c_2=1/\sqrt{2}$ standing for a measurement as follows.
To examine whether the classical feature of the apparatus causes the second order decoherence or not, we consider the whole system in an initial state $$|\psi (0)\rangle =|1_g,1_e\rangle \otimes |\phi (0)\rangle ,$$ where $|\phi (0)\rangle $ is the initial state of the apparatus. In the case with interaction, in stead of defining the equivalent “two-time wave function” in the case of free particle, we define an effective two-time state vector $$|\psi _B(t,t^{\prime })\rangle =\hat{B}(t^{\prime })\hat{B}(t)|\psi
(0)\rangle .$$ to re-write the second order correlation function as $$G[t,t^{\prime },\hat{\rho}(0)]=\langle \psi _B(t,t^{\prime })|\psi
_B(t,t^{\prime })\rangle$$ It is interested that the effective state vector can be evaluated as the superposition $$\begin{aligned}
|\psi _B(t,t^{\prime })\rangle &=&\frac 12e^{i\hat{V}(0,0)t^{\prime }}[\exp
(-i\omega _et^{\prime })e^{-i\hat{V}(1,0)t^{\prime }}e^{i\hat{V}(1,0)t}+ \\
&&\exp (-i\omega _et)e^{i\hat{V}(0,0)t^{\prime }}e^{-i\hat{V}(0,1)t^{\prime
}}e^{i\hat{V}(0,1)t}]e^{-i\hat{V}(1,1)t}|\{0_j\}\rangle \otimes
|0_g,0_e\rangle\end{aligned}$$ of two components with respect to the two paths from the initial two particle state $%
|1_g,1_e\rangle $ to the two particle vacuum $|0_g,0_e\rangle $ . It should be noticed that the effective actions of the apparatus $$\hat{V}(m,n)\equiv \sum_j\hat{V}_j(m,n)=\sum_j\omega _j\hat{a}_j^{\dagger }%
\hat{a}_j+\sum_j(d_e(\omega _j)m+d_g(\omega _j)n)(\hat{a}_j^{\dagger }+\hat{a%
}_j)$$ can label the different paths and thus lead to the higher order quantum decoherence. The above result clearly demonstrates that, in presence of the apparatus, the different probability amplitudes ($\sim $ $\exp (-i\omega
_gt^{\prime })$ and $\exp (-i\omega _et)$) from $|1_g,1_e\rangle $ to $%
|0_g,0_e\rangle $ entangle with the different states ($\frac 12e^{i\hat{V}%
(0,0)t^{\prime }}e^{-i\hat{V}(1,0)t^{\prime }}e^{i\hat{V}(1,0)t}e^{-i\hat{V}%
(1,1)t}|\{0_j\}\rangle $ and $\frac 12e^{i\hat{V}(0,0)t^{\prime }}e^{-i\hat{V%
}(0,1)t^{\prime
}}e^{i\hat{V}(0,1)t}e^{-i\hat{V}(1,1)t}|\{0_j\}\rangle $ ) of the apparatus. This is just physical source of the higher order quantum decoherence.
In the following calculation, the second order correlation function $$G[t,t^{\prime },\hat{\rho}(0)]=\frac 12+\frac{e^{i\omega _e(t-t^{\prime })}}%
4F+\frac{e^{-i\omega _e(t-t^{\prime })}}4F^{*},$$ is expressed explicitly in terms of the decoherence factor $$F=\langle \phi (0)|e^{i\hat{V}(1,1)t}e^{-i\hat{V}(0,1)t}e^{i\hat{V}%
(0,1)t^{\prime }}e^{-i\hat{V}(1,0)t^{\prime }}e^{i\hat{V}(1,0)t}e^{-i\hat{V}%
(1,1)t}|\phi (0)\rangle$$ which determines the extent of coherence or decoherence in the second order case.
Dynamic High-Order Quantum Decoherence
======================================
In the following, to given the factor $F$ explicitly, the normal ordering technique \[12\] is adopted to calculate the second order decoherence factor $F$. The calculation is carried out in six steps.
At the $k$-th step the evolution is dominated by the step-Hamiltonian $$\hat{h}^k={\alpha _1^k}\hat{a}_1^{\dagger }\hat{a}_1+{\alpha _2^k}\hat{a}%
_2^{\dagger }\hat{a}_2+{\beta ^k}(\hat{a}_1^{\dagger }\hat{a}_2+\hat{a}%
_2^{\dagger }\hat{a}_1),{k=1,2,\cdots ,6}$$ during the time period $t_k$. The coefficients $\{\alpha
_1^k,\alpha _2^k,\beta ^k,t_k\}$ take different values in different steps: $$\begin{aligned}
&&\alpha _1^1=\omega _1,\alpha _2^1=\omega _2,\beta ^1=d_e+d_g,t_1=t,
\nonumber \\
&&\alpha _1^2=-\omega _1,\alpha _2^2=-\omega _2,\beta ^2=-d_e,t_2=t,
\nonumber \\
&&\alpha _1^3=\omega _1,\alpha _2^3=\omega _2,\beta ^3=d_e,t_3=t^{\prime },
\nonumber \\
&&\alpha _1^4=-\omega _1,\alpha _2^4=-\omega _2,\beta ^4=-d_g,t_4=t^{\prime
}, \nonumber \\
&&\alpha _1^5=\omega _1,\alpha _2^5=\omega _2,\beta ^5=d_g,t_5=t, \nonumber
\\
&&\alpha _1^6=-\omega _1,\alpha _2^6=-\omega _2,\beta ^6=-d_e-d_g,t_6=t.\end{aligned}$$ Assume that at the k-th step the evolution operator $\hat{u}^k(t)$ can be written in the normal order as $$\hat{u}^k(t)=\aleph \{e^{{A^k}(t)\hat{a}_1^{\dagger }\hat{a}_1+{B^k}(t)\hat{a%
}_2^{\dagger }\hat{a}_2+{C^k}(t)\hat{a}_1^{\dagger }\hat{a}_2+{D^k}(t)\hat{a}%
_2^{\dagger }\hat{a}_1}\}.$$ The advantage of this form is that, when calculating the average of the operator in the coherent state $|\alpha ,\beta \rangle $, we only need to replace the annihilation operators with the corresponding complex values. This evolution operators satisfy the $Schr\ddot{o}dinger$ equation $i\frac
d{dt}\hat{u}^k=\hat{h}^k\hat{u}^k$. To take the expectation values of the above equations in the coherent state $|\alpha ,\beta
\rangle $, the coefficients satisfy the following system of equations: $$\begin{aligned}
i\frac{dA^k}{dt} &=&\alpha _1^k(A^k+1)+\beta ^kD^k, \nonumber \\
i\frac{dB^k}{dt} &=&\alpha _2^k(B^k+1)+\beta ^kC^k, \nonumber \\
i\frac{dC^k}{dt} &=&\alpha _1^kC^k+\beta ^k(B^k+1), \nonumber \\
i\frac{dD^k}{dt} &=&\alpha _2^kD^k+\beta ^k(A^k+1).\end{aligned}$$ The solution of the system of equations is $$\begin{aligned}
A^k+1 &=&e^{-i(\alpha _1^k+\alpha _2^k)t_k/2}(\cos (\Gamma ^kt_k)+\frac{%
i(\alpha _2^k-\alpha _1^k)}{2\Gamma ^k}\sin (\Gamma ^kt_k)), \nonumber \\
B^k+1 &=&e^{-i(\alpha _1^k+\alpha _2^k)t_k/2}(\cos (\Gamma ^kt_k)-\frac{%
i(\alpha _2^k-\alpha _1^k)}{2\Gamma ^k}\sin (\Gamma ^kt_k)), \nonumber \\
C^k=D^k &=&-\frac{i\beta ^k}{\Gamma ^k} e^{-i(\alpha _1^k+\alpha
_2^k)t_k/2}\sin (\Gamma ^kt_k),\end{aligned}$$ where $$\Gamma ^k=\sqrt{(\frac{\alpha _2^k-\alpha _1^k}2)^2+{\beta ^k}^2}$$ Notice that the above results have been before given in ref.\[12\], but the original ones contain some minor misprints. Here, we have corrected them. Then we obtain $$\begin{aligned}
&&e^{-i\hat{h}^kt_k}|\alpha ^{k-1},\beta ^{k-1}\rangle \nonumber \\
&=&|(A^k+1)\alpha ^{k-1}+C^k\beta ^{k-1},(B^k+1)\beta
^{k-1}+D^k\alpha ^{k-1}\rangle
\nonumber \\
&\equiv &|\alpha ^k,\beta ^k\rangle.\end{aligned}$$ From the above equation, it is obvious that, when the apparatus is initially in the product coherent state $|\alpha ^0,\beta
^0\rangle $, after six steps of evolution, the final state of the apparatus remains in a product coherent state $|\alpha ^6(\alpha
^0,\beta ^0),\beta ^6(\alpha ^0,\beta ^0)\rangle $.
To consider the classical feature of the apparatus, two specific initial states will be studied with different classical correspondences. And we give the numerical results respectively thereafter. The first case is that the initial state of the apparatus takes $|\phi (0)\rangle =|0,\beta \rangle $. When the norm of $\beta $ goes to infinity, it corresponds to classical field in some sense. In this case, we can obtain the decoherence factor and therefore the second order correlation function becomes $$G[t,t^{\prime },\hat{\rho}(0)]=\frac 12+(\frac{e^{i\omega _e(t-t^{\prime })}}%
4\langle 0,\beta |\alpha ^6(0,\beta ),\beta ^6(0,\beta )\rangle +h.c.).$$ A typical case of the numerical result of the above equation is given in FIG.$1$.
= =8.3cm = =8.3cm\
=8.3cm =8.3cm\
=8.3cm =8.3cm
In FIG.$1$, we observe that the second order correlation function is an explicit function of both the time interval $t^{\prime }-t$ and the time $t$. With the increasing of time $t$, it obviously oscillate faster and faster. It is all observed that, as the average particle number of the coherent state increases, the second order correlation function decoheres in a shorter time scale. The decoherence rate is independent of the time $t$. The later observation implies that, when the average particle number approaches to infinity, the second order correlation function will decohere in very short time, and the quantum revivals can not be observed in a finite time period.
The second case is that the initial state of the apparatus takes a Fock number state $|\phi (0)\rangle =|0,N\rangle $. When the number $N$ approaches into infinity, it also corresponds to classical field in some sense. In this case, we can obtain the decoherence factor and therefore the second order correlation function $$G[t,t^{\prime },\hat{\rho}(0)]=\frac 12+(\frac{e^{i\omega _e(t-t^{\prime })}}%
4\int \frac{d^2\beta }\pi \langle 0,N|\alpha ^6(0,\beta ),\beta ^6(0,\beta
)\rangle \langle \beta |N\rangle +h.c.).$$ A typical numerical result of the above equation with the same parameters as in FIG. 1. is given in FIG.2.
= =8.3cm = =8.3cm\
=8.3cm =8.3cm\
=8.3cm =8.3cm
In FIG.$2$, we observed the similar phenomena as in FIG.$1$. We would like to emphasize that difference between the number state and the coherent state only manifests in the case that the number is quite small. We can expect that they may gives the same limit for infinite particle number. Indeed, the results are identified by our numerical simulation.
Concluding Remark
=================
In fact, embodying the wave nature of particles in the quantum world, the quantum coherence is usually reflected by the spatial interference of two or more “paths” in terms of single particle wave function. However, the usual quantum coherence phenomenon with the first order interference fringes does not sound very marvellous for the same circumstances can also occur in classical case, such as an usual optical interference. But in association with the Hanburg-Brown-Twiss experiment \[7\], Glauber’s higher order quantum coherence manifests the intrinsically quantum features of coherence beyond the classical analogue. For example, in a quantum system composed by identical particles, the quantum coherence is indeed manifested in the observation of interference fringes reflected not only by the first order correlation functions, but also by higher order ones .
On the other hand, in the present of external quantum system (i.e. an apparatus) interacting with the studied system, the quantum decohernce of the system happens as the disappearance of the first order interference fringes. This decoherence mechanism provides the essential elements in the understanding for quantum measurements and the transition from quantum to classical mechanics. Just based on this conception, our present work extends the above understanding for quantum decoherence in terms of the first order interference to the high order case. With a two mode boson model, we have studied the second order decoherence in the classical limit. Even without the factorization structure and thus the obvious the macroscopic limit, the high-order quantum decoherence still happens in the classical limit, i.e., when the quantum number to infinity. It is concluded that this decoherence process losing the higher order coherence can be also explained as a generalized “which-path” measurement for the defined multi-particle paths.
0.2cm [**Acknowledgement**]{} This work is supported by NSF of China and the knowledged Innovation Programme(KIP) of the Chinese Academy of Science.
Electronic address: suncp@itp.ac.cn
Internet www site: http:// www.itp.ac.cn/suncp
D. Giulini, et. al., [*Decoherence and Appearance of Classical World in Quantum Theory*]{}, Springer Berlin, 1996.
W. H. Zurek, Phys. Today [**44**]{}(10), 36 (1991).
S. Durt, T. Nonn, and G. Rampe, Nature [**395**]{}, 33(1998).
D. L. Zhou, and C. P. Sun, quant-ph/0104038, LANL preprint(2001).
D. L. Zhou, P. Zhang and C. P. Sun, quant-ph/0105088, LNAL preprint(2001).
R. J. Glauber, Phys. Rev. [**130**]{}, 2529(1963); [**131**]{}, 2766(1963).
H. Hanburg-Brow and R. Q. Twiss, Phil. Mag. [**45**]{}, 663(1954); Nature [**178**]{}, 1046(1956); Proc. Roy. Soc. A [**242**]{},300(1957).
M. Cini, Nnovo Cimento, [**B73**]{}, 27 (1983).
X. J. Liu, C. P. Sun, Phys. Lett. A, [**198**]{}, 371 (1995).
M. O. Scully, and M. S. Zubairy, [*Quantum Optics*]{}, Cambridge University press, 1997, pp 97-129.
Y. Shih, Adv. At. Mole. and Opt. Phys. Vol 41(1999), pp 1-42.
W. H. Louisell, [*Quantum Statistical Properties of Radiations*]{}, John Wiley & Sons (1973).
C. P. Sun, Phys. Rev. A 48, 878(1993). C. P. Sun, Chin. J. Phys. [**32**]{}, 7(1994). C. P. Sun, X. X. Yi, and X. J. Liu, Fortschr. Phys. [**43**]{}, 585. C. P. Sun, H. Zhan, and X. F. Liu, Phys. Rev. A [**58**]{}, 1810(1998).
|
---
abstract: 'We consider closed orientable hypersurfaces in a wide class of warped product manifolds which include space forms, deSitter-Schwarzschild and Reissner-Nordström manifolds. By using a new integral formula or Brendle’s Heintze-Karcher type inequality, we present some new characterizations of umbilic hypersurfaces. These results can be viewed as generalizations of the classical Jellet-Liebmann theorem and the Alexandrov theorem in Euclidean space.'
address: 'Department of Mathematical Sciences, Tsinghua University, Beijing [100084]{}, P.R. China'
author:
- Shanze Gao
- Hui Ma
title: Characterizations of umbilic hypersurfaces in warped product manifolds
---
[^1]
Introduction
============
The characterization of hypersurfaces with constant mean curvature in warped product manifolds has attracted much attention recently. There are at least three types of results. Classical Jellet-Liebmann theorem, also referred to as the Liebmann-Süss theorem, asserts that any closed star-shaped (or convex) immersed hypersurface in Euclidean space with constant mean curvature is a round sphere. This has been generalized to a class of warped products by Montiel [@Montiel99]. Similar results are also obtained for hypersurfaces with constant higher order mean curvature or Weingarten hypersurfaces in warped products (see [@Alias-Impera-Rigoli13; @Brendle-Eichmair; @Wu-Xia]).
The classical Alexandrov theorem states that any closed embedded hypersurface of constant mean curvature in Euclidean space is a round sphere. This was generalized to a class of warped product manifolds by Brendle [@Brendle13]. The key step in his proof is the Minkowski type formula and a Heintze-Karcher type inequality, which also works for Weingarten hypersurfaces (c.f. [@Brendle-Eichmair; @Wu-Xia]). In [@Kwong-Lee-Pyo], Kwong-Lee-Pyo proved Alexandrov type results for closed embedded hypersurfaces with radially symmetric higher order mean curvature in a class of warped products.
From a variational point of view, hypersurfaces of constant mean curvature in a Riemannian manifold are critical points of the area functional under variations preserving a certain enclosed volume (see [@Barbosa-doCarmo; @Barbosa-doCarmo-Eschenburg]). Under the assumption of stability, constant mean curvature hypersurfaces in warped products are studied in [@Montiel98; @Veeravalli] etc.
In this paper, we prove Jellet-Liebmann type theorems and an Alexandrov type theorem for certain closed hypersurfaces including constant mean curvature hypersurfaces in some class of warped product manifolds. Throughout this paper, we assume that $\bar{M}^{n+1}=[0,\bar{r})\times_{\lambda}P^{n}$ $(0<\bar{r}\leq \infty)$ is a warped product manifold endowed with a metric $$\bar{g}=dr^{2}+\lambda^{2}(r)g^{P},$$ where $(P,g^{P})$ is an $n$-dimensional closed Riemannian manifold ($n\geq 2$) and $\lambda: [0,\bar{r})\rightarrow [0,+\infty)$ is a smooth positive function, called the warping function. We first consider a hypersurface $x:M^{n}\rightarrow \bar{M}^{n+1}$ immersed in a warped product $\bar{M}^{n+1}$ whose mean curvature $H$ satisfies $$\label{Hphi}
H=\phi(r),$$ where $\phi(r)=x^{*}(\Phi(r))$ and $\Phi(r)$ is a radially symmetric positive function on $\bar{M}$; or $$\label{Halph}
H^{-\alpha}=\langle \lambda\partial_{r},\nu \rangle,$$ where $\nu$ is a normal vector of $M$ and $\alpha>0$ is a constant.
Notice that hypersurfaces satisfying are critical points of the area functional under variations preserving a weighted volume (see Appendix \[variation\]). And hypersurfaces satisfying are self-similar solutions to the curvature flow expanding by $H^{-\alpha}$. Both of them can be regard as generalizations of constant mean curvature hypersurfaces.
Our first main results are Jellet-Liebmann type theorems of these hypersurfaces.
\[nablaH\] Suppose that $(\bar{M}^{n+1}, \bar{g})$ is a warped product manifold satisfying $${\mathrm{Ric}}^{P}\geq (n-1)(\lambda'^{2}-\lambda\lambda'')g^{P},$$ and $x:M\rightarrow \bar{M}$ is an immersion of a closed orientable hypersurface $M^n$ in $\bar{M}$. If $x(M)$ is star-shaped and satisfies $$\label{H<0}
\langle \nabla H,\partial_{r} \rangle\leq 0,$$ then $x(M)$ must be totally umbilic.
If $M$ has constant mean curvature, Theorem \[nablaH\] reduces to the Jellett-Liebmann type theorem proved by Montiel [@Montiel99].
Applying Theorem \[nablaH\] to hypersurfaces satisfying , we obtain the following
\[corHphi\] Under the same assumption of Theorem \[nablaH\], if $x(M)$ is star-shaped and satisfies $$H=\phi(r),$$ where $\phi(r)=x^{*}(\Phi(r))$ and $\Phi(r)$ is a positive non-increasing function of $r$, then $x(M)$ must be totally umbilic.
An Alexandrov type theorem for the above hypersurfaces under the embeddedness assumption was obtained by Kwong-Lee-Pyo [@Kwong-Lee-Pyo].
The following example shows that the non-increasing assumption on $\Phi(r)$ is necessary.
\[ellip\] Let $\Sigma$ be an ellipsoid given by $$\Sigma=\{y\in {\mathbb{R}}^{n+1}|y_{1}^{2}+...+y_{n}^{2}+\frac{y_{n+1}^{2}}{a^{2}}=1\}.$$
The mean curvature of $\Sigma$ is $$H=\frac{a}{n\sqrt{a^{2}+1-r^{2}}}(n-1+\frac{1}{a^{2}+1-r^{2}}).$$
It is easy to check $$\Phi=\frac{a}{n\sqrt{a^{2}+1-r^{2}}}(n-1+\frac{1}{a^{2}+1-r^{2}})$$ is increasing for $r$.
Another application of Theorem \[nablaH\] is about hypersurfaces satisfying .
\[corHalph\] Under the same assumption of Theorem \[nablaH\], if $x(M)$ is strictly convex and satisfies $$H^{-\alpha}=\langle \lambda\partial_{r},\nu \rangle,$$ where $\alpha>0$ is a constant, then $x(M)$ is a slice $\{r_{0}\}\times P$ for some $r_{0}\in (0,\bar{r})$.
Similar results hold for higher order mean curvature under stronger assumptions. Let $\sigma_{k}(\kappa)$ denote the $k$-th elementary symmetric polynomial of principal curvatures $\kappa=(\kappa_{1},...,\kappa_{n})$ of $x(M)$, i.e., $$\sigma_{k}(\kappa)=\sum_{1\leq i_{1}<...<i_{k}\leq n}\kappa_{i_{1}}\cdots\kappa_{i_{k}}.$$ Thus, the $k$-th mean curvature is given by $H_{k}=\frac{1}{\binom{n}{k}}\sigma_{k}(\kappa)$. A hypersurface $x(M)$ is $k$-convex, if, at any point of $M$, principal curvatures $$\kappa\in\Gamma_{k}:=\{\mu\in{\mathbb{R}}^{n}|\sigma_{i}(\mu)>0, \text{ for }1\leq i\leq k\}.$$
\[nablaHk\] Suppose that $\bar{M}^{n+1}=[0,\bar{r})\times_{\lambda} P^{n}$ is a warped product manifold, where $(P,g^{P})$ is a closed Riemannian manifold with constant sectional curvature $\epsilon$ and $$\label{C4>=}
\frac{\lambda(r)''}{\lambda(r)}+\frac{\epsilon-\lambda(r)'^{2}}{\lambda(r)^{2}}\geq 0.$$ Let $x:M\rightarrow \bar{M}$ be an immersion of a closed orientable hypersurface $M^n$ in $\bar{M}$. For any fixed $k$ with $2\leq k\leq n-1$, if $x(M)$ is $k$-convex, star-shaped and satisfies $$\langle \nabla H_{k},\partial_{r} \rangle\leq 0,$$ then $x(M)$ must be totally umbilic.
If we require that the inequality in is strict as in [@Brendle13; @Brendle-Eichmair] and $H_k=\text{constant}$, we obtain
\[corHkconst\] Suppose that $\bar{M}^{n+1}=[0,\bar{r})\times_{\lambda} P^{n}$ is a warped product manifold, where $(P,g^{P})$ is a closed Riemannian manifold with constant sectional curvature $\epsilon$ and $$\label{eq:C4}
\frac{\lambda(r)''}{\lambda(r)}+\frac{\epsilon-\lambda(r)'^{2}}{\lambda(r)^{2}}> 0.$$ Let $x:M\rightarrow \bar{M}$ be an immersion of a closed orientable hypersurface $M^n$ in $\bar{M}$. For any fixed $k$ with $2\leq k\leq n-1$, if $x(M)$ is $k$-convex, star-shaped and $H_k=\text{constant}$, then $x(M)$ is a slice $\{r_0\}\times P$ for some $r_0\in (0,\bar{r})$.
The above corollary implies that the embeddedness condition in Theorem 2 of [@Brendle-Eichmair] is not necessary.
\[corHkphi\] Under the same assumption of Theorem \[nablaHk\], if for any fixed $k$ with $2\leq k\leq n-1$, $x(M)$ is $k$-convex, star-shaped and satisfies $$H_{k}=\phi(r),$$ where $\phi(r)=x^{*}(\Phi(r))$ and $\Phi(r)$ is a positive non-increasing function of $r$, then $x(M)$ must be totally umbilic.
\[corHkalph\] Under the same assumption of Theorem \[nablaHk\], if for any fixed $k$ with $2\leq k\leq n-1$, $x(M)$ is strictly convex and satisfies $$H_{k}^{-\alpha}=\langle \lambda\partial_{r},\nu \rangle,$$ where $\alpha>0$ is a constant, then $x(M)$ is a slice $\{r_{0}\}\times P$ for some $r_{0}\in (0,\bar{r})$.
Now we turn to the warped product manifold $\bar{M}^{n+1}=[0,\bar{r})\times_{\lambda} P^{n}$, where $(P,g^{P})$ is with a closed Riemannian manifold with constant sectional curvature $\epsilon$. As in [@Brendle13; @Wu-Xia], we list four conditions of the warping function $\lambda:[0,\bar{r})\rightarrow [0,+\infty)$:
- $\lambda'(0)=0$ and $\lambda''(0)>0$.
- $\lambda'(r)>0$ for all $r\in(0,\bar{r})$.
- The function $$2\frac{\lambda''(r)}{\lambda(r)}-(n-1)\frac{\epsilon-\lambda'(r)^{2}}{\lambda(r)^{2}}$$ is non-decreasing for $r\in(0,\bar{r})$.
- We have $$\frac{\lambda''(r)}{\lambda(r)}+\frac{\epsilon-\lambda'(r)^{2}}{\lambda(r)^{2}}>0$$ for all $r\in(0,\bar{r})$.
Instead of star-sharpness or convexity, under the embeddedness assumption, we study hypersurfaces satisfying $$\label{uHlambda}
H_{k}^{-\alpha}\lambda'=\langle \lambda\partial_{r},\nu \rangle,$$ and prove the following Alexandrov type theorem.
\[Hklambda\] Suppose that $(\bar{M}, \bar{g})$ is a warped product manifold satisfying conditions (C1)-(C4). Let $x:M\rightarrow \bar{M}$ be an immersion of a connected closed embedded orientable hypersurface $M^n$ in $\bar{M}$. If $H_{k}>0$ and $x(M)$ satisfies $$H_{k}^{-\alpha}\lambda'=\langle \lambda\partial_{r},\nu \rangle,$$ for any fixed $k$ with $1\leq k\leq n$ and $\alpha\geq \frac{1}{k}$, then $x(M)$ is a slice $\{r_{0}\}\times P$ for some $r_{0}\in (0,\bar{r})$.
It is interesting to compare Theorem \[Hklambda\] and Corollary \[corHkalph\] in the special case when $P={\mathbb{S}}^{n}$ and $\lambda(r)=r$, i.e. $\bar{M}$ is Euclidean space ${\mathbb{R}}^{n+1}$. With embeddedness and less convexity requirement (only $H_{k}>0$), Theorem \[Hklambda\] leads to the conclusion including the case when $k=n$.
Throughout the paper, the assumptions for the ambient spaces $\bar{M}$ are satisfied by space forms, the deSitter-Schwarzschild, the Reissner-Nordström manifolds and many other manifolds (c.f. [@Brendle13]).
The paper is organized as follows. In Section \[wprod\], we list some useful properties of warped products. In Section \[intformula\], we derive an integral formula which is the key to the proof of our main theorems. In Section \[proof\], we present the proofs of Theorem \[nablaH\], Theorem \[nablaHk\] and the corollaries. In Section \[embed\], we prove Theorem \[Hklambda\]. In Appendix \[variation\], we show that a hypersurface with a given positive mean curvature function is the critical point of the area functional under variations preserving weighted volume. Throughout the paper, the summation convention is used unless otherwise stated.
Preliminaries of warped products {#wprod}
================================
In this section, we list some basic properties of warped products $(\bar{M}=[0,\bar{r})\times_{\lambda}P^{n}, \bar{g})$ given above (see [@ONeill]).
\[connect\] Suppose $U,V\in \Gamma(TP)$. The Levi-Civita connection $\bar{\nabla}$ of a warped product $(\bar{M}=[0,\bar{r})\times_{\lambda} P, \bar{g})$ satisfies
i) $\bar{\nabla}_{\partial_{r}}\partial_{r}=0$,
ii) $\bar{\nabla}_{\partial_{r}}V=\bar{\nabla}_{V}\partial_{r}=\frac{\lambda'}{\lambda}V$,
iii) $\bar{\nabla}_{V}U=\nabla^{P}_{V}U-\frac{\lambda'}{\lambda}\bar{g}(V,U)\partial_{r}$,
where $\nabla^P$ is the Levi-Civita connection of $(P, g^P)$.
From the preceding proposition, we know that any slice $\{r\}\times P$ in a warped product $\bar{M}=[0,\bar{r})\times_{\lambda} P$ is totally umbilic.
\[curv4\] Suppose $Y_{1},Y_{2},Y_{3},Y_{4}\in \Gamma(TP)$. The $(0,4)$-Riemannian curvature tensor $\overline{{\mathrm{Rm}}}$ of a warped product $(\bar{M}=[0,\bar{r})\times_{\lambda} P, \bar{g})$ satisfies
i) $\overline{{\mathrm{Rm}}}(\partial_{r},Y_{1},\partial_{r},Y_{2})=-\frac{\lambda''}{\lambda}\bar{g}(Y_{1},Y_{2})$,
ii) $\overline{{\mathrm{Rm}}}(\partial_{r},Y_{1},Y_{2},Y_{3})=0$,
iii) $\overline{{\mathrm{Rm}}}(Y_{1},Y_{2},Y_{3},Y_{4})=\lambda^{2}{\mathrm{Rm}}^{P}(Y_{1},Y_{2},Y_{3},Y_{4})-\frac{\lambda'^{2}}{\lambda^{2}}(\bar{g}(Y_{1},Y_{3})\bar{g}(Y_{2},Y_{4})
-\bar{g}(Y_{2},Y_{3})\bar{g}(Y_{1},Y_{4})),
$
where ${\mathrm{Rm}}^{P}$ is the $(0,4)$-Riemannian curvature tensor of $(P, g^P)$.
\[ric\] Suppose $U,V\in \Gamma(TP)$. The Ricci curvature tensor $\overline{{\mathrm{Ric}}}$ of a warped product $(\bar{M}=[0,\bar{r})\times_{\lambda} P, \bar{g})$ satisfies
i) $\overline{{\mathrm{Ric}}}(\partial_{r},\partial_{r})=-n\frac{\lambda''}{\lambda},$
ii) $\overline{{\mathrm{Ric}}}(\partial_{r},V)=0,$
iii) $\overline{{\mathrm{Ric}}}(V,U)={\mathrm{Ric}}^{P}(V,U)-\left(\frac{\lambda''}{\lambda}+(n-1)\frac{\lambda'^{2}}{\lambda^{2}}\right)\bar{g}(V,U),$
where ${\mathrm{Ric}}^{P}$ is the Ricci curvature tensor of $(P, g^P)$.
For the convenience, we introduce the Kulkarni-Nomizu product ${\mathbin{\bigcirc\mspace{-15mu}\wedge\mspace{3mu}}}$. For any two $(0,2)$-type symmetric tensors $h$ and $w$, $h{\mathbin{\bigcirc\mspace{-15mu}\wedge\mspace{3mu}}}w$ is the $4$-tensor given by $$\begin{aligned}
(h{\mathbin{\bigcirc\mspace{-15mu}\wedge\mspace{3mu}}}w)(X_{1},X_{2},X_{3},X_{4})
&=h(X_{1},X_{3})w(X_{2},X_{4})+h(X_{2},X_{4})w(X_{1},X_{3})\\
&\quad-h(X_{1},X_{4})w(X_{2},X_{3})-h(X_{2},X_{3})w(X_{1},X_{4}).\end{aligned}$$ The following result is a corollary of Proposition \[curv4\] through a straightforward calculation and the proof is given for the completeness.
Suppose $(P,g^{P})$ is a Riemannian manifold with constant sectional curvature $\epsilon$. The Riemannian curvature tensor $\overline{{\mathrm{Rm}}}$ of a warped product $\bar{M}=[0,\bar{r})\times_{\lambda} P$ can be expressed as follows: $$\label{eq:Rm}
\overline{{\mathrm{Rm}}}=\frac{\epsilon-\lambda'^{2}}{2\lambda^{2}}\bar{g}{\mathbin{\bigcirc\mspace{-15mu}\wedge\mspace{3mu}}}\bar{g}-\left(\frac{\lambda''}{\lambda}+\frac{\epsilon-\lambda'^{2}}{\lambda^{2}}\right)\bar{g}{\mathbin{\bigcirc\mspace{-15mu}\wedge\mspace{3mu}}}dr^{2}.$$
Let $e_{A},e_{B},e_{C},e_{D}\in \Gamma(T\bar{M})$, $r_{A}=\bar{g}(e_{A},\partial_{r})$ and $e_{A}^{*}=e_{A}-r_{A}\partial_{r}$. Using Proposition \[curv4\], we have $$\begin{aligned}
&\overline{{\mathrm{Rm}}}(e_{A},e_{B},e_{C},e_{D})\\
&=\overline{{\mathrm{Rm}}}(e_{A}^{*},e_{B}^{*},e_{C}^{*},e_{D}^{*})+\overline{{\mathrm{Rm}}}(e_{A}^{*},r_{B}\partial_{r},e_{C}^{*},r_{D}\partial_{r})+\overline{{\mathrm{Rm}}}(e_{A}^{*},r_{B}\partial_{r},r_{C}\partial_{r},e_{D}^{*})\\
&\quad+\overline{{\mathrm{Rm}}}(r_{A}\partial_{r},e_{B}^{*},e_{C}^{*},r_{D}\partial_{r})+\overline{{\mathrm{Rm}}}(r_{A}\partial_{r},e_{B}^{*},r_{C}\partial_{r},e_{D}^{*})\\
&=\lambda^{2}{\mathrm{Rm}}^{P}(e_{A}^{*},e_{B}^{*},e_{C}^{*},e_{D}^{*})-\frac{\lambda'^{2}}{\lambda^{2}}\Big(\bar{g}(e_{A}^{*},e_{C}^{*})\bar{g}(e_{B}^{*},e_{D}^{*})-\bar{g}(e_{B}^{*},e_{C}^{*})\bar{g}(e_{A}^{*},e_{D}^{*})\Big)\\
&\quad-\frac{\lambda''}{\lambda}\Big(r_{B}r_{D}\bar{g}(e_{A}^{*},e_{C}^{*})-r_{B}r_{C}\bar{g}(e_{A}^{*},e_{D}^{*})-r_{A}r_{D}\bar{g}(e_{B}^{*},e_{C}^{*})+r_{A}r_{C}\bar{g}(e_{B}^{*},e_{D}^{*})\Big).\end{aligned}$$
Since $$\begin{aligned}
\bar{g}(e_{A}^{*},e_{C}^{*})=\bar{g}(e_{A},e_{C})-r_{A}r_{C}
=(\bar{g}-dr^{2})(e_{A},e_{C}),\end{aligned}$$ we know $$\begin{aligned}
& \bar{g}(e_{A}^{*},e_{C}^{*})\bar{g}(e_{B}^{*},e_{D}^{*})-\bar{g}(e_{B}^{*},e_{C}^{*})\bar{g}(e_{A}^{*},e_{D}^{*})\\
&\qquad=\frac{1}{2}(\bar{g}-dr^{2}){\mathbin{\bigcirc\mspace{-15mu}\wedge\mspace{3mu}}}(\bar{g}-dr^{2})(e_{A},e_{B},e_{C},e_{D})\end{aligned}$$ and $$\begin{aligned}
& r_{B}r_{D}\bar{g}(e_{A}^{*},e_{C}^{*})-r_{B}r_{C}\bar{g}(e_{A}^{*},e_{D}^{*})-r_{A}r_{D}\bar{g}(e_{B}^{*},e_{C}^{*})+r_{A}r_{C}\bar{g}(e_{B}^{*},e_{D}^{*})\\
&\qquad=(\bar{g}-dr^{2}){\mathbin{\bigcirc\mspace{-15mu}\wedge\mspace{3mu}}}dr^{2}(e_{A},e_{B},e_{C},e_{D}).\end{aligned}$$
Using the sectional curvatures of $(P,g^{P})$ is a constant $\epsilon$, i.e., ${\mathrm{Rm}}^{P}=\frac{\epsilon}{2}g^{P}{\mathbin{\bigcirc\mspace{-15mu}\wedge\mspace{3mu}}}g^{P}$, we have $$\begin{aligned}
\lambda^{2}{\mathrm{Rm}}^{P}(e_{A}^{*},e_{B}^{*},e_{C}^{*},e_{D}^{*})=\frac{\epsilon}{2\lambda^{2}}(\bar{g}-dr^{2}){\mathbin{\bigcirc\mspace{-15mu}\wedge\mspace{3mu}}}(\bar{g}-dr^{2})(e_{A},e_{B},e_{C},e_{D}).\end{aligned}$$
Combining these together, we obtain $$\begin{aligned}
\overline{{\mathrm{Rm}}}=\frac{\epsilon-\lambda'^{2}}{2\lambda^{2}}\bar{g}{\mathbin{\bigcirc\mspace{-15mu}\wedge\mspace{3mu}}}\bar{g}-\left(\frac{\epsilon-\lambda'^{2}}{\lambda^{2}}+\frac{\lambda''}{\lambda}\right)\bar{g}{\mathbin{\bigcirc\mspace{-15mu}\wedge\mspace{3mu}}}dr^{2}.\end{aligned}$$
An integral formula {#intformula}
===================
In this section, we obtain an integral formula by the divergence theorem, which is the key to the proof of our main results.
Let $x:M\rightarrow \bar{M}$ be an immersion of a closed orientable hypersurface $M^n$ into a warped product $\bar{M}^{n+1}=[0,\bar{r})\times_{\lambda} P^{n}$ endowed with a metric $\bar{g}=dr^{2}+\lambda^{2}(r)g^{P}$. Let $\nu$ be a normal vector field of $M$ and $h=(h_{ij})$ denote the second fundamental form with respect to an orthogonal frame $\{e_{1},...,e_{n}\}$ on $M$ defined by $$h_{ij}=\langle \nabla_{i}x,\nabla_{j}\nu \rangle.$$ The principal curvatures $\kappa=(\kappa_{1},...,\kappa_{n})$ are the eigenvalues of $h$. Thus the $k$-th elementary symmetric polynomials of principal curvatures can be expressed as follows $$\sigma_{k}(\kappa(h))=\frac{1}{k!}\delta_{j_{1}...j_{k}}^{i_{1}...i_{k}}h_{i_{1}j_{1}}\cdots h_{i_{k}j_{k}},$$ where $\delta_{j_{1}...j_{k}}^{i_{1}...i_{n}}$ is the generalized Kronecker symbol. Let $\sigma_{k;i}(\kappa)$ denote $\sigma_k(\kappa)$ with $\kappa_{i}=0$ and $\sigma_{k;ij}(\kappa)$, with $i\neq j$, denote the symmetric function $\sigma_k(\kappa)$ with $\kappa_i=\kappa_j=0$.
The following proposition is from a standard calculation (see also [@Brendle-Eichmair; @Kwong-Lee-Pyo]) and the proof is given for the completeness.
\[div\] Under an orthonormal frame such that $h_{ij}=\kappa_{i}\delta_{ij}$, we have the following equality $$\sum_{i}\nabla_{i}(\frac{\partial \sigma_{k}(h)}{\partial h_{ij}})=-\sum_{p\neq j}\bar{R}_{\nu pjp}\sigma_{k-2;jp}(\kappa)$$ for any fixed $j$ and $2\leq k\leq n$.
Let $\tilde{h}=I+th$. Then, $$\sigma_{n}(\tilde{h})=\sigma_{n}(I+th)=\sum_{k=0}^{n}t^{k}\sigma_{k}(h).$$
Using $$\frac{\partial\sigma_{n}(\tilde{h})}{\partial h_{ij}}=t(\tilde{h}^{-1})_{ij}\sigma_{n}(\tilde{h}),$$ $$\sum_{i=1}^{n}\nabla_{i}(\tilde{h}^{-1})_{ij}=-t(\tilde{h}^{-1})_{ip}(\tilde{h}^{-1})_{qj}\nabla_{i}h_{pq},$$ for arbitrary $t$ and the Codazzi equation $$\nabla_{i}h_{pq}=\nabla_{q}h_{pi}+\bar{R}_{\nu pqi},$$ we have $$\label{tk}
\begin{aligned}
&t^{k}\nabla_{i}(\frac{\partial \sigma_{k}(h)}{\partial h_{ij}})=\nabla_{i}(\frac{\partial\sigma_{n}(\tilde{h})}{\partial h_{ij}})\\
&\qquad=t^{2}\sigma_{n}(\tilde{h})\left(-(\tilde{h}^{-1})_{ip}(\tilde{h}^{-1})_{qj}\nabla_{i}h_{pq}+(\tilde{h}^{-1})_{ij}(h^{-1})_{pq}\nabla_{i}h_{pq}\right)\\
&\qquad=t^{2}(\tilde{h}^{-1})_{pq}(\tilde{h}^{-1})_{ij}\sigma_{n}(\tilde{h})\left(-\nabla_{q}h_{pi}+\nabla_{i}h_{pq}\right)\\
&\qquad=t^{2}(\tilde{h}^{-1})_{pq}(\tilde{h}^{-1})_{ij}\sigma_{n}(\tilde{h})\bar{R}_{\nu pqi}.
\end{aligned}$$
Now we choose a local orthonormal frame $\{e_1, \cdots, e_n\}$ such that $h_{ij}=\kappa_{i}\delta_{ij}$. Then $$(\tilde{h}^{-1})_{ij}=\frac{\delta_{ij}}{1+t\kappa_{i}}.$$
Thus, $$\begin{aligned}
&(\tilde{h}^{-1})_{pq}(\tilde{h}^{-1})_{ij}\sigma_{n}(\tilde{h})\bar{R}_{\nu pqi}=-\frac{\bar{R}_{\nu pjp}}{(1+t\kappa_{j})(1+t\kappa_{p})}\prod_{l=1}^{n}(1+t\kappa_{l})\\
&\qquad=-\bar{R}_{\nu pjp}\prod_{l\in\{1,...,n\}\backslash\{j,p\}}(1+t\kappa_{l})\\
&\qquad=-\bar{R}_{\nu pjp}\sum_{k=2}^{n}t^{k-2}\sigma_{k-2;jp}(\kappa).\end{aligned}$$
Combining with , we have $$\sum_{k=1}^{n}t^{k}\nabla_{i}(\frac{\partial \sigma_{k}(h)}{\partial h_{ij}})=-\bar{R}_{\nu pjp}\sum_{k=2}^{n}t^{k}\sigma_{k-2;jp}(\kappa).$$
Comparing the coefficients of $t^{k}$, we have $$\sum_{i}\nabla_{i}(\frac{\partial \sigma_{k}(h)}{\partial h_{ij}})=-\sum_{p\neq j}\bar{R}_{\nu pjp}\sigma_{k-2;jp}(\kappa)$$ for each $k\in\{2,...,n\}$.
Denote $\eta=x^{*}(\int_{0}^{r}\lambda(s)ds)$ and $u=\langle \lambda\partial_{r},\nu \rangle$. We have the following integral formula.
\[key\] Suppose $x(M)$ is a closed hypersurface of $\bar{M}$. The following equality holds $$\begin{aligned}
\int_{M} \{&-(n-k)\langle \nabla\sigma_{k},\lambda\partial_{r} \rangle+((n-k)\sigma_{1}\sigma_{k}-n(k+1)\sigma_{k+1})u\\
&-n\bar{R}_{\nu pjp}\langle \lambda\partial_{r},e_{j} \rangle\sigma_{k-1;jp}\}d\mu=0.\end{aligned}$$
From a straightforward calculation, we have $$\begin{aligned}
&\nabla_{i}(k\sigma_{k}\nabla_{i}\eta-n\frac{\partial\sigma_{k}}{\partial h_{ij}}\nabla_{j}u)=k\langle \nabla\sigma_{k},\nabla\eta \rangle+k\sigma_{k}\Delta\eta-n\nabla_{i}(\frac{\partial\sigma_{k}}{\partial h_{ij}})\nabla_{j}u-n\frac{\partial\sigma_{k}}{\partial h_{ij}}\nabla_{i}\nabla_{j}u\\
&\qquad=k\langle \nabla\sigma_{k},\nabla\eta \rangle+k\sigma_{k}(n\lambda'-\sigma_{1}u)-n\nabla_{i}(\frac{\partial\sigma_{k}}{\partial h_{ij}})\langle \lambda\partial_{r},h_{jl}e_{l} \rangle\\
&\qquad\quad+n\frac{\partial\sigma_{k}}{\partial h_{ij}}(-\lambda'h_{ij}-\langle \lambda\partial_{r},h_{jli}e_{l} \rangle+h_{jl}h_{li}u)\\
&\qquad=k\langle \nabla\sigma_{k},\lambda\partial_{r} \rangle+(n-k)\sigma_{k}\sigma_{1}u-n(k+1)\sigma_{k+1}u-n\nabla_{i}(\frac{\partial\sigma_{k}}{\partial h_{ij}})\langle \lambda\partial_{r},h_{jl}e_{l} \rangle\\
&\qquad\quad-n\langle \lambda\partial_{r},\nabla \sigma_{k} \rangle-n\frac{\partial\sigma_{k}}{\partial h_{ij}}\langle \lambda\partial_{r},e_{l} \rangle\bar{R}_{\nu jli}.\end{aligned}$$
Using Proposition \[div\], we know that $$\begin{aligned}
&-n\nabla_{i}(\frac{\partial\sigma_{k}}{\partial h_{ij}})\langle \lambda\partial_{r},h_{jl}e_{l} \rangle-n\frac{\partial\sigma_{k}}{\partial h_{ij}}\langle \lambda\partial_{r},e_{l} \rangle\bar{R}_{\nu jli}\\
&\qquad=n\bar{R}_{\nu pjp}\sigma_{k-2;jp}\langle \lambda\partial_{r},\kappa_{j}e_{j} \rangle-n\sigma_{k-1;i}\langle \lambda\partial_{r},e_{l} \rangle\bar{R}_{\nu ili}\\
&\qquad=n\bar{R}_{\nu pjp}\langle \lambda\partial_{r},e_{j} \rangle(\sigma_{k-2;jp}\kappa_{j}-\sigma_{k-1;p})\\
&\qquad=-n\bar{R}_{\nu pjp}\langle \lambda\partial_{r},e_{j} \rangle\sigma_{k-1;jp}.\end{aligned}$$
Combining these equalities and using divergence theorem, we finish the proof.
Proof of the main theorems {#proof}
==========================
Using Lemma \[key\] for $k=1$, we know that $$\label{k=1}
\int_{M}\{-n(n-1)\langle \nabla H,\lambda\partial_{r} \rangle+((n-1)\sigma_{1}^{2}-2n\sigma_{2})u-n\overline{{\mathrm{Ric}}}(\nu,\lambda\partial_{r}^{\top})\}d\mu=0,$$ where $\partial_{r}^{\top}$ deontes the tangent part of $\partial_{r}$.
From Newton inequality and $x(M)$ is star-shaped ($\langle \partial_{r},\nu \rangle >0$), we have $$\label{sig2}
((n-1)\sigma_{1}^{2}-2n\sigma_{2})u\geq 0.$$ And the equality of occurs if and only $\kappa_{1}=...=\kappa_{n}$.
Let $\nu^{P}=\nu-\langle \nu,\partial_{r} \rangle\partial_{r}$. Since $$\begin{aligned}
&\overline{{\mathrm{Ric}}}(\nu,\lambda\partial_{r}^{\top})=\overline{{\mathrm{Ric}}}(\nu,\lambda\partial_{r})-\overline{{\mathrm{Ric}}}(\nu,\nu)u\\
&\qquad=u\left(-n\frac{\lambda''}{\lambda}+n\frac{\lambda''}{\lambda}\langle \partial_{r},\nu \rangle^{2}-{\mathrm{Ric}}^{P}(\nu^{P},\nu^{P})+\left(\frac{\lambda''}{\lambda}+(n-1)\frac{\lambda'^{2}}{\lambda^{2}}\right){\lvert \nu^{P} \rvert}^{2}\right)\\
&\qquad=-u\left({\mathrm{Ric}}^{P}(\nu^{P},\nu^{P})+(n-1)\left(\frac{\lambda''}{\lambda}-\frac{\lambda'^{2}}{\lambda^{2}}\right){\lvert \nu^{P} \rvert}^{2}\right)\\
&\qquad=-u\left({\mathrm{Ric}}^{P}(\nu^{P},\nu^{P})+(n-1)\left(\lambda\lambda''-\lambda'^{2}\right)g^{P}(\nu^{P},\nu^{P})\right),\end{aligned}$$ we know $\overline{{\mathrm{Ric}}}(\nu,\lambda\partial_{r}^{\top})\leq 0$ by assumption.
Combining these estimates with $\langle \nabla H,\partial_{r} \rangle\leq 0$, we obtain the left hand side is nonnegative. These implies the inequalities are actually equalities at any point of $M$. Thus, $x(M)$ is totally umbilic.
In the previous proof, if ${\mathrm{Ric}}^{P}>(n-1)(\lambda'^{2}-\lambda\lambda'')g^{P}$, we also obtain $\partial_{r}^{\top}=0$. This means $\partial_{r}$ is the normal vector of $x(M)$ which implies $x(M)$ is a slice.
Under the assumption, it follows from that $$\bar{R}_{\nu pjp}=-\left(\frac{\lambda''}{\lambda}+\frac{\epsilon-\lambda'^{2}}{\lambda^{2}}\right)\langle \partial_{r},\nu \rangle \langle \partial_{r},e_{j} \rangle,$$ for any fixed $p$ and $j\neq p$. Using $$\sum_{p\neq j}\sigma_{k-1;jp}=(n-k)\sigma_{k-1;j},$$ we know $$\begin{aligned}
&-\bar{R}_{\nu pjp}\langle \lambda\partial_{r},e_{j} \rangle\sigma_{k-1;jp}=u\left(\frac{\lambda''}{\lambda}+\frac{\epsilon-\lambda'^{2}}{\lambda^{2}}\right)\langle \partial_{r},e_{j} \rangle^{2}\sigma_{k-1;jp}\\
&=(n-k)u\left(\frac{\lambda''}{\lambda}+\frac{\epsilon-\lambda'^{2}}{\lambda^{2}}\right)\langle \partial_{r},e_{j} \rangle^{2}\sigma_{k-1;j}\geq 0,\end{aligned}$$ where the inequality follows from that $\frac{\lambda''}{\lambda}+\frac{\epsilon-\lambda'^{2}}{\lambda^{2}}\geq 0$, $x(M)$ is star-shaped and $k$-convex.
Similar to the previous proof, we know $$-(n-k)\langle \nabla\sigma_{k},\lambda\partial_{r} \rangle\geq 0$$ and $$((n-k)\sigma_{1}\sigma_{k}-n(k+1)\sigma_{k+1})u\geq 0.$$ But Lemma \[key\] shows that the integral of these terms are zero. Thus, we know all these inequalities are actually equalities which implies $x(M)$ is totally umbilic. The rest of the proof is similar to the previous one.
Since $H_{k}=\text{constant}$, we know $\nabla\sigma_{k}=0$. From the proof of Theorem \[nablaHk\], $$-\bar{R}_{\nu pjp}\langle \lambda\partial_{r},e_{j} \rangle\sigma_{k-1;jp}=0.$$ Combining with the condition , we obtain ${\lvert \partial_{r}^{\top} \rvert}=0$, which implies $x(M)$ is a slice.
From $H_{k}=\phi(r)$, by direct calculation, we have $$\langle \nabla H_{k},\partial_{r} \rangle=\phi'{\lvert \partial_{r}^{\top} \rvert}^{2}.$$
Since $\Phi(r)$ is non-increasing, we know $\phi'(r)\leq 0$. Thus, $$\langle \nabla H_{k},\partial_{r} \rangle\leq 0.$$
By Theorem \[nablaH\] or Theorem \[nablaHk\], we finish the proof.
From $H_{k}^{-\alpha}=u$, we have $$\begin{aligned}
\langle \nabla H_{k},\partial_{r} \rangle=-\frac{1}{\alpha}u^{-\frac{1}{\alpha}-1}\langle \nabla u,\partial_{r} \rangle=-\frac{1}{\alpha}u^{-\frac{1}{\alpha}-1}\lambda\kappa_{i}\langle e_{i},\partial_{r} \rangle^{2}.\end{aligned}$$
Since $x(M)$ is strictly convex, from $u=H_{k}^{-\alpha}$, we know $u>0$. By $\alpha>0$, $$\langle \nabla H_{k},\partial_{r} \rangle\leq 0.$$
From the proof of Theorem \[nablaH\] or Theorem \[nablaHk\], we know $$\langle \nabla H_{k},\partial_{r} \rangle=0.$$ This implies $\partial_{r}$ is the normal vector of $x(M)$, which means $x(M)$ is a slice.
Proof of Theorem \[Hklambda\] {#embed}
=============================
In this section we give the proof of Theorem \[Hklambda\]. By Lemma 2.3 in [@Li-Wei-Xiong], we know $x(M)$ is $k$-convex from $H_{k}>0$. Thus, Maclaurin’s inequality $$\label{Mac}
H_{k}^{\frac{1}{k}}\leq H_{k-1}^{\frac{1}{k-1}}$$ holds. From Brendle’s Heintze-Karcher type inequality established in [@Brendle13] $$\begin{aligned}
\int_{M}ud\mu \leq \int_{M}\frac{\lambda'}{H_{1}}d\mu\end{aligned}$$ and the Minkowski type formula (see [@Brendle-Eichmair]) $$\begin{aligned}
\int_{M}H_{k}ud\mu\geq \int_{M}H_{k-1}\lambda'd\mu,\end{aligned}$$ combining with Maclaurin’s inequality and $H_{k}^{-\alpha}\lambda'=u$, we obtain $$\label{Hkalf}
\int_{M}H_{k}^{-\alpha}\lambda'd\mu \leq \int_{M}\frac{\lambda'}{H_{1}}d\mu \leq \int_{M}H_{k}^{-\frac{1}{k}}\lambda'd\mu$$ and $$\label{Min}
\int_{M}H_{k}^{1-\alpha}\lambda'd\mu\geq \int_{M}H_{k-1}\lambda'd\mu.$$
By Hölder’s inequality, Maclaurin’s inequality and , we have $$\begin{aligned}
\int_{M}H_{k}^{-\frac{1}{k}}\lambda'd\mu &\leq \left(\int_{M}H_{k-1}^{1-p}H_{k}^{-\frac{p}{k}}\lambda'd\mu\right)^{\frac{1}{p}}\left(\int_{M}H_{k-1}\lambda'd\mu\right)^{\frac{p-1}{p}}\\
&\leq \left(\int_{M}H_{k}^{\frac{-kp-1+k}{k}}\lambda'd\mu\right)^{\frac{1}{p}}\left(\int_{M}H_{k}^{1-\alpha}\lambda'd\mu\right)^{\frac{p-1}{p}}.\end{aligned}$$
Choose $p$ such that $p-1+\frac{1}{k}=\alpha$, then $p=\frac{k\alpha+k-1}{k}$. Notice that $p\geq 1$ implies $\alpha\geq \frac{1}{k}$. The above inequality becomes $$\label{Hk}
\int_{M}H_{k}^{-\frac{1}{k}}\lambda'd\mu \leq \left(\int_{M}H_{k}^{-\alpha}\lambda'd\mu\right)^{\frac{k}{k\alpha+k-1}}\left(\int_{M}H_{k}^{1-\alpha}\lambda'd\mu\right)^{\frac{k\alpha-1}{k\alpha+k-1}}.$$
Using Hölder’s inequality, and again as before, we obtain $$\begin{aligned}
\int_{M}H_{k}^{1-\alpha}\lambda'd\mu &\leq \left(\int_{M}H_{k}^{\frac{-1+k+p-pk\alpha}{k}}\lambda'd\mu\right)^{\frac{1}{p}}\left(\int_{M}H_{k}^{1-\alpha}\lambda'd\mu\right)^{\frac{p-1}{p}}.\end{aligned}$$ Equivalently, $$\begin{aligned}
\int_{M}H_{k}^{1-\alpha}\lambda'd\mu &\leq \int_{M}H_{k}^{\frac{-1+k+p-pk\alpha}{k}}\lambda'd\mu.\end{aligned}$$
Now we choose $p$ such that $\frac{-1+k+p-pk\alpha}{k}=-\alpha$. Then $p=\frac{k\alpha-1+k}{k\alpha-1}$. Thus, the above inequality is $$\begin{aligned}
\int_{M}H_{k}^{1-\alpha}\lambda'd\mu &\leq \int_{M}H_{k}^{-\alpha}\lambda'd\mu.\end{aligned}$$
Substituting the above inequality into , we obtain $$\int_{M}H_{k}^{-\frac{1}{k}}\lambda'd\mu \leq \int_{M}H_{k}^{-\alpha}\lambda'd\mu.$$
Combining the above inequality with , we know that the equality of the Heintze-Karcher type inequality occurs. As [@Brendle13], we finish the proof.
{#variation}
Let $ \bar{M}^{n+1}$ be an oriented Riemannian manifold and let $x:M^{n}\rightarrow \bar{M}^{n+1}$ be an immersion of a closed smooth $n$-dimensional manifold $M$ into $\bar{M}^{n+1}$. Suppose that a smooth map $X:(-\epsilon,\epsilon)\times M\rightarrow \bar{M}$ is a normal variation satisfying $$\frac{\partial}{\partial t}X=-f\nu,$$ where $f$ is a smooth function on $M$ and $\nu$ is the unit normal of $X(t, M)$.
We introduce the weighted volume $V: (-\epsilon, \epsilon)\rightarrow \mathbb{R}$ by $$V(t)=\int_{[0,t]\times M}X^{*}(e^{\Psi}d\bar{\mu}),$$ where $d\bar{\mu}$ is a standard volume element of $\bar{M}$ and $\Psi$ is a smooth function on $\bar{M}$. Thus $V(t)$ represents the (oriented) weighted volume sweeping by $M$ on the time interval $[0,t)$. By the same calculations as in [@Barbosa-doCarmo-Eschenburg], we have $$V'(t)=\int_{M}fX^{*}(e^{\Psi})d\mu,$$ where $d\mu$ is the volume element of $M$ with respect to the induced metric.
A variation of $x:M\rightarrow\bar{M}$ is called weighted volume-preserving variation if $V(t)\equiv0$.
Denote $$J(t)=A(t)+nH_{0}V(t),$$ where $A(t)=\int_{M}d\mu$ and $H_{0}=\frac{\int_{M}Hd\mu}{\int_{M}X^{*}(e^{\Psi})d\mu}$.
Then $$\begin{aligned}
J'(0)=\int_{M} nf(-H+H_{0}e^{\psi}) d\mu,\end{aligned}$$ where $\psi$ denotes $x^{*}(\Psi)$.
\[critical\] The following three statements are equivalent:
(i) The mean curvature of $x(M)$ satisfies $H=Ce^{\psi}$ for a constant $C$.
(ii) For all weighted volume-preserving variations, $A'(0)=0$.
(iii) For arbitrary variations, $J'(0)=0$.
It is easy to check $(i)\Rightarrow (iii)$ and $(iii)\Rightarrow (ii)$. Now, we show $(ii)\Rightarrow (i)$. We can choose $f=-He^{-\psi}+H_{0}$ since $\int_{M}fe^{\psi}d\mu=0$. From $$0=J'(0)=\int_{M}n(-He^{-\psi}+H_{0})^{2}e^{\psi}d\mu,$$ we know $-He^{-\psi}+H_{0}=0$. Thus, we finish the proof.
[10]{}
Luis J. Alías, Debora Impera, and Marco Rigoli. Hypersurfaces of constant higher order mean curvature in warped products. , 365(2):591–621, 2013.
J. Lucas Barbosa, Manfredo do Carmo, and Jost Eschenburg. Stability of hypersurfaces of constant mean curvature in [R]{}iemannian manifolds. , 197(1):123–138, 1988.
João Lucas Barbosa and Manfredo do Carmo. Stability of hypersurfaces with constant mean curvature. , 185(3):339–353, 1984.
Simon Brendle. Constant mean curvature surfaces in warped product manifolds. , 117:247–269, 2013.
Simon Brendle and Michael Eichmair. Isoperimetric and [W]{}eingarten surfaces in the [S]{}chwarzschild manifold. , 94(3):387–407, 2013.
Kwok-Kun Kwong, Hojoo Lee, and Juncheol Pyo. Weighted [H]{}siung-[M]{}inkowski formulas and rigidity of umbilical hypersurfaces. , 25(2):597–616, 2018.
Haizhong Li, Yong Wei, and Changwei Xiong. A note on [W]{}eingarten hypersurfaces in the warped product manifold. , 25(14):1450121, 13, 2014.
Sebastián Montiel. Stable constant mean curvature hypersurfaces in some [R]{}iemannian manifolds. , 73(4):584–602, 1998.
Sebastián Montiel. Unicity of constant mean curvature hypersurfaces in some [R]{}iemannian manifolds. , 48(2):711–748, 1999.
Barrett O’Neill. , volume 103 of [*Pure and Applied Mathematics*]{}. Academic Press, Inc. \[Harcourt Brace Jovanovich, Publishers\], New York, 1983.
Alain R. Veeravalli. Stability of compact constant mean curvature hypersurfaces in a wide class of [R]{}iemannian manifolds. , 159:1–9, 2012.
Jie Wu and Chao Xia. On rigidity of hypersurfaces with constant curvature functions in warped product manifolds. , 46(1):1–22, 2014.
[^1]: This work was supported by National Natural Science Foundation of China (Grant No. 11671223 and Grant No. 11831005).
|
---
abstract: 'We present a class of exact vacuum solutions corresponding to de Sitter and warm inflation models in the framework of scalar-tensor cosmologies. We show that in both cases the field equations reduce to planar dynamical systems with constraints. Then, we carry out a qualitative analysis of the models by examining the phase diagrams of the solutions near the equilibrium points.'
author:
- |
V. B. Bezerra$^{1}$, C. Romero$^{1}$, G. Grebot$^{2}$,\
M. E. X. Guimarães$^{2}$, L. P. Colatto$^{3,4}$\
\
\
\
\
\
\
title: 'Remarks on some vacuum solutions of scalar-tensor cosmological models'
---
-1.2cm
Introduction
============
Scalar-tensor theories of gravity[@ST1; @ST2] represent the most natural alternatives to Einstein’s theory of general relativity. The simplest and earliest scalar-tensor theory[@ST1] considered a massless scalar field and was formulated by using as basic metric tensor the physical tensor $\tilde{g}_{\mu \nu }$, to which matter is universally coupled (Jordan-Fierz frame). Later, these theories were generalized[@ST2] by introducing a scalar field self-interaction and a dynamical coupling to matter.
In what concerns cosmology, the presence of a scalar field - which from now on we will call generically [*dilaton*]{}, has gravitational-strength couplings to matter which violate the equivalence principle. To avoid conflicts with experimental tests on the equivalence principle, it is assumed that the dilaton acquires a mass large enough that any deviations from Einstein’s theory are confined on scales that are not sensitive on cosmological scales. However, a lot of work[@kal] has been done in the framework of low energy string theory - which is reminiscent of scalar-tensor theories of gravity - focusing in the case where the scalar field is massless. The mechanism which naturally reconciles a massless dilaton with experimental test was proposed in Ref.[@DP]. Indeed, a massless dilaton is shown to obey a Minimal Coupling Principle, e.g., to decouple from matter by cosmological attraction in much the same way as the generic attractor mechanism of the scalar-tensor theories of gravity[@DN].
On the other hand, recent observational data which contain evidence for an accelerated expansion of the universe[@riess] indicate that this may be induced by scalar fields which appear naturally in scalar-tensor models. Therefore, it is important to consider various possibilities of cosmological scenarios in order to study, for example, the asymptotic behaviour at late times of Friedmann-Robertson-Walker (FRW) cosmological models and to investigate if the universe evolves towards a state indistinguishable or not from the one predicted by general relativity [@santiago].
The aim of this paper is to study vacuum solutions in the context of FRW cosmologies with flat spatial curvature ($k=0$) for the cases of de Sitter models and warm inflation[@freese]. It turns out that these two cases lead to a class of field equations that can be written as planar dynamical systems plus a constraint equation.
This work is outlined as follows. In section 2, we briefly describe the scalar-tensor cosmology. In sections 3 and 4, we find the solutions of the field equations for the de Sitter and warm inflation models, respectively. Finally, in section 5, we present some final remarks concerning our results.
Scalar-Tensor Cosmology: A Brief Review
=======================================
In this section we will make a brief review of the scalar-tensor cosmological models and write out the field equations which we are going to deal with in the next section. We start by considering the most general scalar-tensor theories of gravity in the Jordan-Fierz frame which is given by the action
$${\cal S}= \frac{1}{16\pi} \int d^4 x \sqrt{-\tilde{g}} \left[\tilde{R} {\Phi}
- \frac{\omega({\Phi})}{\Phi}\partial^{\mu}{\ \Phi} \partial_{\mu}{\Phi} - 2{\tilde{\Lambda}}(\Phi)\right] + {\cal S}_{m}[\Psi_m , \tilde{g}_{\mu\nu}] ,
\label{JF}$$
where $\tilde{g}_{\mu\nu}$ is the physical metric, $\tilde{R}$ is the curvature scalar associated to it, ${\tilde{\Lambda}}(\Phi)$ is a cosmological term which corresponds to the scalar field potential and ${\cal S}_{m}$ is the action for general matter fields which, at this point, is left arbitrary.
The physical frame of Jordan-Fierz has the disadvantage of featuring complicated evolution equations for the gravitational and scalar fields and for this reason it is more convenient to work in the Einstein (conformal) frame, in which the scalar and tensor degrees of freedom do not mix. Now, let us define two new variables: $g_{\mu\nu}$, the metric tensor in the Einstein frame, and the scalar field $\phi$. Thus, making the transformation
$$\label{conf}
\tilde{g}_{\mu\nu} = A^2(\phi) g_{\mu\nu},$$
we decouple the two modes of propagations. This relation tells us that the metric tensor in the Einstein frame is conformally related to the physical metric tensor in Jordan-Fierz frame. Besides, by a redefinition of the following quantities $$\begin{aligned}
\Phi &=& \frac{1}{G A^2(\phi)}, \nonumber \\
\Lambda(\phi) & = & A^4(\phi){\tilde{\Lambda}}, \\
\alpha(\phi) & = & \frac{d \ln A(\phi)}{d\phi}, \nonumber\end{aligned}$$ where $G$ is the bare gravitational constant and, by imposing the constraint,
$$\label{alp}
\alpha^2(\phi) = \frac{1}{[2\omega(\phi) + 3]},$$
the new quantities $g_{\mu\nu} , \;\; \phi \;\; \mbox{and} \;\; A(\phi)$ are uniquely defined in terms of the original quantities ${\tilde g}_{\mu\nu} ,
\Phi , \omega(\Phi)$. In the Einstein frame, the action (1) turns into $${\cal S} = \frac{1}{16\pi G} \int d^4x \sqrt{-g} \left[ R - 2g^{\mu\nu}
\partial_{\mu}\phi \partial_{\nu}\phi - 2\Lambda(\phi)\right] + {\cal S}_m
[\Psi_m , A^2(\phi)g_{\mu\nu}], \label{EF}$$
In this new frame the field equations read
$$\begin{aligned}
\label{eqs}
G_{\mu\nu} + g_{\mu\nu}\Lambda(\phi) & = & 8\pi G T_{\mu\nu} + 2(\phi_{,
\mu} \phi_{, \nu} - \frac{1}{2} g_{\mu\nu}\phi^{, \sigma}\phi_{, \sigma} ),
\nonumber \\
\Box_g \phi -\frac{1}{2} \frac{d\Lambda (\phi)}{d\phi} & = & - 4\pi G
\alpha(\phi) T,\end{aligned}$$
where $$T_{\mu\nu} = \frac{2}{\sqrt{-g}}\frac{\delta {\cal S}_m}{\delta g^{\mu\nu}},$$ with $$T^{\mu}_{\nu ; \mu} = \alpha(\phi) T \partial_{\nu} \phi ,$$ which means that the energy-momentum tensor in the conformal frame is no longer conserved, differently from the Jordan-Fierz frame in which the energy-momentum tensor is conserved.
In what follows we will concentrate on the Friedmann-Robertson-Walker (FRW) cosmologies with flat spatial curvature ($k=0$) and perfect-fluid matter distributions. These models are represented by a spacetime with metric
$$\label{frw}
ds^2 = - dt^2 + R^2(t) [ dr^2 + r^2(d\theta^2 + \sin^2 \theta d\varphi^2)],$$
being sourced by an energy-momentum tensor corresponding to a perfect fluid given by
$$T^{\mu \nu }=(\rho +p)u^{\mu }u^{\nu }+pg^{\mu \nu }, \label{em}$$
with $u^{\mu }\equiv \frac{dx^{\mu }}{d{\tau }}$ denoting the 4-velocity of the fluid in the Einstein frame. We can relate quantities such as density and pressure in both frames through the equations
$$\rho = A^4(\phi) \tilde{\rho} \;\;\; \mbox{and} \;\;\; p = A^4(\phi) \tilde{p}.$$
For the FRW models, equations (\[eqs\]) become $$\begin{aligned}
\label{eqs2}
- 3 \frac{\ddot{R}}{R} & = & 4\pi G (\rho + 3p) + 2(\dot{\phi})^2 -
\Lambda(\phi), \nonumber \\
3 \left( \frac{\dot{R}}{R}\right)^2 & = & 8\pi G \rho + (\dot{\phi})^2 +
\Lambda(\phi), \\
\ddot{\phi} + 3\frac{\dot{R}}{R}\dot{\phi} & = & - 4\pi G \alpha(\phi) (\rho
- 3p) - \frac{1}{2} \frac{d\Lambda}{d\phi}. \nonumber\end{aligned}$$
In this work we are going to study general solutions of the above equations for two particular vacuum inflationary models: the de Sitter model (corresponding to $\Lambda(\phi) = \Lambda_0 = \mbox{constant}$) and the warm inflation model [@freese] (corresponding to $\Lambda(\phi) = 3
\beta H^2$, with $H \equiv \dot{R}/R$). Let us define two new variables, $\psi$ and $H$, being given by $$\begin{aligned}
\psi & \equiv & \dot{\phi} \\
H & \equiv & \frac{\dot{R}}{R}\end{aligned}$$
In terms of these two new variables, the equations (\[eqs2\]) for the case of de Sitter model can be written as $$\label{sit1}
\dot{H} = - H^2 -\frac{2}{3}\psi^2 + \frac{\Lambda_0}{3},$$
$$\label{sit2}
\dot{\psi} = - 3H \psi,$$
$$\label{sit3}
H^2 = \frac{\psi^2}{3} + \frac{\Lambda_0}{3},$$
with the last equation being a constraint equation.
On the other hand, for the warm inflation model we have $$\label{warm1}
\dot{H} = (\beta - 1) H^2 -\frac{2}{3}\psi^2,$$
$$\label{warm2}
\dot{\psi} = (2\,\beta- 3)H\,\psi-3\,\beta(\beta-1)\frac{H^3}{\psi},$$
with the constraint equation given by
$$\label{warm3}
H^2 = \frac{\psi^2}{3(1-\beta)}.$$
Thus, for the cases of de Sitter and warm inflation models, the field equations (9) can be written as a planar dynamical system plus a constraint equation in the variables $H$ and $\psi$, as we can conclude from the analysis of Eqs.(\[sit1\])-(\[sit3\]) and (\[warm1\])-(\[warm2\]), respectively.
The de Sitter Model in Scalar-Tensor Cosmologies
================================================
Let us now find analytical solutions of Eqs.(\[sit1\])-(\[sit3\]). From Eq.(\[sit1\]) and the constraint Eq.(\[sit3\]) we get
$$\label{sit4}
\dot{H} = \Lambda_0 - 3 H^2,$$
which can be integrated giving
$$\label{sol1}
H_1= \frac{\sqrt{\frac{\Lambda_0}{3}} \tanh(\sqrt{3 \Lambda_0}(t-t_0)) + H_0}{1 + \sqrt{\frac{3}{\Lambda_0}} H_{0}\tanh(\sqrt{3 \Lambda_0}(t-t_0))},$$
$$\label{sol2}
H_2= \frac{\sqrt{\frac{-\Lambda_0}{3}} \tan(\sqrt{-3 \Lambda_0}(t-t_0)) + H_0}{1 - \sqrt{\frac{-3}{\Lambda_0}} H_{0}\tan(\sqrt{-3 \Lambda_0}(t-t_0))},$$
$$\label{sol3}
H_3=\frac{H_0}{1 + 3H_{0}(t-t_0)},$$
for $\Lambda > 0$, $\Lambda < 0$, $\Lambda = 0$, respectively, with $H_0$ being an integration constant. From Eqs.(\[sit1\])-(\[sit3\]) and (\[sol1\])-(\[sol3\]) we obtain the solutions $\psi(t)$ for each value of $\Lambda_0$. Thus, we have the following solutions $(H_{i}(t),
\psi_{i}^{\pm}=\pm\sqrt{3H_{i}^{2} - \Lambda_0})$, for $i=1,2$, and $(H_{3}(t), \psi_{3}=\pm\sqrt{3}H_{3})$.
Let us note that the constraint equation (\[sit3\]) is compatible with (\[sit1\]) and (\[sit2\]). The same remark applies to the equations (\[warm1\])-(\[warm3\]). We conclude that the constraint equations (\[sit3\]) and (\[warm3\]) are nothing but particular curves of the set of integral curves of the vector fields defined by the right-hand side of Eqs.(\[sit1\]), (\[sit2\]), (\[warm1\]) and (\[warm2\]). It can be directly verified that the first integral of the dynamical system formed by (\[sit1\]) and (\[sit2\]) is given by
$$\label{eq1}
H^2 = a\psi^{2/3} + {\frac{1}{3}}(\psi^2 + \Lambda_0)$$
with $a=0$ corresponding to the constraint equation (\[sit3\]).
Now, if a dynamical system has critical points (equilibrium points), it is often useful to investigate the behaviour of the solutions near these points. For $\Lambda >0$, Eqs.(\[sit1\])-(\[sit3\]) admit four critical points in the phase plane $H\psi $. These points are located at $A(\sqrt{\frac{\Lambda _{0}}{3}}),0)$; $B(-\sqrt{\frac{\Lambda _{0}}{3}},0)$; $C(0,\sqrt{\frac{\Lambda _{0}}{2}})$; $D(0,-\sqrt{\frac{\Lambda _{0}}{2}},0)$. These four points themselves represent solutions of the dynamical system formed by Eqs.(\[sit1\]) and (\[sit2\]), however only $A$ and $B$ satisfy the constraint equation given by (\[sit3\]). Incidentally, note that the solutions represented by $A$ and $B$ are obtained from Eq.(\[sol1\]) by assigning the values $H_{0}=\pm \sqrt{\frac{\Lambda _{0}}{3}}$. Moreover, $A$ and $B$ correspond(in the Einstein frame) to de Sitter cosmological models whose scale factors are given, respectively, by
$$\label{eq2}
R(t)=R_{0}\exp({\sqrt{\frac{\Lambda_0}{3}} t}),$$
$$R(t)=R_{0}\exp ({-\sqrt{\frac{\Lambda _{0}}{3}}t}), \label{eq3}$$
where $R_{0}$ is a constant. For $\Lambda <0$ the system has no critical points, while for $\Lambda =0$ there is only one critical point at the origin O(0,0) of the phase plane, corresponding, modulo a rescaling of the coordinates, to the Minkowski spacetime. In all the above configurations represented by the critical points the scalar field $\phi $ is constant, since $\psi =0$.
We now will draw the phase diagrams corresponding to the solutions given by Eqs.(\[sol1\])-(\[sol3\]) plus those represented by the critical points. The curves appearing in these diagrams represent parametric solutions $(H(t),\psi (t))$ evolving in time, in the Einstein frame. Of particular interest are the constant solutions corresponding to the equilibrium points since all the solutions considered are attracted or repelled by them. The position of the equilibrium points, when they exist, depends on the values assigned to $\Lambda _{0}$.
![ Phase diagram for $\Lambda _{0}>0.$ The critical points $A$ and $B$ represent de Sitter universes.[]{data-label="Fig1"}](hyper1.eps)
The first diagram(see Fig. 1) refers to the case $\Lambda _{0}>0$. In this case we have six solutions represented by the two critical points $(A$ and $B
$, the two curves lying in the upper part of the hyperbolae and the two curves lying in the lower part. As we have seen before, the points $A$ and $B
$ describe expanding and collapsing universes in the Einstein frame (see Eqs.(\[eq2\]) and (\[eq3\])). At this point, it is worth noting that since $\phi =constant$ in all solutions represented by the points $A$, $B$ and $O$, we see from (\[conf\]) that the metric tensor $\tilde{g}_{\mu \nu }$ is obtained from $g_{\mu \nu }$ by simply rescaling the coordinates. Thus, we have essentially the same geometry in both Einstein and Jordan-Fierz frames. It is also easy to see that, when $H_{0}>\sqrt{\frac{3}{\Lambda _{0}}}$ the point $A$ acts as an attractor for the two solutions $(H_{1}(t),\psi
_{1}^{+})t))$ and $(H_{1}(t),\psi _{1}^{-})t))$ when $t\rightarrow \infty $. By reasons of continuity the same qualitative behaviour is carried over into the Jordan-Fierz frame. Quite analogously, when $H_{0}<-\sqrt{\frac{3}{\Lambda _{0}}}$, then both $(H_{1}(t),\psi _{1}^{+}(t))$ and $(H_{1}(t),\psi
_{1}^{-}(t))$ move away from $B$ as time goes by. For $\Lambda =0$ ( Fig. 2 ) we have three solutions, which describe an expanding universe, a collapsing universe and Minkowski spacetime (represented by the equilibrium point at the origin). When $\Lambda <0$ ( Fig.3 ) we have no equilibrium points and in the Einstein frame the solutions appear as bouncing universes possessing an expansion stage followed by a collapsing era.
![ Phase diagram for $\Lambda _{0}=0$. Here the critical points $A$and $B$ representing de Sitter universe merge into the origin (Minkowski spacetime)[]{data-label="Fig2"}](hyper2.eps)
![ Phase diagram for $\Lambda _{0}<0.$ The critical points disappear.[]{data-label="Fig3"}](hyper4.eps)
The Warm Inflation Model in Scalar-Tensor Cosmologies
=====================================================
Let us now consider the case of warm inflation, which corresponds to the system of equations (\[warm1\])-(\[warm3\]). Here, the solutions for $H(t)$ and $\psi(t)$ are easily obtained if we substitute Eq.(\[warm3\]) into (\[warm1\]). This leads to
$$\label{warm4}
\dot H= 3(\beta - 1)H^2$$
the solution of which is given by
$$\label{warm5}
H(t)=\frac{H_0}{1 + 3H_{0}(1 - \beta)(t - t_0)},$$
where $H_0$ is an integration constant. From the constraint equation (\[warm3\]) we have
$$\label{warm6}
\psi(t)= \pm {\sqrt{3( 1 - \beta)} H(t)}.$$
Clearly the constraint equation (\[warm3\]) also implies that $\beta <1$. ( The case $\beta =1$ leads to $\psi =0$ and $H=H_{0}$, which describes a de Sitter universe).
![Phase diagram for the warm inflation model.[]{data-label="Fig4"}](hyper3.eps)
The phase diagram corresponding to the solutions (\[warm5\]) and (\[warm6\]) is shown in Fig.4. These solutions represent expanding and collapsing universes, for $H>0$ and $H<0$, respectively, and Minkowski spacetime ($H=\psi=0$). The expanding universe starts with a big-bang at $t^{*}=t_{0} + \frac{1}{3H_{0}(\beta - 1)}$ and approach Minkowski spacetime as $t \rightarrow \infty$, gradually slowing their expansion rate. On the other hand, the collapsing models start at $t=-\infty$ as Minkowski spacetime and collapse at $t=t^{*}$. As we have noted in the previous section, the same qualitative analysis of the solutions in the vicinity of the origin may again be carried over into the Jordan-Fierz frame.
Final remarks
=============
In this paper we have presented a class of exact solutions corresponding to vacuum solutions of de Sitter and warm inflation models in the context of scalar-tensor cosmology. These two particular models provide a class of field equations that can be written as planar dynamical systems plus a constraint equation.
In the case of the de Sitter model the dynamical system phase diagrams show that if $\Lambda_{0} > 0$ and $\Lambda=0$ there exist solutions corresponding to critical points (de Sitter universes). It just so happens that in these cases the scalar field is constant. This fact allows us to carry over our qualitative analysis of the solutions near the equilibrium points from the Einstein frame to the Jordan-Fierz physical frame. The same remarks applies to the warm inflation model, where we have only one equilibrium point, which corresponds to Minkowski space-time. Finally, we would like to stress that we do not touch here the problem of providing a mechanism to terminate the inflationary phase of the universe, our solutions for both models being valid for all $t$.
Acknowledgments {#acknowledgments .unnumbered}
===============
V.B.B. and C.R. would like to thanks CNPq for partial financial support. V.B.B., M.E.X.G. and L.P.C. would like to thank CAPES in the context of the interinstitutional program PROCAD/CAPES for partial financial support. M.E.X.G. would like to thanks the kind hospitality of the Departamento de Física of the Universidade Federal da Paraíba where part of this work has been developed.
[99]{} M.Fierz, Helv. Phys. Acta [**29**]{}, 128 (1956); P. Jordan, Z. Phys. [**157**]{}, 112 (1959); C. Brans and R. H. Dicke, Phys. Rev. [**24**]{}, 925 (1961).
P. G. Bergmann, Int. J. Theor. Phys. [**1**]{}, 25 (1968). R. V. Wagoner, Phys. Rev. [**D1**]{}, 3209 (1970); K. Nordtvedt, Astrophys. J. [**161**]{}, 1059 (1970).
D. Kalligas, K. Nordtvedt and R. V. Wagoner, in [*Proceedings of the Seventh Marcel Grossmann Meeting on General Relativity*]{} (World Scientific, 1996); A. Serna and J. M. Alimi, [*Phys. Rev. D*]{}[**50**]{}, 7304 (1996); R. V. Wagoner and D. Kalligas, in [*Gravitation and Gravitational Radiation, Les Houches 1995*]{} (Cambridge, 1996); D. I. Santiago, D. Kalligas and R. V. Wagoner, [*Phys. Rev. D*]{}[**56**]{}, 7627 (1997).
T. Damour, A. M. Polyakov, Nucl. Phys. [**B 423**]{}, 532 (1994).
T. Damour, K. Nordtvedt, Phys. Rev. Lett. [**70**]{}, 2217 (1993); Phys. Rev. D [**48**]{}, 3436 (1993).
A. G. Riess et al., [*Astron. J.*]{} [**116**]{} 1009, (1998); S. Perlmutter et al., [*Ap. J.*]{} [**517**]{}, 565 (1998).
D. I. Santiago, D. Kalligas and R. V. Wagoner, [*Phys. Rev. D*]{} [**58**]{}, 124005 (1998).
K. Freese, F. C. Adams, J. A. Frieman and E. Mottola, [*Nucl. Phys.*]{} [**B287**]{}, 797 (1987); J. M. F. Maia and J. A. S. Lima, [*Phys. Rev. D*]{}[**60**]{}, 101301 (1999).
A.D. Polyanin and V. F. Zaitsev, [*Handbook of Exact Solutions for Ordinary Differential Equations*]{}, CRC Press, 1995.
J. S. R. Chrisholm and A. K. Common, J. Phys. A [**20**]{}, 5459 (1987).
|
---
abstract: 'This paper proposes a non-smooth control scheme to achieve finite-time convergence to an eigenstate of the internal Hamiltonian of a given closed quantum system. First, we define finite-time stability for quantum systems, and then present a Lyapunov stability criterion to identify the finite-time stability of quantum systems. Second, we propose a new non-smooth control law for two-level quantum systems by selecting a distance between quantum states as a Lyapunov function and using the Lyapunov stability theory, and prove the existence and uniqueness of solutions of the quantum system under the action of the non-smooth controller in the framework of Bloch vectors. Further, an equivalent transformation is made for the system model and the control target by expressing quantum states in terms of complex exponentials, and the finite-time stability of the system is proved by combining the finite-time Lyapunov stability criterion with the homogeneity theorem. Finally, we perform numerical simulations on a spin-1/2 particle system and demonstrate the effectiveness of the finite-time stabilization control scheme.'
address:
- 'Department of Automation, University of Science and Technology of China, Hefei 230027, PR China'
- 'School of Engineering and Information Technology, University of New South Wales, Canberra ACT 2600, Australia'
author:
- Sen Kuang
- Xiaoke Guan
- Daoyi Dong
title: 'Finite-time stabilization control of quantum systems'
---
,
,
quantum systems, finite-time stability, non-smooth control, quantum control, finite-time convergence
Introduction {#sec1}
============
The frontier technologies such as nanotechnology, nuclear magnetic resonance (NMR), and ultrafast laser make it possible to manipulate matter at the level of a single atom or a single electron, which has been motivating the rise of quantum control. As a new multidisciplinary research field, quantum control promotes the development of quantum computing, quantum communication, quantum optics, quantum chemistry and other fields [@dong2010quantum]. One of important approaches to studying the control of quantum systems is to extend the concepts and methods in control theory to microscopic quantum systems. For instance, quantum optimal control [@dolde2014high; @PhysRevA.95.063418], quantum Lyapunov control [@5409612; @ZHAO20121833; @kuang2017rapid; @8481533], sliding mode control [@dong2012sliding; @DONG20123089], quantum $H^\infty$ control [@4625217; @XIANG20178], structure decomposition of linear quantum systems [@7942122], fault-tolerant quantum control and filtering [@7556290; @GAO2016125], and quantum control based on machine learning [@biamonte2017quantum; @PhysRevA.99.042327; @niu2019universal] have been extensively investigated. The designed control laws based on these methods are usually smooth or discontinuous. Generally, smooth control laws achieve infinite-time convergence of the system state to the target state, while discontinuous control laws may lead to the non-existence of solutions of the system dynamics so that the system stops evolving toward the target state although they can speed up the control process.
In order to incorporate the rapidness and convergence of the control process, the non-smooth control between smooth control and discontinuous control was proposed in 1961 [@dorato1961short] and can be used to achieve finite-time convergence of the system. At present, the design of non-smooth controllers mainly focuses on some systems with special structures, e.g., upper triangular systems [@ding2011global], lower triangular systems [@huang2005global], and integral systems [@hong2002finite]. For upper triangular nonlinear systems, Ref. [@tsinias2001explicit] proposed a non-smooth stabilizing controller by combining the nested saturation method [@teel1996nonlinear] and the Lyapunov method [@mazenc1996adding] and achieved the global stabilization of the system in an infinite time interval. On the basis of [@tsinias2001explicit], Ref. [@ding2011global] designed a global finite-time stabilizing controller, which is suitable for a broader class of upper triangular systems, by further considering the adding-a-power integrator technique [@qian2001continuous]. For lower triangular nonlinear systems, several global finite-time stability results were obtained in [@huang2005global] by using the same adding-a-power integrator technique when the system parameters satisfy certain conditions. Ref. [@hong2006non] considered more universal $P$-normal-form lower triangular systems and designed a more general finite-time stabilizing controller by constructing a Lyapunov function different from that in [@huang2005global]. For integral systems, Ref. [@wang2009finite] designed a globally finite-time stabilizing controller for third-order systems via the adding-a-power integrator technique and gave a method for the selection of parameters contained in the controller. Ref. [@bhat2005geometric] designed a global finite-time stabilizing controller for higher-order systems by considering the Hurwitz stability of the system polynomial. However, it is hard to determine the parameters in the controller. In the above mentioned literature, to prove the finite-time stability of the closed-loop systems, the methods based on the Lyapunov stability theory [@ding2011global; @huang2005global; @tsinias2001explicit; @hong2006non; @wang2009finite] and the homogeneity theory [@bhat2005geometric] have been used. It should be noted that there is no uniform non-smooth control framework for general nonlinear systems without special structures. In this case, dependent on different systems, a special dealing is usually necessary.
Another property that non-smooth control is different from continuously differentiable control or smooth control is that the Lipschitz continuity condition in the usual sense does not hold any longer in the non-smooth control systems. Therefore, the existence and uniqueness of solutions cannot be guaranteed. However, for the aforementioned three types of systems, the existing literatures do not discuss the uniqueness of solutions. This is because that the corresponding closed-loop systems do not satisfy the Lipschitz condition only at the origin. For an other system, e.g., the double integral system in [@bhat1998continuous], the closed-loop system under the action of the designed non-smooth controller may be non-Lipschitz continuous at points other than the origin. In this case, it is essential to discuss the uniqueness of solutions. For example, the notion of transversality has been used to demonstrate the uniqueness of solutions of the closed-loop system at points dissatisfying the Lipschitz continuity [@kawski1989stabilization]. Readers can refer to [@filippov2013differential] and [@agarwal1993uniqueness] for more other methods to show the existence and uniqueness of solutions of non-smooth systems.
This paper considers the non-smooth control problem of quantum systems and aims at achieving the finite-time convergence of a given system to the target state. Since the models of quantum control systems are not of the above special structures, the design methods of finite-time stabilizing controllers and the stability proof methods existing in classical systems cannot be directly applied to quantum systems. The main contributions of this paper are summarized as follows. First, we consider the finite-time convergence control problem of quantum systems. Although the optimal control problem with a fixed terminal time also can be regarded as a finite-time control problem, the exact convergence of the system to the target state within a finite time may not be able to guarantee. Furthermore, the optimal control often suffers from complicated numerical computation problems. Second, the definition of finite-time stability of quantum systems is presented and a finite-time Lyapunov stability criterion is proposed to identify finite-time stability of quantum systems. Third, via the Lyapunov stability theory, we further propose a non-smooth control law (i.e., continuous and non-differentiable control law) with a fractional power factor, where the fractional power factor is a function of the system state. This new control law enables the finite-time convergence of the system. Fourth, for two-level quantum systems, we prove the existence and uniqueness of solutions of the system with the non-smooth control. Due to the non-smoothness of the control law, the vector field of the controlled quantum system no longer satisfies the Lipschitz continuity condition. Therefore, it is important to ensure that the quantum system under the control field has a unique solution, which does not need to be considered in the usual smooth control. Finally, by expressing quantum states in terms of complex exponentials, the homogeneity theory and the finite-time Lyapunov stability criterion are simultaneously used to prove the finite-time stability of the controlled quantum system, i.e., the target state is reached within a finite time.
This paper is organized as follows. Section \[sec2\] introduces the considered quantum system models in the Schrödinger picture and their counterparts in the corresponding Bloch space, and then presents the definition of finite-time stability of quantum systems and a Lyapunov criterion for finite-time stability. In Section \[sec3\], based on the Lyapunov stability theory, we design a non-smooth controller for a two-level quantum system and prove the existence and uniqueness of solutions of the “closed-loop" system under the action of the controller. The finite-time convergence of the “closed-loop" system to the target state, i.e., finite-time stability of the “closed-loop" system, is analyzed and proved in detail in Section \[sec4\]. Section \[sec5\] presents several numerical examples to demonstrate the effectiveness of the proposed finite-time control scheme. Conclusions are presented in Section \[sec6\].
**Notation.** Let $\mathbb{R}$ be the set of real numbers, $\mathbb{R}_+$ be the set of non-negative real numbers, $\nabla$ be a vector differential operator, $\langle\,,\,\rangle$ denote an inner product operation, and $[A,B]$ denote the commutator between $A$ and $B$. The two unit state vectors $|\psi_1\rangle$ and $|\psi_2\rangle$ satisfying $|\psi_1\rangle=e^{i\phi}|\psi_2\rangle$ are called equivalent state vectors, and the set of all equivalence state vectors of $|\psi\rangle$ forms an equivalence class of $|\psi\rangle$. In physics, equivalent state vectors have the same observation meaning, and therefore we can ignore the global phase between equivalent states and regard all states in an equivalence class as the same state. A continuous function is called $\mathrm{C}^0$ smooth; a function with up to $n-$order continuous derivatives is called $\mathrm{C}^n$ smooth; a function with any order continuous derivative is called a smooth function or a $\mathrm{C}^\infty$ function.
Finite-time stability of quantum systems {#sec2}
========================================
In this section, we present several concepts of finite-time stability of quantum systems and the finite-time stability discrimination criteria based on Lyapunov stability theory.
Basic concepts of finite-time stability {#sec2.1}
---------------------------------------
For a general $n$-dimensional closed controllable quantum system, its state is fully described by a unit column vector $|\psi\rangle$ in the Hilbert space defined on $\mathbb{C}^n$ and its dynamics evolution obeys the following Schrödinger equation $$\label{e1}
|\dot{\psi}(t)\rangle=\frac{-i}{\hbar}H{|\psi(t)\rangle}=\frac{-i}{\hbar}\Big(H_0+\sum_{k=1}^rH_ku_k\Big)|\psi(t)\rangle,$$ where $H_0$ is the internal Hamiltonian of the system, $H_k$ is the control Hamiltonian that describes the interaction between the external control fields and the system, $H_0$ and $H_k$ are time-independent; $\hbar$ is the reduced Planck constant and we set $\hbar=1$ in this paper; $u_k$ is an external control field to be designed.
This paper considers the control problem of finite-time convergence to an eigenstate $|\psi_f\rangle$ of the internal Hamiltonian $H_0$ of the system (\[e1\]), that is, the aim is to achieve finite-time convergence to $|\psi_f\rangle$ by designing $u_k$ as a certain non-smooth control law. It should be pointed out that the solutions of the system (\[e1\]) under the non-smooth control is continuously differentiable. Since the quantum system with the non-smooth control no longer satisfies the Lipschitz condition, we can prove the uniqueness of solutions via the method based on the transversality between the system vector field and a non-Lipschitz set [@kawski1989stabilization]. In the Bloch vector framework, the quantum state $|\psi\rangle$ in the Hilbert space can be expressed in terms of the vector $s$ in the Bloch space as $$\label{e2}
{{|\psi\rangle}{\langle\psi|}=\rho={s_0}{\sigma_0}+\frac{1}{2}\sum_{\kappa=1}^{n^2-1}{s_\kappa}{\sigma_\kappa}=\frac{I_n}{n}+\frac{1}{2}\sum_{\kappa=1}^{n^2-1}{s_\kappa}{\sigma_\kappa}},$$ where $\rho$ denotes the density operator, $\{\sigma_\kappa\}_{\kappa=0}^{n^2-1}$ is a set of orthogonal bases of the $n\times n$ complex Hermitian matrix space, $\sigma_0=\frac{I_n}{\sqrt{n}}$, and $s_0=\frac{1}{\sqrt{n}}$. The ($n^2-1$)-dimensional real vector $\left(s_1,\ldots,s_{n^2-1}\right)\triangleq s=\left(\textrm{tr}\left(\rho\sigma_1\right),\ldots,\textrm{tr}\left(\rho\sigma_{n^2-1}\right)\right)\in\mathbb{R}^{n^2-1}\left(n>1\right)$ is called the Bloch vector in a selected basis, and the set of all Bloch vectors forms the Bloch space $\mathcal{B}(\mathbb{R}^{n^2-1})$ [@kimura2003bloch]. For simplicity, we only consider the case of one control field and the generalization to multiple control fields is straightforward. By expressing the quantum state in terms of the Bloch vector $s$, the quantum system (\[e1\]) can be written as [@wang2010analysis] $$\label{e3}
{\dot{s}\left(t\right)=\left(A_0+u_1A_1\right)s\left(t\right)},$$ $$\label{e4}
{\dot{s}_f(t)=A_0s_f(t)},$$ where $s_f(t)$ is the Bloch vector associated with the target state $|\psi_f\rangle$, $A_0$ and $A_1$ are the following antisymmetric matrices $$\label{e5}
{A_0\left(m,n\right)=\mathrm{tr}\left(iH_0\left[\sigma_m,\sigma_n\right]\right)},$$ $$\label{e6}
{A_1\left(m,n\right)=\mathrm{tr}\left(iH_1\left[\sigma_m,\sigma_n\right]\right)}.$$
Denote the system (\[e3\]) as $$\label{e7}
{\dot{s}\left(t\right)=f\left(s\left(t\right)\right),\; s\left(t\right)\in\mathcal{B}\big(\mathbb{R}^{n^2-1}\big)}$$ where $f:\mathcal{B}(\mathbb{R}^{n^2-1})\rightarrow\mathcal{B}(\mathbb{R}^{n^2-1})$ is a continuous function defined on $\mathcal{B}(\mathbb{R}^{n^2-1})$.
Let the initial moment be $t_0=0$. For $t\in\mathbb{R}_+$, if there exists a continuously differentiable function $s(t)$ such that (\[e7\]) holds, then the function $s(t):\mathbb{R}_+\rightarrow\mathcal{B}(\mathbb{R}^{n^2-1})$ is called a solution of the system (\[e7\]). To illustrate the concept of finite-time stability, we assume that the quantum system (\[e7\]) has a unique solution in the space $\mathcal{B}(\mathbb{R}^{n^2-1})$. For any initial vector $s_0\in\mathcal{B}(\mathbb{R}^{n^2-1})$, we denote the unique solution of the system (\[e7\]) as $s(t,s_0)$ ($t\geq0$). Then, $s(t,s_0)$ defines a flow from $\mathbb{R}_{+}\times\mathcal{B}(\mathbb{R}^{n^2-1})$ to $\mathcal{B}(\mathbb{R}^{n^2-1})$ when $s_0$ varies.
Now, we give the definition of finite-time stability of the quantum system (\[e7\]).
\[1\] For the quantum system (\[e7\]), the target vector $s_f$ is said to be finite-time stable if for an arbitrarily given initial vector $s_0\in\mathcal{B}(\mathbb{R}^{n^2-1})$, there exists a continuous function $T(s_0):\mathcal{B}(\mathbb{R}^{n^2-1})\rightarrow[0,\infty)$ such that the unique solution $s(t,s_0)$ of the system (\[e7\]) satisfies $\lim_{t\rightarrow T(s_0)}s(t,s_0)$ and $s(t,s_0)=s_f$ for $t\geq T(s_0)$. Particularly, we define $T(s_f)=0$ when the initial vector is $s_0=s_f$. The continuous function $T(s_0)$ is referred to as the finite convergence time corresponding to $s_0$.
Some properties of the solution $s(t,s_0)$ and the finite convergence time $T(s_0)$ can be summarized in the following proposition.
\[2\] Assume that $s_f$ is the finite-time stable target vector of the system (\[e7\]). Let $s_0\in\mathcal{B}(\mathbb{R}^{n^2-1})$, $t,t_1\in\mathbb{R}_+$, and the finite convergence time function be $T(s_0):\mathcal{B}(\mathbb{R}^{n^2-1})\rightarrow[0,\infty)$ as in Definition \[1\], then the following conclusions hold:\
(i) $s\left(0,s_0\right)=s_0$;\
(ii) $s\left(T\left(s_0\right)+t,s_0\right)=s_f$;\
(iii) $s\left(t,s\left(t_1,s_0\right)\right)=s\left(t+t_1,s_0\right)$;\
(iv) The function $T\left(s_0\right)$ can be written as $$T\left(s_0\right)=\inf\{t\in\mathbb{R}_+:s\left(t,s_0\right)=s_f\};$$ (v) Further, it follows from (ii)-(iv) that $$T\left(s\left(t,s_0\right)\right)=\max\{T\left(s_0\right)-t,0\}.$$
\[3\] Consider the scalar differential equation $$\label{e13}
{\dot{y}\left(t\right)=-k\mathrm{sign}\left(y\left(t\right)\right)|y\left(t\right)|^\alpha}$$ where $\mathrm{sign}(0)=0$, $k>0$, and $\alpha\in(0,1)$.
Since the right-hand side of (\[e13\]) is continuous everywhere and the local Lipschitz condition is always satisfied outside the origin, the system (\[e13\]) has a unique solution for any initial condition $y_0\in\mathbb{R}$. By direct integration, the solution of the system (\[e13\]) can be obtained as $$\label{e14}
\mu\left(t,y_0\right)=
\left\{
\begin{array}{lll}
\mathrm{sign}\left(y_0\right)&\![|y_0|^{1-\alpha}-k\left(1-\alpha\right)t]^{\frac{1}{1-\alpha}},\\
&\quad\quad \big(t<\frac{|y_0|^{1-\alpha}}{k\left(1-\alpha\right)},\,y_0\neq0\big)\\
0,&\quad\quad \big(t\geq\frac{|y_0|^{1-\alpha}}{k\left(1-\alpha\right)},\,y_0\neq0\big)\\
0,&\quad\quad \big(t\geq0,\,y_0=0\big).
\end{array}
\right.$$ It is known from (\[e14\]) that the finite-time convergence function is $T(y_0)=\frac{1}{k(1-\alpha)}|y_0|^{1-\alpha}$. The Lyapunov function $V(y)=y^2$ can be used to prove that the origin of the system (\[e13\]) is globally finite-time stable. Here, we omit the proof for brevity.
Lyapunov theorem of finite-time stability {#sec2.2}
-----------------------------------------
We first give the following comparison lemma:
[@khalil2002nonlinear]\[4\] Let $V$ be a Lyapunov function defined on $\mathbb{R}_{+}\times\mathcal{B}(\mathbb{R}^{n^2-1})$ and assume that $$\label{e15}
%\setcounter{equation}{15}
\dot{V}_E\left(t,m\right)\leq\gamma\left(t,V\left(t,m\right)\right),$$ where $\left(t,m\right)\in\mathbb{R}_{+}\times\mathcal{B}(\mathbb{R}^{n^2-1})$; ${E}$ denotes the differential equation $\dot{x}=F(t,x)$; $\dot{V}_E(t,m)$ represents the time derivative of the Lyapunov function $V$ along any trajectory of the differential equation $E$; $\gamma:\mathbb{R}_{+}\times\mathbb{R}\rightarrow\mathbb{R}$ is a continuous function. Further, we assume that the initial value problem $\dot{m}=\gamma(t,m),m(t_0)=m_0$ has a unique solution $m(t,m_0)$ in the interval $[t_0, T)$, where $0\leq t_0<T\leq+\infty$. Let $x(t)\; (t\in[t_0,T))$ be any solution of $E$ with $V(t_0,x(t_0))\leq m_0$. Then, $V(t,x(t))\leq m(t,m_0)$ holds for every $t\in[t_0,T)$.
Based on Lemma \[4\], we can give the following finite-time stability theorem for the quantum system (\[e7\]).
\[5\] For the quantum system (\[e7\]), suppose that $s_f$ is a target vector and there exists a continuously differentiable function $V:\mathcal{B}(\mathbb{R}^{n^2-1})\rightarrow\mathbb{R}$ such that the following conditions are satisfied:\
(i) $V$ is positive definite;\
(ii) For $s_0\in\mathcal{B}(\mathbb{R}^{n^2-1})$, there exist two positive real numbers $c>0$ and $\alpha\in(0,1)$ such that $$\label{e16}
\dot{V}\left(s\left(t,s_0\right)\right)+c\left(V\left(s\left(t,s_0\right)\right)\right)^\alpha\leq0.$$ Then, the system (\[e7\]) is finite-time stable, that is, it converges to the target vector $s_f$ within a finite time. And the finite convergence time function $T(s_0)$ satisfies $$\label{e17}
{T\left(s_0\right)\leq{\frac{1}{c\left(1-\alpha\right)}V\left(s_0\right)^{1-\alpha}}}.$$
Let us consider (\[e13\]) in Example \[3\]. When $y(t)=V(s(t,s_0))$ and $k=c$, it can be simplified as $$\label{e18}
{\dot{V}\left(s(t,s_0)\right)=-c\left(V\left(s(t,s_0)\right)\right)^\alpha}.$$ For $t\in\mathbb{R}_+$ and $s_0\in\mathcal{B}\big(\mathbb{R}^{n^2-1}\big)$, applying Lemma \[4\] to the differential inequality (\[e16\]) and the scalar differential equation (\[e18\]) yields $$\label{e19}
V\left(s\left(t,s_0\right)\right)\leq{\mu\left(t,V\left(s_0\right)\right)},$$ where $\mu$ can be written from (\[e14\]) as $$\label{e20}
\mu\left(t,V\left(s_0\right)\right)=\left\{
\begin{array}{lll}
[V(s_0)&\!\!^{1-\alpha}-c\left(1-\alpha\right)t]^{\frac{1}{1-\alpha}},\\
&\big(t<\frac{V\left(s_0\right)^{1-\alpha}}{c\left(1-\alpha\right)},\,s_0\ne s_f\big)\\
0,&\big(t\geq\frac{V\left(s_0\right)^{1-\alpha}}{c\left(1-\alpha\right)},\,s_0\ne s_f\big)\\
0,&\big(t\geq0,\,s_0=s_f\big).
\end{array}
\right.$$ According to (\[e20\]), when $t\geq\frac{1}{c(1-\alpha)}\left(V(s_0)\right)^{1-\alpha}$ with $s_0\in\mathcal{B}\big(\mathbb{R}^{n^2-1}\big)$, the right-hand side of is equal to zero and therefore $V\left(s(t,s_0)\right)=0$, that is, $$\label{e21}
%\setcounter{equation}{21}
s\left(t,s_0\right)=s_f.$$ Since $s(t,s_0)$ is continuous, it follows that $\inf\{t\in\mathbb{R}_+:s(t,s_0)=s_f\}>0$ for $s_0\in\mathcal{B}(\mathbb{R}^{n^2-1})\setminus s_f$; and $\inf\{t\in\mathbb{R}_+:s(t,s_0)=s_f\}<\infty$ for $s_0\in\mathcal{B}(\mathbb{R}^{n^2-1})$. Define $T(s_0):\mathcal{B}(\mathbb{R}^{n^2-1})\rightarrow\mathbb{R}_+$ as $\inf\{t\in\mathbb{R}_+:s(t,s_0)=s_f\}$, then Definition \[1\] and Proposition \[2\] guarantee that the system (\[e7\]) is finite-time stable and $s_f$ is the finite-time stable target vector of the system (\[e7\]). Furthermore, it is clear from (\[e19\])-(\[e21\]) that (\[e17\]) holds. $\blacksquare$
Theorem \[5\] can be regarded as a Lyapunov criterion for the finite-time stability of the quantum system (\[e7\]). In addition, the theory based on homogeneity also can be used to determine the finite-time stability of the system. We list some concepts and results associated with homogeneity in the Appendix, and prove the finite-time stability of the quantum system (\[e1\]) with two energy levels by combining Theorem \[5\] and the homogeneity theory in Section \[sec4\].
Design of finite-time convergent controller for two-level quantum systems {#sec3}
==========================================================================
A two-level quantum system may form a two-state qubit as an information unit in quantum communication and quantum computation, and information processing can be achieved by controlling the quantum state. In this section, for two-level quantum systems, we design the control law $u_k$ in (\[e1\]) via the Lyapunov stability theory to realize the finite-time convergence of the system to the eigenstate $|\psi_f\rangle$ of $H_0$.
We assume that the internal and control Hamiltonians of the system (\[e1\]) when $n=2$ are given as $$\begin{aligned}
{H_0=\left[\begin{array}{cc}
1&0\\
0&-1
\end{array}\right]}\label{e22},\\
%\end{equation}
%\begin{equation}
{H_1=\left[\begin{array}{cc}
0&-i\\
i&0
\end{array}\right]}\label{e23}.\end{aligned}$$ Note that the two eigenstates of $H_0$ are $|0\rangle=[1\quad0]^\mathrm{T}$ and $|1\rangle=[0\quad1]^\mathrm{T}$, respectively. Further, we assume that the target state is $|\psi_f\rangle=|1\rangle$ and therefore have $$\label{e24}
{H_0{|\psi_f\rangle}=-{|\psi_f\rangle}}.$$ We choose the following function based on the Hilbert-Schmidt distance as a Lyapunov function [@shuang2007quantum], that is, $$\label{e25}
{V=1-|{\langle\psi_f|\psi\rangle}|^2}.$$ Its first-order time derivative can be calculated as $$\label{e26}
{\dot{V}=-2u_1|{\langle\psi|\psi_f\rangle}|\textrm{Imag}\big[e^{i\angle{\langle\psi|\psi_f\rangle}}{\langle\psi_f|H_1|\psi\rangle}\big]}.$$ To guarantee $\dot{V}\leq0$, we design a control law with a fractional power as $$\label{e27}
{u_1=K\textrm{sign}\left(\phi_\alpha\left({|\psi\rangle}\right)\right)|\phi_\alpha\left({|\psi\rangle}\right)|^\alpha},$$ where $K>0$, $\phi_\alpha(|\psi\rangle)=\textrm{Imag}\left[e^{i\angle{\langle\psi|\psi_f\rangle}}{\langle\psi_f|H_1|\psi\rangle}\right]$, and $\alpha\in(0,1)$. It is easily verified that the control law is non-smooth, namely, continuous and non-differentiable.
We apply the homogeneity criterion for finite-time stability of Lemma \[6\] in the Appendix to prove that the controller (\[e27\]) will achieve the finite-time stability of the system (\[e1\]). To this end, we need to calculate the homogeneous degree of the system. By expressing complex numbers in terms of their complex exponentials, the controlled quantum state can be written as $$\label{e28}
{{|\psi\rangle}=\left[x_1,x_2\right]^\mathrm{T}=r_{1}e^{i\phi_a}{|0\rangle}+r_{2}e^{i\phi_b}{|1\rangle}},$$ where $r_1$ and $r_2$ are non-negative real numbers satisfying $r_{1}^{2}+r_{2}^{2}=1$; $e^{i\phi_a}$ and $e^{i\phi_b}$ are the global phase factors of $|\psi\rangle$ with $\phi_a,\phi_b\in\mathbb{R}$. $\phi_b-\phi_a\triangleq\phi$ is called the relative phase of $|\psi\rangle$. Particularly, we define the phase of $x_j$ as 0 when $x_j=0\;(j=1,2)$.
Considering , we have ${{\langle\psi|\psi_f\rangle}=r_{2}e^{-i\phi_b}}$, $e^{i\angle{\langle\psi|\psi_f\rangle}}$ ${\langle\psi_f|H_1|\psi\rangle}=e^{-i\phi_b}ir_{1}e^{i\phi_a}=ir_{1}e^{i\left(\phi_a-\phi_b\right)}
$, and $\phi_\alpha\left({|\psi\rangle}\right)=\textrm{Imag}\left[e^{i\angle{\langle\psi|\psi_f\rangle}}{\langle\psi_f|H_1|\psi\rangle}\right]=r_{1}\cos\left(\phi_a-\phi_b\right)=r_{1}\cos\phi
$. Thus, (\[e25\])-(\[e27\]) can be written as $$\label{e33}
{V=1-|{\langle\psi_f|\psi\rangle}|^2=r_1^2},$$ $$\label{e34}
\begin{aligned}
\dot{V}&=-2u_1|{\langle\psi|\psi_f\rangle}|\textrm{Imag}\left[e^{i\angle{\langle\psi|\psi_f\rangle}}{\langle\psi_f|H_1|\psi\rangle}\right]\\
%&=-2K|{\langle\psi|\psi_f\rangle}|\left|\textrm{Imag}\left[e^{i\angle{\langle\psi|\psi_f\rangle}}{\langle\psi_f|H_1|\psi\rangle}\right]\right|^{\alpha+1}\\
&=-2Kr_{2}|r_1\cos\phi|^{\alpha+1},
\end{aligned}$$ $$\label{e35}
\begin{aligned}
u_1&=K\textrm{sign}\left(\phi_\alpha\left({|\psi\rangle}\right)\right)|\phi_\alpha\left({|\psi\rangle}\right)|^\alpha\\
&=K\textrm{sign}\left(r_{1}\cos\phi\right)|r_{1}\cos\phi|^\alpha.
\end{aligned}$$
Note that $\dot{V}=0$ will occur in the process of the transition of the system state toward the target state $|\psi_f\rangle$, which means that $r_2=0$ or $r_1=0$ or $\cos\phi=0$. When $r_2=0$, the system is in $|\psi\rangle=[1\quad0]^T$. In this case, it follows from (\[e35\]) that $u_1\neq0$, which shows that the system state is transferring toward the target state $|\psi_f\rangle$. When $r_1=0$, the system is in $|\psi\rangle=|\psi_f\rangle$. In this case, it follows from (\[e33\])-(\[e35\]) that $V=0$, $\dot{V}=0$, and $u_1=0$. Further, considering the positive definiteness of $V$ and the negative definiteness of $\dot{V}$, we know that the system will be stabilized in the equivalence class of the target state after the target state $|\psi_f\rangle$ is reached. For the case of $\cos\phi=0$, we denote the quantum state before the target state is reached and that satisfies $\cos\phi=0$ as $|\psi_q\rangle$, and the corresponding time as $t_q$. Since $u_1=0$ is satisfied at $t_q$ and the two eigenvalues of the internal Hamiltonian are mutually different, the relative phase $\phi$ will continue evolving, that is, there exists $t_1>t_q$ such that $\cos\phi(t)\neq0$ $(t_q<t\leq t_1)$. It can be known from (\[e35\]) that $u_1(t)\neq0$ $(t_q<t\leq t_1)$. Thus, the system state will keep evolving toward the target state $|\psi_f\rangle$ and will not remain at $|\psi_q\rangle$ forever, that is, the transition moment $t_q$ associated with the transition state $|\psi_q\rangle$ forms a zero-measure set. This means that the transition states and the corresponding transition moments do not change the finite-time stability of the control system.
Since the system (\[e1\]) under the action of the controller (\[e35\]) does not satisfy the Lipschitz continuity condition, we show the existence and uniqueness of solutions of two-level quantum systems via a sufficient condition in [@kawski1989stabilization].
\[6\] Under the action of the controller (\[e35\]), the two-level quantum system (\[e1\]) with the Hamiltonians as shown in (\[e22\]) and (\[e23\]) has a unique continuously differentiable solution for every initial state.
In the Bloch spherical coordinate frame (as shown in Fig. 1), any Bloch vector of the two-level quantum system is represented by the vector $s=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$ in the unit sphere of the three-dimensional European space. In this case, the wave function of the system can be written as $$\label{e36}
{|\psi\left(\theta,\phi\right)\rangle}=\big[\begin{array}{c}
\cos\frac{\theta}{2},\,e^{i\phi}\sin\frac{\theta}{2}\end{array}\big]^\mathrm{T},$$ where $\theta\in[0,\pi]$ and $\phi\in[0,2\pi)$ [@fano2017twenty]. The relative phase $\phi$ of $|\psi\rangle$ is the angle between the projection of the vector $s$ on the $x-y$ plane and the positive $x-$axis, and $\theta$ is the angle between the vector $s$ and the positive $z-$axis. The correspondence between the system state (\[e36\]) and the Bloch spherical coordinate $s$ is one-to-one.
![The Bloch vector space of two-level quantum systems.[]{data-label="fig1"}](Bloch.eps){height="4cm"}
For any initial state outside the set $\mathcal{O}=\{|\psi\rangle:\cos\phi=0\}$, the vector field of the system (\[e1\]) with the control law (\[e35\]) is Lipschitz everywhere. The system (\[e1\]) in this case has a unique solution. In what follows, we discuss the solution of the system (\[e1\]) for the initial state in the set $\mathcal{O}$.
When the initial state is given in the set $\{|\psi\rangle:\cos\phi=0\}$, all quantum states satisfying $\cos\phi=0$ form a longitude circle such that $\phi=\frac{\pi}{2}+q\pi(q=\ldots,-1,0,+1,\ldots)$ holds on the Bloch sphere. Since each quantum state of $\{|\psi\rangle:\cos\phi=0\}$ is a transition state $|\psi_q\rangle$, when $|\psi\rangle=|\psi_q\rangle$ at $t_q$, there exists $t_1$ such that the relative phase $\phi$ changes from $\phi(t_0)=\frac{\pi}{2}+q\pi$ to $\phi(t)\neq\frac{\pi}{2}+q\pi\;(q=\ldots,-1,0,+1,\ldots)$ in the time interval $(t_q,t_1]$. Reflected on the Bloch sphere, the evolution trajectory of the system (\[e1\]) in the time interval $[t_0,t_1]$ intersects the longitude circle of $\phi=\frac{\pi}{2}+q\pi\;(q=\ldots,-1,0,+1,\ldots)$ in a non-overlap manner. That is to say, the vector field $f$ of the “closed-loop" system (\[e1\]) is transversal to the non-Lipschitz set $\{|\psi\rangle:\cos\phi=0\}$ on the Bloch sphere. It is known from [@kawski1989stabilization] that the system (\[e1\]) in this case has a unique solution for every initial condition in $\{|\psi\rangle:\cos\phi=0\}$. When the initial state satisfies $r_1=0$, we have $|\psi_0\rangle=|\psi_f\rangle$. In this case, the Lyapunov stability theorem guarantees that the state of the system always stays in $|\psi_0\rangle=|\psi_f\rangle$, that is, the system (\[e1\]) has a unique solution $|\psi_f\rangle$.
To sum up, the system (\[e1\]) has a unique solution for any initial state. $\blacksquare$
\[7\] The notion of transversality is involved in the proof of Theorem \[6\], which is a description of how two objects intersect. For two intersecting curves, if they are not tangent, then these two curves are said to be transversal each other. By the phrase that the two curves are tangent, we mean that the angle between the tangent lines of the two curves at the intersection point is zero. Readers can refer to [@golubitsky2012stable] for more general concepts and criteria of transversality.
Analysis of finite-time stability of two-level quantum control systems {#sec4}
======================================================================
For two-level quantum systems, we have the following finite-time stability theorem.
\[8\] Under the action of the control law (\[e35\]), the system (\[e1\]) with the Hamiltonians in (\[e22\]) and (\[e23\]) is globally finite-time stable, that is, the system will be stabilized in the equivalence class of the target state $|\psi_f\rangle=[0,\,1]^\mathrm{T}$ within a finite time for any initial state.
By considering (\[e28\]) and the relative phase $\phi=\phi_b-\phi_a$ of $|\psi\rangle$, (\[e1\]) can be written as $$\label{e38}
{{\left[\begin{array}{c}\dot{r_1}e^{i\phi_a}+ir_{1}e^{i\phi_a}\dot{\phi}_a\\
\dot{r_2}e^{i\phi_b}+ir_{2}e^{i\phi_b}\dot{\phi}_b\end{array}\right]}\!=\!
-i{\left[\begin{array}{c}r_{1}e^{i\phi_a}\\-r_{2}e^{i\phi_b}\end{array}\right]}
\!-iu_1{\left[\begin{array}{c}-ir_{2}e^{i\phi_b}\\ir_{1}e^{i\phi_a}\end{array}\right]}},$$ that is, $$\label{e40}
\left\{
\begin{array}{l}
\dot{r}_{1}=-u_{1}r_{2}\cos\phi=-Kr_{1}^{\alpha}r_{2}|\cos\phi|^{\alpha+1}\\
r_{1}\dot{\phi}_a=-r_1-u_{1}r_{2}\sin\phi=-r_1-Kr_{1}^{\alpha}r_{2}|\cos\phi|^{\alpha}\sin\phi\\
\dot{r}_{2}=-u_{1}r_{1}\cos\phi=K|r_{1}\cos\phi|^{\alpha+1}\\
r_{2}\dot{\phi}_b=r_2-u_{1}r_{1}\sin\phi=r_2-Kr_{1}^{\alpha+1}|\cos\phi|^{\alpha}\sin\phi.
\end{array}
\right.$$ Since the system (\[e40\]) is obtained by an equivalent transformation of the system (\[e1\]), it is known from Theorem \[6\] that the system (\[e40\]) also has a unique solution. Thus, we can regard $|\cos\phi|^{\alpha+1}$ in (\[e40\]) as a function of time $t$. Let $|\cos\phi|^{\alpha+1}=g(t)$, and then we have $$\label{e41}
\left\{
\begin{array}{l}
\dot{r}_{1}=-Kr_{1}^{\alpha}r_{2}g\left(t\right)\\
\dot{r}_{2}=Kr_{1}^{\alpha+1}g\left(t\right).
\end{array}
\right.$$
The analysis of the transition of the controlled quantum state $|\psi(t)\rangle=[x_1,\, x_2]^\mathrm{T}$ defined on $\mathbb{C}^2$ from the initial state $|\psi_0\rangle$ to the target state $|\psi_f\rangle=[0,\,1]^\mathrm{T}$ is equivalent to the analysis on whether the controlled variable $(r_1,\,r_2)^\mathrm{T}$ defined on $\mathbb{R}_{+}^{2}$ can be stabilized to the target point $[0,\,1]^\mathrm{T}$ from the initial point $[r_1(0),\,r_2(0)]^\mathrm{T}$. Since $r_{1}^{2}+r_{2}^{2}=1$, we only need to consider whether the controlled variable $r_1$ defined on $\mathbb{R}_+$ can be stabilized to the origin $0$ from the initial point $r_1(0)$. Expressing $r_2$ with $r_1$, we have $$\label{e42}
r_2=\left(1-r_{1}^{2}\right)^{\frac{1}{2}}
=1-\sum_{j=1}^\infty\frac{\mathrm{C}_{2j}^{j}}{2^{2j}\times\left(2j-1\right)}r_{1}^{2j}.$$ Substituting (\[e42\]) into the first equation of (\[e41\]) gives $$\label{e43}
\begin{aligned}
\dot{r}_1=&-Kr_{1}^{\alpha}g\left(t\right)+\sum_{j=1}^\infty\frac{\mathrm{C}_{2j}^{j}r_{1}^{2j}Kr_{1}^{\alpha}g\left(t\right)}{2^{2j}\times\left(2j-1\right)}.
\end{aligned}$$ For convenience in analysis, we write (\[e43\]) as $$\label{e44}
\dot{r}_1=f\left(r_1\right)=p_0(r_1)+\sum_{j=1}^\infty p_{j}\left(r_1\right)=\sum_{j=0}^\infty p_{j}\left(r_1\right),$$ where $p_0(r_1)=-Kr_{1}^{\alpha}g(t)$ and $p_{j}(r_1)=\frac{\mathrm{C}_{2j}^{j}Kr_{1}^{\alpha+2j}g(t)}{2^{2j}\times(2j-1)}$ $(j\ge 1)$.
Thus, we only need to prove the system (\[e44\]) is finite-time stable. The proof can be divided into two steps.
**Step 1** The following system defined by $$\label{e45}
{\dot{r}_1=p_0\left(r_1\right)}$$ is finite-time stable.
**Step 2** The system (\[e44\]) is globally finite-time stable.
**Proof of Step 1.**
According to Lemma \[A6\] in the Appendix, in order to prove the finite-time stability of the system (\[e45\]), we only need to verify that the system (\[e45\]) is asymptotically stable and has a negative degree of homogeneity.
*Asymptotic stability.* For the Lyapunov function $V(r_1)=r_1^2$, we calculate its Lie derivative along any trajectory of the system (\[e45\]) and have $$\label{e46}
\begin{aligned}
L_{p_0}V\left(r_1\right)&=\langle\nabla V\left(r_1\right),p_0\left(r_1\right)\rangle\\
&=2r_{1}p_0\left(r_1\right)=-2Kr_{1}^{\alpha+1}g\left(t\right).
\end{aligned}$$ It can be known from (\[e46\]) that the Lyapunov function $V(r_1)$ is non-increasing and $L_{p_0}V(r_1)$ is bounded. The latter implies that $L_{p_0}V(r_1)$ is uniformly continuous, and therefore the Barbalat’s lemma [@khalil2002nonlinear] guarantees that $L_{p_0}V(r_1)\rightarrow0$ as $t\rightarrow\infty$. Considering $g(t)>0$, we have $r_1\rightarrow0$, that is, the system (\[e45\]) is asymptotically stable.
*Negative degree of homogeneity.* According to Definition \[A3\] in the Appendix, when $0<\alpha<1$ and the dilation is taken as $\delta_\varepsilon^1$, the vector field of $p_0(r_1)$ satisfies $$\label{e47}
{p_0\left(\varepsilon r_1\right)=\varepsilon^{\alpha}p_0\left(r_1\right)=\varepsilon^{1+\left(\alpha-1\right)}p_0\left(r_1\right)}.$$ Therefore, the degree of homogeneity of the vector field $p_0(r_1)$ with respect to the dilation $\delta_\varepsilon^1$ is $k_0=\alpha-1<0$. It follows from Lemma \[A6\] in the Appendix that the origin of the system (\[e45\]) is finite-time stable.
**Proof of Step 2.**
For $j=1,2,3,\ldots$, we calculate the degree of homogeneity of the vector field $p_j(r_1)$ in (\[e44\]) with respect to the dilation $\delta_\varepsilon^1$, $k_{j}$, and have $$\label{e48}
\begin{aligned}
p_j\left(\varepsilon
r_1\right)&=\frac{\mathrm{C}_{2j}^{j}}{2^{2j}\times\left(2j-1\right)}K\varepsilon^{\alpha+2j}r_{1}^{\alpha+2j}g\left(t\right)\\
&=\varepsilon^{1+\left(\alpha+2j-1\right)}p_j\left(r_1\right)\\
&=\varepsilon^{1+k_{j}}p_j\left(r_1\right),
\end{aligned}$$ that is, $k_{j}=\alpha+2j-1$ $(j=1,2,3,\ldots)$.
Noticing the facts that the degree of homogeneity of $V(r_1)$ with respect to the dilation $\delta_\varepsilon^1$ is $l_1=2$ and $\langle\nabla V(r_1),p_{j}(r_1)\rangle$ $(j=0,1,2,\ldots)$ is continuous and its degree of homogeneity with respect to $\delta_\varepsilon^1$ is $l_1+k_{j}$, we take $V_1=V(r_1)$ and $V_2=\langle\nabla V(r_1),p_j(r_1)\rangle$ for Lemma \[A5\] of the Appendix. Since $l_1=2>0$ and $l_2=l_1+k_{j}=\alpha+2j+1>0$, Lemma \[A5\] in the Appendix implies $$\label{e49}
\langle\nabla V\left(r_1\right),p_j\left(r_1\right)\rangle\leq-c_{j}V\left(r_1\right)^{\frac{\alpha+2j+1}{2}},$$ where $c_{j}\!=\!-\max_{\{r_1:V(r_1)=1\}}\langle\nabla V(r_1),p_j(r_1)\rangle\!\in\mathbb{R}$ $(j=0,1,2,\ldots)$. Thus, $$\label{e50}
\begin{aligned}
&\langle\nabla V\left(r_1\right),f\left(r_1\right)\rangle\\
\leq&-c_{0}V\left(r_1\right)^{\frac{\alpha+1}{2}}-\cdots-c_{j}V\left(r_1\right)^{\frac{\alpha+2j+1}{2}}-\cdots\\
=&\;V\left(r_1\right)^{\frac{\alpha+1}{2}}\big(-c_0+\mathcal{U}(r_1)\big),
\end{aligned}$$ where $\mathcal{U}(r_1)\triangleq-c_{1}V(r_1)^{\frac{2}{2}}-\cdots-c_{j}V(r_1)^{\frac{2j}{2}}-\cdots$. Since $\frac{2j}{2}>0$ for $j\geq1$, $\mathcal{U}(r_1)$ is a continuous function with $\mathcal{U}(0)=0$.
Now, we show that satisfies the condition in (\[e16\]). Assume that there exists an open neighborhood $\mathcal{V}$ of the origin such that $\mathcal{U}(r_1)<\frac{c_0}{2}$ holds for any $r_1\in\mathcal{V}$. Then, (\[e50\]) can be written as $$\label{e51}
\langle\nabla V\left(r_1\right),f\left(r_1\right)\rangle<-\frac{c_0}{2}V\left(r_1\right)^{\frac{\alpha+1}{2}},$$ where $c_0>0$ and $\frac{\alpha+1}{2}\in(0,1)$. Thus, the condition (\[e16\]) in Theorem \[5\] is satisfied. In view of the positive definiteness of $V(r_1)$, Theorem \[5\] guarantees that the origin is a finite-time stable equilibrium point of the system (\[e44\]).
Next, we verify the existence of the open neighborhood $\mathcal{V}$, that is, there exist $r_1$ such that $\mathcal{U}(r_1)<\frac{c_0}{2}$ holds. Considering that $c_{j}=-\max_{\{r_1:V(r_1)=1\}}\langle\nabla V(r_1),p_j(r_1)\rangle$ $(j=0,1,2,\ldots)$ and $r_1=1$ holds when $V(r_1)=1$, we can calculate $c_0$ and $c_{j}$ $(j\ge 1)$ as $$\label{e52}
{c_0=-\langle\nabla V\left(r_1\right),p_0\left(r_1\right)\rangle=2Kg\left(t\right)},$$ $$\label{e53}
c_{j}=-\langle\nabla V\left(r_1\right),p_j\left(r_1\right)\rangle
=-\frac{2K\mathrm{C}_{2j}^{j}g(t)}{2^{2j}\times\left(2j-1\right)}.$$
It follows from that $$\label{e54}
\begin{aligned}
&\mathcal{U}\left(r_1\right)=-c_1V\left(r_1\right)^{\frac{2}{2}}-\cdots-c_{j}V\left(r_1\right)^{\frac{2j}{2}}-\cdots\\
=&\,2Kg(t)\Big[\frac{1}{2}V(r_1)+\cdots+\frac{\mathrm{C}_{2j}^{j}V(r_1)^j}{2^{2j}\times\left(2j-1\right)}+\cdots\Big]\\
=&\,2Kg(t)\big[1-\left(1-V(r_1)\right)^{\frac{1}{2}}\big].
\end{aligned}$$
Substituting (\[e54\]) and into $\mathcal{U}(r_1)<\frac{c_0}{2}$, we have $r_1<\frac{\sqrt{3}}{2}$, that is, $\mathcal{V}=\{r_1:r_1<\frac{\sqrt{3}}{2}\}$. This shows the existence of $\mathcal{V}$ in (\[e51\]). Further, considering that all the moments $t_q$ corresponding to the transition state $|\psi_q\rangle$ constitute a zero measure set and any non-transition state satisfies $\dot{V}<0$, it is easy to draw the conclusion that $r_1$ always can converge into $\mathcal{V}$ within a finite time for every initial state $r_1(0)\notin\mathcal{V}$.
Thus, the origin of the system (\[e44\]) is a global finite-time stable equilibrium point, that is, $r_1$ can be stabilized to the origin within a finite time, equivalently, the quantum state is stabilized to the equivalence class of the target state $|\psi_f\rangle=[0,\,1]^\mathrm{T}$ within a finite time. $\blacksquare$
\[9\] According to the proof of Theorem \[8\], (\[e51\]) always holds for the system (\[e44\]). Therefore, from Theorem \[5\], we know that the finite convergence time function of the system (\[e44\]) satisfies $T(r_1(0))<\frac{4}{c_{1}(1-\alpha)}V(r_1(0))^{\frac{1-\alpha}{2}}$, where the finite convergence time $T(r_1(0))$ represents the evolution time from the initial state $r_1(0)$ to the origin. It should be noted that the above inequality relation of the finite convergence time function can be obtained only when $r_1(0)\in\mathcal{V}$. When $r_1(0)\notin\mathcal{V}$, the calculation of the finite convergence time relies on the system equation (\[e44\]), and therefore it is hard to give a range of $T(r_1(0))$ restricted by a certain analytical expression. In addition, since $c_1$, $V(r_1(0))$, and $\alpha\in(0,1)$ are bounded, $T(r_1(0))$ is also bounded. However, its value will vary with $\alpha$.
Numerical examples {#sec5}
==================
In order to demonstrate the validity of the non-smooth control law designed in this paper, we choose a spin $\frac{1}{2}$ particle system for numerical simulation experiments. In simulations, we set $K=0.5$ and $\alpha=\frac{2}{3}$ for the control law in (\[e35\]). In addition, we also consider simulation results under the standard Lyapunov control law $u_1^s$ [@shuang2007quantum] and the standard bang-bang Lyapunov control law $u_1^b$ [@kuang2017rapid] to compare the results under the control law in this paper, where $u_1^s$ and $u_1^b$ are $$\label{e55}
{u_1^s=K\textrm{Imag}\big[e^{i\angle{\langle\psi|\psi_f\rangle}}{\langle\psi_f|H_1|\psi\rangle}\big]},$$ $$\label{e56}
{u_1^b=K\textrm{sign}\big(\textrm{Imag}\big[e^{i\angle{\langle\psi|\psi_f\rangle}}{\langle\psi_f|H_1|\psi\rangle}\big]\big)}.$$
Assume that the initial state is given as $|\psi(0)\rangle=[1,\,0]^\mathrm{T}\notin\mathcal{V}$. We choose $K=0.5$ for the standard Lyapunov control law $u_1^s$ and the standard bang-bang Lyapunov control law $u_1^b$. The simulation results are shown in Fig. \[fig2\] and Fig. \[fig3\].
![With the initial state $|\psi(0)\rangle=[1,\,0]^\mathrm{T}$, the evolution curves of the non-smooth control $u_1$, the standard Lyapunov control $u_1^s$, and the standard bang-bang Lyapunov control $u_1^b$.[]{data-label="fig3"}](fig2.eps){width="9cm"}
![With the initial state $|\psi(0)\rangle=[1,\,0]^\mathrm{T}$, the evolution curves of the non-smooth control $u_1$, the standard Lyapunov control $u_1^s$, and the standard bang-bang Lyapunov control $u_1^b$.[]{data-label="fig3"}](fig3.eps){width="9cm"}
The simulations show that the finite convergence time of the system with the initial state $|\psi(0)\rangle=[1,\,0]^\mathrm{T}$ is $t_f\approx9.96$ a.u.. According to Fig. \[fig2\] and simulation data, the population of the target state under the standard Lyapunov control $u_1^s$ is only $97.7\%$ at $t=9.96$ a.u., and the population of the target state under the standard bang-bang Lyapunov control law $u_1^b$ is $96.82\%$ at $t=9.96$ a.u. and no longer changes. It can be seen from Fig. \[fig3\] that the non-smooth control in this paper is continuous and non-differentiable, that is, continuous in the whole time interval but non-differentiable at some points; while the standard Lyapunov control is continuously differentiable in $[0,\infty)$, and the standard bang-bang Lyapunov control is discontinuous.
To verify the estimation relation of the finite convergence time, we also perform simulation experiments for the initial state $|\psi(0)\rangle=[\frac{1}{2},\,\frac{\sqrt{3}}{2}]^\mathrm{T}\in\mathcal{V}$. The simulation results are shown in Fig. \[fig4\], which indicates that the convergence time of the system in this case is $t_f\approx5.33$ a.u.. According to Remark \[9\], the finite convergence time corresponding to the initial state $|\psi(0)\rangle=[\frac{1}{2},\,\frac{\sqrt{3}}{2}]^\mathrm{T}$ satisfies $\mathrm{T}(\frac{1}{2})=5.33\;\mathrm{a.u.}<\frac{4}{c_1(1-\alpha)}V(\frac{1}{2})^{\frac{1-\alpha}{2}}=9.52 \;\mathrm{a.u.}$, which is consistent with the theoretical results.
According to the simulation results in Fig. \[fig2\]-Fig. \[fig4\], the standard Lyapunov control only can achieve the infinite-time convergence of the system and the speed of convergence is slow; the standard bang-bang Lyapunov control can achieve rapid state transition in the early stages of control, but cannot guarantee convergence to the target state; while the non-smooth control proposed in this paper achieves the finite-time convergence of the system to the target state.
![With the initial state $|\psi(0)\rangle=[\frac{1}{2},\,\frac{\sqrt{3}}{2}]^\mathrm{T}$, the evolution curves of population of the target state and the non-smooth control field $u_1$.[]{data-label="fig4"}](fig4.eps){width="9cm"}
Conclusion {#sec6}
==========
To achieve the finite-time stabilization of an arbitrary eigenstate of the internal Hamiltonian of a quantum system, this paper proposed a new non-smooth control scheme. We provided the concept of finite-time stability of quantum systems and corresponding identification criteria. Via the exponential representation of quantum states, an equivalent transformation was performed for the system model and the control target. The Lyapunov stability theory and the homogeneity theorem of finite-time stability were used to prove the finite-time stabilization of two-level quantum systems to the target state. At the same time, the estimation problem of the finite convergence time was discussed. Based on the transversality condition in the Bloch space, we also proved the existence and uniqueness of solutions of the quantum system under the non-smooth control law. The effectiveness of the non-smooth control law was illustrated by numerical examples.
Since the control law designed in this paper contains two adjustable parameters, it is more flexible in the use of the control law. In order to achieve an optimal convergence performance, the optimization of the parameter $\alpha$ needs further research. In addition, this paper only considers the finite-time stabilization control of two-dimensional quantum systems. Extending the scheme to high-dimensional and mixed-state systems is also a topic worth exploring.
Appendix: Homogeneity theory of finite-time stability {#Appendix .unnumbered}
=====================================================
Here, we list some notions related to homogeneity and the finite-time stability criterion based on the homogeneity theory, which can be found in [@bhat2005geometric].
\[A1\] Let $d=\left(d_1,d_2,\ldots,d_{n-1}\right)$ be a set of positive real numbers. For a set of coordinates $r=\left(r_1,r_2,\ldots,r_{n-1}\right)$ in $\mathbb{R}^{n-1}$, define the dilation $\delta_\varepsilon^d$ of $r$ as the following coordinate vector $$\label{e57}
\begin{aligned}
&\delta_\varepsilon^d\left(r\right)=\left(\varepsilon^{d_1}r_1,\ldots,\varepsilon^{d_{n-1}}r_{n-1}\right),\forall\varepsilon>0
\end{aligned}$$ where $d_i$ is the weight of the coordinate $r_i$. The dilation with $d_1=\cdots= d_{n-1}=1$ is called a standard dilation.
\[A2\] A function $V:\mathbb{R}^{n-1}\rightarrow\mathbb{R}$ is said to be homogeneous of degree $m\left(m\in\mathbb{R}\right)$ with respect to $\delta_{\varepsilon}^{d}$ if $$\label{e58}
{V\left(\delta_\varepsilon^d(r)\right)=\varepsilon^{m}V(r),\;\forall r\in\mathbb{R}^{n-1},\forall\varepsilon>0.}$$
\[A3\] A vector field $f(r):\,\mathbb{R}^{n-1}\rightarrow \mathbb{R}^{n-1}$ with $f(r)=(f_1(r),\,\ldots,\,f_{n-1}(r))^\mathrm{T}$ is said to be homogeneous of degree $k\left(k\in\mathbb{R}\right)$ with respect to $\delta_{\varepsilon}^{d}$ if for each $i=1,\ldots,n-1$, $f_i$ is homogeneous of degree $k+d_i$, that is, $$\label{e59}
f_i\left(\delta_\varepsilon^d\left(r\right)\right)=\varepsilon^{k+d_{i}}f_{i}\left(r\right),\;\forall r\in\mathbb{R}^{n-1},\,\forall\varepsilon>0.$$
\[A4\] Assume that the function $f:\,\mathbb{R}^{n-1}\rightarrow \mathbb{R}^{n-1}$ is homogeneous of degree $k\left(k\in\mathbb{R}\right)$ with respect to $\delta_{\varepsilon}^{d}$ and the origin is a locally asymptotically stable equilibrium point. Then, when $m>\max\{-k,0\}$, there exists a Lyapunov function $V$ such that $V$ and its time derivative $\dot{V}$ are homogeneous of degrees $m$ and $m+k$ with respect to $\delta_{\varepsilon}^{d}$, respectively.
\[A5\] Let $V_1$ and $V_2$ be continuous real-valued functions defined on $\mathbb{R}^{n-1}$ and $V_1$ be positive definite. Suppose that $V_1$ and $V_2$ are homogeneous of degrees $l_1>0$ and $l_2>0$ with respect to $\delta_{\varepsilon}^{d}$ respectively. Then, for every $r\in\mathbb{R}^{n-1}$, the following holds: $$\label{e60}
\begin{aligned}
&\Big(\min_{\{z:V_1(z)=1\}}V_2(z)\Big)\big(V_1(r)\big)^{\frac{l_2}{l_1}}
\leq V_2(r)\\
\leq&\;\Big(\max_{\{z:V_1(z)=1\}}V_2(z)\Big)\big(V_1(r)\big)^{\frac{l_2}{l_1}}.
\end{aligned}$$
The following lemma shows the application of the homogeneity theory to the finite-time stability.
\[A6\] Let $f\left(r\right)=\left(f_1\left(r\right),\ldots,f_{n-1}\left(r\right)\right)^\mathrm{T}:\,\mathbb{R}^{n-1}$ $\rightarrow \mathbb{R}^{n-1}$ be a continuous vector function and be homogeneous of degree $k\,\left(k\in\mathbb{R}\right)$ with respect to $\delta_{\varepsilon}^{d}$, where $d=\left(d_1,d_2,\ldots,d_{n-1}\right)$ is a set of positive real numbers and $\varepsilon>0$. Then, the origin is a finite-time stable equilibrium point if and only if it is an asymptotically stable equilibrium point and $k<0$.
[10]{}
D. Dong and I. R. Petersen, “Quantum control theory and applications: A survey,” [*IET Control Theory $\&$ Applications*]{}, vol. 4, no. 12, pp. 2651–2671, 2010.
F. Dolde, V. Bergholm, Y. Wang, I. Jakobi, B. Naydenov, S. Pezzagna, J. Meijer, F. Jelezko, P. Neumann, T. Schulte-Herbr[ü]{}ggen, [*et al.*]{}, “High-fidelity spin entanglement using optimal control,” [*Nature Communications*]{}, vol. 5, p. 3371, 2014.
G. Riviello, R.-B. Wu, Q. Sun, and H. Rabitz, “Searching for an optimal control in the presence of saddles on the quantum-mechanical observable landscape,” [*Phys. Rev. A*]{}, vol. 95, p. 063418, Jun 2017.
X. [Wang]{} and S. G. [Schirmer]{}, “Analysis of effectiveness of [Lyapunov]{} control for non-generic quantum states,” [*IEEE Transactions on Automatic Control*]{}, vol. 55, pp. 1406–1411, June 2010.
S. Zhao, H. Lin, and Z. Xue, “Switching control of closed quantum systems via the [Lyapunov]{} method,” [*Automatica*]{}, vol. 48, no. 8, pp. 1833–1838, 2012.
S. Kuang, D. Dong, and I. R. Petersen, “Rapid [Lyapunov]{} control of finite-dimensional quantum systems,” [*Automatica*]{}, vol. 81, pp. 164–175, 2017.
S. [Kuang]{}, D. [Dong]{}, and I. R. [Petersen]{}, “Lyapunov control of quantum systems based on energy-level connectivity graphs,” [*IEEE Transactions on Control Systems Technology*]{}, to be pulished, DOI: 10.1109/TCST.2018.2871186.
D. Dong and I. R. Petersen, “Sliding mode control of two-level quantum systems,” [*Automatica*]{}, vol. 48, no. 5, pp. 725–735, 2012.
D. Dong and I. R. Petersen, “Notes on sliding mode control of two-level quantum systems,” [*Automatica*]{}, vol. 48, no. 12, pp. 3089 – 3097, 2012.
M. R. [James]{}, H. I. [Nurdin]{}, and I. R. [Petersen]{}, “${H^\infty}$ control of linear quantum stochastic systems,” [*IEEE Transactions on Automatic Control*]{}, vol. 53, no. 8, pp. 1787–1803, 2008.
C. Xiang, I. R. Petersen, and D. Dong, “Coherent robust [$H^\infty$]{} control of linear quantum systems with uncertainties in the [Hamiltonian]{} and coupling operators,” [*Automatica*]{}, vol. 81, pp. 8 – 21, 2017.
G. [Zhang]{}, S. [Grivopoulos]{}, I. R. [Petersen]{}, and J. E. [Gough]{}, “The [Kalman]{} decomposition for linear quantum systems,” [*IEEE Transactions on Automatic Control*]{}, vol. 63, pp. 331–346, Feb 2018.
S. [Wang]{} and D. [Dong]{}, “Fault-tolerant control of linear quantum stochastic systems,” [*IEEE Transactions on Automatic Control*]{}, vol. 62, pp. 2929–2935, June 2017.
Q. Gao, D. Dong, and I. R. Petersen, “Fault tolerant quantum filtering and fault detection for quantum systems,” [*Automatica*]{}, vol. 71, pp. 125 – 134, 2016.
J. Biamonte, P. Wittek, N. Pancotti, P. Rebentrost, N. Wiebe, and S. Lloyd, “Quantum machine learning,” [*Nature*]{}, vol. 549, no. 7671, p. 195, 2017.
R.-B. Wu, H. Ding, D. Dong, and X. Wang, “Learning robust and high-precision quantum controls,” [*Phys. Rev. A*]{}, vol. 99, p. 042327, Apr 2019.
M. Y. Niu, S. Boixo, V. N. Smelyanskiy, and H. Neven, “Universal quantum control through deep reinforcement learning,” [*npj Quantum Information*]{}, vol. 5, no. 1, p. 33, 2019.
P. Dorato, “Short-time stability in linear time-varying systems,” tech. rep., Polytechnic Institute of Brooklyn, New York Microwave Research Institute, 1961.
S. Ding, C. Qian, S. Li, and Q. Li, “Global stabilization of a class of upper-triangular systems with unbounded or uncontrollable linearizations,” [*International Journal of Robust and Nonlinear Control*]{}, vol. 21, no. 3, pp. 271–294, 2011.
X. Huang, W. Lin, and B. Yang, “Global finite-time stabilization of a class of uncertain nonlinear systems,” [*Automatica*]{}, vol. 41, no. 5, pp. 881–888, 2005.
Y. Hong, “Finite-time stabilization and stabilizability of a class of controllable systems,” [*Systems $\&$ Control Letters*]{}, vol. 46, no. 4, pp. 231–236, 2002.
J. Tsinias and M. Tzamtzi, “An explicit formula of bounded feedback stabilizers for feedforward systems,” [*Systems $\&$ Control Letters*]{}, vol. 43, no. 4, pp. 247–261, 2001.
A. R. Teel, “A nonlinear small gain theorem for the analysis of control systems with saturation,” [*IEEE Transactions on Automatic Control*]{}, vol. 41, no. 9, pp. 1256–1270, 1996.
F. Mazenc and L. Praly, “Adding integrations, saturated controls, and stabilization for feedforward systems,” [*IEEE Transactions on Automatic Control*]{}, vol. 41, no. 11, pp. 1559–1578, 1996.
C. Qian and W. Lin, “A continuous feedback approach to global strong stabilization of nonlinear systems,” [*IEEE Transactions on Automatic Control*]{}, vol. 46, no. 7, pp. 1061–1079, 2001.
Y. Hong and J. Wang, “Non-smooth finite-time stabilization for a class of nonlinear systems,” [*Science in China Series F*]{}, vol. 49, no. 1, pp. 80–89, 2006.
Z. Wang, S. Li, and S. Fei, “Finite-time tracking control of a nonholonomic mobile robot,” [*Asian Journal of Control*]{}, vol. 11, no. 3, pp. 344–357, 2009.
S. P. Bhat and D. S. Bernstein, “Geometric homogeneity with applications to finite-time stability,” [*Mathematics of Control, Signals and Systems*]{}, vol. 17, no. 2, pp. 101–127, 2005.
S. P. Bhat and D. S. Bernstein, “Continuous finite-time stabilization of the translational and rotational double integrators,” [*IEEE Transactions on Automatic Control*]{}, vol. 43, no. 5, pp. 678–682, 1998.
M. Kawski, “Stabilization of nonlinear systems in the plane,” [*Systems $\&$ Control Letters*]{}, vol. 12, no. 2, pp. 169–175, 1989.
A. F. Filippov, [*Differential Equations with Discontinuous Righthand Sides: Control Systems*]{}, vol. 18. Springer Science $\&$ Business Media, 2013.
R. P. Agarwal and V. Lakshmikantham, [*Uniqueness and Nonuniqueness Criteria for Ordinary Differential Equations*]{}, vol. 6. World Scientific Publishing Company, 1993.
G. Kimura, “The [B]{}loch vector for [$N$]{}-level systems,” [*Physics Letters A*]{}, vol. 314, no. 5-6, pp. 339–349, 2003.
X. Wang and S. G. Schirmer, “Analysis of [Lyapunov]{} method for control of quantum states,” [*IEEE Transactions on Automatic Control*]{}, vol. 55, no. 10, pp. 2259–2270, 2010.
H. K. Khalil, [*Nonlinear Systems*]{}. New Jersey: Prentice Hall, 2002.
S. Cong and S. Kuang, “Quantum control strategy based on state distance,” [*Acta Automatica Sinica*]{}, vol. 33, no. 1, pp. 28–31, 2007.
G. Fano and S. Blinder, [*Twenty-First Century Quantum Mechanics: Hilbert Space to Quantum Computers*]{}. Springer, 2017.
M. Golubitsky and V. Guillemin, [*Stable Mappings and Their Singularities*]{}, vol. 14. Springer Science & Business Media, 2012.
|
---
abstract: 'In a recent paper, Nguyen, Kuhn, and Esfahani (2018) built a distributionally robust estimator for the precision matrix of the Gaussian distribution. The distributional uncertainty size is a key ingredient in the construction of this estimator. We develop a statistical theory which shows how to optimally choose the uncertainty size to minimize the associated Stein loss. Surprisingly, rather than the expected canonical square-root scaling rate, the optimal uncertainty size scales linearly with the sample size.'
author:
- Jose Blanchet
- Nian Si
bibliography:
- 'mybib.bib'
date:
- ' This Version: '
- '*Management Science and Engineering, Stanford University*'
title: Optimal Uncertainty Size in Distributionally Robust Inverse Covariance Estimation
---
Introduction
============
Motivated by a wide range of problems which require the estimation of the inverse of a covariance matrix, [@nguyen2018distributionally] recently constructed an estimator based on distributionally robust optimization using the Wasserstein distance in Euclidean space. A crucial ingredient is the distributional uncertainty size, which plays the role of a regularization parameter.
In their paper, [@nguyen2018distributionally] show excellent empirical performance of their estimator in comparison to several commonly used estimators (based on shrinkage and regularization). The comparison is based in terms of the corresponding Stein loss (defined in terms of the likelihood, as we shall review). However, no theory is provided as how to choose the distributional uncertainty size.
Our goal is to provide an asymptotically optimal expression for the distributional uncertainty size, in terms of the Stein loss performance, as the sample size increases.
This paper provides interesting insights which validate the empirical observations in [@nguyen2018distributionally]. In particular, in the Introduction of [@nguyen2018distributionally], leading to equation (4), they argue that the distributional uncertainty size, $\rho _{n}$, should scale at rate $\rho _{n}=O\left( n^{-1/2}\right) $ (where $n$ is the sample size) due to the existence of a central limit theorem for the Wasserstein distance for Gaussian distributions. However, the numerical experiments, reported in Section 6.1 of [@nguyen2018distributionally], suggest an optimal scaling of the form $\rho _{n}=O\left( n^{-\kappa }\right) $ where $\kappa >1/2$.
Our main result shows that the asymptotically optimal choice of distributional uncertainty is of the form $\rho _{n}=$ $\rho _{\ast
}n^{-1}\left( 1+o\left( 1\right) \right) $ as $n\rightarrow \infty $, where $\rho _{\ast }>0$ is a constant which is characterized explicitly. Our results therefore validate the empirical findings of [nguyen2018distributionally]{} with $\kappa =1$.
This paper is organized as follows. We review the estimator of [nguyen2018distributionally]{} and state our main result in Section [Section\_Main]{}. We then provide the proof of our result in Section [Proof\_of\_Thm\_1]{}. Numerical experiments are included in Section [Sec\_Numerical]{}, which provide a sense of the non-asymptotic performance of our asymptotically optimal choice.
Basic Notions and Main Result\[Section\_Main\]
==============================================
We now review the basic definitions underlying the estimator from [nguyen2018distributionally]{}. Suppose we have i.i.d. samples $\xi _{i}\sim
\mathcal{N}\left( 0,\Sigma _{0}\right) $ (normally distributed with zero mean and covariance matrix $\Sigma _{0}$), where $\xi _{i}\in \mathbb{R}^{d}$ and $\Sigma _{0}$ is assumed to be strictly positive definite. We write $$\hat{\Sigma}_{n}=\frac{1}{n}\sum_{i=1}^{n}\xi _{i}\xi _{i}^{T},$$and let $\mathbb{\hat{P}}_{n}$ correspond to a distribution with mean zero and covariance matrix $\hat{\Sigma}_{n}$, which we denote as $\mathcal{N}\left( 0,\hat{\Sigma}_{n}\right) $. Throughout our development we use the notation $\left\langle A,B\right\rangle =$ tr$(A^{T}B)$ for any $d\times d$ matrices $A$, $B$, where $A^{T}$ denotes the transpose of $A$. The identity matrix is denoted by $I$. We use $\Rightarrow $ and $\overset{p}{\rightarrow
}$ to denote weak convergence (convergence in distribution) and convergence in probability, respectively. Finally, for two symmetric matrices $A,B\in
\mathbb{R}^{d\times d}$, $A\preceq B$ denotes that $A-B $ is positive semi-definite.
We define the Stein loss as $$L(X,\Sigma _{0})=-\log \det (X\Sigma _{0})+\left\langle X,\Sigma
_{0}\right\rangle -d,$$where $X$ is any estimator of the precision matrix (i.e. the inverse covariance matrix).
Given an uncertainty size $\rho $, let us write $X_{n}^{\ast }(\rho )$ for the distributionally robust estimator proposed in [nguyen2018distributionally]{}; i.e,$$X_{n}^{\ast }(\rho )=\arg \min_{X\succ 0}\left\{ -\log \det X+\sup_{\mathbb{Q}\in \mathcal{P}_{\rho }}\mathbb{E}^{\mathbb{Q}}\left[ \left\langle \xi \xi
^{T},X\right\rangle \right] \right\} , \label{DRO_sol}$$where $\mathcal{P}_{\rho }$ is the set of $d$-dimensional normal distributions with mean zero and which lie within distance $\rho $ measured in the Wasserstein sense, which we define next; see, for example, Chapter 7 in [@villani2003topics] for background on Wasserstein distances and, more generally, optimal transport costs. The Wasserstein distance (more precisely, the Wasserstein distance of order two with Euclidean norm) is defined as follows. First, let $\mathcal{M}_{+}(\mathbb{R}^{d}\times \mathbb{R}^{d})$ be the set of Borel (positive) measures on $\mathbb{R}^{d}\times
\mathbb{R}^{d}$ and define the Wasserstein distance between $\mathbb{\hat{P}}_{n}$ and $\mathbb{Q}$ via$$\begin{aligned}
\mathbb{W}_{2}(\mathbb{\hat{P}}_{n},\mathbb{Q})=\inf_{\pi \in \mathcal{M}_{+}(\mathbb{R}^{d}\times \mathbb{R}^{d})}\left\{ \left( \int \left\Vert
z-w\right\Vert _{2}^{2}\pi \left( \mathrm{d}x,\mathrm{d}w\right) \right)
^{1/2}\right. && \\
:\int_{w\in \mathbb{R}^{d}}\pi \left( \mathrm{d}x,\mathrm{d}w\right) =\mathbb{\hat{P}}_{n}\left( \mathrm{d}x\right) , &&\left. \int_{x\in \mathbb{R}^{d}}\pi \left( \mathrm{d}x,\mathrm{d}w\right) =\mathbb{Q}\left( \mathrm{d}w\right) \right\} .\end{aligned}$$Then $$\mathcal{P}_{\rho }=\left\{ \mathbb{Q}\sim \mathcal{N}\left( 0,\Sigma
\right) \text{ for some }\Sigma :\mathbb{W}_{2}(\mathbb{\hat{P}}_{n},\mathbb{Q})\leq \rho \right\} .$$
In simple terms, $\mathcal{P}_{\rho }$ is the set of probability measures corresponding to a Gaussian distribution which lie within $\rho $ units in the Wasserstein distance from $\mathbb{\hat{P}}_{n}$. It is well known (in fact, an immediate consequence of the delta method) that $n^{1/2}\mathbb{W}_{2}(\mathbb{\hat{P}}_{n},\mathbb{P}_{\infty })\Rightarrow \mathbb{W}$ for some limit law $\mathbb{W}$ which can be explicitly characterized (but not important for our development; see [@rippl2016limit]). This result suggests that $\rho :=\rho _{n}$ should scale in order $O\left(
n^{-1/2}\right) $. It is therefore somewhat surprising that the optimal scaling of $\rho $ for the purpose of minimizing the Stein loss is actually significantly smaller, as the main result of this paper indicates next.
\[Thm\_Main\]Let $$\rho _{n}=\arg \min_{\rho \geq 0}\{\mathbb{E}[L(X_{n}^{\ast }(\rho ),\Sigma
_{0})]\}, \label{rho_n_def}$$then $$\lim_{n \rightarrow \infty} n\rho _{n}=\rho _{\ast },$$for $\rho _{\ast }>0$.
**Remark:** The explicit expression of $\rho _{\ast }$ can be characterized as follows. First, let us consider the weak limit$$Z=\lim_{n\rightarrow \infty }n^{1/2}\left( \hat{\Sigma}_{n}-\Sigma
_{0}\right) ,$$which, by the Central Limit Theorem is a matrix with correlated mean zero Gaussian entries. Then, we have $$\rho _{\ast }=\mathbb{E}\left( \frac{4\text{tr}\left( \Sigma
_{0}^{-2}Z\Sigma _{0}^{-1}Z\right) }{\text{tr}(\Sigma _{0}^{-1})^{1/2}}-\frac{\text{tr}(Z\Sigma _{0}^{-2})^{2}}{\text{tr}(\Sigma _{0}^{-1})^{3/2}}\right) \frac{\text{tr}(\Sigma _{0}^{-1})}{4\text{tr}(\Sigma _{0}^{-2})}.$$Theorem \[Thm\_Main\] indicates that $\rho _{\ast }>0$, which will be verified as a part of the proof of this result.
Proof of Theorem \[Thm\_Main\]\[Proof\_of\_Thm\_1\]
===================================================
We first collect the following observations, which we summarize in the form of propositions and lemmas for which we provide references or corresponding proofs in the appendix [@supplement]. We then use these results to develop the proof of Theorem \[Thm\_Main\].
Auxiliary Results
-----------------
We provide a lemma based on the analytical solution (Theorem 3.1 in [nguyen2018distributionally]{}). [ ]{}
\[lemma\] When $n>d$ and $\rho \leq 1,$ with probability one, we have following Taylor expansions$$\begin{aligned}
\partial X_{n}^{\ast }(\rho )/\partial \rho &=&\hat{A}_{n}+O(\rho ), \\
X_{n}^{\ast }(\rho )^{-1} &=&\hat{\Sigma}_{n}-\hat{\Sigma}_{n}\hat{A}_{n}\hat{\Sigma}_{n}\rho +O(\rho ^{2}),\end{aligned}$$where$$\hat{A}_{n}=-\frac{2}{\sqrt{\text{tr}(\hat{\Sigma}_{n}^{-1})}}\hat{\Sigma}_{n}^{-2},$$Furthermore, the remainder terms satisfy$$\begin{aligned}
\frac{\partial X_{n}^{\ast }(\rho )}{\partial \rho }-\hat{A}_{n} &\succeq
&-\left( \frac{4\hat{M}_{n}+2\hat{M}_{n}^{2}}{\sqrt{\sum_{i=1}^{d}\hat{\lambda}_{i}^{-1}}}\sum_{i=1}^{d}\frac{\hat{v}_{i}\hat{v}_{i}^{T}}{\hat{\lambda}_{i}^{2}}\right) \rho \notag \\
\frac{\partial X_{n}^{\ast }(\rho )}{\partial \rho }-\hat{A}_{n} &\preceq
&\left( \frac{2\hat{M}_{n}^{3}+8\hat{M}_{n}}{\sqrt{\sum_{i=1}^{d}\hat{\lambda}_{i}^{-1}}}\sum_{i=1}^{d}\frac{\hat{v}_{i}\hat{v}_{i}^{T}}{\hat{\lambda}_{i}^{2}}\right) \rho , \label{remainder}\end{aligned}$$and $$-\left( \frac{2\left( 1+\hat{M}_{n}\right) ^{2}}{\sum_{i=1}^{d}\hat{\lambda}_{i}^{-1}}\sum_{i=1}^{d}\frac{\hat{v}_{i}\left( \hat{v}_{i}\right) ^{T}}{\hat{\lambda}_{i}}\right) \rho ^{2}\preceq X_{n}^{\ast }(\rho )^{-1}-\hat{\Sigma}_{n}+\hat{\Sigma}_{n}\hat{A}_{n}\hat{\Sigma}_{n}\rho \preceq \left(
\frac{2\hat{M}_{n}}{\sqrt{\sum_{i=1}^{d}\hat{\lambda}_{i}^{-1}}}\sum_{i=1}^{d}\hat{v}_{i}\left( \hat{v}_{i}\right) ^{T}\right) \rho ^{2},
\label{remainder2}$$where $$\hat{M}_{n}=\frac{8}{\left( \min_{i}\hat{\lambda}_{i}\right) \min \left\{ d,\frac{\sqrt{d}}{\sqrt{\max_{i}\hat{\lambda}_{i}}}\right\} }.$$
From Lemma \[lemma\], we have that $$X_{n}^{\ast }(\rho )^{-1}-\Sigma _{0}=\left( \hat{\Sigma}_{n}-\Sigma
_{0}\right) -\hat{\Sigma}_{n}\hat{A}_{n}\hat{\Sigma}_{n}\rho +O(\rho ^{2}).
\label{rho_decomp2}$$
The first proposition provides standard asymptotic normality results for various estimators.
\[probCLT\]The following convergence results hold
- $\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\xi _{i}\Rightarrow N(0,\Sigma
_{0}),$
- $\sqrt{n}\left( \hat{\Sigma}_{n}-\Sigma _{0}\right) \Rightarrow Z$, where $Z$ is a symmetric matrix of jointly Gaussian random variables with mean zero and $$cov(Z_{i_{1}j_{1}},Z_{i_{2}j_{2}})=\mathbb{E}\xi ^{(i_{1})}\xi ^{(j_{1})}\xi
^{(i_{2})}\xi ^{(j_{2})}-\left( \mathbb{E}\xi ^{(i_{1})}\xi
^{(j_{1})}\right) \left( \mathbb{E}\xi ^{(i_{2})}\xi ^{(j_{2})}\right)
=\sigma _{i_{1},i_{2}}^{2}\sigma _{j_{1},j_{2}}^{2}{}+\sigma
_{i_{1},j_{2}}^{2}\sigma _{j_{1},i_{2}}^{2},$$where $\xi ^{(i)}$ is the i-th entry of $\xi $ and $\sigma
_{i,j}^{2}=cov(\xi ^{(i)}\xi ^{(j)})$.
- $\hat{A}_{n}\overset{p}{\rightarrow }A_{0}$ and $\sqrt{n}\left(
\hat{A}_{n}-A_{0}\right) \Rightarrow Z_{A}$, where $A_{0}=-\frac{2}{\sqrt{\text{tr}(\Sigma _{0}^{-1})}}\Sigma _{0}^{-2}$ and $$Z_{A}=-\frac{\text{tr}\left( \Sigma _{0}^{-1}Z\Sigma _{0}^{-1}\right) \Sigma
_{0}^{-2}}{\text{tr}(\Sigma _{0}^{-1})^{3/2}}+2\frac{\Sigma _{0}^{-1}Z\Sigma
_{0}^{-2}+\Sigma _{0}^{-2}Z\Sigma _{0}^{-1}}{\text{tr}(\Sigma
_{0}^{-1})^{1/2}}.$$
Further, we also have the following observations.
\[large\_zero\]
- $\mathbb{E}\left\langle Z,Z_{A}\right\rangle >0$,
- $\mathbb{E}\left\langle \hat{\Sigma}_{n}-\Sigma _{0},\hat{A}_{n}-A_{0}\right\rangle >0.$
\[temp\] The following convergence in expectation results hold
- $\mathbb{E}\left[ \left\langle \hat{\Sigma}_{n}\hat{A}_{n}\hat{\Sigma}_{n},\hat{A}_{n}\right\rangle \right] \rightarrow \left\langle \Sigma
_{0}A_{0}\Sigma _{0},A_{0}\right\rangle .$
- $\mathbb{E}\left\langle \sqrt{n}\left( \hat{\Sigma}_{n}-\Sigma
_{0}\right) ,\hat{A}_{n}\right\rangle \rightarrow \mathbb{E}\left\langle
Z,A_{0}\right\rangle .$
- $\mathbb{E}\left\langle \sqrt{n}\left( \hat{\Sigma}_{n}-\Sigma
_{0}\right) ,\sqrt{n}\left( \hat{A}_{n}-A_{0}\right) \right\rangle
\rightarrow \mathbb{E}\left\langle Z,Z_{A}\right\rangle .$
The following proposition shows consistency of the estimator.
\[consistency\] For $\rho _{n}$ defined in (\[rho\_n\_def\]), we have $\lim_{n\rightarrow \infty }\rho _{n}=0.$
Using the previous technical results we are ready to provide the proof of Theorem \[Thm\_Main\].
Development of Proof of Theorem \[Thm\_Main\]
---------------------------------------------
The gradient of the Stein loss is given by $$h(X,\Sigma _{0})=\frac{\partial L(X,\Sigma _{0})}{\partial X}=-X^{-1}+\Sigma
_{0}.$$We claim that $\rho _{n}=0$ is not a minimizer. The derivative of loss function with respect to $\rho $ evaluating at $\rho =0$ is $$\left. \frac{\partial L(X_{n}^{\ast }(\rho ),\Sigma _{0})}{\partial \rho }\right\vert _{\rho =0}=\left\langle -\hat{\Sigma}_{n}+\Sigma _{0},\hat{A}_{n}\right\rangle .$$And by Proposition \[large\_zero\], we have $$\mathbb{E}\left\langle -\hat{\Sigma}_{n}+\Sigma _{0},\hat{A}_{n}\right\rangle =-\mathbb{E}\left\langle \hat{\Sigma}_{n}-\Sigma _{0},\hat{A}_{n}-A_{0}\right\rangle <0,$$which shows that $\rho _{n}=0$ is not a minimizer. Furthermore, we have $\lim_{\rho \rightarrow \infty }L(X_{n}^{\ast }(\rho _{n}),\Sigma
_{0})=+\infty $ (see, Proposition 3.5 in [@nguyen2018distributionally]). Therefore, the optimal solution is an interior point, i.e., $\rho _{n}\in
(0,+\infty ).$ Since $\rho _{n}$ is chosen to minimize $\mathbb{E}\left[
L(X_{n}^{\ast }(\rho _{n}),\Sigma _{0})\right] $, we have that $\rho _{n}$ satisfies the first order condition $$\mathbb{E}\left\langle h(X_{n}^{\ast }(\rho _{n}),\Sigma _{0}),\hat{A}_{n}+O(\rho _{n})\right\rangle =0. \label{FOC}$$By plugging (\[rho\_decomp2\]) into (\[FOC\]), we have$$\mathbb{E}\left\langle h(X_{n}^{\ast }(\rho _{n}),\Sigma _{0}),\hat{A}_{n}+O(\rho _{n})\right\rangle =-\mathbb{E}\left\langle \hat{\Sigma}_{n}-\Sigma _{0}-\hat{\Sigma}_{n}\hat{A}_{n}\hat{\Sigma}_{n}\rho _{n}+O(\rho
_{n}^{2}),\hat{A}_{n}+O(\rho _{n})\right\rangle =0, \label{imp_eqn}$$which is equivalent to $$\mathbb{E}\left[ \left\langle \hat{\Sigma}_{n}\hat{A}_{n}\hat{\Sigma}_{n},\hat{A}_{n}\right\rangle \right] \rho _{n}+O(\rho _{n}^{2})=\mathbb{E}\left\langle \hat{\Sigma}_{n}-\Sigma _{0},\hat{A}_{n}+O(\rho
_{n})\right\rangle . \label{FOC2}$$The validity of expanding the expectations follows by applying the uniform integrability results of the upper and lower bounds in (\[remainder\]) and (\[remainder2\]) underlying the proof of Lemma \[temp\].
Now, note that, also by Lemma \[temp\],$$\lim_{n\rightarrow \infty }\mathbb{E}\left[ \left\langle \hat{\Sigma}_{n}\hat{A}_{n}\hat{\Sigma}_{n},\hat{A}_{n}\right\rangle \right] =\left\langle
\Sigma _{0}A_{0}\Sigma _{0},A_{0}\right\rangle =4\text{tr}(\Sigma _{0}^{-2})/\text{tr}(\Sigma _{0}^{-1})>0.$$By multiplying $\sqrt{n}$ on both sides of (\[FOC2\]) and by Slutsky’s lemma (Theorem 1.8.10 in [@lehmann2006theory]), we have$$\lim_{n\rightarrow \infty }\sqrt{n}\left( \mathbb{E}\left[ \left\langle \hat{\Sigma}_{n}\hat{A}_{n}\hat{\Sigma}_{n},\hat{A}_{n}\right\rangle \right] \rho
_{n}+O(\rho _{n}^{2})\right) =\lim_{n\rightarrow \infty }\mathbb{E}\left\langle \sqrt{n}\left( \hat{\Sigma}_{n}-\Sigma _{0}\right) ,\hat{A}_{n}+O(\rho _{n})\right\rangle =\mathbb{E}\left\langle Z,A_{0}\right\rangle
=0.$$The last equality follows from $\mathbb{E}Z=0$ and $A_{0}$ being deterministic. Therefore,$$\lim_{n\rightarrow \infty }\sqrt{n}\rho _{n}=0.$$Furthermore, since $\mathbb{E}\left[ \hat{\Sigma}_{n}-\Sigma _{0}\right] =0$ for every $n,$ we have (once again by Lemma \[temp\]) $$\begin{aligned}
&&\lim_{n\rightarrow \infty }\mathbb{E}\left\langle n\left( \hat{\Sigma}_{n}-\Sigma _{0}\right) ,\hat{A}_{n}+O(\rho _{n})\right\rangle \\
&=&\lim_{n\rightarrow \infty }\mathbb{E}\left\langle n\left( \hat{\Sigma}_{n}-\Sigma _{0}\right) ,\hat{A}_{n}-A_{0}+A_{0}+O(\rho _{n})\right\rangle
\\
&=&\lim_{n\rightarrow \infty }\left\langle n\mathbb{E}\left[ \hat{\Sigma}_{n}-\Sigma _{0}\right] ,A_{0}\right\rangle +\lim_{n\rightarrow \infty }\mathbb{E}\left\langle \sqrt{n}\left( \hat{\Sigma}_{n}-\Sigma _{0}\right) ,\sqrt{n}\left( \hat{A}_{n}-A_{0}\right) +O(\sqrt{n}\rho _{n})\right\rangle
\\
&=&\mathbb{E}\left\langle Z,Z_{A}\right\rangle .\end{aligned}$$By multiplying $n$ on both sides of (\[FOC2\]), we have$$\rho _{\ast }=\lim_{n\rightarrow \infty }n\rho _{n}=\frac{\mathbb{E}\left\langle Z,Z_{A}\right\rangle }{\left\langle \Sigma _{0}A_{0}\Sigma
_{0},A_{0}\right\rangle }=\mathbb{E}\left( \frac{4\text{tr}\left( \Sigma
_{0}^{-2}Z\Sigma _{0}^{-1}Z\right) }{\text{tr}(\Sigma _{0}^{-1})^{1/2}}-\frac{\text{tr}(Z\Sigma _{0}^{-2})^{2}}{\text{tr}(\Sigma _{0}^{-1})^{3/2}}\right) \frac{\text{tr}(\Sigma _{0}^{-1})}{4\text{tr}(\Sigma _{0}^{-2})}>0,$$which is the desired result.
Numerical Experiments\[Sec\_Numerical\]
=======================================
Here we provide various numerical experiments to provide an empirical validation of our theory and the performance of the asymptotically optimal choice of uncertainty size in finite samples.
The first example is in one dimension. The data is sampled from a normal distribution, $N(0,\sigma _{0}^{2})$; i.e, $\Sigma _{0}=\sigma _{0}^{2}$ in the real line. Therefore, $$A_{0}=-2\sigma _{0}^{-3}\text{, }\mathbb{E}\left\langle Z,Z_{A}\right\rangle
=6\sigma _{0}^{-1}.$$Theorem \[Thm\_Main\] indicates that $$\lim_{n\rightarrow \infty }n\rho _{n}=\frac{3}{2}\sigma _{0}.$$
In our numerical example we fix $\sigma _{0}^{2}=10$. We vary the number of data points, $n$, ranging from 10 to 1000. For each $n$, we use $T=5000$ trials to compute empirically the optimal choice of $\rho =\rho _{n}$ in order to minimize the empirical Stein loss. Furthermore, we reformulate the limiting result as $$\rho _{n}=\frac{3}{2}\sigma _{0}/n\Leftrightarrow \log (\rho _{n})=-\log
(n)+\log \left( \frac{3}{2}\sigma _{0}\right) \text{.}$$We then perform a regression on $\log (\rho _{n})$ with respect to $\log (n)
$. Figure \[rho\] gives the relationship between $\rho $ and $n$ and the regression line. We can find that $n\rho _{n}$ is approximately equal to a constant, which is validated by the top right plot. The plots on the left show the qualitative behavior of $\rho _{n}$; the figure on the top left shows a behavior consistent with a decrease of order $O\left( 1/n\right) $, the bottom left plot shows that $n^{1/2}\rho _{n}$ still decreases to zero, indicating that $\rho _{n}$ converges to zero faster than the square-root rate. The regression statistics, corresponding to the regression plot shown in the bottom right of the plot, are shown in Table \[tab\] and $R^{2}=0.97 $.
The theoretical constant $\log \left( 1.5\cdot \sigma _{0}\right) =1.5568$ is very close to the empirical regression intercept $1.5525,$ while the coefficient multiplying $-\log \left( n\right) $ is close to unity. Hence, the empirical result matches perfectly with our theory.
![$\protect\rho _{n}$ VS $n$ for 1-dimension normal distribution[]{data-label="rho"}](rho_n.eps){width="8cm"}
We provide additional examples involving higher dimensions. In the subsequent examples, the data is sampled from a normal distribution $N(0,\Sigma _{0})$, where $\left( \Sigma _{0}\right) _{ij}=10\times
0.5^{|i-j|},$ $1\leq i,j\leq d$. We test the cases corresponding to $d=3$ and $d=5$ in the experiments. Due to computational constraints, we vary the number of data points, $n$, ranging from 20 to 400. For each $n$, we use $T=100$ trials to compute empirically the optimal choice of uncertainty to minimize the empirical Stein loss. Figures \[rho3\] and \[rho5\] show the results for the 3-dimension and 5-dimension cases, respectively. Tables \[tab2\] and \[tab3\] give the regression statistics and $R^{2}=0.97$ in both cases, and the performance is completely analogous to the one dimensional case, thus empirically validating our theoretical results.
![$\protect\rho _{n}$ VS $n$ for 3-dimension normal distribution[]{data-label="rho3"}](rho_n3.eps){width="8cm"}
![$\protect\rho _{n}$ VS $n$ for 5-dimension normal distribution[]{data-label="rho5"}](rho_n5.eps){width="8cm"}
Acknowledgements {#acknowledgements .unnumbered}
================
We gratefully acknowledge support from the following NSF grants 1915967, 1820942, 1838676 as well as DARPA award N660011824028.
Proofs of Auxiliary Results
===========================
Proof of Lemma \[lemma\]
------------------------
We first restate a theorem in [@nguyen2018distributionally].
If $\rho >0$ and $\hat{\Sigma}_{n}$ admits the spectral decomposition $\hat{\Sigma}_{n}=\sum_{i=1}^{d}\hat{\lambda}_{i}\hat{v}_{i}\left( \hat{v}_{i}\right) ^{T}$ with eigenvalues $\hat{\lambda}_{i}$ and corresponding orthonormal eigenvectors $\hat{v}_{i},$ $i\leq d,$ then the unique minimizer of (\[DRO\_sol\]) is given by $X_{n}^{\ast }(\rho
)=\sum_{i=1}^{d}x_{i}^{\ast }\hat{v}_{i}\left( \hat{v}_{i}\right) ^{T},$ where $$\hat{x}_{i}^{\ast }=\gamma ^{\ast }\left[ 1-\frac{1}{2}\left( \sqrt{\hat{\lambda}_{i}^{2}\left( \gamma ^{\ast }\right) ^{2}+4\hat{\lambda}_{i}\gamma
^{\ast }}-\hat{\lambda}_{i}\gamma ^{\ast }\right) \right] , \label{x_eqn}$$and $\gamma ^{\ast }>0$ is the unique positive solution of the algebraic equation $$\left( \rho ^{2}-\frac{1}{2}\sum_{i=1}^{d}\hat{\lambda}_{i}\right) \gamma -d+\frac{1}{2}\sum_{i=1}^{d}\sqrt{\hat{\lambda}_{i}^{2}\gamma ^{2}+4\hat{\lambda}_{i}\gamma }=0. \label{gamma_eqn}$$
Since the underlying covariance matrix is invertible with probability one when $n>d$, we have $\hat{\lambda}_{i}>0$ for $i=1,2,\ldots ,d$. We consider the case $\rho \leq1$. Note that we have the following inequality,$$\sqrt{\hat{\lambda}_{i}^{2}\gamma ^{2}+4\hat{\lambda}_{i}\gamma }-\left(
\hat{\lambda}_{i}\gamma +2\right) =-\frac{4}{\sqrt{\hat{\lambda}_{i}^{2}\gamma ^{2}+4\hat{\lambda}_{i}\gamma }+\left( \hat{\lambda}_{i}\gamma +2\right) }\geq -\frac{2}{\hat{\lambda}_{i}\gamma }.
\label{first_inequ}$$Then, (\[gamma\_eqn\]) gives us $\rho \gamma ^{\ast }\leq \sqrt{\sum_{i=1}^{d}\hat{\lambda}_{i}^{-1}}.$ On the other hand, we have $$\begin{aligned}
\rho ^{2}\gamma ^{\ast } &=&\frac{1}{2}\sum_{i=1}^{d}\left( 2+\hat{\lambda}_{i}\gamma ^{\ast }-\sqrt{\hat{\lambda}_{i}^{2}\left( \gamma ^{\ast }\right)
^{2}+4\hat{\lambda}_{i}\gamma ^{\ast }}\right) \notag \\
&=&\sum_{i=1}^{d}\frac{2}{\sqrt{\hat{\lambda}_{i}^{2}\left( \gamma ^{\ast
}\right) ^{2}+4\hat{\lambda}_{i}\left( \gamma ^{\ast }\right) ^{2}}+\left(
\hat{\lambda}_{i}\gamma ^{\ast }+2\right) } \notag \\
&\geq &\sum_{i=1}^{d}\frac{1}{\hat{\lambda}_{i}\gamma ^{\ast }+2}
\label{large_gamma_inequ} \\
&\geq &d\frac{1}{\left( \max_{i}\hat{\lambda}_{i}\right) \gamma ^{\ast }+2}.
\notag\end{aligned}$$Then, a basic property of the quadratic equation gives us that $$\gamma ^{\ast }\geq \frac{\sqrt{1+\left( \max_{i}\hat{\lambda}_{i}\right)
d/\rho ^{2}}-1}{\left( \max_{i}\hat{\lambda}_{i}\right) }\geq \frac{1}{4}\min \left\{ d/\rho ^{2},\frac{\sqrt{d}/\rho }{\sqrt{\max_{i}\hat{\lambda}_{i}}}\right\} .$$Furthermore, (\[large\_gamma\_inequ\]) also shows that $$\begin{aligned}
\rho ^{2}\gamma ^{\ast } &\geq &\left( \sum_{i=1}^{d}\frac{1}{\hat{\lambda}_{i}\gamma ^{\ast }}\right) \frac{1}{1+2/\left( \left( \min_{i}\hat{\lambda}_{i}\right) \gamma ^{\ast }\right) } \notag \\
&\geq &\frac{1}{\gamma ^{\ast }}\left( \sum_{i=1}^{d}\hat{\lambda}_{i}^{-1}\right) \frac{\min \left\{ d/\rho ^{2},\frac{\sqrt{d}/\rho }{\sqrt{\max_{i}\hat{\lambda}_{i}}}\right\} }{\min \left\{ d/\rho ^{2},\frac{\sqrt{d}/\rho }{\sqrt{\max_{i}\hat{\lambda}_{i}}}\right\} +8/\left( \min_{i}\hat{\lambda}_{i}\right) }. \label{gamma_bdd}\end{aligned}$$By combining all of the above and noticing that $1+x\geq \sqrt{1+x}$ for $x\geq 0$, we have for $\rho \leq 1$ $$\frac{\rho }{\sqrt{\sum_{i=1}^{d}\hat{\lambda}_{i}^{-1}}}+\frac{\hat{M}_{n}}{\sqrt{\sum_{i=1}^{d}\hat{\lambda}_{i}^{-1}}}\rho ^{2}\geq \frac{1}{\gamma
^{\ast }}\geq \frac{1}{\sqrt{\sum_{i=1}^{d}\hat{\lambda}_{i}^{-1}}}\rho ,
\label{gamma}$$where $$\hat{M}_{n}=\frac{8/\left( \min_{i}\hat{\lambda}_{i}\right) }{\min \left\{ d,\frac{\sqrt{d}}{\sqrt{\max_{i}\hat{\lambda}_{i}}}\right\} }.$$By plugging it to (\[gamma\_eqn\]), we have $$\frac{1}{x_{i}^{\ast }}= \frac{\sqrt{\hat{\lambda}_{i}^{2}\left(
\gamma ^{\ast }\right) ^{2}+4\hat{\lambda}_{i}\gamma ^{\ast }}+\hat{\lambda}_{i}\gamma ^{\ast }+2}{2\gamma ^{\ast }}\leq \hat{\lambda}_{i}+2/\gamma ^{\ast }\leq \hat{\lambda}_{i}+\frac{2\rho }{\sqrt{\sum_{i=1}^{d}\hat{\lambda}_{i}^{-1}}}+\frac{2\hat{M}_{n}}{\sqrt{\sum_{i=1}^{d}\hat{\lambda}_{i}^{-1}}}\rho ^{2}.$$For the lower bound of $1/x_{i}^{\ast }$, we have$$1/x_{i}^{\ast }=\frac{\sqrt{\hat{\lambda}_{i}^{2}\left( \gamma ^{\ast
}\right) ^{2}+4\hat{\lambda}_{i}\gamma ^{\ast }}+\hat{\lambda}_{i}\gamma
^{\ast }+2}{2\gamma ^{\ast }}\geq \frac{\sqrt{\hat{\lambda}_{i}^{2}\left(
\gamma ^{\ast }\right) ^{2}+4\hat{\lambda}_{i}\gamma ^{\ast }}}{\gamma
^{\ast }}.$$Then by (\[first\_inequ\]), we have for $\rho \leq 1$ $$\begin{aligned}
\frac{\sqrt{\hat{\lambda}_{i}^{2}\left( \gamma ^{\ast }\right) ^{2}+4\hat{\lambda}_{i}\gamma ^{\ast }}}{\gamma ^{\ast }} &\geq &\hat{\lambda}_{i}+2/\gamma ^{\ast }-\frac{2}{\hat{\lambda}_{i}\left( \gamma ^{\ast
}\right) ^{2}} \\
&\geq &\hat{\lambda}_{i}+\frac{2\rho }{\sqrt{\sum_{i=1}^{d}\hat{\lambda}_{i}^{-1}}}-\frac{2\left( 1+\hat{M}_{n}\right) ^{2}}{\hat{\lambda}_{i}\left(
\sum_{i=1}^{d}\hat{\lambda}_{i}^{-1}\right) }\rho ^{2}\end{aligned}$$Therefore, we conclude that $$\frac{1}{x_{i}^{\ast }}=\hat{\lambda}_{i}+\frac{2\rho }{\sqrt{\sum_{i=1}^{d}\hat{\lambda}_{i}^{-1}}}+O(\rho ^{2}).$$and $$X_{n}^{\ast }(\rho )^{-1}=\hat{\Sigma}_{n}-\hat{\Sigma}_{n}\hat{A}_{n}\hat{\Sigma}_{n}\rho +O(\rho ^{2}),$$where $$\hat{A}_{n}=-\sum_{i=1}^{d}\frac{2\hat{v}_{i}\left( \hat{v}_{i}\right) ^{T}}{\hat{\lambda}_{i}^{2}\sqrt{\sum_{i=1}^{d}\hat{\lambda}_{i}^{-1}}}=-\frac{2}{\sqrt{\text{tr}(\hat{\Sigma}_{n}^{-1})}}\hat{\Sigma}_{n}^{-2}.$$Specifically, the remainder terms satisfy $$-\left( \frac{2\left( 1+\hat{M}_{n}\right) ^{2}}{\sum_{i=1}^{d}\hat{\lambda}_{i}^{-1}}\sum_{i=1}^{d}\frac{\hat{v}_{i}\left( \hat{v}_{i}\right) ^{T}}{\hat{\lambda}_{i}}\right) \rho ^{2}\preceq X_{n}^{\ast }(\rho )^{-1}-\hat{\Sigma}_{n}+\hat{\Sigma}_{n}\hat{A}_{n}\hat{\Sigma}_{n}\rho \preceq \left(
\frac{2\hat{M}_{n}}{\sqrt{\sum_{i=1}^{d}\hat{\lambda}_{i}^{-1}}}\sum_{i=1}^{d}\hat{v}_{i}\left( \hat{v}_{i}\right) ^{T}\right) \rho ^{2}.$$We complete the proof of (\[remainder2\]).
For the the proof of (\[remainder\]), note that (\[gamma\_eqn\]) indicates $$\begin{aligned}
\frac{d\gamma ^{\ast }}{d\rho } &=&\frac{-2\rho \gamma ^{\ast }}{\left( \rho
^{2}-\frac{1}{2}\sum_{i=1}^{d}\hat{\lambda}_{i}\right) +\frac{1}{2}\sum_{i=1}^{d}\hat{\lambda}_{i}\frac{\hat{\lambda}_{i}\gamma ^{\ast }+2}{\sqrt{\hat{\lambda}_{i}^{2}\left( \gamma ^{\ast }\right) ^{2}+4\hat{\lambda}_{i}\gamma ^{\ast }}}} \\
&=&-\frac{2\rho \gamma ^{\ast }}{\rho ^{2}+\sum_{i=1}^{d}\frac{2\hat{\lambda}_{i}}{\sqrt{\hat{\lambda}_{i}^{2}\left( \gamma ^{\ast }\right) ^{2}+4\hat{\lambda}_{i}\gamma ^{\ast }}\left( \hat{\lambda}_{i}\gamma ^{\ast }+2+\sqrt{\hat{\lambda}_{i}^{2}\left( \gamma ^{\ast }\right) ^{2}+4\hat{\lambda}_{i}\gamma ^{\ast }}\right) }}.\end{aligned}$$Since $\hat{\lambda}_{i}\gamma ^{\ast }\leq \sqrt{\hat{\lambda}_{i}^{2}\left( \gamma ^{\ast }\right) ^{2}+4\hat{\lambda}_{i}\gamma ^{\ast }}\leq \hat{\lambda}_{i}\gamma ^{\ast }+2$ and $\rho \gamma ^{\ast }\leq \sqrt{\sum_{i=1}^{d}\hat{\lambda}_{i}^{-1}},$ we have $$-\frac{2\rho \gamma ^{\ast }}{\rho ^{2}+\sum_{i=1}^{d}\frac{\hat{\lambda}_{i}}{\left( \hat{\lambda}_{i}\gamma ^{\ast }+2\right) ^{2}}}\leq \frac{\partial
\gamma ^{\ast }}{\partial \rho }\leq -\frac{2\rho \gamma ^{\ast }}{\rho
^{2}+\sum_{i=1}^{d}\frac{\hat{\lambda}_{i}}{\left( \hat{\lambda}_{i}\gamma
^{\ast }\right) ^{2}}}\leq -\frac{\rho \left( \gamma ^{\ast }\right) ^{3}}{\sum_{i=1}^{d}\hat{\lambda}_{i}^{-1}}.$$Then, by using the bound (\[gamma\]), we further have $$-\frac{\rho \left( \gamma ^{\ast }\right) ^{3}}{\sum_{i=1}^{d}\hat{\lambda}_{i}^{-1}}\leq -\frac{\left( \gamma ^{\ast }\right) ^{2}}{\sqrt{\sum_{i=1}^{d}\hat{\lambda}_{i}^{-1}}}\left( 1-\hat{M}_{n}\rho \right) .$$Furthermore, the proof of Proposition 3.5 in [nguyen2018distributionally]{} indicates that$$\frac{\partial x_{i}}{\partial \gamma ^{\ast }}=1+\hat{\lambda}_{i}\gamma
^{\ast }-\frac{\hat{\lambda}_{i}^{2}\left( \gamma ^{\ast }\right) ^{2}+3\hat{\lambda}_{i}\gamma ^{\ast }}{\sqrt{\hat{\lambda}_{i}^{2}\left( \gamma ^{\ast
}\right) ^{2}+4\hat{\lambda}_{i}\gamma ^{\ast }}}.$$Let $z_{i}=\hat{\lambda}_{i}\gamma ^{\ast }$ for $i=1,2,\ldots ,d.$ We have$$\frac{\partial x_{i}}{\partial \gamma ^{\ast }}=\frac{4z_{i}}{\sqrt{z_{i}^{2}+4z_{i}}\left( \left( 1+z_{i}\right) \sqrt{z_{i}^{2}+4z_{i}}+z_{i}^{2}+3z_{i}\right) }\in \left[ \frac{2z_{i}}{\left( z_{i}+2\right) ^{3}},\frac{2}{z_{i}^{2}}\right] .$$From (\[gamma\_bdd\]), we have$$\sum_{i=1}^{d}\frac{\hat{\lambda}_{i}}{\left( \hat{\lambda}_{i}\gamma ^{\ast
}+2\right) ^{2}}\geq \frac{1}{\left( \gamma ^{\ast }\right) ^{2}}\left(
\sum_{i=1}^{d}\hat{\lambda}_{i}^{-1}\right) \left( \frac{\min \left\{ d/\rho
^{2},\frac{\sqrt{d}/\rho }{\sqrt{\max_{i}\hat{\lambda}_{i}}}\right\} }{\min
\left\{ d/\rho ^{2},\frac{\sqrt{d}/\rho }{\sqrt{\max_{i}\hat{\lambda}_{i}}}\right\} +8/\left( \min_{i}\hat{\lambda}_{i}\right) }\right) ^{2}
\label{bdd_2_2}$$and $$\frac{2z_{i}}{\left( z_{i}+2\right) ^{3}}\geq \frac{2}{\left( \hat{\lambda}_{i}\gamma ^{\ast }\right) ^{2}}\left( \frac{\min \left\{ d/\rho ^{2},\frac{\sqrt{d}/\rho }{\sqrt{\max_{i}\hat{\lambda}_{i}}}\right\} }{\min \left\{
d/\rho ^{2},\frac{\sqrt{d}/\rho }{\sqrt{\max_{i}\hat{\lambda}_{i}}}\right\}
+8/\left( \min_{i}\hat{\lambda}_{i}\right) }\right) ^{3}.$$Therefore, by combining (\[gamma\_bdd\]) and (\[bdd\_2\_2\]), we have for $\rho \leq 1,$ $$\begin{aligned}
\frac{\partial \gamma ^{\ast }}{\partial \rho } &\geq &-\frac{2\rho \left(
\gamma ^{\ast }\right) ^{3}}{\left( \sum_{i=1}^{d}\hat{\lambda}_{i}^{-1}\right) \left( \frac{\min \left\{ d/\rho ^{2},\frac{\sqrt{d}/\rho }{\sqrt{\max_{i}\hat{\lambda}_{i}}}\right\} }{\min \left\{ d/\rho ^{2},\frac{\sqrt{d}/\rho }{\sqrt{\max_{i}\hat{\lambda}_{i}}}\right\} +8/\left( \min_{i}\hat{\lambda}_{i}\right) }+\left( \frac{\min \left\{ d/\rho ^{2},\frac{\sqrt{d}/\rho }{\sqrt{\max_{i}\hat{\lambda}_{i}}}\right\} }{\min \left\{ d/\rho
^{2},\frac{\sqrt{d}/\rho }{\sqrt{\max_{i}\hat{\lambda}_{i}}}\right\}
+8/\left( \min_{i}\hat{\lambda}_{i}\right) }\right) ^{2}\right) } \\
&\geq &-\frac{\rho \left( \gamma ^{\ast }\right) ^{3}}{\left( \sum_{i=1}^{d}\hat{\lambda}_{i}^{-1}\right) }\left( 1+\hat{M}_{n}\rho \right) ^{2} \\
&\geq &-\frac{\rho \left( \gamma ^{\ast }\right) ^{3}}{\left( \sum_{i=1}^{d}\hat{\lambda}_{i}^{-1}\right) }\left( 1+\left( 2\hat{M}_{n}+\hat{M}_{n}^{2}\right) \rho \right) .\end{aligned}$$Similarly, we have for $\rho \leq 1,$$$\frac{2z_{i}}{\left( z_{i}+2\right) ^{3}}\geq \frac{2}{\left( \hat{\lambda}_{i}\gamma ^{\ast }\right) ^{2}}\left( 1-\hat{M}_{n}\rho \right) ^{3}\geq
\frac{2}{\left( \hat{\lambda}_{i}\gamma ^{\ast }\right) ^{2}}\left( 1-\left(
\hat{M}_{n}^{3}+3\hat{M}_{n}\right) \rho \right) .$$Finally,$\ $by combining all of the above together and the chain rule, we have$$-\frac{2}{\hat{\lambda}_{i}^{2}}\frac{\rho \gamma ^{\ast }}{\left(
\sum_{i=1}^{d}\hat{\lambda}_{i}^{-1}\right) }\left( 1+\left( 2\hat{M}_{n}+\hat{M}_{n}^{2}\right) \rho \right) \leq \frac{\partial x_{i}}{\partial \rho
}\leq -\frac{2}{\hat{\lambda}_{i}^{2}\sqrt{\sum_{i=1}^{d}\hat{\lambda}_{i}^{-1}}}\left( 1-\hat{M}_{n}\rho \right) \left( 1-\left( \hat{M}_{n}^{3}+3\hat{M}_{n}\right) \rho \right) .$$After simplification, we have$$-\frac{\left( 4\hat{M}_{n}+2\hat{M}_{n}^{2}\right) }{\hat{\lambda}_{i}^{2}\sqrt{\sum_{i=1}^{d}\hat{\lambda}_{i}^{-1}}}\rho \leq \frac{\partial x_{i}}{\partial \rho }+\frac{2}{\hat{\lambda}_{i}^{2}\sqrt{\sum_{i=1}^{d}\hat{\lambda}_{i}^{-1}}}\leq \frac{\left( 2\hat{M}_{n}^{3}+8\hat{M}_{n}\right) }{\hat{\lambda}_{i}^{2}\sqrt{\sum_{i=1}^{d}\hat{\lambda}_{i}^{-1}}}\rho .$$Furthermore, we have $$-\left( \frac{4\hat{M}_{n}+2\hat{M}_{n}^{2}}{\sqrt{\sum_{i=1}^{d}\hat{\lambda}_{i}^{-1}}}\sum_{i=1}^{d}\frac{\hat{v}_{i}\left( \hat{v}_{i}\right) ^{T}}{\hat{\lambda}_{i}^{2}}\right) \rho \preceq \frac{\partial X_{n}^{\ast }(\rho
)}{\partial \rho }-\hat{A}_{n}\preceq \left( \frac{2\hat{M}_{n}^{3}+8\hat{M}_{n}}{\sqrt{\sum_{i=1}^{d}\hat{\lambda}_{i}^{-1}}}\sum_{i=1}^{d}\frac{\hat{v}_{i}\left( \hat{v}_{i}\right) ^{T}}{\hat{\lambda}_{i}^{2}}\right) \rho .$$This completes the proof.
Proof of Proposition \[probCLT\]
--------------------------------
[ ]{}
The proof follows from the standard central limit theorem (CLT).
Since $\hat{\Sigma}_{n}$ is the average of i.i.d copies $\xi _{i}\xi
_{i}^{T},$ the result follows by CLT.
The first statement follows from the continuous mapping theorem and $\hat{\Sigma}_{n}\overset{p}{\rightarrow }\Sigma _{0}.$ Let $f(\Sigma )=-2\left( \text{tr}(\Sigma ^{-1})\right) ^{-1/2}\Sigma ^{-2}$, where $\Sigma $ is positive-definite matrix. We now expand $f\left( \Sigma
+hA\right) $ for any matrix $A$ as the scalar $h>0$ tends to zero to obtain a representation for the gradient of $f(\Sigma )$, $Df(\Sigma )$. This expansion yields$$\begin{aligned}
f\left( \Sigma +hA\right) &=&-2\left( \text{tr}(\left( \Sigma +hA\right)
^{-1})\right) ^{-1/2}\left( \Sigma +hA\right) ^{-2} \\
&=&-2\left( \text{tr}(\Sigma ^{-1})-\text{tr}\left( h\Sigma ^{-1}A\Sigma
^{-1}\right) +o\left( h\right) \right) ^{-1/2}\left( \left( I+h\Sigma
^{-1}A\right) ^{-1}\Sigma ^{-1}\right) ^{2} \\
&=&-2\text{tr}(\Sigma ^{-1})^{-1/2}\left( 1-h\frac{\text{tr}\left( \Sigma
^{-1}A\Sigma ^{-1}\right) }{\text{tr}(\Sigma ^{-1})}\right) ^{-1/2}\left(
\Sigma ^{-2}-h\Sigma ^{-1}A\Sigma ^{-2}-h\Sigma ^{-2}A\Sigma ^{-1}\right)
+o\left( h\right) \\
&=&-2\text{tr}(\Sigma ^{-1})^{-1/2}\left( 1+h\frac{\text{tr}\left( \Sigma
^{-1}A\Sigma ^{-1}\right) }{2\text{tr}(\Sigma ^{-1})}\right) \left( \Sigma
^{-2}-h\Sigma ^{-1}A\Sigma ^{-2}-h\Sigma ^{-2}A\Sigma ^{-1}\right) +o\left(
h\right) \\
&=&-2\text{tr}(\Sigma ^{-1})^{-1/2}\left( \Sigma ^{-2}+h\frac{\text{tr}\left( \Sigma ^{-1}A\Sigma ^{-1}\right) \Sigma ^{-2}}{2\text{tr}(\Sigma
^{-1})}-h\Sigma ^{-1}A\Sigma ^{-2}-h\Sigma ^{-2}A\Sigma ^{-1}\right)
+o\left( h\right) \\
&=&f\left( \Sigma \right) -h\frac{\text{tr}\left( \Sigma ^{-1}A\Sigma
^{-1}\right) \Sigma ^{-2}}{\text{tr}(\Sigma ^{-1})^{3/2}}+2h\frac{\Sigma
^{-1}A\Sigma ^{-2}+\Sigma ^{-2}A\Sigma ^{-1}}{\text{tr}(\Sigma ^{-1})^{1/2}}+o\left( h\right) ,\end{aligned}$$which, in turn, results in the linear operator satisfying for any $A\in
\mathbb{R}^{d\times d}$$$Df(\Sigma )A=-\frac{\text{tr}\left( \Sigma ^{-1}A\Sigma ^{-1}\right) \Sigma
^{-2}}{\text{tr}(\Sigma ^{-1})^{3/2}}+2\frac{\Sigma ^{-1}A\Sigma
^{-2}+\Sigma ^{-2}A\Sigma ^{-1}}{\text{tr}(\Sigma ^{-1})^{1/2}}. \label{Df}$$After applying the delta method, we have the desired result.
Proof of Proposition \[large\_zero\]
------------------------------------
We first note the following elementary result, which is standard in matrix algebra.
\[trace\_ineqn\]For any $d\times d$ matrices $A,B$ (real valued) we have$$\text{tr}(A^{T}A)\text{tr}(B^{T}B)\geq \text{tr}(A^{T}B)^{2}=\left\vert
\left\langle A,B\right\rangle \right\vert ^{2},$$where strict inequality holds unless $A$ is a multiple of $B$.
By the Cauchy-Schwarz inequality, we have$$\text{tr}(A^{T}A)\text{tr}(B^{T}B)=\left( \sum_{i,j}A_{ij}^{2}\right) \left(
\sum_{i,j}B_{ij}^{2}\right) \geq \left( \sum_{i,j}A_{ij}B_{ij}\right)
^{2}=\left\vert \left\langle A,B\right\rangle \right\vert ^{2}.$$
Now we proceed with the proof of Proposition \[large\_zero\].
It suffices to show that $\left\langle Z,Z_{A}\right\rangle \geq 0$ with probability one and that $\left\langle Z,Z_{A}\right\rangle >0$ with positive probability. Note that$$\left\langle Z,Z_{A}\right\rangle =-\frac{\text{tr}(Z\Sigma _{0}^{-2})^{2}}{\text{tr}(\Sigma _{0}^{-1})^{3/2}}+\frac{4\text{tr}\left( \Sigma
_{0}^{-1}Z\Sigma _{0}^{-2}Z\right) }{\text{tr}(\Sigma _{0}^{-1})^{1/2}}.$$We will show that$$\text{tr}(\Sigma _{0}^{-1})\text{tr}\left( \Sigma _{0}^{-1}Z\Sigma
_{0}^{-2}Z\right) \geq \text{tr}(Z\Sigma _{0}^{-2})^{2} \label{matrix_inequ}$$follows from Lemma \[trace\_ineqn\]. This implies that $\left\langle
Z,Z_{A}\right\rangle \geq 0$. The equality holds if and only if there exists $a\geq 0$ such that [$Z\Sigma _{0}^{-2}Z=aI,$ which is equivalent to ]{}$Z=\sqrt{a}\Sigma _{0}$[. We know that ]{}$Z\neq \sqrt{a}\Sigma
_{0}$ with probability one. Thus, $\left\langle Z,Z_{A}\right\rangle >0$ with probability one.
To show (\[matrix\_inequ\]), we use the Polar factorization (see, for example, Chapter 4.2 in [@golub2012matrix]) for positive definite matrices. That is, we write $\Sigma _{0}^{1/2}\Sigma _{0}^{1/2}=\Sigma _{0}$, where $\Sigma _{0}^{1/2}$ is a symmetric positive definite matrix. Note that we can write $$Z=\Sigma _{0}^{1/2}W\Sigma _{0}^{1/2},$$where $W=\Sigma _{0}^{-1/2}Z\Sigma _{0}^{-1/2}$ is a symmetric matrix. To recover the matrices $A$ and $B$, we let$$A=\Sigma _{0}^{-1/2}\text{, }S=\Sigma _{0}\text{ and }B=WS^{-1/2}.$$Note that$$\begin{aligned}
&&\text{tr}\left( \Sigma _{0}^{-1}Z\Sigma _{0}^{-2}Z\right) =\text{tr}\left(
Z\Sigma _{0}^{-1}\cdot \Sigma _{0}^{-1}Z\Sigma _{0}^{-1}\right) \\
&=&\text{tr}\left( S^{1/2}WS^{1/2}S^{-1/2}S^{-1/2}\cdot
S^{-1/2}S^{-1/2}\left( S^{1/2}WS^{1/2}\right) S^{-1/2}S^{-1/2}\right) \\
&=&\text{tr}\left( S^{1/2}WS^{-1/2}\cdot S^{-1/2}WS^{-1/2}\right) =\text{tr}\left( S^{-1/2}WWS^{-1/2}\right) =\text{tr}\left( B ^{T}B\right) .\end{aligned}$$
Therefore, this verifies that the choice of $B$ is consistent with the use of Lemma \[trace\_ineqn\]. Clearly, $AA^{T}=\Sigma _{0}^{-1}$, thus making this choice also consistent with Lemma \[trace\_ineqn\]. Finally, we have that$$\begin{aligned}
\text{tr}(Z\Sigma _{0}^{-2}) &=&\text{tr}(S^{1/2}WS^{1/2}S^{-1/2}S^{-1/2}\Sigma _{0}^{-1}) \\
&=&\text{tr}(S^{1/2}WS^{-1/2}S^{-1/2}S^{-1/2}) \\
&=&\text{tr}(WS^{-1/2}S^{-1/2})=\text{tr}(S^{-1/2}S^{-1/2}W)=\text{tr}\left(
A^{T}B\right) .\end{aligned}$$The result then follows.
[ ]{}
$$\begin{aligned}
\left\langle \hat{\Sigma}_{n}-\Sigma _{0},\hat{A}_{n}-A_{0}\right\rangle
&=&-2\left\langle \hat{\Sigma}_{n}-\Sigma _{0},\frac{1}{\sqrt{\text{tr}(\hat{\Sigma}_{n}^{-1})}}\hat{\Sigma}_{n}^{-2}-\frac{1}{\sqrt{\text{tr}(\Sigma
_{0}^{-1})}}\Sigma _{0}^{-2}\right\rangle \label{2_inequ} \\
&=&2\left( -\sqrt{\text{tr}(\hat{\Sigma}_{n}^{-1})}-\sqrt{\text{tr}(\Sigma
_{0})}+\frac{\text{tr}(\hat{\Sigma}_{n}^{-1}\Sigma _{0}\hat{\Sigma}_{n}^{-1})}{\sqrt{\text{tr}(\hat{\Sigma}_{n}^{-1})}}+\frac{\text{tr}(\Sigma _{0}^{-1}\hat{\Sigma}_{n}\Sigma _{0}^{-1})}{\sqrt{\text{tr}(\Sigma _{0}^{-1})}}\right) . \notag\end{aligned}$$
By Lemma (\[trace\_ineqn\]) and similar arguments with (1), we have $$\text{tr}(\hat{\Sigma}_{n}^{-1}\Sigma _{0}\hat{\Sigma}_{n}^{-1})\geq \frac{\text{tr}(\hat{\Sigma}_{n}^{-1})^{2}}{\text{tr}(\Sigma _{0}^{-1})}\text{ and
tr}(\Sigma _{0}^{-1}\hat{\Sigma}_{n}\Sigma _{0}^{-1})\geq \frac{\text{tr}(\Sigma _{0}^{-1})^{2}}{\text{tr}(\hat{\Sigma}_{n}^{-1})}.
\label{norm_inequ}$$By plugging (\[norm\_inequ\]) into (\[2\_inequ\]), we have$$\left\langle \hat{\Sigma}_{n}-\Sigma _{0},\hat{A}_{n}-A_{0}\right\rangle
\geq 2\left( -\sqrt{\text{tr}(\hat{\Sigma}_{n}^{-1})}-\sqrt{\text{tr}(\Sigma
_{0})}+\frac{\text{tr}(\hat{\Sigma}_{n}^{-1})^{3/2}}{\text{tr}(\Sigma
_{0}^{-1})}+\frac{\text{tr}(\Sigma _{0}^{-1})^{3/2}}{\text{tr}(\hat{\Sigma}_{n}^{-1})}\right) .$$Consider the function $g:\mathbb{R}_{+}\times \mathbb{R}_{+}\rightarrow \mathbb{R}$, $$g(a,b)=-a-b+\frac{b^{3}}{a^{2}}+\frac{a^{3}}{b^{2}}=\frac{(a^{3}-b^{3})(a^{2}-b^{2})}{a^{2}b^{2}}\geq 0,$$and the equality holds if $a=b.$ Since $\sqrt{\text{tr}(\hat{\Sigma}_{n}^{-1})}=\sqrt{\text{tr}(\Sigma _{0})}$ with probability zero, the desired result follows.
Proof of Lemma \[temp\]
-----------------------
We first collect a few results from linear algebra (see, for example, equation (2.3.3) and (2.3.7) in [@golub2012matrix]).
\[Lem\_Aux\_UI1\]For any $d\times d$ matrix $A$ (real valued) we define $\left\Vert A\right\Vert _{F}^{2}=\left\langle A,A\right\rangle = \mathrm{tr}
\left( A^{T}A\right) $ (the Frobenius norm) and let $\left\Vert A\right\Vert
_{2}^{2}=\left\vert \lambda _{\max }\left( A^{T}A\right) \right\vert $ (where $\lambda _{\max }\left( B\right) $ is the eigenvalue of largest modulus of the matrix $B$). Then, for any $A,B$ matrices of size $d\times d$ with real valued elements we have $$\left\Vert AB\right\Vert _{F}\leq \left\Vert A\right\Vert _{F}\left\Vert
B\right\Vert _{F},\text{ \ }\left\Vert B\right\Vert _{2}\leq \left\Vert
B\right\Vert _{F}.$$
In addition, we have the following properties of the distribution of $\hat{\Sigma}_{n}$, which follows the Wishart law (see, for example, Theorem 13.3.2 in [@anderson2003introduction]).
\[Lem\_Aux\_UI2\]Assume $n>d.$ Let us write $\xi _{i}=C\zeta _{i}$ where $C\in \mathbb{R}^{d\times d}$ and $CC^{T}=\Sigma _{0}$ and put$$S_{n}=C\left( \sum_{i=1}^{n}\zeta _{i}\zeta _{i}^{T}\right) C^{T}.$$Note that $\hat{\Sigma}_{n}=S_{n}/n$. Then, $S_{n}$ follows Wishart distribution with parameters $d,n$ and $\Sigma _{0}$ (denoted as $W_{d}\left( n,\Sigma _{0}\right) $). Equivalently, $W=C^{-1}S_{n}\left(
C^{T}\right) ^{-1}$ is distributed $W_{d}\left( n,I\right) ,$ where $I$ denotes the $d\times d$ identity matrix. Moreover, the eigenvalue distribution of $W$ satisfies $$\begin{aligned}
&&f_{w_{\left( 1\right) },...,w_{\left( d\right) }}\left(
w_{1},...,w_{d}\right) \\
&=&c_{d}\prod\limits_{i=1}^{d}\frac{\exp \left( -w_{i}/2\right) }{2^{n/2}\Gamma \left( \left( n-i+1\right) /2\right) }w_{i}^{\left(
n-d-1\right) /2}\prod\limits_{j>i}\left( w_{j}-w_{i}\right) \mathbb{I}\left(
0<w_{1}<...<w_{d}\right) .\end{aligned}$$where $\Gamma (\cdot )$ is the gamma function, $c_{d}$ is a constant independent of $n$, and $\mathbb{I}(\cdot)$ is the indicator function.
We are now ready to provide the proof of Lemma \[temp\]. By Proposition (\[probCLT\]), Slutsky’s theorem, and the continuous mapping theorem, we have $$\begin{aligned}
&&\left\langle \hat{\Sigma}_{n}\hat{A}_{n}\hat{\Sigma}_{n},\hat{A}_{n}\right\rangle \overset{p}{\rightarrow }\left\langle \Sigma
_{0}A_{0}\Sigma _{0},A_{0}\right\rangle , \\
&&\left\langle \sqrt{n}\left( \hat{\Sigma}_{n}-\Sigma _{0}\right) ,\hat{A}_{n}\right\rangle \Rightarrow \left\langle Z,A_{0}\right\rangle , \\
&&\left\langle \sqrt{n}\left( \hat{\Sigma}_{n}-\Sigma _{0}\right) ,\sqrt{n}\left( \hat{A}_{n}-A_{0}\right) \right\rangle \Rightarrow \left\langle
Z,Z_{A}\right\rangle .\end{aligned}$$ Therefore, to verify Lemma \[temp\], we need to show the uniform integrability of $\left\langle \hat{\Sigma}_{n}\hat{A}_{n}\hat{\Sigma}_{n},\hat{A}_{n}\right\rangle ,$ $\left\langle \sqrt{n}\left( \hat{\Sigma}_{n}-\Sigma _{0}\right) ,\hat{A}_{n}\right\rangle $ and $\left\langle \sqrt{n}\left( \hat{\Sigma}_{n}-\Sigma _{0}\right) ,\sqrt{n}\left( \hat{A}_{n}-A_{0}\right) \right\rangle .$ In turn, it suffices to verify that for some $r>1$ and some $n_{0}<\infty $ we have $$\begin{aligned}
&&\sup_{n\geq n_{0}}\mathbb{E}\left[ \left\vert \left\langle \hat{\Sigma}_{n}\hat{A}_{n}\hat{\Sigma}_{n},\hat{A}_{n}\right\rangle \right\vert ^{r}\right]
<\infty , \\
&&\sup_{n\geq n_{0}}\mathbb{E}\left[ \left\vert \left\langle \sqrt{n}\left(
\hat{\Sigma}_{n}-\Sigma _{0}\right) ,\hat{A}_{n}\right\rangle \right\vert
^{r}\right] <\infty , \\
&&\sup_{n\geq n_{0}}\mathbb{E}\left[ \left\vert \left\langle \sqrt{n}\left(
\hat{\Sigma}_{n}-\Sigma _{0}\right) ,\sqrt{n}\left( \hat{A}_{n}-A_{0}\right)
\right\rangle \right\vert ^{r}\right] <\infty ,\end{aligned}$$(see, for example, Chapter 5 in [@durrett2010probability]).
From Lemma \[Lem\_Aux\_UI2\] we have $$\begin{aligned}
&&f_{w_{\left( 1\right) },...,w_{\left( d\right) }}\left(
w_{1},...,w_{d}\right) \\
&=&c_{d}\prod\limits_{i=1}^{d}\frac{\exp \left( -w_{i}/2\right) }{2^{n/2}\Gamma \left( \left( n-i+1\right) /2\right) }w_{i}^{\left(
n-d-1\right) /2}\prod\limits_{j>i}\left( w_{j}-w_{i}\right) \mathbb{I}\left(
0<w_{1}<...<w_{d}\right) \\
&=&c_{d}\prod\limits_{i=1}^{d}\left( \frac{\exp \left( -w_{i}/2\right) }{2^{n/2}\Gamma \left( \left( n-i+1\right) /2\right) }w_{i}^{\left(
n-d-1\right) /2}w_{i}^{d-i}\prod\limits_{j>i}\left( w_{j}/w_{i}-1\right)
\mathbb{I}\left( 0<w_{1}<...<w_{d}\right) \right) \\
&=&c_{d}\prod\limits_{i=1}^{d}\left( \frac{\exp \left( -w_{i}/2\right) }{2^{n/2}\Gamma \left( \left( n-i+1\right) /2\right) }w_{i}^{\left(
n-i+1\right) /2-1}w_{i}^{{(d-i)/2}}\prod\limits_{j>i}\left(
w_{j}/w_{i}-1\right) \mathbb{I}\left( 0<w_{1}<...<w_{d}\right) \right) \\
&=&c_{d}^{\prime }\left( \prod\limits_{i=1}^{d}f_{\chi _{n-i+1}^{2}}\left(
w_{i}\right) \right) \prod\limits_{i=1}^{d}\left( \frac{w_{i}}{n}\right) ^{{(d-i)/2}}\prod\limits_{j>i}\left[ n^{1/2}\left( w_{j}/w_{i}-1\right) \right]
\mathbb{I}\left( 0<w_{1}<...<w_{d}\right) ,\end{aligned}$$where $f_{\chi _{n-i+1}^{2}}\left( \cdot \right) $ denotes the density of a chi-squared distribution with $n-i+1$ degrees of freedom and $c_{d}^{\prime
} $ is another constant also independent of $n$. The previous identity can be interpreted as follows. Let $W^{\left( n\right) }:=(W_{\left( 1\right)
}^{\left( n\right) },...,W_{\left( d\right) }^{\left( n\right) })$ be the eigenvalues of a $W_{d}\left( n,I\right) $ random matrix, and let $\Lambda
\left( n\right) :=\left( \Lambda _{1}\left( n\right) ,...,\Lambda _{d}\left(
n\right) \right) $ be independent random variables such that $\Lambda
_{i}\left( n\right) \sim \chi _{n-i+1}^{2}$. Then for any positive (and measurable) function $g:\mathbb{R}^{d}\rightarrow \lbrack 0,\infty )$, we have $$\begin{aligned}
&&\mathbb{E}\left[ g\left( W^{\left( n\right) }\right) \right] \notag \\
&=&c_{d}^{\prime }\mathbb{E}\left[ g\left( \Lambda \left( n\right) \right)
\prod\limits_{i=1}^{d}\left( \frac{\Lambda _{i}\left( n\right) }{n}\right) ^{{(d-i)/2}}\prod\limits_{j>i}\left\vert n^{1/2}\left( \Lambda _{j}\left(
n\right) /\Lambda _{i}\left( n\right) -1\right) \right\vert \mathbb{I}\left(
0<\Lambda _{1}\left( n\right) <...<\Lambda _{d}\left( n\right) \right) \right] \notag \\
&\leq &c_{d}^{\prime }\mathbb{E}\left[ g\left( \Lambda \left( n\right)
\right) \prod\limits_{i=1}^{d}\left( \frac{\Lambda _{i}\left( n\right) }{n}\right) ^{{(d-i)/2}}\prod\limits_{j>i}\left\vert n^{1/2}\left( \Lambda
_{j}\left( n\right) /\Lambda _{i}\left( n\right) -1\right) \right\vert \right] \notag \\
&\leq &c_{d}^{\prime }\sqrt{\mathbb{E}\left[ g\left( \Lambda \left( n\right)
\right) ^{2}\right] }\sqrt{\mathbb{E}\left[ \left(
\prod\limits_{i=1}^{d}\left( \frac{\Lambda _{i}\left( n\right) }{n}\right) ^{{(d-i)/2}}\prod\limits_{j>i}\left\vert n^{1/2}\left( \Lambda _{j}\left(
n\right) /\Lambda _{i}\left( n\right) -1\right) \right\vert \right) ^{2}\right] } \notag \\
&\leq &c_{d}^{\prime }\sqrt{\mathbb{E}\left[ g\left( \Lambda \left( n\right)
\right) ^{2}\right] }\mathbb{E}^{1/4}\left( \prod\limits_{i=1}^{d}\left(
\frac{\Lambda _{i}\left( n\right) }{n}\right) ^{2{(d-i)}}\right) \mathbb{E}^{1/4}\left( \prod\limits_{j>i}\left\vert n^{1/2}\left( \Lambda _{j}\left(
n\right) /\Lambda _{i}\left( n\right) -1\right) \right\vert ^{4}\right) ,
\label{Id_A}\end{aligned}$$
where the last two inequalities are obtained by the Cauchy-Schwarz inequality. We will show $\sup_{n\geq n_{0}}\mathbb{E}\left[ \left\vert
\left\langle \hat{\Sigma}_{n}\hat{A}_{n}\hat{\Sigma}_{n},\hat{A}_{n}\right\rangle \right\vert ^{r}\right] <\infty $ to verify the first statement of Lemma \[temp\]. Note that we can simplify $\left\vert
\left\langle \hat{\Sigma}_{n}\hat{A}_{n}\hat{\Sigma}_{n},\hat{A}_{n}\right\rangle \right\vert ^{r}$ as $$\left\vert \left\langle \hat{\Sigma}_{n}\hat{A}_{n}\hat{\Sigma}_{n},\hat{A}_{n}\right\rangle \right\vert ^{r}=\left\vert \text{tr}\left( \hat{A}_{n}\hat{\Sigma}_{n}\hat{A}_{n}\hat{\Sigma}_{n}\right) \right\vert ^{r}=\frac{4^{r}}{\text{tr}\left( \hat{\Sigma}_{n}^{-1}\right) ^{r}}\mathrm{tr}\left( \hat{\Sigma}_{n}^{-2}\right) ^{r}.$$By our definition, we have$$\hat{\Sigma}_{n}^{-1}=\left( C^{T}\right) ^{-1}\left( W/n\right) ^{-1}C^{-1},$$and thus there exist numerical constants $_{1},\bar{c}_{1}>0$ such that $$\underline{c}_{1}\text{tr}\left( \left( W/n\right) ^{-1}\right) \leq \text{tr}\left( \hat{\Sigma}_{n}^{-1}\right) \leq \bar{c}_{1}\text{tr}\left( \left(
W/n\right) ^{-1}\right) .$$Similarly, there exist numerical constants $_{2},\bar{c}_{2}>0 $ such that$$\underline{c}_{2}\text{tr}\left( \left( W/n\right) ^{-2}\right) \leq \text{tr}\left( \hat{\Sigma}_{n}^{-2}\right) \leq \bar{c}_{2}\text{tr}\left( \left(
W/n\right) ^{-2}\right) .$$After using the Cauchy-Schwarz inequality again, we have $$\begin{aligned}
\mathbb{E}\left\vert \left\langle \hat{\Sigma}_{n}\hat{A}_{n}\hat{\Sigma}_{n},\hat{A}_{n}\right\rangle \right\vert ^{r} &\leq &c\mathbb{E}\left[
\sum_{i=1}^{d}\left( \frac{W_{\left( i\right) }^{\left( n\right) }}{n}\right) ^{r}\sum_{i=1}^{d}\left( \frac{n}{W_{\left( i\right) }^{\left(
n\right) }}\right) ^{2r}\right] \notag \\
&\leq &c\sqrt{\mathbb{E}\left[ \sum_{i=1}^{d}\left( \frac{W_{\left( i\right)
}^{\left( n\right) }}{n}\right) ^{2r}\right] }\sqrt{\mathbb{E}\left[
\sum_{i=1}^{d}\left( \frac{n}{W_{\left( i\right) }^{\left( n\right) }}\right) ^{4r}\right] }, \label{g_func}\end{aligned}$$where $c$ is a numerical constant$.$ Therefore, it suffices to show that for any $r>1$ there exists $n_{0}$ such that$$\sup_{n\geq n_{0}}\left( \mathbb{E}\left[ \left( \frac{\Lambda _{j}\left(
n\right) }{n}\right) ^{r}\right] \cdot \mathbb{E}\left[ \left( \frac{n}{\Lambda _{i}\left( n\right) }\right) ^{r}\right] \cdot \mathbb{E}\left[
\left\vert n^{1/2}\left( \Lambda _{j}\left( n\right) /\Lambda _{i}\left(
n\right) -1\right) \right\vert ^{r}\right] \right) <\infty . \label{AU0}$$We know that $\Lambda _{i}\left( n\right) /n$ follows the gamma distribution with shape parameter $\alpha =\left( n-i+1\right) /2$ and scale parameter $\lambda =n/2$. Write $Y_{n}\sim Gamma\left( \alpha ,\lambda \right) $ and note that $$\begin{aligned}
\mathbb{E}\left( \frac{1}{Y_{n}^{r}}\right) &=&\int_{0}^{\infty }\frac{1}{y^{r}}\frac{\exp \left( -\lambda y\right) \lambda ^{\alpha }y^{\alpha -1}}{\Gamma \left( \alpha \right) }\mathrm{d}y \notag \\
&=&\int_{0}^{\infty }\frac{\Gamma \left( \alpha -t\right) \lambda ^{r}\exp
\left( -\lambda y\right) \lambda ^{\alpha -r}y^{\alpha -r-1}}{\Gamma \left(
\alpha \right) \Gamma \left( \alpha -r\right) }\mathrm{d}y \notag \\
&=&\frac{\Gamma \left( \alpha -r\right) \lambda ^{r}}{\Gamma \left( \alpha
\right) }. \label{AU2}\end{aligned}$$It follows from standard properties of the gamma function that lim$_{n\rightarrow \infty }\Gamma \left( \alpha -r\right) \lambda ^{r}/\Gamma
\left( \alpha \right) =1$ (see, for example, Chapter 3 in [hogg1978introduction]{}). After applying exactly the same approach to $\mathbb{E}\left[ \left( \Lambda _{i}\left( n\right) /n\right) ^{r}\right] $, we have $$\mathbb{E}\left[ \left( \Lambda _{i}\left( n\right) /n\right) ^{r}\right] =\frac{\Gamma \left( \alpha +r\right) }{\Gamma \left( \alpha \right) \lambda
^{r}}\rightarrow 1,
\label{AU22}$$as $n\rightarrow \infty .$
Now, we only need to show the third term in (\[AU0\]) is finite. Note that $$\begin{aligned}
&&\mathbb{E}\left[ \left\vert n^{1/2}\left( \Lambda _{j}\left( n\right)
/\Lambda _{i}\left( n\right) -1\right) \right\vert ^{r}\right] \label{AU2C}
\\
&=&\mathbb{E}\left[ \left\vert n^{1/2}\left( \Lambda _{j}\left( n\right)
/\Lambda _{i}\left( n\right) -1\right) \right\vert ^{r}\mathbb{I}\left(
\left\vert \Lambda _{j}\left( n\right) /n-1\right\vert \leq \varepsilon
,\left\vert \Lambda _{i}\left( n\right) /n-1\right\vert \leq \varepsilon
\right) \right] \notag \\
&&+\mathbb{E}\left[ \left\vert n^{1/2}\left( \Lambda _{j}\left( n\right)
/\Lambda _{i}\left( n\right) -1\right) \right\vert ^{r}\mathbb{I}\left(
\left\vert \Lambda _{j}\left( n\right) /n-1\right\vert >\varepsilon \cup
\left\vert \Lambda _{i}\left( n\right) /n-1\right\vert >\varepsilon \right) \right] . \notag\end{aligned}$$It is straightforward to verify (for example by computing moment generating functions of the Gamma distribution) that $$\sup_{n\geq 1}\mathbb{E}\left( n^{r/2}\left\vert \frac{\Lambda _{j}\left(
n\right) }{n}-1\right\vert ^{r}\right) <\infty \label{B1aa}$$for any $r>0$ and further, we can conclude that $$\begin{aligned}
&&\sup_{n\geq 1}\mathbb{E}\left[ \left\vert n^{1/2}\left( \Lambda _{j}\left(
n\right) /\Lambda _{i}\left( n\right) -1\right) \right\vert ^{r}\mathbb{I}\left( \left\vert \Lambda _{j}\left( n\right) /n-1\right\vert \leq
\varepsilon ,\left\vert \Lambda _{i}\left( n\right) /n-1\right\vert \leq
\varepsilon \right) \right] \\
&\leq &\sup_{n\geq 1}\mathbb{E}\left[ \left\vert n^{1/2}\left( \left( \left(
\frac{\Lambda _{j}\left( n\right) }{n}-1\right) -\left( \frac{\Lambda
_{i}\left( n\right) }{n}-1\right) \right) /\left( 1-\epsilon \right) \right)
\right\vert ^{r}\mathbb{I}\left( \left\vert \Lambda _{j}\left( n\right)
/n-1\right\vert \leq \varepsilon ,\left\vert \Lambda _{i}\left( n\right)
/n-1\right\vert \leq \varepsilon \right) \right] \\
&\leq &\frac{2^{r-1}}{\left( 1-\varepsilon \right) ^{r}}\sup_{n\geq 1}\mathbb{E}\left( n^{r/2}\left\vert \frac{\Lambda _{j}\left( n\right) }{n}-1\right\vert ^{r}\right) <\infty .\end{aligned}$$Then, because $\Lambda _{j}\left( n\right) /n$ (being the sum of $n-j+1$ i.i.d. random variables with finite moment generating function) satisfies the large deviations principle (see, for instance, Chapter 2.2 in [largedeviation]{}), we have $$\begin{aligned}
&&\mathbb{E}\left[ \left\vert n^{1/2}\left( \Lambda _{j}\left( n\right)
/\Lambda _{i}\left( n\right) -1\right) \right\vert ^{r}\mathbb{I}\left(
\left\vert \Lambda _{j}\left( n\right) /n-1\right\vert >\varepsilon \right) \right] \notag \\
&\leq &\sqrt{\mathbb{E}\left[ \left\vert n^{1/2}\left( \Lambda _{j}\left(
n\right) /\Lambda _{i}\left( n\right) -1\right) \right\vert ^{2r}\right] }\sqrt{\mathbb{P}\left( \left\vert \Lambda _{j}\left( n\right)
/n-1\right\vert >\varepsilon \right) }. \label{AU3a}\end{aligned}$$Because of our discussion involving the finiteness of the first two factors in (\[AU0\]), we can conclude that when $n>d+8r$, $$\begin{aligned}
\sqrt{\mathbb{E}\left[ \left\vert n^{1/2}\left( \Lambda _{j}\left( n\right)
/\Lambda _{i}\left( n\right) -1\right) \right\vert ^{2r}\right] } &\leq
&n^{r}\sqrt{1+\mathbb{E}\left[ \left( \frac{\Lambda _{j}\left( n\right) }{n}\times \frac{n}{\Lambda _{i}\left( n\right) }\right) ^{2r}\right] } \\
&\leq &n^{r}\sqrt{1+\sqrt{\mathbb{E}\left( \left( \Lambda _{j}\left(
n\right) /n\right) ^{4r}\right) }\times \sqrt{\mathbb{E}\left( \left(
n/\Lambda _{i}\left( n\right) \right) ^{4r}\right) }} \\
&\leq &C_r^{\prime }n^{r}.\end{aligned}$$Notice that from \[AU2\] and \[AU22\], we have $$\sup_{n > d+8r} \mathbb{E}\left( \left( \Lambda _{j}\left(
n\right) /n\right) ^{4r}\right) \times \mathbb{E}\left( \left(
n/\Lambda _{i}\left( n\right) \right) ^{4r}\right)< \infty,$$ which means $C_r^{\prime }$ is a numerical constant independent with $n$. Therefore, the first term in the right hand side of (\[AU3a\]) grows at rate $O\left( n^{r}\right) $, which is polynomial, whereas the second term, due to the large deviations principle invoked earlier converges exponentially fast to zero for each $\varepsilon >0$. This completes the first part of Lemma \[temp\].
For the second part of Lemma \[temp\], note that Lemma \[trace\_ineqn\] implies$$\left\vert \left\langle \sqrt{n}\left( \hat{\Sigma}_{n}-\Sigma _{0}\right) ,\hat{A}_{n}\right\rangle \right\vert ^{2}\leq \left\Vert \sqrt{n}\left( \hat{\Sigma}_{n}-\Sigma _{0}\right) \right\Vert _{F}^{2}\left\Vert \hat{A}_{n}\right\Vert _{F}^{2}.$$Then, for the uniform integrability of $\left\Vert \sqrt{n}\left( \hat{\Sigma}_{n}-\Sigma _{0}\right) \right\Vert _{F}^{2},$ we have $$\left\Vert \sqrt{n}\left( \hat{\Sigma}_{n}-\Sigma _{0}\right) \right\Vert
_{F}^{4}=\left\Vert C\sqrt{n}\left( \frac{1}{n}\sum_{i=1}^{n}\zeta
_{i}^{T}\zeta _{i}-I\right) C^{T}\right\Vert _{F}^{4}\leq \left\Vert \sqrt{n}\left( \frac{1}{n}\sum_{i=1}^{n}\zeta _{i}^{T}\zeta _{i}-I\right)
\right\Vert _{F}^{4}\left\Vert C\right\Vert _{F}^{8},$$by Lemma \[Lem\_Aux\_UI1\]. We denote $\hat{\Psi}^{(n)}=\frac{1}{n}\sum_{i=1}^{n}\zeta _{i}^{T}\zeta _{i}.$ And by the Cauchy-Schwarz inequality, we have$$\left\Vert \sqrt{n}\left( \frac{1}{n}\sum_{i=1}^{n}\zeta _{i}^{T}\zeta
_{i}-I\right) \right\Vert _{F}^{4}=\left( \sum_{i,j}\left( \sqrt{n}\left(
\hat{\Psi}_{ij}^{(n)}-\delta _{ij}\right) \right) ^{2}\right) ^{2}\leq
d^{2}\sum_{i,j}\left( \sqrt{n}\left( \hat{\Psi}_{ij}^{(n)}-\delta
_{ij}\right) \right) ^{4},$$where $\delta _{ij}=\mathbb{I}\{i=j\}.$ Note that$$\begin{aligned}
&&\mathbb{E}\left[ \sum_{i,j}\left( \sqrt{n}\left( \hat{\Psi}_{ij}^{(n)}-\delta _{ij}\right) \right) ^{4}\right] \\
&=&\mathbb{E}\left[ \sum_{i=1}^{d}\left( \sqrt{n}\left( \left( \frac{1}{n}\sum_{k=1}^{n}z_{ik}^{2}\right) -1\right) \right) ^{4}\right] +2\mathbb{E}\left[ \sum_{i=1}^{d}\sum_{j=1+1}^{d}\left( \sqrt{n}\left( \frac{1}{n}\sum_{k=1}^{n}z_{ik}z_{jk}\right) \right) ^{4}\right] \\
&=&d\mathbb{E}\left[ \left( \frac{1}{\sqrt{n}}\sum_{k=1}^{n}\left(
z_{ik}^{2}-1\right) \right) ^{4}\right] +d(d-1)\mathbb{E}\left[ \left( \frac{1}{\sqrt{n}}\sum_{k=1}^{n}z_{ik}z_{jk}\right) ^{4}\right],\end{aligned}$$where $z_{ik}\sim N(0,1)$ are i.i.d random variables. Further, direct calculations give us $$\begin{aligned}
\mathbb{E}\left[ \left( \frac{1}{\sqrt{n}}\sum_{k=1}^{n}\left(
z_{ik}^{2}-1\right) \right) ^{4}\right] &=&\frac{1}{n^{2}}\left( \mathbb{E}\left[ \sum_{k=1}^{n}\left( z_{ik}^{2}-1\right) ^{4}\right] +6\frac{1}{n^{2}}\mathbb{E}\left[ \sum_{k=1}^{n}\sum_{l=k+1}^{n}\left( z_{ik}^{2}-1\right)
^{2}\left( z_{il}^{2}-1\right) ^{2}\right] \right) \\
&=&\frac{60}{n}+\frac{12(n-1)}{n}<\infty ,\end{aligned}$$and$$\mathbb{E}\left[ \left( \frac{1}{\sqrt{n}}\sum_{k=1}^{n}z_{ik}z_{jk}\right)
^{4}\right] =\frac{1}{n^{2}}\mathbb{E}\left[
\sum_{k=1}^{n}z_{ik}^{4}z_{jk}^{4}\right] +6\frac{1}{n^{2}}\mathbb{E}\left[
\sum_{k=1}^{n}\sum_{l=k+1}^{n}z_{ik}^{2}z_{jk}^{2}z_{il}^{2}z_{jl}^{2}\right]
=\frac{9}{n}+\frac{3\left( n-1\right) }{n}<\infty .$$Therefore, we complete the uniform integrability of $\left\Vert \sqrt{n}\left( \hat{\Sigma}_{n}-\Sigma _{0}\right) \right\Vert _{F}^{2}.$
For $\left\Vert \hat{A}_{n}\right\Vert _{F}^{2},$ using the similar argument with (\[g\_func\]), we have $$\left\Vert \hat{A}_{n}\right\Vert _{F}^{2r}=\left\vert \text{tr}(\hat{A}_{n}^{2})\right\vert ^{r}=\frac{4^{r}\mathrm{tr}\left( \hat{\Sigma}_{n}^{-4}\right)
^{r}}{\text{tr}\left( \hat{\Sigma}_{n}^{-1}\right) ^{r}}\leq c_{2}\sqrt{\mathbb{E}\left[ \sum_{i=1}^{d}\left( \frac{W_{\left( i\right) }^{\left(
n\right) }}{n}\right) ^{2r}\right] }\sqrt{\mathbb{E}\left[
\sum_{i=1}^{d}\left( \frac{n}{W_{\left( i\right) }^{\left( n\right) }}\right) ^{8r}\right] }. \label{A_n_bd}$$ From the earlier bounds leading to the analysis of (\[AU0\]), we complete the uniform integrability of $\left\Vert \hat{A}_{n}\right\Vert _{F}^{2}$. Hence, the second part of Lemma \[temp\] follows.
For the third part of Lemma \[temp\], recall in the proof of Proposition \[probCLT\](3), $f(\Sigma )=-\frac{2}{\sqrt{\text{tr}(\Sigma ^{-1})}}\Sigma ^{-2}$ and thus $\hat{A}_n=f\left(\hat{\Sigma}_n \right)$.
The argument is similar to that given to establish (\[AU2C\]). We have argued that $f\left( \cdot \right) $ is smooth around $\Sigma _{0}$, which was the basis for the use of the delta method earlier in our argument. Moreover, note that $\hat{\Sigma}_{n}$ satisfies a large deviations principle. Therefore$$\begin{aligned}
&&\left\vert \left\langle \sqrt{n}\left( \hat{\Sigma}_{n}-\Sigma _{0}\right)
,\sqrt{n}\left( \hat{A}_{n}-A_{0}\right) \right\rangle \right\vert \\
&=&\left\vert \left\langle \sqrt{n}\left( \hat{\Sigma}_{n}-\Sigma
_{0}\right) ,\sqrt{n}\left( f\left( \hat{\Sigma}_{n}\right) -A_{0}\right)
\right\rangle \right\vert \\
&=&\left\vert \left\langle \sqrt{n}\left( \hat{\Sigma}_{n}-\Sigma
_{0}\right) ,\sqrt{n}\left( f\left( \hat{\Sigma}_{n}\right) -A_{0}\right)
\right\rangle \right\vert \mathbb{I}\left( \left\Vert \hat{\Sigma}_{n}-\Sigma _{0}\right\Vert _{F}\leq \varepsilon \right) \\
&&+\left\vert \left\langle \sqrt{n}\left( \hat{\Sigma}_{n}-\Sigma
_{0}\right) ,\sqrt{n}\left( f\left( \hat{\Sigma}_{n}\right) -A_{0}\right)
\right\rangle \right\vert \mathbb{I}\left( \left\Vert \hat{\Sigma}_{n}-\Sigma _{0}\right\Vert _{F}>\varepsilon \right) .\end{aligned}$$By applying Lemma \[trace\_ineqn\] and the fact that $Df\left( \cdot
\right) $ is continuous around $\Sigma _{0}$ (see the expression of $Df\left( \cdot \right) $ in (\[Df\])) we conclude that $$\begin{aligned}
&&\left\vert \left\langle \sqrt{n}\left( \hat{\Sigma}_{n}-\Sigma _{0}\right)
,\sqrt{n}\left( f\left( \hat{\Sigma}_{n}\right) -A_{0}\right) \right\rangle
\right\vert \mathbb{I}\left( \left\Vert \hat{\Sigma}_{n}-\Sigma
_{0}\right\Vert _{F}\leq \varepsilon \right) \\
&\leq &\sup_{\Sigma :\left\Vert \Sigma _{0}-\Sigma \right\Vert _{F}\leq
\varepsilon }\left\vert \left\langle \sqrt{n}\left( \hat{\Sigma}_{n}-\Sigma
_{0}\right) , Df\left( \Sigma \right) \left( \sqrt{n}\left( \hat{\Sigma}_{n}-\Sigma _{0}\right) \right) \right\rangle \right\vert \\
&\leq &c_{0}\left\Vert \sqrt{n}\left( \hat{\Sigma}_{n}-\Sigma _{0}\right)
\right\Vert _{F}^{2},\end{aligned}$$where $c_{0}=\sup_{\Sigma :\left\Vert \Sigma _{0}-\Sigma \right\Vert
_{F}\leq \varepsilon }\left\Vert Df\left( \Sigma \right) \right\Vert
_{op}$ and $\left\Vert Df\left( \Sigma \right) \right\Vert
_{op}$ is the operator norm, which is defined by $$\left\Vert Df\left( \Sigma \right) \right\Vert
_{op}:=\sup\left\{ \left\Vert Df\left( \Sigma \right) A \right\Vert_F: A \in \mathbb{R}^{d\times d} \text{ with }\Vert A \Vert_F = 1 \right\}.$$ Since $Df\left( \cdot
\right) $ is continuous around $\Sigma _{0}$, there exists sufficient small $\epsilon$ such that $c_0$ is finite. $\left\Vert \sqrt{n}\left( \hat{\Sigma}_{n}-\Sigma _{0}\right)
\right\Vert _{F}^{2}$ is proved to be uniformly integrable in the second part of Lemma \[temp\]. On the other hand, we have for $r>1$ $$\begin{aligned}
&&\mathbb{E}\left[ \left\vert \left\langle \sqrt{n}\left( \hat{\Sigma}_{n}-\Sigma _{0}\right) ,\sqrt{n}\left( f\left( \hat{\Sigma}_{n}\right)
-A_{0}\right) \right\rangle \right\vert ^{r}\mathbb{I}\left( \left\Vert \hat{\Sigma}_{n}-\Sigma _{0}\right\Vert _{F}>\varepsilon \right) \right] \\
&\leq &\mathbb{E}\left[ \left\Vert \sqrt{n}\left( \hat{\Sigma}_{n}-\Sigma
_{0}\right) \right\Vert _{F}^{r}\left\Vert \sqrt{n}\left( f\left( \hat{\Sigma}_{n}\right) -A_{0}\right) \right\Vert _{F}^{r}\mathbb{I}\left( \left\Vert
\hat{\Sigma}_{n}-\Sigma _{0}\right\Vert _{F}>\varepsilon \right) \right] \\
&\leq &\sqrt{\mathbb{E}\left( \left\Vert \sqrt{n}\left( \hat{\Sigma}_{n}-\Sigma _{0}\right) \right\Vert _{F}^{2r}\right) }\mathbb{E}^{1/4}\left(
\left\Vert \sqrt{n}\left( f\left( \hat{\Sigma}_{n}\right) -A_{0}\right)
\right\Vert _{F}^{4r}\right) \left( \mathbb{P}\left( \left\Vert \hat{\Sigma}_{n}-\Sigma _{0}\right\Vert _{F}>\varepsilon \right) \right) ^{1/4}.\end{aligned}$$The proof of the second part of Lemma \[temp\] shows $\mathbb{E}\left(
\left\Vert \sqrt{n}\left( \hat{\Sigma}_{n}-\Sigma _{0}\right) \right\Vert
_{F}^{2r}\right) <\infty$ when $r\leq 2.$ Further, we have argued throughout the proof of the first part of Lemma \[temp\] and the proof leading to (\[A\_n\_bd\]) that$$\mathbb{E}\left( \left\Vert \sqrt{n}\left( f\left( \hat{\Sigma}_{n}\right)
-A_{0}\right) \right\Vert _{F}^{4r}\right) \leq c_{3}n^{2r}\left( \mathbb{E}\left( \left\Vert \hat{A}_{n}\right\Vert ^{4r}\right) +\left\Vert
A_{0}\right\Vert ^{4r}\right) \leq O\left( n^{2r}\right) ,$$where $c_{3}$ is a numerical constant only related to $r$ and $d$. However, the large deviations principle gives us $\mathbb{P}\left( \left\Vert \hat{\Sigma}_{n}-\Sigma _{0}\right\Vert _{F}>\varepsilon \right) =O\left( \exp
\left( -cn\right) \right) $ for some $c>0$. Therefore, using Lemmas [trace\_ineqn]{} and \[Lem\_Aux\_UI1\] and the previous estimates we can complete the last part of Lemma \[temp\].
Proof of Proposition \[consistency\]
------------------------------------
Let $X^{\ast }(\rho ):=\arg \min_{X\succ 0}\left\{ -\log \det X+\sup_{\mathbb{Q}\in \mathcal{P}_{\rho }^{\ast }}\mathbb{E}^{\mathbb{Q}}\left[
\left\langle \xi \xi ^{T},X\right\rangle \right] \right\} ,$ where $$\mathcal{P}_{\rho }^{\ast }=\left\{ \mathbb{Q}\sim \mathcal{N}\left(
0,\Sigma \right) \text{ for some }\Sigma :\mathbb{W}_{2}(\mathcal{N}\left(
0,\Sigma _{0}\right) ,\mathbb{Q})\leq \rho \right\} .$$Then, $\arg \min_{\rho \geq 0}\{\mathbb{E}[L(X^{\ast }(\rho ),\Sigma
_{0})]\}=0.$ Since $X_{n}^{\ast }(\rho )$ is a continuous function of $\hat{\Sigma}_{n},$ we have $X_{n}^{\ast }(\rho )\rightarrow $ $X^{\ast }(\rho )$ almost surely for all $\rho \geq 0$ by the continuous mapping theorem. Furthermore, proof of Lemma \[temp\] gives us $\mathbb{E}X_{n}^{\ast }(0)=\mathbb{E}\hat{\Sigma}_{n}^{-1}\rightarrow \Sigma _{0}^{-1}.$ And from Lemmas 1 and 2 in [@cai2015law], we have $$\left\vert \log \det (\hat{\Sigma}_{n}^{-1})-\log \det (\Sigma
_{0}^{-1})\right\vert =\left\vert \sum_{k=1}^{d}\log \left( \frac{1}{n}\chi
_{n-k-1}^{2}\right) \right\vert \rightarrow 0,$$where $\chi _{n}^{2},\ldots ,\chi _{n-p-1}^{2}$ are mutually independent $\chi ^{2}$ distribution with the degree of freedom $n,\ldots ,n-p+1$ respectively. Due to the uniform integrability of $\log \left(\frac{1}{n}\chi
_{n-k-1}^{2}\right)$, we conclude $\mathbb{E}\left[ \log \det \left( X_{n}^{\ast
}(0)\right) \right] \rightarrow \mathbb{E}\left[ \log \det \left( X^{\ast
}(0)\right) \right] .$
By (\[gamma\_eqn\]), (\[large\_gamma\_inequ\]) and (\[x\_eqn\]), we have $x_{i}^{\ast }\leq \gamma ^{\ast }\leq d/\rho ^{2}$ and $$\begin{aligned}
\hat{x}_{i}^{\ast } &=&\gamma ^{\ast }\left[ 1-\frac{1}{2}\left( \sqrt{\hat{\lambda}_{i}^{2}\left( \gamma ^{\ast }\right) ^{2}+4\hat{\lambda}_{i}\gamma
^{\ast }}-\hat{\lambda}_{i}\gamma ^{\ast }\right) \right] \\
&\geq &\frac{1}{\rho ^{2}}\left( \sum_{i=1}^{d}\frac{1}{\hat{\lambda}_{i}\gamma ^{\ast }+2}\right) \frac{1}{\hat{\lambda}_{i}\gamma ^{\ast }+2} \\
&\geq &\frac{1}{\rho ^{2}}\left( \frac{1}{\hat{\lambda}_{i}\gamma ^{\ast }+2}\right) ^{2}.\end{aligned}$$Therefore, we have$$\begin{aligned}
L(X_{n}^{\ast }(\rho ),\Sigma _{0}) &=&-\log \det (X_{n}^{\ast }(\rho
)\Sigma _{0})+\left\langle X_{n}^{\ast }(\rho ),\Sigma _{0}\right\rangle -d
\\
&=&-\log \det (X_{n}^{\ast }(\rho ))+\left\langle X_{n}^{\ast }(\rho
),\Sigma _{0}\right\rangle -\log \det (\Sigma _{0})-d \\
&\geq &d\log \left( \rho ^{2}/d\right) -\log \det (\Sigma _{0})-d.\end{aligned}$$Thus, $\rho _{n}\in \lbrack 0,C]$, for large enough $n$ and a large enough constant $C.$ By proposition 3.5 in [@nguyen2018distributionally], we have $\log \det (X_{n}^{\ast }(\rho ))$ and $\left\langle X_{n}^{\ast }(\rho
),\Sigma _{0}\right\rangle $ decrease with $\rho .$ Then, $\mathbb{E}\left\langle X_{n}^{\ast }(\rho ),\Sigma _{0}\right\rangle \rightarrow
\mathbb{E}\left\langle X^{\ast }(\rho ),\Sigma _{0}\right\rangle $ since $\mathbf{0}\preceq X_{n}^{\ast }(\rho )\preceq X_{n}^{\ast }(0)=\hat{\Sigma}_{n}^{-1}$ and $\hat{\Sigma}_{n}^{-1}$ is uniformly integrable$.$ For $\log
\det (X_{n}^{\ast }(\rho )),$ the upper bound is given by $$\log \det (X_{n}^{\ast }(\rho ))\leq \log \det (\hat{\Sigma}_{n}^{-1}),$$and the lower bound is given by $$\begin{aligned}
\log \det (X_{n}^{\ast }(\rho )) &=&\log \left( \prod\limits_{i=1}^{d}\hat{x}_{i}^{\ast }\right) \\
&\geq &\log \left( \prod\limits_{i=1}^{d}\frac{1}{\rho ^{2}}\left( \frac{1}{\hat{\lambda}_{i}\gamma ^{\ast }+2}\right) ^{2}\right) \\
&\geq &-2d\log \rho -2\sum_{i=1}^{d}\left( \log \left( \hat{\lambda}_{i}\left( d/\rho ^{2}\right) +2\right) \right) \\
&\geq &-2d\log \rho -2\log \det \left( \left( d/\rho ^{2}\right) \hat{\Sigma}_{n}+2I\right) .\end{aligned}$$Due to the uniform integrability of $\log \det \left( \left( d/\rho
^{2}\right) \hat{\Sigma}_{n}+2I\right) $ and $\log \det (\hat{\Sigma}_{n}^{-1}),$ we have $\mathbb{E}\left[ \log \det (X_{n}^{\ast }(\rho ))\right] \rightarrow \mathbb{E}\left[ \log \det (X^{\ast }(\rho ))\right] .$ Finally, by the monotonicity of $\mathbb{E}\left[ \log \det (X_{n}^{\ast
}(\rho ))\right] $ and $\mathbb{E}\left\langle X_{n}^{\ast }(\rho ),\Sigma
_{0}\right\rangle ,$ we have $\mathbb{E}\left[ L(X_{n}^{\ast }(\rho ),\Sigma
_{0})\right] $ converges uniformly; thus, $\rho _{n}\rightarrow \arg
\min_{\rho \geq 0}\{\mathbb{E}[L(X^{\ast }(\rho ),\Sigma _{0})]\}=0.$
|
---
abstract: 'In the presence of a finite interlayer displacement field bilayer graphene has an energy gap that is dependent on stacking and largest for the stable AB and BA stacking arrangements. When the relative orientations between layers are twisted through a small angle to form a moir$\mathrm{\acute{e}}$ pattern, the local stacking arrangement changes slowly. We show that for non-zero displacement fields the low-energy physics of twisted bilayers is captured by a phenomenological helical network model that describes electrons localized on domain walls separating regions with approximate AB and BA stacking. The network band structure is gapless and has of a series of two-dimensional bands with Dirac band-touching points and a density-of-states that is periodic in energy with one zero and one divergence per period.'
author:
- 'Dmitry K. Efimkin'
- 'Allan H. MacDonald'
bibliography:
- 'MoireNetworkBIB.bib'
title: Helical Network Model for Twisted Bilayer Graphene
---
*Introduction*— The electronic structure of bilayer graphene is sensitive to strain, interlayer potential differences, and the stacking arrangement between layers [@BGReview2; @BGReview1]. For the energetically favored Bernal stacking configurations, either $\mathrm{AB}$ or $\mathrm{BA}$, Bloch states have $2 \pi$ Berry phases, quadratic band-touchng, and a gap that opens when a displacement fields is applied by external gates. The gapped state is characterized by nontrivial valley-dependent Chern numbers and supports topological confinement of electrons on domain walls that separate regions with opposite signs of displacement field [@BGdomainMartin; @BGdomainNunez; @BGdomainCosma; @BGdomainCosta] or different stacking arrangements [@MacDonaldZhang; @BGdomainVaezi; @BGdomainKoshino]. The presence of confined electronic states, which occur in helical pairs with opposite propagation directions in opposite valleys, has [@BGdomainExp1; @BGdomainExp2; @BGdomainExp3] been confirmed experimentally. Control of these domain walls and of their intersections has attracted attention recently [@MacDonaldQiao1; @MacDonaldQiao2; @BGdomainPan; @BGdomainWright; @BGdomainRen; @BGdomainMosallanejad] because of its potential relevance for valleytronics [@2DReview].
Whereas an engineering of a network of helical states with tunable geometry is a challenging problem, the triangular one has been recently observed [@2018arXiv180202999H] with help of scanning tunneling spectroscopy (STM) in misoriented graphene bilayers [@MacDonaldBistritzer; @MacDonaldJung; @GBTwist2; @GBTwist3; @GBTwist4; @GBTwist5; @GBTwist6; @GBTwist7; @GBTwist8; @GBTwist9; @BGdomainXintao; @BGtwistExp1; @BGtwistExp2; @BGtwistExp3; @BGtwistExp4; @BGtwistExp5; @BGtwistExp6; @BGtwistExp7; @BGtwistExp8; @BGtwistExp9]. In the presence of a twist local stacking arrangement changes slowly in space in a periodic moir$\mathrm{\acute{e}}$ pattern in which regions with approximate $\mathrm{AB}$ and $\mathrm{BA}$ stacking are separated by domain walls with helical states . The measured local density of states at a domain wall is strongly energy dependent with a single peak within the gap, that demonstrates the importance of an interference between helical states propagating along network. Because the moir$\mathrm{\acute{e}}$ pattern is well developed only when its period greatly exceeds graphene’s lattice constant, theories of its electronic structure [@BGHelicalNetwork; @GBTwistBias1] often employ complicated multi-scale approaches to advantage.
In this Letter, we derive a phenomenological helical network model for the electronic structure of gated bilayer graphene moir$\mathrm{\acute{e}}$s valid in the energy range below the $\mathrm{AB}$ and $\mathrm{BA}$ gaps where only topologically confined domain wall states are present. The model is related to Chalker-Coddington type models [@Network1; @Network2; @Network3] introduced in theories of the quantum Hall effect. The spectrum of the network model consists of a set of minibands connected by Dirac band touching points, which repeats and is gapless. A single period of the model’s band structure is illustrated in Fig. 1.
\[Fig1\] ![Helical model band structure over half of the rhombic Brillouin zone (BZ) defined in Fig. 3-(c). The bands in the other half of the BZ can be obtained by the reflection. The model’s band energies $\epsilon^{n0}_{\mathbf}{q}$ are given by Eq. (\[BandStructure\]) and depend on a single controlling parameter $\alpha$ which was set to $\alpha=1.1$ in this illustration. The bands touch at Dirac points located at high symmetry $\mathrm{K}$, $\mathrm{K}'$ and $\mathrm{\Gamma}$ points. ](Fig1.pdf "fig:"){width="8.5"}
*Moir$\mathrm{\acute{e}}$ pattern and helical states*— To describe the electronic structure of gated bilayer graphene with a small twist angle $\theta\lesssim 1^{\circ}$ [@NoteShift] between layers, we start from the continuum model Hamiltonian derived in Ref. [@MacDonaldBistritzer], which is valid independent of atomic scale commensurability
\[Fig2\] {width="16.4"}
$$\label{HamiltonianBilayer}
H_0=
\begin{pmatrix}
v \sigma_\mathrm{t} {\mathbf}{p}-u & T({\mathbf}{r})
\\ T^+({\mathbf}{r}) & v \sigma_\mathrm{b} {\mathbf}{p}+u
\end{pmatrix}.$$
The Hamiltonian for a valley $K$ acts in the sublattice space $\psi=\{\psi_\mathrm{A}^\mathrm{t},\psi_\mathrm{B}^\mathrm{t}, \psi_\mathrm{A}^\mathrm{b},\psi_{\mathrm{B}}^\mathrm{b}\}$, where $\mathrm{t}$ and $\mathrm{b}$ refer to the top and bottom layer, $v$ is the single-layer Dirac velocity; $\sigma_\mathrm{t(b)}$ is the vector of Pauli matrices rotated by the angle $\pm \theta/2$ in top and bottom layers, and $2 u$ is the potential difference between layers produced by the gates. The spectrum is valley and spin independent, while electronic states in two valleys $K$ and $K'$ transform to each other by the time-reversal transformation. The inter-layer hopping operator is given by $$\label{HamiltonianHybrid}
T({\mathbf}{r})=\frac{w}{3}\sum_{i=1}^3 e^{- i {\mathbf}{k}_i {\mathbf}{r}} T_i,$$ where $w$ is a hybridization energy scale. The vectors ${\mathbf}{k}_1=-k_\mathrm{\theta}{\mathbf}{e}_y$, ${\mathbf}{k}_{2,3}=k_\mathrm{\theta} (\pm\sqrt{3}{\mathbf}{e}_\mathrm{x}+{\mathbf}{e}_\mathrm{y})/2$ all have magnitude equal to the twist-induced separation between the Dirac points of the two-layers, $k_\mathrm{\theta}=2 k_\mathrm{D} \sin(\theta/2)$ where $k_\mathrm{D}=4\pi/3 a_0$ is the magnitude of the Brillouin-zone corner vector of a single layer and $a_0$ is the corresponding Bravais period. The matrices $T_i$ are given by $$T_1=\begin{pmatrix}
1 & 1\\
1 & 1
\end{pmatrix}, \;\;
T_2=\begin{pmatrix}
e^{- i \zeta} & 1\\
e^{ i \zeta} & e^{- i \zeta}
\end{pmatrix}, \;\;
T_3=\begin{pmatrix}
e^{i\zeta} & 1\\
e^{- i \zeta} & e^{i \zeta}
\end{pmatrix},$$ with $\zeta=2\pi/3$. The inter-layer hopping operator in Eq.(\[HamiltonianHybrid\]) is spatially periodic with the period of the moir$\mathrm{\acute{e}}$ pattern $L=a_0/(2\sin(\theta/2))$.
The network model we derive has its widest range of applicability in the large gate voltage regime $\epsilon_\mathrm{L}\ll u \sim w$ where $\epsilon_\mathrm{L}=2\pi \hbar v/L$ is the energy scale of the network mini-bands, as we explain below. In this limit an energy gap $\sim w$ develops around the momentum space ring of radius $p_\mathrm{u}=u/v$ where the the conduction band of the low potential top layer overlaps with the valence band of the high potential bottom layer. At energies $\epsilon \ll w$ the bilayer spectrum can be described by the projected two-band Hamiltonian $$\label{HamiltonianProjected}
H=
\begin{pmatrix}
v (p-p_\mathrm{u}) & t_\mathrm{P}+t_\mathrm{S} \\
t_\mathrm{P}^*+t_\mathrm{S}^* & -v (p-p_\mathrm{u}) \end{pmatrix}.$$ In Eq. \[HamiltonianProjected\] we have separated the tunneling matrix element into two parts, an anisotropic part with $\mathrm{p}$-wave symmetry $t_\mathrm{P}(\phi_{\mathbf}{p},{\mathbf}{r})=[T_\mathrm{BA}e^{- i \varphi_{\mathbf}{p}}-T_\mathrm{AB}e^{ i \varphi_{\mathbf}{p}}]/2$, where $\varphi_{\mathbf}{p}$ is the direction of a momentum ${\mathbf}{p}$, and an isotropic part $t_\mathrm{S}({\mathbf}{r})=-i T_\mathrm{AA}({\mathbf}{r}) \sin(\theta/2)$ independent of $\varphi_{\mathbf}{p}$ that can be neglected [@NoteTd] for $\theta\ll 1$. The resulting local spectrum $\epsilon_{{\mathbf}{p} \pm}=\pm \sqrt{(v p-u)^2+\Delta_{\mathbf}{p}^2}$ has an anisotropic gap $$\label{Gap}
\Delta_{\mathbf}{p}^2=\delta^2_{-} \cos^2[\varphi_{\mathbf}{p}-\Theta]+\delta^2_{+}\sin^2[\varphi_{\mathbf}{p}-\Theta].$$ which achieves minima $|\delta_-|=|(|T_\mathrm{AB}|-|T_\mathrm{BA}|)/2|$ at momentum orientations $\varphi_\mathrm{I}=\Theta$ and $\varphi_\mathrm{II}=\Theta+\pi$, where $\Theta({\mathbf}{r})=(\arg[T_\mathrm{BA}]-\arg[T_\mathrm{AB}])/2$. The gap is maximized at $\delta_+({\mathbf}{r})=(|T_\mathrm{BA}|+|T_\mathrm{AB}|)/2$ at the two perpendicular orientations.
It follows from the preceding analysis that the gap in the local electronic spectrum (\[Gap\]) closes if $|T_\mathrm{AB}|=|T_\mathrm{BA}|$. This condition is satisfied along the domain walls specified by dashed lines in Fig. 2-$\hbox{(a)}$, where we illustrate the spatial pattern of $\delta_-({\mathbf}{r})$. The domain walls separate regions where the inter-layer hybridization is dominated by the $T_\mathrm{AB}$ from regions in which it is dominated by $T_\mathrm{BA}$. The local valley Chern number of Hamiltonian (\[HamiltonianProjected\]) $$C=\int \frac{d {\mathbf}{p}}{4\pi} \;{\mathbf}{d} \left[\frac{\partial {\mathbf}{d}}{\partial p_x} \times \frac{\partial {\mathbf}{d}}{\partial p_y}\right]=\frac{\delta_{-}}{|\delta_{-}|},$$ where ${\mathbf}{d}={\mathbf}{h}/h$ and the vector ${\mathbf}{h}$ is defined by the Pauli matrix expansion of Eq. (\[HamiltonianProjected\]), $H=(\bm{\sigma} \cdot {\mathbf}{h})$. The valley Chern number difference across the domain wall is $C_\mathrm{AB}-C_\mathrm{BA}=2$, guaranteeing that two helical electronic channels are present in the gaps per valley and per spin.
In the vicinity of each domain wall the low-energy states are concentrated around the minima at orientations $\varphi_\mathrm{I(II)}$, which are perpendicular to the domain wall, as illustrated in Fig. 2-$\hbox{(b)}$. The expansion of the Hamiltonian (\[HamiltonianProjected\]) in the vicinity of these minima results in a pair of identical anisotropic Dirac cones with spatially depended mass $\delta_-({\mathbf}{r})$: $$\label{HamiltonianDirac}
H_{D}=\begin{pmatrix}
\delta_-({\mathbf}{r}) & v p_\perp-i v_{||} p_{||} \\
v p_\perp+i v_{||} p_{||}& -\delta_-({\mathbf}{r})
\end{pmatrix}.$$ Here the velocity for momenta $p_\perp$ perpendicular to the domain wall is the single-layer graphene Dirac velocity $v$. The velocity for momenta $p_{||}$ along the domain wall can be approximated by its value at the domain wall center $v_{||}=\delta_+/p_\mathrm{u}\approx 2 w v/3 u$. Each Dirac point carries one half of the valley Chern number $C_\mathrm{D}=\delta_-/2|\delta_-|$, and is responsible for a single helical state. The Dirac mass $\delta_-({\mathbf}{r})$ changes sign across the domain wall and Eq.(\[HamiltonianDirac\]) therefore has a Jackiw-Rebbi [@JackiwRebbi] solution that describes helical electronic states with dispersion $\epsilon_{p_{||}}=v_{||} p_{||}$, and wave function $$\psi_{p_{||}}(r_\perp)= N \begin{pmatrix} 1 \\
i
\end{pmatrix}\exp\left[i \frac{p_{||} r_{||}}{\hbar} - \frac{w L}{\pi \hbar v} \sin^2\left( \frac{\pi r_\perp}{\sqrt{3} L}\right) \right],$$ where $N$ is a normalization factor. The center of AB/BA region, where wave functions of helical states from different domain walls overlap, are distanced at length $r_\perp^0=L/2\sqrt{3}$ from them. The domain wall network is well developed if the overlap of wave functions $|\psi_{p_{||}}(r_\perp^0)|^2|/|\psi_{p_{||}}(0)|^2=\exp[-w/\epsilon_\mathrm{L}]\ll1$ is weak. Here $\epsilon_\mathrm{L}=2\pi \hbar v/L$ is the character energy scale of the moir$\mathrm{\acute{e}}$ pattern.
\[Fig3\]
These helical states are the only electronic degrees of freedom present when $|\epsilon|\ll u, w$. Three sets of parallel domain walls with orientations differing by $120^{\circ}$ surround $\mathrm{AB}$ and $\mathrm{BA}$ regions and intersect at a set of points with local $\mathrm{AA}$ stacking. The considerations we have discussed to this point establish the physical picture we use to motivate our phenomenological helical network model for domain wall states.
*Phenomenological network model*— Our phenomenological helical network model consists of the links and nodes illustrated in Fig. 3-$\hbox{(a)}$ and $\hbox{(b)}$, which connect to form the domain wall pattern. We assume ballistic propagation along links and scattering only at nodes. The dispersion law along links, $\epsilon=v_{||} q$, is consistent with the Jackiw-Rebbi confined mode solution. For $\epsilon_\mathrm{L}\ll w\lesssim u$, the two Dirac cones on opposite sides of the ring at $\varphi_\mathrm{I}$ and $\varphi_\mathrm{II}$ are well separated, allowing scattering between them to be neglected. This simplification allows us to consider a network with a single helical channel per link.
The full domain wall network can be constructed by placing the set of three elementary nodes on a triangular lattice with elementary lattice vectors ${\mathbf}{l}_{1,2}=L (\pm\sqrt{3} {\mathbf}{e}_x + {\mathbf}{e}_y)/2$. The wavefunction amplitudes on links $1$, $2$ and $3$ of the cell centered at ${\mathbf}{R}_{ij}=i {\mathbf}{l}_1 + j {\mathbf}{l}_2$ are denoted by $\psi_{ij}=\left\{\psi_{ij}^1, \psi_{ij}^2,\psi_{ij}^3\right\}$. Each node has three input and three output channels and therefore has a $3\times3$ unitary scattering matrix $T$ whose detailed form depends in a complex way [@NodAnglin] on the spatial profile of the domain walls intersection. We follow a simpler phenomenological approach. By observing that the straight-forward scattering amplitude magnitudes $|T_{11}|=|T_{22}|=|T_{33}|$ and the 240$^{\circ}$ deflection scattering amplitudes $|T_{12}|=|T_{13}|=|T_{21}|=|T_{23}|=|T_{31}|=|T_{32}|$ must be equal due to symmetry, it follows that the unitary matrix $T$ can be parametrized by an angle $\alpha$ ranging between $0$ and $\alpha_\mathrm{M} = \arccos[1/3]$, and $6$ phases $\phi_\mathrm{T}, \phi_1^\mathrm{R},\phi_1^\mathrm{L}, \phi_2^\mathrm{R},\phi_2^\mathrm{L}, \phi_3$ ranging between $0$ and $2\pi$: $T=e^{i \phi_\mathrm{T}} T^\mathrm{L}_\phi \bar{T} T^\mathrm{R}_\phi$, where $\phi_\mathrm{T}$ is the average phase shift; $T^\mathrm{L}_\phi=\mathrm{diag}[e^{i(\phi_2^\mathrm{R} + \phi_1^\mathrm{R}+\phi_3)},
e^{-i\phi_2^\mathrm{L}},e^{-i\phi_1^\mathrm{L}}]$ and $T^\mathrm{R}_\phi=\mathrm{diag}[e^{i(\phi_2^\mathrm{L} + \phi_1^\mathrm{L}-\phi_3)}, e^{-i\phi_2^\mathrm{R}},e^{-i\phi_1^\mathrm{R}}]$ are phase shifts before and after scattering, which are not independent, and $\bar{T}$ is the unitary matrix
$$\label{MatrixT}
\bar{T}=
\begin{pmatrix}
\cos{\alpha} e^{i\chi} & \frac{\sin{\alpha}}{\sqrt{2}} & \frac{\sin{\alpha}}{\sqrt{2}}\\
\frac{\sin{\alpha}}{\sqrt{2}} & -\frac{1+ \cos{\alpha} e^{- i \chi}}{2} & \frac{1- \cos{\alpha} e^{- i \chi}}{2}\\
\frac{\sin{\alpha}}{\sqrt{2}} & \frac{1- \cos{\alpha} e^{- i \chi}}{2} & -\frac{1+ \cos{\alpha} e^{- i \chi}}{2}
\end{pmatrix}.$$
Here $\chi=\arccos[\{3 \cos^2(\alpha)-1\}/2\cos(\alpha)]$. The angle $\alpha$ defines the ratio of scattering probabilities between forward $P_\mathrm{f}$ and deflected $P_\mathrm{d}$ channels by $P_\mathrm{f}/P_\mathrm{d}=2 \cot^2(\alpha)$.
The outgoing and incoming electronic waves at a node are connected by $\psi_\mathrm{out}=e^{-i \phi_\mathrm{E}} T \psi_\mathrm{in}$, where $\psi_\mathrm{out}=(\psi_{i+1,j}^\mathrm{1}, \psi_{i,j-1}^\mathrm{2},\psi_{i,j}^\mathrm{3})$ and $\psi_\mathrm{in}=(\psi_{i,j-1}^\mathrm{1}, \psi_{i,j}^\mathrm{2},\psi_{i+1,j}^\mathrm{3})$. Here $\phi_\mathrm{E}=\epsilon L /\hbar v_{||}$ is the dynamical phase accumulated by electrons while propagating between links. Bloch’s theorem connects wave function amplitudes in different cells by $\psi_{ij}=e^{i {\mathbf}{q}{\mathbf}{R}_{ij}} \bar{\psi}$, where $\bar{\psi}\equiv\{\bar{\psi}^1, \bar{\psi}^2,\bar{\psi}^3\}$ and ${\mathbf}{q}$ is the moir$\mathrm{\acute{e}}$ momentum. The connection between input and output waves can be written as $[\lambda-U_{\mathbf}{q}]\bar{\psi}=0$, and has a nontrivial solution only if $\lambda=e^{i ( \phi_\mathrm{E} - \phi_\mathrm{T})}$ is equal to one of eigenvalues of the matrix
$$\label{MatrixU}
U_{\mathbf}{q}=\left(
\begin{array}{ccc}
\cos{\alpha} e^{\bm{i}(\chi+\phi_1^\mathrm{R}+\phi_2^\mathrm{R}+\phi_1^\mathrm{L}+\phi_2^\mathrm{L} - {\mathbf}{q}{\mathbf}{l}_1 - {\mathbf}{q}{\mathbf}{l}_2 )} & \frac{\sin{\alpha}}{\sqrt{2}} e^{i (\phi_1^\mathrm{R}+ \phi_3 - {\mathbf}{q}{\mathbf}{l}_1 )}
& \frac{\sin{\alpha}}{\sqrt{2}} e^{i (\phi_2^\mathrm{R}+\phi_3)} \\
\frac{\sin{\alpha}}{\sqrt{2}} e^{i (\phi_1^\mathrm{L}-\phi_3)} & -\frac{1+ \cos{\alpha} e^{- \bm{i} \chi}}{2} e^{i ({\mathbf}{q}{\mathbf}{l}_2 - \phi_2^\mathrm{R} - \phi_2^\mathrm{L})} &\frac{1- \cos{\alpha} e^{- \bm{i} \chi}}{2} e^{i ({\mathbf}{q} {\mathbf}{l}_1 +{\mathbf}{q} {\mathbf}{l}_2 - \phi_1^\mathrm{R}- \phi_2^\mathrm{L})} \\
\frac{\sin{\alpha}}{\sqrt{2}} e^{i (\phi_2^\mathrm{L}-\phi_3-{\mathbf}{q} {\mathbf}{l}_2)} & \frac{1- \cos{\alpha} e^{- \bm{i} \chi}}{2} e^{-i (\phi_2^\mathrm{R}+\phi_1^\mathrm{L})} & -\frac{1+ \cos{\alpha} e^{- \bm{i} \chi}}{2} e^{i ({\mathbf}{q}{\mathbf}{l}_1 - \phi_1^\mathrm{R}-\phi_1^\mathrm{L})} \\
\end{array}
\right).$$
It follows that the electronic spectrum consists of groups of three bands $n=-1,0,1$ that repeat in energy with period $\epsilon_\mathrm{L}^{||}=2\pi\hbar v_{||}/L$ and have dispersion $$\label{BandStructure}
\epsilon^{nm}_{\mathbf}{q}=\epsilon_\mathrm{L}^{||}\;\left(\frac{\arg[\lambda^n_{\mathbf}{q}]}{2\pi}+\frac{\phi_\mathrm{T}}{2 \pi}+ m\right).$$ Here $\lambda^n_{\mathbf}{q}$ are the eigenvalues of $U_{\mathbf}{q}$ and $m$ is an integer. The role of the phase $\phi_\mathrm{T}$ is just a rigid shifts of all bands in energy. Since the matrix $U_{\mathbf}{q}$ is also unitary $U_{\mathbf}{q}^+=U_{\mathbf}{q}^{-1}$ and $\det[U_{\mathbf}{q}]=1$, its eigenvalues satisfy $$\label{EigenvalueProblem}
\lambda_{\mathbf}{q}^3-\mathrm{tr}[U_{\mathbf}{q}]\lambda_{\mathbf}{q}^2+\mathrm{tr}[U_{\mathbf}{q}^+]\lambda_{\mathbf}{q}-1=0.$$ The electronic spectrum therefore depends only on
\[Fig4\] {width="7.6"}
$$\label{TrU}
\begin{split}
\mathrm{tr}[U_{\mathbf}{q}]=\cos(\alpha) e^{\bm{i} \chi} e^{i (\Phi_1+\Phi_2-{\mathbf}{q} {\mathbf}{l}_1 -{\mathbf}{q} {\mathbf}{l}_2 )}\\
-\frac{1}{2}\left[1+\cos(\alpha) e^{- i \chi}\right] \left[e^{i ({\mathbf}{q} {\mathbf}{l}_1 - \Phi_1)}+e^{i ({\mathbf}{q} {\mathbf}{l}_2 - \Phi_2)}\right].
\end{split}$$
Here we have introduced phases $\Phi_1=\phi_1^\mathrm{L}+\phi_1^\mathrm{R}$, $\Phi_2=\phi_2^\mathrm{L}+\phi_2^\mathrm{R}$. These phases $\Phi_1$ and $\Phi_2$ can be eliminated by the shift of the momentum space origin, and therefore do not influence the density of states of the network and electronic transport through it. The latter remarkably depend only on $\alpha$, which in turn characterizes the distribution of scattering probability between forward and deflected channels. It has been numerically shown [@QiaoJungPRL2014] that, contrary to classical intuition, because nearby paths have larger wavefunction overlap with the incoming electron, deflection is the more likely outcome. For presentation of results we chose $\alpha=1.1$ corresponding to $P_\mathrm{f}\approx 0.2$ and $P_\mathrm{d}\approx 0.4$.
The first Brillouin zone of the network has a hexagonal shape and is illustrated in Fig.3-c where we also illustrate an equivalent rhombic primitive cell. The spectrum has the mirror symmetry across the $\mathrm{K}\mathrm{K}'$ line since $\mathrm{tr}[U_{q_\mathrm{M}-q_x,q_y}]=\mathrm{tr}[U_{q_\mathrm{M}+q_x,q_y}]$, where ${\mathbf}{q}_\mathrm{M}=2\pi {\mathbf}{e}_\mathrm{x}/\sqrt{3}L$ is the position of the $\mathrm{M}$-point in the Brillouin zone. For presentation of results we have chosen $\phi_\mathrm{T}=\Phi_1=\Phi_2=(\pi-2\arcsin[3\sin\alpha/2\sqrt{2}])/3$ that ensures the discrete rotational symmetry of the network band structure with respect to $120^\circ$ around the $\Gamma$-point. A single period $\epsilon^{n0}_{\mathbf}{q}$ of the repeating band structure is plotted in the half of the rhombic Brillouin zone in Fig. 1, where we see that it is gapless because of Dirac band touching points situated in $\Gamma$, $\hbox{K}$, and $\hbox{K}'$ high symmetry points. Their positions are independent on $\alpha$ and they are separated by momentum $\Delta k_\mathrm{D}=4\pi/3L$ and energy $\Delta \epsilon_\mathrm{D}=\epsilon_\mathrm{L}^{||}/3$. The density of states of the network is presented in Fig. 4 and is periodic with period $\Delta \epsilon_\mathrm{D}$. It is three time smaller than the period of the network band structure $\epsilon_\mathrm{L}^{||}$, that reflects the symmetry between three links in an elementary cell of the model. The single period contains one zero at the Dirac point, and one saddle-point logarithmic divergence. The latter reflects the van Hove singularity due to the presence of saddle points in the network band structure, which are clearly visible in Fig. 1.
In recent experiments [@2018arXiv180202999H] the small twist-angle $\theta=0.245^{\circ}$ has been applied between layers and has resulted in moir$\mathrm{\acute{e}}$ patterns with period $L\approx58\;\mathrm{nm}$. The resulting energy scale of the pattern $\epsilon_\mathrm{L}=2\pi \hbar v/L\approx 72 \; \hbox{meV}$ is comparable with the induced gap $\epsilon_\mathrm{g}\approx 60\; \hbox{meV}$. While the phenomenological network model is still reasonable at energies $\epsilon\ll \epsilon_\mathrm{g}$, the expressions for $v_\mathrm{||}$ and $\Delta \epsilon_\mathrm{D}$ do not directly apply. Our model predicts the periodic set of features in the density of states, whereas only one feature within the gap has been observed [@2018arXiv180202999H]. For the gap $\epsilon_\mathrm{g} \approx 250 \; \hbox{meV}$ achievable in bilayer graphene [@BGgap1; @BGgap2], our model is well applicable in much wider range of energies. Using the hybridization energy $w=400\; \hbox{meV}$ we get that the velocity of helical states $v_{||}=1.6\; 10^6 \; \mathrm{m}/\mathrm{s}$ is larger than the velocity of electrons in graphene $v=10^6 \; \mathrm{m}/\mathrm{s}$. The period of the network is equal to $\epsilon_\mathrm{L}^{||}\approx 115\; \hbox{meV}$ and the the period of density of states $\Delta \epsilon_\mathrm{D}\approx 38 \; \hbox{meV}$. It is much smaller than the gap $\epsilon_\mathrm{g}$ and we expect a set of features due to van Hove singularities of network spectrum to be well resolved in experiments. Alternatively, the condition $\epsilon_\mathrm{L}^{||}\ll\epsilon_\mathrm{g}$ can be achieved at smaller twist angles $\theta$.
To conclude, we have introduced a new phenomenological network model which captures the electronic structure of twisted bilayer graphene in the energy range below the $\mathrm{AB}$ and $\mathrm{BA}$ gaps where only topologically confined domain wall states are present. Motivated by the recent observation of the domain wall network in STM experiments [@2018arXiv180202999H] we have focused on its band structure and density of states. Very recently signatures of the network formation have been found in magneto-transport experiments [@2018arXiv180207317R]. Whereas our model predicts anisotropic transport properties that are approximately periodic in carrier density, the magneto-transport theory is postponed for future work.
*Acknowledgment*. This material is based upon work supported by the Department of Energy under Grant No DE-FG02-ER45118 and by the Welch Foundation under Grant No. F1473.
|
---
author:
- Adrian Iovita
- Annette Werner
title: '$p$-adic height pairings on abelian varieties with semistable ordinary reduction'
---
\[section\] \[theorem\][Lemma]{} \[theorem\][Proposition]{} \[theorem\][Facts]{} \[theorem\][Definition]{} \[theorem\][Corollary]{}
2ex plus0.5ex minus0.5ex 0.em 0.5cm
=81ex =115ex
**Abstract**
We prove that for abelian varieties with semistable ordinary reduction the $p$-adic Mazur-Tate height pairing is induced by the unit root splitting of the Hodge filtration on the first deRham cohomology.
Introduction {#introduction .unnumbered}
============
The main goal of this paper is to prove that the $p$-adic Mazur-Tate height pairing on an abelian variety with semistable ordinary reduction is induced by the unit root splitting of the Hodge filtration on its first deRham cohomology.
$p$-adic height pairings are the ${{\Bbb Q}}_p$-valued counterparts of the real-valued Néron-Tate height pairings on abelian varieties over a global field $F$. As the Néron-Tate pairings they can be decomposed into local contributions, one for each finite place of the ground field $F$. At the places not dividing $p$, these local contributions are basically given by the local Néron heights. Hence only at the places over $p$ something genuinely $p$-adic happens.
If the abelian variety $A$ has semistable ordinary reduction at a place over $p$, there are (at least) three interesting candidates for such a local $p$-adic height pairing on $A$. First of all, there is the Mazur-Tate height defined with splittings of the Poincaré biextension. Then there is Schneider’s “norm-adapted” $p$-adic height, and finally one can use the unit root splitting of the Hodge filtration on the first deRham cohomology to define a $p$-adic height pairing. In the first section of this paper we explain some details of these constructions. In particular we show how to pass from a pairing defined by a splitting the Hodge filtration of the first deRham cohomology to a pairing in the Mazur-Tate style.
It is well-known that if $A$ has [*good*]{} ordinary reduction, the Mazur-Tate height pairing coincides with Schneider’s height pairing, cf. [@mata]. Besides, Coleman has proven in [@co] that in this case the Mazur-Tate pairing is given by the unit root splitting. Hence in the good ordinary reduction case all three definitions give the same pairing.
What happens in the semistable ordinary reduction case? In [@we] it is shown that the Mazur-Tate pairing differs in general from Schneider’s height pairing. Hence the question remains if the height pairing given by the unit root splitting coincides with one of those pairings. We answer this question by showing in Theorem 3.6 that it is equal to the Mazur-Tate pairing.
In the course of the proof we use the Raynaud extension which is a lift of the semiabelian variety in the reduction of $A$ to a semiabelian rigid analytic variety lying over $A$. Its abelian quotient $B$ has good reduction. By Coleman’s result, the Mazur-Tate height on $B$ is given by the unit root splitting. A result of the second author shows how the Mazur-Tate height pairings on $A$ and $B$ are related. Using results of LeStum, the deRham cohomologies of $A$ and $B$ are connected by a diagram involving the deRham cohomology of the Raynaud extension. Besides, Coleman and the first author have shown how the Frobenius on the deRham cohomologies can be described explicitely. We use these facts in section 2 to relate the unit root splittings for $A$ and $B$.
However, the previous results cannot simply be put together to prove the desired Theorem 3.6. The reason is that the step from height pairings defined with splittings of the Hodge filtration to Mazur-Tate height pairings involves the universal vectorial extension of the abelian variety - and we have no result relating the universal vectorial extensions of $A$ and $B$. Hence we go the other way and start with the Mazur-Tate pairing on $A$. We prove in section 3 that it is of an analytic nature. To be precise, we show that it gives rise to a $p$-adic analytic splitting of the rational points of the universal vectorial extension of $A$. Hence the Mazur-Tate height is in fact induced by a splitting of the Hodge filtration on the first deRham cohomology, and we can adapt some ideas from [@mame] to deduce that this is the unit root splitting.
In an appendix we study the Hodge filtration of the first deRham cohomology of a semiabelian variety and show that it is given by its invariant differentials. This answers a question raised in [@ls2].
[**Acknowledgements:** ]{}We are very grateful to Christopher Deninger for his interest in these results. Much of the research on this paper was carried out while the first author was a guest of the SFB 478 in Münster. He is grateful to this institution for its support and hospitality during two visits. The first author was also partially supported by an NSF research grant.
$p$-adic height pairings
========================
In this section we collect various facts about $p$-adic height pairings, thereby fixing our notation.
Let $A_F$ be an abelian variety over a number field $F$, and let $A'_F$ denote its dual abelian variety.
Recall that the classical Néron-Tate height pairing on $A_F$ $${(\; \; , \; \;)}_N: A'_F(F) \times A_F(F) \rightarrow {\mathbb{R}}$$ can be decomposed in a sum of local pairings $${(\; \; , \; \;)}_{N,v}: ({\mbox{Div}}^0(A_{F_v}) \times Z^0(A_{F_v}/F_v))' \rightarrow
{\mathbb{R}},$$ where $v$ runs over the places of $F$, and $F_v$ denotes the $v$-adic completion of $F$. Here $({\mbox{Div}}^0(A_{F_v}) \times Z^0(A_{F_v}/F_v))'$ is the set of all pairs $(D,z)$ such that $D$ is a divisor on $A_{F_v}$, algebraically equivalent to zero, and $z$ is a cycle $\sum_i n_i a_i$ with $a_i \in A_{F_v}(F_v)$ and $\sum_i n_i
=0$, such that the supports of $D$ and $z$ are disjoint. These local pairings can be characterized by a list of axiomatic properties, see [@ne]. For all rational functions $f$ on $A_{F_v}$ we have $(\mbox{div}(f), \sum n_i a_i)_{N,v} = \sum n_i \log|f(a_i)|_v$.
Now we want to investigate $p$-adic height pairings, i.e. pairings $${(\; \; , \; \;)}_p: A'_F(F) \times A_F(F) \rightarrow {{\mathbb{Q}}}_p.$$ The initial datum is a collection of continuous homomorphisms $\rho_v: F_v{^\times}\rightarrow {{\mathbb{Q}}}_p$, one for each finite place $v$ of $F$, such that $\sum_v
\rho_v = 0$. (This substitutes the collection $(\log|\,|_v)$ from the classical case.) For example, we could take $$\rho_v = \left\{ \begin{array}{ll}
\log_p \circ N_{F_v/ {\mathbb{Q}}_p} & \mbox{ if $v|\,p$}\\
\log_p \circ |\;\; |_l \circ N_{F_v/{\mathbb{Q}}_l} & \mbox{ if $v | \! \! /
\,p$,}
\end{array} \right.$$ where $\log_p: {\mathbb{Q}}^\times_p \rightarrow {{\Bbb Q}}_p$ is the branch of the $p$-adic logarithm vanishing on $p$.
Note that for all places $v$ not dividing $p$, any continuous homomorphism $\rho_v$ is unramified, i.e. it maps the elements of absolute value one to zero. Hence it is proportional to the valuation map $v$, and also to $\log|\,|_v$.
Assume for a moment that we can construct a collection of bilinear maps $${(\; \; , \; \;)}_v: ({\mbox{Div}}^0(A_{F_v}) \times Z^0(A_{F_v}/F_v))' \rightarrow
{{\mathbb{Q}}}_p,$$ one for each finite place of $F$, with the following properties:
1\) $({\rm div}f, \sum n_i a_i )_v = \sum n_i \rho_v(f(a_i))$ for all rational functions $f$ on $A_{F_v}$.
2\) $(t_a^\ast D, t_a^\ast z)_v = (D,z)_v$ for all $a\in A_{F_v}(F_v)$ and all pairs $(D,z)$.
3\) If $\rho_v$ is unramified, i.e. $\rho_v(x) = c v(x)$ for some constant $c$, then ${(\; \; , \; \;)}_v$ is proportional to the Néron pairing.
With these data we can construct a global $p$-adic height pairing as follows: For $a \in A_F(F)$ and $a' \in A'_F(F)$ choose a divisor $D$, algebraically equivalent to zero, whose class is $a'$, and whose support contains neither $a$ nor the unit element $ 0 \in A_F(F)$. Then put $$(a',a) = \sum_{v}(D, a-0)_v \in {{\mathbb{Q}}}_p,$$ where we lift $D$ and the cycle $a-0$ to the variety $A_{F_v}$. This is well-defined, i.e. the sum is finite, and independent of the choice of $D$. Using property 2) of the local height pairings one can show that it is bilinear.
One recipe to construct local pairings with the desired properties was given by Mazur and Tate in [@mata].
Let us fix a place $v$ and a continuous homomorphism $\rho_v: F_v{^\times}\rightarrow
{{\mathbb{Q}}}_p$. From now on we will work in a local setting. To save subscripts, let us put $\rho= \rho_v$ and $K = F_v$. Besides, denote by $R$ the ring of integers in $K$ and by $k$ the residue field. Recall that we call $\rho$ unramified, if it maps $R{^\times}$ to zero.
Let ${{\cal A}}$ respectively ${{\cal A}}'$ be Néron models of $A = A_F \times_F K$ respectively $A' = A'_{F} \times_F K$ over $R$. Besides, let $P= P_{A \times A'}$ be the Poincaré biextension expressing the duality between $A$ and $A'$, see [@sga7], VIII, 1.4. $P$ induces an isomorphism $A' \simeq \underline{{\mbox{\rm Ext}}}^1(A, \ {{\mathbb{G}}}_{m})$ of sheaves in the fppf-site over $K$, mapping a point $a' \in A'(T)$ to the restriction $P_{A_T \times \{a'\}}$. We will often identify points in $A'$ with extensions using this isomorphism.
We call a map $\sigma: P(K) \rightarrow {{\mathbb{Q}}}_p$ a $\rho$-splitting if $\sigma$ behaves homomorphically with respect to both group laws and if $\sigma(\alpha x) =
\rho(\alpha) + \sigma(x)$ for all $\alpha \in K{^\times}$ and $x \in P(K)$, see [@mata], 1.4.
For any $\rho$-splitting $\sigma$ we can define a bilinear map ${(\; \; , \; \;)}_{\sigma}:
({\mbox{Div}}^0(A) \times Z^0(A/K))' \rightarrow {{\mathbb{Q}}}_p$ with the properties 1) and 2) as follows: For any divisor $D \in {\mbox{Div}}^0(A)$ let $d$ be the point in $A'(K)$ corresponding to its class. Then $P_{A \times \{d\}}$ is the extension of $A$ by ${\mathbb{G}}_{m}$ corresponding to the class of $D$, hence $D$ gives rise to a rational section $s_D$ of $P_{A \times \{d\}} \rightarrow A$, which is a morphism on the complement of the support of $D$. Hence we can put $$(D,\sum n_i a_i)_{\sigma} = \sum n_i \sigma(s_D(a_i))$$ to get our local pairing. Conversely, by [@mata], 2.2, every local height pairing with 1) and 2) comes in fact from a uniquely defined $\rho$-splitting $\sigma$.
In [@mata] canonical $\rho$-splittings are defined in several cases. We will discuss two of them. The first one is the case that $\rho$ is unramified.
Note that $P$ can be extended to a biextension $P_{{{\cal A}}^0 \times {{\cal A}}'}$ of ${{\cal A}}^0$ and ${{\cal A}}'$ by ${{\mathbb{G}}}_{m,R}$, where ${{\cal A}}^0$ is the identity component of ${{\cal A}}$, see [@sga7], VIII, 7.1.
Since $\rho$ is unramified, there exists a unique $\rho$-splitting $\sigma$ vanishing on $P_{{{\cal A}}^0 \times {{\cal A}}'}(R)$, see [@mata], 1.9. If $\rho$ is equal to $\log|\,|_v$, then the pairing ${(\; \; , \; \;)}_{\sigma}$ coincides with the local Néron pairing ${(\; \; , \; \;)}_{N,v}$ at our fixed place $v$ by [@mata], 2.3.1. Hence we find that for unramified $\rho$ the Mazur-Tate pairing ${(\; \; , \; \;)}_{\sigma}$ fulfills property 3) from above.
Note that this means that it remains to construct local pairings for ramified continuous homomorphisms $\rho: K{^\times}\rightarrow {{\mathbb{Q}}}_p$, which will only involve places $v$ over $p$.
Another case where Mazur and Tate construct a canonical $\rho$-splitting is the case that $A$ has semistable ordinary reduction, i.e. that the formal completion of the special fibre ${{\cal A}}_k$ along the zero section is isomorphic to a product of copies of ${\mathbb{G}}_m^f$ over the algebraic closure of $k$. This is equivalent to the fact that ${{\cal A}}_k^0$ is an extension of an ordinary abelian variety by a torus, see [@mata], 1.1. Note that in particular ${{\cal A}}$ has semistable reduction. Denote by ${{\cal A}}^f$ and ${{\cal A}}'^f$ the formal completions along the zero sections in the special fibres, and by $P_{{{\cal A}}^0 \times {{\cal A}}'}^f$ the formal completion of $P_{{{\cal A}}^0 \times {{\cal A}}'}$ along the preimage of the zero section of ${{\cal A}}^0_k \times
{{\cal A}}'_k$ under the projection map. Then $P_{{{\cal A}}^0 \times {{\cal A}}'}^f$ is a formal biextension of ${{\cal A}}^f$ and ${{\cal A}}'^f$ by ${{\mathbb{G}}}_m^\wedge$, the formal completion of ${{\mathbb{G}}}_m$ along the special fibre. If $A$ has semistable ordinary reduction, $P_{{{\cal A}}^0 \times {{\cal A}}'}^f$ admits a unique trivialization, hence there is a uniquely defined biextension splitting $\tilde{\sigma}: P_{{{\cal A}}^0 \times {{\cal A}}'}^f \rightarrow {{\mathbb{G}}}_m^\wedge$, see [@mata], 5.11.2. Then there exists a unique $\rho$-splitting $\sigma: P(K)
\rightarrow {{\mathbb{Q}}}_p$ such that for all $x \in P_{{{\cal A}}^0 \times {{\cal A}}'}^f(R)$ we have $\sigma(x) = \rho(\tilde{\sigma}(x))$, see [@mata], 1.9. We call the corresponding local height pairing the canonical Mazur-Tate pairing in the semistable ordinary reduction case.
Note that if additionally $\rho$ is unramified, we get the same $\rho$-splitting as defined previously (see [@mata], p.204).
There is also another way of constructing in certain cases for ramified $\rho$ a $\rho$-splitting, namely the norm-adapted Schneider height, cf [@sch]. By [@mata], 1.11.6 it coincides with the Mazur-Tate height if $A$ has good ordinary reduction. A formula for the difference of these two $p$-adic height pairings in the case of semistable ordinary reduction can be found in [@we], 7.2.
Let us now give a brief account of another approach to construct local $p$-adic height pairings, which uses splittings of the Hodge filtration of the first deRham cohomology. (See e.g. [@cogr] or [@za].)
Let $V'$ be the vector group corresponding to $e^* \Omega_{A/K}^1$, where $e$ is the unit section of $A$. By $I'$ we denote the universal vectorial extension of the dual abelian variety $A'$, cf. [@mame], chapter 1, §1. Then $I'$ sits in an exact sequence $$0 \longrightarrow V' \longrightarrow I' \longrightarrow A' \longrightarrow 0,$$ and it is universally repellent with respect to extensions of $A'$ by vector groups, see [@mame]. $I'$ represents the sheaf $\underline{\rm Extrig}_K(A, {{\mathbb{G}}}_{m})$ of rigidified extensions of $A$ by ${{\mathbb{G}}}_{m}$. This is the Zariski sheaf associated to the presheaf mapping $S$ to ${\rm Extrig}_S(A_S, {{\mathbb{G}}}_{m,S})$, the group of isomorphism classes of pairs $(E,t)$, where $E$ is an extension of $A_S$ by ${{\mathbb{G}}}_{m,S}$ and $t$ is a section on the level of the first infinitesimal extensions. To be precise, $t$ is a morphism of $S$-pointed $S$-schemes $t : {\mbox{\rm Inf}}^1(A_S/S) \rightarrow E$ such that the diagram $$\begin{aligned}
\begin{CD}
E @>>> A_S \\
@A{t}AA @AAA \\
{\mbox{\rm Inf}}^1(A_S/S) @= {\mbox{\rm Inf}}^1(A_S/S)
\end{CD}\end{aligned}$$ commutes. Here ${\mbox{\rm Inf}}^1$ is the first infinitesimal neighbourhood of the unit section in the sense of [@ega4], 16.1.2. If we identify $A'$ with the sheaf $\underline{\rm Ext}^1_K(A, {{\mathbb{G}}}_{m})$, the projection $I' \rightarrow A'$ corresponds to the map “forget the rigidification”.
Moreover, by [@mame], chapter 1, §4, there is an isomorphism $\mbox{Lie}\, I'
{\stackrel{\sim}{\longrightarrow}}{H^1_{dR}(A)}$ such that the following diagram commutes $$\begin{aligned}
\begin{CD}
0 @>>> \mbox{Lie}\,V' @>>> \mbox{Lie}\,I' @>>> \mbox{Lie}\,A' @>>> 0 \\
@. @VV{\simeq}V @VV{\simeq}V @VV{\simeq}V @. \\
0 @>>> H^0(A, \Omega^1) @>>> {H^1_{dR}(A)}@>>> H^1(A, {\cal O}) @>>> 0,
\end{CD}\end{aligned}$$ where the left vertical map is obvious and the right one is given by [@mu], p.130, and where the lower horizontal sequence is given by the Hodge filtration on ${H^1_{dR}(A)}$.
For any smooth commutative $K$-group $H$ the $K$-rational points $H(K)$ define a Lie group over $K$ (in the sense of [@bo], chapter III) whose Lie algebra coincides with the algebraic Lie algebra ${\mbox{Lie}} \, H$.
Suppose now that $r: H^1(A, {\cal O}) \rightarrow {H^1_{dR}(A)}$ is a splitting of the Hodge filtration. This induces a splitting $\mbox{Lie} \, A' \rightarrow \mbox{Lie} \, I'$ which can be lifted to a Lie group homomorphism in a suitable neighbourhood of the unit section by [@bo], III, §7, Theorem 3. It is easy to see that this can be extended to a (uniquely determined) splitting $\eta: A'(K) \rightarrow I'(K)$ of the projection $I'(K) \rightarrow A'(K)$ with $\mbox{Lie}\, \eta =r$, cf. [@za], Theorem 3.1.3. With other words, this is a splitting of the forgetful homomorphism $$\mbox{Extrig}(A, {{\mathbb{G}}}_{m}) \rightarrow \mbox{Ext}^1(A, {{\mathbb{G}}}_{m}),$$ i.e. we found a multiplicative way of associating to an extension $X$ of $A$ by ${{\mathbb{G}}}_{m}$ a rigidification. Note that a rigidification on $X$ is the same as a splitting of the corresponding sequence of Lie algebras $$0 \longrightarrow {\mbox{\rm Lie}}\, {{\mathbb{G}}}_{m} \longrightarrow {\mbox{\rm Lie}}\, X \longrightarrow
{\mbox{\rm Lie}}\, A \longrightarrow 0.$$
Now take a divisor $D \in {\mbox{Div}}^0(A)$ whose class gives a point $d \in A'_K(K)$. Then $P_{A \times \{d\}}$, the extension corresponding to $d$, is endowed with a rigidification, which in turn induces a splitting $$t_d: \mbox{Lie}\, P_{A \times \{d\}}(K) = \mbox{Lie}\, P_{A \times \{d\}}
\rightarrow \mbox{Lie}\, {{\mathbb{G}}}_{m} = \mbox{Lie} \, K{^\times}$$ of the Lie algebra sequence corresponding to the extension $P_{A \times \{d\}}$.
We fix again a continuous, ramified homomorphism $\rho: K{^\times}\rightarrow {{\Bbb Q}}_p$. By [@za], p. 319 we have $\rho = \delta \circ \lambda$, where $\delta: K \rightarrow {{\mathbb{Q}}}_p$ is a ${{\mathbb{Q}}}_p$-linear map, and $\lambda: K{^\times}\rightarrow K$ is a branch of the $p$-adic logarithm. Using once more [@bo], III, §7, Theorem 3 we find a uniquely determined homomorphism of Lie groups $$\gamma_d: P_{A \times \{d\}}(K) \rightarrow K$$ such that $K{^\times}\rightarrow P_{A \times \{d\}}(K)
\stackrel{\gamma_d}{\rightarrow}K$ is the homomorphism $\lambda$ and such that $\mbox{Lie}\, \gamma_d = \mbox{Lie}\,
\lambda \circ t_d$, cf. [@za], Theorem 3.1.7. These maps $\gamma_d$ fit together to a $\lambda$-splitting $\gamma$ of $P(K)$. Hence $\delta \circ \gamma$ is a $\rho$-splitting of $P(K)$.
Now we can define a pairing $${(\; \; , \; \;)}_r: ({\mbox{Div}}^0(A) \times Z^0(A/K))' \rightarrow {{\mathbb{Q}}}_p$$ by $(D,\sum n_i a_i)_r = \sum n_i \, \delta\circ \gamma(s_D(a_i))$, where as above $s_D$ is a rational section of $P_{A_{K} \times \{d\}}$ corresponding to $D$. This map is bilinear and has properties 1) and 2). Obviously, ${(\; \; , \; \;)}_r$ is just the height pairing associated to the $\rho$-splitting $\delta \circ\gamma$.
Hence every splitting of the Hodge filtration induces a $\rho$-splitting on $P(K)$ such that the corresponding height pairings are the same. What about the other direction?
Suppose that $\tau: P(K) \rightarrow K$ is a $\lambda$-splitting. Then we can define a splitting $$\eta(\tau): A'(K) \longrightarrow I'(K)$$ of the projection map by associating to every $a' \in A'(K)$ the extension $P_{A \times \{a'\}}$ endowed with the rigidification $
({\mbox{\rm Lie}}\lambda){^{-1}}\circ {\mbox{\rm Lie}}(\tau_{|P_{A \times \{a'\}}(K)}):
{\mbox{\rm Lie}}P_{A \times \{a'\}}(K) {\longrightarrow}
{\mbox{\rm Lie}}K {\stackrel{\sim}{\longrightarrow}}{\mbox{\rm Lie}}K{^\times}$. If $\eta(\tau)$ is analytic (in the sense of [@bo2]), then it induces a Lie algebra splitting $r : {\mbox{\rm Lie}}A' \rightarrow {\mbox{\rm Lie}}I'$, hence a splitting of the Hodge filtration of ${H^1_{dR}(A)}$. This construction is converse to the previous association $r \mapsto \gamma$. We will see in Proposition 3.2 that $\eta(\tau)$ is analytic if $\tau$ is an analytic map.
Splittings of the Hodge filtration of ${H^1_{dR}(A)}$
=====================================================
Let, as in section 1, $K$ be a non-archimedean local field of characteristic $0$ with ring of integers $R$ and residue field $k$. Besides, let $A$ be an abelian variety over $K$ with ordinary semistable reduction, $A'$ its dual abelian variety, and ${{\cal A}}$ respectively ${{\cal A}}'$ their Néron models. We assume that the torus parts in the reductions of ${{\cal A}}^0$ and ${{\cal A}}'_0$ are split. For our purposes, this is no restriction, since height pairings are compatible with base change.
Let us recall some facts about the rigid analytic uniformization of $A$ and $A'$. There is an extension of algebraic groups over $K$ $$0 \longrightarrow T\stackrel{g}{\longrightarrow} G \stackrel{p}{\longrightarrow}
B \longrightarrow 0,$$ such that $T$ is a split torus of dimension $t$ over $K$, and $B$ is an abelian variety over $K$ with good reduction. There is also a rigid analytic homomorphism $\pi: G{^{an}}\rightarrow A{^{an}}$ inducing a short exact sequence of rigid analytic groups over $K$ $$0 \longrightarrow \Gamma^{an} \stackrel{i}{\longrightarrow} G^{an}
\stackrel{\pi}{\longrightarrow} A{^{an}}\longrightarrow 0,$$ where $\Gamma$ is the constant group scheme corresponding to a free ${\mathbb{Z}}$-module $\Gamma$ of rank $t$. (See [@bolu], section 1, and [@ray].)
Let $\Gamma'$ be the character group of $T$. We denote the corresponding constant $K$-group scheme also by $\Gamma'$. Fix a dual abelian variety $(B', {P_{B \times B'}})$ of $B$, where ${P_{B \times B'}}$ is the Poincaré biextension expressing the duality. Then $G$ corresponds to a homomorphism $\phi': \Gamma' \rightarrow B'$ (see e.g. [@sga7], VIII, 3.7).
The embedding $i: \Gamma \rightarrow G$ induces a homomorphism $\phi: \Gamma
\stackrel{i}{\rightarrow} G \stackrel{p}{\rightarrow}B$, which gives us an extension $G'$ (again by [@sga7], VIII, 3.7) $$0 \longrightarrow T' \longrightarrow G' \stackrel{p'}{\longrightarrow} B'
\longrightarrow 0,$$ where $T'$ is the split torus of dimension $t$ over $K$ with character group $\Gamma$. There is a short exact sequence $$0 \longrightarrow \Gamma^{' an} \stackrel{i'}{\longrightarrow} G^{' an}
\stackrel{\pi'}{\longrightarrow} A^{' an} \longrightarrow 0.$$
In [@ls1] and [@ls2] it is shown how the rigid analytic uniformization of $A$ can be used to describe the deRham cohomology of $A$. Let us first fix some notation.
Let $X$ be a commutative $K$-group variety, hence smooth over $K$. The space of invariant differentials ${\mbox{\rm Inv}}(X)$ of $X$ is the space of sections of $e^\ast \Omega^1_{X/K}$, where $e:
\mbox{\rm Spec}(K) \rightarrow X$ is the unit section. Note that ${\mbox{\rm Inv}}(X)$ can be identified with the space of global differentials $\omega \in \Gamma(X, \Omega_X^1)$ satisfying $m^\ast \omega = p_1^\ast \omega
+ p_2^\ast \omega$, where $m,p_1,p_2: X \times X \rightarrow X$ denote multiplication and projections, respectively. Besides, all invariant differentials are closed.
If $Z$ is a rigid analytic $K$-group variety, we define ${\mbox{\rm Inv}}(Z)$ in the same way, using the rigid analytic differentials $\Omega^1_{Z/K}$ as defined e.g. in [@bkkn]. There is a natural GAGA-isomorphism ${\mbox{\rm Inv}}(X) \simeq
{\mbox{\rm Inv}}(X^{an})$.
Let us denote by $H^1_{dR}(X)$, respectively, $H^1_{dR}(Z)$ the first deRahm cohomology group of the algebraic variety $X$, respectively, the rigid analytic variety $Z$. Hence $H^1_{dR}(X)$ ist the first hypercohomology of the complex $(0 \rightarrow {\cal O}_{X} \rightarrow \Omega^1_{X/K} \rightarrow \Omega^2_{X/K} \rightarrow
\ldots)$ and, similarly, $H^1_{dR}(Z)$ is the first hypercohomology of the complex $(0 \rightarrow {\cal O}_{Z} \rightarrow \Omega^1_{Z/K} \rightarrow \Omega^2_{Z/K}
\rightarrow \ldots )$. By [@ki], there is a GAGA-isomorphism $$H^1_{dR}(X) \simeq H^1_{dR}(X^{an}),$$ which we tacitely use to identify these groups.
By [@ls1] we have a commutative diagram with exact rows and columns: $$\begin{aligned}
\begin{CD}
@. 0 @. 0 @. @.\\
@. @VVV @VVV @.@.\\
~~~ \quad 0 @>>> {\mbox{\rm Inv}}(B) @>>> {H^1_{dR}(B)}@>{\delta}>> H^1(B, {\cal{O}}_{B}) @>>> 0\\
@. @VVV @VV{p^\ast}V @V{\simeq}V{\beta}V @.\\
(1) \quad 0 @>>> {\mbox{\rm Inv}}(G) @>>> {H^1_{dR}(G)}@>>> H @>>> 0 \\
@. @VVV @V{g^\ast}VV @.@.\\
@. {\mbox{\rm Inv}}(T) @>>{\simeq}> {H^1_{dR}(T)}@.@.\\
@. @VVV @VVV @. @.\\
@.0 @. 0 @. @.
\end{CD}\end{aligned}$$ where $H$ is by definition the vector space making the middle horizontal sequence exact. The first row of this diagram is again induced by the Hodge filtration on ${H^1_{dR}(B)}$. We show in the appendix that in fact the middle row is also induced by the Hodge filtration on ${H^1_{dR}(G)}$.
Besides, we have a commutative diagram with exact rows and columns for ${H^1_{dR}(A)}$ (see for example [@coleman_iovita].)
$$\begin{aligned}
\begin{CD}
@. @. 0 @. 0 @. \\
@. @. @VVV @VVV @. \\
@.@. Hom(\Gamma,K) @>{=}>> Hom(\Gamma,K) @.\\
@. @. @VVV @VVV @.\\
(2) \quad 0 @>>> {\mbox{\rm Inv}}({A}) @>>> {H^1_{dR}(A)}@>>> H^1(A,{\cal{O}}) @>>> 0\\
@. @V{\alpha}V{\simeq}V @VV{\pi^\ast}V @VV{\gamma}V @.\\
~~~ \quad 0 @>>> {\mbox{\rm Inv}}({G}) @>>> {H^1_{dR}(G)}@>>> H @>>> 0\\
@. @. @VVV @VVV @. \\
@. @. 0 @. 0 @.
\end{CD}\end{aligned}$$
Here the middle row is again induced by the Hodge filtration on ${H^1_{dR}(A)}$.
With these two diagrams we can lift any splitting $r: {H^1(B,{\cal{O}})}\rightarrow {H^1_{dR}(B)}$ of the Hodge filtration of ${H^1_{dR}(B)}$ to a splitting $L(r) : {H^1(A,{\cal{O}})}\rightarrow {H^1_{dR}(A)}$ of the Hodge filtration of ${H^1_{dR}(A)}$. First lift $r$ to a splitting $$p^\ast \circ r \circ \beta{^{-1}}: H \rightarrow {H^1_{dR}(G)}$$ of the middle row in diagram $(1)$. Take the corresponding splitting $s: {H^1_{dR}(G)}\rightarrow {\mbox{\rm Inv}}(G)$ in the other direction, and lift this to $$\alpha{^{-1}}\circ s \circ \pi^\ast: {H^1_{dR}(A)}\rightarrow {\mbox{\rm Inv}}(A)$$ with the help of diagram $(2)$.
Then let $L(r): {H^1(A,{\cal{O}})}\rightarrow {H^1_{dR}(A)}$ be the splitting in the other direction. A diagram chase shows that $L(r)$ is the unique splitting of the Hodge filtration of ${H^1_{dR}(A)}$ making the following diagram commutative: $$\begin{aligned}
\begin{CD}
@. {H^1_{dR}(A)}@<{L(r)}<< {H^1(A,{\cal{O}})}\\
@. @VV{\pi^\ast}V @VV{\gamma}V \\
@. {H^1_{dR}(G)}@. H\\
@. @AA{p^\ast}A @VV{\beta{^{-1}}}V \\
@.{H^1_{dR}(B)}@<{r}<< {{H^1(B,{\cal{O}})}}\\
\end{CD}\end{aligned}$$
For all $X\in \{A,G,T,B,\Gamma \}$, the $K$-vector space $H^1_{dR}(X)$ (which is by definition $Hom(\Gamma, K)$ if $X=\Gamma$) can be endowed with a ($K$-linear) Frobenius operator $$\varphi _X\colon H^1_{dR}(X)\rightarrow H^1_{dR}(X)$$ (which is canonical for all $X$’s except for $X=A$, when it depends on the choice of a branch of log on $K^*$.) By definition $\varphi _{\Gamma }=id$, $\varphi _T$= multiplication by $q=p^{[k:{\bf F}_p]}$. See [@coleman_iovita] for properties of these maps.
Let now $X\in\{A,B,G\}$, then we put $H(X)=H$ if $X=G$ and $H(X)=H^1(X,{{\cal O}})$ if $X=B$ or $A$. For each $X\in\{A,G,B\}$ we define the unit root subspace, $W_X
\subset H^1_{dR}(X)$ to be the subspace on which $\varphi _X$ acts with slope $0$. Let us remark that although $\varphi_A$ depends on the choice of a branch of the $p$-adic logarithm on $K^*$, $W_A$ is canonical (see [@illusie] or [@iovita].)
So let us define $r_X$ to be the unique splitting $H(X) \rightarrow H^1_{dR}(X)$ such that Im$(r_X)\subset
W_X$. By $s_X: H^1_{dR}(X) \rightarrow {\mbox{\rm Inv}}(X)$ we denote the corresponding splitting in the other direction. Either of $s_X$ or $r_X$ will be called “the unit root splitting" of (X).
For each $X\in\{A,G,B\}$ we have
i\) $\mbox{Im}(r_X)=W_X$ (which justifies the uniqueness in the definition above.)
ii\) $s_X$ is the composition $$H^1_{dR}(X){\longrightarrow}H^1_{dR}(X)/W_X\stackrel{i_X^{-1}}{{\longrightarrow}}{\mbox{\rm Inv}}(X).$$
[**Proof:** ]{} i) Let us recall a few simple facts about slope decomposition of Frobenius modules (see for example [@zink] or [@berthelot]). First, the unit root subspace is functorial with respect to Frobenius morphisms, i.e. with respect to $K$-linear maps which commute with the Frobenii. Then the unit root subspace functor is left exact. Let us now prove the lemma. Since $r_X$ is a section, it is an injective $K$-linear map. By assumption, $A$ has ordinary reduction, which implies that $B$ has ordinary reduction, i.e. dim$_K(W_A)=$dim$_K(H(A))$ and dim$_K(W_B)=$dim$_K(H(B))$. Therefore Im$(r_A)=W_A$ and Im$(r_B)=W_B$. Moreover, from the exact sequence of Frobenius modules $$0{\longrightarrow}H^1_{dR}(B){\longrightarrow}H^1_{dR}(G){\longrightarrow}H^1_{dR}(T){\longrightarrow}0,$$ we get an exact sequence of $K$-vector spaces $$0{\longrightarrow}W_B{\longrightarrow}W_G{\longrightarrow}(H^1_{dR}(T))^{slope=0}.$$ As the last vector space is $0$, we get that dim$_K(W_G)=$dim$_K(W_B)$ which implies that Im$(r_G)=W_G.$
ii\) Follows by definition from i).$\Box$
Let $r_B: H^1(B,{{\cal O}}) \rightarrow {H^1_{dR}(B)}$ be the unit root splitting on $B$. Then $L(r_B)$ is equal to $r_A$, the unit root splitting of ${H^1_{dR}(A)}$. Hence $r_A$ is the unique splitting of the Hodge filtration of ${H^1_{dR}(A)}$ making the following diagram commutative: $$\begin{aligned}
\begin{CD}
@. {H^1_{dR}(A)}@<{r_A}<< {H^1(A,{\cal{O}})}\\
@. @VV{\pi^\ast}V @VV{\gamma}V \\
(3)\quad \quad @. {H^1_{dR}(G)}@. H\\
@. @AA{p^\ast}A @VV{\beta{^{-1}}}V \\
@.{H^1_{dR}(B)}@<{r_B}<< {{H^1(B,{\cal{O}})}}\\
\end{CD}\end{aligned}$$
[**Proof:** ]{} As $p^\ast$ is a morphism of Frobenius modules we have that $p^\ast (W_B)\subset W_G$. Therefore $p^\ast r_B\beta^{-1}(H)
\subset W_G$ and as $p^\ast r_B\beta^{-1}$ is a section it follows that $p^\ast r_B\beta^{-1}=r_G$.
Now we want to prove that $\alpha^{-1} s_G\pi^\ast=s_A.$ For this let us remark that $\pi^\ast$ is a morphism of Frobenius modules, therefore $\pi^\ast(W_A)\subset W_G$ hence we have the following diagram: $$\begin{array}{cccccc}
{\mbox{\rm Inv}}(A)&\stackrel{(i_A)^{-1}}{{\longleftarrow}}&H^1_{dR}(A)/W_A&
{\longleftarrow}&H^1_{dR}(A)\\
\alpha\downarrow&&\downarrow\pi^*&&\downarrow\pi^*\\
{\mbox{\rm Inv}}(G)&\stackrel{(i_G)^{-1}}{{\longleftarrow}}&H^1_{dR}(G)/W_G&
{\longleftarrow}&H^1_{dR}(G)
\end{array}$$ Both small squares are commutative therefore the large rectangle is commutative as well, so we have $\alpha s_A=s_G\pi^\ast$.
It remains to check uniqueness. Two splittings of the Hodge filtration which both make our diagram commutative differ by a homomorphism ${H^1(A,{\cal{O}})}\rightarrow \mbox{\rm Inv}(A)$, which becomes zero after composition with $\pi^\ast$. Since $\pi^\ast$ is an isomorphism on invariant differentials our original map must also be zero.$\Box$
The Mazur-Tate height corresponds to the unit root splitting
============================================================
Let $F$ be a rigid analytic group functor over $K$, i.e. a contravariant functor from the category of rigid analytic $K$-varieties to the category of groups. We denote by $K[\epsilon]= K[T]/(T^2)$ the ring of dual numbers over $K$. Then the Lie algebra associated to $F$ is defined as $$L(F) = \mbox{\rm ker}(F(\mbox{Sp}\,{{K[\epsilon]}}) \longrightarrow F(\mbox{Sp}\, K)) .$$
If $Z$ is a rigid analytic $K$ group variety, its Lie algebra is defined as the Lie algebra of the corresponding group functor, i.e. ${\mbox{\rm Lie}}(Z) = \mbox{\rm ker}
(Z({{K[\epsilon]}}) \rightarrow Z( K))$. We have a natural duality between ${\mbox{\rm Inv}}(Z)$ and ${\mbox{\rm Lie}}(Z)$. Note that if $Z = X{^{an}}$ for some algebraic $K$-variety $X$ then we have a GAGA- isomorphism ${\mbox{\rm Lie}}(X{^{an}}) \simeq {\mbox{\rm Lie}}\, X$, compatible with the dual isomorphism ${\mbox{\rm Inv}}(X{^{an}}) \simeq {\mbox{\rm Inv}}(X)$.
Now we can define a rigidification of an exact sequence of rigid analytic $K$-groups as a splitting of the corresponding sequence of Lie algebras. Note that for an abelian variety $A$ over $K$ the GAGA-isomorphism ${\mbox{\rm Ext}}^1(A, {{\mathbb{G}}}_{m}) \simeq {\mbox{\rm Ext}}^1(A{^{an}}, {{\mathbb{G}}}_{m}{^{an}})$ can be continued to an isomorphism $\mbox{\rm Extrig}(A, {{\mathbb{G}}}_{m}) \simeq \mbox{\rm Extrig}(A{^{an}}, {{\mathbb{G}}}_{m}{^{an}})$. (When we use Ext groups for rigid analytic groups, we will always work in the site of all rigid $K$-varieties endowed with their Grothendieck topologies.)
Note further that the $K$-rational points of any smooth rigid analytic $K$-group $Z$ form a Lie group over $K$ in the sense of [@bo], III, §1. Its Lie algebra coincides with the Lie algebra of $Z$ as defined above. Rigid analytic morphisms of $K$-varieties give rise to analytic maps in the sense of [@bo2], i.e. on $K$-rational points they are locally given by converging power series. (See also [@se1].) From now on we mean analytic in this sense when we talk about analytic maps. Whenever we consider rigid analytic objects, we will say so explicitely.
The uniformization maps $\pi$ and $\pi'$ which we introduced in the preceeding section induce isomorphisms on rigid analytic open subgroups $$G{^{an}}\supset \overline{G} \simeq \overline{A} \subset A{^{an}}$$ respectively $$G{^{' an}}\supset \overline{G}' \simeq \overline{A}' \subset A{^{' an}}.$$
By [@we], 3.1, we have a (uniquely determined) isomorphism of biextensions of $G{^{an}}$ and $G{^{' an}}$ by ${{\mathbb{G}}}_{m}{^{an}}$ $$\theta: (\pi \times \pi')^\ast {P_{A \times A'}}{^{an}}{\longrightarrow} (p \times
p')^\ast {P_{B \times B'}}{^{an}}.$$
We fix a continuous, ramified homomorphism $\rho: K{^\times}\rightarrow {{\mathbb{Q}}}_p$. Recall that by [@za], p. 319, there is a non-zero ${{\Bbb Q}}_p$-linear map $\delta: K \rightarrow
{{\Bbb Q}}_p$ and a branch $\lambda$ of the $p$-adic logarithm such that $\rho = \delta
\circ \lambda$.
Let $J'$ be the universal vectorial extension of $B$, and let $\eta_B: B'(K)
\rightarrow J'(K)$ be the unique splitting such that ${\mbox{\rm Lie}}\,\eta_B = r_B$, the unit root splitting on $B$. By [@co], Theorem 3.3.1, the corresponding height pairing ${(\; \; , \; \;)}_{r_B}$ coincides with the Mazur-Tate pairing on $B$.
Now we consider the Mazur-Tate pairing on $A$. Let $\tau_A$ denote the $\lambda$-splitting defined by the Mazur-Tate condition $\tau_A(x) = \lambda(\tilde{\sigma} (x))$ on $P^f_{{{\cal A}}^0 \times {{\cal A}}'}(R)$, where $\tilde{\sigma}:P^f_{{{\cal A}}^0 \times {{\cal A}}'} \rightarrow {{\mathbb{G}}}_m^\wedge $ is defined via the unique formal trivialization of $P_{{{\cal A}}^0 \times {{\cal A}}'}^f$. Define $\tau_B$ in an analogous way. Then $\sigma_A = \delta \circ \tau_A$ and $\sigma_B
= \delta \circ \tau_B$ are the $\rho$-splittings giving rise to the Mazur-Tate height pairings on $A$ respectively $B$.
Fix some $a' \in \overline{A}'(K)$ and consider $x \in P_{\overline{A} \times
\{a'\}}(K)$. The same argument as in [@we], 7.1 shows that we have $$\tau_B (pr \circ \theta(j(x))) = \tau_A(x)$$ where $j: P_{\overline{A} \times \{a'\} }{^{an}}\hookrightarrow (\pi \times
\pi')^\ast {P_{A \times A'}}{^{an}}$ is induced by $\overline{A}\simeq \overline{G} \hookrightarrow
G$ and $\overline{A}'\simeq \overline{G}' \hookrightarrow G'$ and where $pr: (p
\times p')^\ast {P_{B \times B'}}{^{an}}\rightarrow {P_{B \times B'}}{^{an}}$ is the natural projection.
This implies that the Mazur-Tate pairing on $A$ corresponds to the following section $\eta_A: A'(K) \rightarrow I'(K)$ of the projection: For $a'$ in the open subgroup ${\overline}{A}'$ denote its preimage in $\overline{G}'$ by $g'$, and put $b'=p'(g')$. Then $\theta$ induces an isomorphism $\pi^\ast P_{A \times \{a'\}}{^{an}}\simeq p^\ast P_{B \times \{b'\}}{^{an}}$. The map $\eta_B$ endows $ P_{B \times
\{b'\}}$ with a rigidification. This induces a rigidification on $\pi^\ast P_{A
\times \{a'\}}$, and hence on $P_{A \times \{a'\}}$, since $\pi$ is an isomorphism in a neighbourhood of the unit element. Then $\eta_A(a')$ is the extension $ P_{A
\times \{a'\}}$ together with this rigidification. Since $I'$ is an extension of $A'$ by a vector group, $\eta_A$ can be extended uniquely to the whole of $A'(K)$. Our next goal is to prove that $\eta_A$ is analytic as a map of $K$-varieties. As a first step we show the following lemma.
The $\lambda$-splitting $\tau_A: P(K) \rightarrow
K$ which is induced by the formal splitting $P_{{{\cal A}}^0 \times {{\cal A}}'}^f \rightarrow
{{\mathbb{G}}}^\wedge_m$ is an analytic map.
[**Proof:** ]{} Let us first show that under the natural inclusion $P_{{{\cal A}}^0\times {{\cal A}}'}^f(R)$ may be identified with an open subset of $P(K)$ (in the Lie group topology as in [@bo2].) This follows from a more general fact, namely let ${{\frak X}}$ be separated and of finite type over $R$ and let ${\overline{\frak X}}$ denote its special fiber. Let $Z\subset {\overline{\frak X}}$ be a closed subscheme. Denote by ${\hat{\frak X}}_Z$ the formal completion of ${{\frak X}}$ along $Z$ and by $X={{\frak X}}_K$ the generic fiber of ${{\frak X}}$. Let us denote by $X^{an}$ the rigid analytic variety corresponding to $X$ and by ${\hat{\frak X}}={\hat{\frak X}}_{{\overline{\frak X}}}$ the formal completion of ${{\frak X}}$ along ${\overline{\frak X}}$. Then, ${\hat{\frak X}}_Z$ is a formal scheme whose affine opens are formal spectra of $R$-algebras which are quotients of algebras of the form: $R\langle
T_1,...T_r\rangle [[X_1,...,X_s]]$. To such a formal scheme one attaches canonically (see [@berth], 0.2) a rigid analytic space, $({\hat{\frak X}}_Z)_K$ together with a specialization map $sp: ({\hat{\frak X}}_Z)_K \rightarrow {\hat{\frak X}}_Z$ such that the following diagram is commutative $$\begin{array}{ccc}
({\hat{\frak X}}_Z)_K&\stackrel{sp}{\longrightarrow}&{\hat{\frak X}}_Z\\
i\ \cap&&\downarrow\\
({\hat{\frak X}})_K&\stackrel{sp}{\longrightarrow}&{\hat{\frak X}}\\
j\ \cap\\
X^{an}
\end{array}$$ by [@berth], 0.2.7. Here $({\hat{\frak X}})_K$ denotes the rigid analytic generic fiber of the formal scheme ${\hat{\frak X}}$ which is topologically of finite type. Moreover, $i(({\hat{\frak X}}_Z)_K)=sp^{-1}(Z)$ is an admissible open of $({\hat{\frak X}})_K$ and $j$ is an open immersion. Therefore, $({\hat{\frak X}}_Z)_K$ can be identified with an admissible open of $X^{an}$, so that $({\hat{\frak X}}_Z)_K(K)$ is an open subset of $X(K)$. Let us now remark that ${\hat{\frak X}}_Z(R)=({\hat{\frak X}}_Z)_K(K)$, which proves that ${\hat{\frak X}}_Z(R)$ is an open subset of $X(K)$.
In order to finish the proof of the lemma note that $P^f_{{{\cal A}}^0\times {{\cal A}}'}(R)$ may be identified with the set of points of $P(R)$ which project to points on ${{\cal A}}^0(R)\times {{\cal A}}'(R)$ whose specialization is $(0,0)$ on ${{\cal A}}_k^0(k)\times {{\cal A}}_k'(k)$. As $P^f_{{{\cal A}}^0\times {{\cal A}}'}$ is the trivial biextension, we fix formal sections $\tilde{\sigma}: P^f_{{{\cal A}}^0\times {{\cal A}}'}{\longrightarrow}{\mathbb{G}}_m^\wedge$ and $s: {{\cal A}}^f\times {{\cal A}}'^f {\longrightarrow}P^f_{{{\cal A}}^0\times {{\cal A}}'}$.
Let us now recall how the map $\tau_A :P(K){\longrightarrow}K$ is defined. Let $x\in P(K)$ and let $(a,a')\in A(K)\times A'(K)={{\cal A}}(R)\times {{\cal A}}'(R)$ be its projection. For suitable integers $m$ and $n$ we have $(ma,na')\in {{\cal A}}^f(R)\times {{\cal A}}'^f(R)$, so let $y=s(ma,na')\in
P^f_{{{\cal A}}^0\times {{\cal A}}'}(R)\subset P(K)$. Actually, $y$ is an analytic function of $x$. Then the elements $(m,n)x$ (biextension multiplication) and $y$ of $P(K)$ differ by a unique element $c\in K^*$, and let us note that $c$ is also analytic as a function of $x$ as the biextension operations are algebraic, hence analytic. Now we have $$\tau_A(x)=\frac{1}{mn} \left(\lambda(\tilde{\sigma}y)+\lambda(c)\right).$$ Therefore $\tau_A$ is analytic.$\Box$
We will now prove in general that analytic $\lambda$-splittings lead to analytic splittings of the projection map $I'(K) \rightarrow A'(K)$.
Let $\tau: P(K)
\rightarrow K$ be an analytic $\lambda$-splitting à la Mazur and Tate, where $\lambda: K{^\times}\rightarrow K$ is a branch of the $p$-adic logarithm. Recall from section 1 that $\tau$ induces a splitting $\eta = \eta(\tau):
A'(K) \rightarrow I'(K)$ by associating to $a'$ the extension $P_{A \times \{a'\}}$ endowed with the rigidification given by ${\mbox{\rm Lie}}(\tau_{|P_{A \times \{a'\}}})$. Then $\eta$ is an analytic map.
[**Proof:** ]{} Choose Zariski open neighbourhoods $U \subset A$ and $U' \subset A'$ of the unit sections, such that the ${{\mathbb{G}}}_m$-torsor $P$ is trivial over $U
\times U'$, i.e. we have a ${{\mathbb{G}}}_m$-equivariant isomorphism $\varphi: P_{U
\times U'} {\stackrel{\sim}{\longrightarrow}}{{\mathbb{G}}}_{m} \times U \times U'$ over $U \times U'$. We can correct $\varphi$ by a suitable morphism $U' \rightarrow {{\mathbb{G}}}_{m}$ to achieve that $\varphi \circ e_{P/A'}: U' \rightarrow {{\mathbb{G}}}_{m } \times U \times U'$ is the map $u' \mapsto (1,1,u')$. Here $e_{P/A'}$ is the unit section of $P$ regarded as an $A'$-group. Then the map $$h: U \times U' \stackrel{1 \times id}{\longrightarrow} {{\mathbb{G}}}_{m} \times U
\times U' \stackrel{\varphi{^{-1}}}{\longrightarrow} P_{U \times U'}$$ is a $U'$-morphism compatible with the unit sections over $U'$. Therefore it induces a section ${\mbox{\rm Inf}}^1(U \times U' / U') \rightarrow {\mbox{\rm Inf}}^1(P_{U \times
U'}/U')$ of the map ${\mbox{\rm Inf}}^1(P_{U \times U'}/U') \rightarrow {\mbox{\rm Inf}}^1(U \times U' /
U')$ given by the projection $P_{U \times U'} \rightarrow U \times U'$. Here ${\mbox{\rm Inf}}^1$ denotes the first infinitesimal neighbourhood with respect to the unit section. Now $U
\times U'$ is an open subset of $A \times U'$ containing the image of the unit section over $U'$. Hence ${\mbox{\rm Inf}}^1(U \times U' /U') \simeq {\mbox{\rm Inf}}^1(A \times U'/U')$. Similarly, we have ${\mbox{\rm Inf}}^1(P_{U \times U'}/U') \simeq {\mbox{\rm Inf}}^1(P_{A \times U'}/U')$. Hence we get a commutative diagram $$\begin{aligned}
\begin{CD}
{\mbox{\rm Inf}}^1(A \times U'/U') @>>> {\mbox{\rm Inf}}^1(P_{A \times U'}/U')\\
@VVV @VVV \\
A \times U' @<<< P_{A \times U'},\\
\end{CD}\end{aligned}$$ which induces a rigidification on the extension $P_{A \times U'}$ of $A \times U'$ by ${{\mathbb{G}}}_{m,U'}$ over $U'$.
In this way we get an element in $I'(U')$ projecting to the natural inclusion $U'
\hookrightarrow A'$ in $A'(U')$, i.e. a local section $s: U' \rightarrow I'$ of the projection map $I'\rightarrow A'$. Since $s$ is a morphism of schemes, the corresponding map on $K$-rational points $s: U'(K) \rightarrow I'(K)$ is analytic.
For every $x \in U'(K)$ its image $s(x)$ corresponds to the extension $P_{A \times
\{x\}}$ together with the rigidification induced by the map $h(-,x): U
\rightarrow P_{U \times \{x\}}$. It is easy to see that this rigidification can also be described as follows: The isomorphism of ${{\mathbb{G}}}_m$-torsors $\varphi: P_{U \times U'} \rightarrow
{{\mathbb{G}}}_{m} \times U \times U'$ induces a $K{^\times}$-equivariant analytic map $\psi:
P_{U \times U'}(K) \rightarrow K{^\times}$, which in turn induces for all $x \in U'$ an analytic map $\psi_x: P_{U \times \{x\}}(K) \rightarrow K{^\times}$ respecting the unit sections. Hence the corresponding map on Lie algebras splits the Lie algebra sequence associated to $P_{A \times \{x\}}$, and gives the desired rigidification.
Since $\tau: P(K) \rightarrow K$ is a $\lambda$-splitting, we have $\tau(\alpha x) =
\lambda(\alpha) + \tau(x)$ for all $\alpha \in K{^\times}$ and $x \in P(K)$. An analogous formula holds for $\lambda \circ \psi$. By hypothesis, the difference $ \tau -\lambda \circ \psi: P_{U \times U'}(K) \rightarrow K$ is a $K{^\times}$-invariant analytic map, hence it factors over some analytic map $\theta:
U(K)
\times U'(K)
\rightarrow K$. Since $\tau$ and $\lambda \circ\psi$ map the unit section over $U'$ to zero, we have $\theta(1,x) = 0$ for all $x \in U'(K)$.
For all $x \in U(K)$ we now have two rigidifications on the extension $P_{A \times \{x\}}$. One comes from the point $s(x) \in I'(K)$. As shown above, it is given by the map $\psi_x$. The other one comes from the point $\eta(x) \in I'(K)$, by definition it is given by the Lie algebra map corresponding to the Lie group homomorphism $\tau: P_{A \times \{x\}}(K) \rightarrow K$, where we always identify ${\mbox{\rm Lie}}(K{^\times}) \simeq {\mbox{\rm Lie}}(K)$ by means of ${\mbox{\rm Lie}}\lambda$.
A straightforward calculation shows that for all $x \in U'(K)$ these two rigidifications on $P_{A \times \{x\}}$ differ by the invariant differential $\omega_x$ corresponding to the Lie algebra map $${\mbox{\rm Lie}}\,\theta(-,x): {\mbox{\rm Lie}}\, A \longrightarrow {\mbox{\rm Lie}}\,K \simeq K.$$
This means that the sections $\eta: U'(K) \rightarrow I'(K)$ and $s: U'(K) \rightarrow I'(K)$ of the projection map differ by $$\begin{aligned}
\omega: U'(K) & \longrightarrow & {\mbox{\rm Inv}}(A)\\
x & \longmapsto & \omega_x.
\end{aligned}$$ We claim that the map $\omega$ is analytic. As the claim has a local nature let us fix $x_0\in U'(K)$ and choose neighbourhoods $V \subset U(K)$ of $1$ and $V' \subset U'(K)$ of $x_0$ such that $\theta$ is given by a convergent power series $$\begin{aligned}
\lefteqn{\theta (u_1,...,u_r;x_1,...,x_s)}\\
= & \sum_{i_1,...,i_r,j_1,...,j_s\ge 0}
a_{i_1,...,i_s;j_1,...,j_s}(u_1-1_1)^{i_1}...(u_r-1_r)^{i_r}(x_1-x_{0,1})^
{j_1}...(x_s-x_{0,s})^{j_s}\end{aligned}$$ on $V \times V'$. Here $(1_1,...,1_r)$ and $(x_{0,1},
...,x_{0,s})$ are the local coordinates of $1$ and $x_0$, respectively. A simple calculation shows that under the identification $\mbox{Hom}_K({\mbox{\rm Lie}}(A), K))\simeq K^r$ given by the local coordinates $u_1,\ldots, u_r$ we have $$\begin{aligned}
\lefteqn{
{\mbox{\rm Lie}}(\theta)(-, x_1,...,x_s)} \\
= (&\sum_{j_1,...,j_s\ge 0}a_{1,0,...,0;j_1,...,j_s}
(x_1-x_{0,1})^{j_1}...(x_s-x_{0,s})^{j_s},...,\\
~ &\sum_{j_1,...,j_s\ge 0}
a_{0,...,0,1;j_1,...,j_s}(x_1-x_{0,1})^{j_1}...(x_s-x_{0,s})^{j_s}),\end{aligned}$$ which proves the claim.
Since the difference between $\eta$ and $s$ is an analytic map, we find that $\eta$ is analytic on $U'(K)$, hence everywhere.$\Box$
The section $\eta_A: A'(K) \rightarrow I'(K)$ is analytic.
Therefore we know now that the Mazur-Tate $\lambda$-splitting $\tau_A$ is induced by a splitting of the Hodge filtration, namely by ${\mbox{\rm Lie}}\, \eta_A$. Recall from the end of section 1 that this means that the Mazur-Tate height pairing coincides with the $p$-adic height pairing induced by ${\mbox{\rm Lie}}\, \eta_A$. Hence it remains to identify ${\mbox{\rm Lie}}\,\eta_A$ with the unit root splitting. We need some technical preparations.
Let $X$ be a scheme over the base scheme $S$ or a rigid analytic variety over a rigid analytic base variety $S$. Then we denote by ${\mbox{\rm Pic}}^\natural(X)=
{\mbox{\rm Pic}}^\natural(X/S)$ the group of invertible sheaves on $X$ endowed with an integrable connection. For a morphism $X \rightarrow Y$ of $S$-schemes or rigid analytic varieties over $S$ we have a natural map ${\mbox{\rm Pic}}^\natural(Y) \rightarrow
{\mbox{\rm Pic}}^\natural(X)$.
Let $(U_i)_i$ be a (Zariski or admissible rigid analytic) covering of $X$ such that the invertible sheaf ${{\cal L}}$ is trivial on each $U_i$ with transition functions $f_{ij} \in \Gamma(U_i \cap U_j, {{\cal O}}^\times)$. Then a connection $\nabla$ on ${{\cal L}}$ gives rise to a collection of forms $\omega_i \in\Gamma(U_i, \Omega_{X/S}^1)$ such that $\omega_i - \omega_j = d f_{ij}/f_{ij}$ on $U_i \cap U_j$. Conversely, every such collection of forms defines a connection. $\nabla$ is integrable iff all the $\omega_i$ are closed forms. In this way one sees that there is a functorial isomorphism $${\mbox{\rm Pic}}^\natural(X) \simeq {\mathbb{H}}^1(X,\Omega^\times_{X/S}),$$ where $\Omega^\times_{X/S}$ is the complex of sheaves $$\Omega^\times_{X/S} = ({{\cal O}}_X^\times \stackrel{d \log}{\longrightarrow}
\Omega_{X/S}^1 \longrightarrow \Omega_{X/S}^2 \longrightarrow \ldots).$$
Now let $X$ be a smooth rigid analytic $K$-group. There is a natural functorial homomorphism $$\mbox{\rm Extrig}_K(X, {{\mathbb{G}}}_{m}{^{an}}) \longrightarrow {\mbox{\rm Pic}}^\natural(X)$$ defined as follows. First of all, note that the rigidifications of the extension $$0 \longrightarrow {{\mathbb{G}}}_{m}{^{an}}\longrightarrow Z \longrightarrow X
\longrightarrow 0$$ correspond bijectively to invariant 1-forms $\omega$ on $Z$ which restrict to the form $dt/t$ on ${{\mathbb{G}}}_{m}{^{an}}$ (where $t$ is the standard parameter on ${{\mathbb{G}}}_{m}{^{an}}$), cf. [@za], p.323.
Starting with an extension $Z$ and an invariant $1$-form $\eta$ on $Z$ restricting to $dt/t$ on ${{\Bbb G}}_m^{an}$ we take a covering $(U_i)_i$ of $X$ such that the ${{\mathbb{G}}}_{m}{^{an}}$-torsor $Z$ is trivial over $U_i$, i.e. we have local sections $s_i: U_i \rightarrow Z$ of the projection $Z \rightarrow X$.
Now we put $\omega_i= s_i^\ast \eta \in \Gamma(U_i, \Omega^1_{X/K})$. Let ${{\cal L}}$ be the sheaf of sections of the line bundle associated to the torsor $Z$. Then, tautologically, the sections $f_{ij}\in \Gamma(U_i \cap U_j, {{\cal O}}^\times)$ such that $f_{ij} s_j = s_i$ on $U_i \cap U_j$ are transition functions for ${{\cal L}}$. One can easily check that the $\omega_i$ define a connection on ${{\cal L}}$. All $\omega_i$ are closed since $\eta$ is closed as an invariant differential. Hence our connection is integrable.
The following diagram commutes: $$\begin{aligned}
\begin{CD}
\overline{A}'(K) @>{\eta_A}>> \mbox{\rm Extrig}(A{^{an}}, {{\mathbb{G}}}_{m}{^{an}}) @>>>
{\mbox{\rm Pic}}^\natural(A{^{an}})\\
@AA{\simeq}A @VVV @VV{\pi^\ast}V\\
\overline{G}'(K) @. \mbox{\rm Extrig}(G{^{an}}, {{\mathbb{G}}}_{m}{^{an}}) @>>>
{\mbox{\rm Pic}}^\natural(G{^{an}})\\
@VVV @AAA @AA{p^\ast}A \\
B'(K) @>{\eta_B}>> \mbox{\rm Extrig}(B{^{an}}, {{\mathbb{G}}}_{m}{^{an}}) @>>> {\mbox{\rm Pic}}^\natural
(B{^{an}}).\\
\end{CD}\end{aligned}$$
[**Proof:** ]{} The right part is commutative by functoriality, the left one by definition of $\eta_A$.$\Box$
Let $X$ be a commutative rigid analytic group over $K$. Then we have a functorial isomorphism of $K$-vector spaces $$L ({\mbox{\rm Pic}}^\natural(X)) {\stackrel{\sim}{\longrightarrow}}H^1_{dR}(X),$$ where $L ({\mbox{\rm Pic}}^\natural(X))$ is the Lie algebra of the group functor $S \mapsto {\mbox{\rm Pic}}^\natural(X \times S)$.
[**Proof:** ]{} This can be proved as in the algebraic case (see [@me], 2.6.8). Let $\Omega^\bullet$ be the complex $ (0 \rightarrow {\cal O}_X \rightarrow
\Omega_{X/K} \rightarrow \Omega_{X/K}^2 \rightarrow \ldots)$ on $X$. If we put $X_\epsilon = X \times_K {{K[\epsilon]}}$ and denote by $p: X_\epsilon \rightarrow X$ the projection, there is a split exact sequence of complexes of abelian sheaves on $X$: $$0 \longrightarrow \Omega^\bullet_{X/K} \longrightarrow p_\ast
\Omega_{X_\epsilon/{{K[\epsilon]}}}^\times \longrightarrow \Omega_{X/K}^\times
\longrightarrow 0.$$ Taking first hypercohomologies, our claim follows.$\Box$
Now we are able to prove the following theorem
The Mazur-Tate height pairing on $A$ coincides with the height pairing defined by the unit root splitting $r_A$ of ${H^1_{dR}(A)}$.
[**Proof:** ]{} It suffices to show that $r_A = {\mbox{\rm Lie}}\, \eta_A$.
Comparing $\eta_A$ with an algebraic splitting as in the proof of 3.2 we see that we can pass from the diagram in Lemma 3.4 to the corresponding Lie algebra diagram. Using Lemma 3.5, we get a commutative diagram $$\begin{aligned}
\begin{CD}
{\mbox{\rm Lie}}\,\overline{A}'(K) @>{\mbox{\rm \small Lie} \, \eta_A}>> L
({\mbox{\rm Pic}}^\natural(A{^{an}})) @>{\simeq}>> {H^1_{dR}(A)}\\
@VV{\simeq}V @VVV @V{\pi^\ast}VV\\
{\mbox{\rm Lie}}\, \overline{G}'(K) @. L( {\mbox{\rm Pic}}^\natural(G{^{an}})) @>{\simeq}>> {H^1_{dR}(G)}\\
@VVV @AAA @A{p^\ast}AA \\
{\mbox{\rm Lie}}\, B'(K) @>>> L ({\mbox{\rm Pic}}^\natural(B{^{an}})) @>{\simeq}>> {H^1_{dR}(B)}.\\
\end{CD}\end{aligned}$$ Here the upper horizontal map is a section of the natural projection ${H^1_{dR}(A)}\rightarrow {H^1(A,{\cal{O}})}\simeq {\mbox{\rm Lie}}\, \overline{A}'(K)$ and the lower horizontal map is induced by ${\mbox{\rm Lie}}\,\eta_B = r_B$. Besides, the diagram $$\begin{aligned}
\begin{CD}
{H^1(A,{\cal{O}})}@>{\simeq}>> {\mbox{\rm Lie}}\,\overline{A}'(K)\\
@VVV @VV{\simeq}V\\
H @. {\mbox{\rm Lie}}\,\overline{G}'(K) \\
@VVV @VVV \\
{H^1(B,{\cal{O}})}@>{\simeq}>> {\mbox{\rm Lie}}\, B'(K)\\
\end{CD}\end{aligned}$$ commutes by [@ls2], 6.7. Hence ${\mbox{\rm Lie}}\,\eta_A$ makes diagram $(3)$ in Theorem 2.2 commutative, and our claim follows from this result.$\Box$
Appendix
========
In this section we will prove that the Hodge filtration of the first de Rham cohomology group of a semiabelian variety over $K$ is given by its invariant differentials. This answers a question raised in [@ls2].
Let us first recall some notations from section 2. Let $K$ be a finite extension of ${\Bbb Q}_p$, $T=({\Bbb G}_m)^t$ a split torus over $K$, $B$ an abelian variety over $K$ with good reduction and $G$ a semiabelian variety over $K$ defined by the extension of algebraic groups over $K$: $$(4)\quad 0{\longrightarrow}T\stackrel{g}{{\longrightarrow}} G \stackrel{p}{{\longrightarrow}} B{\longrightarrow}0.$$ We will prove
For $X\in \{T,G,B\}$ the image of ${\mbox{\rm Inv}}(X)$ in $H^1_{dR}(X)$ can be naturally identified with the Hodge filtration on $H^1_{dR}(X)$.
Note that this result is well known for $X=B$.
As $T$ and $G$ are smooth schemes over $K$ which are not proper, let us first briefly review the Hodge theory for non-proper varieties.
Let $X$ be a smooth scheme over $K$, $Y$ a smooth and proper scheme over $K$ and $X\stackrel{i_X}{{\longrightarrow}}Y$ an open embedding. Let $Z=Y-X$ and suppose that $Z$ is a divisor with normal crossings over $K$. Denote by $\Omega^i_Y(\log Z)$ the sheaf of $i$-th differential forms on $Y$ with logarithmic poles along $Z$. The first deRham cohomology of $Y$ with logarithmic poles along $Z$ is defined as $H^i_{dR}(Y,\log(Z)) = {\Bbb H}^i(\Omega_Y^\bullet(\log Z))$. Then we have (see [@deligne]):
1\) The inclusion $\Omega_Y^\bullet (\log Z)\subset (i_X)_*\Omega_X^\bullet$ induces an isomorphism $$H^i_{dR}(X)\cong H^i_{dR}(Y,\log(Z)).$$
2\) The spectral sequence $$E_1^{i,j}=H^j(Y, \Omega^i_Y(\log Z)) ==> H^{i+j}_{dR}(Y, \log Z)=
H^{i+j}_{dR}(X)$$ degenerates at $E_1$ and induces the Hodge filtration.
3\) The Hodge filtration thus defined on $H^i_{dR}(X)$ does not depend on the compactification of $X$.
As a consequence of the above we have that the first step of the Hodge filtration on $H^1_{dR}(X)$, denoted $F^0_X$, is given by $$F^0_X=\mbox{Ker}(d:H^0(Y, \Omega_Y^1(\log Z)){\longrightarrow}H^0(Y, \Omega^2_Y(\log Z))).$$
Let us now prove the theorem.
[**Proof:** ]{} a) We will first prove the statement for $X=T$. Consider the embedding $T=({\Bbb G}_m)^t\stackrel{i_T}{{\longrightarrow}} {({\Bbb P}_K^1)^t}$, where each ${{\Bbb G}}_m$ is embedded naturally in ${\Bbb P}^1_K$, after a parameter was chosen. This embedding may be seen as a smooth compactification of $T$. Moreover the group law: $T\times T\longrightarrow T$ extends naturally to a morphism $T\times {({\Bbb P}_K^1)^t}\longrightarrow {({\Bbb P}_K^1)^t}$, i.e. endows ${({\Bbb P}_K^1)^t}$ with a natural action of $T$. (See [@serre].)
Let $Z:={({\Bbb P}_K^1)^t}-T$. It is clearly a divisor with normal crossings and so, by the above we have $H^1_{dR}(T)=H^1_{dR}({({\Bbb P}_K^1)^t}, \log Z)$. As the $1$-forms in ${\mbox{\rm Inv}}(T)$ are closed and are logarithmic when considered in ${{\Bbb P}}:={({\Bbb P}_K^1)^t}$, we have an inclusion $${\mbox{\rm Inv}}(T){\longrightarrow}F^0_T=\mbox{Ker}(d:H^0({{\Bbb P}}, \Omega^1_{{\Bbb P}}(\log Z){\longrightarrow}H^0({{\Bbb P}},\Omega^2_{{\Bbb P}}(\log Z))).$$ To show that the map is surjective let us fix parameters $z_1,z_2,...,z_t$ on each factor of ${{\Bbb P}}={({\Bbb P}_K^1)^t}$ compatible with the parameters on the factors of $T=({{\Bbb G}}_m)^t$. Then an element $\omega\in H^0({{\Bbb P}}, \Omega^1_{{\Bbb P}}(\log Z))$ has the form $$\omega=\sum_{n=1}^t f_n\frac{dz_n}{z_n},$$ where $f_n\in H^0({{\Bbb P}}, {{\cal O}}_{{\Bbb P}})=K$. Therefore $\omega$ is closed and $\omega\in {\mbox{\rm Inv}}(T)$.
b\) Let us now prove the theorem for $X=G$. We will first briefly recall how one obtains a good compactification of $G$ (see [@serre]). Let’s recall the short exact sequence (4) at the beginning of this section. As torus torsors are locally trivial in the Zarski topology, we may find a finite open, affine cover $\{U_i\}_i$ of $B$ such that $$p^{-1}(U_i)\cong U_i\times T\stackrel{1 \times i_T}{{\longrightarrow}}U_i\times {({\Bbb P}_K^1)^t}.$$ Now embed each $U_i\times T\subset U_i\times {({\Bbb P}_K^1)^t}$ and notice that the isomorphisms used to glue the opens $\{U_i\times T\}_i$ extend naturally and define gluing data for the set $\{U_i\times {({\Bbb P}_K^1)^t}\}_i$. We glue the schemes $\{U_i\times {({\Bbb P}_K^1)^t}\}_i$ along these isomorphisms and obtain a proper scheme $M$ over $K$ such that the following hold
i\) The natural map $G\stackrel{i_G}{{\longrightarrow}} M$ is an open embedding.
ii\) We have a natural morphism $M{\longrightarrow}B$ and the torsor structure of $G$ over $B$ naturally extends to a stucture of principal fiber-space of $M$ over $B$ of fiber type ${({\Bbb P}_K^1)^t}$.
iii\) Let $V_i:=U_i\times {({\Bbb P}_K^1)^t}\subset M$. Then if $Z:=M-G$ we have $Z|_{V_i}=
U_i\times ({({\Bbb P}_K^1)^t}-T)=U\times Z_i$ which is a divisor with normal crossings in $V_i$ for every $i$. Therefore $Z$ is a divisor with normal crossings in $M$.
As a consequence, we have $H^1_{dR}(G)=H^1_{dR}(M, \log Z)$ and as the invariant differentials on $G$ are closed we have a natural inclusion $${\mbox{\rm Inv}}(G){\longrightarrow}\mbox{Ker}(d: (i_G)_*\Omega^1_{G}(M){\longrightarrow}(i_G)_*\Omega^2_{G}(M)).$$ Let us fix $\{Y_j\}_j$ a standard open affine cover of ${({\Bbb P}_K^1)^t}$ (compatible with the parameters $z_1,z_2,...,z_t$ chosen above.) Then $\{U_i\times Y_j\}_{i,j}$ is an open affine cover of $M$ and let $Z_{ij}:=Z\cap (U_i\times Y_j)=U_i\times (Y_j-T)$. Let $\omega\in {\mbox{\rm Inv}}(G)$. As both $\Omega_M^1(\log Z)$ and $(i_G)_*\Omega^1_{G}$ are coherent sheaves on $M$, in order to show that $\omega\in \Omega_M^1(\log Z)(M)$ it would be enough to show that $$\omega|_{U_i\times Y_j}\in \Omega^1_M(\log Z)(U_i\times Y_j)=\Omega^1_{U_i\times
Y_j}(\log Z_{ij}).$$ Let us fix $i,j$ and denote by $p_1,p_2$ the projections from $U_i\times Y_j$ to its factors (in this order). Then we have a natural isomorphism $$\Omega^1_{U_i\times Y_j}(\log Z_{ij})\cong p_1^*\Omega^1_{U_i}\times p_2^*
\Omega_{Y_j}(\log (Y_j-T)).$$ Moreover, $$\omega|_{U_i\times Y_j}\in ((i_G)_*\Omega^1_{G})(U_i\times Y_j)=
\Omega^1_{U_i\times (Y_j\cap T)}=p_1^*\Omega^1_{U_i}\times
p_2^*\Omega^1_{Y_j\cap T}.$$ So $\omega|_{U_i\times Y_j}=(\omega_1, \omega_2)$, with $\omega_1\in
p_1^*\Omega_{U_i}^1$ and $\omega_2=p_2^*((g^*\omega)|_{Y_j\cap T}).$ As $\omega\in {\mbox{\rm Inv}}(G)$ it follows that $g^*\omega\in {\mbox{\rm Inv}}(T)$ and we have proved at a) above that $g^*\omega$ is logarithmic, i.e. that $g^*\omega|_{Y_j\cap T}\in \Omega^1_{Y_j}(\log (Y_j-T))$.
This shows that we have a natural injection: $${\mbox{\rm Inv}}(G){\longrightarrow}F^0_G.$$ Now we claim that the exact sequence $(4)$ induces an exact sequence $$(5)\quad 0{\longrightarrow}F^0_B{\longrightarrow}F^0_G{\longrightarrow}F^0_T{\longrightarrow}0.$$ To see this let us recall that for all $X\in \{T,G,B\}$ we have isomorphisms as Gal$(\overline{K}/K)$-modules $$H^1_{et}(X_{\overline{K}}, {\Bbb Q}_p(1))\cong V_p(X),$$ where $V_p(X):=(\stackrel{\mbox{lim}}{\leftarrow}
X_K[p^n](\overline{K}))\otimes_{{\Bbb Z}_p}
{\Bbb Q}_p$ is the Tate module of $X_K$. The exact sequence $(4)$ also produces an exact sequence of Gal$(\overline{K}/K)$-modules (see [@ls2]) $$0{\longrightarrow}V_p(B){\longrightarrow}V_p(G){\longrightarrow}V_p(T){\longrightarrow}0.$$ Using Fontaine’s theory and the results in [@tsuji] we have a commutative diagram of [*filtered modules*]{} with exact rows: $$\begin{array}{ccccccccc}
0&{\longrightarrow}&D_{dR}(V_p(B))&{\longrightarrow}&D_{dR}(V_p(G))&{\longrightarrow}&D_{dR}(V_p(T))&{\longrightarrow}&0\\
&&\downarrow\cong&&\downarrow\cong&&\downarrow\cong\\
0&{\longrightarrow}&H^1_{dR}(B)&{\longrightarrow}&H^1_{dR}(G)&{\longrightarrow}&H^1_{dR}(T)&{\longrightarrow}&0
\end{array}$$ This implies the exactness of the sequence $(5)$ which implies that dim$_K(F^0_G)=$dim$_K({\mbox{\rm Inv}}(G))$ (we have used that the theorem is true for $T$ and $B$). Therefore the natural injection ${\mbox{\rm Inv}}(G){\longrightarrow}F^0_G$ is an isomorphism.
[xxxxxxx]{}
R. Berger, R. Kiehl, E. Kunz, H.-J. Nastold: Differentialrechnung in der analytischen Geometrie. Lecture Notes in Mathematics [**38**]{}. Springer 1967.
P. Berthelot: [*Cohomologie rigide et cohomologie rigide a support propre*]{} (http://www.maths.univ-rennes1.fr/berthelo/)
P. Berthelot: [*Slopes of Frobenius in crystalline cohomology*]{}, Math. Notes, Princeton University Press, 1978
S. Bosch, W. Lütkebohmert: [*Degenerating abelian varieties*]{}, Topology [**30**]{} (1991) 653-698.
N. Bourbaki: Groupes et algèbres de Lie. Chapitres 2 et 3. Hermann 1968. N. Bourbaki: Variétés différentiables et analytiques. Fascicule de résultats. Hermann 1971.
R. Coleman: [*The universal vectorial biextension and $p$-adic heights*]{}. Invent. math. [**103**]{} (1991) 631-650. R. Coleman, A. Iovita:[*Frobenius and monodromy operators on curves and abelian varieties*]{}. Duke Math. J. [**97**]{} (1999).
R. Coleman, B. Gross: [*$p$-adic heights on curves*]{}. In: Algebraic number theory. Adv. Stud. Pure Math [**17**]{}. Academic Press 1989, 73-81.
P. Deligne: [*Théorie de Hodge II*]{}. Publ. Math. IHES [**49**]{} (1971) 5-58.
A. Grothendieck: Éléments de géométrie algébrique IV. Publ. Math. IHES [**20**]{} (1964), [**24**]{} (1965), [**28**]{} (1966), [**32**]{} (1967).
L. Illusie: [*Reduction semi-stable ordinaire, cohomologie étale $p$-adique et cohomology de de Rham d’après Bloch-Kato et Hyodo, Appendice à l’exposé IV*]{} Asterisque [**223**]{} (1994) 209-220
A. Iovita: [*Formal sections and de Rham cohomology of semistable abelian varieties*]{}, Israel J. of Math. [**120**]{} (2000) 429-447.
R. Kiehl:[*Die de Rham Kohomologie algebraischer Mannigfaltigkeiten über einem bewerteten Körper*]{}. Publ. Math. IHES [**33**]{} (1967) 5 -20.
B. Le Stum: [*Cohomologie rigide et varietés abéliennes.*]{} C.R. Acad. Sc. Paris, t. 303, Série I [**20**]{} (1986) 989-992.
B. Le Stum: Cohomologie rigide et varietés abéliennes. Thèse. Université de Rennes I 1985.
B. Mazur, W. Messing: Universal extensions and one dimensional crystalline cohomology. Lecture Notes in Mathematics [**370**]{}. Springer 1974.
B. Mazur, J. Tate: [*Canonical height pairings via biextensions*]{}, In: Arithmetic and Geometry. Papers dedicated to I.R. Shafarevich. Progress in Mathematics [**35**]{}. Birkhäuser 1983, 195-235.
W. Messing: [*The universal vectorial extension of an abelian variety by a vector group*]{}. Symp. Math. [**XI**]{} (1973) 359-372.
D. Mumford: Abelian Varieties. Oxford University Press 1970.
A. Néron: [*Quasi-fonctions et hauteurs sur les variétés abéliennes*]{}. Ann. Math. [**82**]{} (1965) 249-331.
M. Raynaud: [*Variétés abéliennes et géométrie rigide*]{}. In: Actes, Congrès intern. math. 1970. Tome 1, 473-477.
P. Schneider: [*$p$-adic height pairings I.*]{} Invent. math. [**69**]{} (1982) 401-409.
J.-P. Serre: Lie algebras and Lie groups. W.A. Benjamin 1965.
J-P. Serre: [*Quelques propriétés des groupes algébriques commutatifs*]{}, Société Mathématique de France, Astérisque 69-70 (1979) 191-202.
A. Grothendieck et al.: Séminaire de géométrie algébrique du Bois-Marie 1967/69, Groupes de monodromie en géométrie algébrique I. Lecture Notes in Mathematics [**288**]{}. Springer 1972.
T. Tsuji: [*$p$-adic [é]{}tale cohomology and crystalline in the semi-stable reduction case*]{}. Invent. Math. [**137**]{} (1999) 233–411.
A. Werner: [*Local heights on Abelian varieties and rigid analytic uniformization*]{}. Documenta Math. [**3**]{} (1998) 301-319.
Y.I. Zarhin:[*$p$-adic heights on abelian varieties*]{}, in: Séminaire de Théorie des Nombres Paris 1997-88. Birkhäuser Progress in Mathematics [**81**]{} (1990) 317-341.
Th. Zink: [*Cartiertheorie kommutativer formaler Gruppen*]{}. Teubner Texte zur Mathematik [**68**]{}. Leipzig 1984.
\
|
---
abstract: 'Temporal sentence grounding in videos aims to detect and localize one target video segment, which semantically corresponds to a given sentence. Existing methods mainly tackle this task via matching and aligning semantics between a sentence and candidate video segments, while neglect the fact that the sentence information plays an important role in temporally correlating and composing the described contents in videos. In this paper, we propose a novel semantic conditioned dynamic modulation (SCDM) mechanism, which relies on the sentence semantics to modulate the temporal convolution operations for better correlating and composing the sentence-related video contents over time. More importantly, the proposed SCDM performs dynamically with respect to the diverse video contents so as to establish a more precise matching relationship between sentence and video, thereby improving the temporal grounding accuracy. Extensive experiments on three public datasets demonstrate that our proposed model outperforms the state-of-the-arts with clear margins, illustrating the ability of SCDM to better associate and localize relevant video contents for temporal sentence grounding. Our code for this paper is available at **** .'
author:
- |
Yitian Yuan[^1]\
Tsinghua-Berkeley Shenzhen Institute\
Tsinghua University\
`yyt18@mails.tsinghua.edu.cn`\
Lin Ma\
Tencent AI Lab\
`forest.linma@gmail.com`\
Jingwen Wang\
Tencent AI Lab\
`jaywongjaywong@gmail.com`\
Wei Liu\
Tencent AI Lab\
`wl2223@columbia.edu`\
Wenwu Zhu\
Tsinghua University\
`wwzhu@tsinghua.edu.cn`\
title: |
Semantic Conditioned Dynamic Modulation\
for Temporal Sentence Grounding in Videos
---
Introduction
============
Detecting or localizing activities in videos [@lin2017single; @wang2014action; @singh2016multi; @yuan2016temporal; @gavrilyuk2018actor; @tran2018a; @liu2019multi; @feng2018video; @feng2019spatio] is a prominent while fundamental problem for video understanding. As videos often contain intricate activities that cannot be indicated by a predefined list of action classes, a new task, namely temporal sentence grounding in videos (TSG) [@Hendricks2017Localizing; @gao2017tall], has recently attracted much research attention [@chen2018temporally; @zhang2018man; @chen2019localizing; @liu2018attentive; @chen2019SAP; @chen2019weaklysupervised; @yuan2019thumbnailgeneration]. Formally, given an untrimmed video and a natural sentence query, the task aims to identify the start and end timestamps of one specific video segment, which contains activities of interest semantically corresponding to the given sentence query.
Most of existing approaches [@gao2017tall; @Hendricks2017Localizing; @liu2018attentive; @chen2019SAP] for the TSG task often sample candidate video segments first, then fuse the sentence and video segment representations together, and thereby evaluate their matching relationships based on the fused features. Lately, some approaches [@chen2018temporally; @zhang2018man] try to directly fuse the sentence information with each video clip, then employ an LSTM or a ConvNet to compose the fused features over time, and thus predict the temporal boundaries of the target video segment. While promising results have been achieved, there are still several problems that need to be concerned.
First, previous methods mainly focus on semantically matching sentences and individual video segments or clips, while neglect the important guiding role of sentences to help correlate and compose video contents over time. For example, the target video sequence shown in Figure \[fig:intro\] mainly expresses two distinct activities “*woman walks cross the room*” and “*woman reads the book on the sofa*”. Without referring to the sentence, these two distinct activities are not easy to be associated as one whole event. However, the sentence clearly indicates that “*The woman takes the book across the room to read it on the sofa*”. Keeping such a semantic meaning in mind, persons can easily correlate the two activities together and thereby precisely determine the temporal boundaries. Therefore, how to make use of the sentence semantics to guide the composing and correlating of relevant video contents over time is very crucial for the TSG task. Second, activities contained in videos are usually of diverse visual appearances, and present in various temporal scales. Therefore, the sentence guidance for composing and correlating video contents should also be considered in different temporal granularities and dynamically evolve with the diverse visual appearances.
In this paper, we propose a novel semantic conditioned dynamic modulation (SCDM) mechanism, which leverages sentence semantic information to modulate the temporal convolution processes in a hierarchical temporal convolutional network. The SCDM manipulates the temporal feature maps by adjusting the scaling and shifting parameters for feature normalization with referring to the sentence semantics. As such, the temporal convolution process is activated to better associate and compose sentence-related video contents over time. More specifically, such a modulation dynamically evolves when processing different convolutional layers and different locations of feature maps, so as to better align the sentence and video semantics under diverse video contents and various granularities. Coupling SCDM with the temporal convolutional network, our proposed model naturally characterizes the interaction behaviors between sentence and video, leading to a novel and effective architecture for the TSG task.
![ The temporal sentence grounding in videos (TSG) task. Our proposed SCDM relies on the sentence to modulate the temporal convolution operations, which can thereby temporally correlate and compose the various sentence-related activities (highlighted in red and green) for more accurate grounding results. []{data-label="fig:intro"}](intro_new.pdf){width="100.00000%"}
Our main contributions are summarized as follows. (1) We propose a novel semantic conditioned dynamic modulation (SCDM) mechanism, which dynamically modulates the temporal convolution procedure by referring to the sentence semantic information. In doing so, the sentence-related video contents can be temporally correlated and composed to yield a precise temporal boundary prediction. (2) Coupling the proposed SCDM with the hierarchical temporal convolutional network, our model naturally exploits the complicated semantic interactions between sentence and video in various temporal granularities. (3) We conduct experiments on three public datasets, and verify the effectiveness of the proposed SCDM mechanism as well as its coupled temporal convolution architecture with the superiority over the state-of-the-art methods.
Related Works
=============
Temporal sentence grounding in videos is a new task introduced recently [@gao2017tall; @Hendricks2017Localizing]. Some previous works [@gao2017tall; @Hendricks2017Localizing; @liu2018attentive; @xu2019multilevel; @chen2019SAP; @ge2019mac] often adopted a two-stage multimodal matching strategy to solve this problem. They sampled candidate segments from a video first, then integrated the sentence representation with those video segments individually, and thus evaluated their matching relationships through the integrated features. With the above multimodal matching framework, Hendricks *et al.* [@Hendricks2017Localizing] further introduced temporal position features of video segments into the feature fusion procedure; Gao *et al.* [@gao2017tall] established a location regression network to adjust the temporal position of the candidate segment to the target segment; Liu *et al.* [@liu2018attentive] designed a memory attention mechanism to emphasize the visual features mentioned in the sentence; Xu *et al.* [@xu2019multilevel] and Chen *et al.* [@chen2019SAP] proposed to generate query-specific proposals as candidate segments; Ge *et al.* [@ge2019mac] investigated activity concepts from both videos and queries to enhance the temporal sentence grounding.
Recently, some other works [@chen2019SAP; @zhang2018man] proposed to directly integrate sentence information with each fine-grained video clip unit, and then predicted the temporal boundary of the target segment by gradually merging the fusion feature sequence over time in an end-to-end fashion. Specifically, Chen *et al.* [@chen2018temporally] aggregated frame-by-word interactions between video and language through a Match-LSTM [@wang2016machine]. Zhang *et al.* [@zhang2018man] adopted the Graph Convolutional Network (GCN) [@kipf2017semi] to model relations among candidate segments produced from a convolutional neural network.
Although promising results have been achieved by existing methods, they all focus on better aligning semantic information between sentence and video, while neglect the fact that sentence information plays an important role in correlating the described activities in videos. Our work firstly introduces the sentence information as a critical prior to compose and correlate video contents over time, subsequently sentence-guided video composing is dynamically performed and evolved in a hierarchical temporal convolution architecture, in order to cover the diverse video contents of various temporal granularities.
![An overview of our proposed model for the TSG task, which consists of three fully-coupled components. The multimodal fusion fuses the entire sentence and each video clip in a fine-grained manner. Based on the fused representation, the semantic modulated temporal convolution correlates sentence-related video contents in the temporal convolution procedure, with the proposed SCDM dynamically modulating temporal feature maps with reference to the sentence. Finally, the position prediction outputs the location offsets and overlap scores of candidate video segments based on the modulated features. Best viewed in color.[]{data-label="fig:framework"}](scin_6.pdf){width="1.0\columnwidth"}
The Proposed Model
==================
Given an untrimmed video $V$ and a sentence query $S$, the TSG task aims to determine the start and end timestamps of one video segment, which semantically corresponds to the given sentence query. In order to perform the temporal grounding, the video is first represented as $\mathbf{V} = \left\{ \mathbf{v}_t \right\}_{t=1}^{T}$ clip-by-clip, and accordingly the query sentence is represented as $\mathbf{S} = \left\{ \mathbf{s}_n \right\}_{n=1}^{N}$ word-by-word.
In this paper, we propose one novel model to handle the TSG task, as illustrated in Figure \[fig:framework\]. Specifically, the proposed model consists of three components, namely the multimodal fusion, the semantic modulated temporal convolution, and the position prediction. Please note that the three components fully couple together and can therefore be trained in an end-to-end manner.
Multimodal Fusion
-----------------
The TSG task requires to understand both the sentence and video. As such, in order to correlate their corresponding semantic information, we first let each video clip meet and interact with the entire sentence, which is formulated as: $$% \setlength{\abovedisplayskip}{3pt}
% \setlength{\belowdisplayskip}{3pt}
\mathbf{\mathbf{f}}_t = \text{ReLU} \left(\mathbf{W}^f \big( \mathbf{v}_t \| \mathbf{\overline{s}} \big) + \mathbf{b}^f \right),
\label{eq:multi_modal_fusion}$$ where $\mathbf{W}^f$ and $\mathbf{b}^f$ are the learnable parameters. $\mathbf{\overline{s}}$ denotes the global sentence representation, which can be obtained by simply averaging the word-level sentence representation $\mathbf{S}$. With such a multimodal fusion strategy, the yielded representation $\mathbf{F}=\{\mathbf{f}_t\}_{t=1}^T \in \mathcal{R}^{T \times d_f}$ captures the interactions between sentence and video clips in a fine-grained manner. The following semantic modulated temporal convolution will gradually correlate and compose such representations together over time, expecting to help produce accurate temporal boundary predictions of various scales.
Semantic Modulated Temporal Convolution
---------------------------------------
As aforementioned, the sentence-described activities in videos may have various durations and scales. Therefore, the fused multimodal representation $\mathbf{F}$ should be exploited from different temporal scales to comprehensively characterize the temporal diversity of video activities. Inspired by the efficient single-shot object and action detections [@liu2016ssd; @lin2017single], the temporal convolutional network established via one hierarchical architecture is used to produce multi-scale features to cover the activities of various durations. Moreover, in order to fully exploit the guiding role of the sentence, we propose one novel semantic conditioned dynamic modulation (SCDM) mechanism, which relies on the sentence semantics to modulate the temporal convolution operations for better correlating and composing the sentence-related video contents over time. In the following, we first review the basics of the temporal convolutional network. Afterwards, the proposed SCDM will be described in details.
### Temporal Convolutional Network
Taking the multimodal fusion representation $\mathbf{F}$ as input, the standard temporal convolution operation in this paper is denoted as $\text{Conv}(\theta_k,\theta_s,d_h)$, where $\theta_k$, $\theta_s$, and $d_h$ indicate the kernel size, stride size, and filter numbers, respectively. Meanwhile, the nonlinear activation, such as ReLU, is then followed with the convolution operation to construct a basic temporal convolutional layer. By setting $\theta_k$ as 3 and $\theta_s$ as 2, respectively, each convolutional layer will halve the temporal dimension of the input feature map and meanwhile expand the receptive field of each feature unit within the map. By stacking multiple layers, a hierarchical temporal convolutional network is constructed, with each feature unit in one specific feature map corresponding to one specific video segment in the original video. For brevity, we denote the output feature map of the $k$-th temporal convolutional layer as $\mathbf{A}_k = \{ \mathbf{a}_{k,i} \}_{i=1}^{T_k} \in \mathcal{R}^{T_k \times d_h}$, where $T_k = T_{k-1} / 2$ is the temporal dimension, and $\mathbf{a}_{k,i} \in \mathbf{R}^{d_h}$ denotes the $i$-th feature unit at the the $k$-th layer feature map.
### Semantic Conditioned Dynamic Modulation
Regarding video activity localization, besides the video clip contents, their temporal correlations play an even more important role. For the TSG task, the query sentence, presenting rich semantic indications on such important correlations, provides crucial information to temporally associate and compose the consecutive video contents over time. Based on the above considerations, in this paper, we propose a novel SCDM mechanism, which relies on the sentence semantic information to dynamically modulate the feature composition process in each temporal convolutional layer. Specifically, as shown in Figure \[fig:compare\](b), given the sentence representation $\mathbf{S} = \{ \mathbf{s}_n\}_{n=1}^N$ and one feature map extracted from one specific temporal convolutional layer $\mathbf{A} = \{\mathbf{a}_i\}$ (we omit the layer number here), we attentively summarize the sentence representation to $\mathbf{c}_i$ with respect to each feature unit $\mathbf{a}_i$: $$\setlength{\abovedisplayskip}{3pt}
\setlength{\belowdisplayskip}{3pt}
\begin{split}
\rho_i^n = \text{softmax} \big(\mathbf{w}^\top \text{tanh} \left( \mathbf{W}^s \mathbf{s}_n + \mathbf{W}^a \mathbf{a}_i + \mathbf{b} \right) \big), \qquad
\mathbf{c}_i = \sum_{n=1}^N \rho_i^n \mathbf{s}_n,
\end{split}
\label{eq:sentence_attention}$$
[r]{}[0.45]{} {width="0.26\textheight"}
where $\mathbf{w}$, $\mathbf{W}^s$, $\mathbf{W}^a$, and $\mathbf{b}$ are the learnable parameters. Afterwards, two fully-connected (FC) layers with the `tanh` activation function are used to generate two modulation vectors $\gamma^c_i \in \mathbf{R}^{d_h}$ and $\beta^c_i \in \mathbf{R}^{d_h}$, respectively: $$\begin{split}
\gamma^c_i &= \text{tanh}(\mathbf{W}^{\gamma} \mathbf{c}_i + \mathbf{b}^{\gamma}), \\
\beta^c_i &= \text{tanh}(\mathbf{W}^{\beta} \mathbf{c}_i + \mathbf{b}^{\beta}),
\end{split}
\label{eq:beta_gamma_generation}$$ where $\mathbf{W}^{\gamma}$, $\mathbf{b}^{\gamma}$, $\mathbf{W}^{\beta}$, and $\mathbf{b}^{\beta}$ are the learnable parameters. Finally, based on the generated modulation vectors $\gamma^c_i$ and $\beta^c_i$, the feature unit $\mathbf{a}_i$ is modulated as: $$\begin{split}
\hat{\mathbf{a}_i} = \gamma^c_i \cdot \frac{\mathbf{a}_i - \mu(\mathbf{A})}{ \sigma({\mathbf{A})}} + \beta^c_i.
\end{split}
\label{eq:anchor_update}$$
With the proposed SCDM, the temporal feature maps, yielded during the temporal convolution process, are meticulously modulated by scaling and shifting the corresponding normalized features under the sentence guidance. As such, each temporal feature map will absorb the sentence semantic information, and further activate the following temporal convolutional layer to better correlate and compose the sentence-related video contents over time. Coupling the proposed SCDM with each temporal convolutional layer, we thus obtain the novel semantic modulated temporal convolution as shown in the middle part of Figure \[fig:framework\]. **Discussion.** As shown in Figure \[fig:compare\], our proposed SCDM differs from the existing conditional batch/instance normalization [@vries2017modulating; @dumoulin2017a], where the same $\gamma^c$ and $\beta^c$ are applied within the whole batch/instance. On the contrary, as indicated in Equations (\[eq:sentence\_attention\])-(\[eq:anchor\_update\]), our SCDM dynamically aggregates the meaningful words with referring to different video contents, making the yielded $\gamma^c$ and $\beta^c$ dynamically evolve for different temporal units within each specific feature map. Such a dynamic modulation enables each temporal feature unit to be interacted with each word to collect useful grounding cues along the temporal dimension. Therefore, the sentence-video semantics can be better aligned over time to support more precise boundary predictions. Detailed experimental demonstrations will be given in Section \[ablation\_section\].
Position Prediction
-------------------
Similar to [@liu2016ssd; @lin2017single] for object/action detections, during the prediction, lower and higher temporal convolutional layers are used to localize short and long activities, respectively. As illustrated in Figure \[fig:anchor\], regarding a feature map with temporal dimension $T_k$, the basic temporal span for each feature unit within this feature map is $1/T_k$. We impose different scale ratios based on the basic span, and denote them as $r \in R = \{0.25,0,5,0.75,1.0\}$. As such, for the $i$-th feature unit of the feature map, we can compute the length of the scaled spans within it as $r/T_k$, and the center of these spans is $(i+0.5)/T_k$. For the whole feature map, there are a total number of $T_k \cdot |R|$ scaled spans within it, with each span corresponding to a candidate video segment for grounding.
[r]{}[0.45]{} {width="0.26\textheight"}
Then, we impose an additional set of convolution operations on the layer-wise temporal feature maps to predict the target video segment position. Specifically, each candidate segment will be associated with a prediction vector $p=(p^{over},\triangle c, \triangle w)$, where $p^{over}$ is the predicted overlap score between the candidate and ground-truth segment, and $\triangle c$ and $\triangle w$ are the temporal center and width offsets of the candidate segment relative to the ground-truth. Suppose that the center and width for a candidate segment are $\mu^c$ and $\mu^w$, respectively. Then the center $\phi^c$ and width $\phi^w$ of the corresponding predicted segment are therefore determined by: $$\setlength{\abovedisplayskip}{4pt}
\setlength{\belowdisplayskip}{4pt}
\phi^c = \mu^c + \alpha^c \cdot \mu^w \cdot \triangle c, \qquad
\phi^w = \mu^w \cdot exp(\alpha^w \cdot \triangle w),
\label{eq:prediction}$$ where $\alpha^c$ and $\alpha^w$ both are used for controlling the effect of location offsets to make location prediction stable, which are set as 0.1 empirically. As such, for a feature map with temporal dimension $T_k$, we can obtain a predicted segment set $\Phi_k$ = {$(p_j^{over},\phi^c_j,\phi^w_j)$}$_{j=1}^{T_k \cdot |R|}$. The total predicted segment set is therefore denoted as $\mathbf{\Phi} = \{\Phi_k \}_{k=1}^K$, where $K$ is the number of temporal feature maps.
Training and Inference
----------------------
**Training:** Our training sample consists of three elements: an input video, a sentence query, and the ground-truth segment. We treat candidate segments within different temporal feature maps as positive if their tIoUs (temporal Intersection-over-Union) with ground-truth segments are larger than 0.5. Our training objective includes an overlap prediction loss $L_{over}$ and a location prediction loss $L_{loc}$. The $L_{over}$ term is realized as a cross-entropy loss, which is defined as: $$\setlength{\abovedisplayskip}{4pt}
\setlength{\belowdisplayskip}{4pt}
\begin{split}
L_{over} = \sum_{z \in \{pos,neg\}} - \frac{1}{N_{z}} \sum_{i}^{N_{z}} g_i^{over} \log(p_i^{over}) + (1-g_i^{over})\log(1-p_i^{over}),
\end{split}
\label{eq:loss_over}$$ where $g^{over}$ is the ground-truth tIoU between the candidate and target segments, and $p^{over}$ is the predicted overlap score. The $L_{loc}$ term measures the Smooth $L_1$ loss [@girshick2015fast] for positive samples: $$\setlength{\abovedisplayskip}{3pt}
\setlength{\belowdisplayskip}{3pt}
\begin{split}
L_{loc} = \frac{1}{N_{pos}} \sum_{i}^{N_{pos}} SL_{1} (g_i^c - \phi_i^c) + SL_{1} (g_i^w - \phi_i^w),
\end{split}
\label{eq:loss_loc}$$ where $g^c$ and $g^w$ are the center and width of the ground-truth segment, respectively.
The two losses are jointly considered for training our proposed model, with $\lambda$ and $\eta$ balancing their contributions: $$\setlength{\abovedisplayskip}{3pt}
\setlength{\belowdisplayskip}{3pt}
\begin{split}
L_{all} = \lambda L_{over} + \eta L_{loc}.
\end{split}
\label{eq:loss_all}$$
**Inference:** The predicted segment set $\mathbf{\Phi}$ of different temporal granularities can be generated in one forward pass. All the predicted segments within $\mathbf{\Phi}$ will be ranked and refined with non maximum suppression (NMS) according to the predicted $p^{over}$ scores. Afterwards, the final temporal grounding result is obtained.
Experiments
===========
Datasets and Evaluation Metrics
-------------------------------
We validate the performance of our proposed model on three public datasets for the TSG task: [TACoS]{} [@Regneri2013Grounding], [Charades-STA]{} [@gao2017tall], and [AcitivtyNet Captions]{} [@krishna2017dense]. The TACoS dataset mainly contains videos depicting human’s cooking activities, while Charades-STA and ActivityNet Captions focus on more complicated human activities in daily life. For fair comparisons, we adopt “[R@n, IoU@m]{}” as our evaluation metrics as in previous works [@chen2018temporally; @gao2017tall; @zhang2018man; @wu2018multi; @liu2018attentive]. Specifically, “R@n, IoU@m” is defined as the percentage of the testing queries having at least one hitting retrieval (with IoU larger than $m$) in the top-$n$ retrieved segments.
Implementation Details
----------------------
Following the previous methods, 3D convolutional features (C3D [@tran2015learning] for TACoS and ActivityNet, and I3D [@carreira2017quo] for Charades-STA) are extracted to encode videos, with each feature representing a 1-second video clip. According to the video duration statistics, the length of input video clips is set as 1024 for both ActivityNet Captions and TACoS, and 64 for Charades-STA to accommodate the temporal convolution. Longer videos are truncated, and shorter ones are padded with zero vectors. For the design of temporal convolutional layers, 6 layers with {32, 16, 8, 4, 2, 1} temporal dimensions, 6 layers with {512, 256, 128, 64, 32, 16} temporal dimensions, and 8 layers with {512, 256, 128, 64, 32, 16, 8, 4} temporal dimensions are set for Charades-STA, TACoS, and ActivityNet Captions, respectively. All the first temporal feature maps will not be used for location prediction, because the receptive fields of the corresponding feature units are too small and are too rare to contain target activities. To save model memory footprint, the SCDM mechanism is only performed on the following temporal feature maps which directly serve for position prediction. For sentence encoding, we first embed each word in sentences with the Glove [@pennington2014glove], and then employ a Bi-directional GRU to encode the word embedding sequence. As such, words in sentences are finally represented with their corresponding GRU hidden states. Hidden dimension of the sentence Bi-directional GRU, dimension of the multimodal fused features $d_f$, and the filter number $d_h$ for temporal convolution operations are all set as 512 in this paper. The trade-off parameters of the two loss terms $\lambda$ and $\eta$ are set as 100 and 10, respectively.
Compared Methods
----------------
We compare our proposed model with the following state-of-the-art baseline methods on the TSG task. **CTRL** [@gao2017tall]: Cross-model Temporal Regression Localizer. **ACRN** [@liu2018attentive]: Attentive Cross-Model Retrieval Network. **TGN** [@chen2018temporally]: Temporal Ground-Net. **MCF** [@wu2018multi]: Multimodal Circulant Fusion. **ACL** [@ge2019mac]: Activity Concepts based Localizer. **SAP** [@chen2019SAP]: A two-stage approach based on visual concept mining. **Xu *et al.*** [@xu2019multilevel]: A two-stage method (proposal generation + proposal rerank) exploiting sentence re-construction. **MAN** [@zhang2018man]: Moment Alignment Network. We use **Ours-SCDM** to refer our temporal convolutional network coupled with the proposed SCDM mechanism.
[max width = ]{}
[ccccccccc]{} & & (lr)[2-5]{} (lr)[6-9]{} &
---------
R@1,
IoU@0.3
---------
: Performance comparisons on the TACoS and Charades-STA datasets (%).[]{data-label="tab:performance_comparison_TC"}
&
---------
R@1,
IoU@0.5
---------
: Performance comparisons on the TACoS and Charades-STA datasets (%).[]{data-label="tab:performance_comparison_TC"}
&
---------
R@5,
IoU@0.3
---------
: Performance comparisons on the TACoS and Charades-STA datasets (%).[]{data-label="tab:performance_comparison_TC"}
&
---------
R@5,
IoU@0.5
---------
: Performance comparisons on the TACoS and Charades-STA datasets (%).[]{data-label="tab:performance_comparison_TC"}
&
---------
R@1,
IoU@0.5
---------
: Performance comparisons on the TACoS and Charades-STA datasets (%).[]{data-label="tab:performance_comparison_TC"}
&
---------
R@1,
IoU@0.7
---------
: Performance comparisons on the TACoS and Charades-STA datasets (%).[]{data-label="tab:performance_comparison_TC"}
&
---------
R@5,
IoU@0.5
---------
: Performance comparisons on the TACoS and Charades-STA datasets (%).[]{data-label="tab:performance_comparison_TC"}
&
---------
R@5,
IoU@0.7
---------
: Performance comparisons on the TACoS and Charades-STA datasets (%).[]{data-label="tab:performance_comparison_TC"}
CTRL (C3D) [@gao2017tall] &18.32 &13.30 &36.69 &25.42 &23.63 &8.89 &58.92 &29.52
MCF (C3D) [@wu2018multi] &18.64 &12.53 &37.13 &24.73 &- &- &- &-
ACRN (C3D) [@liu2018attentive] &19.52 &14.62 &34.97 &24.88 &- &- &- &-
SAP (VGG) [@chen2019SAP] &- &18.24 &- &28.11 &27.42 &13.36 &66.37 &38.15
ACL (C3D) [@ge2019mac] &24.17 &20.01 &**42.15** &30.66 &30.48 &12.20 &64.84 &35.13
TGN (C3D) [@chen2018temporally] &21.77 &18.90 &39.06 &31.02 &- &- &- &-
Xu et al. (C3D) [@xu2019multilevel] &- &- &- &- &35.60 &15.80 &79.40 &45.40
MAN (I3D) [@zhang2018man] &- &- &- &- - &46.53 &22.72 &**86.23** &53.72
**Ours-SCDM** (\*) &**26.11** &**21.17** &40.16 &**32.18** &**54.44** &**33.43** &74.43 &**58.08**
\*: We adopt C3D [@tran2015learning] features to encode videos on the TACoS and ActivityNet Captions datasets, and I3D [@carreira2017quo] features on the Charades-STA dataset for fair comparisons. Video features adopted by other compared methods are indicated in brackets. VGG denotes VGG16 [@simonyan2015very] features.
[max width = ]{}
[ccccccc]{} Method &R@1,IoU@0.3 &R@1,IoU@0.5 &R@1,IoU@0.7 &R@5,IoU@0.3 &R@5,IoU@0.5 &R@5,IoU@0.7 TGN (INP\*) [@chen2018temporally] &45.51 &28.47 &- &57.32 &43.33 &-
Xu et al. (C3D) [@xu2019multilevel] &45.30 &27.70 &13.60 &75.70 &59.20 &38.30
**Ours-SCDM** (C3D) &**54.80** &**36.75** &**19.86** &**77.29** &**64.99** &**41.53**
\*: INP denotes Inception-V4 [@szegedy2017inception] features.
Performance Comparison and Analysis
-----------------------------------
Table \[tab:performance\_comparison\_TC\] and Table \[tab:performance\_comparison\_anet\] report the performance comparisons between our model and the existing methods on the aforementioned three public datasets. Overall, Ours-SCDM achieves the highest temporal sentence grounding accuracy, demonstrating the superiority of our proposed model. Notably, for localizing complex human activities in Charades-STA and ActivityNet Captions datasets, Ours-SCDM significantly outperforms the state-of-the-art methods with 10.71% and 6.26% absolute improvements in the R@1,IoU@0.7 metrics, respectively. Although Ours-SCDM achieves lower results of R@5,IoU@0.5 on the Charades-STA dataset, it is mainly due to the biased annotations in this dataset. For example, in Charades-STA, the annotated ground-truth segments are 10s on average while the video duration is only 30s on average. Randomly selecting one candidate segment can also achieve competing temporal grounding results. It indicates that the Recall values under higher IoUs are more stable and convincing even considering the dataset biases. The performance improvements under the high IoU threshold demonstrate that Ours-SCDM can generate grounded video segments of more precise boundaries. For TACoS, the cooking activities take place in the same kitchen scene with some slightly varied cooking objects (*e.g.*, chopping board, knife, and bread, as shown in the second example of Figure \[fig:quality\]). Thus, it is hard to localize such fine-grained activities. However, our proposed model still achieves the best results, except slight worse performances in R@5,IoU@0.3.
The main reasons for our proposed model outperforming the competing models lie in two folds. First, the sentence information is fully leveraged to modulate the temporal convolution processes, so as to help correlate and compose relevant video contents over time to support the temporal boundary prediction. Second, the modulation procedure dynamically evolves with different video contents in the hierarchical temporal convolution architecture, and therefore characterizes the diverse sentence-video semantic interactions of different granularities.
Ablation Studies {#ablation_section}
----------------
In this section, we perform ablation studies to examine the contributions of our proposed SCDM. Specifically, we re-train our model with the following four settings.
- **Ours-w/o-SCDM**: SCDM is replaced by the plain batch normalization [@ioffe2015batch].
- **Ours-FC**: Instead of performing SCDM, one FC layer is used to fuse each temporal feature unit with the global sentence representation $\mathbf{\overline{s}}$ after each temporal convolutional layer.
- **Ours-MUL**: Instead of performing SCDM, element-wise multiplication between each temporal feature unit and the global sentence representation $\mathbf{\overline{s}}$ is performed after each temporal convolutional layer.
- **Ours-SCM**: We use the global sentence representation $\mathbf{\overline{s}}$ to produce $\gamma^c$ and $\beta^c$ without dynamically changing these two modulation vectors with respect to different feature units.
Table \[tab:ablation\] shows the performance comparisons of our proposed full model Ours-SCDM w.r.t. these ablations on the Charades-STA dataset (please see results on the other datasets in our supplemental material). Without considering SCDM, the performance of the model Ours-w/o-SCDM degenerates dramatically. It indicates that only relying on multimodal fusion to exploit the relationship between video and sentence is not enough for the TSG task. The critical sentence semantics should be intensified to guide the temporal convolution procedure so as to better link the sentence-related video contents over time. However, roughly introducing sentence information in the temporal convolution architecture like Ours-MUL and Ours-FC does not achieve satisfying results. Recall that temporal feature maps in the proposed model are already multimodal representations since the sentence information has been integrated during the multimodal fusion process. Directly coupling the global sentence representation $\mathbf{\overline{s}}$ with temporal feature units could possibly disrupt the visual correlations and temporal dependencies of the videos, which poses a negative effect on the temporal sentence grounding performance. In contrast, the proposed SCDM mechanism modulates the temporal feature maps by manipulating their scaling and shifting parameters under the sentence guidance, which is lightweight while meticulous, and still achieves the best results.
[r]{}[0.53]{}
[ccccc]{} Method &
---------
R@1,
IoU@0.5
---------
&
---------
R@1,
IoU@0.7
---------
&
---------
R@5,
IoU@0.5
---------
&
---------
R@5,
IoU@0.7
---------
Ours-w/o-SCDM &47.52 &26.91 &69.85 &49.35
Ours-FC &46.33 &25.94 &68.96 &49.81
Ours-MUL &49.08 &28.77 &72.68 &51.02
Ours-SCM &53.07 &31.41 &71.71 &54.57
**Ours-SCDM** &**54.44** &**33.43** &**74.43** &**58.08**
In addition, comparing Ours-SCM with Ours-SCDM, we can find that dynamically changing the modulation vectors $\gamma^c$ and $\beta^c$ with respect to different temporal feature units is beneficial, with R@5,IoU@0.7 increasing from 54.57% of Ours-SCM to 58.08% of Ours-SCDM. The SCDM intensifies meaningful words and cues in sentences catering for different temporal feature units, with the motivation that different video segments may contain diverse visual contents and express different semantic meanings. Establishing the semantic interaction between these two modalities in a dynamic way can better align the semantics between sentence and diverse video contents, yielding more precise temporal boundary predictions.
[r]{}[0.53]{}
Method Run-Time Model Size Memory Footprint
-------------------------- ---------- ------------ ------------------
CTRL [@gao2017tall] 2.23s 22M 725MB
ACRN [@liu2018attentive] 4.31s 128M 8537MB
Ours-SCDM 0.78s 15M 4533MB
Model Efficiency Comparison
---------------------------
Table \[tab:table2\] shows the run-time efficiency, model size (\#param), and memory footprint of different methods. Specifically, “Run-Time” denotes the average time to localize one sentence in a given video. The methods with released codes are run with one Nvidia TITAN XP GPU. The experiments are run on the TACoS dataset since the videos in this dataset are relatively long (7 minutes on average), and are appropriate to evaluate the temporal grounding efficiency of different methods. It can be observed that ours-SCDM achieves the fastest run-time with the smallest model size. Both CTRL and ACRN methods need to sample candidate segments with various sliding windows in the videos first, and then match the input sentence with each of the segments individually. Such a two-stage architecture will inevitably influence the temporal sentence grounding efficiency, since the matching procedure through sliding window is quite time-consuming. In contrast, Ours-SCDM adopts a hierarchical convolution architecture, and naturally covers multi-scale video segments for grounding with multi-layer temporal feature maps. Thus, we only need to process the video in one pass of temporal convolution and then get the TSG results, and achieve higher efficiency. In addition, SCDM only needs to control the feature normalization parameters and is lightweight towards the overall convolution architecture. Therefore, ours-SCDM also has smaller model size.
Qualitative Results
-------------------
Some qualitative examples of our model are illustrated in Figure \[fig:quality\]. Evidently, our model can produce accurate segment boundaries for the TSG task. Moreover, we also visualize the attention weights (defined in Equation (\[eq:sentence\_attention\])) produced by SCDM when it processes different temporal units. It can be observed that different video contents attentively trigger different words in sentences so as to better align their semantics. For example, in the first example, the words “walking” and “open” obtain higher attention weights in the ground-truth segment since the described action indeed happens there. While in the other region, the word attention weights are more inclined to be an even distribution.
![ Qualitative prediction examples of our proposed model. The rows with green background show the ground-truths for the given sentence queries, and the rows with blue background show the final location prediction results. The gray histograms show the word attention weights produced by SCDM at different temporal regions.[]{data-label="fig:quality"}](main_vertical_other_new.pdf){width="1.0\columnwidth"}
![ $t$-SNE projections of temporal feature maps yielded by the models Ours-w/o-SCDM and Ours-SCDM. Each temporal feature unit within these feature maps is represented by its corresponding video clip in the original video. Video clips marked with red color are within ground-truth video segments.[]{data-label="fig:tsne"}](tsne_main.pdf){width="0.9\columnwidth"}
In order to gain more insights of our proposed SCDM mechanism, we visualize the temporal feature maps produced by the variant model Ours-w/o-SCDM and the full-model Ours-SCDM. For both of the trained models, we extract their temporal feature maps, and subsequently apply $t$-SNE [@maaten2008visualizing] to each temporal feature unit within these maps. Since each temporal feature unit corresponds to one specific location in the original video, we then assign the corresponding video clips to the positions of these feature units in the $t$-SNE embedded space. As illustrated in Figure \[fig:tsne\], temporal feature maps of two testing videos are visualized, where the video clips marked with red color denote the ground-truth segments of the given sentence queries. Interestingly, it can be observed that through SCDM processing, video clips within ground-truth segments are more tightly grouped together. In contrast, the clips without SCDM processing are separated in the learned feature space. This demonstrates that SCDM successfully associates the sentence-related video contents according to the sentence semantics, which is beneficial to the later temporal boundary predictions. More visualization results are provided in the supplemental material.
Conclusion
==========
In this paper, we proposed a novel semantic conditioned dynamic modulation mechanism for tackling the TSG task. The proposed SCDM leverages the sentence semantics to modulate the temporal convolution operations to better correlate and compose the sentence-related video contents over time. As SCDM dynamically evolves with the diverse video contents of different temporal granularities in the temporal convolution architecture, the sentence described video contents are tightly correlated and composed, leading to more accurate temporal boundary predictions. The experimental results obtained on three widely-used datasets further demonstrate the superiority of the proposed SCDM on the TSG task.
Acknowledgement
===============
This work was supported by National Natural Science Foundation of China Major Project No.U1611461 and Shenzhen Nanshan District Ling-Hang Team Grant under No.LHTD20170005.
[10]{}
Joao Carreira and Andrew Zisserman. Quo vadis, action recognition? a new model and the kinetics dataset. In [*Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition*]{}, pages 6299–6308, 2017.
Jingyuan Chen, Xinpeng Chen, Lin Ma, Zequn Jie, and Tat-Seng Chua. Temporally grounding natural sentence in video. In [*Proceedings of the 2018 Conference on Empirical Methods in Natural Language Processing*]{}, pages 162–171, 2018.
Jingyuan Chen, Lin Ma, Xinpeng Chen, Zequn Jie, and Jiebo Luo. Localizing natural language in videos. In [*Thirty-Third AAAI Conference on Artificial Intelligence*]{}, 2019.
Shaoxiang Chen and Yu-Gang Jiang. Semantic proposal for activity localization in videos via sentence query. In [*AAAI Conference on Artificial Intelligence*]{}. IEEE, 2019.
Zhenfang Chen, Lin Ma, Wenhan Luo, and Kwan-Yee K Wong. Weakly-supervised spatio-temporally grounding natural sentence in video. In [*The 57th Annual Meeting of the Association for Computational Linguistics (ACL)*]{}, 2019.
Harm de [Vries]{}, Florian [Strub]{}, Jeremie [Mary]{}, Hugo [Larochelle]{}, Olivier [Pietquin]{}, and Aaron C. [Courville]{}. Modulating early visual processing by language. , pages 6594–6604, 2017.
Vincent [Dumoulin]{}, Jonathon [Shlens]{}, and Manjunath [Kudlur]{}. A learned representation for artistic style. , 2017.
Yang Feng, Lin Ma, Wei Liu, and Jiebo Luo. Spatio-temporal video re-localization by warp lstm. In [*Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition*]{}, pages 1288–1297, 2019.
Yang Feng, Lin Ma, Wei Liu, Tong Zhang, and Jiebo Luo. Video re-localization. In [*Proceedings of the European Conference on Computer Vision (ECCV)*]{}, pages 51–66, 2018.
Jiyang Gao, Chen Sun, Zhenheng Yang, and Ram Nevatia. Tall: Temporal activity localization via language query. In [*Proceedings of the IEEE International Conference on Computer Vision*]{}, pages 5267–5275, 2017.
Kirill Gavrilyuk, Amir Ghodrati, Zhenyang Li, and Cees G. M. Snoek. Actor and action video segmentation from a sentence. In [*Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition*]{}, pages 5958–5966, 2018.
Runzhou Ge, Jiyang Gao, Kan Chen, and Ram Nevatia. Mac: Mining activity concepts for language-based temporal localization. In [*2019 IEEE Winter Conference on Applications of Computer Vision*]{}, pages 245–253. IEEE, 2019.
Ross Girshick. Fast r-cnn. In [*Proceedings of the IEEE International Conference on Computer Vision*]{}, pages 1440–1448, 2015.
Lisa Anne Hendricks, Oliver Wang, Eli Shechtman, Josef Sivic, Trevor Darrell, and Bryan Russell. Localizing moments in video with natural language. In [*Proceedings of the IEEE International Conference on Computer Vision*]{}, 2017.
Sergey [Ioffe]{} and Christian [Szegedy]{}. Batch normalization: Accelerating deep network training by reducing internal covariate shift. In [*Proceedings of the International Conference on Machine Learning*]{}, pages 448–456, 2015.
Thomas N. [Kipf]{} and Max [Welling]{}. Semi-supervised classification with graph convolutional networks. , 2017.
Ranjay [Krishna]{}, Kenji [Hata]{}, Frederic [Ren]{}, Li [Fei-Fei]{}, and Juan Carlos [Niebles]{}. Dense-captioning events in videos. In [*Proceedings of the IEEE International Conference on Computer Vision*]{}, pages 706–715, 2017.
Tianwei [Lin]{}, Xu [Zhao]{}, and Zheng [Shou]{}. Single shot temporal action detection. In [*Proceedings of the 25th ACM international conference on Multimedia*]{}, pages 988–996, 2017.
Meng Liu, Xiang Wang, Liqiang Nie, Xiangnan He, Baoquan Chen, and Tat-Seng Chua. Attentive moment retrieval in videos. In [*The 41st International ACM SIGIR Conference on Research & Development in Information Retrieval*]{}, pages 15–24. ACM, 2018.
Wei [Liu]{}, Dragomir [Anguelov]{}, Dumitru [Erhan]{}, Christian [Szegedy]{}, Scott E. [Reed]{}, Cheng-Yang [Fu]{}, and Alexander C. [Berg]{}. Ssd: Single shot multibox detector. , pages 21–37, 2016.
Yuan Liu, Lin Ma, Yifeng Zhang, Wei Liu, and Shih-Fu Chang. Multi-granularity generator for temporal action proposal. In [*Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition*]{}, pages 3604–3613, 2019.
Laurens van der Maaten and Geoffrey Hinton. Visualizing data using t-sne. , 9(Nov):2579–2605, 2008.
Jeffrey Pennington, Richard Socher, and Christopher Manning. Glove: Global vectors for word representation. In [*Proceedings of the 2014 conference on empirical methods in natural language processing*]{}, pages 1532–1543, 2014.
Michaela Regneri, Marcus Rohrbach, Dominikus Wetzel, Stefan Thater, Bernt Schiele, and Manfred Pinkal. Grounding action descriptions in videos. , 1:25–36, 2013.
Karen Simonyan and Andrew Zisserman. Very deep convolutional networks for large-scale image recognition. In [*ICLR 2015 : International Conference on Learning Representations 2015*]{}, 2015.
Bharat Singh, Tim K Marks, Michael Jones, Oncel Tuzel, and Ming Shao. A multi-stream bi-directional recurrent neural network for fine-grained action detection. In [*Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition*]{}, pages 1961–1970, 2016.
Christian Szegedy, Sergey Ioffe, Vincent Vanhoucke, and Alexander A Alemi. Inception-v4, inception-resnet and the impact of residual connections on learning. In [*Thirty-First AAAI Conference on Artificial Intelligence*]{}, 2017.
Du Tran, Lubomir Bourdev, Rob Fergus, Lorenzo Torresani, and Manohar Paluri. Learning spatiotemporal features with 3d convolutional networks. In [*Proceedings of the IEEE International Conference on Computer Vision*]{}, pages 4489–4497, 2015.
Du Tran, Heng Wang, Lorenzo Torresani, Jamie Ray, Yann Lecun, and Manohar Paluri. A closer look at spatiotemporal convolutions for action recognition. In [*Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition*]{}, pages 6450–6459, 2018.
Limin Wang, Yu Qiao, and Xiaoou Tang. Action recognition and detection by combining motion and appearance features. , 1(2):2, 2014.
Shuohang Wang and Jing Jiang. Machine comprehension using match-lstm and answer pointer. , 2016.
Aming [Wu]{} and Yahong [Han]{}. Multi-modal circulant fusion for video-to-language and backward. In [*International Joint Conference on Artificial Intelligence*]{}, pages 1029–1035, 2018.
Huijuan Xu, Kun He, L Sigal, S Sclaroff, and K Saenko. Multilevel language and vision integration for text-to-clip retrieval. In [*AAAI Conference on Artificial Intelligence*]{}, volume 2, page 7, 2019.
Jun Yuan, Bingbing Ni, Xiaokang Yang, and Ashraf A Kassim. Temporal action localization with pyramid of score distribution features. In [*Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition*]{}, pages 3093–3102, 2016.
Yitian Yuan, Lin Ma, and Wenwu Zhu. Sentence specified dynamic video thumbnail generation. In [*27th ACM International Conference on Multimedia.*]{}, 2019.
Da Zhang, Xiyang Dai, Xin Wang, Yuan-Fang Wang, and Larry S Davis. Man: Moment alignment network for natural language moment retrieval via iterative graph adjustment. , 2018.
[^1]: This work was done while Yitian Yuan was a Research Intern at Tencent AI Lab.
|
---
abstract: 'We introduce *random directed acyclic graph* and use it to model the information diffusion network. Subsequently, we analyze the *cascade generation model* (CGM) introduced by Leskovec et al. [@leskovec-blog]. Until now only empirical studies of this model were done. In this paper, we present the first theoretical proof that the sizes of cascades generated by the CGM follow the power-law distribution, which is consistent with multiple empirical analysis of the large social networks. We compared the assumptions of our model with the Twitter social network and tested the goodness of approximation.'
author:
- |
Karol Wgrzycki\
\
Piotr Sankowski\
\
- |
Andrzej Pacuk\
\
Piotr Wygocki\
\
bibliography:
- 'sigproc.bib'
title: 'Why Do Cascade Sizes Follow a Power-Law?'
---
Introduction
============
Each day billions of instant messages, comments, articles, blog posts, emails, tweets and other various mediums of communication are exchanged in the reciprocal, social relations. The study of the information propagation through network is more and more demanded. Such models of propagation are used to minimize transmission costs, enhance the security and prevent information leaks or predict a propagation of malicious software among the users [@moro].
When considering state-of-the-art models of information diffusion, the underlying network structure of a transmission is constructed based on the known connections (e.g., the graph of followers in the Twitter network). Here, we have discovered that the graph of information dissemination has noteworthy features, unexploited in the previous works. It has been well known that more active individuals in the network have more acquaintances [@seismic]. We have conducted experiments confirming those observations in the information diffusion network and showed its underlying structure.
Our model of the information diffusion network explains power-law (or Pareto) distribution of the number of informed nodes (cascade size). This is a major improvement over the state-of-the-art models, which give predictions inconsistent with the real data [@cikm2014]. Random power-law graphs may sufficiently describe the follower-followee relation in the social network [@brach], but these graphs may not necessarily characterize the medium of an information propagation. The results of our study will allow researchers to enhance their models of the information transmission and will enable them to develop a framework to validate these models.
Related Work
------------
Up to the best of our knowledge this work presents the first theoretical analysis of the cascade size distribution using *cascade generation* to model the spread of the information in the social networks. The first experimental analysis has been conducted by [Leskovec et al. [@leskovec-blog]]{}. They proposed a cascade generation model and simulated it on the dataset of blog links. Since then, the research on the cascade sizes has become a fruitful field (for more references see [@rogers-diffusion]).
Through study on an epidemiology and a solid-state physics, different models such as SIR (susceptible-infectious-recovered) or SIS (susceptible-infectious-susceptible) has been employed to model the dynamics of spread of the information. However, all of these models assume that everyone in the population is in contact with everyone else [@bayley], which is unrealistic in large social networks.
The classical example of the modified spreading process incorporates the effect of a *stifler* [@wlosi]. *Stiflers* never spread the information even if they were exposed to it multiple times. Nevertheless, *stiflers* can actively convert other spreaders or susceptible nodes into *stiflers*. That complicated logic may lead to the elimination of the epidemic threshold [^1] and has been actively developed [@brach2].
In 2002 [Watts [@watts]]{} proposed exact solution of the global cascade sizes on an arbitrary random graph. Notwithstanding, this process of the information propagation called the threshold model is utterly different from *cascade generation* by [Leskovec et al. [@leskovec-blog]]{} and does not fully explain the dynamics of modern social networks like the Twitter.
[Iribarren et al. [@moro]]{} have developed the similar model, where an integro-differential equations have been introduced. That equations describe the cascade sizes when the number of messages send by a node is described by the Harris discrete distribution.However, the general solution to their equations is not known and merely solutions for nontrivial cases (e.g., superexponential processes [@moro]) has been considered. Our discoveries provide much simpler method and lucidly explain, that the underlying social graph is far more complex than just the graph of followers.
Results similar to ours were also obtained in the study on the bias of traceroute sampling. [Achlioptas et al. [@traceroute1]]{} characterize the degree distribution of a BFS tree for a random graph with a given degree distribution. Their explanation why the degree distribution under traceroute sampling exhibits power-law motivated researchers to study bias in P2P systems [@traceroute3] and network discovery [@traceroute4]. Their research also resulted in the development of the new tools in the social networks sampling [@traceroute2].
In the seminal paper of [Leskovec et al. [@leskovec-dvd]]{}, the cascade size distribution in the network of recommendation has been analyzed. [Leskovec et al. [@leskovec-dvd]]{} showed that the product purchases follow a “long tail” distribution, where a significant fraction of sold items were rarely sold items. They fit the data to the power-law distribution and discovered, that the parameters may differ for distinct networks (remarkably, the power-law exponent was close to $-1$ for DVD recommendation cascades).
One of the greatest issue in analyzing the cascade size distribution is the lack of a good theoretical background for this process. Recently, researchers [@hypertext2016] has presented new models, designed to fit real distribution of cascade sizes. Moreover, goodness of fit test against CGM model has been conducted. It turns out, that CGM works well for small cascades, however it is unreliable for the large ones. The introduction of time dependent parameters significantly improves predictions for the large cascades [@hypertext2016].
In future, theoretical studies on cascades size distribution could explain phenomena in Gossip-based routing [@iwanicki], rumor virality and influence maximization [@max-influence], recurrence of the cascades [@recur] or assist in forecasting rumors [@forecast]. Currently the research in this area is purely empirical and we need more theoretical models to understand the process of information propagation [@hypertext2016].
Modeling Information Cascades {#model}
=============================
Intuitively, information cascades are generated by the following process: one individual passes the information to all its acquaintances. Then in each round newly informed nodes randomly decide to pass it to their acquaintances. This process continues until no new individuals are informed. The graph generated by the spread of the information is called the cascade.
The *cascade generation model* (CGM) established by [@leskovec-blog] introduces a single parameter $\alpha$ that measures how infectious a passed information is. More precisely, $\alpha$ is the probability that the information will be passed to the acquaintance.
According to [Leskovec et al. [@leskovec-blog]]{}, the cascade is generated by the following:
1. Uniformly at random pick a starting point of the cascade and add it to the set of *newly informed* nodes.
2. Every *newly informed* node independently with the probability $\alpha$ informs their direct neighbors.
3. Let *newly informed* be the set of nodes that has been informed for the first time in step $2$ and add them to the generated cascade.
4. Repeat steps $2$ and $3$ until *newly informed* set is empty.
In this model we assume that all nodes have an identical impact ($\alpha = {\ensuremath{\mathrm{const}}}$) on their neighbors and all generated cascades are trees, since we pick a single, initial node. It is not a major problem, since the most of cascades are trees [@leskovec-blog].
![The log-log plot of the cascade size distribution on the Twitter dataset. It follows a power-law with an exponent $-2.3$.[]{data-label="fig:twitter_reach_distribution"}](twitter_reach_distribution.pdf){width="50.00000%"}
![The log-log plot of the cascade size distribution predicted by the simulation of CGM using the Twitter followers graph. Note the phase transition, absent in real data.[]{data-label="fig:model_const_distribution"}](model_const_distribution.pdf){width="50.00000%"}
Recently, information diffusion models have been evaluated on large social networks and it has been observed [@greg; @cikm2014] that the actual cascade size distribution is inconsistent with the distribution predicted by the model. First, the simulation registered an obvious phase transition (the cascades are either extremely large or are smaller than $100$) (see Figure \[fig:model\_const\_distribution\]). No such gap has been observed in the real data (see Figure \[fig:twitter\_reach\_distribution\]). Second, the probability of the large cascade is intolerably high in simulations. [^2]
There has been many attempts to readjust the *cascade generation model* [@ghosh; @cikm2014; @lerman], but as we have discussed above, those attempts oversimplify the process or incorrectly describe the distribution of cascades sizes. We introduce novel observations concerning the underlying social network and based on them we introduce a theoretical model of information propagation.
Information Transmission
------------------------
The underlying network of rumor spreading is unknown. Even if we would have the network of all social contacts, the rumor could propagate through the mass media with a random interaction or even evolve in time. Because of that, when analyzing the social media we focus on the network of information propagation. The same technique has been used by [Leskovec et al. [@leskovec-blog]]{}, but their network was generated by links in blogs. To observe a non-trivial structure of the social network we considered actions generated by replying to messages. Such replies in the Twitter microblogging network are called *retweets*. We analyzed a set of over 500 millions tweets and the retweets from a $10\%$ sample of all tweets from May 19 to May 30 2013. Each retweet contains identifiers of the cited and replying users. Based on that, we generate a directed graph of the information transmission (same as [@seismic; @cikm2014]). We use data published in [@hypertext2016; @twitter-data] and publish our code and experimental results on [@rdag-code].
It is well known that the degree distribution of the generated graph follows a power-law [@brach]. However, this characterization does not necessarily describe the network of the information transmission. The intuition is that the nodes with greater degree are more active. Hence, when information spreads through the underlying network the distribution of spreading nodes should prefer nodes with higher degree, in consequence inflating the probability for these nodes.
Moreover, we have observed the hierarchical structure of the graph of retweets. The probability that a popular blogger replies to the message of an unpopular one is extremely small.
To confirm our intuition, we have determined the distribution of neighboring degrees for all nodes with a given degree. As shown on Figure \[fig:twitter\_retweets\_followers\_degrees\] each degree has a distinct distribution of neighbors’ degrees (implementation is available on [@rdag-code]). It means that it is more likely that a node is followed by some popular nodes when node itself is popular. Moreover, there is a pattern: the probability decreases with degree (for followers with degree greater than the followed node). Based on this observation we will model the aforementioned distribution as an approximated step function. This observation is consistent with the state-of-the-art analysis [@leskovec-blog] and the most of cascades are “tree-like” or “stars”. According to our knowledge no further research has been conducted for examining distributions of neighbors degrees for node with a given degree (only cumulative degree for every cascade has been studied [@leskovec-blog]).
![The degree distribution of the followers aggregated by the followed degree. Note the inflection point increases with a followed degree.[]{data-label="fig:twitter_retweets_followers_degrees"}](followed_follower_degree_twitter_retweets_graph.pdf){width="50.00000%"}
Undoubtedly, the process of information transmission in social networks is far more complex to be modeled by simulations just on the random power-law graphs. Because of the hierarchical structure and the relation of activity with node degree effects, we propose the model where the information is spread only to the nodes with lower degree with approximately uniform distribution.
Random DAG Generator
--------------------
One of the basic methods for generating random graphs has been introduced in 1960 by [Erdős and Rényi]{} [@erdos]. In a nutshell: for a given set of vertices, all edges have the same probability of being present or absent in a graph. This model of the graph is not suited for modeling the social networks because its degree distribution does not follow a power-law. The distribution of degrees for [Erdős-Rényi ]{}model is binomial.
![A diffusion network generated by the random DAG algorithm (see Algorithm \[generate\_dag\]).[]{data-label="fig:sample_graph"}](sample_graph.pdf){width="50.00000%"}
This model is used in the theoretical research for modeling interactions between networks and propagation of catastrophes [@catastrofic]. Even though [Erdős-Rényi ]{}model does not characterize connections between nodes, we believe it can represent the process of information spreading. We will propose the intuitive variation of [Erdős-Rényi ]{}model for directed graphs.
According to [Leskovec et al. [@leskovec-blog]]{} the cascades very rarely express cycles and can be modeled as a tree-like structure. Even though this graphs do not model relationships in the social network, the directed acyclic graphs (DAG) are appropriate structure of the information propagation in the social network.
We introduce the procedure that generates the random directed acyclic graphs (DAG) and prove that propagating information in CGM regime results in cascade sizes obeying the power-law.
Let us denote by a random graph generated by the <span style="font-variant:small-caps;">RandomDAG($n,p$)</span> (see Algorithm \[generate\_dag\]).
$G \gets \text{empty graph with vertices } 1,2, \ldots, n$ with probability $p$ add directed edge $(j,i)$ to $G$ **return** $G$
The final $n$-vertices graph is acyclic since all edges $(j,i)$ obey $j>i$ (see Figure \[fig:sample\_random\_dag\]).
Any DAG can be generated by the <span style="font-variant:small-caps;">RandomDAG($n,p$)</span> when $p \in (0,1)$. If we label all vertices from $n$ to $1$ in the topological order, the graph $G$ will consist only of edges $(j,i)$ that $j > i$. Finally, all edges obeying $j > i$ can be present in the graph with independent probability $p$.
\(1) at (-3, 0) [ 1 ]{}; (2) at (-1.5, 0) [ 2 ]{}; (3) at ( 0, 0) [ 3 ]{}; (4) at ( 1.5, 0) [ 4 ]{}; (5) at ( 3, 0) [ 5 ]{};
\(1) edge (2); (2) edge (3); (3) edge (4); (4) edge (5);
\(1) edge \[bend left\] (4); (2) edge \[bend left\] (4); (3) edge \[bend left\] (5);
\(1) edge \[bend right\] (5); (1) edge \[bend right\] (3); (2) edge \[bend right\] (5);
[Leskovec et al. [@leskovec-blog]]{} suggested that in the real information diffusion network, the small and simple graphs will occur more often than the complex, non trivial DAGs. This is exactly the case in <span style="font-variant:small-caps;">RandomDAG($n,p$)</span>.
### Degree Distribution of a Random DAG
The distribution of in-degrees in the satisfies:
$$\label{formula_rdag_degrees}
{\mathbb{P}\left[ \mathrm{indeg}(v) = k \right]} =
\frac{1}{n} \sum^{n-1}_{i=k} \binom{i}{k} p^k (1-p)^{i-k}
.$$
Similarly to the [Erdős-Rényi ]{}graph, the in-degree distribution of a given vertex $i$ is binomial but with different parameters for each node. As a consequence, it leads to a step-function like shape (see Figure \[fig:random\_degree\_dist\]).
![The in-degree distribution of the graphs, according to Formula .[]{data-label="fig:random_degree_dist"}](model_random_degree_plots.pdf){width="50.00000%"}
The in-degree distribution is almost uniform when the in-degree is lower than $n p$. Naturally, this is merely a simple approximation of the true network of a probable information transmission. Still, based on observations in Section \[information-transmission\] is the more accurate model than the follower-followee graph.
### Comparison with [Erdős-Rényi ]{}Model
The random DAG, in contrast to the standard [Erdős and Rényi]{} graph is a directed graph. It means that it can model one-way communication and the distribution of degrees. Next key difference is the hierarchical structure (i.e., the node $n$ will not follow the node with the lower label). These differences enable the random DAG model to produce cascades with the power-law distribution of sizes.
Analysis
========
In this section, we will formally analyze the introduced model. Subsequently, we will quantitatively describe a process of the information diffusion on a random DAG by determining the cascade size distribution.
Recall $p$ to be the probability of an edge in a ${\mbox{\textsf{rdag}$(n, p)$}}$ and $\alpha$ to be the average infectiousness of an information. Then set $\beta = 1-p\alpha$ for simplicity. Hence, $\beta$ is the probability that the informed user will not spread the information through a given edge.
Now we will determine the probability $P_{n,k}$, that the cascade reaches $k$ distinct vertices in a graph with $n$ nodes, commencing from the vertex no. $1$. Clearly $P_{1,1} = 1$ and $P_{n,k} = 0$ when $k > n$ or $k \le 0$. For a remaining case assume the cascade has size $k$ and consider two distinct states of the $n$-th node:
- $n$ was . Then, at least one of the other $k-1$ informed nodes had passed it to the $n$-th node, so the probability is $(1-\beta^{k-1}) \cdot P_{n-1,k-1}$.
- $n$ was . It can happen only when none of the other $k$ informed nodes passed the information to $n$-th node. The probability of such event equals $\beta^k \cdot P_{n-1,k}$.
Hence, we obtain a formula when $1 \le k \le n$:
$$P_{n,k} = \beta^k \cdot P_{n-1,k} + (1-\beta^{k-1}) \cdot P_{n-1,k-1},$$
To determine the distribution of the cascade size, we assume that an information shall commence in any node with an equal probability. Because the process of propagating the information in ${\mbox{\textsf{rdag}$(n, p)$}}$ starting from node $1$ is identical to propagating it from node $i$ in ${\mbox{\textsf{rdag}$(n+i-1, p)$}}$, the distribution is:
$${\mathbb{P}\left[ |\emph{Informed}| = k \right]} = S_{n,k} = \frac{1}{n} \sum_{i=1}^{n} P_{i,k}.$$
Now, we have the exact equation for the cascade size distribution. This equation does not have a simple form. However, we can ask what will happen when the number of nodes in graph is large. Let us recall that two series $x_n, y_n$ are *asymptotically equivalent* when: $$x_n \sim y_n \text{ iff } \lim_{n \to \infty}\frac{x_n}{y_n} = 1.$$
The cascade size distribution satisfies Theorem \[asymptotic-cascade\].
$$\label{formula_s_nk}
S_{n,k} \sim \frac{1}{n(1-\beta^k)}$$
\[asymptotic-cascade\]
Let us denote $\widetilde{S}_{n,k} = n S_{n,k}$. We need to prove that: $$\label{proveit}
A_k := \lim_{n \to \infty} \widetilde{S}_{n,k} = \frac{1}{1-\beta^k}
.$$
We will prove it by induction. For $k=1$:
$$\begin{aligned}
\widetilde{S}_{n,1} & = & \sum_{i=1}^n P_{i,1} = P_{1,1} + \sum^{n-1}_{i=1} P_{i+1,1}
= 1 + \sum^{n-1}_{i=1} \beta P_{i,1} \\
& = & 1 + \beta \widetilde{S}_{n-1,1} = 1 + \beta + \beta^2 + \ldots +
\beta^{n-1}
.\end{aligned}$$
Hence, $\widetilde{S}_{n,1}$ is the sum of the geometric series:
$$\widetilde{S}_{n,1} = \frac{1-\beta^n}{1-\beta} \to \frac{1}{1-\beta}
.$$
For $k > 1$: $$\widetilde{S}_{n,k} = \sum_{i=1}^n P_{i,k} = \sum_{i=1}^n \beta^k P_{i-1,k} + \sum_{i=1}^n (1-\beta^{k-1}) P_{i-1,k-1}
.$$
So $\widetilde{S}_{n,k}$ obeys the recursive formula:
$$\label{eq1}
\widetilde{S}_{n,k} = \beta^k \widetilde{S}_{n-1,k} + (1-\beta^{k-1}) \widetilde{S}_{n-1,k-1}
.$$
Technical induction shows, that the $\widetilde{S}_{n,k}$ is bounded and increasing in respect to $n$, so $A_k = \lim_{n
\to \infty} \widetilde{S}_{n,k}$ exists. Hence, we can take a limit on both sides of Equation \[eq1\] and obtain:
$$A_k = \beta^k A_k + (1-\beta^{k-1}) A_{k-1}
,$$
$$\label{Ak_recursive}
A_k = \frac{1-\beta^{k-1}}{1-\beta^k} A_{k-1}
.$$
Finally, by unwinding the recursive Formula , for each $k>0$ we obtain (recall that $A_1 = \frac{1}{1-\beta}$): $$A_k = \frac{1-\beta^{k-1}}{1-\beta^k}
\frac{1-\beta^{k-2}}{1-\beta^{k-1}}
\cdots \frac{1}{1-\beta}
= \frac{1}{1-\beta^k}.$$
Hence, we have proved an asymptotic Formula of the cascade size distribution.
Approximation for the Large Networks
------------------------------------
Recall, that $\beta = 1 - p\alpha = 1 - \epsilon$. Because $p$ and $\alpha$ are extremely small, $\beta$ is close to 1. Taking the Laurent series of our function we get:
$$\frac{1}{1-(1-\epsilon)^k} = \frac{1}{k\epsilon} + \frac{k-1}{2k} + O(\epsilon)
.$$
The social networks have an extremely large number of nodes (e.g., the Twitter network has about $300$ millions distinct users [@brach]). On the other hand, new information reaches only few nodes (the cascade size distribution is believed to be a power-law for $k$ smaller than $10\,000$) [@cikm2014; @leskovec-blog]. Then, because the element $\frac{k-1}{2 k n}$ is insignificant when $n$ is that large, for $k \ll \frac{1}{p\alpha} \ll n$ we get:
$$S_{n,k} \sim \frac{1}{n(1-(1-p\alpha)^k)} = \frac{1}{k n p \alpha} +
\frac{k-1}{2k n} + O(\frac{p \alpha}{n})$$
Hence, the distribution of cascade size:
$$\label{exact-formula}
{\mathbb{P}\left[ |\mathrm{Informed}| = k \right]} \approx \frac{1}{n p \alpha } k^{-1} + \mathrm{const}$$
in the first-order perturbation follows the power-law. Due to the low number of the large cascades, the distribution of sizes is unknown for $k$ close to $\frac{1}{p\alpha}$. In that case, one should use an exact formula.
Goodness of the Approximation
-----------------------------
On the Figure \[fig:analysis\] we have presented a comparison between approximation and exact formula for the distribution of cascade sizes. For a relatively small cascade size $k$ the slope of a distribution matches ideally.
![The log-log plot of an exact formula $S_{n,k}$, asymptotic bound $\frac{1}{n(1-\beta^k)}$ and the power-law distribution with exponent $-1$. []{data-label="fig:analysis"}](analisys_snk_power_law.pdf){width="50.00000%"}
How large cascades can we model using the aforementioned assumptions? The number of nodes $n$ in the Twitter network is approximately 300 millions. In [@seismic] the average infectiousness $\alpha$ of the information on Twitter is said to be of order of $0.01$. According to [Leskovec et al. [@leskovec-blog]]{}, the number of edges in a cascade is proportional to $n^{1.03}$. So the parameter $p$ should be approximately $p \propto \frac{n^{1.03}}{n^2}$, since the number of possible edges is $n^2$. The largest rumor in our dataset has roughly $70\,000$ informed nodes, hence the lower bound for parameter $p$ is of order of $\frac{(7\cdot10^4)^{1.03}}{(7\cdot10^4)^2} \approx 2\cdot10^{-5}$.
Still, $\frac{1}{\alpha p} \approx 5 \cdot 10^6 \ll 3 \cdot 10^8 \approx n$. So, for the Twitter network we can model the cascades with sizes $k \ll 5 \cdot 10^6$. Remarkably, it is enough, since approximately $10^{-5}$ of Twitter rumors have the size greater than $10^4$.
### Goodness of Fit
K-S test comparing the power-law distribution and the real cascade size distribution [@twitter-data] is $0.0145$. At [@hypertext2016], the the K-S test comparing the cascade size distribution on the variant of CGM with real cascade distribution is $0.0447$. It means, that our model can potentially improve the test value by $3\%$ in comparison to the CGM on the graph of retweets.
Other Schemes of Information Diffusion
--------------------------------------
One would state that *cascade generation model* is counterintuitive for microblogging services such as the Twitter. In fact, it is more intuitive that every follower of the spreader eventually will be informed. So every follower, after being informed for the first time ought to make exactly one decision (with probability $\alpha$): whether to pass the information to all of its acquaintances simultaneously (previously the information was passed to each of its followers independently and each follower may had multiple opportunities to become a spreader). In such a case, the Formula is exactly the same (for further proof see Appendix \[model-many\]).
Conclusion and Future Work {#conclusion}
==========================
The graph of the information diffusion is utterly different from the global network of social connections. In contrast to multiple previous approaches, we model the cascade of information propagation as the *random directed acyclic graph* and we show that in the scheme of CGM the distribution of information popularity is asymptotically equivalent to:
$${\mathbb{P}\left[ |\textit{Informed}| = k \right]} \sim \frac{1}{n(1-\beta^k)}
,$$
where $n$ is the number of nodes and $\beta$ is a parameter dependent on both infectiousness of average information and density of the cascade. We show that for a sufficiently big number of nodes this distribution follows the power-law, what is consistent with real world observations. We hope that an introduction of this framework will inspire the theoretical affords to model and describe the information diffusion in the large social networks. The analysis of the graph showed that the cascade size distribution follows the power-law ${\mathbb{P}\left[ |Informed|=k \right]} \propto
k^\gamma$ for $\gamma=-1$. [Leskovec et al. [@leskovec-dvd]]{} presented the rich family of data with power-law exponent close to $-1$. [Leskovec et al. [@leskovec-dvd]]{} suggest that the information propagation in the network of recommendations may produce cascades with desired exponent. However, in the real data, the parameter $\gamma$ can be completely different (e.g., see Figure \[fig:twitter\_reach\_distribution\] of the Twitter cascade size distribution with power-law exponent $\gamma=-2.3$). To adopt our model to those cases, one can customize a distribution of random cascades (we assumed a fairly simple method to generate them). We believe that adaptation of cascade shape distribution to the experimental data will readjust the $\gamma$ parameter (for a start one can use the shape distribution provided by [Leskovec et al. [@leskovec-blog]]{}). Further enhancements might also be achieved by adapting the information diffusion scheme to a particular society (similarly to Appendix \[model-many\]). To encourage other researchers to apply our model in practice, we publish the code used to generate all figures and the results on [@rdag-code].
Still, we need more experiments and research to answer what type of social networks the random directed acyclic graphs model and how richer set of features (e.g., spatio-temporal features) influences the cascade size distribution.
Acknowledgments
===============
This work was partially supported by NCN grant UMO-2014/13/B/ST6/01811, ERC project PAAl-POC 680912, ERC project TOTAL 677651, FET IP project MULTIPLEX 317532 and polish funds for years 2013-2016 for co-financed international projects.
Dependant Passing of the Information {#model-many}
====================================
In this model we count spreaders who pass the information to all of theirs followers with probability $\alpha$. These followers will receive the information and might become new spreaders.
For clarity, we will use the notation from Section \[analysis\] and the proof of Theorem \[formula\_s\_nk\].
Analogously to previous analysis, we obtain the recursive formula: $$\begin{aligned}
P_{1,1} &=& \alpha, \\
P_{n,k} &=& 0, \text{ when } k > n, \\
P_{n,k} &=& \big(1-(1-\beta^k)\alpha\big) \cdot P_{n-1,k} + \alpha (1 - \beta^{k-1})
\cdot P_{n-1,k-1} .\end{aligned}$$
Rest of the proof is almost identical to the proof of Theorem \[formula\_s\_nk\].
For $k=1$, we have: $$\lim_{n \to \infty} \widetilde{S}_{n,1} =
\lim_{n \to \infty} \frac{1-\big(1-(1-\beta^k)\alpha\big)^n}{1-\beta} =
\frac{1}{1-\beta}
.$$
For $k>1$, when $n \to \infty$: $$\begin{aligned}
\lim_{n \to \infty} \widetilde{S}_{n,k} = A_k
& = & \big(1-(1-\beta^k)\alpha\big)
A_k + \\
& & + \alpha (1-\beta^{k-1}) A_{k-1}
.\end{aligned}$$
By subtracting the expression on both sides:
$$A_k - A_k (1 - (1-\beta^k)\alpha) = A_{k-1} \alpha (1-\beta^{k-1})
.$$
And after simplification we get:
$$A_k (1-\beta^k)\alpha = A_{k-1} (1-\beta^{k-1}) \alpha$$
Hence, we have obtained the same formula as in Theorem \[formula\_s\_nk\]: $$A_k = A_{k-1} \frac{1-\beta^k}{1-\beta^{k-1}}$$
and finally obtain: $$\lim_{n\to\infty} \widetilde{S}_{n,k} = \frac{1}{1-\beta^k}
.$$
[^1]: Epidemic threshold determines whether the global epidemic occurs or the disease simply dies out.
[^2]: According to [@cikm2014] the probability that the cascade will have a size greater than $2\,500$ is $0.00008$ for the real data and $0.0134$ for a simulation.
|
---
abstract: 'Starting with a short map $f_0:I\to \mathbb R^3$ on the unit interval $I$, we construct random isometric map $f_n:I\to \mathbb R^3$ (with respect to some fixed Riemannian metrics) for each positive integer $n$, such that the difference $(f_n - f_0)$ goes to zero in the $C^0$ norm. The construction of $f_n$ uses the Nash twist. We show that the distribution of $ n^{3/2} (f_n -f_0)$ converges (weakly) to a Gaussian noise measure.'
address:
- |
Statistics and Mathematics Unit, Indian Statistical Institute\
203, B.T. Road, Calcutta 700108, India.\
e-mail:amites@isical.ac.in\
- |
Statistics and Mathematics Unit, Indian Statistical Institute\
203, B.T. Road, Calcutta 700108, India.\
e-mail:mahuya@isical.ac.in\
author:
- Amites Dasgupta
- Mahuya Datta
title: |
Nash twist and Gaussian noise measure for\
isometric $C^1$ maps
---
[^1]
introduction
============
The problem of associating a measure to the solution space of a differential equation has been mentioned by Gromov in an interview with M. Berger [@berger]. Our point of interest lies in the space of isometric immersions of a Riemannian manifold $(M,g)$ into a Euclidean space ${\mathbb R}^q$ with the canonical metric $h$. In 1954, Nash proved that if a manifold $M$ with a Riemannian metric $g$ can be embdedded in a Euclidean space ${\mathbb R}^q$, $q > n+1$, then one can construct a large class of isometric $C^1$ embeddings ([@nash]). If the initial embedding $f_0:M\to {\mathbb R}^q$ is strictly short, that is if $g-f^*h$ is a Riemannian metric on $M$ then the isometric embeddings can be made to lie in an arbitrary $C^0$ neighbourhood of the initial embedding. In the following year, Kuiper showed that the bound can be improved to $q\geq n+1$ ([@kuiper]). The Nash process is an iterative process; each stage of the iteration consists of several small steps each of which involves a choice of a rapidly oscillating function defining a perturbation, called a Nash twist. Starting with the short map $f_0$, one constructs a sequence $\{f_n\}$ of short immersions, where $f_n$ is obtained from $f_{n-1}$ possibly through infinite steps each involving a Nash twist. Successive Nash twists performed on $f_n$ results in a correction to the induced metric $f_n^*h$. These corrections do not yield an isometric immersion at any stage but each $f_n$ still remains strictly short; however, $f_{n+1}$ is better than $f_n$ in the sense that the induced metric $f_{n+1}^*h$ is closer to $g$ than that in the previous stage. The Nash twist is a controlled perturbation - the $C^1$ distance between any two consecutive maps $f_n$ and $f_{n+1}$ remains bounded by the distance between $g$ and the induced metric $f_n^*h$; furthermore $f_n$ can be made to lie in an arbitrary $C^0$ neighbourhood of $f_0$. As a result the sequence converges to a $g$-isometric $C^1$ immersion.
Nash-Kuiper theory was later generalised by Gromov into the theory of convex integration ([@gromov]). Camillo De Lellis and László Székelyhidi, Jr., have briefly mentioned about the probabilistic approach to convex integration ([@lellis]) by pointing to the fact that Convex integration can be seen as a control problem: at each step of the iteration, one has to choose an admissible perturbation, consisting essentially of a (plane-)wave direction and a frequency. In the present article we shall consider the domain space of the maps to be 1-dimensional. In dimension 1, the solution to the $C^1$-isometric embedding problem does not require an infinite Nash process. Isometric maps $f:\mathbb I\to {\mathbb R}^3$ with respect to the standard Riemannian metrics can be obtained simply by integrating a curve in the 2-sphere and this reduces the Nash process to a single stage. However, Nash twists play an important role in controlling the distance between the initial and the perturbed map - it can produce isometric immersions within an arbitrary $C^0$ neighbourhood of the initial embedding $f_0$. In order to keep the solutions sufficiently close to the original embedding, Nash introduced a periodic function of high frequency (or rapidly oscillating function) under the integration process. We may remark here that in higher dimension, each step in the Nash process can be reduced to a parametric version of the 1-dimensional Nash process described above. The $C^0$-closeness will then translate into $C^\perp$-closeness (refer to [@gromov Pp. 170]). However, the problem in dimension greater than 1 is considerably more difficult and we plan to take it up in future.
It is indeed the case that as the frequency in the Nash twist goes to infinity the distance between the initial short map and the resulting isometric immersion goes to zero in the $C^0$-norm. This motivates us to study the distribution of $f-f_0$ with respect to an appropriate measure on the space of isometric immersions $f:{\mathbb I}\to {\mathbb R}^3$.
We naturally incorporate a randomness in the Nash twist which translates into a Gaussian noise measure for the difference function $f-f_0$. For each positive integer $n$, we construct random functions $f_n$ such that the difference $(f_n(.) - f_0(.))$ goes to zero (in $C^0$ norm). We scale it up and examine the distribution. We show that the distribution of $ n^{3/2} (f_n - f_0)$ converges (weakly) to a Gaussian noise measure. Thus the random solutions $f_n(.)$ can be thought of as distributed like $f_0+n^{-3/2}$(Gaussian noise) for large $n$. In Theorem \[main\] we state this rigorously identifying the weak limit of $n^{3/2}\int_0^t(f_n - f_0)(s)\,ds$ as a Gaussian process. Section 2.2 is devoted to the proof which requires essentially weak convergence of random walks. In the last section, we compare the above process with a class of extensively studied Gaussian processes.
Notation and main result
========================
Let $M$ be a smooth manifold with a Riemannian metric $g$. The isometric immersions $f:M\to {\mathbb R}^q$ are solutions to the following system of partial differential equations: $$\langle\frac{\partial f}{\partial u_i},\frac{\partial f}{\partial u_j}\rangle = g_{ij},\ \ \ i,j=1,2,\dots,n,$$ where $u_1,u_2,\dots u_n$ is a local coordinate system on $M$, $g_{ij}$, $i,j=1,2,\dots,n$, are the matrix coefficients of $g$ and the $\langle\ ,\ \rangle$ denotes the inner product on $\mathbb R^q$.
To motivate the concept of randomness and measure we consider the following simple case where the domain space $M$ is the unit interval $[0,1]$ and hence an arbitrary metric on $M$ is of the form $g\,dt^2$, where $g : [0, 1] \rightarrow \mathbb{R}_+$ is a smooth positive function on $[0,1]$. The shortness condition on a smooth regular curve $f_0 : [0, 1] \rightarrow
\mathbb{R}^3$ then translates into the pointwise inequality $0<||\partial_u f_0|| < \sqrt{g}$ on $[0,1]$. Given such an $f_0$ we want to find a function $f_n
: [0, 1] \rightarrow \mathbb{R}^3$ such that $||\partial_u f_n|| = \sqrt{g}$ (which means that $f_n$ is isometric) and the $C^0$-distance between $f_n$ and $f_0$ decreases with $n$. The following considerations illustrate the *Nash twist* in dimension 1. Suppose that $(X,Y,Z)$ is the Frenet-Serret frame along $f_0$ (assuming that such a frame exists at all points $u\in [0,1]$), where $X$ is the unit tangent along the curve. Then $(Y,Z)$ span a plane field $J$ along $f_0$ perpendicular to $X$. Consider the curve $Y(u)\cos 2\pi s+Z(u)\sin 2\pi s$, $0 \leq s \leq 1$, on $J(u)$ for each fixed $u\in [0,1]$. Then with $r^2 = g - ||\partial_u f_0||^2$ the function $$\partial_u f_0 + r(u) (Y(u)\cos 2\pi s+Z(u)\sin 2\pi s)$$ has the required euclidean norm $\sqrt{g}$ and over $s \in [0, 1]$ integrates to $\partial_u f_0$ (the convex integration condition). Now, a Nash twist of $f_0$ is given by $$f_n(t) = f_0(0) + \int_0^t
\{ \partial_u f_0(u) + r(u) (Y(u)\cos 2\pi nu+Z(u)\sin 2\pi nu) \} du,$$ where $n$ connects with the frequency of the periodic functions, namely $\cos 2\pi nu$ and $\sin 2\pi nu$, mentioned in the previous section. Clearly, $f_n$ is a solution of the isometry equation since $(Y(u),Z(u))$ is an orthonormal basis of $J(u)$. We want to show that the function (or the difference curve) $\int_0^t r(u) (Y(u)\cos 2\pi nu+Z(u)\sin 2\pi nu) du$ is uniformly small over $[0, 1]$. We do this for the two integrals separately with some notational abuse.
Applying integration by parts we get $$\begin{aligned}
\int_0^1 r(u) \cos {2\pi nu}\, du
&=& \frac{1}{2\pi n}r(u)\sin 2\pi nt - \frac{1}{2\pi n}\int_0^t r'(u)\sin 2\pi nu\, du\end{aligned}$$ which in absolute value is bounded by const.$\frac{1}{2n\pi}$ as $r$ is a smooth function on the interval $[0,1]$. The same estimates also apply to $\int_0^t r(u)\sin {2\pi nu}\, du$ and thus we conclude uniform closeness of $f_n$ and $f_0$.
This difference curve can also be considered for a random path by changing the function $H_n(u) = nu$ over random choices. The $H_n$ in the above example can be obtained by integrating the constant function $h_n=n$ over $[0,1]$. Instead we take $h_n=\pm n$ on each subinterval $(k/n, (k+1)/n]$ independently with equal probability. These choices are actually explicitly mentioned by Gromov except for the probability part. Then integration of the resulting function will give a random function $H_n(\omega, u)$ which is the graph of a simple random walk. To see this calculation we consider a sequence of independent and identically distributed random variables $X_k$ which take the values $\pm 1$ with equal probability on a probability space $(\Omega,\mathcal F, P)$, where $\Omega$ can be taken as the infinite product space $\{-1,+1\}^{\mathbb N}$ and consider the function $h_n(\omega, x) = nX_k, (k-1)/n \leq x < k/n$. Therefore, each subinterval of length $1/n$ contributes $\pm 1$ and hence $$H_n(\omega, u) =
\int_0^u h_n(\omega, x) dx = S_k \pm n(u - (k/n)), \ \ k/n \leq u < (k+1)/n,$$ where $S_k = X_1 +
\cdots + X_k$. The random sum $S_n(\omega, t)$ can be interpreted as the random walk obtained by linearly joining $S_k$, $1\leq k\leq n$.
If we now consider the components of the random difference curve $$\int_0^t r(u) [Y(u) \cos {2\pi H_n(\omega, u)}+Z(u) \sin {2\pi H_n(\omega, u)}] du,$$ $C^0$-closeness will follow in the same way. However from the viewpoint of probability, when a sequence of random variables $V_n$ converges almost surely to a random variable $V$, many times the difference $V_n - V$, after scaling, converges in a suitable sense to a nontrivial limiting random variable. When the convergence is weak convergence, the resulting distribution of the limit is a measure, on a suitable space, associated to the sequence ([@billingsley]).
In our case the graph of $\int_0^t e^{2\pi iH_n(\omega, u)} du$ (omitting $r(u), Y(u), Z(u)$ for simplicity now) looks similar (in a probabilistic sense of considering all possible paths) over equal intervals and is independent over disjoint intervals. However, the limit of its normalization is not a function but a random distribution (in the sense of generalized functions), it is called the Brownian white noise measure. To keep track of the random difference curve one tracks its rescaled integral, which converges weakly to Brownian motion. A rigorous formulation (bringing in $r(u)$, $Y(u)$, $Z(u)$) is the following
The sequence of processes $2 \pi n^{3/2} \int_0^t (f_n-f_0) \,ds, 0 \leq t \leq 1$, converges weakly to $$\int_0^t r(u) Z(u) dW(u) - \int_0^t \int_0^s \partial_u (r(u) Z(u)) dW(u)\, ds, 0 \leq t \leq 1,$$ as $n \rightarrow \infty$, where $W(\cdot)$ denotes the Wiener measure on $C[0, 1]$, the space of real valued continuous functions on the interval $[0,1]$. \[main\]
In this sense, the limit of the scaled random difference curve is locally the Gaussian noise measure $r(t) Z(t) dW(t) - (\int_0^t \partial_u (r(u) Z(u)) dW(u))\, dt$ (see [@mitoma]). The rate of convergence may also be interesting from the probabilistic point of view.
Proof of the main result
========================
We first keep track of the integral (omitting $r(u), Y(u), Z(u)$) $$\label{wn}
\int_0^s e^{2 \pi i H_n(u)} du = \int_0^s e^{2 \pi i n u } du =
(1/2\pi n)[ \sin (2 \pi ns) - i \{\cos (2 \pi n s) -
1 \}], 0 \leq s \leq 1.$$ The main observation about this function on the right is that over each $(k/n, (k+1)/n]$ interval the imaginary part is the graph of $1 - \cos (2 \pi n x), 0 < x \leq 1/n$, and the real part is the graph of $\sin (2 \pi n x), 0 < x \leq 1/n$. The periodic behavior along with the factor $(1/2\pi n)$ indicates the $C^0$-closeness as $s$ varies in $[0, 1]$.
Plugging in $H_n(\omega, u)$ described in the previous section the corresponding integral $\int_0^s e^{2 \pi i H_n(\omega, u)} du$ differs from (\[wn\]) in randomly inverting the imaginary part of the graph over each $(k/n, (k+1)/n]$ interval. To see this consider $s \in (k/n, (k+1)/n]$. Then $$\int_{\frac{k}{n}}^s e^{2 \pi i (S_k/n \pm n(u - k/n)} du
= (1/2\pi n)[ \sin (2 \pi n(s - k/n)) \pm (-i) \{\cos (2 \pi n (s - k/n)) - 1 \} ].$$ Noting that the integral over each $(k/n, (k+1)/n]$ is zero and using the periodicity of $\sin$ and $\cos$ functions we have established that the graph of $\int_0^s e^{2 \pi i H_n(\omega, u)} du$ is obtained by randomly inverting the imaginary part of the graph of (\[wn\]) over each $(k/n, (k+1)/n]$ interval. We now prove that $$2 \pi n^{3/2} \int_0^t \{ \int_0^s \sin{2 \pi H_n(\omega, u)} du\} ds,\ 0 \leq t \leq 1,$$ converges weakly to Brownian motion as $n \rightarrow \infty$ and $$2 \pi n^{3/2} \int_0^t \{ \int_0^s \cos{2 \pi H_n(\omega, u)} du\} ds,\ 0 \leq t \leq 1,$$ converges to the zero process.
To see the exact form of this (random) function we note that, as proved, the function $\int_0^s e^{2 \pi i H_n(\omega, u)} du$ has a graph which is the graph of (\[wn\]) with the imaginary part randomly inverted over each $[k/n, (k+1)/n)$ interval. The integral of the (periodic) function in (\[wn\]) over each $[k/n, (k+1)/n)$ interval is $ (i/2\pi n^2)$ (the real part integrates to contribute zero). Thus the integral of the function $\int_0^s e^{2 \pi i H_n(\omega, u)} du$ is $\pm i (1/2\pi n^2)$ over the same interval, the $\pm$ sign coming from the random inverting. Written explicitly, for $k/n \leq t < (k+1)/n$, $$\begin{array}{rcl}\int_0^t\{\int_0^s e^{2 \pi i H_n(\omega, u)} du\}ds & = & \sum_{j = 0}^{k - 1}\int _{j/n}^{(j+1)/n}\{\int_0^s e^{2 \pi i
H_n(\omega, u)} du\}ds\\ && + \int _{k/n}^{t}\{\int_0^s e^{2 \pi i
H_n(\omega, u)} du\}ds \\
& = & \frac{i}{2\pi n^2}\sum_{j = 1}^{k} X_j + O(\frac{1}{n^2}),\end{array}$$ where $X_j$ are independent $\pm 1$ random variables. Note that the integral from $k/n$ to $t$ adds a continuous function of order $O(\frac{1}{n^2})$ to the random walk obtained from the $X_j$’s. Multiplying by $2 \pi n^{3/2}$ we get the weak convergence to Brownian motion. In this sense the random difference $\int_0^s e^{2 \pi i H_n(\omega, u)} du, 0 \leq s \leq 1,$ when normalized converges to the generalized derivative of Brownian motion, called Brownian white noise.
For the general case, we consider (with some abuse of notation) $$\int_0^s r(u) e^{2 \pi i H_n(\omega, u)} du, 0 \leq s \leq 1,$$ where $r$ is a sufficiently differentiable function. In this case, depending on $r$, after two integrations we get back a different random walk minus the area under another random walk, and consider weak convergence again.
We shall deal with the real and the imaginary part of the integral separately. As observed before the randomness has no role to play in the real part. A straightforward calculation shows that
$\begin{array}{rcl}\int_0^s r(u) \cos(2 \pi H_n(\omega, u))du & = & \int_0^s r(u) \cos(2 \pi nu)du\\
& = &\frac{1}{2n\pi}r(s)\sin 2n\pi s - \frac{1}{2n\pi}\int_0^s r'(u)\sin 2n\pi u \,du\\
& = &\frac{1}{2n\pi}r(s)\sin 2n\pi s + \frac{1}{4n^2\pi^2}r'(s)\cos 2n\pi s\\
& - & \frac{1}{4n^2\pi^2}r'(0)- \frac{1}{4n^2\pi^2}\int_0^s r''(u)\cos 2n\pi u \,du
\end{array}$
Therefore,
$\int_0^t\{\int_0^s r(u) \cos(2 \pi H_n(\omega, u))du\}\,ds = \frac{1}{2\pi n}\int_0^t r(s) \sin (2 \pi ns)\, ds + O(\frac{1}{n^2})$.
Since the first term on the right hand side is $O(1/n^2)$ it follows that $$\lim_{n\to\infty} n^{3/2}\int_0^t\{\int_0^s r(u) \cos{2 \pi H_n(\omega, u)} du\}\,ds=0.$$ Thus the real part of the integral when scaled by $n^{3/2}$ converges to zero uniformly.
To deal with the imaginary part of the integral, for $l/n\leq s<(l+1)/n$, we split it as follows $$\begin{aligned}
\label{area}
\sum_{k = 0}^{[ns] - 1} \int_{k/n}^{(k+1)/n} r(u) \sin (2\pi H_n(\omega, u)) du + \int_{l/n}^s r(u) \sin (2\pi H_n(\omega, u)) du,\end{aligned}$$ and then writing $r(u) = r(k/n) + (r(u) - r(k/n))$ on the subinterval $(k/n, (k+1)/n]$ we get the following for (\[area\]):
$$\begin{aligned}
\label{area2}
& & \sum_{k = 0}^{[ns] - 1} \int_{k/n}^{(k+1)/n} (r(u) - r(k/n)) \sin (2\pi H_n(\omega, u)) du \nonumber \\
&+&r(l/n)\int_{l/n}^s \sin(2\pi H_n(\omega,u)) du + \int_{l/n}^s (r(u) - r(l/n)) \sin (2\pi H_n(\omega, u)) du.\end{aligned}$$
We denote the function (represented by the sum) on the first row by $\psi_1(s)$ and the two terms on the second row by $\psi_2(s)$ and $\psi_3(s)$ respectively. Since $n^{3/2}\int_0^t \psi_2(s) ds$ gives a random walk with steps $\pm\frac{1}{\sqrt{n}}r(k/n)$ over the interval $[k/n,(k+1)/n)$, $n^{3/2}\int_0^t \psi_2(s) ds$ converges weakly to $\frac{1}{2\pi}\int_0^t r(s) dW(s)$. Next we consider the part $\psi_1$. The summands of $\psi_1$ are not necessarily zero as $r$ is non-constant. Substituting $u=k/n+z/n$ in the $k$-th summand and disregarding the $\pm$ signs, we get $$\frac{1}{n} \int_0^1 [r\left(\frac{k}{n} + \frac{z}{n}\right) - r\left(\frac{k}{n}\right)] \sin (2 \pi z) dz.$$ To clearly understand this contribution divide each integral $[k/n,(k+1)/n)$ into the parts where the sine function has the same sign (to apply the mean value theorem for integrals for some $z_1 \in (0, 1/2)$). Thus, $$\begin{aligned}
&& \frac{1}{n} \int_0^1 [r\left(\frac{k}{n} + \frac{z}{n}\right) - r\left(\frac{k}{n}\right)] \sin (2 \pi z) dz \\
&=& \frac{1}{n} \int_0^{1/2} [r\left(\frac{k}{n} + \frac{z}{n}\right) - r\left(\frac{k}{n}\right)] \sin (2 \pi z) dz \\
& + & \frac{1}{n}\int_{1/2}^1 [r\left(\frac{k}{n} + \frac{z}{n}\right) - r\left(\frac{k}{n}\right)] \sin (2 \pi z) dz \\
&=& \frac{1}{n} \int_0^{1/2} [r\left(\frac{k}{n} + \frac{z}{n}\right) - r\left(\frac{k}{n} + \frac{z}{n}+\frac{1}{2n}\right)] \sin (2 \pi z)\, dz \\
&=& \frac{1}{n\pi} \left[r\left(\frac{k}{n} + \frac{z_1}{n}\right) - r\left(\frac{k}{n} + \frac{z_1}{n}+\frac{1}{2n}\right)\right] \\
&=& - \frac{1}{2n^2\pi} r^{\prime} \left(\frac{k}{n} + \frac{z_1}{n}+\frac{\theta}{2n}\right),\end{aligned}$$ where $0<\theta<1$. If we add these integrals after multiplying each them by $\pm 1$ from random inversions, and scale the sum by $n^{3/2}$ then it corresponds to a random walk converging weakly to $$- \frac{1}{2\pi} \int_0^s r^\prime(u) dW(u).$$ The continuous random curve $\psi_1+\psi_3$ matches this random walk at the points $k/n$ and is otherwise at a distance at most $$O(\frac{1}{n}|r(k/n + 1/2n) - r(k/n)|) = O(1/n^2)$$ from it. Thus after scaling by $n^{3/2}$ the random curve $\psi_1+\psi_3$ converges weakly to the same limit and the integral of $\psi_1+\psi_3$ converges weakly to $-\frac{1}{2\pi}\int_0^t \{\int_0^s r^\prime(u) dW(u)\} ds$ by the continuous mapping theorem. This completes the proof of the main result. For completeness we indicate how the random walk $$\sum_{k=1}^{[nt_1]} r(k/n)Y_k\label{int_psi_2}$$ (here $Y_k$ are i.i.d. $\pm\frac{1}{\sqrt{n}}$ random variables) and the area under the random walk $$- \sum_{i=1}^{[ns]} r'(i/n)Y_i\label{area_random}$$ up to time $t_2$ converges jointly in distribution, after which tightness on product space can be used to conclude weak convergence on $C[0,1]\times C[0,1]$. To obtain the area under (\[area\_random\]), each random walk height is multiplied by $1/n$ and then added. The interchange of summation gives the area as $$-\sum_{i=0}^{k-1} (\frac{k-i}{n}) r'(i/n) Y_i,\label{int_psi_1+psi_3}$$ where $k=[nt_2]$. From (\[int\_psi\_2\]) and (\[int\_psi\_1+psi\_3\]) the limiting joint finite dimensional distribution follows. The limit of the expression in (\[int\_psi\_1+psi\_3\]) is seen to be $-\frac{1}{2\pi}\int_0^t(t-u)r'(u)\, dW(u)$ which equals $-\frac{1}{2\pi}\int_0^t \{\int_0^s r^\prime(u) dW(u)\} ds$. Now the sum of the two processes (\[int\_psi\_2\]) and (\[int\_psi\_1+psi\_3\]) converges weakly. $\Box$
Concluding remarks:
===================
It is seen from the proof of Theorem \[main\] that the random Nash twist on the initial short curve $f_0$ to obtain an increase of $r^2(u)\,du^2$ to the induced metric $f_0^*h$, leads to a process whose structure is similar to the following process (refer to [@hida Theorem 6.3]) $$X(t)=W(t)-\int_0^t\int_0^s\ell(s,u)dW(u)\,ds,$$ where $\ell(s,u)$ is a Volterra kernel with appropriate conditions. In our case, componentwise we need a Gaussian process $\int_0^t r(u)\,dW(u)$ and a function $r'(u)/r(u)$ to replace $W(t)$ and $\ell(s,u)$ in the above formula.
In higher dimension the difference metric $g-f_0^*h$ can be written as a finite sum of monomials $r^2 d\varphi^2$, where $\varphi$ is a rank 1 function. Applying Nash twist along (Im$Df_0)^\perp$, one is able to add $r^2 d\varphi^2$ only approximately. To look into the problem of $C^0$ distance one may need several independent Brownian motions in different directions and we refer to Theorem 3.2.5 of Kallianpur and Xiong ([@kallianpur]) for such a construction. However a precise formulation combining various directions is not clear and we hope to explore these aspects in future.
*Acknowledgement*: The second author is greatly indebted to Misha Gromov for sharing his insight on the subject during a visit of the author to IHES.
[99]{} M. Berger: Encounter with a Geometer, Part I, *Notices of the Amer. Math. Soc.*,**47** (2000), 183-194. P. Billingsley: *Convergence of probability measures*. Wiley Series in Probability and Statistics. John Wiley and Sons, Inc., New York, 1999. M. Gromov: [*Partial Differential Relations*]{}, Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge Band 9. Springer-Verlag, 1986. Hida, Takeyuki; Hitsuda, Masuyuki: *Gaussian processes*. (English summary) Translated from the 1976 Japanese original by the authors. Translations of Mathematical Monographs, 120. American Mathematical Society, Providence, RI, 1993.
G. Kallianpur, J. Xiong: Stochastic differential equations in infinite-dimensional spaces. IMS Lecture Notes Monogr. Ser. 26, 1995. N.H. Kuiper, $C^1$-isometric embeddings, [*I Proc. Koninkl. Nederl. Ak. Wet. A*]{}, [**58**]{}(1955) 545 – 556. C. De Lellis, L. Székelyhidi, Jr.: The h-principle and the equations of fluid dynamics. *Bull. Amer. Math. Soc*. **49**, 347-375, 2012. Mitoma, Itaru Tightness of probabilities on $C([0,1]);\mathcal S')$ and $D([0,1]);\mathcal S')$. *Ann. Probab*. **11** (1983), no. 4, 989-999. J. Nash: $C^1$-isometric embeddings, *Annals of Math*. [**60**]{}(1954) 383–396.
[^1]: 2010 Mathematics Subject Classification: 60F17, 60H05
|
---
abstract: |
The past, present and future of cosmic microwave background (CMB) anisotropy research is discussed, with emphasis on the Boomerang and Maxima balloon experiments. These data are combined with large scale structure (LSS) information derived from local cluster abundances and galaxy clustering and high redshift supernova (SN1) observations to explore the inflation-based cosmic structure formation paradigm. Here we primarily focus on a simplified inflation parameter set, $\{\omega_b,\omega_{cdm},\Omega_{tot}, \Omega_Q,w_Q,n_s,\tau_C,
\sigma_8\}$. After marginalizing over the other cosmic and experimental variables, we find the current CMB+LSS+SN1 data gives $\Omega_{tot} = 1.04\pm 0.05$, consistent with (non-baroque) inflation theory. Restricting to $\Omega_{tot}=1$, we find a nearly scale invariant spectrum, $n_s =1.03 \pm 0.07$. The CDM density, $\omega_{cdm}=0.17 \pm 0.02$, is in the expected range, but the baryon density, $\omega_b\equiv \Omega_b {\rm h}^2 =
0.030\pm 0.004$, is slightly larger than the current $0.019\pm
0.002$ Big Bang Nucleosynthesis estimate. Substantial dark (unclustered) energy is inferred, $\Omega_Q \approx 0.68 \pm
0.05$, and CMB+LSS $\Omega_Q$ values are compatible with the independent SN1 estimates. The dark energy equation of state, parameterized by a quintessence-field pressure-to-density ratio $w_Q$, is not well determined by CMB+LSS ($w_Q < -0.3$ at 95% CL), but when combined with SN1 the resulting $w_Q < -0.7$ limit is quite consistent with the $w_Q$=$-1$ cosmological constant case. Though forecasts of statistical errors on parameters for current and future experiments are rosy, rooting out systematic errors will define the true progress.
address: |
1. CIAR Cosmology Program, Canadian Institute for Theoretical Astrophysics,\
60 St. George St., Toronto, ON M5S 3H8, Canada\
2. See Jaffe 2000 [@jaffe00] for the full author and institution list.\
[CITA-2000-64, in Proc. CAPP-2000 (AIP), eds. R. Durrer, J. Garcia-Bellida, M. Shoposhnikov ]{}
author:
- 'J. Richard Bond$^{1}$, Dmitry Pogosyan$^{1}$, Simon Prunet$^{1}$, Kris Sigurdson$^{1}$ and the MaxiBoom Collaboration$^{2}$'
title: '[The Quintessential CMB, Past & Future]{}'
---
\^ [C]{} \#1[[\#1 ]{}]{} \#1[to 0pt[\#1]{}]{}
CMB Analysis: Past, Present and Future {#cmb-analysis-past-present-and-future .unnumbered}
======================================
The CMB is a nearly perfect blackbody of $2.725 \pm 0.002\, K$ [@matherTcmb], with a $3.372 \pm 0.007\, mK$ dipole associated with the 300 $\kms$ flow of the earth in the CMB, and a rich pattern of higher multipole anisotropies at tens of $\mu$K arising from fluctuations at photon decoupling and later. Spectral distortions from the blackbody associated with starbursting galaxies detected in the COBE FIRAS and DIRBE data are due to stellar and accretion disk radiation being downshifted into the infrared by dust then redshifted into the submillimetre; they have energy about twice all that in optical light, about a tenth of a percent of that in the CMB. The spectrally well-defined Sunyaev-Zeldovich (SZ) distortion associated with Compton-upscattering of CMB photons from hot gas has not been observed with FIRAS, but only at high resolution along lines-of-sight through dozens of clusters — with very high signal-to-noise though. The FIRAS 95% CL upper limit of $ 6.0 \times 10^{-5}$ of the energy in the CMB is compatible with the $\lta 10^{-5}$ expected from clusters, groups and filaments in structure formation models, and places strong constraints on the allowed amount of earlier energy injection, ruling out mostly hydrodynamic models of LSS.
[**Upper Limit Experiments from the 70s & 80s:**]{} The story of the experimental quest for anisotropies is a heroic one.[^1] The original 1965 Penzias and Wilson discovery paper quoted angular anisotropies below $10\%$, but by the late sixties $10^{-3}$ limits were reached, by Partridge and Wilkinson and by Conklin and Bracewell. As calculations of baryon-dominated adiabatic and isocurvature models improved in the 70s and early 80s, the theoretical expectation was that the experimentalists just had to get to $10^{-4}$, as they did, Boynton and Partridge in 73. The only signal found was the dipole, hinted at by Conklin and Bracewell in 73, but found definitively in Berkeley and Princeton balloon experiments in the late 70s, along with upper limits on the quadrupole. Throughout the 1980s, the upper limits kept coming down, punctuated by a few experiments widely used by theorists to constrain models: the small angle 84 Uson and Wilkinson and 87 OVRO limits, the large angle 81 Melchiorri limit, early (87) limits from the large angle Tenerife experiment, the small angle RATAN-600 limits, the $7^\circ$-beam Relict-1 satellite limit of 87, and Lubin and Meinhold’s 89 half-degree South Pole limit, marking a first assault on the peak.
These upper limit experiments were highly useful, in particular to rule out adiabatic baryon-dominated models. In the early 80s, dark matter dominated universes lowered theoretical predictions by about an order of magnitude. In the 84 to mid-90s period, many groups developed codes to solve the perturbed Boltzmann–Einstein equations when dark matter was present. Armed with these pre-COBE computations, plus the LSS information of the time, a number of very interesting models fell victim to the data: scale invariant isocurvature cold dark matter models in 86, large regions of parameter space for isocurvature baryon models in 87, inflation models with radically broken scale invariance leading to enhanced power on large scales in 87-89, CDM models with a decaying ($\sim
{\rm keV}$) neutrino if its lifetime was too long ($\gta 10 {\rm
yr}$) in 87 and 91. Also in this period there were some limited constraints on “standard” CDM models, restricting $\Omega_{tot}$, $\Omega_B$, and the amplitude parameter $\sigma_8$. ($\sigma_8^2$ is a bandpower for density fluctuations on a scale associated with rare clusters of galaxies, $8\hmpc$, where ${\rm h}=H_0/(100 \kms
\mpc^{-1})$.)
[**Post-DMR Experiments:**]{} The now familiar motley pattern of anisotropies associated with $2 \le \ell \lta 20$ multipoles at the $30 \mu K$ level revealed by COBE at $7^\circ$ resolution was shortly followed by detections, and a few upper limits (UL), at higher $\ell$ in 19 other ground-based (gb) or balloon-borne (bb) experiments — most with many fewer resolution elements than the 600 or so for COBE. Some predated in design and even data delivery the 1992 COBE announcement. Proceeding from the period we began analyzing them, we have the intermediate angle SP91 (gb), the large angle FIRS (bb), both with strong hints of detection before COBE, then, post-COBE, more Tenerife (gb), MAX (bb), MSAM (bb), white-dish (gb, UL), argo (bb), SP94 (gb), SK93-95 (gb), Python (gb), BAM (bb), CAT (gb), OVRO-22 (gb), SuZIE (gb, UL), QMAP (bb), VIPER (gb) and Python V (gb). A list valid to April 1999 with associated bandpowers is given in [@bjk9800], and are referred here as 4.99 data. They showed evidence for a first peak [@bjk9800], although it was not well localized. Within limited parameter sets, good constraints on $n_s$, some on $\Omega_{tot}$ and $\Omega_\Lambda$ could be given, when LSS was added.
[**The Present, TOCO, BOOMERANG & MAXIMA:**]{} The picture dramatically improved this year, as results were announced first in summer 99 from the ground-based TOCO experiment in Chile [@toco98], then in November 99 from the North American balloon test flight of Boomerang [@mauskopf99]. These two additions improved peak localization and gave evidence for $\Omega_{tot}\sim
1$. Then in April 2000 results from the first CMB long duration balloon (LDB) flight, Boomerang [@debernardis00], were announced, followed in May 2000 by results from the night flight of Maxima [@MAXIMA1]. Boomerang’s best resolution was $10^\prime$, about 40 times better than that of COBE, with tens of thousands of resolution elements. Maxima had a similar resolution but covered an order of magnitude less sky.
Boomerang carried a 1.2m telescope with 16 bolometers cooled to 300 mK in the focal plane aloft from McMurdo Bay in Antarctica in late December 1998, circled the Pole for 10.6 days and landed just 50 km from the launch site, only slightly damaged. In [@debernardis00], maps at 90, 150 and 220 GHz showed the same spatial features and the intensities were shown to fall precisely on the CMB blackbody curve. The fourth frequency channel at 400 GHz is dust-dominated. Fig. \[fig:mapCLdat\] shows the 150 GHz map derived using only one of the 16 bolometers. Although Boomerang altogether probed 1800 square degrees, only the region in the rectangle covering 440 square degrees was used in the analysis described in [@lange00; @jaffe00] and this paper. Fig. \[fig:mapCLdat\] also shows the 124 square degree region of the sky (in the Northern Hemisphere) that Maxima-1 probed. Though Maxima was not an LDB, it did so well because its bolometers were cooled even more than Boomerang’s, to 100 mK, leading to higher sensitivity per unit observing time, it had a star camera so the pointing was well determined, and, further, all frequency channels were used in creating its map.
[**Primary CMB Processes and Soundwave Maps at Decoupling:**]{} Both Boomerang and Maxima were designed to measure the [*primary*]{} anisotropies of the CMB, those which can be calculated using linear perturbation theory. What we see in Fig. \[fig:mapCLdat\] are, basically, two images of soundwave patterns that existed about 300,000 years after the Big Bang, when the photons were freed from the plasma. The visually evident structure on degree scales is even more apparent in the power spectra of the Fourier transform of the maps, which show a dominant (first acoustic) peak, a less prominent (or non-existent) second one, and the possible hint of a third one from Maxima. Fig. \[fig:mapCLdat\] also shows that the quite heterogeneous 4.99+TOCO+Boomerang-NA mix of CMB data is very consistent with what Boomerang-LDB and Maxima show.
The images are actually a projected mixture of dominant and subdominant physical processes through the photon decoupling “surface”, a fuzzy wall at redshift $z_r \sim 1100$, when the Universe passed from optically thick to thin to Thomson scattering over a comoving distance $\sim 10 \hmpc$. Prior to this, acoustic wave patterns in the tightly-coupled photon-baryon fluid on scales below the comoving “sound crossing distance” at decoupling, $\lta
100 \hmpc$ ($\lta 100 \kpc$ physical), were viscously damped, strongly so on scales below the $\sim 10 \hmpc$ thickness over which decoupling occurred. After, photons freely-streamed along geodesics to us, mapping (through the angular diameter distance relation) the post-decoupling spatial structures in the temperature to the angular patterns we observe now as the [*primary*]{} CMB anisotropies. The maps are images projected through the fuzzy decoupling surface of the acoustic waves (photon bunching), the electron flow (Doppler effect) and the gravitational potential peaks and troughs (“naive” Sachs-Wolfe effect) back then. Free-streaming along our (linearly perturbed) past light cone leaves the pattern largely unaffected, except that temporal evolution in the gravitational potential wells as the photons propagate through them leaves a further $\Delta T$ imprint, called the integrated Sachs-Wolfe effect. Intense theoretical work over three decades has put accurate calculations of this linear cosmological radiative transfer on a firm footing, and there is a speedy, publicly available and widely used code for evaluation of anisotropies in a variety of cosmological scenarios, “CMBfast” [@cmbfast], including the latest hydrogen/helium recombination evaluations, and with extensions to more cosmological models added by a variety of researchers.
Of course there are a number of nonlinear effects that are also present in the maps. These [*secondary*]{} anisotropies include weak-lensing by intervening mass, Thompson-scattering by the nonlinear flowing gas once it became “reionized” at $z \sim 20$, the thermal and kinematic SZ effects, and the red-shifted emission from dusty galaxies. They all leave non-Gaussian imprints on the CMB sky.
[**The Future, beyond 2000:**]{} We are only at the beginning of the high precision CMB era. HEMT-based interferometers are already in place taking data: the VSA (Very Small Array) in Tenerife, the CBI (Cosmic Background Imager) in Chile, DASI (Degree Angular Scale Interferometer) at the South Pole, where the bolometer-based single dish ACBAR experiment will operate this year. Other LDBs will be flying within the next few years: Arkeops, Tophat, Beast/Boost; and in 2001, Boomerang will fly again, this time concentrating on polarization. As well, MAXIMA will fly as the polarization-targeting MAXIPOL. In April 2001, NASA will launch the all-sky HEMT-based MAP satellite, with $12^\prime$ resolution. Further downstream, in 2007, ESA will launch the bolometer+HEMT-based Planck satellite, with $5^\prime$ resolution.
Secondary anisotropies are also being targeted with new instruments. SZ anisotropies have been probed by single dishes, the OVRO and BIMA mm arrays, and the Ryle interferometer. A number of planned HEMT-based interferometers being built are more ambitious: AMI (Britain), the JCA (Chicago), AMIBA (Taiwan), MINT (Princeton). As well other kinds of bolometer-based experiments will be used to probe the SZ effect, including the CSO (Caltech submm observatory) with BOLOCAM on Mauna Kea, ACBAR at the South Pole, the LMT (large mm telescope) in Mexico, and the LDB BLAST. Anisotropies from dust emission from high redshift galaxies are being targeted by the JCMT with the SCUBA bolometer array, the OVRO mm interferometer, the CSO, the SMA (submm array) on Mauna Kea, the LMT, the ambitious US/ESO ALMA mm array in Chile, the LDB BLAST, and ESA’s FIRST satellite. About $50\%$ of the submm background has so far been identified with sources that SCUBA has found.
[**The CMB Analysis Pipeline:**]{} Analyzing Boomerang and other experiments involves a pipeline that takes (1) the timestream in each of the bolometer channels coming from the balloon plus information on where it is pointing and turns it into (2) spatial maps for each frequency characterized by average temperature fluctuation values in each pixel (Fig. \[fig:mapCLdat\]) and a pixel-pixel correlation matrix characterizing the noise, from which various statistical quantities are derived, in particular (3) the temperature power spectrum as a function of multipole (Fig. \[fig:mapCLdat\]), grouped into bands, and two band-band error matrices which together determine the full likelihood distribution of the bandpowers [@bjk9800]. Fundamental to the first step is the extraction of the sky signal from the noise, using the only information we have, the pointing matrix mapping a bit in time onto a pixel position on the sky.
There is generally another step in between (2) and (3), namely separating the multifrequency spatial maps into the physical components on the sky: the primary CMB, the thermal and kinematic Sunyaev-Zeldovich effects, the dust, synchrotron and bremsstrahlung Galactic signals, the extragalactic radio and submillimetre sources. The strong agreement among the Boomerang maps indicates that to first order we can ignore this step, but it has to be taken into account as the precision increases. The Fig. \[fig:mapCLdat\] map is consistent with a Gaussian distribution, thus fully characterized by just the power spectrum. Higher order (concentration) statistics (3,4-point functions, ) tell us of non-Gaussian aspects, necessarily expected from the Galactic foreground and extragalactic source signals, but possible even in the early Universe fluctuations. For example, though non-Gaussianity occurs only in the more baroque inflation models of quantum noise, it is a necessary outcome of defect-driven models of structure formation. (Peaks compatible with Fig. \[fig:mapCLdat\] do not appear in non-baroque defect models, which now appear unlikely.) Though great strides have been made in the analysis of Boomerang and Maxima, there is intense needed effort worldwide now to develop new fast algorithms to deal with the looming megapixel datasets of LDBs and the satellites [@bcjk99szapudi00].
Cosmic Parameter Estimation {#cosmic-parameter-estimation .unnumbered}
===========================
[**Parameters of Structure Formation:**]{} For this paper, we adopt a restricted set of 8 cosmological parameters, augmenting the basic 7 used in [@lange00; @jaffe00], $\{\Omega_\Lambda,\Omega_{k},\omega_b,\omega_{cdm}, n_s,\tau_C,
\sigma_8\}$, by one. The vacuum or dark energy encoded in the cosmological constant $\Omega_\Lambda$ is reinterpreted as $\Omega_Q$, the energy in a scalar field $Q$ which dominates at late times, which, unlike $\Lambda$, could have complex dynamics associated with it. $Q$ is now often termed a quintessence field - see http://feynman.princeton.edu/ steinh/ “Quintessence? - an overview” for a pedagogical introduction. One popular phenomenology is to add one more parameter, $w_Q = p_Q /\rho_Q$, where $p_Q$ and $\rho_Q$ are the pressure and density of the $Q$-field, related to its kinetic and potential energy by $\rho_Q
= \dot{Q}^2/2+(\nabla Q)^2/2+V(Q)$, $ p_Q = \dot{Q}^2/2 -(\nabla
Q)^2/6-V(Q)$. Thus $w_Q=-1$ for the cosmological constant. Spatial fluctuations of $Q$ are expected to leave a direct imprint on the CMB for small $\ell$, typically smaller than Boomerang or Maxima are sensitive to. We ignore this complication here. As well, as long as $w_Q$ is not exactly $-1$, it will vary with time, but the data will have to improve for there to be sensitivity to this, and for now we can just interpret $w_Q$ as an appropriate time-average of the equation of state. The curvature energy $\Omega_k \equiv
1-\Omega_{tot}$ also can dominate at late times, as well as affecting the geometry.
We use only 2 parameters to characterize the early universe primordial power spectrum of gravitational potential fluctuations $\Phi$, one giving the overall power spectrum amplitude ${\cal
P}_{\Phi}(k_n)$, and one defining the shape, a spectral tilt $n_s
(k_n) \equiv 1+d\ln {\cal P}_{\Phi}/d \ln k$, at some (comoving) normalization wavenumber $k_n$. We really need another 2, ${\cal
P}_{GW}(k_n)$ and $n_t(k_n)$, associated with the gravitational wave component. In inflation, the amplitude ratio is related to $n_t$ to lowest order, with ${\cal O}(n_s-n_t)$ corrections at higher order, [@bh95]. There are also useful limiting cases for the $n_s-n_t$ relation. However, as one allows the baroqueness of the inflation models to increase, one can entertain essentially any power spectrum (fully $k$-dependent $n_s(k)$ and $n_t(k)$) if one is artful enough in designing inflaton potential surfaces. As well, one can have more types of modes present, scalar isocurvature modes (${\cal P}_{is}(k_n),n_{is}(k)$) in addition to, or in place of, the scalar curvature modes (${\cal
P}_{\Phi}(k_n),n_{s}(k)$). However, our philosophy is consider minimal models first, then see how progressive relaxation of the constraints on the inflation models, at the expense of increasing baroqueness, causes the parameter errors to open up. For example, with COBE-DMR and Boomerang, we can probe the GW contribution, but the data are not powerful enough to determine much. Planck can in principle probe the gravity wave contribution reasonably well.
We use another 2 parameters to characterize the transport of the radiation through the era of photon decoupling, which is sensitive to the physical density of the various species of particles present then, $\omega_j \equiv \Omega_j {\rm h}^2$. We really need 4: $\omega_b$ for the baryons, $\omega_{cdm}$ for the cold dark matter, $\omega_{hdm}$ for the hot dark matter (massive but light neutrinos), and $\omega_{er}$ for the relativistic particles present at that time (photons, very light neutrinos, and possibly weakly interacting products of late time particle decays). For simplicity, though, we restrict ourselves to the conventional 3 species of relativistic neutrinos plus photons, with $\omega_{er}$ therefore fixed by the CMB temperature and the relationship between the neutrino and photon temperatures determined by the extra photon entropy accompanying $e^+ e^- $ annihilation. Of particular importance for the pattern of the radiation is the (comoving) distance sound can have influenced by recombination (at redshift $z_r= a_r^{-1}-1$), $r_s =
6000/\sqrt{3} \mpc \int_{0}^{\sqrt{a_r}} (\omega_m + \omega_{er}
a^{-1})^{-1/2} (1+ \omega_b a/(4\omega_\gamma /3))^{-1/2}\
d\sqrt{a}$, where $\omega_\gamma = 2.46 \times 10^{-5}$ is the photon density, $\omega_{er} = 1.68 \omega_\gamma$ for 3 species of massless neutrinos and $\omega_m \equiv
\omega_{hdm}+\omega_{cdm}+\omega_b$.
The angular diameter distance relation, ${\cal R} =\{d_k {\rm sinh} (\chi_r/d_k), \chi_r, d_k {\rm sin} (\chi_r/d_k)\}$, where $\chi_r = 6000 \mpc \int_{\sqrt{a_r}}^{1} (\omega_m + \omega_Q
a^{-6w_Q} +\omega_k a)^{-1/2}\ d\sqrt{a}$ is the comoving distance to recombination, $d_k =3000 |\omega_k|^{-1/2} \mpc$ is the curvature scale and the 3 cases are for negative, zero and positive mean curvature, adds dependence upon $\omega_{k}$, $\omega_Q$ and $w_Q$ as well as on $\omega_m$. The location of the first acoustic peak $L_{Pk}$ is proportional to the ratio of ${\cal R}$ to $r_s$, hence depends upon $\omega_b$ through the sound speed as well. Thus $L_{Pk}$ defines a functional relationship among these parameters, a [*degeneracy*]{} [@degeneracies] that would be exact except for the integrated Sachs-Wolfe effect, associated with the change of $\Phi$ with time if $\Omega_Q$ or $\Omega_k$ is nonzero. (If $\dot{\Phi}$ vanishes, the energy of photons coming into potential wells is the same as that coming out, and there is no net impact of the rippled light cone upon the observed $\Delta T$.)
Our 7th parameter is an astrophysical one, the Compton “optical depth” $\tau_C$ from a reionization redshift $z_{reh}$ to the present. It lowers ${\cal
C}_\ell$ by $\exp(-2\tau_C)$ at the high $\ell$’s probed by Boomerang. For typical models of hierarchical structure formation, we expect $\tau_C \lta 0.2$. It is partly degenerate with $\sigma_8$ and cannot be determined at this precision by CMB data now.
The LSS also depends upon our parameter set: the most important combination is the wavenumber of the horizon when the energy density in relativistic particles equals the energy density in nonrelativistic particles: $k_{Heq}^{-1} \approx 5 \Gamma^{-1}
\hmpc$, where $\Gamma \approx \Omega_m {\rm h}
\Omega_{er}^{-1/2}$. Instead of ${\cal P}_\Phi (k_n)$ for the amplitude parameter, we often use ${\cal C}_{10}$ at $\ell =10$ for CMB only, and $\sigma_8^2$ when LSS is added. When LSS is considered in this paper, it refers to constraints on $\Gamma +
(n_s-1)/2$ and $\ln \sigma_8^2$ that are obtained by comparison with the data on galaxy clustering and cluster abundances [@lange00].
When we allow for freedom in $\omega_{er}$, the abundance of primordial helium, tilts of tilts ($dn_{\{s,is,t\}}(k_n)/d\ln k,
...$) for 3 types of perturbations, the parameter count would be 17, and many more if we open up full theoretical freedom in spectral shapes. However, as we shall see, as of now only 3 or 4 combinations can be determined with 10% accuracy with the CMB. Thus choosing 8 is adequate for the present; 7 of these are discretely sampled[@database], with generous boundaries, though for drawing cosmological conclusions we adopt a weak prior probability on the Hubble parameter and age: we restrict ${\rm h}
$ to lie in the 0.45 to 0.9 range, and the age to be above 10 Gyr.
[**The First Peak and $\Omega_{tot}$, $\Omega_Q$ and $w_Q$:**]{} For given $\omega_m$ and $\omega_b$, we show the lines of constant $L_{Pk}\propto {\cal R}/r_s$ in the $\Omega_{tot}$–$\Omega_Q$ plane for $w_Q$=$-1$ in Fig. \[fig:OmOL\], and in the $w_Q$–$\Omega_Q$ plane for $\Omega_{tot}$=1 in Fig. \[fig:wQOQ\], using the formulas given above and in [@degeneracies]. Our current best estimate [@bnu2K] of $L_{Pk}$, using all current CMB data, is $212\pm 7$, obtained by forming $\exp<\ln L_{Pk}>$, where the average and variance of $\ln
L_{Pk}$ are determined by integrating over the probability-weighted database described above, restricted here to the $\tau_C=0$ part. With just the prior-CMB data the value was $224\pm 25$, showing how it has localized. The numbers change a bit depending upon exactly what database or functional forms one averages over. The constant $L_{Pk}$ lines look rather similar to the contours shown in the right panel, showing that the ${\cal
R}/r_s$ degeneracy plays a large role in determining the contours. The contours hug the $\Omega_{tot}=1$ line more closely than the allowed $L_{Pk}$ band does for the maximum probability values of $\omega_m$ and $\omega_b$, because of the shift in the allowed $L_{Pk}$ band as $\omega_m$ and $\omega_b$ vary in this plane.
=3.5in =5.0in
=3.5in =5.0in
=3.5in =3.5in
=3.5in =3.5in
[**Marginalized Estimates of our Basic 8 Parameters:**]{} Table \[tab:exptparams\] shows there are strong detections with only the CMB data for $\Omega_{tot}$, $\omega_b$ and $n_s$ in the minimal inflation-based 8 parameter set. The ranges quoted are Bayesian 50% values and the errors are 1-sigma, obtained after projecting (marginalizing) over all other parameters. With Maxima, $\omega_{cdm}$ begins to localize, but much more so when LSS information is added. Indeed, even with just the COBE-DMR+LSS data, $\omega_{cdm}$ is already localized. That $\Omega_Q$ is not well determined is a manifestation of the $\Omega_{tot}$–$\Omega_Q$ near-degeneracy discussed above, which is broken when LSS is added because the CMB-normalized $\sigma_8$ is quite different for open cf. pure $Q$-models. Supernova at high redshift give complementary information to the CMB, but with CMB+LSS (and the inflation-based paradigm) we do not need it: the CMB+SN1 and CMB+LSS numbers are quite compatible. In our space, the Hubble parameter, ${\rm h}= (\sum_j (\Omega_j{\rm h}^2
))^{1/2}$, and the age of the Universe, $t_0$, are derived functions of the $\Omega_j{\rm h}^2$: representative values are given in the Table caption. CMB+LSS does not currently give a useful constraint on $w_Q$, though $w_Q \lta -0.7$ with SN1.
cmb +LSS +SN1 +SN1+LSS
------------------------- ------------------------ ------------------------ ------------------------ ------------------------
$\Omega_{tot}$ variable $w_Q=-1$ CASE
$\Omega_{tot}$ $1.09^{+.07}_{-.07}$ $1.08^{+.06}_{-.06}$ $1.04^{+.06}_{-.05}$ $1.04^{+.05}_{-.04}$
$\Omega_b{\rm h}^2$ $.031^{+.005}_{-.005}$ $.031^{+.005}_{-.005}$ $.031^{+.005}_{-.005}$ $.031^{+.005}_{-.005}$
$\Omega_{cdm}{\rm h}^2$ $.17^{+.06}_{-.05}$ $.14^{+.03}_{-.02}$ $.13^{+.05}_{-.05}$ $.15^{+.03}_{-.02}$
$n_s$ $1.05^{+.09}_{-.08}$ $1.04^{+.09}_{-.08}$ $1.05^{+.10}_{-.09}$ $1.06^{+.08}_{-.08}$
$\Omega_{Q}$ $0.48^{+.20}_{-.26}$ $0.63^{+.08}_{-.09}$ $0.72^{+.07}_{-.07}$ $0.70^{+.04}_{-.05}$
$\Omega_{tot}$ =1 $w_Q=-1$ CASE
$\Omega_b{\rm h}^2$ $.030^{+.004}_{-.004}$ $.030^{+.003}_{-.004}$ $.030^{+.004}_{-.004}$ $.030^{+.003}_{-.004}$
$\Omega_{cdm}{\rm h}^2$ $.19^{+.06}_{-.05}$ $.17^{+.02}_{-.02}$ $.16^{+.03}_{-.03}$ $.17^{+.01}_{-.02}$
$n_s$ $1.02^{+.08}_{-.07}$ $1.03^{+.08}_{-.07}$ $1.03^{+.08}_{-.07}$ $1.04^{+.07}_{-.07}$
$\Omega_{Q}$ $0.58^{+.17}_{-.27}$ $0.66^{+.04}_{-.06}$ $0.71^{+.06}_{-.07}$ $0.69^{+.03}_{-.05}$
$\Omega_{tot}$ =1 $w_Q$ variable CASE
$\Omega_b{\rm h}^2$ $.030^{+.004}_{-.004}$ $.030^{+.004}_{-.004}$ $.030^{+.004}_{-.004}$ $.030^{+.004}_{-.004}$
$\Omega_{cdm}{\rm h}^2$ $.17^{+.06}_{-.05}$ $.16^{+.02}_{-.03}$ $.14^{+.04}_{-.03}$ $.17^{+.01}_{-.02}$
$n_s$ $1.01^{+.08}_{-.07}$ $1.02^{+.07}_{-.06}$ $1.01^{+.07}_{-.07}$ $1.03^{+.07}_{-.06}$
$\Omega_{Q}$ $0.56^{+.17}_{-.25}$ $0.59^{+.08}_{-.10}$ $0.74^{+.06}_{-.08}$ $0.68^{+.03}_{-.05}$
$w_{Q}$ (95%) $< -0.29$ $<-0.33$ $< -0.69$ $<-0.73$
: Cosmological parameter values and their 1-sigma errors are shown, determined after marginalizing over the other $6$ cosmological and $4^{+}$ experimental parameters, for B98+Maxima-I+prior-CMB and the weak prior used in [@lange00; @jaffe00] ($0.45 \le {\rm h} \le 0.9$, age $>
10$ Gyr). The LSS prior was also designed to be weak. The detections are clearly very stable if extra “prior” probabilities for LSS and SN1 are included. (Indeed, they are stable to inclusion of stronger priors — except if the BBN-derived $0.019
\pm 0.002$ is imposed [@lange00].) Similar tables for B98+DMR are given in [@lange00] and for B98+MAXIMA-I+DMR in [@jaffe00]. If $\Omega_{tot}$ is varied, but $w_Q=-1$, parameters derived from our basic 8 come out to be: age=$13.2\pm 1.3$ Gyr, ${\rm h}=0.70 \pm 0.09$, $\Omega_m=0.35\pm .06$, $\Omega_b=0.065 \pm .02$. Restriction to $\Omega_{tot}=1$ and $w_Q=-1$ yields: age=$11.6\pm 0.4$ Gyr, ${\rm
h}=0.80\pm .04$, $\Omega_m=0.31\pm .03$, $\Omega_b=0.05 \pm .005$; allowing $w_Q$ to vary yields quite similar results. []{data-label="tab:exptparams"}
[**The Influence of Light Massive Neutrinos:**]{} In [@bnu2K], we considered what happens as we let $\Omega_{m\nu}/\Omega_m$, the fraction of the matter in massive neutrinos, vary from 0 to 0.3, for Boomerang+Maxima+prior-CMB+LSS when the weak-H+age + $\Omega_{tot}=1$ prior probability is adopted. Until Planck precision, the CMB data by itself will not be able to strongly discriminate this ratio. Adding HDM does have a strong impact on the CMB-normalized $\sigma_8$ and the shape of the density power spectrum (effective $\Gamma$ parameter), both of which mean that when LSS is included, adding some HDM to CDM is strongly preferred in the absence of $\Omega_Q$. However, though more (cold+hot) dark matter is preferred at the expense of less dark energy, significant $\Omega_Q$ is still required [@mnu]. The $\omega_b$ and $n_s$ likelihood curves are essentially independent of $\Omega_{m\nu}/\Omega_m$.
[**The Future, Forecasts for Parameter Eigenmodes:**]{} We can also forecast dramatically improved precision with further analysis of Boomerang and Maxima, future LDBs, MAP and Planck. Because there are correlations among the physical variables we wish to determine, including a number of near-degeneracies beyond that for $\Omega_{tot}$–$\Omega_Q$ [@degeneracies], it is useful to disentangle them, by making combinations which diagonalize the error correlation matrix, “parameter eigenmodes” [@bh95; @degeneracies]. For this exercise, we will add $\omega_{hdm}$ and $n_t$ to our parameter mix, but set $w_Q$=$-1$, making 9. (The ratio ${\cal P}_{GW}(k_n)/{\cal P}_\Phi (k_n)$ is treated as fixed by $n_t$, a reasonably accurate inflation theory result.) The forecast for Boomerang based on the 440 sq. deg. patch with a single 150 GHz bolometer used in the published data is 3 out of 9 linear combinations should be determined to $\pm
0.1$ accuracy. This is indeed what we get in the full analysis CMB only for Boomerang+DMR. If 4 of the 6 150 GHz channels are used and the region is doubled in size, we predict 4/9 could be determined to $\pm 0.1$ accuracy. The Boomerang team is still working on the data to realize this promise. And if the optimistic case for all the proposed LDBs is assumed, 6/9 parameter combinations could be determined to $\pm 0.1$ accuracy, 2/9 to $\pm 0.01$ accuracy. The situation improves for the satellite experiments: for MAP, we forecast 6/9 combos to $\pm 0.1$ accuracy, 3/9 to $\pm 0.01$ accuracy; for Planck, 7/9 to $\pm
0.1$ accuracy, 5/9 to $\pm 0.01$ accuracy. While we can expect systematic errors to loom as the real arbiter of accuracy, the clear forecast is for a very rosy decade of high precision CMB cosmology that we are now fully into.
Mather, J.C. ApJ 512, 511 (1999).
Miller, A.D. ApJ [524]{}, L1 (1999) TOCO.
Mauskopf, P. ApJ Lett [536]{}, L59, (2000) BOOM-NA.
de Bernardis, P. Nature 404, 995 (2000), astro-ph/00050087, http://www.physics.ucsb.edu/ boomerang/, and these proceedings.
Hanany, S. ApJ Lett., submitted (2000), astro-ph/0005123, http://cfpa.berkeley.edu/maxima
Bond, J.R., in [*Cosmology and Large Scale Structure*]{}, Les Houches Session LX, eds. R. Schaeffer J. Silk, M. Spiro & J. Zinn-Justin (Elsevier Science Press, Amsterdam), pp. 469-674 (1996).
Lange, A. PRD, in press (2000), astro-ph/0005004.
Jaffe, A.H. PRL, in press (2000), astro-ph/0007333.
Seljak, U. & Zaldarriaga, M. ApJ, 469, 437 (1996) for CMBFAST; Seeger, S., Sasselov, D. & Scott, D. ApJ Lett., 523, L1 (1999) for recombination.
Bond, J.R., Jaffe, A.H. & Knox, L., PRD 57, 2117 (1998), astro-ph/9708203; ApJ 533, 19 (2000), astro-ph/9808264
Bond, J.R., Crittenden, R., Jaffe, A.H. & Knox, L., Computing in Science and Engineering 1, 21 (1999), astro-ph/9903166, and references therein; Szapudi, I., Prunet, S., Pogosyan, D., Szalay, A. & Bond, J.R. ApJ Lett, in press (2000), astro-ph/0010256
Efstathiou, G. & Bond, J.R., Mon. Not. R. Astron. Soc. [304]{}, 75 (1999), where many other near-degeneracies between cosmological parameters are also discussed.
Bond, J.R., Pogosyan, D., Prunet, S. & the MaxiBoom Collaboration, Proc. Neutrino 2000, ed. Law, J., Simpson, J. (Elsevier) (2001); Bond, J.R. & the MaxiBoom Collaboration, Proc. IAU Symposium 201, ed. A. Lasenby & A. Wilkinson (PASP) (2001)
Perlmutter, S., Turner, M. & White, M. PRL 83, 670 (1999), from which the $w_Q$–$\Omega_Q$ SN1 likelihood function was taken, courtesy of Saul Perlmutter; see also Wang, L. astro-ph/9901388
The simplest interpretation of the superKamiokande data on atmospheric $\nu_\mu$ is that $\Omega_{\nu_\tau} \sim 0.001$, about the energy density of stars in the universe, which implies a cosmologically negligible effect. Degeneracy between $\nu_\mu$ and $\nu_\tau$ could lead to the cosmologically very interesting $\Omega_{m\nu}\equiv \Omega_{hdm}
\sim .1$, although the coincidence of closely related energy densities for baryons, CDM, HDM and dark energy required would be amazing.
The specific discrete parameter values used for the ${\cal C}_\ell$-database in this analysis were: ($\Omega_Q
=$ 0,.1,.2,.3,.4,.5,.6,.7,.8,.9,1.0,1.1), ($\Omega_k =$ .9,.7,.5,.3,.2,.15,.1,.05,0,-.05,-.1,-.15,-.2,-.3,-.5) & ($\tau_c
=$0, .025, .05, .075, .1, .15, .2, .3, .5) when $w_Q=$ –1; ($\Omega_Q =$ 0,.1,.2,.3,.4,.5,.6,.7,.8,.9), ($w_Q=$ -1,-.9,-.8,-.7,-.6,-.5,-.4,-.3,-.2,-.1,-.01) & ($\tau_c =$0, .025, .05, .075, .1, .15, .2, .3, .5) when $\Omega_k$=0. For both cases, ($\omega_c =$ .03, .06, .12, .17, .22, .27, .33, .40, .55, .8), ($\omega_b =$ .003125, .00625, .0125, .0175, .020, .025, .030, .035, .04, .05, .075, .10, .15, .2), ($n_s = $1.5, 1.45, 1.4, 1.35, 1.3, 1.25, 1.2, 1.175, 1.15, 1.125, 1.1, 1.075, 1.05, 1.025, 1.0, .975, .95, .925, .9, .875, .85, .825, .8, .775, .75, .725, .7, .65, .6, .55, .5), $\sigma_8^2$ was continuous, and there were 4 experimental parameters, calibration and beam uncertainties, for Boomerang and Maxima, as well as other calibration parameters for some of the prior-CMB experiments.
[^1]: Space constraints preclude adequate referencing here, but these are given in [@bh95; @lange00; @jaffe00].
|
---
abstract: 'We study a class of tame theories $T$ of topological fields and their extension $T_{\delta}^*$ by a generic derivation. The topological fields under consideration include henselian valued fields of characteristic 0 and real closed fields. For most examples, we show that the associated expansion by a generic derivation has the open core property (i.e., there are no new open definable sets). In addition, we show various transfer results between tame properties of $T$ and $T_\delta^*$, including relative elimination of field sort quantifiers, NIP, distality and elimination of imaginaries, among others. As an application, we derive consequences for the corresponding theories of dense pairs. In particular, we show that the theory of pairs of real closed fields (resp. of $p$-adically closed fields and real closed valued fields) admits a distal expansion. This gives a partial answer to a question of P. Simon.'
address:
- |
TU Dresden\
Fachrichtung Mathematik\
Institut für Algebra\
01062 Dresden.
- |
Department of Mathematics (De Vinci)\
UMons\
20, place du Parc 7000 Mons, Belgium
author:
- 'Pablo [Cubides Kovacsics]{}'
- 'Françoise [Point]{}$^{(\dagger)}$'
bibliography:
- '../biblio.bib'
title: Topological fields with a generic derivation
---
[^1]
Introduction {#introduction .unnumbered}
============
The study of topological fields with a derivation has been traditionally divided in two main branches. The first branch, as studied in [@ADH; @rideau; @R1980; @Scanlon], treats the case where some compatibility between the derivation and the topology is assumed (*e.g.*, continuity). The second branch, as studied in [@guzy-point2010; @GP12; @point2011; @singer1978; @Tressl], deals with the case where no such compatibility is required but rather a *generic* behaviour of the derivation occurs. An example of such a generic behaviour arises in existentially closed ordered differential fields, a class studied and axiomatized by Singer in [@singer1978]. Each branch seems to tackle different aspects of differential fields and has its own applications.
The purpose of this article is to further develop the study of generic derivations and show that many tame properties of theories of topological fields transfer to their expansions by such derivations. Examples of the topological fields under consideration include real closed fields and henselian valued fields of characteristic 0. We adopt a uniform treatment and development of such topological fields in the spirit of Mathews [@M] and Pillay [@pillay87], which we consider interesting on its own. As an application of generic derivations, we derive consequences for the corresponding theories of dense pairs of topological fields (as studied in [@berenstein-vassiliev2010; @F; @Macintyre; @Robinson; @Dries1998], to mention a few), supporting the idea that this framework is a useful tool to study such pairs of structures.
The following section gathers a more detailed overview of our main results.
Main results {#sec:mainresults}
============
The article is divided into two main parts. The first part is devoted to the study a particular class of theories of topological fields which we call *open theories of topological fields*. Informally, an open theory of topological fields is a first order topological theory of fields in the sense of Pillay [@pillay87] (i.e., the topology is uniformly definable) in which definable sets are finite boolean combinations of Zariski closed sets and open sets. This being said, we will allow multi-sorted structures in our setting and restrict the above conditions to the field sort. The formal definition will be given in Section \[sec:topopen\]. Examples include complete theories of henselian valued fields of characteristic 0 and the theory of real closed fields.
We show various tameness properties for open theories of topological fields including the fact that the topological dimension defines a dimension function in the sense of van den Dries [@vandendries1989] (later Corollary \[cor:dim1\]) and that they eliminate the field sort quantifier $\exists^\infty$ (Proposition \[prop:conseAA1\]). Of special interest for us is a cell decomposition theorem analogous to the recent cell decomposition theorem proven for dp-minimal topological structures by P. Simon and E. Walsberg in [@simon-walsberg2016]. This corresponds to the following theorem:
Let $T$ be an open theory of topological fields and $K$ be a model of $T$. Let $X$ be a definable subset of $K^n$. There are finitely many definable subsets $X_{i}$ with $X=\bigcup X_{i}$ such that $X_{i}$ is, up to permutation of coordinates, the graph of a definable continuous $m_i$-correspondence $f\colon U_{i}\rightrightarrows K^{n-d_i}$, where $U_{i}$ is a definable open subset of $K^{d_i}$, for some $0\leqslant d_i\leqslant n$, $m_{i}\geq 1$.
Correspondences are simply multi-valued functions. A crucial input of the proof of the previous theorem consists in showing that a definable correspondence on an open set is continuous almost everywhere (i.e., outside of a set of lower dimension). This is the content of Proposition \[prop:acont\]. When the topology on a model $K$ of $T$ is given by a valuation and $\Gamma$ is the value group of $K$, a similar result is proven for $\Gamma$-valued correspondences (see Proposition \[prop:ae\]). The proof presented here closely follows Simon and Walsberg’s argument, adapting it to the present setting. It is worthy to point out that, in contrast with other cell decompositions results for topological fields such as in Mathews [@M], Simon and Walsberg’s proof is almost purely combinatorial and does not make use of an implicit function theorem on definable functions.
The results of this first part will also play an essential role in the second part of the article.
Let $T$ be a first order topological $\cL$-theory of fields, again in the sense of Pillay. Let $\cL_\delta$ denote the extension of $\cL$ by a symbol $\delta$ for a derivation, and $T_\delta$ be the theory $T$ together with axioms stating that $\delta$ is a derivation on the field sort. The second part of the article focuses on the study of models of an $\cL_\delta$-extension $T_\delta^*$ of $T_\delta$. Informally, models of $T_\delta^*$ satisfy the following property: for any unary differential polynomial $P$, if the ordinary polynomial associated with $P$ has a regular solution $a$, then one can find differential solutions of $P$ arbitrarily close to $a$. A derivation $\delta$ on a model $K$ of $T$ is called *generic* if $(K,\delta)$ is a model of $T_\delta^*$. The above property implies that the derivation is highly non-continuous.
When $T$ is the theory of real closed fields, the theory $T_\delta^*$ corresponds to the theory of closed ordered differential fields $\CODF$ as originally introduced and axiomatized by M. Singer in [@singer1978]. The idea behind $\CODF$ has been generalized to many different contexts including work by M. Tressl [@Tressl] and N. Solancki [@solanki] in the framework of large fields, and by N. Guzy and the second author in [@guzy-point2010; @GP12]. As in [@guzy-point2010; @GP12], we will closely follow Singer’s original axiomatization. The main difference in the present setting with respect to previous work is the explicit allowance of multi-sorted languages. This permits us to include complete theories of henselian valued fields of characteristic 0 by studying them in a multi-sorted language as defined by J. Flenner [@flenner], where they admit relative quantifier elimination.
Most of our results concerning topological fields with a generic derivation are proven in the particular case when $T$ is an open theory of topological fields as defined in part 1. For such a theory $T$, we show several transfer results from $T$ to $T_\delta^*$. Some of these results, such as the transfer of quantifier elimination, NIP or distality, were known in the one-sorted case and we present adapted arguments in the multi-sorted setting. New results include the following transfer of elimination of imaginaries, whose proof is based on an unpublished argument of M. Tressl in the case of $\CODF$.
Let $T$ be an open theory of topological fields. Let $\G$ be a collection of sorts of $\cL^{eq}$ and $\cL^{\G}$ denote the restriction of $\cL^{eq}$ to the sorts in $\G$. Suppose that $T$ admits elimination of imaginaries in $\cL^\G$ and that the theory $T_\delta^*$ has $\cL$-open core. Then the theory $T^*_{\delta}$ admits elimination of imaginaries in $\cL_{\delta}^\G$.
Recall that $T_\delta^*$ has $\cL$-open core if every open $\cL_\delta$-definable set is already $\cL$-definable. We provide a general criterion, both in the ordered and valued case, to show that $T_\delta^*$ has $\cL$-open core (see later Theorem \[thm:opencore\]). Here the cell decomposition theorem proven in the first part plays a crucial role. Using this criterion, we show that the theory $T_\delta^*$ has $\cL$-open core in the following cases:
Let $T$ be either: $\ACVF_{0,p}, \RCVF, \PCF_d$ or the $\cL_\RV$-theory of a henselian valued field of characteristic 0, as defined in [@flenner], with value group either a $\Z$-group or a divisible ordered group. Then, the theory $T_\delta^*$ has $\cL$-open core.
As a consequence of the previous two theorems we obtain the following corollary:
Let $\cL^\G$ denote the geometric language of valued fields as defined in [@HHM2006]. The theories $(\ACVF_{0,p})_\delta^*$, $\RCVF_\delta^*$ and $\PCF_\delta^*$ have elimination of imaginaries in $\cL_\delta^\G$.
One can also use the above transfer of elimination of imaginaries to give another proof of the fact that $\CODF$ has elimination of imaginaries. This result was first proved by the second author in [@point2011] and later reproved in [@BCP]. The argument here presented corresponds to Tressl’s argument.
Last but not least, we illustrate how the theory $T_{\delta}^*$ provides a useful setting to study dense pairs of models of a one-sorted open $\cL$-theory of topological fields $T$. Let $\cL_P$ be the expansion of $\cL$ by a unary relation $P$ and let $T_P$ be the theory of dense elementary pairs of models of $T$. If $K\models T_{\delta}^*$, then the pair $(K,C_{K})$, where $C_{K}$ is the subfield of constants of $K$, is a dense elementary pair of models of $T$ (see later Lemma \[lem:elext\]). Using this observation, we derive various consequences for the theory $T_P$. Among them, we show that if $T_{\delta}^*$ has $\cL$-open core, then $T_{P}$ has $\cL$-open core (Theorem \[thm:opencorepairs\]), providing a new proof that the theory $T_P$ has $\cL$-open core when $T$ is either $\RCF$, $\ACVF_{0,p}$, $\PCF_d$ and $\RCVF$. We also deduce that the theory $T_P$ admits a distal expansion (namely $T_\delta^*$) whenever $T$ is a distal theory (see later Corollary \[cor:dist\_expansion\]). In particular, we show that $T_P$ admits a distal expansion when $T$ is $\RCF$, $\PCF_d$ and $\RCVF$. It is worthy to note that even when $T$ is a distal theory, the theory $T_P$ is not in general distal [@HN2017]. Our result gives a positive answer to a particular case of a question of P. Simon who asked if the theory of dense pairs of an o-minimal structure (extending a group) has a distal expansion (see [@nell2018] for a discussion). T. Nell provided a positive answer in the case of ordered vector fields [@nell2018]. Our result extends to pairs of real closed fields.
The paper is laid out as follows. Open theories of topological fields are studied in Section \[sec:topopen\]: dimension properties are considered in Section \[sec:dimension\]; correspondences are studied in Sections \[sec:correspon\] and \[sec:correspon\_gamma\]; and the cell decomposition theorem is presented in Section \[sec:celldecomp\]. Topological fields with a generic derivation are introduced in Section \[sec:dpmingen\]: consistency results are presented in Section \[sec:consistency\]; relative quantifier elimination is given in Section \[sec:relQE\] and its consequences are gathered in Section \[sec:conse\_QE\]. In Section \[sec:EI\] we show the transfer of elimination of imaginaries under the assumption of the open core property. The applications to dense pairs are presented in Section \[sec:app\]. Finally, the open core property is studied in Section \[sec:opencore\]. Some transfer proofs which were known in the one-sorted case (such as the transfer of NIP and distality) are gathered in the Appendix.
Acknowledgements {#acknowledgements .unnumbered}
----------------
We would like to thank Marcus Tressl for encouraging discussions and for sharing with us his proof strategy to show elimination of imaginaries in $\CODF$. It is precisely his strategy what we adapted in the the general setting. We would also like to thank Arno Fehm and Philip Dittmann for interesting discussions around henselian valued fields, and the Institute of Algebra of the Technische Universität Dresden for its hospitality during a visit of the second author in May 2019.
Open expansions of topological fields {#sec:topopen}
=====================================
Preliminaries {#sec:prelim}
-------------
### Model theory
We will follow standard model theoretic notation and terminology. Lower-case letters like $a, b, c$ and $x,y,z$ will usually denote finite tuples and we let $\ell(x)$ denote the length of $x$. We will sometimes use $\bar{x}$ to denote a tuple $\bar{x}=(x_1,\ldots, x_n)$ where each $x_i$ is a tuple. Let $\cL$ be a possibly multi-sorted language and $M$ be an $\cL$-structure. For a sort $S$ in $\cL$, we let $S(M)$ denote the elements of $M$ of sort $S$. For a single variable $x$, we let $S_x$ denote the sort of the variable $x$. Given a tuple of variables $x=(x_1,\ldots,x_n)$ we let $S_x(M)=S_{x_1}(M)\times \cdots\times S_{x_n}(M)$. For an $\cL$-formula $\varphi(x)$ we let $\varphi(M)$ denote the set $$\{a\in S_x(M) : M\models \varphi(a)\}.$$ We let $\cL(M)$ denote the extension of $\cL$ by constants for all elements in $M$. By an $\cL$-definable set of $M$ we mean definable with parameters, that is, of the form $\varphi(M)$ for an $\cL(M)$-formula $\varphi$. Given a complete $\cL$-theory $T$ we let $\mathbb{U}$ denote a monster model of $T$.
We let $\acl$ denote the model-theoretic algebraic closure operator on $M$. Given a sort $S$ in $\cL$, we let $\acl_S$ denote the model-theoretic algebraic closure restricted to $S$, that is, for any subset $C\subseteq S(M)$, we let $\acl_S(C)=\acl(C)\cap S(M)$. Note that $\acl_S$ is a closure operator on $S(M)$.
For a subset $X\subseteq R\times T$ where $R$ and $T$ are finite products of sorts in $\cL$, and for $a\in R$, the fiber of $X$ over $a$ is denoted by $X_{a}\coloneqq\{b\in T : (a, b)\in X\}$.
### Topological fields {#sec:topfields}
Throughout this article, every topological field will assumed to be non-discrete and Hausdorff.
Let $K$ be a field and $\tau$ be a topology on $K$ making it into a topological field. The topological closure of a set $X\subseteq K^n$ will be denoted by $\overline{X}$ and its interior by $\Int(X)$. The frontier of $X$, denoted $\mathrm{Fr}(X)$, is equal to the set $\overline{X}\setminus X$. The topological dimension of a non-empty subset $X\subseteq K^n$, denoted $\Dim(X)$, is defined as the maximal $\ell\leqslant n$ such that there is a projection $\pi\colon K^n\rightarrow K^{\ell}$ such that $\Int(\pi(X))\neq \emptyset$ (and equal to $-1$ if $X=\emptyset$).
We let $\cL_{\mathrm{ring}}$ denote the language of rings $\{\cdot,+,-,0,1\}$ and $\cL_{\mathrm{field}}\coloneqq\cL_{\mathrm{ring}}\cup\{{\,}^{-1}\}$ denote the language of fields. We treat every field is an $\cL_{\mathrm{field}}$-structure by extending the multiplicative inverse to 0 by $0^{-1}=0$. Let $\cL$ be a (possibly multi-sorted) language extending the language of rings and suppose $M$ is an $\cL$-structure. We say $\tau$ is an *$\cL$-definable field topology* if there is an $\cL$-formula $\chi_\tau(x,z)$ with $x$ a single variable of field sort such that $\{ \chi_\tau(M,a): a\in S_z(M)\}$ is a basis of neighbourhoods of 0. For example, if $M$ is an ordered field and the order is $\cL$-definable, then the order topology on $M$ is an $\cL$-definable field topology. Similarly, if $(M,v)$ is a valued field and the relation $\{(x,y)\in M^2 : v(x)\leqslant v(y)\}$ is $\cL$-definable, then the valuation topology on $M$ is an $\cL$-definable field topology.
When $K$ is a dp-minimal field, the following result of W. Johnson [@johnson18] guarantees the existence of a definable field topology
\[thm:johnson\]Let $(K,+,\cdot, \cdots)$ be an infinite field, possibly with extra structure. Suppose $K$ is dp-minimal but not strongly minimal. Then $K$ can be endowed with a non-discrete Hausdorff definable field topology such that any definable subset of $K$ has finite boundary. Furthermore, the topology is always induced either by a non-trivial valuation or an absolute value.
For more on dp-minimal fields, we refer to reader to [@jahnke_simon_walsberg_2017] and [@johnson18].
Open expansion of topological fields
------------------------------------
We will work in a first-order setting of topological fields which follows the same spirit of [@vandendries1989 Section 2], [@M] and [@guzy-point2010]. The main new ingredient of the present account is that we explicitly allow multi-sorted structures. Let $\cL_r$ be a relational extension of $\cL_{\mathrm{field}}$. For the rest of the article we will work in a possibly multi-sorted language $\cL$ extending $\cL_{r}$ such that $\cL$ and $\cL_r$ coincide on the field sort, and every new sort is a sort in $\cL_r^\eq$. When $\cL$ is multi-sorted and $K$ is an $\cL$-structure, we will abuse of notation and identify $K$ with the field sort and write any other sort of $K$ as $S(K)$ (for $S$ a sort in $\cL$).
Let $K_0$ be a field of characteristic 0 endowed with an $\cL$-structure and an $\cL$-definable field topology. Let $T$ be its $\cL$-theory. Any such theory will be called an *$\cL$-theory of topological fields*. They are first order theories of topological structures in the sense of Pillay [@pillay87]. We will further impose the following two conditions on $T$ which will be hereafter referred to as *assumption $(\mathbf{A})$*:
1. $T$ eliminates field sort quantifiers in $\cL$ and
2. for every tuple $x$ of field sort variables, every field sort quantifier free $\cL$-formula $\varphi(x)$ is equivalent modulo $T$ to a formula $$\bigvee_{i\in I}\bigwedge_{j\in J_i} P_{ij}(x)=0 \wedge \theta_i(x)$$ where $I$ is a finite set, each $J_i$ is a finite set (possibly empty), $P_{ij}\in \Q[x]\setminus \{0\}$ and $\theta_i(x)$ defines an open set in every model of $T$.
Any $\cL$-theory of topological fields satisfying assumption $(\mathbf{A})$ will be called an *open $\cL$-theory of topological fields*. Note that any open $\cL$-theory of topological fields $T$ is a complete $\cL$-theory.
For $K$ a model of $T$, when $\cL$ is one-sorted and both the relations of $\cL_r$ and their complement are interpreted in $K$ by the union of an open set and a Zariski closed set, such a model $K$ is also a *topological $\cL$-field* as defined in [@guzy-point2010].
### Examples
The theory $T=\mathrm{Th}(K_0)$ is an open $\cL$-theory of topological fields in the following cases:
1. when $K_0$ is a real closed field and $\cL$ is $\cL_{\mathrm{of}}$ the language of ordered fields ${\cL_{\mathrm{field}}\cup\{<\}}$. The definable topology is given by the order topology. We use in this case $\RCF$ for $T$.
2. When $K_0$ is an algebraically closed valued field of characteristic 0 and $\cL$ is the one-sorted language of valued fields $\cL_{\Div}=\cL_{\mathrm{field}}\cup\{\Div\}$. The definable topology corresponds to the valuation topology. We use in this case $\ACVF_{0,p}$ for $T$, where $p$ is a prime number or 0.
3. When $K_0$ is a real closed valued field and $\cL=\cL_{\mathrm{ovf}}$ is the language of ordered valued fields $\cL_{\mathrm{of}}\cup\{\Div\}$. We use in this case $\RCVF$ for $T$. The definable topology corresponds to both the order and the valuation topology (which coincide).
4. When $K_0$ is a $p$-adically closed field of $p$-rank $d$ and $\cL$ is $$\cL_{p, d}\coloneqq\cL_{\mathrm{field}}\cup \{\Div, c_{1},\cdots,c_{d}\}\cup \{P_{n}: n\geqslant 1\}$$ as defined in [@pre-ro-84]. The definable topology corresponds to the valuation topology. We use in this case $\PCF_d$ for $T$.
5. When $K_0$ is a henselian valued field of characteristic 0 and $\cL$ is the multi-sorted $\cL_\RV$-language as defined in [@flenner] (having $\cL_{\mathrm{field}}$ in the field-sort). The definable topology corresponds again to the valuation topology. Examples include classical fields such as $\mathbb{C}(\!(t)\!)$, $\mathbb{R}(\!(t)\!)$ and more generally any Hahn power series field $k(\!(t^\Gamma)\!)$, where $k$ is a field of characteristic 0 and $\Gamma$ is an ordered abelian group.
\[rem:sorts\] If $T=\mathrm{Th}(K_0)$ is an open $\cL$-theory of topological fields, then $T^{\eq}$ is an $\cL^{\eq}$-open theory of topological fields. In fact, if $(K_0,\cL')$ is an extension by definitions of a reduct of $(K_0,\cL^{\eq})$, then the $\cL'$-theory of $K_0$ is an $\cL'$-open theory of topological fields. For example, if $K_0$ is an algebraically closed valued field of characteristic $0$, its theory in the two-sorted language of valued fields with a new sort for the value group also satisfies assumption $(\mathbf{A})$.
\[rem:notdpmin\] Observe that most but not all theories in Examples \[examples\] are dp-minimal. Indeed, while theories in (1)-(4) are dp-minimal, there are various henselian fields of equicharacteristic 0 which are not dp-minimal. By a result of F. Delon [@delon81] combined with results of Y. Gurevich and P. H. Schmitt [@gurevich_schmitt], the Hahn valued field $k(\!(t^\Gamma)\!)$ is NIP if and only if $k$ is NIP (as a pure field). Even assuming NIP, by a result of A. Chernikov and P. Simon in [@chernikov_simon], when $k$ is algebraically closed, the field $k(\!(t^\Gamma)\!)$ is dp-minimal if and only if $\Gamma$ is dp-minimal. However, there are ordered abelian groups which are not dp-minimal, as follows by a characterization of pure dp-minimal ordered abelian groups due to F. Jahnke, P. Simon and E. Walsberg in [@jahnke_simon_walsberg_2017 Proposition 5.1].
Is there any open $\cL$-theory of topological fields whose topology does not come from an order or a valuation?
In the remainder of the section we prove various tameness properties of open theories of topological fields. To begin with, we will prove that such theories eliminate the field sort quantifier $\exists^\infty$ and are *algebraically bounded* in the sense of [@vandendries1989 Definition 2.6]. This implies that $\acl_K$ induces a dimension function on definable sets in the sense of [@vandendries1989] which we show that agrees with the topological dimension (Section \[sec:dimension\]). In particular, when $\cL$ is a one-sorted language, $T$ is a *geometric theory* in the sense of [@berenstein-vassiliev2010]. We will finish the section by showing that definable functions (and more generally definable correspondences) are continuous almost everywhere and that definable sets are finite unions of correspondences as in the cell decomposition theorem proved by Simon and Walsberg for non-strongly minimal dp-minimal fields in [@simon-walsberg2016 Proposition 4.1] (see Section \[sec:celldecomp\]).
We start with some notation preliminaries together with some basic but crucial lemmas from commutative algebra.
Some auxiliary lemmas from commutative algebra {#sec:commalg}
----------------------------------------------
Through this section, $K$ will denote any field of characteristic 0.
Let $x=(x_1,\ldots, x_n)$ be a tuple of variables and $y$ be a single variable. We will need to present Zariski closed subsets of $K^{n+1}$ as finite unions of locally Zariski closed sets with further properties on formal derivatives. It will thus be useful to work with presentations of ideals rather than with the ideals themselves. Let us introduce some notation.
Throughout Section \[sec:commalg\], we let $\cA$ be a finite subset of $K[x,y]$ and $R\in K[x,y]$. We let $$\begin{aligned}
\cA^y&\coloneqq\{P\in \cA\mid \deg_y(P)>0\}, \\
\deg_y(\cA)&\coloneqq\max\{\deg_y(P)\mid P\in\cA\}, &\\
\cA_{\max}^y&\coloneqq\{P\in \cA^y\mid \deg_y(P)=\deg_y(\cA)\}, \text { and } &\\
d_{\cA}&\coloneqq\sum_{P\in \cA_{\max}^y} \deg_y(P). \end{aligned}$$ We let the $\cL_{\mathrm{ring}}(K)$-formula $Z_\cA(x,y)$ be $$\bigwedge_{P\in \cA} P(x,y)=0.$$ Thus, the algebraic subset of $K^{n+1}$ defined by $\cA$ corresponds to $Z_\cA(K)$. For $P\in K[x,y]$, we let $Z_P$ denote $Z_{\{P\}}$. For an element $R\in K[x,y]$ we let $Z_\cA^R(x,y)$ be $$\label{eq:locclosed}
Z_\cA(x,y)\wedge R(x,y)\neq 0.$$
\[lem:degY\] The locally Zariski closed subset $Z_\cA^R(K)$ is the union of finitely many sets $Z_\cB^{S}(K)$ such that
1. $|\cB^y| \leqslant 1$;
2. $S\in K[x,y]$;
3. $\deg_y(\cB)\leqslant \deg_y(\cA)$.
If $|\cA^y|=1$ there is nothing to prove, so suppose that $|\cA^y|>1$. By induction, it suffices to show that $Z_\cA^R(K)$ is the union of finitely many sets of the form $Z_\cB^S(K)$ with $S\in K[x,y]$ and $\cB$ such that $\deg_y(\cB)\leqslant \deg_y(\cA)$ and $|\cB^y|<|\cA^y|$. We proceed by a second induction on $d_{\cA}$. Let $P_1\in \cA_{\max}^y$ and $P_2\in \cA^y$ be such that $P_1\neq P_2$. If $P_{2}=\sum_{i=0}^m c_i(x)y^i$ with $c_i\in K[x]$, by Euclid’s algorithm (see [@jacobson Lemma 2.14]), there is a positive integer $\ell$ and $Q, R_1\in K[x,y]$ such that $c_m^\ell P_{1} = P_{2}Q +R_{1}$. In addition, $\deg_y(R_{1})<\deg_y(P_{2})$. Setting $\cB\coloneqq\cA\cup\{c_m\}$, we have $$Z_\cA^R(K) = Z_\cA^{c_mR}(K) \cup Z_\cB^R(K).$$ Letting $\cA_1\coloneqq(\cA\setminus \{P_1\})\cup\{R_{1}\}$, we have that $Z_\cA^{c_mR}(K) = Z_{\cA_1}^{c_mR}(K)$. Since $d_{\cA_1}<d_{\cA}$, the result for $Z_{\cA_1}^{c_mR}$ follows by induction, and hence for $Z_{\cA}^{c_mR}$. For $Z_\cB^R(K)$, setting $$\cB_1=(\cA\setminus \{P_2\})\cup \{c_m, \sum_{i=0}^{m-1}c_i(x)y^i\},$$ we have $Z_\cB^{R}(K)=Z_{\cB_1}^R(K)$. Since $d_{\cB_1}<d_{\cB}\leqslant d_{\cA}$, the result for $Z_\cB^R(K)$ follows by induction, which completes the proof.
\[lem:degY2\] Suppose that $|\cA^y|=1$. Then, the locally Zariski closed set $Z_\cA^R(K)$ is a finite union of sets of the form $Z_\cB^S(K)$ where $S\in K[x,y]$ and either $\cB\subseteq K[x]$, or $\cB^y=\{P\}$ and $\frac{\partial}{\partial y} P$ divides $S$.
Let $P$ be the unique element of $\cA^y$ and write it as $P(x,y)=\sum_{i=0}^m c_i(x)y^i$ for $c_i\in K[x]$. We proceed by induction on $\deg_y(\cA)=\deg_y(P)$. First, note that for $\cA_0=\cA\cup\{\frac{\partial}{\partial y}P\}$ and $S=(\frac{\partial}{\partial y}P)R$ we have $$Z_\cA^R(K) = Z_\cA^S(K) \cup Z_{\cA_0}^R(K).$$ Since $Z_\cA^S(K)$ has already the desired form, it suffices to show that $Z_{\cA_0}^R$ is the union of finitely many locally closed sets as in the statement. By Euclid’s algorithm, there is a positive integer $\ell$ and $Q,R_1\in K[x,y]$ such that $c_m^\ell P = (\frac{\partial}{\partial y} P)Q + R_1$ with $\deg_y(R_1)<\deg_y(\frac{\partial}{\partial y} P)$. Setting $\cA_1\coloneqq\cA_0\cup\{c_m\}$, we have $$Z_{\cA_0}^R(K) = Z_{\cA_0}^{c_mR}(K) \cup Z_{\cA_1}^R(K).$$ As before, letting $\cA_2\coloneqq(\cA_0\setminus \{P\})\cup\{R_1\}$, we have that $Z_{\cA_0}^{c_mR}(K) = Z_{\cA_2}^{c_mR}(K)$. Moreover, $d_{\cA_2}<d_{\cA_0}$. By Lemma \[lem:degY\], $Z_{\cA_2}^{c_mR}(K)$ is the union of finitely many locally closed sets $Z_{\cB}^S$ with $|\cB^y|=1$, $S\in K[x,y]$ and $d_{\cB}\leqslant d_{\cA_2}<d_{\cA_0}$, so the result follows by induction for each such set. It remains to show the result for $Z_{\cA_1}^R(K)$. Setting $$\cA_3\coloneqq(\cA\setminus \{P\})\cup \{c_m, \sum_{i=0}^{m-1} c_i(x)y^i, \sum_{i=1}^{m-1} i c_i(x)y^{i-1} \},$$ we have $Z_{\cA_1}^{R}(K) = Z_{\cA_3}^{R}(K)$. In addition, $\deg_y(\cA_3)<\deg_y(\cA_1)$. The result follows again by Lemma \[lem:degY\] and the induction hypothesis.
\[cor:goodform\] Every locally Zariski closed subset of $K^{n+1}$ can be written as a union of sets of the form $Z_{\cB}^{S}(K)$ where $S\in K[x,y]$ and either $\cB\subseteq K[x]$, or $\cB^y=\{P\}$ and $\frac{\partial}{\partial y} P$ divides $S$.
Direct consequence of Lemmas \[lem:degY\] and \[lem:degY2\].
\[cor:niceform1\] Let $T$ be an open $\cL$-theory of topological fields and $K$ be a model of $T$. Then every $\cL$-definable subset of $K^{n+1}$ is defined by $$\bigvee_{j\in J} Z_{\cA_j}^{S_j}(x,y) \wedge \theta_j(x,y)$$ where $x=(x_0,\ldots, x_{n-1})$, $y$ a single variable, and for each $j\in J$, $\theta_j$ is an $\cL(K)$-formula that defines an open subset of $K^n$, $S_j\in K[x,y]$, and either
1. $\cA_j\subseteq K[x]\setminus\{0\}$ or
2. $\cA_j\subseteq K[x,y]\setminus\{0\}$, $\cA_j^y=\{P_j\}$ and $\frac{\partial}{\partial y} P_j$ divides $S_j$.
Follows from assumption $(\mathbf{A})$ and Corollary \[cor:goodform\].
Topological dimension and the algebraic closure {#sec:dimension}
-----------------------------------------------
Through this section, we let $T$ be an open $\cL$-theory of topological fields and $K$ be a model of $T$.
Recall that an integral domain $D$ is *algebraically bounded* (in the sense of [@vandendries1989]) if for every definable subset $X\subseteq D^{n+1}$ there exist non-zero polynomials $P_1,\ldots,P_m\in D[x_1,\ldots, x_n,y]$ such that for every $a\in D^n$, if $X_a$ is finite, then $X_a\subseteq Z_{P_i(a,y)}(D)$ for some $i\in \{1,\ldots,m\}$. Being algebraically bounded implies that the algebraic dimension $\mathrm{alg}\Dim$, in the sense of van den Dries [@vandendries1989 Lemma 2.3], defines a *dimension function* on definable subsets of $D$ [@vandendries1989 Proposition 2.7]. Given a definable set $X\subseteq K^n$, $\mathrm{alg}\Dim(X)$ is the maximal integer $k$ for which there is $a\in X(\mathbb{U})$ such that the field extension $K(a)|K$ has transcendence degree $k$. When $\acl_K$ has the exchange property, we let $\Dim_{\mathrm{acl}_K}$ denote the induced dimension function.
\[prop:conseAA1\] The field sort of every model of $T$ is algebraically bounded. The algebraic dimension $\mathrm{alg}\Dim$ on the field sort coincides with $\Dim_{\mathrm{acl}_K}$ and defines a dimension function in the sense of [@vandendries1989]. In particular, $T$ eliminates the field sort quantifier $\exists^{\infty}$. When $\cL$ is a one-sorted language, $T$ is thus a geometric theory.
Since open sets are infinite, algebraic boundedness follows directly from assumption $(\mathbf{A})$. It also follows from assumption $(\mathbf{A})$ that $\mathrm{alg}\Dim$ coincides with $\Dim_{\mathrm{acl}_K}$. That $\mathrm{alg}\Dim$ defines a dimension function as defined in [@vandendries1989] follows by [@vandendries1989 Proposition 2.15]. The remaining properties are straightforward.
\[lem:int\_zar\] Let $P(x)\in K[x]\setminus\{0\}$ with $x=(x_1,\ldots,x_n)$. Then $\dim(Z_P(K))<n$.
By induction on $n$. For $n=1$, we have that $Z_{P}(K)$ is finite and the result is clear. Suppose the result holds for all $k<n+1$, let $y$ be a single variable and $P\in K[x,y]$. Write $P$ as $\sum_{i=0}^d c_i(x)y^i$ where $c_i\in K[x]$. Suppose for a contradiction that $Z_P(K)$ contains an open set $U\times V$ where $U\subseteq K^n$ and $V\subseteq K$. For $a\in U$, if $\bigvee_{i=1}^d c_i(a)\neq 0$, then the fiber $Z_P(K)_a$ would be finite, contradicting that it contains the infinite set $V$. Therefore, for every $a\in U$ we have $\bigwedge_{i=0}^d c_i(a)= 0$. But this contradicts the induction hypothesis since for every $i\in\{0,\ldots,d\}$ $$U\subseteq \{a\in K^n: \bigwedge_{i=0}^d c_i(a)= 0\} \subseteq Z_{c_j}(K) \subseteq K^n, {\rm where}\;\; c_{j}\neq 0.$$
\[cor:dens\_zar\] Let $P\in K[x]\setminus\{0\}$ with $x=(x_1,\ldots,x_n)$. Then the set $D(P)\coloneqq K^n\setminus Z_P(K)$ is open and dense in $K^n$ with respect to the ambient topology.
\[prop:conseAA2\] For every $n\geqslant 1$ and every definable subset $X\subseteq K^n$, $\Dim(X)=\Dim_{\mathrm{acl}_K}(X)$.
Suppose $X$ is defined over $C\subseteq K$ and that $\Dim_{\mathrm{acl}_K}(X)=k$. Let $a\in X(\mathbb{U})$ be such that $(a_{i_1},\ldots,a_{i_k})$ is an algebraically independent tuple over $C$. Let $I=\{i_1,\ldots,i_k\}$ and $\pi_I\colon K^n\to K^k$ be its corresponding projection. Letting $\pi(x)\coloneqq(x_{i_1},\ldots,x_{i_k})$, by assumption $(\mathbf{A})$, $\pi_I(X)$ is defined by $$\bigvee_{i\in I}\bigwedge_{j\in J_i} P_{ij}(\pi_I(x))=0 \wedge \theta_i(\pi_I(x)),$$ where each $P_{ij}$ is a non-zero polynomial with coefficients in $C$ and $\theta_i(\pi_I(x))$ defines an open subset of $K^k$. If for every $i\in I$, the set $J_i\neq\emptyset$, then $(a_{i_1},\ldots,a_{i_k})$ would be algebraically dependent over $C$. Thus, there is $i\in I$ such that $J_i=\emptyset$. This shows $\Int(\pi_I(X))\neq\emptyset$ and therefore $\Dim_{\mathrm{acl}_K}(X)\leqslant \Dim(X)$. Conversely, suppose $\Dim(X)=d$ and let $\pi_I\colon K^n\to K^d$ be such that $\pi_I(X)$ has non-empty interior. It suffices to show that the open set $\Int(\pi_I(X))$ contains (in $\mathbb{U}$) an algebraically independent tuple. This follows by compactness and Corollary \[cor:dens\_zar\].
\[cor:dim1\] The topological dimension satisfies the following properties for definable sets $X, Y\subseteq K^n$:
1. $\Dim(X)=0$ if and only if $X$ is finite and non-empty,
2. $\Dim(X\cup Y)=\max(\Dim(X),\Dim(Y))$.
3. $\Dim(\mathrm{Fr}(X))<\Dim(X)=\Dim(\overline{X})$,
4. $\Dim$ is additive, that is: for a non-empty definable set $X\subseteq K^{m+n}$ and $d \in \{0,1, \ldots, n\}$, $$\dim \bigcup_{a \in X(d)} X_a= \dim(X(d)) + d,$$ where $X(d) = \{ a \in K^m : \dim X_a = d \}$ and is definable.
Points (1), (2) and (4) follows by Proposition \[prop:conseAA1\]. Point (3) follows by Proposition \[prop:conseAA1\] and [@vandendries1989 Proposition 2.23].
Uniform structures {#sec:uniform_struc}
------------------
In Sections \[sec:correspon\] and \[sec:celldecomp\] we will closely follow various results from [@simon-walsberg2016]. For the reader’s convenience, and to make easier the comparison with [@simon-walsberg2016], we will recall part of their setting and notation.
A *basis* for a uniform structure on a set $A$ is a collection $\mathcal{B}$ of subsets of $A^2$ satisfying the following:
1. the intersection of the elements of $\mathcal{B}$ is the diagonal of $A^2$;
2. if $U \in\mathcal{B}$ and $(x,y)\in U$, then $(y,x)\in U$;
3. for all $U, V\in \mathcal{B}$ there is $W \in\mathcal{B}$ such that $W \subseteq U\cap V $;
4. for all $U \in\mathcal{B}$ there is a $W \in\mathcal{B}$ such that $W\circ W\subseteq U$, where $$W\circ W=\{(x,z)\in A^2: (\exists y\in A)(x,y)\in W, (y,z)\in W\}.$$
The collection $\mathcal{B}$ induces a topology on $A$ by setting as a neighbourhood basis for $a\in A$, the collection $\{U[a] : U\in \mathcal{B}\}$ where $$U[a]\coloneqq\{x\in A : (a,x)\in U\}.$$
Suppose $M$ is an $\cL$-structure for some first-order language $\cL$. Let $S$ be a sort in $\cL$ and suppose $S(M)=A$. We say $\mathcal{B}$ is a *definable uniform structure on $S$* (or a *definable basis for a uniform structure on $S$*) if there is an $\cL$-formula $\varphi(x,y,z)$ with $\ell(x)=\ell(y)=1$ variables of sort $S$ such that $$\mathcal{B}=\{\varphi(M, c) : c\in D\subseteq S_z(M)\}$$ for some $0$-definable set $D$.
Let $K$ be a field endowed with an $\cL$-structure and an $\cL$-definable topology $\tau$. Let $\chi_\tau(x,z)$ be an $\cL$-formula defining a basis of open neighbourhoods of 0. The collection $\mathcal{B}_K=\{W_c : c\in S_z(K)\}$ where $$W_c\coloneqq\{(a,b)\in K^2 : K\models \chi_\tau(a-b,c)\}$$ is a definable uniform structure on $K$ having $\tau$ as its induced topology.
We will also need to equip certain ordered abelian groups extended by an infinitely large element $\infty$ with a definable uniform structure. Let $\Gamma\coloneqq(\Gamma,+,-,0,<)$ be an ordered abelian group and $\Gamma_\infty\coloneqq\Gamma\cup \{\infty\}$ for $\infty $ a new element satisfying, for all $\gamma\in \Gamma$:
1. $\gamma<\infty$,
2. $\gamma+\infty=\infty+\gamma=\infty$,
3. $\infty+\infty=\infty$,
4. $-\infty=\infty$.
Let $\cL_{\ogr}=\{+,-,0,<\}$ be the language of ordered groups and $\cL_{\ogr}^\infty$ be $\cL_{\ogr}$ extended by a new constant symbol $\infty$. Let $\cL$ be a language extending $\cL_{\ogr}^\infty$ and $\Gamma_\infty$ be an ordered abelian group equipped with an $\cL$-structure. For $\gamma\in \Gamma_\infty$ we let $|\gamma|$ denote $\gamma$ if $\gamma\geqslant 0$ or $-\gamma$ otherwise. Consider the following $\cL_{\ogr}$-definable family $$\mathcal{B}_\Gamma\coloneqq\{W_{\gamma,\xi}: \gamma,\xi\in \Gamma, 0< \gamma,\xi\}$$ where $$W_{\gamma,\xi}\coloneqq\{(x,y)\in \Gamma_\infty^2: |x-y|<\gamma \vee (x=\infty \wedge \xi<y) \vee (y=\infty \wedge \xi<x)\}$$
\[lem:uni\_structure\_gamma\] Let $\Gamma$ be an ordered abelian group which is either divisible or discrete. Then, the collection $\mathcal{B}_\Gamma$ is a definable uniform structure on $\Gamma_\infty$.
Conditions (1)-(3) are straightforward and hold for any group $\Gamma$.
For condition (4), suppose first $\Gamma$ is divisible and fix two strictly positive elements $\gamma,\xi\in \Gamma$. Let $\gamma'$ be such that $0<2\gamma'<\gamma$ and $\xi'\coloneqq\xi+\gamma'$. The reader may check that $W_{\gamma',\xi'}\circ W_{\gamma',\xi'}\subseteq W_{\gamma,\xi}$. Now suppose $\Gamma$ is discrete and let $1$ denote the minimal strictly positive element of $\Gamma$. Then, for any strictly positive elements $\gamma,\xi\in \Gamma$ we have that $W_{1,\xi}\circ W_{1,\xi}\subseteq W_{\gamma,\xi}$.
The induced topology on $\Gamma_\infty$ by $\mathcal{B}_\Gamma$ is the order topology extended by open sets of the form $(\gamma,\infty]$ for every $\gamma\in \Gamma$.
Almost continuity of definable correspondences {#sec:correspon}
----------------------------------------------
In the absence of finite Skolem functions, we need to deal with the more general concept of correspondence which we now recall (see also [[@simon-walsberg2016 Section 3.1]]{}).
\[def:correspondence\] A [*correspondence*]{} $f\colon E\rightrightarrows K^\ell$ consists of a definable set $E$ together with a definable subset $\mathrm{graph}(f)$ of $E\times K^\ell$ such that $$0< \vert\{y\in K^\ell : (x,y)\in \mathrm{graph}(f)\}\vert<\infty, \text{ for all } x\in E.$$ The set $\{y\in K^\ell : (x,y)\in \mathrm{graph}(f)\}$ is also denoted by $f(x)$. For a positive integer $m$, we say $f$ is an *$m$-correspondence* if $|f(x)|=m$ for all $x\in E$. The correspondence $f$ is [*continuous*]{} at $x\in E$ if for every open set $V\subseteq K^\ell$ containing $f(x)$, there is an open neighbourhood $U$ of $x$ such that $f(U)\subseteq V$.
Note that a $1$-correspondence can be trivially identified with a function. The following lemma is a reformulation of [@simon-walsberg2016 Lemma 3.1].
\[lem:corre\_local\_cont\] Let $U \subseteq K^n$ be open and let $f \colon U \rightrightarrows M^\ell$ be a continuous $m$-correspondence. Every $a\in U$ has a neighbourhood $V$ such that there are continuous functions $g_1,\ldots,g_m\colon V \to M^\ell$ such that $\Graph(g_i) \cap \Graph(g_j)=\emptyset$ when $i\neq j$ and $$\Graph(f_{|V})= \Graph(g_1) \cup \cdots \cup \Graph(g_m).$$ In addition, if $f$ is definable, we can further choose $V$ and the functions $g_i$ to be definable.
\[convention\] Let $f\colon U\subseteq K^m\rightrightarrows K^{n}$ be a correspondence. If $m=0$, we identify $\mathrm{graph}(f)$ with a finite subset of $K^{n}$. If $U$ is an open subset and $n=0$, then we identify $\mathrm{graph}(f)$ with the set $U$.
\[convention2\] Given a definable set $X$, we say that a property holds *almost everywhere on $X$* if there is a definable subset $Y\subseteq X$ such that $\dim(X\setminus Y)<\dim(X)$ and the property holds on $Y$.
The following result is a reformulation of [@simon-walsberg2016 Proposition 3.7] in which we isolate the components of its proof in an axiomatic way. Recall that a family of sets $F$ is said to be *directed* if for every $A,B\in F$ there is $C\in F$ such that $A\cup B\subseteq C$.
\[prop:continuity\] Let $T$ be an $\cL$-theory of topological fields (not necessarily satisfying assumption $(\mathbf{A})$) and $K$ be a model of $T$. Suppose $K$ satisfies the following properties
1. if $A$ is a definable open subset of $K^n$ and $B$ is a definable subset of $A$ which is dense in $A$, then the interior of $B$ is dense in $A$; in particular, $\dim(A\setminus B)<\dim(A)$.
2. if $A, A_{1},\ldots, A_{k}$ are definable subsets of $K^n$, $A$ is open and $A=\bigcup_{i=1}^{k} A_{i}$, then, $\Int(A_{i})\neq \emptyset$ for some $1\leqslant i\leqslant k$.
3. if $C\subseteq K^{m+n}$ is a definable set such that the definable family $\{C_{a} : a\in K^m\}$ is a directed family and $\bigcup_{a\in K^m} C_{a}$ has non-empty interior, then, there is $a\in K^m$ such that $C_a$ has non-empty interior.
4. There is no infinite definable discrete subset of $K^{n}$.
Then, for $V \subseteq K^n$ a definable open set, every definable correspondence $f\colon V \rightrightarrows K^\ell$ is continuous on an open dense subset of $V$, and thus is continuous almost everywhere on $V$.
The proof is a word by word analogue after replacing [@simon-walsberg2016 Lemma 2.6] by condition (1), [@simon-walsberg2016 Corollary 2.7] by condition (2), [@simon-walsberg2016 Lemma 3.5] by condition (3) and [@simon-walsberg2016 Lemma 3.6] by condition (4).
We will now show that all four conditions in Proposition \[prop:continuity\] hold for open theories of topological fields. Note that condition (2) already follows from Corollary \[cor:dim1\]. For the remaining of this section we assume $T$ is an open $\cL$-theory of topological fields and let $K$ be a model of $T$.
\[lem:condi1\] If $A$ is a definable open subset of $K^n$ and $B$ is a definable subset of $A$ which is dense in $A$, then the interior of $B$ is dense in $A$. In particular, $\dim(A\setminus B)<\dim(A)$.
It suffices to show that $\Int(B)\neq\emptyset$. By assumption [**(A)**]{}, $B$ is defined by a formula of the form $\bigvee_{i\in I}\bigwedge_{j\in J_i} P_{ij}(x)=0 \wedge \theta_i(x)$, where $P_{ij}\in K[x]\setminus \{0\}$ and $\theta_i(x)$ defines an open set in every model of $T$. Suppose that for every $i\in I$, the set of indices $J_i$ is non-empty. Therefore, the algebraic dimension of $B$ is strictly less than $n$. Since the topological and the algebraic dimension coincide, $\dim(B)<n$. By Corollary \[cor:dim1\], $\dim(A\setminus B)=n$ which implies there is an open subset $U\subseteq A$ disjoint from $B$. This contradicts the density of $B$ in $A$. Then, there must be $i\in I$ such that $J_i=\emptyset$, hence $\Int(B)\neq\emptyset$.
\[lem:condi3\] Suppose $C\subseteq K^{m+n}$ is a definable set inducing a definable family $\{C_{a} : a\in K^m\}$ which is directed. If $\bigcup_{a\in K^m} C_{a}$ has non-empty interior, then there is $a\in K^m$ such that $C_a$ has non-empty interior.
Let $\varphi(x,y)$ with $\ell(x)=m$ and $\ell(y)=n$ be an $\cL(K)$-formula defining $C$. Let $Y\subseteq K^n$ denote the definable set $\bigcup_{a\in K^m} C_a$. By hypothesis, $\Int(Y)\neq\emptyset$. Since the family $\{C_a: a\in K^m\}$ is directed, we may assume there are infinitely many different $C_{a}$ in the family (as otherwise the result follows directly from Corollary \[cor:dim1\]).
For $y=(y_1,\ldots, y_n)$, we let $\tilde y$ denote the tuple $(y_1,\ldots, y_{n-1})$. By Corollary \[cor:niceform1\] applied to the formula $\varphi(x,y)$ with respect to the variable $y_{n}$, $\varphi(x,y)$ is equivalent to a finite disjunction $\bigvee_{i\in I} \varphi_i$ where each $\varphi_i$ is of the form $Z_{\cA_i}^{S_i}(x,y)\wedge \theta_i(x,y)$ where $\theta_i(K)$ defines an open subset of $K^{m+n}$ and either
1. $\cA_i\subset K[x, \tilde y]$, or
2. $\cA_i^{y_{n}}=\{P_i\}$ and $\frac{\partial }{\partial y_{n}} P_i$ divides $S_i$.
Collect all the subformulas of the disjunction of form (1) (resp. form (2)) and denote by $\varphi^{1}(x,y)$ (resp. $\varphi^{2}(x,y)$) their disjunction. We have that $\varphi(x,y)=\varphi^1(x,y)\vee \varphi^2(x,y)$.
Note that if $\cA_i=\emptyset$ for some $i\in I$, then each fiber $C_a$ contains the open set $\theta_i(a,K)$ and has therefore non-empty interior. Thus, we may assume that $\cA_i\neq \emptyset$ for all $i\in I$. We proceed by induction on $n$. Let $d$ be the maximum of the degrees (in $y_{n}$) of the polynomials occurring in all $\cA_i$’s.
Assume $n=1$. Suppose first that $\cA_i\subseteq K[x]$ for some $i\in I$. Then, the fiber $C_a$ contains an open set whenever $Z_{\cA_i}^S(a,K)\neq \emptyset$. If $Z_{\cA_i}(a, K)= \emptyset$ for every $a\in K^m$, then we remove the corresponding member from the disjunction. Therefore, we are left with the case where $\varphi(x,y)=\varphi^2(x,y)$. We show this case cannot happen. First, note that in this situation each fiber $C_a$ has finite cardinality bounded by $d|I|$. Since the family $\{C_a:a\in K^m\}$ is directed, there is $a_{0}\in K^m$ such that $\varphi(a_{0}, K)=Y$. But this contradicts that $Y$ contains an open set (and is thus infinite). This concludes the case $n=1$.
Now assume $n>1$. Let $\pi\colon K^{n}\to K^{n-1}$ denote the projection onto the first $n-1$ coordinates. For $(a,u)\in K^{m}\times K^{n-1}$ we denote by $C_{a,u}$ the fiber $(C_a)_u=\{b\in K: (a,u,b)\in C\}$. By the form of each formula $\varphi_i$, each fiber $C_{a,u}$ either contains a non-empty open subset or is finite (and bounded by $d|I|$). We uniformly partition the projection $\pi(C_a)$ of each fiber $C_a$ into sets $\pi(C_a)_1$ and $\pi(C_a)_2$ where $$\begin{aligned}
& \pi(C_a)_1\coloneqq\{u\in K^{n-1}: C_{a,u} \text{ contains an open set }\}\text{ and } \\
& \pi(C_a)_2\coloneqq\{u\in K^{n-1}: |C_{a,u}|\leqslant d|I|\}. \end{aligned}$$ Since the definition is uniform, we have $$\pi(Y)= \pi(\bigcup_{a\in K^m} C_a)= \bigcup_{a\in K^m} \pi(C_a) = \bigcup_{a\in K^m} \pi(C_{a})_1 \cup \bigcup_{a\in K^m} \pi(C_{a})_2.$$ As $Y$ contains an open set, so does $\pi(Y)$. Therefore, by Corollary \[cor:dim1\], either $$\pi(Y)_2\coloneqq\bigcup_{a\in K^m} \pi(C_{a})_2\setminus (\bigcup_{a\in K^m} \pi(C_{a})_1) \text{ or } \pi(Y)_1\coloneqq\bigcup_{a\in K^m} \pi(C_{a})_1,$$ contains an open set.
The set $\pi(Y)_1$ must contain an open set.
Suppose for a contradiction $\dim(\pi(Y)_1)<n-1$. Partition $Y$ into $Y_1\cup Y_2$ where $$Y_i=\{(u,b)\in K^n: u\in \pi(Y)_i\} \text{ for $i=1,2$ }.$$ By Corollary \[cor:dim1\], $Y_1$ or $Y_2$ contains an open subset. By construction, we have that $\pi(Y_1)=\pi(Y)_1$ and, by assumption, $\dim(\pi(Y)_1)<n-1$. Therefore $\dim(Y_1)<n$. Hence we must have that $\dim(Y_2)=n$. By the additivity of the dimension function (Corollary \[cor:dim1\]), there must be $u\in \pi(Y_2)$ such that the fiber $(Y_2)_u$ is infinite. In particular, there are $k\in \N$ and $a_1,\ldots, a_k\in K^m$ such that $\bigcup_{j=1}^k C_{a_j,u}$ has cardinality bigger than $d|I|$. Since the family $\{C_{a,u}: a\in K^m\}$ is directed, there is $a\in K^m$ such that $\bigcup_{j=1}^k C_{a_j,u}\subseteq C_{a,u}$, which contradicts the fact that the fiber $C_{a,u}$ has cardinality smaller or equal than $d|I|$. This completes the claim.
Consider the directed family $\{\pi(C_{a})_1: a\in K^m\}$. By the claim, $\pi(Y)_1=\bigcup_{a\in K^m} \pi(C_{a})_1$ contains a non-empty open set. Therefore, by induction, there is $a\in K^m$ such that $\pi(C_{a})_1$ contains an open subset, say $U\subseteq K^{n-1}$. For each $i\in I$, set $$U_i\coloneqq\{u\in U : \dim(Z_{\mathcal{A}_i}^{S_i}(a,u,K)\cap\theta_i(a,u,K))=1 \}.$$ Given that $U\subseteq \pi(Y)_1$, we have that $U=\bigcup_{i\in I} U_i$. Then, by Corollary \[cor:dim1\], there is $i\in I$ such that $U_i$ contains an open set, say $V\subseteq U_i$. By the definition of $U_i$, the set $Z_{\mathcal{A}_i}^{S_i}(a,u,K)\cap \theta_i(a,u,K)$ is infinite for every $u\in V$. Thus, since $\mathcal{A}_i\neq \emptyset$, we must have $\mathcal{A}_i\subseteq K[x,\tilde y]$. But in this situation $Z_{\mathcal{A}_i}(a,K)=K^n$. Indeed, consider $\mathcal{B}=\mathcal{A}_i$ as a set of polynomials in $K[x,\tilde y]$. Then $Z_\mathcal{B}(a,K)\subseteq K^{n-1}$ contains an open set, namely, $V$. By Lemma \[lem:int\_zar\], $Z_{\mathcal{B}}(a,K)=K^{n-1}$, and thus $Z_{\mathcal{A}_i}(a,K)=K^n$ (as a subset of $K^n$). Therefore, the fiber $C_a$ contains the open set $$\{(u,b)\in V\times K: S_i(a,u,b)\neq 0 \wedge \theta_i(a,u,b)\}.$$ This is indeed open, as the set defined by the formula $S_i(x,\tilde y,y_n)\neq 0\wedge \theta_i(x,\tilde y,y_n)$ defines a non-empty open subset of $K^{m+n}$ by Corollary \[cor:dens\_zar\].
\[lem:condi4\] There is no infinite definable discrete subset of $K^{n}$.
The proof goes by induction on $n$ exactly as the proof of [@simon-walsberg2016 Lemma 3.6]. The base case follows directly straightforward by assumption $(\mathbf{A})$. The inductive case follows word by word the proof in [@simon-walsberg2016].
\[prop:acont\] Suppose $T$ is an open $\cL$-theory of topological fields and let $K$ be a model of $T$. For a definable open set $V \subseteq K^n$, every definable correspondence $f\colon V \rightrightarrows K^\ell$ is continuous almost everywhere.
This follows from Proposition \[prop:continuity\] where conditions (1)-(4) correspond respectively to Lemma \[lem:condi1\], Corollary \[cor:dim1\], Lemma \[lem:condi3\] and Lemma \[lem:condi4\].
### Almost continuity of definable functions to the value group {#sec:correspon_gamma}
In this section we let $K$ be a model of an open $\cL$-theory of topological fields, where $\cL$ is a language extending the language of valued fields $\cL_\Div$ containing a sort for the value group $\Gamma_\infty$. We will prove a result analogous to Proposition \[prop:acont\] for definable functions $f\colon U\subseteq K^n \to \Gamma_\infty$, where $U$ is an open set. We will show such a result when:
1. there is a model $K'$ of $T$ for which $\Gamma(K')$ is a divisible ordered abelian group in which every infinite $\cL$-definable set has an accumulation point;
2. there is a model $K'$ of $T$ with $\Gamma(K')=\Z$.
Observe that when $\Gamma(K')$ is a pure divisible ordered abelian group, every infinite definable set contains a non-empty interval, and therefore $(\dagger)$ is satisfied. Similarly, when $\Gamma(K')$ is a pure $\Z$-group, then $(\dagger\dagger)$ is satisfied. This covers most examples listed in Examples \[examples\]. In addition, by Lemma \[lem:uni\_structure\_gamma\], we have an $\cL$-definable uniform structure on $\Gamma_\infty$ in both contexts.
Since $\Gamma_\infty$ is totally ordered, every $\Gamma_\infty$-valued definable correspondence can be definably decomposed into finitely many definable $\Gamma_\infty$-valued functions. Thus, we do not need to work with definable correspondences but only with definable functions.
\[prop:ae\] $K$ be a model of an open $\cL$-theory of topological fields, where $\cL$ is a language extending the language of valued fields $\cL_\Div$ containing a sort for the value group $\Gamma_\infty$. Assume $T$ satisfies either $(\dagger)$ or $(\dagger\dagger)$. Then, for every $\cL$-definable open set $V\subseteq K^n$, any $\cL$-definable function $f\colon V \to \Gamma_\infty$ is continuous almost everywhere.
Since the property stated in the proposition is an elementary property, we may suppose $K$ is a model of $T$ as in $(\dagger)$ (resp. $(\dagger\dagger)$). The proof follows the same strategy as in [@simon-walsberg2016 Proposition 3.7]. However, since there are two uniform structures at play, namely the uniform structure on $K$ and the uniform structure on $\Gamma_\infty$, we include a proof for the reader’s convenience. We let $\cB_K$ be the uniform structure of $K$ and $\cB_\Gamma$ be the uniform structure of $\Gamma$ (as defined in \[sec:uniform\_struc\]). Suppose for a contradiction there are a definable open set $V\subseteq K^n$ and a definable function $f\colon V\to \Gamma_\infty$ which is discontinuous at every point in $V$. Let $n$ be minimal with this property.
Assume $n=1$. Consider the definable set $B\subseteq \cB_\Gamma\times V$ of pairs $(W,a)$ such that for each open neighbourhood $U$ of $a$, there exists $b\in U$, such that $(f(a),f(b)) \notin W$. Observe that the family of fibers $\{B_W\colon W\in \cB_\Gamma\}$ is a directed definable family. Indeed, for any $W_1, W_2\in\cB_\Gamma$, taking $W_3\in \cB_\Gamma$ such that $W_3\subseteq W_1\cap W_2$, we obtain $B_{W_1}\cup B_{W_2}\subset B_{W_3}$. By assumption on $V$ and $f$, $V=\bigcup_{W\in\cB_\Gamma} B_W$. Therefore, by Lemma \[lem:condi3\], there is $W\in \cB_\Gamma$ such that $\Int(B_W)\neq \emptyset$. Let $O$ be an open definable subset of $B_W$. Consider the definable set $f(O)\subseteq \Gamma_{\infty}$. If $f(O)$ is finite, then $f$ would be constant (and hence continuous) on an open definable subset of $O$, which contradicts the assumption. So suppose $f(O)$ is infinite. If $\Gamma$ satisfies $(\dagger)$, then $f(O)$ has an accumulation point $\gamma\in \Gamma_\infty$. If $\Gamma=\Z$, $f(O)$ must be either coinitial or cofinal. By considering $-f$, we may assume without loss of generality $f(O)$ is cofinal. In this case, set $\gamma=\infty$. Let $W'\in \cB_\Gamma$ be such that $W'\circ W'\subseteq W$ and consider the set $A=\{a\in O\colon (f(a),\gamma)\in W'\}$. In both cases, by the choice of $\gamma$, the set $A$ is an infinite definable subset of $O$ and therefore contains a non-empty open subset $O'$. For $a\in O'$, since $O'\subset B_W$, there is $b\in O'$ such that $(f(a),f(b))\notin W$. However $(f(a),\gamma)\in W'$ and $(f(b),\gamma)\in W'$, so $(f(a),f(b))\in W$, a contradiction.
Suppose $n\geqslant 2$. We may assume $V$ is an open box of the form $V_{0}\times V_{1}$, where $V_{0}\subseteq K^{1}$ and $V_{1}\subseteq K^{n-1}$. For each $\bar y\in V_{1}$, let $f_{\bar y}\colon V_{0} \to \Gamma_{\infty}$, where $f_{\bar y}\coloneqq f(t,\bar y)$. By the minimality of $n$, $f_{\bar y}$ is continuous almost everywhere for each $\bar y\in V_1$. Thus, by additivity of dimension (Corollary \[cor:dim1\]), $$\dim(\{(t,\bar y)\colon f_{\bar y} \text { is discontinuous at $t$ }\} < n.$$ By replacing $V_0$ and $V_1$ by smaller open definable subsets, we may suppose $f_{\bar y}$ is continuous on $V_0$ for every $\bar y\in V_1$. Defining $B\subseteq \cB_\Gamma\times V$ as for $n=1$, we may assume that there is $W\in \cB_\Gamma$ such that $B_W$ has non-empty interior and for every $a\in B_W$ and every neighbourhood $U$ of $a$ there is $b\in U$ such that $(f(a),f(b))\notin W$. Let $W'\in \cB_\Gamma$ be such that $W'\circ W'\subset W$. Consider the definable family $C\subseteq \cB_K\times (V_0\times V_1)$ formed of pairs $(O,(t,\bar b))$ such that if $t'\in O[t]$, then $(f_{\bar y}(t),f_{\bar y}(t')) \in W'$. Similarly as for $n=1$, the definable family $\{C_O : O\in \cB_K\}$ is directed and covers $V_0\times V_1$. By Lemma \[lem:condi3\], there is an $O\in \cB_K$ such that $C_O$ has non-empty interior. Possibly replacing $V_0\times V_1$ by smaller open subsets, we assume both that $V_0\times V_1\subset B_O$ and $V_0\times V_0\subseteq O$. Fix $t\in V_0$ and let $f^{t}\colon V_1\to \Gamma_\infty$ be the function $f^t(y)=f(t,y)$. By the minimality of $n$, there is $\bar z\in V_{1}$ such that $f^{t}$ is continuous at $\bar z$. By continuity, possibly shrinking $V_{1}$, we may assume that $(f^{t}(\bar y),f^{t}(\bar z))\in W'$ for all $\bar y\in V_{1}$. Now, if $(s,\bar y)\in V_{0}\times V_{1}$, then $(f(t,\bar y),f(t,\bar z))\in W'$. In addition, since $s\in O[t]$ and $(s,\bar y)\in O$, we obtain that $(f(t,\bar y),f(s,\bar y))\in W'$. Therefore, $(f(t,\bar z),f(s,\bar y))\in W$. This contradicts the assumption as there are arbitrary close elements $(s, \bar y)$ to $(t,\bar z)$ such that $(f(t,\bar z),f(s,\bar y))\notin W$.
One may also replace in the above proposition the assumption that $K$ is a model of an open $\cL$-theory of topological fields by assuming that the theory of $K$ is dp-minimal. As shown in [@simon-walsberg2016], all conditions used in the proof are also satisfied under the assumption of dp-minimality.
Cell decomposition {#sec:celldecomp}
------------------
We finish this section with the cell decomposition theorem for open $\cL$-theories of topological fields. It is an exact analogue of the cell decomposition [@simon-walsberg2016 Proposition 4.1] proved for dp-minimal fields.
\[prop:cell-decomposition\] Let $T$ be an open $\cL$-theory of topological fields and $K$ be a model of $T$. Let $X$ be a definable subset of $K^n$. There are finitely many definable subsets $X_{i}$ with $X=\bigcup X_{i}$ such that $X_{i}$ is, up to permutation of coordinates, the graph of a definable continuous $m_i$-correspondence $f\colon U_{i}\rightrightarrows K^{n-d_i}$, where $U_{i}$ is a definable open subset of $K^{d_i}$, for some $0\leqslant d_i\leqslant n$, $m_{i}\geq 1$.
One can argue exactly as in the proof of [@simon-walsberg2016 Proposition 4.1] after replacing [@simon-walsberg2016 Lemma 2.3] by Proposition \[prop:conseAA2\], [@simon-walsberg2016 Corollary 2.7] by Corollary \[cor:dim1\] and [@simon-walsberg2016 Proposition 3.7] by Proposition \[prop:acont\]. We include here an alternative argument.
We proceed by induction on $(n,\dim(X))$. If $n=1$, the statement is directly implied by assumption [**(A)**]{}. If $\dim(X)=0$, then $X$ is finite and the statement also holds. This shows the base case $(n,0)$ for each $n$. Suppose the result has been shown for all $(k,m)$ with $k<n+1$ and that $\dim(X)>0$.
Let $x=(x_1,\ldots,x_n)$, $y$ be a single variable and $\varphi(x,y)$ be an $\cL(K)$-formula defining $X\subseteq K^{n+1}$. By Corollary \[cor:goodform\], $X$ is a finite union of sets defined by formulas of the form $$\label{eq:varphi_form}
Z_{\cA}^S(x,y) \wedge \theta(x,y)$$ where $S\in K[x,y]$ and either $\cA\subset K[x]$, or $\cA^{y}=\{P\}$ and $\partial_{y} P)$ divides $S$. Without loss of generality, we may assume $X$ is defined by a formula as in . If $\cA=\emptyset$, then $X$ is open and we are done, so suppose $\cA\neq\emptyset$. We split in cases depending on whether $\cA\subseteq K[x]$ or $\cA^{y}=\{P\}$.
*Case 1:* Suppose $\cA\subseteq K[x]$. Let $\cB$ denote $\cA$ as a subset of $K[x]$, so that $Z_\cB(K)$ is a subset of $K^n$. By induction hypothesis, the set $Z_\cB(K)$ decomposes into finitely many $\{Y_i\}_{i\in I}$ each of which is the graph of a continuous definable correspondence $g_i\colon V_i\rightrightarrows K^{n-m_i}$. For each $i\in I$, consider the set $W_i\coloneqq(V_i\times K)\cap \theta(K)$ and the correspondence $h_i\colon W_i \rightrightarrows K^{n-m_i}$ defined by $h_i(a,b)\coloneqq g_i(a)$. Each $W_i$ is non-empty and open. Moreover, $h_i$ is continuous and definable. It is easy to check that $X$ is the union over $I$ of the graphs of the correspondences $h_i$, up to permutation of variables.
*Case 2:* Suppose $\deg_y(P)=d$ and consider for each $i\in \{1,\ldots, d\}$ the definable set $$Y_{i}\coloneqq\{a\in K^n : \exists^{=i} y \varphi(a,y)\}.$$ Let $\pi\colon K^{n+1}\to K^n$ denote the projection onto the first $n$ coordinates. By the case assumption, $\pi(X)=\bigcup_{1\leqslant i\leqslant d} Y_i$. It suffices to show the result for each $\pi^{-1}(Y_i)\cap X$. So fix some $i\in\{1,\ldots, d\}$. By induction hypothesis, $Y_{i}$ is a finite union of sets $\{Z_j\}_{j\in J}$ each of which is the graph of a continuous definable $\ell_j$-correspondence $g_j\colon V_j\rightrightarrows K^{n-m_j}$. Once more, it suffices to show the result for $\pi^{-1}(Z_j)\cap X$ for each $j\in J$, so fix $j\in J$. If $m_j=0$, then $V_j$ is finite and therefore $\pi^{-1}(Z_j)\cap X$ is finite too, so we are done. Suppose $m_j>0$ and consider the $i\ell_j$-correspondence $h_j\colon V_j\rightrightarrows K^{n-m_j+1}$ whose graph precisely corresponds to $\pi^{-1}(Z_j)\cap X$. It remains to take care of continuity. By Proposition \[prop:acont\], $h_j$ is continuous on an open dense subset $U_j$ of $V_j$. Since the set $\pi^{-1}(V_j)\cap X$ is the union of $\pi^{-1}(U_j)\cap X$ and $\pi^{-1}(V_j\setminus U_j)\cap X$, it suffices to show the result for these two sets. The former is the graph of a continuous definable correspondence. For the latter, note that $\dim(V_j\setminus U_j)<\dim(U_j)$, which implies that $$\dim(\pi^{-1}(V_j\setminus U_j)\cap X)<\dim(\pi^{-1}(V_j)\cap X) \leqslant \dim(X).$$ Thus, the result holds for $\pi^{-1}(V_j\setminus U_j)\cap X$ by the induction hypothesis.
Theories of topological fields with a generic derivation {#sec:dpmingen}
========================================================
Let $T$ be an $\cL$-theory of topological fields and $\cL_\delta$ be the language $\cL$ extended by a symbol for a derivation (in the field sort). The second part of the article focuses on the study of an $\cL_\delta$-extension $T_\delta^*$ of $T$. The derivation $\delta$ of any model of $T_\delta^*$ will be called a *generic derivation*. Every such a derivation is highly non-continuous. The theory $T_\delta^*$ will be defined in Section \[sec:Tdelta\]. In Section \[sec:consistency\] we show various examples of theories $T$ for which the theory $T_\delta^*$ is consistent. Then, in Sections \[sec:relQE\] and \[sec:conse\_QE\] we further investigate the theory $T_\delta^*$ when $T$ is an open $\cL$-theory of topological fields, showing that many tame properties transfer from $T$ to $T_\delta^*$.
Before defining $T_\delta^*$, let us fix some notation and recall the needed background on differential algebra.
Differential algebra background
-------------------------------
Let $(K,\delta)$ be a differential field of characteristic 0, that is, a field $K$ of characteristic 0 endowed with an additive morphism $\delta\colon K\to K$ which satisfies Leibnitz’s rule $\delta(ab)=\delta(a)b+a\delta(b)$. Such a function is called a *derivation on $K$*. We let $C_K$ denote the field of constants of $K$, namely, $C_K\coloneqq\{a\in K :\delta(a)=0\}$. It is a subfield of $K$.
For $m\geqslant 0$ and $a\in K$, we define $$\delta^m(a)\coloneqq\underbrace{\delta\circ\cdots\circ\delta}_{m \text{ times}}(a), \text{ with $\delta^0(a)\coloneqq a$,}$$ and $\bar{\delta}^m(a)$ as the finite sequence $(\delta^0(a),\delta(a),\ldots,\delta^m(a))\in K^{m+1}$. Similarly, given an element $a=(a_1,\ldots,a_n)\in K^n$, we will write $\bar{\delta}^m(a)$ to denote the element $(\bar{\delta}^m(a_1),\ldots,\bar{\delta}^m(a_n))\in K^{n(m+1)}$. For notational clarity, we will sometimes use $\J_m$ instead of $\bd_m$, especially concerning the image of subsets of $K^n$. For example, when $A\subseteq K$, we will use the notation $\J_m(A)$ for $\{\bar{\delta}^m(a):\;a\in A\}$ instead of $\bd^m(A)$. Likewise for $A\subseteq K^n$, $\J_m(A)\coloneqq\{\bar{\delta}^m(a):\;a\in A\}\subseteq K^{n(m+1)}$.[^2]
We will always assume our tuples of variables are ordered, as for example $x=(x_0,\ldots,x_n)$. Moreover, as a convention, given a variable $y$, the tuple $(x,y)$ is ordered such that $y$ is bigger than $x_n$ (and similarly when $y$ is an ordered tuple of variables).
Given $x=(x_0,\ldots,x_n)$, we let $K\{x\}$ be the ring of differential polynomials in $n+1$ differential indeterminates $x_{0},\ldots, x_{n}$ over $K$, namely it is the ordinary polynomial ring in formal indeterminates $\delta^j(x_{i})$, $0\leqslant i\leqslant n$, $j\in \N$, with the convention $\delta^0(x_{i})\coloneqq x_{i}$. We extend the derivation $\delta$ to $K\{x\}$ by setting $\delta(\delta^i(x_j))=\delta^{i+1}(x_j)$. By a rational differential function we simply mean a quotient of differential polynomials.
### Order and separant of a differential polynomial {#sec:order_sep}
For $P(x)\in K\{x\}$ and $0\leqslant i\leqslant n$, we let $\ord_{x_i}(P)$ denote the *order of $P$ with respect to the variable $x_i$*, that is, the maximal integer $k$ such that $\delta^k(x_i)$ occurs in a non-trivial monomial of $P$ and $-1$ if no such $k$ exists. We let *the order of $P$* be $$\ord(P)\coloneqq\max\{\ord_{x_i}(P) : 0\leqslant i\leqslant n\}.$$ Similarly, for a finite subset $\cA$ of $K\{x\}$, we let $$\ord_{x_i}(\cA)\coloneqq\max\{\ord_{x_i}(P) : P \in \cA\} \text{ and } \ord(\cA)\coloneqq\max\{\ord(P): P\in \cA\}.$$ For $R\in K\{x\}$, we write $\ord_{x_i}(\cA,R)$ for $\ord_{x_i}(\cA\cup\{R\})$.
Suppose $\ord(P)=m$. For $\bar{x}=(\bar{x}_0,\ldots,\bar{x}_n)$ a tuple of variables with $\ell(\bar{x}_i)=m+1$, we let $P^*\in K[\bar{x}]$ denote the corresponding ordinary polynomial such that $P(x)=P^*(\bar{\delta}^m(x))$. Suppose $\ord_{x_n}(P)=m\geqslant 0$. Then, there are (unique) differential polynomials $c_i\in K\{x\}$ such that $\ord_{x_n}(c_i)<m$ and
$$\label{eq:diffpol}
P(x)=\sum_{i=0}^d c_i(x)(\delta^m(x_n))^i.$$
The separant $s_{P}$ of $P$ is defined as $s_{P}\coloneqq\frac{\partial}{\partial \delta^m(x_{n})}P\in K\{x\}$. We extend the notion of separant to arbitrary polynomials with an ordering on their variables in the natural way, namely, if $P\in K[x]$, the separant of $P$ corresponds to $s_P\coloneqq\frac{\partial}{\partial x_{n}}P\in K[x]$. By convention, we induce a total order on the variables $\delta^j(x_{i})$ by declaring that $$\delta^k(x_{i})< \delta^{k'}(x_{j}) \Leftrightarrow
\begin{cases}
i<j \\
i=j \text{ and } k<k'.
\end{cases}$$ This order makes the notion of separant for differential polynomials compatible with the extended version for ordinary polynomials, *i.e.*, $s_{P^*} = s_P^*$.
### Minimal differential polynomials {#sec:gen_poly}
Let $F\subseteq K$ be an extension of differential fields. Recall that an ideal $I$ of $F\{x\}$ is a differential ideal if for every $P\in I$, $\delta(P)\in I$. For $a\in K$, let $I(a,F)$ denote the set of differential polynomials in $F\{x\}$ vanishing on $a$. The set $I(a,F)$ is a prime differential ideal of $F\{x\}$. Let $P\in I(a,F)$ be a differential polynomial of minimal degree among the elements of $I(a,F)$ having minimal order. Any such differential polynomial is called a *minimal differential polynomial of $a$ over $F$*. Let $\langle P\rangle$ denote the differential ideal generated by $P$ and $I(P)\coloneqq\{Q(x)\in F\{ x\}: s_{P}^{\ell}Q\in \langle P\rangle$ for some $\ell\in \N\}$.
\[lem:generic\] If $P$ is a minimal differential polynomial for $a$ over $F$, then $I(a,F)=I(P)$
See [@marker1996 Section 1].
### Rational prolongations {#sec:rat_prolong}
We define an operation on $K\{x\}$ sending $P\mapsto P^\delta$ as follows: for $P$ written as in $$P(x)\mapsto P^\delta(x) = \sum_{i=0}^d \delta(c_i(x)) (\delta^m(x_n))^i.$$ A simple calculation shows that $$\label{eq:diffpol2}
\delta(P(x)) = P^\delta(x) + s_P(x)\delta^{m+1}(x_n).$$
\[lemdef:ratioprolong\] Let $x=(x_0,\ldots,x_n)$ be a tuple of variables and $y$ be a single variable. Let $P\in K\{x,y\}$ be a differential polynomial such that $m=\ord_y(P)\geqslant 0$. There is a sequence of rational differential functions $(f_i^P)_{i\geqslant 1}$ such that for every $a\in K^{n+1}$ and $b\in K$ $$K\models [P(a,b)=0 \wedge s_P(a,b)\neq 0] \to \delta^{m+i}(b)=f_i^P(a,b).$$ In addition, each $f_i^P$ is of the form $$f_i^P(x,y) = \frac{Q_i(x,y)}{s_P(x,y)^{\ell_i}},$$ where ${\ell_i}\in \N$, $\ord_{y}(Q_i)=\ord_y(P)$ and $$\ord_{x_j}(Q_i)=
\begin{cases}
\ord_{x_j}(P)+i & \text{ if $\ord_{x_j}(P)\geqslant 0$ }\\
-1 & \text{ otherwise }
\end{cases}$$ We call the sequence $(f_i^P)_{i\geqslant 1}$ the *rational prolongation along $P$*.
It suffices to inductively define the polynomials $Q_i$. By , if $\delta(P(x,y))=0$ we obtain that $$\delta^{m+1}(y) = \frac{-P^{\delta}(x,y)}{s_P(x,y)},$$ Setting $Q_1=-P^{\delta}$, the rational differential function $f_1=\frac{Q_1}{s_P}$ satisfies the required property. Now suppose $Q_{i}$ has been defined and that $f_{i}=\frac{Q_i}{(s_P)^{\ell_i}}$ satisfies $\delta^{m+i}(y)=f_i(x,y)$. By applying $\delta$ on both sides we obtain $$\begin{aligned}
\label{eq:prolongation}
\delta^{m+i+1}(y) & = \frac{\delta(Q_i(x,y))s_P(x,y) - Q_i(x,y)\delta(s_P(x,y))}{s_P(x,y)^{2\ell_i}}.
% & = \frac{R_1(\bd(a))+R_2(\bd(a))\delta^{n+1}(a)}{s_P(\bd(a))^{2\ell_i}}\\
% & = \frac{Q_1(\bd(a))(R_1(\bd(a))+R_2(\bd(a)))}{s_P(\bd(a))^{\ell_{i+1}}},\\ \end{aligned}$$ By replacing instances of $\delta^{m+i}(y)$ in $\delta(Q_i(x,y))$ and $\delta(s_P(x,y))$ by $f_i(x,y)$, we obtain in the numerator a differential polynomial of order $m$ with respect to $y$. Setting $Q_{i+1}$ as such numerator shows the result. The last assertion is a straightforward calculation.
\[not:prolong\] For an integer $d\geqslant 0$ and a tuple of variables $x$, we define the tuple of variables $x(d)$ by induction on $d$ as follows: $x(0)\coloneqq x$ and $x(d+1)\coloneqq (x(d), u_d)$, where $u_{d}$ is a variable. We will assume that if $x$ and $y$ are disjoint tuples of variables, then $x(d)$ and $y(d)$ are also disjoint.
\[not:prolong2\] Let $x=(x_0,\ldots, x_n)$ and $y$ be a single variable. Let $P\in K\{x,y\}$ be a differential polynomial of order $m$ and let $(f_i^P)_{i\geqslant 1}$ be its rational prolongation along $P$. Let $\bar{x}=(\bar{x}_0,\ldots,\bar{x}_n)$ where $\bar{x}_i=(x_{i0},\ldots,x_{im})$ and $\bar{y}=(y_0,\ldots,y_m)$. For every $d\geqslant 0$, we let $\lambda_{P}^d (\bar{x}(d), \bar{y}(d))$ be the $\cL(K)$-formula: $$P^*(\bar{x}, \bar{y}))=0 \wedge s_{P}^*(\bar{x}, \bar{y}))\neq 0 \wedge \bigwedge_{i\geqslant 1}^d y_{m+i}=(f_i^P)^*(\bar{x}(i), \bar{y}).$$
### Kolchin closed sets {#sec:kolchin}
Let $x$ be a tuple of variables with $\ell(x)=n$. Similarly as in Section \[sec:commalg\], for a finite subset $\cA$ of $K\{x\}$ and $R\in K\{x\}$, we let $\cZ_\cA(x)$ denote the $\cL_\delta(K)$-formula $$\bigwedge_{P\in \cA} P(x)=0,$$ and $\cZ_\cA^R(x)$ denote the $\cL_\delta(K)$-formula $$\bigwedge_{P\in \cA} P(x)=0 \wedge R(x)\neq 0.$$ Recall that a subset $X\subseteq K^n$ is called *Kolchin closed* if there is a finite subset $\cA\subseteq K\{x\}$ such that $X=\cZ_\cA(K)$. It is called *locally Kolchin closed* if $X=\cZ_{\cA}^R(K)$ for some $R\in K\{x\}$.
For the rest of Section \[sec:kolchin\], we let $x=(x_1,\ldots,x_n)$, $y$ be a single variable, $\cA$ be a finite subset of $K\{x,y\}$ and $R\in K\{x,y\}$. We let $$\begin{aligned}
k_\cA & \coloneqq
\begin{cases}
-1 & \text{if $\cA\subseteq K\{x\}$ } \\
\min\{\ord_y(Q) : Q\in \cA\setminus K\{x\} \} & \text{otherwise. }
\end{cases} \end{aligned}$$ For an integer $k\geqslant -1$, we let $\cA(k) \coloneqq \{P\in \cA : \ord_y(Q)=k \}$.
\[lem:minord1\] The set $\cZ_\cA^R(K)$ is the union of finitely many sets $\cZ_\cB^{S_\cB}(K)$ such that $\ord_y(S_\cB)\leqslant \ord_y(\cA,R)$, $\ord_{x_i}(\cB,S_\cB)\leqslant \ord_{x_i}(\cA,R)$ for each $1\leqslant i\leqslant n$, and either
1. $\ord_y(\cB)<\ord_y(\cA)$ or
2. $\ord_y(\cB)=\ord_y(\cA)$, $\cB(\ord_y(\cB))=\{P_\cB\}$ and $s_{P_\cB}$ divides $S_\cB$.
Letting $m=\ord(\cA)$ and $d=\ord_y(\cA)$, the result follows by Corollary \[cor:goodform\] applied to the polynomial ring $K[\bd^m(x),\bd^d(y)]$ with respect to the variable $\delta^{d}(y)$. Note that since we work in an ordinary polynomial ring, the order in $x_i$ cannot increase.
\[lem:minord2\] The set $\cZ_\cA^R(K)$ is the union of finitely many sets $\cZ_\cB^{S_\cB}(K)$ such that $\ord_y(S_\cB)\leqslant \ord_y(\cA,R)$, $\ord_{x_i}(\cB,S_\cB)\leqslant \ord_{x_i}(\cA,R)$ for each $1\leqslant i\leqslant n$, and either
1. $\cB\subseteq K\{x\}$ or
2. $\cB(k_\cB)=\{P_\cB\}$ and $s_{P_\cB}$ divides $S_\cB$.
We proceed by induction on $\ord_y(\cA)$. If $\ord_y(\cA)=-1$, then $\cA\subseteq K\{x\}$ and there is nothing to show. By Lemma \[lem:minord1\], we may suppose there is $P\in \cA$ such that $\cA(\ord_y(\cA))=\{P\}$ and $s_{P}$ divides $R$. Letting $\cD=\cA\setminus \{P\}$, we have that $\ord_y(\cD)<\ord_y(\cA)$. Thus, by induction, $$\cZ_\cD^R(K)=\bigcup_{j\in J} \cZ_{\cB_j}^{S_j}$$ where $J$ is a finite set and for each $j\in J$ the set $\cZ_{\cB_j}^{S_j}$ satisfies the conclusion of the lemma. The result follows by setting $\cC_j\coloneqq\cB_j\cup\{P\}$, $T_j\coloneqq s_PS_j$ and noting that $$Z_\cA^R(K)=\bigcup_{j\in J} Z_{\cC_j}^{T_j}(K).$$ We let the reader verify that the order bounds hold for this family of locally Kolchin closed sets. Note that if $\cB_j\subseteq K\{x\}$ then $P_{\cC_j}=P$, and otherwise $P_{\cC_j}=P_{\cB_j}$.
\[lem:subst\] Suppose there is a unique $P\in \cA(k_\cA)$ and that $0\leqslant \ord_{y}(P)<\ord_y(\cA)$. Then, there is a finite subset $\cB$ of $K\{x,y\}$ such that
1. $\ord_{y}(\cB)=\ord_y(P)$,
2. $\ord_{x_{i}}(\cB)\leqslant \ord_{x_i}(\cA)+\ord_y(\cA)-\ord_y(P)$, for each $1\leqslant i\leqslant n$,
and if $s_P$ divides $R$, then $\cZ_{\cA}^{R}(K)=\cZ_{\cB}^{R}(K)$.
For notational simplicity, let $m=\ord(\cA)$, $d=\ord_y(\cA)$ and $k=\ord_y(P)$. Consider, for each $Q\in \cA$ with $\ord_y(Q)>k$, the rational differential function $$T(x,y)=Q^*(\bd^m(x),\bd^k(y), f_1^{P}(x,y), \ldots, f_{d-k}^{P}(x,y)),$$ which arises by replacing all occurrences of $\delta^{k+i}(y)$ by the rational differential function $f_i^{P}(x, y)$ for each $1\leqslant i\leqslant d-k$. By Lemma-Definition \[lemdef:ratioprolong\], $f_{i}^P=\frac{Q_i}{s_P^{\ell_i}}$ with $\ord_{y}(Q_{i})\leqslant k$ and for each $1\leqslant j\leqslant n$, $$\ord_{x_{j}}(Q_{i})=
\begin{cases}
\ord_{x_{j}}(P)+i & \text{ if $\ord_{x_{j}}(P)\geqslant 0$,} \\
-1 & \text{ otherwise. }
\end{cases}$$ Clearing out the denominator by multiplying $T$ by the required power of $s_{P}$, we obtain a differential polynomial $\widetilde{Q}(x, y)$ with $\ord_{y}(\widetilde Q) \leqslant k$ and $$\ord_{x_j}(\widetilde{Q})\leqslant \ord_{x_j}(\cA)+d-k = \ord_{x_j}(\cA)+\ord_y(\cA)-\ord_y(P)$$ for each $1\leqslant j\leqslant n$. Define $\cB$ as $$\cB\coloneqq\{Q\in \cA: \ord_y(Q)\leqslant k\} \cup \{\widetilde{Q} : Q\in \cA, \ord_y(Q)>k\}.$$ By construction, $\cB$ satisfies conditions (1)-(2). The last statement follows by Lemma-Definition \[lemdef:ratioprolong\].
\[lem:minord3\] The set $\cZ_\cA^R(K)$ is the union of finitely many sets $\cZ_\cB^{S_\cB}(K)$ such that $\ord_y(S_\cB)\leqslant \ord_y(\cA,R)$, $\ord_{x_i}(\cB,S_\cB)\leqslant \ord_{x_i}(\cA,R)+\ord_y(\cA,R)$ for each $1\leqslant i\leqslant n$, and either
1. $\cB\subseteq K\{x\}$ or
2. there is a unique $P\in \cB$ of non-negative order in $y$, $\ord_y(P_\cB)\leqslant \ord_y(\cA,R)$ and $s_{P_\cB}$ divides $S_\cB$.
We proceed by induction on $\ord_y(\cA)$. If $\ord_y(\cA)=-1$ there is nothing to show. So suppose $\ord_y(\cA)\geqslant 0$. By Lemma \[lem:minord2\], we may suppose that there is $P\in \cA$ such that $\cA(k_\cA)=\{P\}$ and $s_{P}$ divides $R$. If $k_\cA=\ord_y(\cA)$, then we are done. Otherwise, if $k_\cA<\ord_y(\cA)$, by Lemma \[lem:subst\] there is a finite subset $\cC$ of $K\{x,y\}$ such that $\cZ_{\cA}^{R}(K)=\cZ_{\cC}^{R}(K)$ and
1. $\ord_{y}(\cC)=\ord_y(P)$,
2. $\ord_{x_{i}}(\cC)\leqslant \ord_{x_i}(\cA)+\ord_y(\cA)-\ord_y(P)$, for each $1\leqslant i\leqslant n$,
Since $\ord_y(\cC)<\ord_y(\cA)$ we can apply the induction hypothesis to $\cZ_\cC^R(K)$. So suppose $\cZ_\cC^R(K)$ is a finite union of locally Kolchin closed sets $\cZ_\cB^{s_\cB}(K)$ as in the statement. Let us show that each $\cZ_\cB^{S_\cB}(K)$ satisfies the needed bounds with respect to $\cZ_\cA^R(K)$. First, by (1) and (2) $$\begin{aligned}
\ord_{x_i}(\cB,S_\cB) & \leqslant \ord_{x_i}(\cC,R)+\ord_y(\cC,R) \\
& \leqslant (\ord_{x_i}(\cA,R)+\ord_y(\cA,R)-\ord_y(P))+\ord_y(P) \\
& = \ord_{x_i}(\cA,R)+\ord_y(\cA,R). & \end{aligned}$$ Similarly, $$\ord_y(S_\cB)\leqslant \ord_y(\cC,R)\leqslant \ord_{y}(\cA,R).$$ Finally, if $k_\cB\geqslant 0$, then $$\ord_y(P_\cB)\leqslant \ord_y(\cC,R)\leqslant \ord_{y}(\cA,R).$$
The theory $T_\delta^*$ {#sec:Tdelta}
-----------------------
Let $T$ be an $\cL$-theory of topological fields. Let $\cL_\delta$ be the language $\cL$ extended by a unary field sort function symbol $\delta$. Denote by $T_\delta$ the $\cL_{\delta}$-theory $T$ together with the usual axioms of a derivation, namely, $$\begin{cases}
\forall x\forall y(\delta(x+y)=\delta(x)+\delta(y))\\
\forall x\forall y(\delta(xy)=\delta(x)y+x\delta(y)).
\end{cases}$$
\[not:etoile\] Let $x$ be a tuple of field sort variables and $w$ be a tuple of variables of other sorts. Let $\varphi(x,w)$ be a field sort quantifier free $\cL_\delta$-formula. Then, there is an $\cL$-formula $\psi$ such that $$T_\delta \models \forall x \forall w (\varphi(x,w) \leftrightarrow \psi(\bd^m(x), w)).$$ Note that we use here the assumption that the restriction of $\cL$ to the field sort is a relational extension of $\cL_\mathrm{field}$. We define the *order of $\varphi$* as the minimal integer $m$ such that $\varphi$ is equivalent to $\psi(\bd^m(x),w)$ for a field sort quantifier free $\cL$-formula $\psi$. Even if $\psi$ is not unique, we will denote some (any) such $\cL$-formula by $\varphi^*$.
We will now describe a scheme of $\cL_{\delta}$-axioms generalizing the axiomatization of closed ordered differential fields ($\CODF$) given by M. Singer in [@singer1978]. Let $\chi_\tau(x,z)$ be an $\cL$-formula providing a basis of neighbourhoods of 0. Abusing of notation, when $x$ is a tuple of field sort variables $x=(x_1,\ldots, x_n)$ we let $\chi_\tau(x,z)$ denote the formula $$\bigwedge_{i=1}^n \chi_\tau(x_i,z).$$
\[def:theoryT\_delta\] The $\cL_\delta$-theory $T_{\delta}^*$ is the union of $T_{\delta}$ and the following scheme of axioms $(\mathrm{DL})$: given a model $K$ of $T_\delta$, $K$ satisfies $(\mathrm{DL})$ if for every differential polynomial $P(x)\in K\{x\}$ with $\ell(x)=1$ and $\ord_x(P)=m\geqslant 1$, for field sort variables $y=(y_0,\ldots,y_m)$ it holds in $K$ that $$\forall z\big((\exists y(P^*(y)=0 \land s_P^*(y)\ne 0 )
\rightarrow \exists x\big(P(x)=0\land s_P(x)\ne 0\wedge
\chi_\tau(\bd^m(x)-y, z)\big)\big).$$
As usual, by quantifying over coefficients, the axiom scheme $(\mathrm{DL})$ can be expressed in the language $\cL_\delta$.
The theory $\RCF_\delta^*$ is $\CODF$.
Consistency {#sec:consistency}
-----------
The main result of this section is Theorem \[thm:consistency\] which shows that if $T$ is a complete theory of henselian valued fields of characteristic 0, then $T_\delta^*$ is consistent. As a consequence we obtain the consistency of $T_\delta^*$ for all theories $T$ described in Examples \[examples\]. For some of such theories, the consistency of $T_\delta^*$ has already been proved. Indeed, the consistency of $\CODF$ was proved in [@singer1978] and was later generalized in [@guzy-point2010] to a broader class of theories. Although we will follow a very similar strategy to the known proofs, our argument is based on henselizations rather than using explicitly a notion of largeness (or topological largeness) for the fields under consideration.
Let us start by a general criterion to show that $T_\delta^*$ is consistent.
\[prop:criterion\] Let $T$ be a complete $\cL$-theory of topological fields and $\chi_\tau(x,z)$ be the $\cL$-formula defining a basis of neighbourhoods of 0. Suppose that for every model $K$ of $T$ and every derivation $\delta$ on $K$ the following holds
1. for every $P\in K\{x\}$ ($\ell(x)=1$) of order $m\geqslant 1$ for which there is $a\in K^{m+1}$ such that $P^*(a)=0$ and $s_P^*(a)\neq 0$, there is a differential field extension $(F,\delta)$ of $(K,\delta)$ such that $F$ is in addition an $\cL$-elementary extension of $K$ and there is $b\in F$ such that $P(b)=0$, $s_P(b)\neq 0$ and for every $c\in S_z(K)$ $$F\models \chi_\tau(\bd^m(b) - a, c).$$
Then, for every model $K$ of $T$ and every derivation $\delta$ on $K$, there is an extension $K\prec_\cL L$ and an extension of $\delta$ to $L$ making $(L,\delta)$ into a model of $T_\delta^*$. In particular, $T_\delta^*$ is consistent.
Fix some model $K$ of $T$ and some derivation $\delta$ on $K$. We use the following two step construction to build $(L,\delta)$.
*Step 1:* We construct an $\cL$-elementary extension $K\prec_\cL F_K$ and a derivation on $F_K$ extending $\delta$ as follows. Let $(P_i)_{i< \lambda}$ be an enumeration of all differential polynomials $P_i\in K\{x\}$ with $\ord(P_i)=m_i\geqslant 1$ for which there is $a_i\in K^{m_i+1}$ such that $P_i^*(a_i)=0$ and $s_{P_i}^*(a_i)\neq 0$. Consider the following chain $(F_i, \delta_i)_{i<\lambda}$ defined by
1. $F_0\coloneqq K$,
2. $(F_{i+1}, \delta_{i+1})$ is given by condition $(\ast)$ with respect to $(F_i,\delta_i)$ and $P_i$, that is, $F_i\prec_\cL F_{i+1}$, $\delta_{i+1}$ extends $\delta_i$ and there is $b_i\in F_{i+1}$ such that $P(b_i)=0$, $s_P(b_i)\neq 0$ and for every $c\in S_z(K)$ $$F_{i+1}\models \chi_\tau(\bd^m(b_i) - a_i, c).$$
Let $F_K\coloneqq \bigcup_{i<\lambda} F_i$ and, abusing notation, let $\delta$ denote the union of the derivations $\delta_i$. Observe that indeed $K\prec_\cL F_K$ and $(K,\delta)\subseteq (F_K,\delta)$ is an extension of differential fields.
*Step 2:* Define a chain $(L_i)_{i<\omega}$ where $(L_0,\delta)\coloneqq (K,\delta)$, and $L_{i+1}$ corresponds to the differential field $(F_{L_i},\delta)$ obtained in Step (1) with respect to $(L_i,\delta)$, so that $L_i\prec_\cL L_{i+1}$. Let $L\coloneqq \bigcup_{i<\omega} L_i$ and again, abusing of notation, $\delta$ denote the union of their derivations. By construction, $K\prec_\cL L$. It remains to show that $L$ satisfies the axiom scheme (DL). Let $P\in L\{x\}$ be a differential polynomial of order $m\geqslant 1$ for which there is $a\in L^{m+1}$ such that $P^*(a)=0$ and $s_P^*(a)\neq 0$. Fix $c\in S_z(L)$ and let $i<\omega$ be such that $a\in L_i^{m+1}$, $c\in S_z(L_i)$ and $P\in L_i\{x\}$. By Step (1), there is $b\in L_{i+1}\subseteq L$ such that $$L_{i+1}\models \chi_\tau(\bd^m(b) - a, c),$$ which shows the result.
\[thm:consistency\] Let $T$ be a complete $\cL_{\Div}$-theory of henselian valued fields of characteristic 0. For every model $K$ of $T$ and every derivation $\delta$ on $K$, there is an extension $K\prec_\cL L$ and an extension of $\delta$ to $L$ making $(L,\delta)$ into a model of $T_\delta^*$. In particular, $T_\delta^*$ is consistent.
By Proposition \[prop:criterion\], it suffices to show condition $(\ast)$ above defined. Let $\chi_\tau(x,z)$ be the $\cL_\Div$-formula $v(x)>v(z)\wedge z\neq 0$ and let $(K,v)$ be a model of $T$ equipped with a derivation $\delta$. Let $\Gamma^v$ denote the value group of $(K,v)$. Suppose $P\in K\{x\}$ is a differential polynomial of order $m\geqslant 1$ for which there is $a=(a_0,\ldots, a_m)\in K^{m+1}$ such that $P^*(a)=0$ and $s_P^*(a)\neq 0$. Let $t=(t_0,\ldots, t_{m})$ be a tuple of new variables and consider the (ordinary) polynomial $$Q(x)=P^*(a_0-t_0, \ldots, a_{m-1}-t_{m-1}, x)$$ in $K(t_0,\ldots t_{m-1})[x]$. Let $w\colon K(t)\to \Z^n_\infty$ denote the $t$-adic valuation, that is, the iterated composition of the $t_i$-adic valuation such that $0<w(t_0)\ll w(t_1)\ll\cdots\ll w(t_m)$. Let $\Z^n\overrightarrow{\times} \Gamma^v$ denote the lexicographic extension of $\Gamma_{v}$ by $\Z^n$ and $v_t\colon K(t) \to (\Z^n \overrightarrow{\times} \Gamma^v)_\infty$ denote the composite valuation which sends a polynomial $R(t)= \sum_{i\in I} a_it^i$ (in multi-index notation) to the pair $(w(R), v(a_{w(R)}))$.
Note that $w$ is a coarsening of $v_t$. Let $F=(K(t), w)^h$ and $L=(K(t), v_t)^h$ be their corresponding henselizations. Without loss of generality, we may suppose that $F\subseteq L$ and that for all $a\in F$ $$w(a)>0 \Leftrightarrow v_t(a)>\Gamma^v.$$ Let us show that there is $c\in F$ such that $Q(c)=0$ and $w(c-a_m)>0$. The reduction $\widetilde{Q}$ of $Q$ in $F$ corresponds to $P^*(a_0,\ldots,a_{m-1},x)\in K[x]$. By assumption, $\widetilde{Q}(a_m)=0$ and $\frac{\partial}{\partial x} \widetilde{Q}(a_m)\neq 0$. Then, by Hensel’s lemma, there is $c\in F$ such that $Q(c)=0$ and $w(c-a_m)>0$ (equivalently $v_t(c-a_m)>\Gamma^v$). This implies both that $c\notin K$ and that $\frac{\partial}{\partial x}Q(c)\neq 0$. We extend $\delta$ to the subfield $K(t_0,\ldots, t_{m-1}, c)\subseteq F$ by inductively setting
1. $\delta(t_i)=\delta(a_i)+t_{i+1}-a_{i+1}$ ,
2. $\delta(t_{m-1}) = c$.
Note that since $Q(c)=0$ and $\frac{\partial}{\partial x} Q(c)\neq 0$, the derivative of $c$ is already determined by the rational prolongation $f_1^Q(c)$. Setting $b\coloneqq a_0-t_0$, we have that $$P(b)=P^*(\bd^m(b))=P^*(a_0-t_0, \ldots, a_{m-1}-t_{m-1},c)=Q(c) = 0.$$ Similarly, $s_P(b)\neq 0$. In addition, for every $e\in K^\times$ $$\label{eq:valuationbig}
v_t(\bd^m(b) -a) = \min\{ v_t(t_0), \ldots, v_t(t_{m-1}), v_t(c-a_m)) > v(e).$$ Extend the derivation from $K(t_0,\ldots,t_{m-1},b)$ to $L$ (such an extension always exists by [@lang Theorem 5.1]). Let $K^*$ be a saturated $\cL_\Div$-elementary extension of $K$ and $c_0,\ldots,c_m\in K^*$ be such that $$\Gamma^v< v(c_0) \ll v(c_1) \ll \ldots \ll v(c_m).$$ Let $g\colon (K(t_0,\ldots,t_m), v_t) \to (K^*,v)$ be the (unique) $\cL_\Div$-embedding over $K$ sending $t_i\to c_i$. Then $g$ extends to an $\cL_\Div$-embedding $h\colon (L,v_t)\to (K^*,v)$. Equip $K(c_0,\ldots, c_m)$ with the induced derivation $\delta$ from $h$ and extend it to $K^*$. Then $P(h(b))=0$, $s_P(h(b))\neq 0$ and since $h$ is an embedding of valued fields it follows from $$v(\bd^m(h(b)) -a) > v(e),$$ for every $e\in K^\times$, which completes the result.
\[cor:cons\_otherlang\] Let $T$ be an $\cL_\Div$-complete theory of henselian valued fields of characteristic 0 and $K$ be a model of $T$. Let $(K,\cL)$ be an extension by definitions of some reduct of $(K,\cL_\Div^{\eq}$ and let $T'$ be the complete $\cL$-theory of some (any) model of $T$. Then the $\cL_\delta$-theory $(T')_\delta^*$ is consistent.
\[cor:cons\_examples\] Let $T$ be any theory from Examples \[examples\]. Then $T_\delta^*$ is consistent.
Except for $\CODF$, all examples in Examples \[examples\] correspond to theories of henselian valued fields of characteristic 0 are extensions by definitions of a reduct of their $\cL_\Div^{\eq}$ expansions. Thus, the result follows by Corollary \[cor:cons\_otherlang\]. Note that the consistency of $\RCVF_\delta^*$ implies the consistency of $\CODF$, as every model of $\RCVF_\delta^*$ is a model of $\CODF$ (the valuation topology and the order topology induce the same topology on any model of $\RCVF$).
\[rem:closedness\] Let $(K,v)$ be a valued field of characteristic 0 endowed with a derivation $\delta$. Let $(K^h,v)$ be the henselization of $(K,v)$. Note that the derivation extends (uniquely) to $K^h$. Let $T$ be the theory of $(K^h,v)$. Theorem \[thm:consistency\] implies that $(K,v,\delta)$ embeds as an $\cL_{\Div,\delta}$-structure into a model of $T_\delta^*$.
\[rem:consistency\] Note that if $T_\delta^*$ is consistent, then every model $K$ of $T$ embeds as an $\cL$-structure into a model of $T_\delta^*$. Indeed, take a model $K'$ of $T_\delta^*$. Then the reduct of $K'$ to $\cL$ is a model of $T$, and since $T$ is complete $K\equiv_\cL K'$. The result follows by Keisler-Shelah’s theorem. A similar argument will be used later in Section \[sec:DenseP2\].
Relative quantifier elimination {#sec:relQE}
-------------------------------
For the rest of Section \[sec:dpmingen\] we let $T$ be an open $\cL$-theory of topological fields and assume $T_\delta^*$ is a consistent theory. As shown in Corollary \[cor:cons\_examples\], all theories $T$ listed in Examples \[examples\] satisfy such an assumption.
We will need the following classical consequence of the axiom scheme (DL).
\[fact:density\] Let $K$ be a model of $T_{\delta}^{*}$. Let $O$ be an open subset of $K^n$. Then there is $a\in K$ such that $\bar \delta^{n-1}(a)\in O$.
\[thm:QE\] The theory $T_{\delta}^{*}$ eliminates field sort quantifiers in $\cL_{\delta}$.
Let $\Sigma$ denote the set of field sort quantifier-free $\cL_\delta$-formulas, $x,y$ be field sort tuples of variables with $\ell(y)=1$ and $\varphi(x,y)$ be a formula in $\Sigma$. Let $K_1, K_2$ be two models of $T_\delta^{*}$ and $b_i\in K_i^{\ell(x)}$ be tuples which have the same $\Sigma$-type (i.e., they satisfy the same formulas in $\Sigma$). Let $F_i$ denote the differential subfield of $K_i$ generated by $b_i$. The assumption on the tuples $b_1$ and $b_2$ implies there is an isomorphism of differential fields $\sigma\colon F_1\to F_2$ fixing $\Q$ and sending $\bd^\ell(b_1)$ to $\bd^\ell(b_2)$ for every $\ell\geqslant 0$. Moreover, by elimination of field sort quantifiers in $T$, we may suppose $F_i$ algebraically closed in $K_i$. Suppose there is $a\in K_1$ such that $K_1\models \varphi(b_1, a)$. We must show that there is $c\in K_2$ such that $K_2\models \varphi(b_2, c)$. Let $m$ be the order of $\varphi$ and $\varphi^*(\bar{x},\bar{y})$ be the field sort quantifier-free $\cL$-formula such as in Notation \[not:etoile\]. By assumption $(\mathbf{A})$, the formula $\varphi^*(\bar{x},\bar{y})$ is equivalent to a finite disjunction of formulas of the form $$\label{eq:intersection1}
\bigwedge_{i\in I} P_i(\bar{x}, \bar{y})=0 \wedge \theta(\bar{x},\bar{y}),$$ where $P_i\in \Q[\bar{x}, \bar{y}]\setminus\{0\}$, $I$ possibly empty and $\theta(\bar{x},\bar{y})$ defines an open set (in every model of $T$). As existential quantifiers commute with disjunctions, we may suppose $\varphi^*$ is already a conjunction as in . We split into cases depending on whether $a$ is differentially transcendental over $F_1$ or not:
*Case 1:* Suppose $a$ is differentially algebraic (but not algebraic) over $F_1$. Let $P\in F_1\{x\}$ be a minimal differential polynomial for $a$ over $F_1$ order $k\geqslant 1$. Since $P$ is minimal, we must have both $k\leqslant m$ and that $s_P(a)\neq 0$. For $d=m-k$, we have then $$\label{eq:minimality1}
K_1\models \lambda_P^{d}(\bd^{m+d}(b_1),\bd^m(a)).$$ Let $P^\sigma$ (resp. $P_i^\sigma$) denote the corresponding polynomial over $F_2$ in which every coefficient of $P$ (resp. $P_i$) is replaced by its image under $\sigma$. Since $I(a, F_1)=I(P)$ (by Lemma \[lem:generic\]), $P_i\in I(P)$ for each $i\in I$. Our assumption on $b_1$ and $b_2$ implies that $P_i^\sigma\in I(P^\sigma)$. Therefore, it suffices to show that there is $c\in K_2$ such that $$K_2 \models P^\sigma(c)=0 \wedge s_{P^\sigma}(c)\neq 0 \wedge \theta(\bd^m(b_2), \bd^m(c)),$$ as this will also imply that $K_2\models \bigwedge_{i\in I} P_i(b_2,c)=0$. By assumption $(\mathbf{A})$ and , there is $\bar{e}=(e_0,\ldots,e_m)\in K_2^{m+1}$ such that $K_2\models\lambda_{P^\sigma}^{d}(\bd^{m+d}(b_2),\bar{e})$ (note that $\lambda_{P^\sigma}^d$ is an $\cL(K)$-formula). Letting $\widehat{e}=(e_0,\ldots,e_k)$, the previous formula yields that $e_{k+i}=(f_i^{P^\sigma})^*(\bd^{m+i}(b_2),\widehat{e})$ for all $1\leqslant i\leqslant d$, where $(f_i^{P^\sigma})_{i\geqslant 1}$ is the rational prolongation of $P^\sigma$. Since $(P^\sigma)^*(\widehat{e})=0$ and $s_{P^\sigma}^*(\widehat{e})\neq 0$, the axiom scheme (DL) implies there is $c\in K_2$ such that $P^\sigma(c)=0$ and $s_{P^\sigma}(c)\neq 0$. Moreover, by the continuity of the functions $(f_i^{P^\sigma})^*$, we may further suppose that $\theta(\bd^m(b_2), \bd^m(c))$ holds. This completes Case 1.
*Case 2:* Suppose $b$ is differentially transcendental over $F_1$ so $I=\emptyset$. Since the set $\varphi^*(\bd^m(b_2),K_2)$ is an open subset of $K_2^{m+1}$, the result follows directly form Lemma \[fact:density\].
Consequences of quantifier elimination {#sec:conse_QE}
--------------------------------------
\[cor:complete\] The theory $T_\delta^*$ is complete.
Let $\varphi$ be an $\cL_\delta$-sentence. By Theorem \[thm:QE\], we may suppose $\varphi$ has no variable of field sort. Therefore, since the constants of $\cL$ belong to the subfield of constants of $K$, every $\cL_\delta$-term in $\varphi$ is equal modulo $T_\delta^*$ to an $\cL$-term. Then, $\varphi$ is equivalent modulo $T_\delta^*$ to an $\cL$-sentence and the result follows from the completeness of $T$.
Let us recall some transfer results which (essentially) follow from Theorem \[thm:QE\].
\[thm:delta-dim\] The $\cL_\delta$-definable subsets of the field sort can be endowed with a dimension function as defined by van den Dries in [@vandendries1989].
\[thm:NIP\] If $T$ is NIP, then $T_\delta^*$ is NIP.
\[thm:cherni\] If $T$ is distal, then $T_\delta^*$ is distal.
\[prop:QE\_infty\] The theory $T_\delta^*$ eliminates the field sort quantifier $\exists^\infty$.
For the reader’s convenience, proofs of Theorems \[thm:NIP\] and \[thm:cherni\] will be given in the multi-sorted setting in the Appendix (the arguments are *mutatis-mutandis*, essentially the same). Theorem \[prop:QE\_infty\] was proven for $\CODF$ by the second author in [@point2011] and the result is to our knowledge new in the general setting. Its proof is a little bit more involved and will be given at the end of this section.
It is worthy to mention that other model-theoretic properties such as the existence of prime models or dp-minimality do not transfer from $T$ to $T_\delta^*$. Indeed, M. Singer showed in [@singer1978] (see also [@Fr]) that $\CODF$ has no prime models (while $\RCF$ has) and Q. Brouette showed in his thesis [@brouette2015] that $\CODF$ is not dp-minimal (while $\RCF$ is).
Before, proving Theorem \[prop:QE\_infty\], let us start by showing some consequences of relative quantifier elimination on definable sets.
\[cor:induced\] Let $K$ be a model of $T_\delta^*$ and $S$ be a sort of $\cL$ different from the field sort. Then every $\cL_\delta$-definable subset $X\subseteq S(K)^n$ is $\cL$-definable.
Let $z=(z_1,\ldots,z_n)$ be $S$-sorted variables. By Theorem \[thm:QE\], there are a tuple $x$ of field sort variables, a field sort quantifier free $\cL_\delta(K)$-formula $\varphi(x,z)$ (possibly with other non-field sort parameters) and $a\in K^{\ell(x)}$, such that $X$ is defined by $\varphi(a,z)$. Since $\varphi(x,z)$ has no field quantifiers, $X$ is also defined by $\varphi^*(\bd^m(a),z)$, where $m$ is the order of $\varphi$.
The following is a simple but important corollary of Theorem \[thm:QE\] that will be implicitly used hereafter.
Every $\cL_\delta$-definable set $X\subseteq K^n$ is of the form $\J_m^{-1}(Y)$ for a field sort quantifier-free $\cL$-definable set $Y\subseteq K^{n(m+1)}$.
\[def:order\] Let $X\subseteq K^n$ be an $\cL_\delta$-definable set. The *order of $X$*, denoted by $o(X)$, is the smallest integer $m$ such that $X=\J_m^{-1}(Y)$ for some field-sort quantifier-free $\cL$-definable set $Y\subseteq K^{n(m+1)}$.
Note that $o(X)=0$ if and only if $X$ is $\cL$-definable. We will finish by showing that $T_\delta^*$ eliminates the field quantifier $\exists^\infty$. To prove this we need the following technical lemma, which is a parametric version of the density of differential points (Lemma \[fact:density\]). This lemma will also play a crucial role in Section \[sec:continuous\] to describe $\cL_{\delta}$-correspondences.
\[lem:deltaniceform\] Let $K$ be a model of $T_\delta^*$. Let $\varphi(x,y)$ be an $\cL_\delta(K)$-formula where $x=(x_1,\ldots, x_{n})$ and $y$ is a single variable. Let $m$ be the order of $\varphi$. Then $\varphi$ is equivalent to a finite disjunction of $\cL_\delta(K)$-formulas of the form $$\cZ_{\cA}^{S}(x,y) \wedge \theta(\bd^{m}(x), \bd^m(y)),$$ where $\theta$ is an $\cL(K)$-formula which defines an open subset of $K^{(n+1)(m+1)}$ and either $\cA\subseteq K\{x\}$ or $\cA$ contains only one differential polynomial $P$ of non-negative order in $y$ and $s_P$ divides $S$. In addition, $\ord_{x_i}(\cA,S)\leqslant 2m$ for $1\leqslant i\leqslant n$ and $\ord_y(\cA,S)\leqslant m$.
By Theorem \[thm:QE\] we may suppose $\varphi$ has no field sort quantifiers. Consider the $\cL(K)$-formula $\varphi^*(\bar{x},\bar{y})$ where $\bar{x}=(\bar{x}_0,\ldots,\bar{x}_{n-1})$ with $\ell(\bar{x}_i)=m+1$ and $\bar{y}=(y_0,\ldots,y_m)$. By assumption ([**[A]{}**]{}), $\varphi^*(\bar{x},\bar{y})$ is equivalent to a disjunction of $\cL$-formulas of the form $$Z_\cA(\bar{x},\bar{y})\wedge \theta(\bar{x}, \bar{y}),$$ where $\theta$ defines an open subset of $K^{(n+1)(m+1)}$ and $\cA\subseteq K[\bar{x},\bar{y}]$. Define $$\cA'\coloneqq \{Q(\bd^m(x),\bd^m(y)) : Q\in \cA\}.$$ By definition we have that $\varphi$ is equivalent to the corresponding disjunction of $\cL_\delta(K)$-formulas of the form $$\cZ_{\cA'}(x,y)\wedge \theta(\bd^m(x), \bd^m(y)).$$ By Lemma \[lem:minord3\], the formula $\cZ_{\cA'}(x,y)$ is equivalent (modulo $T_\delta$) to a finite disjunction of formulas of the form $\cZ_\cB^{S_\cB}(x,y)$ such that $\ord_y(S_\cB)\leqslant m$, $\ord_{x_i}(\cB,S_\cB)\leqslant 2m$ for each $1\leqslant i\leqslant n$, and either
1. $\cB\subseteq K\{x\}$ or
2. there is a unique $P\in \cB$ of non-negative order in $y$, $\ord_y(P_\cB)\leqslant m$ and $s_{P_\cB}$ divides $S_\cB$.
Then $\varphi(x,y)$ is equivalent to the disjunction to the corresponding disjunction of $\cL_\delta(K)$-formulas $$\cZ_\cB^{S_\cB}(x,y)\wedge \theta(\bd^{m}(x), \bd^{m}(y)).$$
\[lem:envelop\] Let $K$ be a model of $T_{\delta}^*$ and $X$ be an $\cL_{\delta}$-definable subset of $K^{n+1}$ of order $m$. Then, for $d=2m$, there is an $\cL$-definable subset $Y\subset K^{(n+1)(d+1)}$ such that
1. $X=\J_{d}^{-1}(Y)$ and
2. for every $a\in K^n$ and $c\in K^{d+1}$ such that $(\bd^d(a),c)\in Y$ it holds that for every open neighbourhood $W$ of $c$ there is $b\in K$ such that $\bd^m(b)\in W$ and $(\bd^d(a),\bd^d(b))\in Y$.
In particular, for every $a\in K^n$ such that if $X_{a}$ is finite, $\vert X_{a}\vert=\vert Y_{\bd^d(a)}\vert$.
Let $\varphi(x,y)$ be an $\cL_\delta(K)$-formula of order $m$ defining $X$ where $x=(x_1,\ldots, x_n)$ and $y$ is a single variable. By Lemma \[lem:deltaniceform\], $\varphi(x,y)$ is equivalent, modulo $T_\delta^*$, to a finite disjunction of the form $$\bigvee_{j\in J} \cZ_{\cA_j}^{S_j}(x,y) \wedge \theta_j(\bd^{m}(x), \bd^m(y)),$$ where for each $j\in J$, $\theta_j$ is an $\cL(K)$-formula which defines an open subset of $K^{(n+1)(m+1)}$ and either $\cA_j\subseteq K\{x\}$ or $\cA_j\subseteq K\{x,y\}$, it only contains one differential polynomial $P_j$ of non-negative order in $y$ and $s_{P_j}$ divides $S_j$. In addition, $\ord_{x_i}(\cA_j,S_j)\leqslant 2m$ for $1\leqslant i\leqslant n$ and $\ord_y(\cA_j,S_j)\leqslant m$. For each $j\in J$, let $\tilde{\theta}_j(\bar{x}(m), \bar{y}(m))$ be the $\cL(K)$-formula $\theta(\bar{x},\bar{y})\wedge S_j^*(\bar{x}(m),\bar{y})\neq 0$. Note that $\tilde{\theta}_j$ defines an open subset of $K^{(n+1)(d+1)}$. For each $j\in J$, we define by cases an $\cL(K)$-formula $\psi_j(\bar{x}(m),\bar{y}(m))$ depending on whether $\cA\subseteq K\{x\}$ or not:
1. if $\cA_{j}\subseteq K\{x\}$ then $\psi_j(\bar{x}(m),\bar{y}(m))$ is $$Z_{\cA_{j}}(\bar x(m), \bar{y})\wedge \tilde{\theta}_j(\bar{x}(m),\bar{y}(m)).$$
2. otherwise, letting $k_j=\ord_{y}(P_j)$ we define $\psi_j(\bar{x}(m),\bar{y}(m))$ as $$\lambda_{P_{j}}^{m-k_j}(\bar{x}(m),\bar{y})\wedge \tilde{\theta}_{j}(\bar{x}(m),\bar{y}(m)).$$
Let $\psi(\bar{x}(m),\bar{y}(m))$ be the disjunction $\bigvee_{j\in J} \psi_j(\bar{x}(m),\bar{y}(m))$ and $Y$ be the subset of $K^{(n+1)(d+1)}$ defined by $\psi$. Let us show (1). The inclusion $\J_d^{-1}(Y)\subseteq X$ is clear. The converse follows by noting that for each $j\in J$ $$T_\delta^* \models \forall x \forall y (\cZ_{\cA_j}^{S_j}(x,y) \to \lambda_{P_{j}}^{m-k_j}(\bd^d(x), \bd^m(y))).$$ It remains to show (2).
Fix $a\in K^n$ and $c=(c_0,\ldots,c_d)\in K^{d+1}$ such that $(\bd^d(a),c)\in Y$. Let $j\in J$ be such that $\psi_j(\bd^d(a),c)$ holds. We split in cases. If $\psi_j$ is as in $(i)$, then the result follows from Lemma \[fact:density\]. So suppose $\psi_j$ is as in $(ii)$. Let $W$ be an open neighbourhood of $c$. Without loss of generality, we may suppose there is $V$ an open neighbourhood of $\bd^d(a)$ such that $V\times W\subseteq \tilde\theta_j(K)$. By the continuity of the functions $f_{i}^{P_{\ell}}$ (see Lemma-Definition \[lemdef:ratioprolong\]), we may shrink $V$ to a smaller open neighbourhood of $\bd^d(a)$ and find an open neighbourhood $W_{1}$ of $(c_{0},\ldots,c_{k_{j}})$ such that, letting $U\coloneqq V\times W_{1}$ $$W_1\times f_1^{P_j}(U) \times \ldots \times f_{d-k_j}^{P_j}(U) \subseteq W.$$ By the scheme (DL), we can find a differential tuple $\bd^{k_j}(b)\in W_{1}$ such that $$K\models P_{j}(\bd^m(a),\bd^{k_j}(b))=0\wedge\frac{\partial}{\partial y_{k_j}} P_{j}(\bd^m(a),\bd^{k_j}(b))\neq 0.$$ Since $\delta^{k_j+i}(b)=f_{i}^{P_{j}}(\bd^{m+i}(a),\bar \delta^{k_j}(b))$ for each $i\in \{1,\ldots, d-k_{j}\}$, we have that $(\bd^d(a), \bd^d(b))\in V\times W$, and hence in $\tilde \theta_\ell(K)$. This shows that $\psi_j(\bd^d(a),\bd^d(b))$ holds, so $(\bd^d(a),\bd^d(b))\in Y$.
The last statement follows directly from part (2) and the fact the topology is Hausdorff.
We have now all tools to show Theorem \[prop:QE\_infty\].
Let $K$ be a model of $T_{\delta}^*$. Let $X\subseteq K^{n+1}$ be an $\cL_{\delta}$-definable set of order $m$. Let $d=2m$ and $Y\subseteq K^{(n+1)(d+1)}$ be the $\cL$-definable set given by Lemma \[lem:envelop\]. Since $T$ eliminates $\exists^{\infty}$, there is a finite bound $n_{Y}$ such that for any $n(d+1)$-tuple of elements $\bar e$ of $K$, either $Y_{\bar e}$ is infinite or has cardinality $\leqslant n_{Y}$. By Lemma \[lem:envelop\], if $X_a$ is finite for $a\in K^n$, then $\vert X_{a}\vert=\vert Y_{\bd^d(a)}\vert\leqslant n_Y$, so the same bound shows the result for $X$.
Transfer of elimination of imaginaries {#sec:EI}
======================================
In this section, following a proof strategy due to M. Tressl to show elimination of imaginaries in $\CODF$, we show how to transfer elimination of imaginaries from $T$ to $T_\delta^*$ under the additional assumption on $T_\delta^*$ of having *$\cL$-open core*. For background facts on the elimination of imaginaries, we refer to [@tent-ziegler2012 Section 8.4]. Let us start by recalling the definition of open core. Throughout this section we let $T$ be an open $\cL$-theory of topological fields and assume $T_\delta^*$ is consistent. We let $K$ be a model of $T$.
\[def:open\_core\]Let $\tilde \cL$ be an extension of $\cL$ and $\tilde K$ be an $\tilde \cL$-expansion of $K$. We say $\tilde K$ has *$\cL$-open core* if every $\tilde \cL$-definable open subset is $\cL$-definable. An $\tilde{\cL}$-theory $\tilde T$ extending $T$ has $\cL$-open core if every model of $\tilde T$ has $\cL$-open core.
We will use the following three properties satisfied by the topological dimension on $\cL$-definable sets $X, Y\subseteq K^n$:
1. $\Dim(X)=0$ if and only if $X$ is finite and non-empty,
2. $\Dim(X\cup Y)=\max(\Dim(X),\Dim(Y))$,
3. $\Dim(\mathrm{Fr}(X))<\Dim(X)=\Dim(\overline{X})$.
(see Corollary \[cor:dim1\]).
Before proving the main theorem of this section, we will give a useful characterization of the $\cL$-open core property for $T_\delta^*$.
\[def:circ1\] Let $X \subseteq K^n$ be a non-empty field sort quantifier-free $\cL_\delta$-definable set. Given a positive integer $m$ and an $\cL$-definable set $Z\subseteq K^{n(m+1)}$, we call the triple $(X, Z, m)$ a *linked triple* if
1. $X=\J_m^{-1}(Z)$ and
2. $\overline{Z}=\overline{\J_{m}(X)}$.
Note that the integer $m$ occurring in a linked triple might be bigger than $o(X)$. However, as the next proposition shows, in our setting one can always take $m=o(X)$.
\[thm:fermeture\] The theory $T_{\delta}^*$ has $\cL$-open core if and only if for every $\cL_\delta$-definable set $X$, there is an integer $m$ and an $\cL$-definable set $Z\subseteq K^{n(m+1)}$, such that $(X,Z,m)$ is a linked triple. In addition, if $T_{\delta}^*$ has $\cL$-open core, for every $X$ there is a linked triple of the form $(X,Z,o(X))$.
Let $X \subseteq K^n$ be an $\cL_\delta$-definable set. By Theorem \[thm:QE\], we may assume $X$ is defined by a field-sort quantifier-free $\cL_\delta(K)$-formula.
$(\Rightarrow)$ Let $Y\subseteq K^{(o(X)+1)n}$ be an $\cL$-definable set such that $X=\J_{o(X)}^{-1}(Y)$. The subset $\overline{\J_{o(X)}(X)}$ is both closed and $\cL_{\delta}$-definable and so it is $\cL$-definable by the $\cL$-open core. Consider the $\cL$-definable set $Z\coloneqq Y\cap \overline{\J_{o(X)}(X)}$. Since $\J_{o(X)}(X)\subseteq Z\subseteq \overline{\J_{o(X)}(X)}$, both properties (1) and (2) are easily shown. This also shows the last assertion of the proposition.
$(\Leftarrow)$ It suffices to show that $\overline{X}$ is $\cL$-definable. By assumption there is an integer $m$ and an $\cL$-definable set $Z$ such that $(X, Z, m)$ is a linked triple. Let $\pi\colon K^{n(m+1)}\to K^n$ be the projection sending each block of $m+1$ coordinates to its first coordinate, that is, $$\pi(x_{1,0},\ldots,x_{1,m}, x_{2,0},\ldots,x_{2,m}, \ldots, x_{n,0},\ldots,x_{n,m}) = (x_{1,0},x_{2,0},\ldots, x_{n,0}).$$ We leave as an exercise to show that $\overline{X}=\overline{\pi(\overline{Z})}$. The results follows since, as $Z$ is $\cL$-definable, so is $\overline{\pi(\overline{Z})}$.
\[lem:order\_EI\] Let $X\subseteq K^n$ be field sort quantifier-free $\cL_\delta$-definable set. If $m_1\leqslant m_2$ then $$\Dim(\overline{\J_{m_1}(X)})\leqslant \Dim(\overline{\J_{m_2}(X)}).$$
Let $\pi\colon \overline{\J_{m_1}(X)}\to K^\ell$ be a projection such that $\pi(\overline{\J_{m_1}(X)})$ has non-empty interior. Then, letting $\rho$ denote the projection from $\overline{\J_{m_2}(X)}$ onto $\overline{\J_{m_1}(X)}$, we have that $\pi\circ\rho(\overline{\J_{m_2}(X)})$ has non-empty interior.
Let $\G$ be a collection of sorts of $\cL^{\mathrm{eq}}$. We let $\cL^{\G}$ denote the restriction of $\cL^{\eq}$ to the field sort together with the new sorts in $\G$. Given an automorphism $\sigma$ and a set $X$, we say that $X$ is *$\sigma$-invariant* if $\sigma$ fixes $X$ setwise.
\[thm:EI\] Suppose that the $\cL_\delta$-theory $T_\delta^*$ has $\cL$-open core and that $T$ admits elimination of imaginaries in $\cL^\G$. Then the theory $T^*_{\delta}$ admits elimination of imaginaries in $\cL_{\delta}^\G$.
Fix a sufficiently saturated model $K$ of $T_\delta^*$. Let $X\subseteq K^n$ be a non-empty $\cL_{\delta}$-definable set. It suffices to show that $X$ has a code in $\cL_{\delta}^\G$ (that is, an element $e\in \G$ such that $\sigma(e)=e$ if and only if $\sigma(X)=X$ for every $\cL_\delta$-automorphism $\sigma$ of $K$). Observe that every $\cL$-definable set has a code in $\cL^\G$, and therefore a code in $\cL_\delta^\G$, as the $\cL_\delta$-automorphism group of $K$ is a subgroup of the $\cL$-automorphism group of $K$. Consider the set $\widetilde{X}\supseteq X$ defined by $$\widetilde{X}\coloneqq \J_{o(X)}^{-1}(\overline{\J_{o(X)}(X)}).$$ Since $T_\delta^*$ has $\cL$-open core, the set $\overline{\J_{o(X)}(X)}$ is $\cL$-definable. We proceed by induction on $\Dim(\overline{\J_{o(X)}(X)})$. If $\Dim(\overline{\J_{o(X)}(X)})= 0$, then $X$ is finite (by (D1)) and in particular $\cL$-definable, so it has a code in $\cL_{\delta}^\G$. Alternatively, one may use that every finite definable set has a code modulo the theory of fields [@marker1996 Lemma 3.2.16]. To show the inductive step we need the following claim:
\[cla:dim\] $\Dim(\overline{\J_{o(X)}(\widetilde{X}\setminus X)}) < \Dim(\overline{\J_{o(X)}(X)})$.
Suppose the claim holds. Since $o(\widetilde{X}\setminus X)\leqslant o(X)$, by Lemma \[lem:order\_EI\], we have that $$\Dim(\overline{\J_{o(\widetilde{X}\setminus X)}(\widetilde{X}\setminus X)}) \leqslant \Dim(\overline{\J_{o(X)}(\widetilde{X}\setminus X)}).$$ Therefore, by Claim \[cla:dim\] and the induction hypothesis, let $e_{1}$ be a code for $\widetilde{X}\setminus X$. By the previous observation, let $e_{2}$ be a code for $\overline{\J_{o(X)}(X)}$ (which is $\cL$-definable by the $\cL$-open core hypothesis). It is an easy exercise to show that $e=(e_{1},e_{2})$ is a code for $X$.
It remains to prove the claim. By the $\cL$-open core assumption and Proposition \[thm:fermeture\], let $(X, Z, o(X))$ be a linked triple. Applying (D3), we have $$\begin{aligned}
\Dim(\overline{\J_{o(X)}(\widetilde{X}\setminus X)}) & = \Dim(\overline{\J_{o(X)}(\J_{o(X)}^{-1}(\overline{\J_{o(X)}(X)})\setminus X)}) \\
& = \Dim(\overline{\J_{o(X)}(\{x\in K^n : \J_{o(X)}(x)\in \overline{Z}\}\setminus X)}) \\
& = \Dim(\overline{\J_{o(X)}(\{x\in K^n : \J_{o(X)}(x)\in \overline{Z}\setminus Z\})}) \\
& \leqslant \Dim(\overline{Z}\setminus Z) < \Dim(\overline{Z})= \Dim(\overline{\J_{o(X)}(X)}). \end{aligned}$$
We will later show in Section \[sec:opencore\] that $T_\delta^*$ has $\cL$-open core for most $\cL$-theories $T$ given in Examples \[examples\] (including all henselian valued fields of characteristic 0 having a value group which is either divisible or a $\Z$-group). As a corollary we obtain the following.
\[cor:EI\_examples\] Let $\G$ denote the geometric language of valued fields. The theories $\ACVF_\delta^*$, $\RCVF_\delta^*$ and $\PCF_\delta^*$ have elimination of imaginaries in $\cL_\delta^\G$.
Let $T$ be either $\ACVF$, $\RCVF$ or $\PCF_d$. The theory $T$ has elimination of imaginaries in $\cL^\G$ by results of Haskell, Hrushovski and Macpherson for $\ACVF$ [@HHM2006], of Mellor for $\RCVF$ [@mellor2006], and of Hrushovski, Martin and Rideau for $\PCF_d$ [@HMR2018]. By Corollary \[cor:opencore\], $T_\delta^*$ has $\cL$-open core. The result follows by Theorem \[thm:EI\].
The fact that $\CODF$ has $\cL$-open core and eliminates imaginaries was first proved in [@point2011] by different methods. The following proof strategy is precisely Tressl’s unpublished argument.
\[cor:EI\_CODF\] The theory $\CODF$ has elimination of imaginaries in $\cL_\delta$.
By Corollary \[cor:opencore\], $\CODF$ has $\cL$-open core. The result follows by Theorem \[thm:EI\].
Applications to dense pairs {#sec:app}
===========================
The study of pairs of models of a given complete theory is a classical topic in model theory. Early results by A. Robinson [@Robinson] showed completeness (and model-completeness) of the theories of pairs of algebraically closed fields and *dense* pairs of real-closed fields, that is, pairs in which the smaller field is dense in the larger one. In [@Macintyre], A. Macintyre recasted Robinson’s results in an abstract setting which also encompassed dense pairs of $p$-adically closed fields. In another direction, L. van den Dries [@Dries1998] studied dense pairs of models of an o-minimal theory expanding the theory of ordered abelian groups, also generalizing some of Robinson’s results.
New developments have encompassed these results in different abstract frameworks. Two such frameworks are the theory of lovely pairs of geometric structures developed by A. Berenstein and E. Vassiliev [@berenstein-vassiliev2010], and the theory of dense pairs of theories with existential matroids developed by A. Fornasiero [@F]. In this section we will study the theory $T_P$ of dense pairs of models of a one-sorted $\cL$-open theory of topological fields $T$. Our goal is to show that such theory is closely related with the theory $T_\delta^*$. In Section \[sec:DenseP1\], we will define the theory $T_P$ and show how it fits into the two above mentioned abstract frameworks. In Section \[sec:DenseP2\], we will show how to use $T_{\delta}^*$ to deduce properties of $T_P$. Although most of the results gather in this section concerning $T_P$ are known, the proofs and methods will put in evidence the interesting connexion between the model theory of dense pairs and generic derivations.
Dense pairs of models of $T$ {#sec:DenseP1}
----------------------------
Let us start by recalling Fornariero’s setting in [@F]. Given that the literature of the model theory of pairs is quite extensive, we will unify references and cite [@F] even if particular cases of cited results where proven before by many different authors. Let $T$ be a complete one-sorted geometric $\cL$-theory $T$ extending the theory of fields (not necessarily an open $\cL$-theory of topological fields). Fornasiero considers more generally the case where $T$ admits an *existential matroid* (see [@F Definition 3.25]), but we will not need this level of generality in the present paper. Let $\Dim_\mathrm{acl}$ denote the dimension function induced by the algebraic closure $\acl$. Given a model $M$ of $T$ and a definable subset $X\subseteq M$, we say that $X$ is [*dense*]{} if $X\cap U\neq \emptyset$ for every $M$-definable subset $U$ of $M$ such that $\Dim_\mathrm{acl}(U)=1$ (see [@F Definition 7.1]). Now we are ready to define the theory of dense pairs of models of $T$. We work in the language of pairs $\cL_P$ defined as $\cL_{P}\coloneqq\cL\cup\{P\}$ for $P$ a new unary predicate. The theory $T_{P}$ of dense pairs of models of $T$ is defined as the $\cL_P$-theory of pairs $(K, P(K))$ such that $K\models T$, $P(K)$ is $\acl$-closed and dense in $K$ (in the above sense). Equivalently (by [@F Lemma 7.4]), it corresponds to the $\cL_P$-theory of pairs $(K, P(K))$ such that $K\models T$, $P(K)\prec_\cL K$ and $P(K)$ is dense in $K$. Among various model-theoretic results which are proven in [@F] about the theory $T_P$, what plays a crucial role in this section is the fact that $T_P$ is a complete theory [@F Theorem 8.3].
Let us now recall the framework of lovely pairs of geometric theories introduced by A. Berenstein and E. Vassiliev [@berenstein-vassiliev2010]. Let $M$ be a model of $T$ and $N\subseteq M$ be a proper subset of $M$. The pair $(M, N)$ is said to be a *lovely pair* (see also [@berenstein-vassiliev2010 Definition 2.3]) if
1. $N=\acl(N)$ and
2. for every $A\subseteq M$ with $\acl(A)=A$ and $\dim_\mathrm{acl}(A)\in \N$, and for every non-algebraic type $q\in S_{1}(A)$,
1. there exists an element $a\in N$ realizing $q$,
2. there exists an element $a\in M$ realizing $q$ with $a\notin \acl(A\cup N)$.
Berenstein and Vassiliev showed that all lovely pairs of models of $T$ are elementarily equivalent [@berenstein-vassiliev2010 Corollary 2.9] and gave an explicit axiomatization of their common $\cL_P$-theory which we will denote by $T_{\mathrm{LP}}$ [@berenstein-vassiliev2010 Theorem 2.10]. They also showed that $\vert T\vert^+$-saturated models of $T_{\mathrm{LP}}$ are lovely pairs. It is not difficult to show that a lovely pair of models of $T$ is a model of $T_P$ and therefore, in the light of the previous results, the theories $T_{\mathrm{LP}}$ and $T_{P}$ coincide.
Observe that when $T$ is a one-sorted $\cL$-open expansion of topological fields, by Proposition \[prop:conseAA1\], $T$ is geometric and therefore, $T_P$ is complete. Note also that in view of Part (2) of Proposition \[prop:conseAA2\], the notion of density above defined coincides with the topological notion of density.
Dense pairs and generic derivations {#sec:DenseP2}
-----------------------------------
Throughout this section we suppose $T$ is a one-sorted $\cL$-open expansion of topological fields for which $T_\delta^*$ is consistent. The connection of $T_P$ with the theory $T_\delta^*$ arises via the field of constants $C_K$ of a model $K$ of $T_\delta^*$. One can readily observe that when $K\models \CODF$, the pair $(K, C_K)$ is a dense pair of real-closed fields. The following lemma shows this holds in general for $T_\delta^*$. It was proven in [@BCPP2018 Corollary 1.7] under the assumption that the language $\cL=\cL_{\mathrm{ring}}$ and that the theory $T$ is a model-complete theory of large fields. It can also be deduced from [@F Lemma 7.4] and from [@berenstein-vassiliev2010 Lemma 2.5]. For the convenience of the reader, we give a proof here, following the last two references.
\[fact:model\] Let $K$ be a model of $T^*_{\delta}$ and $C_K$ be the constant subfield of $K$. Then $(K,C_K)$ is a model of $T_P$ and if $K$ is $\vert T\vert^+$-saturated, then $(K,C_{K})$ is a lovely pair of models of $T$.
Let $K\models T_{\delta}^*$. Then, a direct consequence of the scheme (DL) is that $C_{K}\neq K$. Since $C_{K}$ is topologically dense in $K$ [@guzy-point2010 Lemma 3.12], $C_{K}$ is dense. So it remains to show that $C_{K}\models T$. We apply Tarski-Vaught test. Let $\varphi(x,\bar y)$ be an $\cL$-formula and let $\bar b\in C_{K}$. By hypothesis ${\bf(A)}$ on $T$, $\varphi(K,\bar b)$ is a finite union of finite subsets and open sets. Since $C_{K}$ is algebraically closed in $K$, either $\varphi(K,\bar b)\subset C_{K}$ or contains an open subset. Since $C_{K}$ is topologically dense in $K$, we get the result.
\[lem:elext\] Every model $(K,F)$ of $T_P$ has an $\cL_P$-elementary extension $(K^*,F^*)$ such that there is a generic derivation on $K^*$ with constant field $F^*$.
Since $T_P$ is complete, by Lemma \[fact:model\], there is a model $K'$ of $T^*_{\delta}$ with constant field $F'$ such that $(K',F')\equiv_{\cL_P} (K,F)$. By Keisler-Shelah’s isomorphism theorem, there is a set $I$ and an ultrafilter $\cF$ on $I$ such that $$(K,F)\preccurlyeq_{\cL_P} (K,F)^I/\mathcal{F}=:(K^*,F^*)\cong_{\cL_P} (K^{'*},F^{'*})\coloneqq(K',F')^I/\mathcal{F} \succcurlyeq_{\cL_P} (K',F').$$ Since $K'$ is a model of $T_\delta^*$, we have that $K^{'*}$ is also a model of $T_\delta^*$ with constant field $F^{'*}=F^I/\mathcal{F}$. Hence, the isomorphism $(K^*,F^*)\cong_{\cL_P} (K^{'*},F^{'*})$ induces on $K^*$ an $\cL_\delta$-structure making of $K^*$ a model of $T_\delta^*$ with constant field $F^{'}$.
Let us now show how the previous results allows us to transfer properties of $T_\delta^*$ to $T_P$.
\[cor:delta\_exist\_inf\] The theory $T_{P}$ eliminates $\exists^{\infty}$.
This follows directly from Theorem \[prop:QE\_infty\].
\[thm:opencorepairs\] Assume that the theory $T_{\delta}^*$ has $\cL$-open core. Then the theory $T_P$ has $\cL$-open core.
By Lemma \[lem:elext\], let $(K^*,F^*)$ be an $\cL_P$-elementary extension such that $K^*$ is a model of $T_\delta^*$ with constant field $F^*$. Let $\varphi(x,y)$ be an $\cL_P$-formula such that for $a\in K^{|y|}$, $\varphi(x,a)$ defines the open set $U$. Then $\varphi(K^*,a)$ defines an open set $U(K^*)$ in $(K^*)^n$. Since $K^*$ is a model of $T_\delta^*$ and every $\cL_P$-formula defines a set which is $\cL_\delta$-definable by replacing $P(t)$ by the formula $\delta(t)=0$, $U(K^*)$ is $\cL_\delta$-definable. Now by assumption, we have that $U(K^*)$ is definable by $\psi(x,c)$ where $\psi(x,z)$ is an $\cL$-formula and $c\in (K^*)^{\ell(z)}$. Then we have that $$(K^*,F^*)\models (\forall x)(\varphi(x,a)\leftrightarrow \psi(x,c)),$$ and quantifying over $c$, we have that $$(K,F) \models (\exists c)(\forall x)(\varphi(x,a)\leftrightarrow \psi(x,c)),$$ which shows that $U$ is $\cL$-definable.
In Section \[sec:opencore\] we will show that most theories $T_{\delta}^*$ corresponding to $T$ as in Examples \[examples\] have $\cL$-open core (see later Theorems \[cor:opencore\] and \[thm:CODF\_opencore\]). As a corollary we obtain the following result shown by Hieronymi and Boxall [@Boxall2012 Corollary 3.4] and Fornasiero [@F Theorem 13.11].
\[cor:opencorepairs\] Let $T$ be either $\RCF$, $\PCF_d$, $\RCVF$ or $\ACVF_{0,p}$. Then $T_{P}$ has $\cL$-open core.
We finish this section with some remarks on distality. By a result of P. Hieronymi and T. Nell in [@HN2017], the theory of dense pairs of an o-minimal expansion of an ordered group is not distal. A natural question posed by P. Simon asks whether the theory of such pairs always admits a distal expansion [@nell2018 Question 1]. In [@nell2018], T. Nell provided a positive answer to the question for the theory of dense pairs of ordered vector spaces. A simple consequence of our analysis is that the theory of dense elementary pairs of real closed fields admits a distal expansion, namely, the theory $\CODF$. More generally, the following is a direct consequence of Theorem \[thm:distality\] and Lemma \[lem:elext\].
\[cor:dist\_expansion\] If the theory $T$ is distal, then $T_\delta^*$ is a distal expansion of $T_P$. In particular, $T_P$ admits a distal expansion when $T$ is $\RCF$, $\PCF_d$ or $\RCVF$.
As a consequence of results of A. Chernikov and S. Starchenko in [@chernikov-star2018], definable relations in models of $T_P$ satisfied the so called strong Erdős-Hajnal property (see [@chernikov-star2018 Definition 1.6, Theorem 6.10 (3)]).
\[cor:distality\_erdos\] If $T$ is distal, then definable relations in models of $T_P$ satisfy the strong Erdős-Hajnal property. This holds in particular for models of $T_P$ when $T$ is $\RCF$, $\PCF_d$ or $\RCVF$.
This follows from Corollary \[cor:dist\_expansion\] and [@chernikov-star2018 Corollary 4.8].
Open core {#sec:opencore}
=========
We will prove in this section that $T_\delta^*$ has $\cL$-open core for some theories $T$ listed in Examples \[examples\]. The proof strategy has two main steps. The first one consists in showing that continuous $\cL_\delta$-definable functions (and more generally continuous $\cL_\delta$-definable correspondences) are in fact $\cL$-definable. The second one consists in associating to every closed $\cL_\delta$-definable set $X\subseteq K^n$ a continuous $\cL_\delta$-definable function which “measures” the distance of a point in $K^n$ to $X$. Combining both steps, one recovers $X$ as the elements in $K^n$ of “distance 0”. We will carry out these steps in the following two main contexts:
1. [**(Ordered)**]{} the definable topology $\tau$ on $K$ comes from a total order;
2. [**(Valued)**]{} the definable topology $\tau$ on $K$ comes from a valuation $v\colon K\to \Gamma_\infty$. Moreover we assume $T$ satisfies either $(\dagger)$ or $(\dagger\dagger)$ as defined in Section \[sec:correspon\_gamma\]. For simplicity, we assume in this case both the value group and the valuation are part of the language $\cL$.
The strategy described above was devised by M. Tressl for $\CODF$. Various new ideas were needed to be included in order to adapt it to the present setting. In particular, the fact that open theories of topological fields do not necessarily have finite Skolem functions naturally led us to consider the more general case of continuous definable correspondences. Furthermore, in the case of valued fields, continuous definable functions to the value group also needed to be treated. Through the section, we let $T$ be an open $\cL$-theory of topological fields and assume $T_\delta^*$ is consistent. We will add (Ordered) or (Valued) to indicate we are respectively in one of the above contexts.
\[thm:continuity\_L\] Let $K\models T_\delta^*$. Let $X\subseteq K^n$ be an $\cL$-definable set and $f\colon X\rightrightarrows K$ be an $\cL_\delta$-definable $\ell$-correspondence. If $f$ is continuous, then it is $\cL$-definable.
\[thm:continuity\_L2\] Let $K$ be a model of $T_\delta^*$. Let $X\subseteq K^n$ be an $\cL$-definable set and $f\colon X\to \Gamma_{\infty}$ be an $\cL_\delta$-definable function. If $f$ is continuous, then it is $\cL$-definable.
The proof of the previous theorems will be given in Section \[sec:continuous\].
In order to associate a function $d_X$ to every $\cL_\delta$-definable closed set $X$ in the above cases, we further need the following results concerning definable completeness on either the field sort or the value group sort (when it applies). Recall that for a first order language $\cL_0$, a totally ordered $\cL_0$-structure $M$ is *$\cL_0$-definably complete* if every bounded $\cL_0$-definable set $X\subseteq M$ has an infimum and a supremum in $M$.
\[thm:def-complete2\] Let $K$ be a model of $T_\delta^*$. If $\Gamma_\infty$ is $\cL$-definably complete, then it is also $\cL_\delta$-definably complete.
It is a straightforward consequence of Corollary \[cor:induced\]. In fact, this also holds even without assuming $(\dagger)$ or $(\dagger\dagger)$.
To show definable completeness in the ordered case we will use the following lemma which is equivalent to having the open core for definable sets in one variable. It hints to a potential proof of the open core which will follow directly from the axiomatization of $T_\delta^*$ without specifying what type of topology (ordered, valued, etc.) comes with the theory $T$.
\[lem:closure\] Let $K$ be a model of $T_{\delta}^*$ and $X\subseteq K$ be an $\cL_{\delta}$-definable subset of order $m$. Then, for $d=2m$, there is an $\cL$-definable subset $Y\subset K^{d+1}$ such that $(X,Y,d)$ is a linked triple. In particular, if $X$ is open then $X$ is $\cL$-definable.
By Lemma \[lem:envelop\], there is an $\cL$-definable set $Y\subseteq K^{d+1}$ with $d=2m$ such that $X=\J_d^{-1}(Y)$ and $\overline{Y}=\overline{\J_d(X)}$ (this second property follows by Part (2) of Lemma \[lem:envelop\]). This shows that $(X, Y, d)$ is a linked triple. Letting $\pi\colon K^{d+1}\to K$ denote the projection onto the first coordinate, we have that $\overline{\pi(\overline{Y})}=\overline{X}$, so $\overline{X}$ is $\cL$-definable. This shows the last statement of the lemma.
\[thm:def-complete1\] Let $K$ be a model of $T_\delta^*$. If $K$ is $\cL$-definably complete, then it is also $\cL_\delta$-definably complete.
Let $X\subseteq K$ be an $\cL_\delta$-definable set. By Lemma \[lem:closure\], $\overline{X}$ is $\cL$-definable. Now, $\overline{X}$ is bounded since $X$ is bounded and, in addition, $X$ and $\overline{X}$ have the same supremum (resp. same infimum).
We have all needed tools to associate the function $d_X$ to an $\cL_\delta$-definable set $X$ and prove the $\cL$-open core for $T_\delta^*$ in both contexts.
Let $K$ be a model of $T_\delta^*$ and $X\subseteq K^n$ be an $\cL_\delta$-definable set.
1. [**(Ordered)**]{} Assume $K$ is $\cL$-definably complete. The function $d_X\colon K^n\to K$ is defined as $$d_X(a)\coloneqq\inf_{b\in X} \sum_{i=1}^n (a_{i}-b_{i})^2.$$
2. [**(Valued)**]{} Assume $\Gamma_\infty$ is $\cL$-definably complete. The function $d_X\colon K^n\to \Gamma_\infty$ is defined as $$d_X(a)\coloneqq\inf_{b\in X} \{v(a_{1}-b_{1}),\ldots, v(a_n-b_n)\}.$$
Propositions \[thm:def-complete1\] and \[thm:def-complete2\] ensure that the function $d_X$ is well defined in each case. As a consequence we obtain:
\[thm:opencore\] The theory $T_\delta^*$ has $\cL$-open core, whenever
1. [**(Ordered)**]{} the field sort is $\cL$-definably complete;
2. [**(Valued)**]{} the value group sort is $\cL$-definably complete.
Let $K$ be a model of $T_\delta^*$. As having $\cL$-open core is an elementary property, it suffices to show that every closed $\cL_{\delta}$-definable set $X\subseteq K^n$ is $\cL$-definable. We split in cases.
[**(Ordered)**]{} By Proposition \[thm:def-complete1\], $d_X$ is a well-defined $\cL$-definable function. The function $d_X$ is continuous, so by Theorem \[thm:continuity\_L\], $d_X$ is $\cL$-definable. Since $X$ is closed, $X=\{a\in K^n : d_X(a)=0\}$, and hence it is $\cL$-definable.
[**(Valued)**]{} By Proposition \[thm:def-complete1\], $d_X$ is a well-defined $\cL$-definable function. As in the previous case, $d_X$ is continuous, so by Theorem \[thm:continuity\_L2\], $d_X$ is $\cL$-definable. Finally, $X$ being closed implies that $X=\{a\in K^n : d_X(a)=\infty\}$. Hence, $X$ is $\cL$-definable.
\[cor:opencore\] Let $T$ be one of the following theories: $\ACVF_{0,p}, \RCVF, \PCF_d$ or the $\cL_\RV$-theory of $k(\!(t^\Gamma)\!)$ for $k$ a field of characteristic 0 and $\Gamma$ either $\Z$-group or a divisible group. Then, the theory $T_\delta^*$ has $\cL$-open core.
Let $K$ be a model of $T_\delta^*$ and $\Gamma$ be its value group. In all cases, the $\cL$-theory $T$ satisfies either $(\dagger)$ or $(\dagger\dagger)$. Note that by Ax-Kochen/Eršov, if $\Gamma$ is a $\Z$-group, then $k(\!(t^\Gamma)\!)$ has an elementary substructure of the form $k(\!(t^{\Z})\!)$. Moreover, in each case, $\Gamma_\infty$ is $\cL$-definably complete. The result follows by Theorem \[thm:opencore\].
\[thm:CODF\_opencore\] The theory $\CODF$ has $\cL$-open core.
Since $\RCF$ is definably complete, the result follows by Theorem \[thm:opencore\]. Alternatively, this follows from Theorem \[cor:opencore\]. Indeed, every model of $\CODF$ embeds into a model of $\RCVF_\delta^*$, and hence, the $\cL$-open core of $\RCVF_\delta^*$ implies the result for $\CODF$.
The proof given in [@point2011] uses the fact that $\CODF$ is $\cL_\delta$-definable complete together with the following criterion due to A. Dolich, C. Miller and C. Steinhorn [@DMS]: any expansion of a densely ordered abelian group has “o-minimal open core” if it eliminates the quantifier $\exists^{\infty}$ and is definably complete. Since $\CODF$ eliminates the quantifier $\exists^{\infty}$ (Theorem \[prop:QE\_infty\]), the result follows.
Continuous $\cL_{\delta}$-definable functions and correspondences {#sec:continuous}
-----------------------------------------------------------------
In this section we prove Theorems \[thm:continuity\_L\] and \[thm:continuity\_L2\]. We will also need the following two lemmas showing that $\cL_\delta$-definable correspondences with an $\cL$-definable domain are essentially compositions of $\cL$-definable correspondences with the derivation.
\[prop:honest\] Let $K$ be a model of $T_\delta^*$. Let $X\subseteq K^n$ be an $\cL$-definable set and ${f\colon X\rightrightarrows K}$ be an $\cL_\delta$-definable $\ell$-correspondence with $\ell\geqslant 1$. Then, there are $d\in \N$, an $\cL$-definable set $Y\subset K^{n(d+1)}$ and an $\cL$-definable $\ell$-correspondence $F\colon Y\rightrightarrows K$, such that for every $x\in X$ $$f(x)=F(\bar{\delta}^d(x)).$$
Let $\tilde X\coloneqq \Graph(f)$. By Lemma \[lem:envelop\], there is a natural number $d$ and an $\cL$-definable subset $Z\subset K^{(n+1)(d+1)}$ such that given any $(a,b)\in K^n\times K$, $(\bd^d(a),\bd^d(b))\in Z$ if and only if $(a,b)\in \Graph(f)$.
Moreover if $\tilde X_{a}$ is finite, then $\vert \tilde X_{a}\vert=\vert Z_{\bd^d(a)}\vert$. Let $\pi$ be the projection from $K^{(n+1)(d+1)}$ to $K^{n(d+1)}$ and let $Y\coloneqq\pi(Z)\cap \{\bar x\in K^{n(d+1)}: \exists^{=\ell} \bar{y}\; (\bar x, \bar{y})\in Z_{\bar x}\}$. Define $F\colon Y \to K$ by $F(\bar e)\coloneqq\pi_{1}(Z_{\bar e})$, where $\pi_{1}$ is the projection on the first coordinate. Let $a\in K^n$, then $F(\bd^d(a))=\pi_{1}(Z_{\bd^d(a)})=X_{a}$.
The following is an analogous result in the valued context.
\[prop:honest2\] Let $K$ be a model of $T_\delta^*$. Let $X\subseteq K^n$ be an $\cL$-definable set and $f\colon X\rightrightarrows \Gamma_{\infty}$ be an $\cL_\delta$-definable correspondence. Then there are $m\in \N$ and an $\cL$-definable $\ell$-correspondence $F\colon Y\rightrightarrows \Gamma_{\infty}$, $Y\subset K^{n(m+1)}$, such that for every $x\in X$ $$f(x)=F(\bar{\delta}^m(x)).$$
For $x=(x_1,\ldots, x_n)$ and $\xi$ a single variable (varying in $\Gamma_\infty$), let $\varphi(x,\xi)$ be an $\cL_\delta(K)$-formula defining $f$. Let $\varphi^{*}(\bar{x}, \xi)$ be the corresponding $\cL(K)$-formula with $\bar{x}=(\bar{x}_1,\ldots, \bar{x}_n)$, $\ell(\bar{x}_i)=m+1$. Consider the $\cL$-definable set: $$Y\coloneqq\{\bar u\in K^{n(m+1)}: (u_{1},\ldots,u_{n})\in X \wedge \exists^{=\ell} \xi \varphi^*(\bar u, \xi)\}.$$ and the $\ell$-correspondence $F(\bar u,\xi)$ defined by the $\cL(K)$-formula $$\psi_{\ell}(\bar u,\xi)\coloneqq\varphi^*(\bar u,\xi) \wedge \bar x\in Y.$$ For $x\in X$ and $\xi\in f(x)$, we have that $\varphi(x,\xi)$ holds, so $\varphi^*(\bar \delta^m(x),\xi)$ too. Since $\vert f(x)\vert=\ell$, we get that $\psi(\bar \delta^m(x),\xi)$ holds.
We have all tools to show Theorems \[thm:continuity\_L\] and \[thm:continuity\_L2\].
We proceed by induction on $\dim(X)$, the case $\dim(X)=0$ being clear. By cell decomposition (Theorem \[prop:cell-decomposition\]) and the induction hypothesis (possibly changing $n$ and $\ell$), we may suppose that $X$ is open in $K^n$.
By Proposition , let $F\colon Y\subseteq K^{n(m+1)}\rightrightarrows K$ be an $\cL$-definable $\ell$-correspondence such that for all $x\in X$ and $y\in K$ $$y\in f(x) \Leftrightarrow y\in F(\bar{\delta}^m(x)).$$ If $m=0$ there is nothing to show, so suppose $m>0$.
Let $\pi\colon K^{n(m+1)}\to K^n$ be the projection sending each block of $(m+1)$ tuples to its first element. Without loss of generality, we may suppose that $\pi(Y)=X$.
Given $x=(x_{1},\ldots,x_{n})\in K^n$ and $z=(z_1,\ldots,z_n)\in K^{nm}$, we let $(x,z)_\pi$ denote the element $$(x_1,z_1,\ldots, x_n,z_n)\in K^{n(m+1)}.$$ In particular, $\pi((x,z)_\pi)=x$ for all $x\in K^n$.
\[1\] We have $\overline{Y}=\overline{\pi^{-1}(X)}$.
Since $\J_m(X)\subseteq Y\subseteq \pi^{-1}(X)$, it suffices to show that $\J_m(X)$ is dense in $\pi^{-1}(X)$. Let $(x,z)_\pi\in \pi^{-1}(X)$ with $z=(z_1,\ldots,z_n)\in K^{nm}$. Let $U$ an open neighbourhood of 0 such that $(x,z)_\pi+ U^{n(m+1)}\subseteq \pi^{-1}(X)$. Since $X$ is open, by Lemma \[fact:density\], for each $i\in\{1,\ldots,n\}$ there is $u_i\in x_i+U$ such that $\bd^m(u_i)\in (x_i,z_i)+U^{m+1}$. Letting $u=(u_1,\ldots,u_n)$, we have that $\bd^m(u)\in (x,z)_\pi+U^{n(m+1)}$. This shows the claim.
By the claim, $\dim(Y)=n(m+1)$. Indeed $$\dim(Y)=\dim(\overline{Y})=\dim(\overline{\pi^{-1}(X)})=n(m+1),$$ where the last equality holds since $X$ is open. Define $$\tilde Y\coloneqq\left\{\bar{x}\in Y \middle| \begin{array}{l}
\text{there is an open set $V\subseteq Y$ of $\bar{x}$ such that} \\
%\text{$|F(\bar{z})|=\ell$ for all $\bar{z}\in V$, and
\text{$F|_V$ is continuous}
\end{array}\right\}.$$ By Proposition \[prop:acont\], it holds that $$\dim\left(Y\setminus \tilde Y\right)<\dim(Y).$$
\[3\] The following holds $$\dim\left(X\setminus \pi(\tilde Y)\right)<\dim(X).$$
Suppose for a contradiction this is not the case. Therefore there is an $\cL$-definable open set $U\subseteq X$ such that $U\cap \pi(\tilde Y)=\emptyset$. This implies that $$\pi^{-1}(U)\subseteq Y\setminus \tilde Y,$$ and therefore that $$n(m+1)=\dim(\pi^{-1}(U))\leqslant \dim\left(Y\setminus \tilde Y\right)<\dim(Y)=n(m+1),$$ a contradiction. This shows the claim.
By Claim \[3\] and the induction hypothesis, we may suppose $\pi(\tilde Y)=X$. The theorem follows directly from the following final claim.
\[cl:final\] For $x\in X$ and all $y$ such that $(x,y)_\pi\in \tilde Y$, $F((x,y)_\pi)=f(x)$.
Suppose for a contradiction this is not the case and let $y$ be such that $(x,y)_\pi\in \tilde Y$ but $F((x,y)_\pi)\neq f(x)$. Therefore, there is $z\in F((x,y)_\pi)\setminus f(x)$ (since both are $\ell$-correspondences). Let $U$ be an open neighbourhood of 0 such that $z+U$ is disjoint from $f(x)+U$. By the definition of $\tilde Y$, let $V\subseteq U$ be an open neighbourhood of 0 such that $(x,y)_\pi+V^{n(m+1)}\subseteq \tilde Y$ and $F|_{(x,y)_\pi+V^{n(m+1)}}$ is continuous. By Lemma \[lem:corre\_local\_cont\], we may assume (possibly shrinking $V$) that $\Graph(F_{\vert V})$ is the disjoint union of the graphs of $\ell$ continuous definable functions $g_{1},\ldots,g_{\ell}$ from $(x,y)_\pi+V^{n(m+1)}$ to $K$. Suppose without loss of generality that $((x,y)_\pi,z)\in \Graph(g_{1})$. By the continuity of $f$, let $U_0\subseteq V$ be an open neighbourhood of 0 such that $f(x+U_0^n)\subseteq f(x)+U$. Let $V_0\subseteq U_{0}$ be such that, $g_{1}\vert (x,y)_\pi+V_{0}^{n(m+1)}\subset z+U$. By Lemma \[fact:density\], there is $w\in X$ such that $\bd^m(w)\in (x,y)_\pi+V_0^{n(m+1)}$. Since $F(\bd^m(w))=f(w)$, there is $z'\in f(w)$ such that $z'\in z+U$, which contradicts that $z+U$ and $f(x)+U$ are disjoint.
The proof is an immediate analogue of the proof of Theorem \[thm:continuity\_L\], replacing Proposition \[prop:honest\] by \[prop:honest2\], Proposition \[prop:acont\] by \[prop:ae\] and noting that a stronger version of Lemma \[lem:corre\_local\_cont\] holds in this context since the graph of a definable $\Gamma_\infty$-valued $\ell$-correspondence is the disjoint union of the graphs of $\ell$ definable $\Gamma_\infty$-valued functions (even globally).
Classical transfers
===================
Through this section we let $T$ be an open $\cL$-theory of topological fields. Let $\bU$ be a monster model of $T_\delta^*$ and $A$ be some small subset. We let $\langle A \rangle$ be $A$ together with the differential closure of the elements of $A$ in the field sort.
\[lem:types\] Let $x$ be a tuple of variables of field sort and let $z$ be a tuple of variables of other sorts. Then for $a\in S_x(\bU)$ and $b\in S_z(\bU)$, the $\cL_\delta$-type $tp_\delta(a,e/A)$ is determined by the infinite sequence of $\cL$-types $\{tp(\bd^m(a), e / \langle A\rangle): m\in \N\}$.
This follows by relative quantifier elimination (Theorem \[thm:QE\]) and the fact that for every field sort quantifier free $\cL_\delta$-formula $\varphi(x,z)$ over $A$, the formula $\varphi^*$ as defined in Notation \[not:etoile\] is an $\cL$-formula over $\langle A\rangle$.
\[cor:indiscernibles\] Let $x$ and $z$ be as in the previous lemma. Let $(a_i, e_i)_{i\in I}$ be a sequence where $a_i\in S_x(\bU)$ and $e_i\in S_z(\bU)$. Then the sequence is $\cL_\delta$-indiscernible sequence over $A$ if and only if for each $m\in \N$, the sequence $(\bd^m(a_i), e_i)_{i\in I}$ is $\cL$-indiscernible over $\langle A\rangle$.
\[thm:ANIP\] $T$ is NIP if and only if $T_\delta^*$ is NIP.
Suppose $T_\delta^*$ is not NIP. Let $\varphi(x,z;y,w)$ be a partitioned $\cL_\delta$-formula with IP where $x, y$ are tuples of field sort and $z,w$ are tuples of other sorts. By Theorem \[thm:QE\], we may assume that $\varphi$ has no field sort quantifiers. Then, since $\varphi$ has IP so does the $\cL$-formula $\varphi^*$ in $T$. The converse is clear, since being NIP is preserved by reducts (see Remark \[rem:consistency\]).
To show the transfer of distality, a dividing line introduced by P. Simon in [@simon2013], we will use the following equivalent definition of distality which appears in [@HN2017]. In the following definition we let $\cL$ be any first order language, $T$ be a complete $\cL$-theory and $\bU$ be a monster model of $T$.
Let $\varphi(x_1,\ldots,x_n;y)$ be a partitioned $\cL$-formula, where $x_i$, $1\leqslant i\leqslant n$ is a $p$-tuple of variables and $y$ is a $q$-tuple of variables, $p, q>0$. Then $\varphi$ is distal (in $T$) if for every $b\in \bU^q$, and every indiscernible sequence $(a_i)_{i\in I}$ in $\bU^p$ such that
1. $I=I_1+c+I_2$, where both $I_1, I_2$ are (countable) infinite dense linear orders without end points and $c$ is a single element with $I_1<c<I_2$,
2. the sequence $(a_i)_{i\in I_1+I_2}$ in $\bU^p$ is indiscernible over $b$,
then $\bU\models \varphi(a_{i_1},\ldots,a_{i_n};b)\leftrightarrow \varphi(a_{j_1},\ldots,a_{j_n};b)$ with $i_1<\ldots<i_n$, $j_1<\ldots<j_n$ in $I$.
A theory $T$ is distal if every formula is distal in $T$.
The transfer of distality from $T$ to $T_\delta^*$ is an unpublished result of A. Chernikov. The converse has not been, to our knowledge, observed before. Note that since distality is not preserved under reducts, the converse implication is not straightforward as in Theorem \[thm:ANIP\]. Examples of distal open $\cL$-theories of topological fields include $\RCF$, $\PCF_d$ and $\RCVF$. In contrast, the theory $\ACVF_{0,p}$ is not distal (see [@simon2013]).
\[thm:distality\] $T$ is distal if and only if $T_\delta^*$ is distal.
Let us check that in $T_\delta^*$ every formula is distal. Let $\varphi(x_1,z_1,\ldots,x_n,z_n;y,w)$ be a partitioned $\cL_{\delta}$-formula where each $x_i$ is a $p$-tuple of field sort variables, each $z_i$ is a $q$-tuple of variables of fixed sorts $S_1,\ldots, S_q$ (none being the field sort), $y$ is a tuple of field sort variables and $w$ is a tuple of other sorts. Let $\bU$ be a monster model of $T_\delta^*$. By Theorem \[thm:QE\], we may assume $\varphi$ has no field sort quantifiers. Let $m$ be the order of $\varphi$ Take an $\cL_{\delta}$-indiscernible sequence $(a_i, e_i)_{i\in I}$ in $\bU$ where $(a_i,e_i)\in S_{x_1}(\bU)\times S_{z_1}(\bU)$ and $I=I_1+c+I_2$ with $I_1 , I_2$ infinite dense linear orders without end points. Let $(b,d)$ be a tuple in $S_y(\bU)\times S_w(\bU)$, and assume that $(a_i,e_i)_{i\in I_1+I_2}$ is $\cL_{\delta}$-indiscernible over $(b,d)$. Then, by Corollary \[cor:indiscernibles\], the sequence $(\bar{\delta}^{m}(a_i), e_i)_{i\in I}$ (resp. $(\bar{\delta}^{m}(a_i),e_i)_{i\in I_1+I_2}$) is $\cL$-indiscernible (resp. $\cL$-indiscernible over $B$ where $B=\{\bd^m(b):m\in \N\}\cup \{d\})$) for every $m\in \N$. Since $T$ is distal, the partitioned $\cL$-formula $$\varphi^*(\bar{x}_1,z_1, \ldots, \bar{x}_n, z_n; \bar{y},w)$$ is distal, which easily implies the distality of $\varphi$.
For the converse, suppose $\varphi(x_1,z_1,\ldots, x_n,z_n; y, w)$ is an $\cL$-formula which is not distal in $T$. Consider the $\cL_\delta$-formula $\psi$ $$\varphi(x_1,z_1,\ldots, x_n,z_n; y, w) \wedge \bigwedge_{j=1}^n \delta(x_i)=0 \wedge \delta(y)=0.$$ Let $A\subseteq \bU$ be such that all elements of the field sort are in the constant field $C_\bU$ of $\bU$. Let $(a_i,e_i)_{i\in I}$ be an $\cL$-indiscernible sequence over $A$, where $a_i\in S_{x_1}(\bU)$ and $e_i\in S_{z_1}(\bU)$. If $a_i\in C_\bU$ for each $i\in I$, then by Corollary \[cor:indiscernibles\], we have that $(a_i,e_i)_{i\in I}$ is also $\cL_\delta$-indiscernible over $A$. Then if $(a_i, e_i)_{i\in I}$ and $(b,d)\in S_y(\bU)\times S_w(\bU)$ are a counterexample for the distality of $\varphi$ in $T$, the same witnesses show that $\psi$ is not distal in $T_\delta^*$.
[^1]: ($\dagger$) Research director at the Fonds National de la Recherche Scientifique (FNRS-FRS)
[^2]: This way we avoid expressions like $(\bd^m)^{-1}(A)$ which might lead to confusion, and simply write $\J_m^{-1}(A)$.
|
---
author:
- |
T. Banks[^1]\
Department of Physics and SCIPP\
University of California, Santa Cruz, CA 95064\
E-mail:
title: Breaking SUSY on the Horizon
---
**Introduction**
=================
In [@tbfolly] I proposed that the breaking of Supersymmetry (SUSY) in the world we observe is correlated with a nonzero value of the cosmological constant. Crucial elements of this conjecture were the claim that Poincare invariant theories of gravity had to be exactly supersymmetric, and the claim that the cosmological constant is an input parameter, determined by the finite number of quantum states necessary to describe the universe[^2]. It follows that the gravitino mass is a function of $\Lambda$, vanishing as $\Lambda \rightarrow 0$ and the number of states goes to infinity. As in any critical phenomenon, one may expect that classical estimates of the critical exponents are not correct. Thus I proposed that the classical formula $m_{3/2} \sim \Lambda^{(1/2)}$ (in Planck units) might be replaced by $m_{3/2} \sim
\Lambda^{(1/4)}$ in a correct quantum mechanical calculation. The latter formula has been known for years to predict TeV scale superpartners when the cosmological constant is near the current observational bounds.
Unfortunately, I was not able to come up with even a crude argument for the validity of this conjecture. Indeed, in a recent note on the phenomenology of cosmological SUSY breaking (CSB) [@scpheno], I entertained the hypothesis that various terms in the low energy effective Lagrangian scaled with different powers of $\Lambda$. I am happy to report that this state of affairs has changed. Where there was nothing, there is now a waving hand. That is, there is a set of plausible sounding arguments about the interaction of particles with the cosmological horizon that reproduces the critical exponent $1/4$ for the gravitino mass. I do not pretend that these arguments are definitive, but I do hope that they are approximately correct.
In my initial thinking about this problem, I suggested that "Feynman Diagrams” describing virtual black hole production and decay were responsible for the anomalous relation between the gravitino mass and the cosmological constant. I soon realized that this was unlikely to make sense. Although production of black holes by high energy few particle collisions has probability of order one, the probability that the decay products of a large black hole will reassemble themselves in spacetime so that they can be absorbed by the particle that emitted the high energy virtual lines , and contribute to its mass renormalization, is infinitesimally small. While this argument involves an extrapolation between onshell and offshell processes, it convinced me that black holes were not the answer. Simultaneously, I realized that most of the states in dS space, could not be described as black holes in a single observer’s horizon volume. Rather they should be thought of as black holes sprinkled among many different static horizon volumes, causally disconnected from each other. According to cosmological complementarity [@bfs], a single observer sees these as states near his cosmological horizon. This observation led to the paper which follows.
The calculation I will present is very heuristic and unconventional and one might be led to ask why more conventional methods of quantum field theory in curved spacetime do not lead to hints of this behavior. In fact, no calculations have been done in the conventional framework, and there are many indications that any such calculation would suffer from a variety of divergences. I will outline some of the problems in an appendix.
The Low Energy Effective Lagrangian
===================================
According to the conjecture of [@tbfolly][@wf], Asymptotically dS (AsdS) spaces have a finite number of quantum states. In a universe with a finite number of states, there can never be precisely defined observables, because any exact measurement presupposes the existence of an infinite classical measuring apparatus. With a finite number of states (all of which have at least mutual gravitational interactions) there is no way to make a precise separation between observed system and measuring apparatus, nor any possibility of exactly neglecting the quantum nature of the measuring apparatus. Thus there should be no mathematically defined observables in dS spacetime.
Nonetheless, we know (though we do not yet know why or how) that when the cosmological constant is small, there should be an approximate notion of scattering matrix and (at low enough energy) of an effective Lagrangian. The hypotheses of [@tbfolly] make this more precise. The cosmological constant is a tunable parameter, and in the limit that it vanishes there is a SUSic theory of quantum gravity in asymptotically flat spacetime. This theory has a well defined S-matrix. There should be an object for finite $\Lambda$ which converges to this S-matrix. Indeed, by analogy to critical phenomena, one might expect that there are a plethora of different unitary operators in the finite $\Lambda$ Hilbert space, which all converge to the same S-matrix. However, again by analogy to critical phenomena, we might expect that several terms in the asymptotic expansion of the S matrix around $\Lambda = 0$ are [*universal*]{}. I believe that it is in these universal terms that the “physics of AsdS space” lies. Everything else will be ambiguous.
I expect that the validity of this asymptotic expansion is highly nonuniform, both in energy and particle number. This is a consequence of the postulated finite number of states of the AsdS universe. The best convergence is to be expected for low energy, localizable processes, which do not explore most of the spacetime. These are the processes described by the low energy effective Lagrangian. The gravitino mass is the coefficient of a term in this Lagrangian, and has no more exact definition.
In the $\Lambda \rightarrow 0 $ limit the effective Lagrangian should become SUSic and of course have vanishing cosmological constant. This indicates [@scpheno] that a complex (discrete) R symmetry is also restored in this limit. In [@scpheno] I argued that the limiting theory had to be a four dimensional $N=1$ SUGRA with a massless chiral or vector multiplet which will be eaten by the gravitino when $\Lambda$ is turned on. From the point of view of the effective Lagrangian, SUSY breaking should be spontaneous and triggered by explicit R breaking terms, all of which vanish as some power of $\Lambda$. Among these is a constant in the superpotential, which guarantees that the cosmological constant take on its fixed input value. From the low energy point of view this looks like fine tuning. From a fundamental point of view it is merely a device for assuring that the low energy theory describes a system with the correct number of states (once one implements the Bekenstein-Hawking bound).
The reason that R breaking is explicit while SUSY breaking must be spontaneous is that the R symmetry is discrete, while SUSY is an infinitesimal local gauge symmetry. Explicit SUSY breaking can be made to look spontaneous by doing a local SUSY transformation and declaring that the local SUSY parameter is a Goldstino field. The possibility of restoring SUSY by tuning the cosmological constant to zero implies that the Goldstino must come from a standard linearly realized SUSY multiplet, which appears in the low energy Lagrangian. No such arguments are available for the discrete R symmetry. The picture of low energy SUSY breaking which thus emerges consists of a SUSic, R symmetric theory, in which SUSY is spontaneously broken once R violating terms are added to the Lagrangian. The R violation should be attributed to interaction of the local degrees of freedom with the cosmological horizon.
The purpose of this paper is to estimate the size of the R breaking terms in the low energy effective Lagrangian. In [@scpheno] I also had to invoke Fayet-Iliopoulos (FI) D terms for some $U(1)$ groups . I have since learned from Ann Nelson that such terms can be generated by nonperturbative physics in low energy gauge theory[@anetal] . This physics in turn depends on the existence of certain terms in the superpotential; terms which could be forbidden by an R symmetry. In addition, I have found models not considered in [@scpheno], where dynamical SUSY breaking is triggered by the addition of R violating terms to an otherwise SUSic theory. I will therefore assume that our task is just to estimate the breaking of R symmetry by the horizon. As we will see, this is dominated by the lightest R-charged particle in the bulk. In many models of low energy physics, this will be the gravitino, and I will assume that this is the case.
Horizonal Breaking of R Symmetry
================================
The basic process by which the horizon can effect the low energy effective Lagrangian is described by Feynman diagrams like that of Fig. 1. A gravitino line emerges from a vertex localized near the origin of some static coordinate in dS space, propagates to the horizon, and after interacting with the degrees of freedom there, returns to the vertex. The dominance of diagrams with gravitinos is a consequence of our attempt to calculate R violating vertices and our assumption that the gravitino is the lightest R charged particle. The dominance of diagrams with a single gravitino propagating to the horizon will become evident below.
In field theory, the effective Lagrangian induced by a diagram like Fig. 1 will have a factor[^3]
$$\delta{\cal{L}} \sim e^{- 2 m_{3/2} R} R^{-4},$$ where $R$ is the spacelike distance to the horizon. This factor comes from the two propagators and an integral over the point where the gravitino lines touch the horizon. The gravitino is assumed massive because we know that SUSY is broken in dS space.
1.5in
(400,100)(-100,-40)
(100,20)(100,0,360)
(80,20)(100,120)
(120,20)(100,120)
(100,20)(19.8,180,360)
(80,20)(120,20)
(100,.2)(100,-19.8)
(86.1,6.1)(72.3,-7.7)
(113.9,6.1)(127.7,-7.7)
\
.3in
Our hope to overcome this field theoretic suppression comes from the fact that the horizon has $e^{(RM_P)^2 \over 4}$ states. We thus want to estimate how many of these states the gravitino line interacts with. Since the entropy of the horizon is extensive, this is, crudely, the amount of horizon area the gravitino sees. The thermal nature of Hawking radiation from the horizon suggests that the gravitino interacts in much the same way, with most of the horizon states it comes in contact with. Since the gravitino is massive and the horizon a null surface, it can only propagate along the null surface for a proper time of order $1/m_{3/2}$.
During its contact with the horizon the gravitino is interacting with a Planck density of degrees of freedom. Rather than free propagation, we should imagine that it performs a random walk with Planck length step over the surface of the horizon, covering a distance of order $m_{3/2}^{- {1\over 2}}$ and an area of order $1/m_{3/2}$.
We now make our major assumption, which is that the diagram of Fig. 1 gets a coherent contribution from a number of states of order $e^{a \over m_{3/2}}$ that the gravitino encounters as it wanders over the horizon. For small gravitino mass, this might be an extremely tiny fraction of the total number of states localized in the area of the horizon explored by the gravitino. Thus, to exponential accuracy, the contributions to the R breaking part of the low energy effective Lagrangian are of order $$\delta {\cal L} \sim e^{- 2 m_{3/2} R + {a M_P\over m_{3/2}}}$$
We imagine the calculation of this diagram to be part of a self consistent calculation of the gravitino mass, in the spirit of Nambu and Jona-Lasinio [@NJL]. That is, the interaction with the horizon of a gravitino of a certain mass causes R breaking, which gives rise to SUSY breaking and a gravitino mass. If $m_{3/2}/M_P > ({2RM_P \over a})^{- {1\over 2}}$ then our calculation gives an $R$ breaking effective Lagrangian which falls exponentially with $RM_P$ . Thus, the horizon contribution is totally negligible. On the other hand, we know of no other contribution to the mass which is this large, for large $RM_P$. Thus masses in Planck units greater than $ ({2R M_P \over a})^{-
{1\over 2}}$ are not self consistent.
If $m_{3/2}/M_P < ({2RM_P \over a})^{- {1\over 2}}$, the equation predicts an exponentially growing breaking of $R$ symmetry, and a correspondingly huge gravitino mass, so again the assumption is inconsistent. Notice that this, in particular, rules out the classical formula , $m_{3/2} \sim R^{-1}$. The only self consistent formula, to leading order in $RM_P$, is $m_{3/2} = M_P
({2RM_P \over a})^{- {1\over 2}}$. Taking $R$ of order the Hubble radius of the observable universe, we get a gravitino mass of order $10^{- 11.5} GeV$. This corresponds to a scale for the splitting in nongravitational SUSY multiplets of order $5-6$ TeV. This is the scaling of the gravitino mass conjectured in [@tbfolly].
It may appear that our solution for $m_{3/2}$ is not consistent at the power law order in $RM_P$. That is to say, if the $R$ dependence of $\delta\cal{L}$ is given by the above equation, it does not give rise to a gravitino mass of order $\Lambda^{1/4} =
R^{-1\over 2}$However, corrections to the parameter $a$ of the form $\delta a \sim b {\rm ln (RM_P) \over (RM_P)^{1/2}}$ can remedy this difficulty. Alternatively, (or in addition) nonleading, logarithmic terms in the self consistent formula for $m_{3/2}$ for fixed $a$ can have the same effect. Notice that although at present we have no way of estimating such corrections, self consistency requires them to be present in precisely the right amounts. The dominant exponential terms in the gravitino mass relation are self consistent only for $m_{3/2}\sim
\Lambda^{1/4}$.
It should now also be clear why diagrams with more than one gravitino line propagating to the horizon, or with any heavier particle replacing the gravitino, are subdominant. These have a larger negative term in the exponential, but no larger enhancement from the number of states.
Our calculation clearly rests squarely on the assumption that we can get a contribution of order $e^A$ from interacting with an area $A$ of a system. This sounds peculiar when heard with the ears of local field theory. If, in the spirit of the Membrane paradigm, we modeled the physics of the horizon by a cutoff field theory we would again find $e^{A \sigma}$ states in an area $A$, where $\sigma$ is the entropy density of the field theory. Yet the interactions of a probe concentrated in an area $A$ would be expected to renormalize the effective action of the probe by terms of order $A$. This follows from general clustering and locality arguments in field theory.
However, there is another way that such a field theoretic model fails to capture the physics of horizons. Consider the case of a black hole. A field theory model would predict an energy density as well as an entropy density. The total energy of the black hole would then be of order its area, much larger than its mass.
A somewhat better model of a horizon may be obtained by considering a system of fermions on a two sphere, coupled to an external $U(1)$ gauge field[^4]. The field configuration on the sphere is that produced by a magnetic monopole of very large charge. All fermions are in the lowest Landau level and we tune the magnetic charge so that this is completely full. We can choose linear combinations of the single particle wave functions so that each fermion is localized in a quantum of area on the sphere. Now imagine giving each fermion a two valued ”isospin" quantum number, on which neither the horizon’s Hamiltonian nor its coupling to the external probe depend. The probe is coupled to the position coordinates of the fermions, in a local manner. The system has $2^A$ degenerate states in area $A$ and the probes effective action will be renormalized by an amount $\propto 2^A$.
Discussion
==========
The handwaving nature of the arguments I have presented is probably unavoidable at this stage of our understanding of quantum gravity in de Sitter space. To do better, we must first construct a complete quantum model of de Sitter space, presumably a quantum system with a finite number of states. This construction must recognize the approximate nature of any theory in dS space. There should not be precisely defined, gauge invariant observables, corresponding to the fact that no precise self-measurements can be carried out in a quantum system with a finite number of states. Rather we should look for approximations to the Super Poincare Generators and S-matrix. We should then understand how to identify an approximate notion of low energy effective Lagrangian, which describes some of the physics of the full quantum S-matrix. The latter in particular is a difficult task. Even in Matrix Theory, and AdS/CFT, where there is a precisely defined quantum theory, we can only find the low energy effective Lagrangian by computing the S-matrix and taking limits[^5].
Further development of the kind of mongrel argument used in this note, in which properties of the low energy effective Lagrangian and the horizon, are used as separate constructs which have to fit into a self consistent picture, depends on refinement of our understanding of horizons. It may be that this can be achieved by studying Schwarzchild black holes, and assuming that the local properties of dS horizons are similar. Black holes are objects in asymptotically flat space and precise mathematical questions about their properties can be formulated. It is intriguing that we have already found that a picture of horizon dynamics in terms of a cutoff local quantum field theory on the horizon is inconsistent both with the large breaking of SUSY conjectured in [@tbfolly] and with the entropy/mass relation of the black hole. In this context, it is worth pointing out that for those near extremal black holes where a field theoretic counting of entropy is successful, the field theory does not live on the horizon, and some of the horizon coordinates are quantum operators. This suggests, as does the Landau level model of the last section, that horizons are described by a noncommutative geometry.
On a more phenomenological note, the present calculation sheds some light on a possibility that I raised in [@scpheno]. I suggested that different R breaking terms in the low energy effect Lagrangian might scale with different powers of the cosmological constant. There is no hint of that possibility in the calculation I presented. That is, the powers of the dS radius do not depend at all on the external legs of the Feynman diagram, which would distinguish between different R breaking operators. On the other hand, if the FI D term is generated, as in [@anetal] from low energy dynamics , which is itself triggered by the existence of R breaking terms, the FI D term might depend on both the explicit R breaking scale and the dynamical scale of a low energy SUSY gauge theory. I will leave the exploration of the latter question, as well as of alternative models of low energy physics, to another paper.
It is worth pointing out that large effects of the type I have calculated would not occur in FRW cosmologies which asymptote to SUSic universes, at least if one follows the rules that I have advocated here. The holographic screen, analogous to the cosmological horizon, for such a universe is future null infinity. It is an infinite spacelike distance away from any finite point on the worldline of a timelike observer. The effect I have calculated is larger than any which might have been found in local physics, but it still vanishes as the spacelike distance to the holographic screen goes to infinity. SUSY breaking in such spacetimes will be dominated by local physics.
A somewhat more puzzling situation is presented by a hypothetical universe which stays in a dS phase for a very long time (60 gazillion years, to use technical language) but then asymptotes to a SUSic state. One can easily invent (fine tuned) models of quintessence which have this property. I think the answer here is that the effective Lagrangian of a local timelike observer in such a universe is time dependent. It will initially exhibit larger than normal SUSY breaking, and then become rapidly SUSic. The puzzle that remains is how the transition is made and what a convenient set of holographic screens for such a spacetime might be.
Finally, I want to respond to a question I have been asked many times in the context of lectures on [@tbfolly]: why doesn’t CSB also imply large corrections to other calculations in low energy effective field theory? The answer has been given before and does not really depend on the calculation in this paper, but perhaps it obtains new force from that calculation. The basic idea of CSB is that $\Lambda$ is a variable parameter and that the $\Lambda = 0$ theory has a SUSic, R symmetric low energy effective Lagrangian. All terms in this Lagrangian obviously have a finite $\Lambda\rightarrow 0$ limit. CSB is the theory of how the R violating terms (and perhaps an FI D term that is induced by them) depend on $\Lambda$ in the flat space limit. It is calculating quantities that are parametrically smaller than the SUSic, R symmetric terms in the Lagrangian. Our calculation did not lead to any terms, which diverge in the limit. Such behavior is not compatible with the self consistency of the induced gravitino mass. Thus, any corrections to the preexisting terms in the Lagrangian take the form of small (suppressed by a positive power of the cosmological constant) corrections to their finite, SUSic, values.
Appendix
========
In order to discuss the question of SUSY breaking in dS space we first have to realize dS space as a solution of a SUGRA Lagrangian. This restricts us to minimal supergravity in 4 dimensions. The simplest Lagrangian with a dS solution contains one chiral supermultiplet in addition to the SUGRA multiplet, and is characterized by a holomorphic superpotential $W(Z) $ and a Kahler Potential $K(Z,Z^*
)$. In fact, away from zeroes and singularities of $W$, these can be combined into a single function. We will not make this combination because we start out from the assumption of an R symmetric minimum where $W$ vanishes. The quantum field theory associated with this Lagrangian is not renormalizable and must be supplemented with a cutoff procedure, with a cutoff of order the Planck scale. Ensuring that the cutoff procedure is invariant under superdiffeomorphisms is a complicated technical problem to which there is no known solution. I will assume that this can be solved.
The presumed stationary point of $W$, with $W =0$ gives a Minkowski spacetime in which can compute a gauge invariant S-matrix, and extract from it, order by order in perturbation theory, a gauge invariant effective action. If we perturb the superpotential by small explicit R breaking terms that lead to a dS minimum with small cosmological constant, we would hope that, at least to some order in the perturbation, the gauge invariant effective action is still a valid physical quantity. Indeed, apart from the problem of superdiffeomorphism invariant regulators, the standard background field action of DeWitt seems to provide us with the required perturbatively gauge invariant quantity, even for a dS background. When I refer to particle masses, I mean coefficients in this gauge invariant action. Note that the action is globally dS invariant, so that even if one insists that the global isometries are gauge transformations, the effective action should still be meaningful. In order to tune the cosmological constant to be much smaller than the SUSY breaking scale, $F = |DW|$ we must, generally, add an explicit R breaking constant to $W$.
Now consider loop corrections to the effective action. These are logarithmically UV divergent at one loop[^6]. Higher loop calculations will have higher powers of the logarithm, and there is no small expansion parameter. In previous discussions of this problem, I have argued that this shows UV sensitivity of the calculation of UV effects, and then invoked the UV/IR connection to claim that physics [*above*]{} the Planck scale will renormalize the gravitino mass by amounts that depend on the cosmological constant.
It now appears more probable that the signal for large breaking of SUSY, within the framework of field theory, is the infamous IR divergence problem of dS space[@IR]. The transverse, traceless part of the graviton propagator grows logarithmically at large distances. In an initial version of [@tbfolly] I considered the possibility that these divergences might be the mechanism responsible for the anomalous scaling behavior of the gravitino mass. I rejected this mechanism because there was confusion in the literature as to whether the IR divergences could appear in gauge invariant physical quantities. It was only later that I came to the conclusion that there were no mathematically precise, gauge invariant quantities in dS space. IR divergences have certainly been shown to appear in quantities that appear to be perturbatively gauge invariant. I would speculate that the calculation of the gravitino mass renormalization will similarly have IR problems.
It is not at all clear that any kind of resummation of these divergences should give a finite answer, much less an answer that agrees with the calculation of this paper. Our calculation had an explicit cutoff on the number of states, that is absent in QFT in dS space. So the IR divergence of the naive low energy theory may just be an indication that the true quantum theory of dS space is not well approximated by field theory. At issue here is whether there is a sort of duality between the description by a single static observer, of UV processes localized near his cosmological horizon (this is the description used in the foregoing paper), and an IR description of the same physics using quantum field theory in the global dS space time. None of our experience with black hole physics gives us guidance here, since we do not yet have an adequate description of the physics of an infalling observer.
If one believes in such a duality, he would be motivated to recover our result by calculations in quantum field theory, that is, by trying to resum the IR divergences that I believe would appear in a global computation of the gravitino mass. On the other hand, for a given observer, it may be that only the description of low energy processes localized far from his/her cosmological horizon is well approximated by quantum field theory. At the moment, I do not know which of these two points of view is correct, though the lack of a natural IR cutoff in the field theory calculation suggests that one is unlikely to reproduce the correct physics by simply resumming field theory diagrams. Unfortunately, the calculation of loop corrections to the gravitino mass in the perturbative approach to quantum gravity in dS space, is a daunting exercise. One may hope to compute the one loop contribution, but a systematic analysis to all orders in perturbation theory, seems difficult.
This work supported in part by the U.S. Department of Energy under grant DE-FG03-92ER40689. I would like to thank M.Dine, W. Fischler and O. Narayan for discussions.
[19]{} T. Banks, [*Cosmological Breaking of Supersymmetry?*]{}, hep-th/0007146
T. Banks, [*The Phenomenology of Cosmological SUSY Breaking*]{}, hep-ph/0203066
W. Fischler, [*Talk at the 60th Birthday Celebration for G. West, June 2000.*]{} M. Dine, A. Nelson, Y. Nir, Y. Shirman, [*New Tools for Low Energy Dynamical Supersymmetry Breaking*]{}, Phys. Rev. D53, (1996), 2658, hep-ph/9507378 Y. Nambu, G. Jona-Lasinio, Phys. Rev. 122, 345, (1961). T. Banks, W. Fischler, [*M Theory Observables for Cosmological Space Times*]{}, hep-th/0102077; L. Susskind, [*Twenty Years of Debate With Stephen*]{}, Contribution to Stephen Hawking’s 60th Birthday Celebration, hep-th/0204027. N.P. Myhrvold,Phys.Rev. D28 (1983) 2439; L.H. Ford, Phys.Rev. D31 (1985) 710;E.Mottola, Phys.Rev. D31 (1985) 754 , Phys.Rev. D33 (1986) 1616, Phys.Rev. D33 (1986) 2136; P.Mazur, E.Mottola, Nucl.Phys. B278 (1986) 694 ; I.Antoniadis, E.Mottola J.Math.Phys. 32 (1991) 1037 ,I.Antoniadis, J.Iliopoulos, T.Tomaras, Phys.Rev.Lett. 56 (1986) 1319 ; N.Tsamis, R.Woodard,Commun.Math.Phys. 162 (1994) 217 , Class.Quant.Grav. 11 (1994) 2969 B.Allen, C.Turyn, Nucl.Phys. B292 (1987) 813 ; E. Floratos, J.Iliopoulos, T.Tomaras,Phys.Lett. B197 (1987) 373 ; B.Rhatra, Phys.Rev. D31 (1985) 1931 ; B.Allen, Phys. Rev. D34 (1986) 3670; J.Polchinski, Phys.Lett. B219 (1989) 251 I.Antoniadis, Floratos, J.Iliopoulos, T.Tomaras, Nucl. Phys. B462 (1996), 451.
[^1]: On leave from NHETC, Rutgers U., Piscataway, NJ
[^2]: One can play with the idea of a meta-theory in which universes with different numbers of states are generated by some random or deterministic process, and the number characterizing our world is picked out by anthropic or number theoretic criteria. Such a theory could never be subjected to experimental test, and so appears somewhat futile.
[^3]: All of the calculations of this section are done in four dimensions.
[^4]: I would like to thank O. Narayan for discussions about quantum Hall systems.
[^5]: In perturbative string theory we can also derive the Lagrangian from the world sheet renormalization group, an intriguing hint, which has not had any echoes in nonperturbative physics.
[^6]: It is often said that in minimal four dimensional SUSY, the cosmological constant is quadratically divergent. It is hard to understand how this could be compatible with the general structure of SUSic effective Lagrangians for positive cosmological constant. The positive term in the effective potential is proportional to the square of the gravitino mass. Thus, a divergent positive cosmological constant would imply a divergent gravitino mass. S.Thomas has informed me that if the calculation is done with SUSic Pauli-Villars regulators, the one loop vacuum energy is only logarithmically divergent.
|
---
abstract: 'We discuss theoretical AGB predictions for hydrogen-deficient PG 1159 stars and Sakurai’s object, which show peculiar enhancements in He, C and O, and how these enhancements may be understood in the framework of a very late thermal pulse nucleosynthetic event. We then discuss the nucleosynthesis origin of rare subclasses of presolar grains extracted from carbonaceous meteorites, the SiC AB grains showing low $^{12}$C/$^{13}$C in the range 2 to 10 and the very few high-density graphite grains with $^{12}$C/$^{13}$C around 10.'
author:
- 'R. Gallino$^{1,2}$, O. Straniero$^2$, E. Zinner$^3$, M. Jadhav$^3$, L. Piersanti$^2$, S. Cristallo$^{2,4}$, & S. Bisterzo$^1$'
title: 'Nucleosynthesis origin of PG 1159 stars, Sakurai’s object and of rare subclasses of presolar grains'
---
H-deficient stars
=================
PG 1159 stars: extremely hot Post-AGB stars
-------------------------------------------
The hydrogen-deficiency in extremely hot post-AGB stars of spectral class PG 1159, which includes about 40 stars, with $T_{\rm eff}$ ranging from 75000 and 200000 K and log $g$ from 5.5 to 7.5, is probably caused by a very late thermal pulse in the He shell (VLTP, Sch[ö]{}nberner 1979, Iben 1984) while the post-AGB star is in the hot WD cooling sequence. Because of the high $T$ $_{\rm eff}$ in PG 1159 stars, all species are highly ionized and, hence, most metals are only accessible by UV spectroscopy. A passionate work has been conducted in the last 20 years by Klaus Werner and collaborators (Werner & Herwig 2006, Werner et al. 2009, 2010 and references therein). In Table 1 we report in particular the range of peculiar abundances of He, C, N, O estimated in PG 1159 stars, where the mass fraction of He ranges between 0.30 and 0.85, C between 0.15 and 0.40, N between 0.001 and 0.01, and O between 0.02 and 0.2. The energy released by the VLTP forces the stellar radius to inflate and the star to cool and proceed back toward a born-again AGB. At the maximum extension of the convective thermal instability the very small residual and inactivated H shell is likely engulfed by the pulse and severly depleted, so that the usually hidden He-, C-rich and s-rich He intershell is eventually exposed to the photosphere. PG 1159 stars are seemingly descendants of \[WC\] stars, which show similar HeCNO peculiarities.
Sakurai’s object (V4334 Sgr)
----------------------------
Sakurai’s object (V4334 Sgr) was discovered in 1996 and soon recognized to be the central star of an old planetary nebula (6000 yr ago). It recently underwent a VLTP and is now a hydrogen-deficient born-again AGB star. The estimated mass fractions He/C/O = 0.90/0.07/0.03 and other element abundances (Asplund et al. 1998, their Figure 6) are reported in Table 1 and compared with PG1159 stars. This composition is based on the choice of the best adopted atmospheric model with C/He $\approx$ 0.10 (by number). Note that there is a “carbon problem” for Sakurai’s object similar to that encountered for the hydrogen-deficient RCrB stars. Indeed, the spectroscopic C abundance derived from CI lines is about 0.6 dex smaller than the one deduced from the selected model atmosphere. The same is the case for \[Fe/H\]. However, the relative abundance ratios, like \[El/Fe\], are scarcely dependent on the C/He choice. In Fig. 1 left panel we compare the observed \[El/Fe\] data of Sakurai’s object with predicted abundances in the He intershell at the last thermal pulse for an AGB model of $M^{\rm AGB}_{ini}$ = 3 $M_\odot$, \[Fe/H\] = $-$0.3, case ST/4.5. Given the uncertainty of the initial metallicity, we also plot in the figure a similar predicted \[El/Fe\] distribution for $M^{\rm AGB}_{ini}$ = 3 $M_\odot$, \[Fe/H\] = $-$0.5 and case ST/12 (dashed line). In this case, Sc and Rb appear better reproduced, but the reverse is true for C and Cu. As to Sr, its abundance in October 1996 was overestimated. The presence of carbon dust buffers around Sakurai’s object may in general introduce a noticeable uncertainty in spectroscopic abundances. V 605 Aql (Nova Aql 1919) is a second star having likely suffered a VLTP about 90 yr ago. Clayton et al. (2006) estimated He/C/O = 0.54/0.40/0.05. Both V106 Aql and Sakurai’s object showed peculiar rapid declines and fading characteristic of episodic carbon dust emission, as in the case of R CrB stars. However, as discussed below, several R CrB stars likely originated in a completely different way, as binary WD mergers.
H ingestion and partial burning in the He intershell
----------------------------------------------------
In order to produce a consistent amount of N and to achieve the low $^{12}$C/$^{13}$C observed in Sakurai’s object, ingestion and burning of hydrogen in a TP is impossible when the H shell is still active. It may work when a very thin H envelope is left after the star leaves the AGB (Herwig et al. 1999, Miller Bertolami et al. 2006). A quite low $^{12}$C/$^{13}$C $\le$ 10 results while a consistent amount of $^{14}$N is built up. During the AGB phase, standard elemental mass fractions in the He intershell after a thermal pulse are He/C/O = 0.75/0.20/0.005. Higher C and O abundances may be obtained by including an efficient overshoot at the base of the convective thermal pulse in the TP-AGB phase (Herwig et al. 1997). Alternatively, proper account should be given to the peeling effect by mass loss both at the tip of the AGB and in the early phase of the post-AGB track. There, a “superwind” of up to several 10$^{-5}$ $M_\odot$/yr has been measured. Lawlor & MacDonald (2006) introduced these effects in their stellar evolution code in an wide spectrum of initial masses and metallicities. Another important factor is the thickness of the He buffer, which decreases with increasing the CO core mass, that is with the initial stellar mass. The authors showed that chemical peculiarities observed in stars having suffered the VLTP do not strictly require overshoot in the AGB phase. One should also consider that the bottom of the VLTP is degenerate, different from what occurs during the AGB phase, with the possibility to further increase C and O.
FG Sge
-------
The peculiar FG Sge has also been assumed to have recently suffered a VLTP. Over the last 120 years FG Sge evolving from a hot post-AGB star to a present cool and born-again AGB. Figure 1 (panel right) shows the FG Sge spectroscopic data by Gonzalez et al. (1998) (full triangles) compared with theoretical predictions (upper curve) in the He-intershell after the last thermal pulse for an AGB initial mass of 1.5 $M_\odot$, metallicity \[Fe/H\] = $-$0.3 and the $^{13}$C pocket choice ST $\times$ 1.3. Note the huge \[ls/Fe\], \[hs/Fe\] and \[Pb/Fe\], of the order of 3 dex each. The high value \[Eu/Fe\] = 2 dex observed is s-process Eu, in agreement with the typical s-process expectation \[La/Fe\]$_s$ $\approx$ 1 dex. Moreover, Gonzalez et al. estimated $^{12}$C/$^{13}$C $>$ 10, and provided first spectroscopic evidence of H-deficiency. Jeffery & Sch[ö]{}nberner (2006) reanalyzed all extant spectroscopic data and atmospheric parameters, raising doubts on the huge s-process abundances derived by Gonzalez et al. (1998). The lower $T_{\rm eff}$ = 5500 K chosen for the model atmosphere brought Jeffery & Sch[ö]{}nberner to conclude that the s-process element abundances are more that one order of magnitude less than inferred, most likely inherited already during its previous AGB phase. They concluded that FG Sge suffered a late thermal pulse (LTP), not a VLTP, then evolving back to a born-again AGB. In the lower curve of Figure 1, we compare the envelope AGB model prediction at the last thermal pulse. Note that the two indicators of the s-process distribution, \[hs/ls\] and \[Pb/hs\], would remain unaltered. The spectroscopic heavy elements have been reduced by 1.5 dex (empty triangles), what corresponds to a typical factor of 30 of dilution of He intershell material mixed with the envelope. Note that predicted \[C/Fe\] would better compare with a LTP solution.
The R CrB stars
---------------
![Permil variation with respect to solar of Ca isotopes of graphite presolar grains g-34 and g-40 compared with He intershell predictions of two different AGB models (adapted from Jadhav et al. 2008). []{data-label="fig2"}](gallino_fig3.eps "fig:"){width="6cm"} ![Permil variation with respect to solar of Ca isotopes of graphite presolar grains g-34 and g-40 compared with He intershell predictions of two different AGB models (adapted from Jadhav et al. 2008). []{data-label="fig2"}](gallino_fig4.eps "fig:"){width="6cm"}
So far about 50 R CrB have been discovered. Their atmospheres are extremely hydrogen deficient and carbon rich. Another distinctive feature of some R CrB stars is the enormous F detection, in the range 1,000 to 8,000 times solar for \[Fe/H\] in the range $-$0.5 to $-$2.0 (Pandey et al. 2008). Such drastic $^{18}$O and $^{19}$F excesses indicate that the merging of a CO-WD with a He-WD gives rise to partial He burning and production of $^{18}$O via $\alpha$-capture on $^{14}$N, accompanied by $^{18}$O(p,$\gamma$)$^{19}$F. Detailed nucleosynthesis calculations for these peculiar objects are not easy however.
Presolar grains
===============
A subclass of presolar SiC grains discovered in carbonaceus meteorites, the SiC of type AB (4 to 5 % of all presolar SiC grains) are characterized by very low $^{12}$C/$^{13}$C ratios, in the range 2 to 10. Mainstream SiC (covering 93% of all presolar SiC grains), show higher $^{12}$C/$^{13}$C ratios, from $\sim$10 to 100 (solar ratio is 89), averaging at around 60 (Zinner 1998, 2008). While mainstream SiC likely originated in low mass AGB stars of around solar metallicity, the stellar origin of SiC AB grains is still enigmatic. These grains clearly show the signature of H burning in the CNO cycle and H burning is also indicated by their relatively high inferred $^{26}$Al/$^{27}$Al ratios (Amari et al. 2001). However, the low $^{12}$C/$^{13}$C ratios are difficult to reconcile with the condition C$>$O, necessary for SiC condensation. J-type carbon stars and born-again AGBs like Sakurai’s object have been proposed as sources of AB grains. Despite SiC AB grains show low $^{12}$C/$^{13}$C, the permil variations of $^{29}$Si/$^{28}$Si and $^{30}$Si/$^{28}$Si with respect to solar is indistinguishable from mainstream SiC that reach maximum values of $\approx$200 and $\approx$150, respectively. Instead, far higher permil variations are predicted in the He intershell. This indicates that SiC AB grains are incompatible with an origin in born-again AGBs like Sakurai’s object, unless one speculates that the grains formed in a cool circumstellar disk, after having been mixed with previously ejected material.
Very rare high-density graphite grains have been discovered with the signature of He intershell in the trace elements Ca and Ti (Jadhav et al. 2008). A couple of examples are reported in the two panels of Fig. 3, for the grains g-34 and g-40. A similar exceptional permil variation has been detected for both Ca and Ti in grain g-9. Also trace Mg and Si are present but they show essentially normal isotopic composition, maybe related with isotopic equilibration with solar material.
R.G. thanks the INAF-Osservatorio di Teramo for financial support.
<span style="font-variant:small-caps;">**[References]{}**</span>
Amari, S, et al. (2001), ApJ, 559, 643\
Asplund, M., Gustafsson, B., Rao, N. K., & Lambert, D. L. (1998), A&A, 332, 651\
Clayton, G. C., et al. (2006), ApJ, 646, L69\
Gonzalez, G., et al. (1998), ApJS, 114, 133\
Herwig, F., Bl[ö]{}cker, T., Sch[ö]{}nberner, D., & El Eid, M. (1997), A&A, 324, L81\
Herwig, F., Blcker, T., Langer, N., & Driebe, T. (1999), A&A, 349, L5\
Iben, I., Jr (1984), ApJ, 277, 333\
Jadhav, M., et al. (2008), ApJ, 682, 1479\
Jeffery, C. S., & Sch[ö]{}nberner, D. (2006), A&A, 459, 885\
Miller Bertolami, M. M. et al. (2006),A&A, 449, 313\
Pandey, G., Lambert, D. L., & Rao, N. K. (2008), ApJ, 674, 1068\
Sch[ö]{}nberner, D. (1979), A&A, 79, 108\
Werner, K., & Herwig, F., (2006), PASP, 118, 183\
Werner, K., Rauch, T., & Kruk, J. W. (2010), ApJ, 719, L32\
Werner, K., et al. (2009), Astrophys. Space Sci., 320, 159\
Zinner, E. (1998), Annu. Rev. Earth Planet. Sci. 26, 147\
Zinner, E. (2008), PASA, 25, 7\
|
---
abstract: 'We investigate a one dimensional quantum mechanical model, which is invariant under translations and dilations but does not respect the conventional conformal invariance. We describe the possibility of modifying the conventional conformal transformation such that a scale invariant theory is also invariant under this new conformal transformation.'
author:
- 'S.-H. Ho[^1]'
title: Alternative conformal quantum mechanics
---
It is known that a scale invariant theory does not always enjoy conventional conformal symmetry, where the configuration variables transform in the conventional way, rendering them to be primary operators (invariant under conformal transformations at the origin) [@Jackiw:2011vz; @ElShowk:2011gz]. We examine these issues for a theory in one time dimension, i.e. quantum mechanics for a single quantum variable $q(t)$.
The theory is governed by a one-dimensional Lagrangian $L=L(q, \dot{q}; t)$. When it takes the form \[eq1\] L=f(x) q\^[-2]{},where $x \equiv \dot{q}q$ and $f(x)=x^n$ with a real number $n$ [^2], it enjoys time translation and scale invariance
\[eq2\] \[eq2a\]& & \_T q =\
\[eq2b\]& & \_D q = t - q but is conventionally conformally invariant \[eq2c\]& & \_C q =t\^2 - t q
only when [@Jackiw:1972cb; @de; @Alfaro:1976je; @Jackiw:1980mm] \[eq3\] L=\^2 - . The associated conserved quantities can be obtained by Noether’s theorem
\[eq4\] \[eq4a\] H && = p\^2 + , p=\
\[eq4b\] D && =tH - (q+q)=tH-(qp+pq)\
\[eq4c\] K && =-t\^2 H +2tD +.
In this paper we describe the possibility of changing the conformal transformation law (\[eq2c\]) for $q$, such that conformal invariance holds for the entire family of Lagrangians (\[eq1\]). Time translation and dilatations remain conventional, leading to constants of motion
\[eq5\] \[eq5a\] H = && p-f(x)q\^[-2]{}\
\[eq5b\] D = && tH - (qp+pq) .
But the conformal generator is changed[^3].
To begin, we assume that a conserved conformal generator $K$ exists and together with $H$ and $D$ generates the group $SO(2,1)$:
\[eq6\] \[eq6a\] i = && + H,\
\[eq6b\] i = && - K,\
\[eq6c\] i = && 2 D.
The Casimir operator $\mathcal{C}$ of $SO(2,1)$ is \[eq7\] =(KH+HK)-D\^2 . From (\[eq6\]) and (\[eq7\]), we can derive the general form of conformal generator $K$ for given $H$ and $D$ : \[eq8\] K=&& -H\^[-1]{} K H +2 H\^[-1]{} D\^2 +2H\^[-1]{}\
=&& -(+KH\^[-1]{})H+2 (+DH\^[-1]{})D+2H\^[-1]{}\
=&&-( -H\^[-1]{}(2 i D)H\^[-1]{}+KH\^[-1]{})H+2 (-H\^[-1]{}(i )+D H\^[-1]{})D+2H\^[-1]{}\
=&& 2iH\^[-1]{}D-K-2iH\^[-1]{}D+2D H\^[-1]{}D+2 H\^[-1]{}\
=&& -K +2 D H\^[-1]{} D+2 H\^[-1]{} Therefore, the expression of $K$ is [@ref5] \[eq9\] K=D H\^[-1]{}D + H\^[-1]{}. The addition to $K$ of the $\mathcal{C}/H$ term reflects the fact that the $SO(2,1)$ (\[eq6\]) algebra is unchanged when the generators are modified according to H && H+\
K && K+ provided $ab=0$. We allow for the conformal modification, so we set $a=0$. Because of this ambiguity, $\mathcal{C}$ parametrizes an entire family of $K$.
First we consider the conventional case $L_2=\frac{\dot{q}^2}{2}$ and $H=\frac{p^2}{2}$ (the interacting case $H=\frac{p^2}{2}+\frac{\lambda}{2q^2}$ is too difficult to treat quantum mechanically owing to $H^{-1}$ in (\[eq9\])). We can write the dilatation generator (\[eq5b\]) and conformal generator (\[eq9\]) as \[eq10\] D=&& tH - (qp+pq) = tH+D\_0\
\[eq11\] K=&&DH\^[-1]{}D+ =(tH+D\_0)H\^[-1]{}(tH+D\_0)+\
=&&t\^2 H+2 t D\_0+D\_0 H\^[-1]{} D\_0 + = - t\^2 H+2tD +K\_0 where $D_0$ and $K_0$ are the generators at $t=0$: \[eq12\] D\_0 && - (qp+pq)\
\[eq13\] K\_0 && D\_0 H\^[-1]{} D\_0 + . From (\[eq13\]), we can derive $K_0$: \[eq14\] K\_0=++ = +(+) . In order for $q$ to be a primary operator, we choose $\mathcal{C}=-\frac{3 \hbar^2}{16}$ and $K_0=\frac{q^2}{2}$. The transformation law of $q$ in (\[eq2\]) can be reproduced by commutation and the usual formulas emerge:
\[eq15\] \[eq15a\] && \_T q = = p=\
\[eq15b\] && \_D q = = t - q\
\[eq15c\] && \_C q = = t\^2 - t q
Next we derive $K$ for the special case $n=4/3$. We show in Appendix \[Appendix A\], where we examine arbitrary $n$, that $n=4/3$ is the only other case with a primary operator. We start from the Lagrangian \[eq16\] L\_[4/3]{} =\^[4/3]{}q\^[-2/3]{} From (\[eq5\]), we have the classical Hamiltonian \[eq16-1\] H=p\^4 q\^2, p=\^[1/3]{}q\^[-2/3]{}. We then write (\[eq16-1\]) and dilatation generators in a quantum mechanical way with the ordering[^4]:
\[eq17\] \[eq17a\] H=&& p\^2 q\^2 p\^2 ,\
\[eq17b\] D=&& tH-(pq+qp)tH+D\_0
From (\[eq13\]), $K_0$ is given by \[eq18\] K\_0 && = D\_0 H\^[-1]{} D\_0+\
=&& p\^[-2]{}+ ( +) By choosing Casimir $\mathcal{C}= - \frac{3 \hbar^2}{16}$ we can eliminate the $H^{-1}$ term in $K_0$ and $K$ (note the standard case also has $\mathcal{C}= - \frac{3 \hbar^2}{16}$):
\[eq19\] \[eq19a\] K\_0 = && p\^[-2]{}\
\[eq19b\] K = && -t\^2 H +2 t D + p\^[-2]{} .
Since $\left[ K_0, p \right]=0$, the primary operator in the present case is canonical momentum $p$.
We now have the explicit expressions of $H$, $D$ and $K$ so we can reproduce the transformation law for $q$:
\[eq20\] \[eq20a\] \_T q =&& = ( p\^2 q\^[2]{} p + p q\^[2]{} p\^2 )\
\[eq20b\] \_D q = && =t -q\
\[eq20c\] \_C q =&&\
=&&t\^2 -t q-2 p\^[-3]{} Substituting (\[eq16-1\]) into (\[eq20c\]), we obtain the classical transformation law for $q$: \[eq20d\] \_C q = t\^2 -tq -2 \^[-1]{} q\^2
Next we derive the classical form of the new conformal constant of motion by applying Noether’s theorem to the Lagrangian (\[eq16\]). The variation of the Langrangian under (\[eq20d\]) can be written as a total time-derivative without using equations of motion: \[eq22\] \_C L\_[4/3]{} && = \_C + \_C q\
&& = (t\^2\^[4/3]{}q\^[-2/3]{}-4 \^[-2/3]{}q\^[4/3]{}) X which indicates that (\[eq20c\]) is a symmetry transformation of $L_{4/3}$ (\[eq16\]). The constant of motion is \[eq23\] K= \_C q - X = && p\^4 q\^2 - tq + p\^[-2]{}. This is the classical version of (\[eq19b\]).
Conformal generator for arbitrary $n>1$ {#Appendix A}
=======================================
In this appendix, we derive the conformal generator $K$ for Lagrangian in (\[eq1\]) \[eqL\] L=\^nq\^[n-2]{} for arbitrary $n>1$ and two different orderings of the Hamiltonian $H$ which have the same classical transformation law of $q$ with different Casimirs $\mathcal{C}$.
$H=\frac{N}{N+1} p^{\frac{1}{2}\left(1+\frac{1}{N}\right)}q^{-1+\frac{1}{N}} p^{\frac{1}{2}\left(1+\frac{1}{N}\right)}$, $N \equiv n-1$
---------------------------------------------------------------------------------------------------------------------------------------
We start from the classical Hamiltonian $H$:
\[eqB0\] \[eqB0a\] H=&& p\^[1+]{} q\^[-1+]{} ,\
\[eqB0b\] p=&& \^N q\^[N-1]{}.
Then we write (\[eqB0\]) and time dilatation operator $D$ associated with Lagrangian (\[eqL\]) quantum mechanically with the ordering
\[eqB1\] \[eqB1a\] H=&& p\^[(1+)]{} q\^[-1+]{} p\^[(1+)]{}\
\[eqB1b\] D=&& t H - (pq+qp) t H+D\_0, and using (\[eq11\]) for conformal operator $K$: \[eqB1c\] K=&&t\^2 H+2tD\_0 +K\_0.
After a straightforward calculation, the $K_0$ is: \[eqB2\] K\_0&& = D\_0 H\^[-1]{} D\_0 +\
=&& p\^[(1-)]{} q\^[3-]{} p\^[(1-)]{}+( (-)+) . By choosing Casimir $\mathcal{C}$ \[eqB4\] =&& - (-), we eliminate the $H^{-1}$ term in $K_0$: \[eqB3\] K\_0=&& p\^[(1-)]{} q\^[3-]{} p\^[(1-)]{}. Note from (\[eqB3\]) that only $n=2$ $(N=1)$ and $n=4/3$ $(N=1/3)$ produce a primary operator.
The transformation law for $q$ can be reproduced by computing
\[eqB5\] \[eqB5a\] \_T q =&&\
=&& ( p\^[(1+)]{} q\^[-1+]{} p\^[(-1+)]{}+p\^[(-1+)]{} q\^[-1+]{} p\^[(1+)]{})\
\[eqB5b\] \_D q = && = t-q\
\[eqB5c\] \_C q =&&\
=&& t\^2 -tq + p\^[-(1+)]{}(p q\^[3-]{}+q\^[3-]{} p) p\^[-(1+)]{} We then obtain the classical transformation law of $q$ by using (\[eqB0b\]): \[eqB5d\] \_C q = && t\^2 -tq+ (1-) \^[-1]{}q\^2
$H=\frac{N}{N+1} q^{\frac{1}{2}\left(-1+\frac{1}{N}\right)} p^{1+\frac{1}{N}}q^{\frac{1}{2}\left(-1+\frac{1}{N}\right)}$
-------------------------------------------------------------------------------------------------------------------------
We change (\[eqB1a\]) to \[eqB7\] H= q\^[(-1+)]{} p\^[1+]{} q\^[(-1+)]{} . The generator $K_0$ is: \[eqB8\] K\_0 && =D\_0 D\_0+\
=&& q\^[(3-)]{} p\^[1-]{} q\^[(3-)]{} +( (4-)+) . The choice of Casimir $\mathcal{C}$ \[eqB9\] = - (4-) , leads to \[eqB10\] K\_0= q\^[(3-)]{} p\^[1-]{} q\^[(3-)]{}.
Following the same procedure, we obtain the classical transformation law of $q$
\[eqB11\] \[eqB11a\] \_T q =&&\
\[eqB11b\] \_D q = && t-q\
\[eqB11c\] \_C q = && t\^2 -tq+ (1-) \^[-1]{}q\^2
which are the same as (\[eqB5\]).
We can see the variation of Lagrangian (\[eqL\]) under transformation (\[eqB11c\]) \[eqB12\] \_C L=&& \_C + \_C q\
=&&(t\^2 L - \^[N-1]{}q\^[N+1]{} ) is a total time-derivative. The constant of motion is \[eqB13\] K=&& \_C q -(t\^2 L - \^[N-1]{}q\^[N+1]{} )\
=&& p\^[1+]{} q\^[-1+]{} -tpq + p\^[1-]{}q\^[3-]{} This is classical form of (\[eqB1c\]) with $K_0$ given in (\[eqB3\]) (and (\[eqB10\])).
Another ordering of Hamiltonian for $n=4/3$ {#Appendix B}
===========================================
In this appendix, we use for $H$ an expression, which is ordered differently from (\[eq17a\]): \[eqA1\] H= q p\^4 q. Using (\[eq13\]), we obtain \[eqA2\] K\_0= p\^[-2]{} + (+) . Hence, we choose Casimir to be \[eqA4\] =. and have \[eqA3\] K\_0= p\^[-2]{} . Comparing (\[eq19a\]) with (\[eqA3\]), we observe the two $K_0$’s are the same in the two orderings. The Casimirs differ because of reordering.
I am grateful to Professor Roman Jackiw for providing to me the main idea of this work and continuous and patient guidance. This work is supported by the National Science Council of R.O.C. under Grant number: NSC98-2917-I-564-122.
[99]{}
R. Jackiw and S. Y. Pi, J. Phys. A [**44**]{}, 223001 (2011) \[arXiv:1101.4886 \[math-ph\]\]. S. El-Showk, Y. Nakayama and S. Rychkov, arXiv:1101.5385 \[hep-th\].
R. Jackiw, Phys. Today 25, 23 (1972) .
V. de Alfaro, S. Fubini and G. Furlan, Nuovo Cim. A [**34**]{}, 569 (1976).
R. Jackiw, Annals Phys. [**129**]{}, 183 (1980). [**201**]{}, 83 (1990). Formula (\[eq9\]) with $\mathcal{C}=0$ was privately communicated by A. Strominger via R. Jackiw. See also
J. Kumar, JHEP [**9904**]{}, 006 (1999) \[arXiv:hep-th/9901139\].
[^1]: E-mail address: shho@mit.edu
[^2]: The form of Lagraingian
[^3]: One may consider the more general case where $q$ has an arbitrary scale dimension $d$. Then a scale invariant Lagrangian becomes $L'=\dot{q}^n q^{\frac{1-n}{d}-n}$. This provides a complicated generalization which I have not studied.
[^4]: There are various ways to order the Hamiltonian as a Hermitian operator. Certainly different orderings give different $K$ and Casimir $\mathcal{C}$, but will result in the same classical transformation law $\delta_C q$. In the Appendix \[Appendix B\], we will consider another ordering of $H= \frac{1}{4} q p^4 q$.
|
---
author:
- Marek Golasiński and Juno Mukai
title: Gottlieb groups of spheres
---
[**Abstract.**]{} This paper takes up the systematic study of the Gottlieb groups $G_{n+k}(\S^n)$ of spheres for $k\le 13$ by means of the classical homotopy theory methods. The groups $G_{n+k}(\S^n)$ for $k\le 7$ and $k=10,12,13$ are fully determined. Partial results on $G_{n+k}(\S^n)$ for $k=8,9,11$ are presented as well. We also show that $[\iota_n,\eta^2_n\sigma_{n+2}]=0$ if $n=2^i-7$ for $i\ge 4$.
[^1]
Introduction {#introduction .unnumbered}
============
The Gottlieb groups $G_k(X)$ of a pointed space $X$ have been defined by Gottlieb in [@G] and [@G1]; first $G_1(X)$ and then $G_k(X)$ for all $k\ge 1$. The higher Gottlieb groups $G_k(X)$ are related in [@G1] and [@G2] to the existence of sectioning fibrations with fiber $X$. For instance, if $G_k(X)$ is trivial then there is a homotopy section for every fibration over the $(k+1)$-sphere $\S^{k+1}$, with fiber $X$.
This paper grew out of our attempt to develop techniques in calculating $G_{n+k}(\S^n)$ for $k\le 13$ and any $n\ge 1$. The composition methods developed by Toda [@T] are the main tools used in the paper. Our calculations also deeply depend on the results of [@H-M], [@K] and [@M2].
Section 1 serves as backgrounds to the rest of the paper. Write $\iota_n$ for the homotopy class of the identity map of $\S^n$. Then, the homomorphism $$P' : \pi_k(\S^n)\longrightarrow\pi_{k+n-1}(\S^n)$$ defined by $P'(\alpha)=[\iota_n,\alpha]$ for $\alpha\in\pi_k(\S^n)$ [@Hilton1] leads to the formula $G_k(\S^n)=\mbox{ker}\,P'$, where $[-,-]$ terms the standard Whitehead product. So, our main task is to consult first [@Hilton1], [@Hil], [@M1], [@M2], [@Thomeier] and [@T] about the order of $[\iota_n,\alpha]$ and then to determine some Whitehead products in unsettled cases as well. In the light of Serre’s result [@Serre], the $p$-primary component of $G_{2m+k}(\S^{2m})$ vanishes for any odd prime $p$, if $2m\ge k+1$ (Proposition \[Gnp0\]).
Let $EX$ be the suspension of a space $X$ and denote by $E: \pi_k(X)\to\pi_{k+1}(EX)$ the suspension map. Write $\eta_2\in\pi_3(\S^2)$, $\nu_4\in\pi_7(\S^4)$ and $\sigma_8
\in\pi_{15}(\S^8)$ for the Hopf maps, respectively. We set $\eta_n=E^{n-2}\eta_2\in\pi_{n+1}(\S^n)$ for $n\ge 2$, $\nu_n=E^{n-4}\nu_4\in\pi_{n+3}(\S^n)$ for $n\ge 4$ and $\sigma_n=E^{n-8}\sigma_8\in\pi_{n+7}(\S^n)$ for $n\ge 8$. Write $\eta^2_n=\eta_n\circ\eta_{n+1}$, $\nu^2_n=\nu_n\circ\nu_{n+3}$ and $\sigma^2_n=\sigma_n\circ\sigma_{n+7}$. Section 2 is a description of $G_{n+k}(\S^n)$ for $k\le 7$. To reach that for $G_{n+6}(\S^n)$, we make use of Theorem \[neq\] partially extending the result of [@K-M]: [*$[\iota_n, \nu^2_n] = 0$ if and only if $n\equiv 4,5,7\ (\bmod\ 8)$, $n = 2^i - 3$ or $n = 2^i - 5$ for $i\ge 4$;*]{} for the proof of which Section 3 and Section 4 are devoted.
Section 5 devotes to proving Mahowald’s claim: $[\iota_n,\sigma_n]\ne 0$ for $n\equiv 7\ (\bmod\ 16)\ge 23$.
Section 6 takes up computations of $G_{n+k}(\S^n)$ for $k=10,12,13$ and partial ones of $G_{n+k}(\mathbb{S}^n)$ for $k=8,9,11$. In a repeated use of [@M2], we have found out the triviality of the Whitehead product : $$[\iota_n,\eta^2_n\sigma_{n+2}]=0, \ \mbox{if}\;
n=2^i-7\ (i\ge 4),$$ which corrects thereby [@M2] for $n=2^i-7$.
The authors thank Professor M. Mimura for suggesting the problem and fruitful conversations, Professors H. Ishimoto, I. Madsen, M. Mahowald, Y. Nomura and N. Oda for helpful informations on the orders of the Whitehead products $[\iota_n,\nu^2_n]$ and $[\iota_n,\sigma_n]$.
The authors are also very grateful to Professor H. Toda for informing the order of the Whitehead product $[\iota_{2n},[\iota_{2n},\alpha_1(2n)]]$ for $n\ge
2$ (Proposition \[Toda\]), where $\alpha_1(2n)$ is a generator of the $3$-primary component of $\pi_{2n+3}(\mathbb{S}^{2n})$.
Preliminaries on Gottlieb groups
================================
Throughout this paper, spaces, maps and homotopies are based. We use the standard terminology and notations from the homotopy theory, mainly from [@T]. We do not distinguish between a map and its homotopy class.
Let $X$ be a connected space. The $k$-th [*Gottlieb group*]{} $G_k(X)$ of $X$ is the subgroup of the $k$-th homotopy group $\pi_k(X)$ consisting of all elements which can be represented by a map $f : \S^k\to X$ such that $f\vee\mbox{id}_X : \S^k\vee
X\to X$ extends (up to homotopy) to a map $F : \S^k\times X\to X$. Define $P_k(X)$ to be the set of elements of $\pi_k(X)$ whose Whitehead product with all elements of all homotopy groups is zero. It turns out that $P_k(X)$ forms a subgroup of $\pi_k(X)$ and, by [@G1 Proposition 2.3], $G_k(X)\subseteq P_k(X)$. Recall that $X$ is said to be a $G$-[*space*]{} ([*resp.*]{} $W$-[*space*]{}) if $\pi_k(X)=G_k(X)$ ([*resp.*]{} $\pi_k(X)=P_k(X)$) for all $k$.
Given $\alpha\in\pi_k(\S^n)$ for $k\ge 1$, we deduce that $\alpha\in G_k(\S^n)$ if and only if $[\iota_n,\alpha]=0$. In other words, consider the map $$P' : \pi_k(\S^n)\longrightarrow
\pi_{k+n-1}(\S^n)$$ defined by $P'(\alpha)=[\iota_n,\alpha]$ for $\alpha\in\pi_k(\S^n)$. Then, this leads to the formula $$G_k(\S^n)=\mbox{ker}\,P'.$$
Write now $\sharp$ for the order of a group or its any element. Then, from the above interpretation of Gottlieb groups of spheres, we obtain
\[lead\] If $\pi_k(\S^n)$ is a cyclic group for some $k\ge 1$ with a generator $\alpha$ then $G_k(\S^n)=(\sharp[\iota_n,\alpha])\pi_k(\S^n)$.
Since $\S^n$ is an H-space for $n=3,7$, we have $$G_k(\S^n) = \pi_k(\S^n)\ \mbox{for}\ k\ge 1,
\ \mbox{if} \ n =3,7.$$
We recall the following result from [@Hil] and [@WG] needed in the sequel.
\[pro\] If $\xi\in\pi_m(X)$, $\eta\in\pi_n(X)$, $\alpha\in\pi_k(\S^m)$, $\beta\in\pi_l(\S^n)$ and if $[\xi,\eta]=0$ then $[\xi\circ\alpha,\eta\circ\beta]=0.$
Let $\alpha\in\pi_{k+1}(X)$, $\beta\in\pi_{l+1}(X)$, $\gamma\in\pi_m(\S^k)$ and $\delta\in\pi_n(\S^l)$.
Then $[\alpha\circ E\gamma,\beta\circ E\delta]=[\alpha,\beta]\circ E(\gamma\wedge\delta).$
If $\alpha\in\pi_k(\S^2)$ and $\beta\in\pi_l(\S^2)$ then $[\alpha,\beta]=0$ unless $k=l=2$.
$[\beta,\alpha] = (-1)^{(k+1)(l+1)}[\alpha,\beta]$ for $\alpha\in\pi_{k+1}(X)$ and $\beta\in\pi_{l+1}(X)$.
In particular, $2[\alpha,\alpha]=0$ for $\alpha\in\pi_n(X)$ if $n$ is odd.
If $\alpha_1,\alpha_2\in\pi_{p+1}(X)$, $\beta\in\pi_{q+1}(X)$ and $p\ge 1$, then $[\alpha_1+\alpha_2,\beta]=$
$[\alpha_1,\beta]+[\alpha_2,\beta]$ and $[\beta,\alpha_1+\alpha_2]=[\beta,\alpha_1]
+[\beta,\alpha_2]$.
$E[\alpha,\beta]=0$ for $\alpha\in\pi_k(X)$ and $\beta\in\pi_l(X)$.
$3[\alpha,[\alpha,\alpha]]=0$ for $\alpha\in\pi_{n+1}(X)$.
Let $G_k(X;p)$ and $\pi_k(X;p)$ be the $p$-primary components of $G_k(X)$ and $\pi_k(X)$ for a prime $p$, respectively. But for $X=\S^n$, recall the notation from [@T]: $$\pi^n_k= \left\{ \begin{array}{ll}
\pi_n(\S^n), & \mbox{if $k=n$};\\
E^{-1}\pi_{2n}(\S^{n+1};2),& \mbox{if $k=2n-1$};\\
\pi_k(\S^n;2),& \mbox{if $k\not=n,2n-1.$}
\end{array}
\right.$$
As it is well-known, $[\iota_n,\iota_n]=0$ if and only if $n=1,3,7$ and $\sharp[\iota_n,\iota_n]=2$ for $n$ odd and $n\ne 1,3,7$, and it is infinite provided $n$ is even. Thus, we have reproved the result [@G1] that $G_n(\S^n)=\pi_n(\S^n)\cong\Z$ for $n=1,3,7$, $G_n(\S^n)=2\pi_n(\S^n)\cong 2\Z$ for $n$ odd and $n\not=1,3,7$, and $G_n(\S^n)=0$ for $n$ even, where $\Z$ denotes the additive group of integers. It is easily obtained that $G_k(\S^n)=P_k(\S^n)$ for all $k,n$ [@La Theorem I.9]. In other words, on the level of spheres the class of $G$-spaces coincides with that of $W$-spaces.
Let now $n$ be odd. Then, by Lemma \[pro\]:(4), (5) and (7), $[\iota_n, [\iota_n,\iota_n]]=0$. Furthermore, by Lemma \[pro\]:(1), (4) and (5), $[2\iota_n,\iota_n]=0$ implies that $[\iota_n,2\alpha]=0$ for any $\alpha\in\pi_k(\S^n)$, that is, $2\pi_k(\S^n)\subset G_k(\S^n)$ and thus $G_k(\S^n;p)=\pi_k(\S^n;p)$ for any odd prime $p$. In the light of [@Liulevicius], we also know $$\label{Lv}
\sharp[\iota_{2n}, [\iota_{2n}, \iota_{2n}]]=3, \ \mbox{if} \ n\geq 2.$$ Whence, Lemma \[pro\] and (\[Lv\]) yield the results proved partially in [@GG].
\[CC\] [*(1)*]{} $3[\iota_n,\iota_n]\in G_{2n-1}(\S^n)$. In particular, $a[\iota_n,\iota_n]\in$
$G_{2n-1}(\S^n)$ for $n$ even if and only if $a\equiv 0\ (\bmod\ 3)$.
[*(2)*]{} If $k\ge 3$ then $G_k(\S^2)=\pi_k(\S^2)$.
[*(3)*]{} If $n$ is odd and $n\not=1,3,7$ then $2\pi_k(\S^n)\subset G_k(\S^n)$. In particular,
$G_k(\S^n;p)=\pi_k(\S^n;p)$ for any odd prime $p$ and $k\ge 1$.
[*(4)*]{} $G_k(\S^n)=\pi_k(\S^n)$ provided that $E : \pi_{k+n-1}(\S^n)\to\pi_{k+n}(\S^{n+1})$ is
a monomorphism.
We note that $P'$ and the homomorphism $$P :\pi_{k+n+1}(\mathbb{S}^{2n+1})
\longrightarrow\pi_{k+n-1}(\mathbb{S}^n)\ (k\le 2n-2)$$ in the EHP sequence are related as follows: $$P'=P\circ E^{n+1}\;\mbox{for}\;k\le 2n-2.$$
Let $SO(n)$ be the rotation group and $J:\pi_k(SO(n))\to\pi_{n+k}(\S^n)$ be the $J$-homomorphism and $\Delta: \pi_k(\S^n)\to\pi_{k-1}(SO(n))$ the connecting map associated with the fibration $SO(n+1)\stackrel{SO(n)}{\longrightarrow}\S^n$. We recall $$\label{P'}
P'=J\circ\Delta$$ and so, $$\label{JDel} \mbox{ker} \{\Delta: \pi_k(\S^n)\to\pi_{k-1}(SO(n))\}\subset G_k(\S^n).$$
In virtue of [@Serre Chapter IV] ([@T (13.1)]), Serre’s isomorphism $$\label{Serre}
\pi_{i-1}(\S^{2m-1};p)\oplus\pi_i(\S^{4m-1};p)\cong\pi_i(\S^{2m};p)$$ is given by the correspondence $(\alpha, \beta)\mapsto E\alpha
+ [\iota_{2m}, \iota_{2m}]\circ\beta$.
We show
\[Gnp0\] Let $p$ be an odd prime and $n$ even.
If $n\ge k+1$ then $G_{n+k}(\S^n;p)=0$.
Suppose that $\pi_{n+k}(\S^n;p)$ is cyclic with a generator $\theta$, $\pi_{n+k}(\S^n;p)=$
$E\pi_{n+k-1}(\S^{n-1};p)$ and that $E^{n-1}: \pi_{n+k}(\S^n;p)\to\pi_{2n+k-1}(\S^{2n-1};p)$
is an epimorphism. Then, $G_{n+k}(\S^n;p)=\{\sharp(\mbox{\em ker}\,E^{n-1})\theta\}$.
[**Proof.**]{} By the Freudenthal suspension theorem, $E: \pi_{n+k}(\S^n)\to\pi_{n+k+1}(\S^{n+1})$ is an epimorphism, if $n=k+1$ and an isomorphism, if $n\ge k+2$. For the case $n=k+1$, by the EHP sequence $$\pi_{2n-2}(\S^{n-1}){{\buildrel E \over \longrightarrow}}\pi_{2n-1}(\S^n){{\buildrel H \over \longrightarrow}}\pi_{2n-1}(\S^{2n-1})$$ we get $$\pi_{2n-1}(\S^n)=E\pi_{2n-2}(\S^{n-1})\oplus\Z\{\alpha\},$$ where $\alpha=\eta_2,\nu_4,\sigma_8$ according as $n=2,4,8$ and $\alpha=[\iota_n,\iota_n]$ for $n$ otherwise. So, by the EHP sequence $$\pi_{2n+1}(\S^{2n+1}){{\buildrel P \over \longrightarrow}}\pi_{2n-1}(\S^n){{\buildrel E \over \longrightarrow}}\pi_{2n}(\S^{n+1})$$ and by the Freudenthal suspension theorem, $$\sharp\beta=\sharp(E^m\beta) \ \mbox{for any} \ \beta\in\pi_{n+k}(\S^n;p) \
\mbox{and} \ m\ge 1.$$ Hence, by Lemma \[pro\].(2) and (\[Serre\]), we obtain $$\sharp\beta=\sharp[\iota_n,\beta].$$ This leads to (1).
\(2) is easily obtained from (\[Serre\]) and the proof is complete. $\square$
The notation $\pi_{n+m}(\S^n)=\{\alpha_n\}\ (\{\alpha(n)\}, \ resp.)$ means that there exist some $k\ge 1$ and an element $\alpha_k\ (\alpha(k),\ resp.)\in\pi_{k+m}(\S^k)$ satisfying $\alpha_n=
E^{n-k}\alpha_k \ (\alpha(n)=E^{n-k}\alpha(k), \ resp.)$ for $n\ge k$. For the $p$-primary component with any prime $p$, the notation is available.
Hereafter, we omit the reference [@T] unless otherwise stated. Now, we know that $\pi_{n+3}(\S^n;3)=\{\alpha_1(n)\}\cong\Z_3$ and $\pi_{n+7}(\S^n;3)=\{\alpha_2(n)\}\cong\Z_3$ for $n\ge 3$. Write $\{-, -, -\}_n$ for the Toda bracket, where $n\ge 0$ and $\{-, -, -\}=\{-, -, -\}_0$. We recall that there exists the element $\beta_1(5)\in\pi_{15}(\S^5)$ satisfying $\beta_1(5)\in\{\alpha_1(5),\alpha_1(8),\alpha_1(11)\}_1$, $3\beta_1(5)=-\alpha_1(5)\alpha_2(8)$ and that $\pi_{n+10}(\S^n;3)=\{\beta_1(n)\}\cong\Z_9$ for $n=5,6$ and $\cong\Z_3$ for $n\ge 7$.
Let $\Omega^2\S^{2m+1}=\Omega(\Omega \S^{2m+1})$ be the double loop space of $\S^{2m+1}$ and $Q^{2m-1}_2
=\Omega(\Omega^2\S^{2m+1},\S^{2m-1})$ the homotopy fiber of the canonical inclusion (the double suspension map) $i: \S^{2m-1}\to\Omega^2\S^{2m+1}$. Then, the following result and its proof have been shown by Toda [@T4].
\[Toda\] Let $n$ be even and $n\ge 4$. Then $[\iota_n,[\iota_n,\alpha_1(n)]]\ne 0$ if and only if $n\ne 4$ and $n\equiv 1\ (\bmod\ 3)$.
[**Proof.**]{} It is well-known that $[\iota_n,[\iota_n,\iota_n]]\in
E\pi_{3n-3}(\S^{n-1})$. So, by the $(\bmod\ 3)$ EHP sequence [@T3 (2.1.3), Proposition 2.1], we have $$[\iota_n,[\iota_n,\iota_n]]=\pm EP(i(n-1)), \ \mbox{where} \ i(n-1) \ \mbox{is
a generator of} \ \pi_{3n-3}(Q^{n-1}_2;3).$$ By the naturality [@T3 (2.1.5)], we obtain $[\iota_n,[\iota_n,\alpha_1(n)]]
=\pm EP(a_1(n-1))$, where $a_1(n-1)=i(n-1)\alpha_1(3n-3)\in\pi_{3n}(Q^{n-1}_2;3)$. By [@T3 Proposition 4.4], $(\frac{n}{2}+1)a_1(n-1)=HP(i(n+1))$. So, $P(a_1(n-1))=\pm PHP(i(n+1))=0$ if $n\not\equiv 1\ (\bmod\ 3)$. For the case $n=4$, the assertion is trivial.
Next, assume that $n\ne 4$ and $n\equiv 1\ (\bmod\ 3)$. Then, by [@T2 Theorem 10.3], there exists an element $v\in\pi_{3n-4}(\S^{n-3})$ satisfying $H(v)=b(n-5)$ and $E^2v=P(a_1(n-1))$, where $b(n-5)=i(n-5)\beta_1(3n-15)$. Furthermore, by [@T2 Proposition 5.3.(ii)], we obtain $P(a_2(n-3))=3v$, where $a_2(n-3)=i(n-3)\alpha_2(3n-9)$. So, by the $(\bmod\ 3)$ EHP sequence, we have $P(a_1(n-1))\ne 0$. This implies the sufficient condition and completes the proof. $\square$
Gottlieb groups of spheres with stems for $k\le 7$
==================================================
According to [@Hilton1], [@Hil], [@K-M], [@M1], [@Thomeier] and [@T], we know the following results: $$=0 \ \mbox{if and only if} \ n\equiv 3\, (\bmod\
4)\;\mbox{or}\;n=2,6;\label{weta}$$ $$=0 \ \mbox{if and only if} \ n\equiv 2,3\, (\bmod\ 4)\;
\mbox{or} \; n=5.\label{weta2}$$ Hence, Lemma \[lead\] completely determines $G_{n+k}(\S^n)$ for $k=1,2$.
We recall that $\pi^3_6=\{\nu'\}\cong\Z_4$. Write $\omega$ for a generator of the $J$-image $J\pi_3(SO(3))=\pi_6(\S^3)\cong\Z_{12}$ satisfying $\omega=\nu'+\alpha_1(3)$. We recall the relation $\pm[\iota_4,\iota_4]=2\nu_4-\Sigma\nu'$. By abuse of notation, $\nu_n$ represents a generator of $\pi^n_{n+3}$ and $\pi_{n+3}(\S^n)$ for $n\ge 4$, respectively. Then, $\pi_7(\S^4)=\{\nu_4,E\omega\}\cong\Z\oplus \Z_{12}$, $\pi_{n+3}(\S^n)=\{\nu_n\}\cong\Z_{24}$ for $n\ge 5$. By [@B-B], $[\iota_4,\nu_4]=2\nu^2_4$. In the light of Lemma \[pro\].(2) and the relation $\nu'\nu_6 = 0$, we obtain $$[\iota_4, E\nu'] = [\iota_4, \iota_4]\circ E(\iota_3\wedge\nu')
= (2\nu_4 - E\nu')\circ 2\nu_7 = 4\nu^2_4.$$ So, we have $2E\nu' = \eta^3_4\in G_7(\S^4)$. Consequently, by Corollary \[CC\].(1) and Proposition \[Gnp0\], $$G_7(\S^4) = \{3[\iota_4,\iota_4], 2E\nu'\}\cong 3\Z\oplus\Z_2.$$ In the light of [@K-M], [@M1], [@Thomeier], [@T], Corollary \[CC\].(3) and Proposition \[Gnp0\], we know the following:$$\begin{gathered}
\label{wnu}
\sharp[\iota_n,\nu_n] =\left\{\begin{array}{ll}
1,&\quad\mbox{if}\;n\equiv 7\; (\bmod\ 8), \; n = 2^i - 3\;
\mbox{for} \; i\ge 3;\\
2,&\quad\mbox{if} \; n\equiv 1,3,5\; (\bmod\ 8)\ge 9 \;
\mbox{and}\;n\neq 2^i - 3;\\
12,&\quad\mbox{if} \;n\equiv 2\;(\bmod\ 4)\ge 6, n = 12;\\
24,&\quad \mbox{if} \;n\equiv 0\;(\bmod\ 4)\ge 8 \ \mbox{unless} \; 12.
\end{array}
\right.\end{gathered}$$ Thus, Lemma \[lead\] leads to a complete description of $G_{n+3}(\S^n)$.
Now, we recall the following relations: $$\eta_n\nu_{n+1}=0 \ \mbox{for} \ n\ge 5 \ \mbox{and} \
\nu_n\eta_{n+3}=0 \ \mbox{for} \ n\ge 6.$$ By the relation $\nu'\eta_6=\eta_3\nu_4$, we have $[\iota_4,\nu_4\eta_7]=[\iota_4,(E\nu')\eta_7]
=[\iota_5,\nu_5\eta_8]=0$. Hence, by the group structures of $\pi_{n+k}(\S^n)$ for $k = 4,5$, we get
\[prop3\] $G_{n+4}(\S^n)=\pi_{n+4}(\S^n)$; $G_{n+5}(\S^n)=\pi_{n+5}(\S^n)$ unless $n=6$ and $G_{11}(\S^6)=3\pi_{11}(\S^6)\cong 3\Z$.
In the next two sections, we will prove the following result partially extending that of [@K-M Theorem 1.3].
\[neq\]$[\iota_n, \nu^2_n] = 0$ if and only if $n\equiv 4,5,7\ (\bmod\ 8)$, $n = 2^i - 3$ or $n = 2^i - 5$ for $i\ge 4$.
We recall that $\pi_{10}(\S^4)=\{\nu^2_4,\alpha_1(4)\alpha_1(7),
\nu_4\alpha_1(7)\}\cong\Z_8\oplus(\Z_3)^2$. By [@B-B] and the relation $\alpha_1(7)\alpha_1(10)=0$, we get that $[\iota_4,\nu_4\alpha_1(7)]=[\iota_4,\alpha_1(4)\alpha_1(7)]=[\iota_4,\iota_4](\alpha_1(7)\alpha_1(10))=0$. Recall also that $\pi^5_{12}=\{\sigma'''\}\cong\Z_2$, $\pi^6_{13}=\{\sigma''\}\cong\Z_4$, $\pi^7_{14}=\{\sigma'\}\cong\Z_8$, where $E\sigma'''=2\sigma''$, $E\sigma''=2\sigma'$ and $E^2\sigma'=2\sigma_9$. By [@B-B] and [@T], we obtain $$[\iota_5,\sigma''']=[\iota_5,\iota_5]\circ E^4\sigma'''=0,
[\iota_6,\sigma'']=[\iota_6,\iota_6]\circ E^5\sigma''
= 4([\iota_6,\iota_6]\circ\sigma_{11})$$ and $2[\iota_6,\sigma'']\ne 0.$ We recall the relation $\pm[\iota_8,\iota_8]=2\sigma_8-E\sigma'$. In $\pi^8_{22}=\Z_{16}\{\sigma^2_8\}\oplus\Z_8\{(E\sigma')\sigma_{15}\}\oplus\Z_4\{\kappa_8\}$, we have $[\iota_8,E\sigma']=2[\iota_8,\iota_8]\sigma_{15}=\pm 2(2\sigma^2_8-(E\sigma')\sigma_{15})$ and in view of [@B-B], we obtain $[\iota_8,\sigma_8]=[\iota_8,\iota_8]\circ\sigma_{15}=
\pm(2\sigma^2_8-(E\sigma')\sigma_{15})$. We know that $\pi_{n+7}(\S^n;5)=\{\alpha'_1(n)\}$ $\cong\Z_5$ for $n\ge 3$. Thus, by Corollary \[CC\], Proposition \[Gnp0\] and Theorem \[neq\], we obtain
\[ToMa\] $G_{n+6}(\S^n) = \pi_{n+6}(\S^n)$ if $n\equiv 4,5,7\ (\bmod\ 8)$, $n = 2^i - 3$ or $n = 2^i - 5$ and $G_{n+6}(\S^n) = 0$ otherwise.
Furthermore, $G_{n+7}(\S^n)=0$ if $n=4,6$, $G_{12}(\S^5)=\pi_{12}(\S^5)$, $G_{15}(\S^8)=\{3[\iota_8,\iota_8],4E\sigma'\}
\cong 3\Z\oplus\Z_2$.
Let $H: \pi_k(\S^n)\to \pi_k(\S^{2n-1})$ be the Hopf homomorphism. Then, by [@Ad] and [@Oda Proposition 4.5], there exists an element $\gamma\in\pi^{n-7}_{2n-8}$ satisfying$$\label{des7}
[\iota_n,\iota_n]=E^7\gamma \ \mbox{and} \ H\gamma=\sigma_{2n-15}, \
\mbox{if} \ n\equiv 7\ (\bmod\ 16)\ge 23.$$ According to Mahowald [@M5] and (\[des7\]), we obtain
\[Mah\] $[\iota_n,\sigma_n]\ne 0$, if $n\equiv 7\ (\bmod\ 16)\ge 23$. It desuspends seven dimensions whose Hopf invariant is $\sigma^2_{2n-15}$.
By abuse of notation, $\sigma_n$ represents a generator of $\pi^n_{n+7}$ and $\pi_{n+7}(\S^n)$ for $n\ge 8$, respectively. Combining the results of [@M1], [@M2], [@T], Corollary \[CC\].(3), Proposition \[Gnp0\] and Theorem \[Mah\], $$\begin{gathered}
\label{wsigma}
\sharp[\iota_n,\sigma_n]=\left\{\begin{array}{ll}
240,&\ \mbox{if} \ n\ \mbox{is even and}\ n\ge 10;\\
2,&\ \mbox{if} \ \mbox{$n$ is odd and $n\ge 9$ unless $n=11$}\\
& \ \mbox{or $n\equiv 15$}\ (\bmod\ 16);\\
1,&\ \mbox{if} \ n=11 \ \mbox{or} \ n\equiv 15\ (\bmod\ 16).
\end{array}
\right.\end{gathered}$$ Whence, by means of Lemma \[lead\], the groups $G_{n+7}(\S^n)$ for $n\ge 9$ have been fully described as well.
Proof of Theorem \[neq\], part I
================================
Denote by $i_n(\mathbb{R}) : SO(n-1)\hookrightarrow SO(n)$ and $p_n(\mathbb{R}) : SO(n)\to\S^{n-1}$ the inclusion and projection maps, respectively. Hereafter, we use the following exact sequence induced from the fibration $SO(n+1)\stackrel{SO(n)}{\longrightarrow}\S^n$: $$(\mathcal{SO}^n_k) \hspace{3mm} \pi_{k+1}(\S^n){{\buildrel \Delta \over \longrightarrow}}\pi_k(SO(n))
{{\buildrel i_* \over \longrightarrow}}\pi_k(SO(n+1)){{\buildrel p_* \over \longrightarrow}}\pi_k(\S^n)
\longrightarrow\cdots,$$ where $i=i_{n+1}(\R)$ and $p=p_{n+1}(\R)$.
Since $SO(n)\cong
SO(n-1)\times\S^{n-1} \
\mbox{for} \ n=4,8,$ we get that $$\label{37}
\Delta\pi_{k+1}(\S^n)=0, \ \mbox{if} \ n=3,7.$$ By the exact sequence $(\mathcal{SO}^n_n)$ and the fact that $\pi_n(SO(n))\cong\Z$ for $n\equiv 3\ (\bmod\ 4)$ [@K], we have $$\label{Deta3}
\Delta\eta_n=0, \ \mbox{if} \ n\equiv 3\ (\bmod\ 4).$$ We recall the formula [@K Lemma 1] $$\label{DeE}
\Delta(\alpha\circ E\beta) = \Delta\alpha\circ\beta.$$ By (\[Deta3\]) and (\[DeE\]), $$\label{Deta32}
\Delta(\eta^2_n)=0, \ \mbox{if} \ n\equiv 3\ (\bmod\ 4).$$ Denote by $V_{n,k}$ the Stiefel manifold consisting of $k$-frames in $\R^n$ for $k\le n-1$. Then, we show
\[50\] [*(1)*]{} $\Delta(\nu^2_n)=0, \ \mbox{if} \ n\equiv 5\
(\bmod\ 8);$
[*(2)*]{} $\Delta(\nu^2_{4n})=0$, if $n$ is odd.
[**Proof.**]{} Since $\pi_7(SO(5))\cong\Z$ [@K], $\Delta : \pi_8(\S^5)\to\pi_7(SO(5))$ is trivial and $\Delta\nu_5=0$. So, by (\[DeE\]), $\Delta(\nu^2_5)=0$. Let now $n\equiv 5\ (\bmod\ 8)\ge 13$. We consider the exact sequence $(\mathcal{SO}^n_{n+5})$: $$\pi_{n+6}(\S^n)\stackrel{\Delta}{\to}\pi_{n+5}(SO(n))
\stackrel{i_*}{\to}\pi_{n+5}(SO(n+1))\to 0.$$ By [@B-M], we obtain $$\pi_{n+5}(SO(n))\cong\pi_{n+5}(SO)\oplus\pi_{n+6}(V_{n+8,8}).$$ In the light of [@H-M], $\pi_{n+6}(V_{n+8,8})\cong\Z_8$ and by [@Bott], $\pi_{n+5}(SO)=0$. So, $\pi_{n+5}(SO(n))\cong\Z_8$. By [@K], $\pi_{n+5}(SO(n+1))\cong\Z_8$. From the fact that $\pi_{n+6}(\S^n)=\{\nu^2_n\}\cong\Z_2$, we obtain $\Delta(\nu^2_n)=0$, and hence (1) follows.
We obtain $\pi_9(SO(4))\cong\pi_9(SO(3))\oplus\pi_9(\S^3)\cong(\Z_3)^2$, and so $\Delta(\nu^2_4)=0$. Let now $n\ge 3$. Then, we consider the exact sequence $(\mathcal{SO}^{4n}_{4n+5}$): $$\pi_{4n+6}(\S^{4n})\stackrel{\Delta}{\to}\pi_{4n+5}(SO(4n))
\stackrel{i_*}{\to}\pi_{4n+5}(SO(4n+1))\to 0.$$ By [@K], $\pi_{4n+5}(SO(4n+1))\cong\Z_2$. By [@KM], ${i_{13}(\R)}_*: \pi_{17}(SO(12))\to\pi_{17}(SO(13))\cong\Z_2$ is an isomorphism, and hence $\Delta(\nu^2_{12})=0$. Let $n$ be odd and $n\ge 5$. In the light of [@B-M], $$\pi_{4n+5}(SO(4n))\cong\pi_{4n+5}(SO)\oplus\pi_{4n+6}(V_{4n+8,8}), \
\mbox{if} \ n\ge 4.$$ By means of [@Bott] and [@H-M], $
\pi_{4n+5}(SO)\cong\Z_2$ and $\pi_{4n+6}(V_{4n+8,8}) = 0$. Hence, we obtain $\Delta(\nu^2_{4n}) = 0$ if $n$ is odd with $n\ge 5$. This leads to (2) and completes the proof. $\square$
[@K-M Theorem 1.3] suggests the non-triviality of $[\iota_n,\nu^2_n]$ for $n\equiv 0,1,2,$ $3,6\ (\bmod\ 8)\ge 6$ and [@No Proposition 3.4] gives an explicit proof of its non-triviality for $n\equiv 2\ (\bmod\ 4)\ge 6$.
By Lemma \[pro\].(1) and (\[wnu\]), we have $[\iota_n,\nu^2_n]=0$ if $n\equiv 7\ (\bmod\ 8)$ or $n=2^i - 3$ for $i\ge 3$. In virtue of Lemma \[50\] and by (\[P’\]), we get that $$\label{5}[\iota_n,\nu^2_n] = 0, \ \mbox{if} \ n\equiv 5\
(\bmod\ 8)$$
and
$$\label{4}
[\iota_n, \nu^2_n] = 0, \ \mbox{if} \ n\equiv 4\ (\bmod\ 8).$$
Given elements $\alpha\in\pi_{n+k}(\S^n)$ and $\beta\in\pi_{n+k}(SO(n+1))$ satisfying $p_{n+1}(\R)\beta=\alpha$, then $\beta$ is called a lift of $\alpha$ and we write $$\beta=[\alpha].$$ For $k\le n-1$, set $i_{k,n}=i_n(\R)\circ\cdots\circ i_{k+1}(\R)$.
Let now $n\equiv 0 \; (\bmod\ 4)\ge 8$. By [@B-M], [@Bott] and [@H-M], $\pi_{2n+3}(SO(2n-2))\cong\Z\oplus\Z_4$. In the exact sequence $(\mathcal{SO}^{2n-3}_{2n+3})$, $p_{2n-2}(\R)_\ast: \pi_{2n+3}(SO(2n-2))
\to\pi_{2n+3}(\S^{2n-3} )$ is an epimorphism by Lemma \[50\].(1). So, the direct summand $\mathbb{Z}_4$ of $\pi_{2n+3}(SO(2n-2))$ is generated by $[\nu^2_{2n-3}]$. By [@K], $\pi_{2n+3}(SO(2n+1))\cong\Z\oplus\Z_2$ and $\pi_{2n+3}(SO(2n+2))\cong\Z$. It follows from $(\mathcal{SO}^{2n+1}_{2n+3})$ that the direct summand $\Z_2$ of $\pi_{2n+3}(SO(2n+1))$ is generated by $\Delta\nu_{2n+1}$. By [@K], $\pi_{2n+3}(SO(2n+k-1))\cong\Z\oplus\Z_2$ for $0\le k\le 2$. Hence, by use of $(\mathcal{SO}^{2n+k-1}_{2n+3})$ for $-1\le k\le 2$, $(i_{2n-2,2n+1})_*: \pi_{2n+3}(SO(2n-2))
\to \pi_{2n+3}(SO(2n+1))$ is an epimorphism and we get the relation $$(i_{2n-2,2n+1})_\ast[\nu^2_{2n-3}]=\Delta\nu_{2n+1}.$$ Thus, we conclude
\[26\] $E^3J[\nu^2_{2n-3}]=[\iota_{2n+1},\nu_{2n+1}]$, if $n\equiv 0\ (\bmod\ 4)\ge 8$.
Next, we need
\[Deps0\] Let $n\equiv 3\ (\bmod\ 4)\ge 7$. Then,
[*(1)*]{} $\{\Delta\iota_n,\eta_{n-1},2\iota_n\}=0$;
[*(2)*]{} $\Delta(E\{\eta_{n-1},2\iota_n,\alpha\})=0$, where $\alpha\in\pi_k(\S^n)$ is an element satisfying $2\iota_n\circ\alpha=0$.
[**Proof.**]{} By the properties of Toda brackets and the fact that $2\pi_{n+1}(SO(n+1))=0$, if $n\equiv 3\ (\bmod\ 4)\ge 7$ [@K], we obtain $$i_{n+1}(\R)\circ\{\Delta\iota_n,\eta_{n-1},2\iota_n\}
=-\{i_{n+1}(\R),\Delta\iota_n,\eta_{n-1}\}\circ 2\iota_{n+1}
\subset 2\pi_{n+1}(SO(n+1))=0.$$ It follows from $(\mathcal{SO}^n_{n+1})$ and (\[Deta32\]) that ${i_{n+1}(\R)}_*: \pi_{n+1}(SO(n))\to\pi_{n+1}(SO(n+1))$ is a monomorphism. This leads to (1).
By (\[DeE\]) and (1), for any $\beta\in\{\eta_{n-1},2\iota_n,\alpha\}$, we obtain $$\Delta(E\beta)\in\Delta\iota_n\circ\{\eta_{n-1},2\iota_n,\alpha\}
=-\{\Delta\iota_n,\eta_{n-1},2\iota_n\}\circ E\alpha=0.$$ This leads to (2) and completes the proof. $\square$
We recall that $\varepsilon_{n-1}\in\{\eta_{n-1},2\iota_n,\nu^2_n\}$ and $\mu_{n-1}\in\{\eta_{n-1},2\iota_n,E^{n-5}\sigma'''\}$ for $n\ge 5$. So, by (\[37\]) and Lemma \[Deps0\].(2), we obtain
\[Deps\] Let $n\equiv 3\ (\bmod\ 4)$. Then,
[*(1)*]{} $\Delta\varepsilon_n=0$;
[*(2)*]{} $\Delta\mu_n=0$.
Hereafter, we use often the EHP sequence of the following type: $$(\mathcal{PE}^n_{n+k}) \hspace{3ex}
\pi^{2n+1}_{n+k+2}{{\buildrel P \over \longrightarrow}}\pi^n_{n+k}{{\buildrel E \over \longrightarrow}}\pi^{n+1}_{n+k+1}.$$
It is well-known that $$H[\iota_n,\iota_n]=0 \ \mbox{for} \ n \ \mbox{odd, and} \
H[\iota_n,\iota_n]=\pm 2\iota_{2n-1} \ \mbox{for} \ n \ \mbox{even}.$$ So, by [@T Proposition 2.5], we obtain
\[HP\] Suppose that $\pi^{2n+1}_{n+k+2}=E^3\pi^{2n-2}_{n+k-1}$ and $\pi^{2n-1}_{n+k}
=E\pi^{2n-2}_{n+k-1}$. Then $HP(\pi^{2n+1}_{n+k+2})=0$ for $n$ odd and $HP(\pi^{2n+1}_{n+k+2})=2\pi^{2n-1}_{n+k}$ for $n$ even.
Suppose that $\Delta\alpha = 0$ for $\alpha\in\pi_k(\S^{n-1})$. Then, by [@WG0], we obtain $$\label{HJ}
H(J[\alpha])=\pm E^n\alpha.$$
Now, we show
[**I. if .**]{}
In virtue of [@T Theorem 10.3] and its proof, $[\iota_9,\nu^2_9]=\bar{\nu}_9\nu^2_{17}\ne 0$.
Let $n\equiv 0\ (\bmod\ 4)\ge 8$. Assume that $E^3(J[\nu^2_{2n-3}]\circ\nu_{4n+1})
=0\in\pi^{2n+1}_{4n+7}$. Then, by use of $(\mathcal{PE}^{2n}_{4n+6})$ and the fact that $(i_{2n-2,2n})_\ast[\nu^2_{2n-3}]$ generates the direct summand $\Z_2$ of $\pi_{2n+3}(SO(2n))$, we obtain $E^2(J[\nu^2_{2n-3}]\circ\nu_{4n+1})=8a[\iota_{2n},\sigma_{2n}]$ for $a\in\{0,1\}$. By means of [@T Proposition 11.11.i)], there exists an element $\beta\in\pi^{2n-2}_{4n+4}$ such that $P(8\sigma_{4n+1}) = E^2\beta$ and $H\beta\in\{2\iota_{4n-5},
\eta_{4n-5},8\sigma_{4n-4}\}_2$. By the properties of Toda brackets, we see that $$\begin{aligned}
\{2\iota_{4n-5},\eta_{4n-5},8\sigma_{4n-4}\}_2
\subset\{2\iota_{4n-5},\eta_{4n-5},8\sigma_{4n-4}\}\subset\\
\{2\iota_{4n-5},0,4\sigma_{4n-4}\}=2\iota_{4n-5}\circ\pi^{4n-5}_{4n+4}
+\pi^{4n-5}_{4n-3}\circ 4\sigma_{4n-3}=0.\end{aligned}$$ So, there exists an element $\beta'\in\pi^{2n-3}_{4n+3}$ such that $\beta
=E\beta'$. Hence, $E^2(J[\nu^2_{2n-3}]\circ\nu_{4n+1})=aE^3\beta'$.
In virtue of Lemma \[pro\].(1) and (\[weta\]), $[\iota_{2n-1},
\eta_{2n-1}\sigma_{2n}]=0$. In the light of (\[P’\]) and Example \[Deps\].(1), $[\iota_{2n-1},
\varepsilon_{2n-1}]=0$, and so $P\pi^{4n-1}_{4n+7}=0$. Therefore, by $(\mathcal{PE}^{2n-1}_{4n+5})$, $$E:\pi^{2n-1}_{4n+5}\to\pi^{2n}_{4n+6} \ \mbox{is a
monomorphism if} \ n\equiv 0\ (\bmod\ 4)\ge 8.$$ Hence, $E(J[\nu^2_{2n-3}]\circ\nu_{4n+1})=aE^2\beta'$. By use of $(\mathcal{PE}^{2n-2}_{4n+4})$ and (\[HJ\]), we have a contradictory relation $\nu^3_{4n-5} = 0$. Thus, we get $[\iota_{2n+1},\nu^2_{2n+1}]
=E^3(J[\nu^2_{2n-3}]\circ\nu_{4n+1})\ne 0$.
We note the relation $$\label{pDeio}
p_n(\R)(\Delta\iota_n)=(1+(-1)^n)\iota_{n-1}.$$ Let $n\equiv 0\ (\bmod\ 8)\ge 8$. By use of $(\mathcal{SO}^{n-1}_{n+1})$ and [@K], we get that $i_n(\R)_*: \pi_{n+1}(SO(n-1))\to\pi_{n+1}(SO(n))$ is a monomorphism. So, we obtain $$\label{Denu7}
\Delta\nu_{n-1}=0,\ \mbox{if} \ n\equiv 0\ (\bmod\ 8)\ge 8.$$ Hence, by Lemma \[50\].(2), $\nu_{n-1}$ and $\nu^2_{n-4}$ are lifted to $[\nu_{n-1}]\in\pi_{n+2}(SO(n))$ and $[\nu^2_{n-4}]
\in\pi_{n+2}(SO(n-3))$, respectively. We show the following
\[KT\] Let $n\equiv 0\ (\bmod\ 8)\ge 16$. Then, for some odd $x$, $$2[\nu_{n-1}]-\Delta\nu_n=x(i_{n-3,n})_*[\nu^2_{n-4}].$$
[**Proof.**]{} By use of $(\mathcal{SO}^{n-k}_{n+2})$ for $2\le k\le 4$, Lemma \[50\] and [@K], we see that $(i_{n-3,n-1})_*:\pi_{n+2}(SO(n-3))\to\pi_{n+2}(SO(n-1))\cong\Z_8$ is an isomorphism and $\pi_{n+2}(SO(n-3))=\{[\nu^2_{n-4}]\}$. In virtue of [@K], $\pi_{n+2}(SO(n+1))\cong\Z_8$ and $\pi_{n+2}(SO(n))\cong\Z_{24}
\oplus\Z_8$. So, by $(\mathcal{SO}^{n-k}_{n+2})$ for $k=0,1$, we get $\pi_{n+2}(SO(n))=\{\Delta\nu_n,[\nu_{n-1}]\}$. By (\[pDeio\]), we obtain $p_n(\R)(\Delta\nu_n)=2\nu_{n-1}$, and hence $2[\nu_{n-1}]-\Delta\nu_n\in\Im\ (i_n(\R)_*:\pi_{n+2}(SO(n-1))
\to\pi_{n+2}(SO(n)))$, where $i_n(\R)_*$ is a split monomorphism. Since $\sharp( 2[\nu_{n-1}]-\Delta\nu_n)=8$, we have the required relation and this completes the proof. $\square$
The relation in [@T Lemma 11.17] is regarded as the $J$-image of that in Lemma \[KT\].
Now, we present a proof of the non-triviality of $[\iota_n, \nu^2_n]$ in the case $n\equiv 0\ (\bmod\ 8)\ge 8$.
[**II. $\mbox{\boldmath $[\iota_n,\nu^2_n]\ne 0$}$ if $\mbox{\boldmath $n\equiv 0\ (\bmod\ 8)\ge 8$}$.**]{}
By [@T (7.19), Theorem 7.7], $[\iota_8,\nu^2_8]=\nu_8\sigma_{11}\nu_{18}\ne 0$. Let $n\equiv 0 \ (\bmod\ 8)\ge 16$. In the light of [@B-M], [@Bott] and [@H-M], $\pi_{n+5}(SO(n))\cong(\Z_2)^2$. So, by (\[DeE\]) and Lemma \[KT\], $$\Delta(\nu^2_n)=(i_{n-3,n})_*([\nu^2_{n-4}]\nu_{n+2})$$ and hence $[\iota_n,\nu^2_n]=E^3(J[\nu^2_{n-4}]\circ\nu_{2n-1})$.
Assume that $E^3(J[\nu^2_{n-4}]\circ\nu_{2n-1})=0$. Then, $E^2(J[\nu^2_{n-4}]\circ\nu_{2n-1})\in P\pi^{2n-1}_{2n+6}=\{[\iota_{n-1},\sigma_{n-1}]\}$. By [@T Prposition 11.11.ii)], it holds $\pi^{2n-3}_{2n+5}\subset E^2\pi^{n-4}_{2n+1}$ and in virtue of (\[des7\]), we have $E^2(J[\nu^2_{n-4}]\circ\nu_{2n-1})=aE^7(\gamma\sigma_{2n-10})$ for $a\in\{0,1\}$. So, by using $(\mathcal{PE}^{n-2-k}_{2n+3-k})$ for $k=0,1$, we get that $$\begin{aligned}
J[\nu^2_{n-4}]\circ\nu_{2n-1}-aE^5(\gamma\sigma_{2n-10})-E\beta
\in P\pi^{2n-5}_{2n+4}\end{aligned}$$ for some $\beta\in\pi^{n-4}_{2n+1}$. Hence, Lemma \[HP\] and (\[HJ\]) imply a contradictory relation $\nu^3_{2n-7}=0$, and thus $[\iota_n,\nu^2_n]\ne 0$.
We note that Nomura has a different proof from II.
Proof of Theroem \[neq\], part II
=================================
Let $\omega_n(\mathbb{R})\in\pi_{n-1}(O(n))$, $\omega_n(\C)
\in\pi_{2n}(U(n))$ and $\omega_n(\H)\in\pi_{4n+2}(Sp(n))$ be the characteristic elements for the orthogonal $O(n)$, unitary $U(n)$ and symplectic $Sp(n)$ groups, respectively. We note that $\omega_n(\R)=\Delta\iota_n$ and $\sharp(\Delta\iota_n)=2 \ \mbox{for odd} \ n\ge 9$.
Let $r_n: U(n)\to SO(2n)$ and $c_n: Sp(n)\to SU(2n)$ be the canonical maps, respectively. Set $i_n(\C): U(n-1)\hookrightarrow U(n)$ for the inclusion map. As it is well-known, $$i_{2n+1}(\R)r_n\omega_n(\C)=\omega_{2n+1}(\mathbb{R})
\;\;\mbox{and}\;\;
i_{2n+1}(\C)c_n\omega_n(\H)=\omega_{2n+1}(\C).$$ Let $$\tau'_{2n}=r_n\omega_n(\C)\in\pi_{2n}(SO(2n))\;\mbox{and}\;
\bar{\tau}'_{4n}=r_{2n}c_n\omega_n(\H)\in\pi_{4n+2}(SO(4n)).$$ It is well-known that $$p_{2n}(\R)\tau'_{2n}=(n-1)\eta_{2n-1}\;\mbox{and}\;
{p_{4n}(\R)}\bar{\tau}'_{4n}=\pm(n+1)\nu_{4n-1}\ \mbox{for}\ n\ge 2.$$ Whence, we obtain
\[ap\] If $n$ is even and $n\ge 4$ then $i_{n+1}(\R)\tau'_n
=\Delta\iota_{n+1}$ and $p_n(\R)\tau'_n=(\frac{n}{2}-1)\eta_{n-1}$;
If $n\equiv 0\ (\bmod\ 4)\ge 8$ then $(i_{n,n+2})\bar{\tau}'_n
=\tau'_{n+2}$ and $p_n(\R)\bar{\tau}'_n=\pm(\frac{n}{4}+1)\nu_{n-1}$.
By use of $(\mathcal{SO}^{4n+1}_{4n+2})$, Lemma \[ap\] and [@K], we obtain $$\label{fr0}
\Delta(\eta^2_{4n+1})=4i_{4n+1}(\R)\bar{\tau}'_{4n}, \;\mbox{if}\, n\ge 2.$$ So, by $(\mathcal{SO}^{4n}_{4n+2})$, we have $\tau'_{4n}\eta^2_{4n}-4\bar{\tau}'_{4n}=a\Delta\nu_{4n}$ for $a\in\{0,1,\ldots,23\}$. Composing $p_{4n}(\R)$ with this relation, using the relation $\eta^3_{4n-1}=12\nu_{4n-1}$, (\[DeE\]), (\[pDeio\]) and Lemma \[ap\], $a$ is even and $$\label{fr}
\tau'_{4n}\eta^2_{4n}\equiv 4\bar{\tau}'_{4n}\ (\bmod\ 2\Delta\nu_{4n}),\;\mbox{if}\;
n\ge 2.$$
Set $\tau_{2n}=J\tau'_{2n}\in\pi_{4n}(\S^{2n})\;\mbox{and}\;
\bar{\tau}_{4n}=J\bar{\tau}'_{4n}\in\pi_{8n+2}(\S^{4n})$. Then, we note that $$\label{tau1}
E\tau_{2n}=[\iota_{2n+1},\iota_{2n+1}], H\tau_{2n}=(n-1)\eta_{4n-1}$$ and $$\label{tau2}
E^3\bar{\tau}_{4n}=[\iota_{4n+3},\iota_{4n+3}],
H\bar{\tau}_{4n}=\pm(n+1)\nu_{8n-1}$$ By (\[fr0\]), we have $$\label{4Ebt}[\iota_{4n+1},\eta^2_{4n+1}]
=4E\bar{\tau}_{4n}.$$
Let $\iota_X$ be the identity class of a space $X$. Denote by $\P^n(2)$ the Moore space of type $(\Z_2,n-1)$ and by $i_n:\S^{n-1}\hookrightarrow\P^n(2)$, $p_n:\P^n(2)\to\S^n$ the inclusion and collapsing maps, respectively. We recall from [@T1] that $$\label{ietap}
2\iota_{\P^n(2)}=i_n\eta_{n-1}p_n, \;\mbox{if}\; n\ge 3.$$ Let $\bar{\eta}_n\in[\P^{n+2}(2),\S^n]\cong\Z_4$ and for $n\ge 3$ be an extension and a coextension of $\eta_n$, respectively. We note that $$\label{exteta}
\bar{\eta}_n\in\{\eta_n,2\iota_{n+1},p_{n+1}\}, \;\mbox{if}\; n\ge 3$$ and $$\label{tildeta}
\tilde{\eta}_n\in\{i_{n+1},2\iota_n,\eta_n\}, \;\mbox{if}\; n\ge 3.$$ We have $$\label{ieta2p}
2\bar{\eta}_n=\eta^2_np_{n+2}\;\;\mbox{and}\;\;
2\tilde{\eta}_n=i_{n+1}\eta^2_n,\; \mbox{if}\; n\ge 3.$$ We recall that $\bar{\eta}_n\tilde{\eta}_{n+1}=\pm E^{n-3}\nu'$ for $n\ge 3$. Furthermore, we recall that $\pi_{n+8}(\S^n)=\{\varepsilon_n\}\cong\Z_2$ for $3\le n\le 5$ and $\varepsilon_3=\{\eta_3,E\nu',\nu_7\}$. We need
\[eps\] $\varepsilon_n=\{\eta_n\bar{\eta}_{n+1}, \tilde{\eta}_{n+2}, \nu_{n+4}\}$ if $n\ge 7$.
[**Proof.**]{} By (\[tildeta\]), we obtain $$\tilde{\eta}_7\circ\nu_9\in\{i_8, 2\iota_7, \eta_7\}\circ\nu_9
= -(i_8\circ\{2\iota_7, \eta_7, \nu_8\})\subset i_8\circ\pi_{12}(\S^7)=0.$$ So, we can take $$\varepsilon_5=\{\eta_5, 2\nu_6, \nu_9\}=\{\eta_5, \bar{\eta}_6\tilde{\eta}_7,
\nu_9\}=\{\eta_5\bar{\eta}_6, \tilde{\eta}_7, \nu_9\}$$ and $$\varepsilon_n\in\{\eta_n\bar{\eta}_{n+1}, \tilde{\eta}_{n+2}, \nu_{n+4}\}, \
\mbox{if} \ n\ge 5.$$ The indeterminacy of this bracket is $$\eta_n\bar{\eta}_{n+1}\circ\pi_{n+8}(\P^{n+3}(2))+\pi_{n+5}(\S^n)\circ\nu_
{n+5}.$$ For $n\ge 5$, by use of the homotopy exact sequence of a pair $(\P^{n+3}(2),\S^{n+2})$, we obtain $\pi_{n+8}(\P^{n+3}(2))=\{i_{n+3}\nu^2_{n+2}\}$, and so $\bar{\eta}_{n+1}\circ\pi_{n+8}(\P^{n+3}(2))=\{\eta_{n+1}\nu^2_{n+2}\}=0.$ Hence, the indeterminacy is trivial for $n\ge 7$ and this completes the proof.
$\square$
By [@K], $\pi_{4n}(SO(4n))\cong(\Z_2)^3 \ \mbox{or} \ (\Z_2)^2,\ \mbox{if}\ n\ge 2.$ So, $$\label{ordtau}
\sharp\tau'_{4n}=2,\ \mbox{if}\ n\ge 2.$$ Now, we prove
\[hard\] $\tau'_{n-1}\eta_{n-1}\varepsilon_n\equiv 0\,(\bmod\ \tau'_{n-1}\nu^3_{n-1})$ if $n\equiv 1\ (\bmod\ 8)\ge 17$.
[**Proof.**]{} By (\[DeE\]), (\[fr\]), (\[ieta2p\]) and the relation $\nu_{n-1}\eta_{n+2}=0$ for $n\ge 7$, we have $\tau'_{n-1}\eta^2_{n-1}\bar{\eta}_{n+1}=0$ if $n\equiv 1\ (\bmod\
8)\ge 9$. In virtue of Lemma \[eps\], we obtain $$\begin{aligned}
\tau'_{n-1}\eta_{n-1}\varepsilon_n
=\tau'_{n-1}\eta_{n-1}\circ\{\eta_n\bar{\eta}_{n+1}, \tilde{\eta}_{n+2},
\nu_{n+4}\}\\
=-\{\tau'_{n-1}\eta_{n-1}, \eta_n\bar{\eta}_{n+1}, \tilde{\eta}_{n+2}\}
\circ\nu_{n+5}.\end{aligned}$$ By (\[ordtau\]) and noting that $2\iota_{SO(n-1)}\circ\alpha=2\alpha$ for any $\alpha\in\pi_{n+5}(SO(n-1))$, we obtain $$2\{\tau'_{n-1}\eta_{n-1},\eta_n\bar{\eta}_{n+1},\tilde{\eta}_{n+2}\}
=-\{2\iota_{SO(n-1)}, \tau'_{n-1}\eta_{n-1},\eta_n\bar{\eta}_{n+1}\}
\circ\tilde{\eta}_{n+3}.$$ We have $$\{2\iota_{SO(n-1)}, \tau'_{n-1}\eta_{n-1},\eta_n\bar{\eta}_{n+1}\}
\subset[\P^{n+4}(2),SO(n-1)].$$ By [@K], $\pi_{n+3}(SO(n-1))=0$ if $n\equiv 1\ (\bmod\ 8)\ge 9$. So, $[\P^{n+4}(2),SO(n-1)]=\pi_{n+4}(SO(n-1))\circ p_{n+4}$. In virtue of $(\mathcal{SO}^{n-k}_{n+4})$ for $k=1,2,$ [@B-M], [@Bott], [@H-M] and (\[Denu7\]), we see that $$\pi_{n+4}(SO(n-1))=\{\Delta(\nu_{n-1}),[\nu_{n-2}]\}\circ\nu_{n+1}\cong(\Z_2)^2.$$ Hence, we obtain $[\P^{n+4}(2),SO(n-1)]\circ\tilde{\eta}_{n+3}=
\pi_{n+4}(SO(n-1))\circ\eta_{n+4}=0$. This leads to the relation $$2\{\tau'_{n-1}\eta_{n-1},\eta_n\bar{\eta}_{n+1},\tilde{\eta}_{n+2}\}=0.$$ By [@B-M], [@Bott] and [@H-M], we see that $$\pi_{n+5}(SO(n))\cong\Z_{16}\oplus\Z_2, \ \mbox{if}\ n\equiv 1\
(\bmod\ 8)\ge 17,$$ where the direct summand $\Z_2$ is generated by $\Delta(\nu^2_n)$ and $$\label{new1}
\pi_{n+5}(SO(n-1))\cong\left\{\begin{array}{ll}
(\Z_{16})^2\oplus\Z_2\oplus\Z_{15},&\mbox{if} \ n\equiv 1\ (\bmod\ 16)\ge 17;\\
\Z_{32}\oplus\Z_8\oplus\Z_2\oplus\Z_{15},&\mbox{if} \ n\equiv 9\ (\bmod\ 16)
\ge 25,\end{array}\right.$$ where the direct summand $\Z_2$ is generated by $\tau'_{n-1}\nu^2_{n-1}$. Thus, for $n\equiv 1\ (\bmod\ 16)\ge 17$, we obtain $$\{\tau'_{n-1}\eta_{n-1},\eta_n\bar{\eta}_{n+1},\tilde{\eta}_{n+2}\}
\subset\{\tau'_{n-1}\nu^2_{n-1}\}+8\pi_{n+5}(SO(n-1)),$$ and so $\tau'_{n-1}\eta_{n-1}\varepsilon_n\equiv 0\
(\bmod\ \tau'_{n-1}\nu^3_{n-1})$. For $n\equiv 9\ (\bmod\ 16)\ge 25$, by $(\mathcal{SO}^{n-1}_{n+5})$, we get $$\pi_{n+5}(SO(n-1);2)=\{\beta,\Delta\sigma_{n-1}-2\beta,\tau'_{n-1}\nu^2_{n-1}\}\cong
\Z_{32}\oplus\Z_8\oplus\Z_2,$$ where $\beta$ is such an element that $16\beta=8\Delta\sigma_{n-1}$ and $i_n(\R)\beta$ is a generator of the direct summand $\Z_{16}$ of $\pi_{n+5}(SO(n))$. Then, by (\[DeE\]) and the relation $\sigma_{n-1}\nu_{n+6}=0$ for $n\ge 13$ [@T (7.20)], we have $$4(\Delta\sigma_{n-1}-2\beta)\circ\nu_{n+5}
=4\Delta(\sigma_{n-1}\nu_{n+6})+\beta\circ 8\nu_{n+5}=0.$$ Consequently, we obtain the relation $\tau'_{n-1}\eta_{n-1}\varepsilon_n
\equiv 0\,(\bmod\ \tau'_{n-1}\nu^3_{n-1})$ if $n\equiv 9\ (\bmod\ 16)\ge
25$ and this completes the proof. $\square$
Next, we show
\[n48\] If $n\equiv 0, 1\ (\bmod\ 4)\ge 8$ then $[\iota_n,\alpha]\ne 0$ for $\alpha=\varepsilon_n, \bar{\nu}_n$ and $\eta_n\sigma_{n+1}$.
[**Proof.**]{} We show $[\iota_n,\varepsilon_n]\ne 0$. Let $n\equiv 0 \ (\bmod\ 4)\ge 8$. By [@T Proposition 11.10.i)], there exists an element $\beta\in\pi^{n-1}_{2n+6}$ such that $E\beta=[\iota_n,\varepsilon_n]$ and $H\beta=\eta_{2n-3}\varepsilon_{2n-2}$. Assume that $[\iota_n,\varepsilon_n]=0$. Then, by $(\mathcal{PE}^{n-1}_{2n+6})$, we have $\beta\in P\pi^{2n-1}_{2n+8}$. This induces a contradictory relation $\eta_{2n-3}\varepsilon_{2n-2}=0$, and hence $[\iota_n,\varepsilon_n]\not=0$. Next, consider the case $n\equiv 1\ (\bmod\ 4)
\ge 9$. Then, by (\[tau1\]), $[\iota_n,\varepsilon_n]=E(\tau_{n-1}\varepsilon_{2n-2})$ and $H(\tau_{n-1}\varepsilon_{2n-2})=\eta_{2n-3}\varepsilon_{2n-2}$. Assume that $[\iota_n,\varepsilon_n]=0$. Then, $(\mathcal{PE}^{n-1}_{2n+6})$ and Lemma \[HP\] lead to a contradictory relation $\eta_{2n-3}\varepsilon_{2n-2}=0$, and so $[\iota_n,\varepsilon_n]\ne 0$. For other elements, the argument goes ahead similarly. $\square$
By (\[JDel\]) and Lemma \[n48\], $\Delta: \pi_{n+8}(\S^n)\to\pi_{n+7}(SO(n))$ is a monomorphism, if $n\equiv 0,1\ (\bmod\ 4)\ge 12$. So, by $(\mathcal{SO}^n_{n+8})$, we obtain the exact sequence $$\begin{gathered}
\label{exact1}
\pi_{n+9}(\S^n){{\buildrel \Delta \over \longrightarrow}}\pi_{n+8}(SO(n)){{\buildrel i_* \over \longrightarrow}}\pi_{n+8}(SO(n+1))
{{\buildrel \over \longrightarrow}}0,\\
\mbox{if} \ n\equiv 0,1\ (\bmod\ 4)\ge 12.\\\end{gathered}$$ We recall from [@M2] and [@T] the following: $$\label{myu}
\sharp[\iota_n,\mu_n]=\left\{\begin{array}{ll}
1,&\ \mbox{if} \ n=2,6\ \mbox{or} \ n\equiv 3\ (\bmod\ 4);\\
2,&\ \mbox{if} \ n\equiv 0,1,2\ (\bmod\ 4)\ge 4\ \mbox{unless} \ n=6,
\end{array}\right.$$
$$\label{emyu}
\sharp[\iota_n,\eta_n\mu_{n+1}]=\left\{\begin{array}{ll}
1,& \ \mbox{if} \ n=5 \ \mbox{or} \ n\equiv 2, 3\ (\bmod\ 4);\\
2,&\ \mbox{if} \ n\equiv 0, 1\ (\bmod\ 4)\ge 4 \ \mbox{unless} \ n=5,
\end{array}\right.$$
$$\label{eta2s}
\sharp[\iota_n,\eta^2_n\sigma_{n+2}]=\left\{\begin{array}{ll}
1,&\ \mbox{if} \ n\equiv 2,3\ (\bmod\ 4)\ge 6;\\
2,&\ \mbox{if} \ n\equiv 0\ (\bmod\ 4)\ge 8
\end{array}\right.$$
and $$(\ast) \hspace{7ex} \sharp[\iota_n,\eta^2_n\sigma_{n+2}]=2, \ \mbox{if} \
n\equiv 1\ (\bmod\ 8)\ge 17.$$
Now, we conclude
\[hard2\] [*(1)*]{} $[\iota_n,\eta_n\varepsilon_{n+1}]\equiv 0\,(\bmod\
[\iota_n,\nu^3_n])$ if $n\equiv 1\ (\bmod\ 8)\ge 9$;
[*(2)*]{} $[\iota_n,\nu^3_n]=0$ if $n\equiv 5\ (\bmod\ 8)$ and $[\iota_n,\eta_n\varepsilon_{n+1}]=[\iota_n,\eta^2_n\sigma_{n+2}]=0$ provided $n\equiv 5\ (\bmod\ 8)\ge 13$ unless $n\equiv 53\ (\bmod\ 64)$.
[**Proof.**]{} We have $[\iota_9,\eta_9\varepsilon_{10}]
=\eta_9\sigma_{10}\eta_{17}\varepsilon_{18}+\sigma_9\eta^2_{16}\varepsilon_{18}
=\eta^2_9\sigma_{11}\varepsilon_{18}+4\sigma_9\nu_{16}\sigma_{19}$ $=0$. For the case $n\equiv 1\ (\bmod\ 8)\ge 17$, (1) is a direct consequence of Lemma \[hard\].
By (\[DeE\]) and Lemma \[50\].(1), we have $\Delta(\nu^3_n)=0$. So, the first assertion of (2) holds. In the light of [@Mimura], the second assertion of (2) holds for $n=13$. Let $n\equiv 5\ (\bmod\ 8)\ge 21$. We consider the exact sequence (\[exact1\]). By [@B-M], [@Bott] and [@H-M], we see that $$\begin{gathered}
\label{7SO}
\pi_{n+8}(SO(n+1))\cong\left
\{\begin{array}{ll}
\Z_4\oplus\Z_2,&\ \mbox{if} \ n\equiv 5\ (\bmod\ 32)\ge 37;\\
(\Z_4)^2,&\ \mbox{if} \ n\equiv 21\ (\bmod\ 32);\\
\Z_4,&\ \mbox{if} \ n\equiv 13\ (\bmod\ 16)
\end{array}\right.\end{gathered}$$ and $$\pi_{n+8}(SO(n))\cong\left \{\begin{array}{ll}
\Z_4\oplus(\Z_2)^2,& \ \mbox{if} \ n\equiv 5\ (\bmod\ 32)\ge 37;\\
(\Z_4)^2\oplus\Z_2,& \ \mbox{if} \ n\equiv 21\ (\bmod\ 64);\\
\Z_8\oplus\Z_4\oplus\Z_2,& \ \mbox{if} \ n\equiv 53\ (\bmod\ 64);\\
\Z_4\oplus\Z_2,& \ \mbox{if} \ n\equiv 13\ (\bmod\ 16).
\end{array}\right.$$ By (\[myu\]) and (\[emyu\]), $[\iota_n,\mu_n]\ne 0$ and $[\iota_n,\mu_n]\eta_{2n+8}\ne 0$. So, by the group structures of $\pi_{n+8}(SO(n+k))$ for $k=0,1$, we get that $\Delta\mu_n$ is taken as a generator of the direct summand $\Z_2$ of $\pi_{n+8}(SO(n))$. By (\[DeE\]) and (\[fr0\]), we obtain $$\Delta(\eta^2_n\sigma_{n+2})=4i_n(\R)\bar{\tau}'_{n-1}\sigma_{n+1}$$ and hence $$\Delta(\eta^2_n\sigma_{n+2})=\left\{\begin{array}{ll}
0,& \ \mbox{if} \ n\not\equiv 53\ (\bmod\ 64);\\
4i_n(\R)\bar{\tau}'_{n-1}\sigma_{n+1}\ne 0,& \ \mbox{if} \ n\equiv 53\
(\bmod\ 64).
\end{array}\right.$$ This leads to the second assertion of (2) and the proof is complete. $\square$
By [@Oda Proposition 4.2], $$[\iota_n,\eta_n\varepsilon_{n+1}]\equiv 0\,
(\bmod\ [\iota_n,\eta^2_n\sigma_{n+2}]), \ \mbox{if} \ n\equiv 1\ (\bmod\
8)\ge 9.$$ Thus,$$\label{eeps}
[\iota_n,\eta_n\varepsilon_{n+1}]=0, \quad \mbox{if} \quad n\equiv 1\ (\bmod\ 8)
\ge 9.$$
Next, we prove
\[NK\] Let $n\equiv 1\ (\bmod\ 4)\ge 5$. Then $E(\bar{\tau}_{2n-2}\nu^2_{4n-2})=[\iota_{2n-1},\bar{\nu}_{2n-1}]$ if and only if $[\iota_{2n+1},\nu^2_{2n+1}] = 0$.
[**Proof.**]{} By (\[tau2\]), $E^3(\bar{\tau}_{2n-2}\nu^2_{4n-2})
=[\iota_{2n+1},\nu^2_{2n+1}]$. This induces the necessary condition.
Now, suppose that $[\iota_{2n+1},\nu^2_{2n+1}] = 0$. Then, by $(\mathcal{PE}^{2n}_{4n+6})$, $E^2(\bar{\tau}_{2n-2}\nu^2_{4n-2})\in P\pi^{4n+1}_{4n+8}\cong\Z_{16}$. We assume that $E^2(\bar{\tau}_{2n-2}\nu^2_{4n-2})=8aP(\sigma_{4n+1})$ for $a\in\{0,1\}$. By [@T Proposition 11.11.ii)], there exists an element $\beta\in\pi^{2n-2}_{4n+4}$ such that $$P(8\sigma_{4n+1}) =E^2\beta\quad \mbox{and}\quad
H\beta\in\{\eta_{4n-5}, 2\iota_{4n-4}, 8\sigma_{4n-4}\}_2.$$ We recall that $$\begin{aligned}
\mu_{4n-5}
\in\{\eta_{4n-5}, 2\iota_{4n-4}, 8\sigma_{4n-4}\}_2 \
(\bmod\ \nu^3_{4n-5},\eta_{4n-5}\varepsilon_{4n-4}).&\end{aligned}$$ Thus, we get $$H\beta=\mu_{4n-5}+x\nu^3_{4n-5}+y\eta_{4n-5}\varepsilon_{4n-4}\ (x,y\in\{0,1\}).$$ By using $(\mathcal{PE}^{2n-1}_{4n+5})$ and the assumption, we have $$E(\bar{\tau}_{2n-2}\nu^2_{4n-2})-aE\beta
\in P\pi^{4n-1}_{4n+7}
= \{P(\bar{\nu}_{4n-1}), P(\varepsilon_{4n-1})\}.$$ By (\[tau1\]), $P(\bar{\nu}_{4n-1})
=E(\tau_{2n-2}\bar{\nu}_{4n-4})$ and $P(\varepsilon_{4n-1})=E(\tau_{2n-2}\varepsilon_{4n-4})$. So, by $(\mathcal{PE}^{2n-2}_{4n+4})$, $$\bar{\tau}_{2n-2}\nu^2_{4n-2}-a\beta-b\tau_{2n-2}\bar{\nu}_{4n-4}
-c\tau_{2n-2}\varepsilon_{4n-4}\in P\pi^{4n-3}_{4n+6}\ (b,c\in\{0,1\}).$$ Applying $H: \pi^{2n-2}_{4n+4}\to\pi^{4n-5}_{4n+4}$ to this equation, using (\[tau1\]), Lemma \[HP\] and the relation $\eta_{4n-5}\bar{\nu}_{4n-4}=\nu^3_{4n-5}$, we obtain $$\nu^3_{4n-5}+a(\mu_{4n-5}+x\nu^3{4n-5}+y\eta_{4n-5}\varepsilon_{4n-4})+b\nu^3_{4n-5}+c\eta_{4n-5}\varepsilon_{4n-4}=0.$$
By the group structure of $\pi^{4n-5}_{4n+4}$, $a=c=0$ and $b=1$, and so $E(\bar{\tau}_{2n-2}\nu^2_{4n-2})=E(\tau_{2n-2}\bar{\nu}_{4n-4})$. Whence the proof is complete. $\square$
Since $\nu_n\eta_{n+3} = 0$ and $\bar{\nu}_n\eta_{n+8}=\nu^3_n$ for $n\ge 6$, Lemma \[NK\] implies
\[NK2\] If $[\iota_{8n+3}, \nu^2_{8n+3}] = 0$, then $[\iota_{8n+1},\nu^3_{8n+1}]=0$.
Now, we show
[**III. $\mbox{\boldmath $[\iota_n,\nu^2_n] = 0$}$ if $\mbox{\boldmath $n = 2^i - 5\ (i\ge 4)$}$.**]{}
We recall the Mahowald element $\eta'_i\in\pi^S_{2^i}(\S^0)$ for $i\ge 3$ [@M3]. We set $\eta'_{i-1,m}=\eta'_{i-1}$ on $\S^m$ for $m=2^{i-1}-2$ with $i\ge 4$, that is, $\eta'_{i-1,m}\in\pi_{2^{i-1}+m}(\S^m)$. It satisfies the relation $H(\eta'_{i-1,m})=\nu_{2m-1}$. Then, the assertion follows directly from [@Ba].
Finally, we show
[**IV. $\mbox{\boldmath $[\iota_n,\nu^2_n]\ne 0$}$ if $\mbox{\boldmath $n\equiv 3\ (\bmod\ 8)\ge 19$}$ unless $\mbox{\boldmath $n=2^i-5$}$.**]{}
By III and Corollary \[NK2\], we obtain $$[\iota_n,\nu^3_n]=0, \quad \mbox{if} \quad n=2^i-7\ (i\ge 4).$$ Hence, from (\[eeps\]) and the relation $\eta^2_n\sigma_{n+2}=\nu_n^3+\eta_n\varepsilon_{n+1}$, $$[\iota_n,\eta^2_n\sigma_{n+2}]=0,\ \mbox{if}\ n=2^i-7\,(i\ge 4).$$
Let $n\equiv 1\ (\bmod\ 8)\ge 17$. Considering the exact sequence (\[exact1\]), in virtue of [@B-M], [@Bott] and [@H-M], we obtain $$\pi_{n+8}(SO(n))\cong\Z_2\oplus\Z_2\oplus\Z_8\quad\mbox{and}\quad
\pi_{n+8}(SO(n+1))\cong\Z_2\oplus\Z_4.$$ By (\[myu\]), (\[emyu\]) and ($\ast$), we know that $$[\iota_n,\mu_n]\ne 0, [\iota_n,\mu_n]\eta_{2n+8}\ne 0$$ and $$[\iota_n,\eta^2_n\sigma_{n+2}]\ne 0.$$ So, by (\[4Ebt\]), we get the relation $$4E(\bar{\tau}_{n-1}\sigma_{2n})=[\iota_n,\eta^2_n\sigma_{n+2}]\ne 0.$$ By $(\ast)$ and (\[eeps\]), we obtain $$[\iota_n, \nu^3_n]=[\iota_n,\eta^2_n\sigma_{n+2}]\ne 0,\ \mbox{if}\ n\equiv 1\
(\bmod\ 8)\ge 17 \ \mbox{and}\ n\ne 2^i-7.$$ Thus, by Corollary \[NK2\], we obtain the assertion.
We are in a position to assert that Mahowald’s result [@M2] should be stated as follows.
Let $n\equiv 1\ (\bmod\ 8)\ge 9$. Then $[\iota_n,\eta^2_n\sigma_{n+2}]\ne 0$ if and only if $n\ne 2^i-7$.
Proof of $[\iota_{16s+7},\sigma_{16s+7}]\ne 0$ for $s\ge 1$
===========================================================
We give a proof of the first part of Theorem \[Mah\]. First of all, let $n\equiv 2\ (\bmod\ 4)\ge 10$. Then, by use of $(\mathcal{SO}^n_n)$, Lemma \[ap\].(1) and [@K], we obtain $\pi_n(SO(n))=\{\tau'_n\}\cong\Z_4$ and $$\label{deta}
2\tau'_n=\Delta\eta_n, \ \mbox{if}\ n\equiv 2\ (\bmod\ 4)\ge 10.$$
We recall from [@T p.95-6] the construction of the element $\kappa_7\in\pi_{21}(\S^7)$. It is a representative of a Toda bracket $$\{\nu_7,E\alpha,E^2\beta\}_1,$$ where $\alpha=\bar{\eta}_9\in[\P^{11}(2),\S^9]$ is an extension of $\eta_9$ and $\beta=\widetilde{\bar{\nu}}_9\in\pi_{18}(\P^{10}(2))$ is a coextension of $\bar{\nu}_9$ satisfying $\alpha\circ E\beta=0$. Furthermore, $\kappa_n=E^{n-7}\kappa_7$ for $n\ge 7$ and set $\widetilde{\bar{\nu}}_n=E^{n-9}\widetilde{\bar{\nu}}_9$ for $n\ge 9$. Then, we can take $$\kappa_n\in\{\nu_n,\bar{\eta}_{n+3},\widetilde{\bar{\nu}}_{n+4}\} \;\mbox{for}\; n\ge 7.$$ By [@K], $\pi_{n+4}(SO(n+k))\cong\Z\oplus\Z_2$ for $k=1,2$ if $n\equiv 7\ (\bmod\ 8)$. And, by $(\mathcal{SO}^{n+2}_{n+4})$, the direct summand $\Z_2$ of $\pi_{n+4}(SO(n+2))$ is generated by $\Delta\nu_{n+2}$. So, the non-triviality of $[\nu_n]\eta_{n+3}\in\pi_{n+4}(SO(n+1))$ induces the relation ${i_{n+2}(\R)}_*([\nu_n]\eta_{n+3})=\Delta\nu_{n+2}$. Because of the fact that $[\iota_{n+2},\nu^2_{n+2}]\ne 0$, this induces a contradictory relation $0=\Delta\nu^2_{n+2}\ne 0$. Hence, we obtain $$[\nu_n]\eta_{n+3}=0, \ \mbox{if} \ n\equiv 7\ (\bmod\ 8).$$ Next, by [@K], $$\{[\nu_n],\eta_{n+3},2\iota_{n+4}\}\subset\pi_{n+5}(SO(n+1))=0, \; \mbox{if} \;
n\equiv 7\ (\bmod\ 8).$$ So, by (\[exteta\]), we have $[\nu_n]\bar{\eta}_{n+3}\in\{[\nu_n],\eta_{n+3},
2\iota_{n+4}\}\circ p_{n+5}=0$ and hence we can define a lift of $\kappa_n$ for $n\equiv 7\ (\bmod\ 8)$, as follows: $$[\kappa_n]\in\{[\nu_n],\bar{\eta}_{n+3},\widetilde{\bar{\nu}}_{n+4}\}\subset\pi_{n+14}(SO(n+1))
\;\mbox{for}\; n\equiv 7\ (\bmod\ 8).$$
Let $n\equiv 7\ (\bmod\ 8)\ge 15$. By use of $(\mathcal{SO}^{n-4}_{n-4})$, $(\mathcal{SO}^{n-k}_{n-3})$ for $k=2,3,5$, $(\mathcal{SO}^{n-l}_{n-2})$ for $2\le l\le 5$ and [@K], we obtain $$\pi_{n-4}(SO(n-4))=\{\beta\}\cong\Z; \
\pi_{n-3}(SO(n-4))=\{[\eta^2_{n-5}]\}\cong\Z_2;$$ $$\pi_{n-2}(SO(n-4))=\{[\eta^2_{n-5}]\eta_{n-3},\Delta\nu_{n-4}\}\cong(\Z_2)^2;$$ $$\pi_{n-4}(SO(n-3))=\{i_{n-3}(\R)\beta,\Delta\iota_{n-3}\}\cong(\Z)^2;$$ $$\pi_{n-3}(SO(n-3))=\{[\eta_{n-4}],\Delta\eta_{n-3}\}\cong(\Z_2
)^2;$$ $$\pi_{n-2}(SO(n-3))=\{[\eta_{n-4}]\eta_{n-4},\Delta\eta^2_{n-3}\}\cong(\Z_2)^2;$$ $$\pi_{n-2}(SO(n-2))=\{\Delta\eta_{n-2}\}\cong(\Z_2)^2,$$ where $\beta$ is a generator of $\pi_{n-4}(SO(n-4))$ and$$\label{Delta2}
\Delta\eta_{n-3}
=i_{n-3}(\R)[\eta^2_{n-5}].$$ We need $$\label{id}
\{p_n(\R),i_n(\R),\Delta\iota_{n-1}\}\ni\iota_{n-1} \ (\bmod \ 2\iota_{n-1}) \
\mbox{for} \ n\ge 9.$$ Since $\sharp[\eta_{n-4}]=2$ for $n\equiv 7\ (\bmod\ 8)$, $[\eta_{n-4}]$ is lifted to $\overline{[\eta_{n-4}]}\in[\P^{n-2}(2),SO(n-4)]$. Since $p_{n-3}(\R)\beta=0$, we obtain $$\label{beeta}
\beta\eta_{n-4}=0\in\pi_{n-3}(SO(n-4)).$$ So, by (\[Delta2\]) and (\[id\]), we get $$\begin{gathered}
\label{lifter}
[\eta_{n-4}]
\in\{i_{n-3}(\R),\Delta\iota_{n-4},\eta_{n-5}\}\,(\bmod \ i_{n-3}(\R)\circ\pi_{n-3}(SO(n-4))\\
+\pi_{n-4}(SO(n-3))\circ\eta_{n-4}{}=\{\Delta\eta_{n-3}\}) \;\mbox{for}\;n\equiv 7\ (\bmod\ 8)\ge 15.\end{gathered}$$ By the same reason as (\[37\]), we obtain $\Delta(\bar{\eta}_3)=0\in[\P^4(2),SO(3)]$. Let $n\equiv 7\ (\bmod\ 8)\ge 15$. Then, by Lemma \[Deps0\].(1) and (\[exteta\]), we obtain $$\Delta(\bar{\eta}_{n-4})=\Delta\iota_{n-4}\circ\bar{\eta}_{n-5}
\in-\{\Delta\iota_{n-4},\eta_{n-5},2\iota_{n-4}\}\circ p_{n-3}=0.$$ So, $\bar{\eta}_{n-4}$ is lifted to $[\bar{\eta}_{n-4}]\in[\P^{n-2}(2),SO(n-3)]$ for $n\equiv 7\ (\bmod\ 8)$. We note $[\bar{\eta}_{n-4}]\in\{i_{n-3}(\R),\Delta\iota_{n-4},\bar{\eta}_{n-5}\}$ for $n\equiv 7\ (\bmod\ 8)\ge 15$. We show
\[extlift\] Let $n\equiv 7\ (\bmod\ 8)\ge 15$.
$\overline{[\eta_{n-4}]}\in\{i_{n-3}(\R),\Delta\iota_{n-4},
\bar{\eta}_{n-5}\} \ (\bmod\ \{\Delta\bar{\eta}_{n-3}\}+\pi_{n-2}(SO(n-3))\circ p_{n-2}+K)$, where $K={i_{n-3}(\R)}_*[\P^{n-2}(2),SO(n-4)]+\pi_{n-4}(SO(n-3))
\circ\bar{\eta}_{n-4}$.
${i_{n-2}(\R)}_*K\subset \{(\Delta\eta_{n-2})p_{n-2}\}$.
[**Proof.**]{} By use of the cofiber sequence $\S^{n-3}\,{{\buildrel i_{n-2} \over \longrightarrow}}\,\P^{n-2}(2)\,{{\buildrel p_{n-2} \over \longrightarrow}}\,\S^{n-2}$, (\[ietap\]) and (\[lifter\]), we get that $\overline{[\eta_{n-4}]}\in\{i_{n-3}(\R),\Delta\iota_{n-4},
\bar{\eta}_{n-5}\} \ (\bmod\ \{\Delta\bar{\eta}_{n-3}\}+\pi_{n-2}(SO(n-3))\circ p_{n-2}+K)$ and that $[\P^{n-2}(2),SO(n-4)]=\{\overline{[\eta^2_{n-5}]},$ $(\Delta\nu_{n-4})p_{n-2}\}\cong\Z_4\oplus\Z_2$, where $\overline{[\eta^2_{n-5}]}$ is an extension of $[\eta^2_{n-5}]$ and $2\overline{[\eta^2_{n-5}]}
=[\eta^2_{n-5}]\eta_{n-3}p_{n-2}$. Hence, by (\[Delta2\]), we see that $${i_{n-4,n-2}}_*\overline{[\eta^2_{n-5}]}\in i_{n-2}(\R)\circ\{\Delta\eta_{n-3},
2\iota_{n-3},p_{n-3}\}=$$ $$-\{i_{n-2}(\R),\Delta\eta_{n-3},2\iota_{n-3}\}
\circ p_{n-2}.$$ Since $\{i_{n-2}(\R),\Delta\eta_{n-3},2\iota_{n-3}\}\subset\pi_{n-2}(SO(n-2))
=\{\Delta\eta_{n-2}\}$, we have ${i_{n-4,n-2}}_*[\P^{n-2}(2),SO(n-4)]\subset
\{(\Delta\eta_{n-2})p_{n-2}\}$.
By (\[exteta\]) and (\[beeta\]), we have $\beta\bar{\eta}_{n-4}\in\{\beta,\eta_{n-4},
2\iota_{n-3}\}\circ p_{n-2}\subset\pi_{n-2}(SO(n-2))\circ p_{n-2}$. Hence, we obtain ${i_{n-2}(\R)}_*(\pi_{n-4}(SO(n-3))\circ\bar{\eta}_{n-4})
\subset\{(\Delta\eta_{n-2})p_{n-2}\}$. This completes the proof. $\square$
We show
\[hard3\] $(i_{n-7,n-1})_*[\kappa_{n-8}]=\Delta\bar{\nu}_{n-1}$ if $n\equiv 7\ (\bmod\ 8)\ge 15$.
[**Proof.**]{} By the group structures of $\pi_{n-5}(SO(n-7+k))$ for $0\le k\le 3$ [@K], we have $(i_{n-7,n-4})_*[\nu_{n-8}]=\Delta\iota_{n-4}$, and so $$(i_{n-7,n-1})_*[\kappa_{n-8}]\in(i_{n-4,n-1})_*\{\Delta\iota_{n-4},
\bar{\eta}_{n-5},\widetilde{\bar{\nu}}_{n-4}\}.$$ By Lemma \[extlift\], we obtain $$\begin{aligned}
{i_{n-3}(\R)}_*\{\Delta\iota_{n-4},\bar{\eta}_{n-5},\widetilde{\bar{\nu}}_{n
-4}\}
=-\{i_{n-3}(\R),\Delta\iota_{n-4},\bar{\eta}_{n-5}\}
\circ\widetilde{\bar{\nu}}_{n-3}\equiv\\
\overline{[\eta_{n-4}]}\circ\widetilde{\bar{\nu}}_{n-3}
\in\{[\eta_{n-4}],2\iota_{n-3},\bar{\nu}_{n-3}\}
(\bmod \ [\eta_{n-4}]\circ\pi_{n+6}(\S^{n-3})\\
+\pi_{n-2}(SO(n-3))\circ\bar{\nu}_{n-2}+K\circ\widetilde{\bar{\nu}}_{n-3}).\end{aligned}$$ From the relation $i_{n-2}(\R)[\eta_{n-4}]=\Delta\iota_{n-2}$, we see that $$\begin{aligned}
(i_{n-7,n-1})_*[\kappa_{n-8}]
&\in&-i_{n-1}(\R)\circ\{\Delta\iota_{n-2},2\iota_{n-3},\bar{\nu}_{n-3}\}\\
&=&\{i_{n-1}(\R),\Delta\iota_{n-2},2\iota_{n-3}\}\circ\bar{\nu}_{n-2}.\end{aligned}$$ We note ${i_{n-2}(\R)}_*(K\circ\widetilde{\bar{\nu}}_{n-3})\subset
\{\Delta\eta_{n-2}\} \circ\bar{\nu}_{n-3}=\{\Delta\nu^3_{n-2}\}=0$ by Lemma \[extlift\] and (\[5\]). Since $\{i_{n-1}(\R),\Delta\iota_{n-2},2\iota_{n-3}\}\equiv \Delta\iota_{n-1} \
(\bmod\ 2\Delta\iota_{n-1})$ by (\[id\]), we have $$\{i_{n-1}(\R),\Delta\iota_{n-2},2\iota_{n-3}\}\circ\bar{\nu}_{n-2}=
\Delta\bar{\nu}_{n-1}.$$ This completes the proof. $\square$
We can take $[\bar{\nu}_n]\in\{[\nu_n],\eta_{n+3},\nu_{n+4}\}$ for $n\equiv 7\ (\bmod\ 8)$. And we obtain $2[\bar{\nu}_n]=0$ and $2[\kappa_n]\equiv[\bar{\nu}_n]\nu^2_{n+8}\ (\bmod\
[\nu_n]\zeta_{n+3})$. By using these facts and the group structures of $\pi_{n+k}(SO(n+1))$ for $k=11,12$ and $n\equiv 7 \ (\bmod\ 8)\ge 15$, we obtain
[*A lift $[\kappa_n]\in\pi_{n+14}(SO(n+1))$ of $\kappa_n$ is taken so that its order is two for $n\equiv 7\ (\bmod\ 8)\ge 15$.*]{}
Let $n\equiv 2\ (\bmod \ 4)\ge 6$. By the relation $4\zeta_n=\eta^2_n\mu_{n+2}$, Lemma \[pro\].(1) and (\[weta2\]), $4[\iota_n,\zeta_n]=0$. So, by the relation $H[\iota_n,\zeta_n] =\pm 2\zeta_{2n-1}$, we obtain $$\label{zeta6}
\sharp[\iota_n,\zeta_n]=4,\ \mbox{if}\ n\equiv 2\ (\bmod\ 4)\ge 6.$$ By [@Oda Proposition 4.2], there exists an element $M_t\in\pi^{8t+8}_{16t+18}$ for $t\ge 0$ such that $$\label{des3}
[\iota_{8t+11},\iota_{8t+11}]=E^3(M_t) \ \mbox{and} \ HM_t=\nu_{16t+15}
\ (t\ge 0).$$ Hereafter, we fix $n=16s+7\ge 23$. By [@M1], there exists a lift $[\sigma_{n-8}]\in\pi_{n-1}(SO(n-7))$ of $\sigma_{n-8}$. By use of the exact sequences $(\mathcal{SO}^{n-k}_{n-1})$ for $6\le k\le 8$, by the fact that $\sharp\Delta\sigma_{n-7}=240$, $\Delta\nu^2_{n-6}\ne 0$ and by (\[new1\]), we obtain the following: $$\pi_{n-1}(SO(n-7))=\{[\sigma_{n-8}],\Delta\sigma_{n-7},\tau'_{n-7}\nu^2_{n-7}\}
\cong(\Z_{16})^2\oplus\Z_2\oplus\Z_{15}.$$ By use of $(\mathcal{SO}^{n-k}_{n-1})$ for $1\le k\le 5$ and by [@B-M], [@Bott], [@H-M] and [@K], we see that $\pi_{n-1}(SO(n-k))=\{(i_{n-7,n-k})[\sigma_{n-8}]\}
\cong\Z_{4k}$ for $k=1,2,4$. Therefore, $(i_{n-7,n-1})[\sigma_{n-8}]=\pm\tau'_{n-1}$, $(i_{n-7,n})[\sigma_{n-8}]=\Delta\iota_n$ and $\gamma$ in (\[des7\]) is taken as $\gamma=J[\sigma_{n-8}]$. By (\[DeE\]) and (\[deta\]), $\Delta(\eta_{n-1}\sigma_n)=2(\tau'_{n-1}\sigma_{n-1})$. Hence, by (\[P’\]) and (\[deta\]), $2(E^6\gamma)=[\iota_{n-1},\eta_{n-1}]$ and $2E^6(\gamma\sigma_{2n-8})=[\iota_{n-1},\eta_{n-1}\sigma_n]$. Assume that $E^7(\gamma\sigma_{2n-8})=[\iota_n,\sigma_n]=0$. Then, by $(\mathcal{PE}^{n-1}_{2n+5})$ and Lemma \[hard3\], $E^6(\gamma\sigma_{2n-8})=
[\iota_{n-1},\bar{\nu}_{n-1}]=E^6J[\kappa_{n-7}]$. By $(\mathcal{PE}^{n-2}_{2n+4})$, we have $$E^5(\gamma\sigma_{2n-8}-J[\kappa_{n-7}])\in P\pi^{2n-3}_{2n+6}.$$ By Corollary \[hard2\].(2) and its proof, $P(\nu^3_{2n-3})=0$, $P\mu_{2n-3}
\ne 0$ and $$P(\eta^2_{2n-3}\sigma_{2n-1})=\left\{\begin{array}{ll}
0,&\mbox{if} \ n\not\equiv 53\ (\bmod\ 64);\\
4E(\bar{\tau}_{n-3}\sigma_{2n-4}),&\mbox{if} \ n\equiv 53\ (\bmod\ 64).
\end{array}\right.$$ So, for $b$ and $x\in\{0,1\}$, we have $$E^5(\gamma\sigma_{2n-8}-J[\kappa_{n-7}])
=4bE(\bar{\tau}_{n-3}\sigma_{2n-4})+xP\mu_{2n-3}.$$ By [@T Proposition 11.10.ii)], there exists an element $\beta\in\pi^{n-3}_{2n+3}$ such that $P\mu_{2n-3}=E\beta$ and $H\beta
=\eta_{2n-7}\mu_{2n-6}$. Then, by $(\mathcal{PE}^{n-3}_{2n+3})$, we have $$E^4(\gamma\sigma_{2n-8}-J[\kappa_{n-7}])
-4b\bar{\tau}_{n-3}\sigma_{2n-4}-x\beta\in P\pi^{2n-5}_{2n+5}.$$ This induces a relation $x\eta_{2n-7}\mu_{2n-6}=0$. Hence, $x=0$ and we can set $$E^4(\gamma\sigma_{2n-8}-J[\kappa_{n-7}])
-4b\bar{\tau}_{n-3}\sigma_{2n-4}=yP(\eta_{2n-5}\mu_{2n-4})\;\mbox{for}\;y\in\{0,1\}.$$ Since $H\bar{\tau}_{n-3}=\nu_{2n-7}$ and $\nu_{2n-7}\sigma_{2n-4}=0$, we have $\bar{\tau}_{n-3}\sigma_{2n-4}
=E\xi$ for an elements $\xi\in\pi^{n-4}_{2n+2}$. By [@T Proposition 11.10.i)], there exists an element $\beta'\in\pi^{n-4}_{2n+2}$ such that $P(\eta_{2n-5}\mu_{2n-4})=E\beta'$ and $H\beta'=\eta^2_{2n-9}\mu_{2n-7}$. So, we have $$E^3(\gamma\sigma_{2n-8}-J[\kappa_{n-7}])-4b\xi-y\beta'
\in P\pi^{2n-7}_{2n+4}.$$ This leads to a relation $y\eta^2_{2n-9}\mu_{2n-7}=0$, and hence $y=0$. Therefore, by (\[des3\]), we obtain ($n=16s+7$) $$E^3(\gamma\sigma_{2n-8}-J[\kappa_{n-7}]
-eM_{2s-1}\zeta_{2n-12})-4b\xi=0\ (e\in\{0,1\}).$$ We consider the EHP sequence $$\pi^{n-5}_{2n+1}\stackrel{E}{\longrightarrow}
\pi^{n-4}_{2n+2}\stackrel{H}{\longrightarrow}\pi^{2n-9}_{2n+2}\stackrel{P}{\longrightarrow}\pi^{n-5}_{2n}.$$ By (\[zeta6\]), $H\xi=4z\zeta_{2n-9}$ for $z\in\{0,1\}$, and so there exists an element $\xi'\in\pi^{n-5}_{2n+1}$ satisfying $E\xi'=2\xi$. Since $\pi^{2n-11}_{2n+1}=\pi^{2n-13}_{2n}=0$, there exists an element $\xi''\in\pi^{n-7}_{2n-1}$ satisfying $E^2\xi''=\xi'$. Hence, we have $$E^2(\gamma\sigma_{2n-8}-J[\kappa_{n-7}]-eM_{2s-1}\zeta_{2n-12})
-2b\xi'\in P\pi^{2n-9}_{2n+3}=0,$$ $$E(\gamma\sigma_{2n-8}-J[\kappa_{n-7}]-eM_{2s-1}\zeta_{2n-12})
-2bE\xi''\in P\pi^{2n-11}_{2n+2}=0$$ and $$\gamma\sigma_{2n-8}-J[\kappa_{n-7}]-eM_{2s-1}\zeta_{2n-12}-2b\xi''
\in P\pi^{2n-13}_{2n+1}.$$ We note $H(M_{2s-1}\zeta_{2n-12})=\nu_{2n-15}\zeta_{2n-12}=0$. Then, the last relation induces a contradictory relation $\sigma^2_{2n-15}
=\kappa_{2n-15}$. Thus, we obtain the non-triviality of $[\iota_n,\sigma_n]$ if $n\equiv 7\ (\bmod\ 16)\ge 23$.
By Lemma \[hard3\], we have $[\iota_n,\bar{\nu}_n]=E^6J[\kappa_{n-7}]$ if $n\equiv 6\ (\bmod\ 8) \ge 14$. By the parallel arguments to the above, we obtain
\[newbar\] $[\iota_n,\bar{\nu}_n]\neq 0$, if $n\equiv 6\ (\bmod\ 8)\ge 14$.
Gottlieb groups of spheres with stems for $8\le k\le 13$
========================================================
We know that $\pi_{n+8}(\S^n)=\{\varepsilon_n\}\cong\Z_2$ for $n=4,5$ and that $[\iota_4,\varepsilon_4]=(E\nu')\varepsilon_7\ne 0$, $[\iota_5,\varepsilon_5]=\nu_5\eta_8\varepsilon_9\ne 0$. It is easy to show that $G_{16}(\S^8)=\{(E\sigma')\eta_{15},\sigma_8\eta_{15}+\bar{\nu}_8
+\varepsilon_8\}\cong(\Z_2)^2$ and $G_{17}(\S^9)=\{[\iota_9,\iota_9]\}\cong\Z_2$. So, by Lemma \[n48\], we get $$G_{n+8}(\S^n)=0, \quad \mbox{if} \quad n\equiv 0,1\ (\bmod\ 4)\ge 4 \ \mbox{unless} \ n=8,9.$$ Let $n\equiv 3\ (\bmod\ 4)\ge 11$. Then, by Lemma \[pro\].(1) and (\[weta\]), $[\iota_n,\eta_n\sigma_{n+1}]=0$. In virtue of (\[P’\]) and Example \[Deps\].(1), we obtain $[\iota_n,\varepsilon_n]=0$. Thus, $$G_{n+8}(\S^n)=\pi_{n+8}(\S^n), \quad \mbox{if} \quad n\equiv 3\ (\bmod\ 4).$$
Now, we show the following
\[eps2\] Let $n\equiv 2\ (\bmod\ 8)\ge 10$. Then $\Delta\varepsilon_n=0$ and $[\iota_n,\bar{\nu}_n]=[\iota_n,\eta_n\sigma_{n+1}]\ne 0$.
Let $n\equiv 6\ (\bmod\ 8)\ge 14$. Then $\Delta\varepsilon_n
=2a(i_n(\R))[\nu^2_{n-2}]\nu_{n+4}$ for $a\in\{0,1,2,3\}$. And the order of $\Delta\pi_{n+8}(\S^n)$ is four or two according as $n\equiv 22\ (\bmod\ 32)$ or not.
[**Proof.**]{} Let $n\equiv 2\ (\bmod\ 4)\ge 10$. Then, by the fact that $\pi_{n+1}(SO(n))\cong\Z$ [@K], we have $\tau'_n\eta_n=0$. So, by (\[DeE\]), (\[ieta2p\]) and (\[deta\]), we obtain $$\Delta(\eta_n\bar{\eta}_{n+1})=2\tau'_n\circ\bar{\eta}_n
=\tau'_n\circ\eta^2_np_{n+2}=0.$$ Therefore, by Lemma \[eps\], we get $$\begin{aligned}
\Delta\varepsilon_n=\Delta\iota_n\circ\varepsilon_{n-1}
&=&\Delta\iota_n\circ\{\eta_{n-1}\bar{\eta}_n,\tilde{\eta}_{n+1},\nu_{n+3}\}\\
&=&-\{\Delta\iota_n,\eta_{n-1}\bar{\eta}_n,\tilde{\eta}_{n+1}\}\circ\nu_{n+4}.\end{aligned}$$ We have $$\{\Delta\iota_n,\eta_{n-1}\bar{\eta}_n,\tilde{\eta}_{n+1}\}
\subset\pi_{n+4}(SO(n)).$$ In virtue of [@B-M], [@Bott] and [@H-M], $\pi_{n+4}(SO(n))\cong\Z_{8d}$, where $d=2$ or $1$ according as $n\equiv 2\ (\bmod\ 8)\ge 10$ or $n\equiv 6\ (\bmod\ 8)\ge 14$. Noting the relation $4\tilde{\eta}_{n+1}=0$, we obtain $$\begin{aligned}
4\{\Delta\iota_n,\eta_{n-1}\bar{\eta}_n,\tilde{\eta}_{n+1}\}
&=&-\Delta\iota_n\circ\{\eta_{n-1}\bar{\eta}_n,\tilde{\eta}_{n+1},
4\iota_{n+3}\}\\
&\subset&-\Delta\iota_n\circ\pi_{n+4}(\S^{n-1})=0.\end{aligned}$$ This induces $\Delta\varepsilon_n\in (2d)(\pi_{n+4}(SO(n))\circ\nu_{n+4})$. Since $4\pi_{n+7}(SO(n))=0$ by [@B-M], [@Bott] and [@H-M], we obtain the first assertion of (1).
Let $n\equiv 6\ (\bmod\ 8)\ge 14$. By the exact sequences $(\mathcal{SO}^{n+k}_{n+4})$ for $k=-2,-1$ and Lemma \[50\] we get that $i_n(\R)_\ast: \pi_{n+4}(SO(n-1))
\to\pi_{n+4}(SO(n))$ is an isomorphism and $\pi_{n+4}(SO(n-1))=\{[\nu^2_{n-2}]\}
\cong\Z_8$. This leads to the first assertion of (2).
We recall from [@M2] that $\sharp[\iota_n,\eta_n\sigma_{n+1}]=2$ if $n\equiv 2\ (\bmod\ 8)\ge 10$. So, by the first half, we obtain the second half of (1).
By (\[JDel\]) and (\[wsigma\]), $\Delta : \pi_{n+7}(\S^n)\to
\pi_{n+6}(SO(n))$ is a monomorphism for even $n\ge 10$. So, by $(\mathcal{SO}^n_{n+7})$, we have the exact sequence: $$\pi_{n+8}(\S^n){{\buildrel \Delta \over \longrightarrow}}\pi_{n+7}(SO(n)){{\buildrel i_* \over \longrightarrow}}\pi_{n+7}(SO(n+1))
{{\buildrel \over \longrightarrow}}0.$$ By [@B-M], [@Bott] and [@H-M], we know that $$\pi_{n+7}(SO(n+1))\cong\left
\{\begin{array}{ll}
(\Z_2)^2,&\ \mbox{if} \ n\equiv 6\ (\bmod\ 16)\ge 22;\\
\Z_2,&\ \mbox{if} \ n\equiv 14\ (\bmod\ 16)\\
\end{array}\right.$$ and by (\[7SO\]), $$\pi_{n+7}(SO(n))\cong\left \{\begin{array}{ll}
\Z_4\oplus\Z_2,& \ \mbox{if} \ n\equiv 6\ (\bmod\ 32)\ge 38;\\
(\Z_4)^2,& \ \mbox{if} \ n\equiv 22\ (\bmod\ 32);\\
\Z_4,& \ \mbox{if} \ n\equiv 14\ (\bmod\ 16).
\end{array}\right.$$ Hence, we obtain the second half of (2). This completes the proof. $\square$
Now, by Lemma \[eps2\].(1), $$[\iota_n,\varepsilon_n]=0 \ \mbox{and} \ [\iota_n,\bar{\nu}_n]
=[\iota_n,\eta_n\sigma_{n+1}]\ne 0, \ \mbox{if} \ n\equiv2\,(\bmod\ 8)\ge 10.$$ Whence, we conclude that $$G_{n+8}(\S^n)=\{\varepsilon_n\}\cong\Z_2, \
\mbox{if} \ n\equiv 2\ (\bmod\ 8)\ge 10.$$ Next, by (\[JDel\]) and Lemma \[eps2\].(2), we obtain $$G_{n+8}(\S^n)\ne 0, \ \mbox{if} \ n\equiv 6\,(\bmod\ 8)\ge 14 \
\mbox{unless} \ n\equiv 22\ (\bmod\ 32).$$ By [@Mimura], [@Oda2] and [@T], we obtain $G_{14}(\S^6;2)=\pi^6_{14}$ and $G_{n+8}(\S^n)=\{\eta_n\sigma_{n+1}\}\cong\Z_2$ if $n=14,22$. Since $[\iota_6,[\iota_6,\alpha_1(6)]]=0$ by Proposition \[Toda\], we obtain $G_{14}(\mathbb{S}^6;3)=\pi_{14}(\mathbb{S}^6;3)$. Thus, we have shown
\[8\] The group $G_{n+8}(\S^n)$ is equal to the following group: $0$ if $n\equiv 0,1\ (\bmod\ 4)\ge 4$ unless $n=8,9$; $\pi_{n+8}(\S^n)$ if $n=6$ or $n\equiv 3\ (\bmod\ 4)$; $\{\varepsilon_n\}\cong\Z_2, \ \mbox{if} \ n\equiv 2\ (\bmod\ 8)\ge 10$. Moreover, $G_{n+8}(\S^n)\ne 0$ if $n\equiv 6\ (\bmod\ 8)\ge 14$ unless $n\equiv 22\ (\bmod\ 32)$, $G_{16}(\S^8)=\{(E\sigma')\eta_{15},\sigma_8\eta_{15}
+\bar{\nu}_8+\varepsilon_8\}\cong(\Z_2)^2$, $G_{17}(\S^9)=\{[\iota_9,\iota_9]\}\cong\Z_2$ and $G_{n+8}(\S^n)=\{\eta_n\sigma_{n+1}\}\cong\Z_2$ if $n=14,22$.
Finally, we propose
$[\iota_n,\eta_n\sigma_{n+1}]=0$ and $G_{n+8}(\S^n)=\{\eta_n\sigma_{n+1}\}\cong\Z_2$, if $n\equiv 6\ (\bmod\ 8)\ge 14$.
Obviously, we obtain $G_{15}(\S^6)=\pi_{15}(\S^6)$ and $G_{19}(\S^{10})=\{3[\iota_{10},\iota_{10}],$ $
\nu^3_{10},\eta_{10}\varepsilon_{11}\}\cong 3\Z\oplus(\Z_2)^2$. Let $n\equiv 2\ (\bmod\ 4)\ge 14$. Then, by (\[deta\]), $$[\iota_n,\eta^2_n\sigma_{n+2}]=[\iota_n,\eta_n\varepsilon_{n+1}]=0.$$ By (\[myu\]), $[\iota_n,\mu_n]\ne 0$. Whence, we obtain $$G_{n+9}(\S^n)=\{\nu^3_n,\eta_n\varepsilon_{n+1}\}\cong(\Z_2)^2,
\; \mbox{if} \; n\equiv 2\ (\bmod\ 4)\ge 14.$$
Let now $n\equiv 3\ (\bmod\ 4)\ge 11$. Then, by Lemma \[pro\].(1) and (\[weta\]), $[\iota_n,\eta^2_n\sigma_{n+2}]
=[\iota_n,\eta_n\varepsilon_{n+1}]=0$ and by Example \[Deps\].(2), $[\iota_n,\mu_n]=0$. Whence, we obtain $$G_{n+9}(\S^n)=\pi_{n+9}(\S^n), \; \mbox{if} \; n\equiv 3\ (\bmod\ 4).$$
It is easily seen that $G_{13}(\S^4)=\{\nu^3_4\}\cong\Z_2$. Let $n\equiv 4\ (\bmod\ 8)\ge 12$. By Lemma \[pro\].(1) and (\[4\]), we have $[\iota_n,\nu^3_n]=0$. In the light of (\[myu\]) and (\[eta2s\]), $[\iota_n,\eta_n\varepsilon_{n+1}]=[\iota_n,\eta^2_n\sigma_{n+2}]\ne 0$ and $[\iota_n,\mu_n]\ne 0$. Assume that $P(\alpha_{2n+1}+\mu_{2n+1})=0$ for $\alpha_{2n+1}=\eta_{2n+1}\varepsilon_{2n+2}$ or $\eta^2_{2n+1}\sigma_{2n+3}$. By [@T Proposition 11.10.i)], there exists an element $\beta\in\pi^{n-1}_{2n+7}$ satisfying $E\beta=0$ and $H\beta=\eta_{2n-3}(\alpha_{2n-2}+\mu_{2n-2})=\eta_{2n-3}\mu_{2n-2}$. On the other hand, $(\mathcal{PE}^{n-1}_{2n+7})$ and Lemma \[HP\] imply a contradictory relation $H\beta=0$. So, $[\iota_n,\alpha_n]\ne [\iota_n,\mu_n]$ and hence $$G_{n+9}(\S^n)=\{\nu^3_n\}\cong\Z_2, \quad \mbox{if} \quad n\equiv 4\ (\bmod\ 8).$$
Obviously, we obtain $G_{18}(\S^9)=\{\sigma_9\eta^2_{16},\nu^3_9,\eta_9\varepsilon_{10}\}\cong(\Z_2)^3$. Let now $n\equiv 1\ (\bmod\ 8)\ge 17$. By (\[myu\]), $[\iota_n,\mu_n]\ne 0$ and by (\[eeps\]), $[\iota_n,\eta_n\varepsilon_{n+1}]=0$. In the light of IV, $[\iota_n,\eta^2_n\sigma_{n+2}]=0$ if $n=2^i-7$ for $i\ge 4$ and $[\iota_n, \nu^3_n]=[\iota_n,\eta^2_n\sigma_{n+2}]\ne 0$ if $n\equiv 1\ (\bmod\ 8)\ge 17$ and $n\ne 2^i-7$. By the parallel argument to the case $n\equiv 4\ (\bmod\ 8)$, we get $[\iota_n,\eta^2_n\sigma_{n+2}]\ne[\iota_n,\mu_n]$ [@T Proposition 11.10.ii)]. So, we obtain $$G_{n+9}(\S^n)=\left\{\begin{array}{ll}
\{\eta_n\varepsilon_{n+1}\}\cong\Z_2, \
\mbox{if} \ n\equiv 1\ (\bmod\ 8)\ge 17 \ \mbox{and} \ n\ne 2^i-7;\\
\{\eta_n\varepsilon_{n+1},\eta^2_n\sigma_{n+2}\}\cong(\Z_2)^2, \
\mbox{if} \ n=2^i-7 (i\ge 5).
\end{array}
\right.$$
Obviously, we obtain $G_{14}(\S^5)=\{\nu^3_5,\eta_5\varepsilon_6\}\cong(\Z_2)^2$. Let $n\equiv 5\ (\bmod\ 8)\ge 13$. By Corollary \[hard2\].(2) and (\[myu\]), $\nu^3_n\in G_{n+9}(\S^n)$ and $\mu_n\not\in G_{n+9}(\S^n)$. Furthermore, by Corollary \[hard2\].(2), $\eta_n\varepsilon_{n+1}\in G_{n+9}(\S^n)$ unless $n\equiv 53\ (\bmod\ 64)$. So, we obtain $$\begin{aligned}
G_{n+9}(\S^n)=\{\nu^3_n,\eta_n\varepsilon_{n+1}\}
\cong(\Z_2)^2, \ \mbox{if}\ n\equiv 5\ (\bmod\ 8)
\ \mbox{and} \ n\not\equiv \ 53 \ (\bmod\ 64).\end{aligned}$$
At the end, we use the following: $$\zeta_n\in\{2\iota_n,\eta_n,\alpha_{n+1}\}_2 \ (\bmod \ 2\zeta_n)\ \mbox{for} \
\alpha_{n+1}=\eta^2_{n+1}\sigma_{n+3}\ \mbox{or} \ \eta_{n+1}\varepsilon_{n+2}, \
\mbox{if} \ n\ge 11.$$ Let $n\equiv 0\ (\bmod\ 8)\ge 16$. By [@T Proposition 11.11.i)], there exists an element $\beta\in\pi^{n-2}_{2n+6}$ such that $[\iota_n,\alpha_n]=E^2\beta$ and $H\beta\in\
\{2\iota_{2n-5},\eta_{2n-5},$ $\alpha_{2n-4}\}_2
\ni\zeta_{2n-5}\ (\bmod\ 2\zeta_{2n-5})$. Assume that $[\iota_n,$ $\alpha_n]=0$. Then, $(\mathcal{PE}^{n-1}_{2n+7})$ and (\[des7\]) induce a relation $E(\beta-aE^6(\gamma\eta_{2n-10}\mu_{2n-9}))=0$ for $a\in\{0,1\}$. Hence, by $(\mathcal{PE}^{n-2}_{2n+6})$ and Lemma \[HP\], we have a contradictory relation $\zeta_{2n-5}\in 2\pi^{2n-5}_{2n+6}$. Whence, we get that $[\iota_n,\alpha_n]\ne 0$. In the light of (\[myu\]) and (\[emyu\]), we know $[\iota_n,\mu_n]\ne 0$ and $[\iota_n,\mu_n]\eta_{2n+8}\ne 0$. This implies that $[\iota_n,\alpha_n]\ne[\iota_n,\mu_n]$ and $[\iota_n,\nu^3_n]\ne[\iota_n,\mu_n]$. It is easy to show that $[\iota_8,\nu^3_8]=\eta_8\bar{\varepsilon}_9$ and $G_{17}(\S^8)=\{(E\sigma')\eta^2_{15},\sigma_8\eta^2_{15}+\nu^3_8+\eta_8\varepsilon_9\}
\cong(\Z_2)^2$. By [@M-M-O] and [@T], we obtain $G_{25}(\S^{16})=0$. By [@No2 4.14], there exists an element $\tau_1\in\pi^{n-6}_{2n+2}$ such that $$[\iota_n,\nu^3_n]=E^6\tau_1,\ H\tau_1=\eta_{2n-13}\kappa_{2n-12}, \
\mbox{if} \ n\equiv 0\ (\bmod\ 8)\ge 16.$$ Assume that $[\iota_n,\nu^3_n]=0$. Then, by $(\mathcal{PE}^{n-1}_{2n+7})$ and (\[des7\]), we have $E^5(\tau_1-aE^2(\gamma\eta_{2n-10}\mu_{2n-9}))=0$ for $a\in\{0,1\}$. So, by $(\mathcal{PE}^{n-2}_{2n+6})$, we have $E^4(\tau_1-aE^2(\gamma\eta_{2n-10}\mu_{2n-9}))\in P\pi^{2n-3}_{2n+8}
=\{[\iota_{n-2},\zeta_{n-2}]\}$. By applying $H : \pi^{n-2}_{2n+6}\to\pi^{2n-5}_{2n+6}$ to this relation and by (\[zeta6\]), $E^4(\tau_1-aE^2(\gamma\eta_{2n-10}\mu_{2n-9}))=0$. By the fact that $\pi^{2n-5}_{2n+7}=\pi^{2n-7}_{2n+6}=0$, we obtain $E^2(\tau_1-aE^2(\gamma\eta_{2n-10}\mu_{2n-9}))=0$. Hence, by $(\mathcal{PE}^{n-5}_{2n+3})$ and (\[des3\]), we have $$E(\tau_1-aE^2(\gamma\eta_{2n-10}\mu_{2n-9}))\in P\pi^{2n-9}_{2n+5}
=E^3M_t\circ\{\sigma^2_{2n-11},\kappa_{2n-11}\}$$ for $n=8t+16$. By $(\mathcal{PE}^{n-6}_{2n+2})$, we obtain $$\tau_1-E^2(a\gamma\eta_{2n-10}\mu_{2n-9}+bM_t\sigma^2_{2n-14}
+cM_t\kappa_{2n-14})\in P\pi^{2n-11}_{2n+4}$$ with $b,c\in\{0,1\}$. This induces a contradictory relation $\eta_{2n-13}\kappa_{2n-12}
\in 2\pi^{2n-13}_{2n+2}$. Thus, we conclude that $$[\iota_n,\nu^3_n]\ne 0,\ \mbox{if} \ n\equiv 0\ (\bmod\ 8)\ge 16.$$ Summing the above, we get
\[9\] The group $G_{n+9}(\S^n)$ is equal to the following group: $\pi_{n+9}(\S^n)$ if $n=6$ or $n\equiv 3\ (\bmod\ 4)$; $\{\nu^3_n,\eta_n\varepsilon_{n+1}\}\cong(\Z_2)^2$ if $n\equiv 2 \ (\bmod\ 4)\ge 14$, $n=2^i-7$ for $i\ge 5$ or $n\equiv 5 \ (\bmod\ 8)$ unless $n\equiv 53\ (\bmod\ 64)$; $\{\nu^3_n\}\cong\Z_2$ if $n\equiv 4\ (\bmod\ 8)$; $\{\eta_n\varepsilon_{n+1}\}\cong\Z_2$ if $n\equiv 1\ (\bmod\ 8)
\ge 17$ and $n\ne 2^i-7$; $0$ if $n\equiv 0\ (\bmod\ 8)\ge 16$. Moreover, $G_{17}(\S^8)=\{(E\sigma')\eta^2_{15},
\sigma_8\eta^2_{15}+\nu^3_8+\eta_8\varepsilon_9\}\cong(\Z_2)^2$, $G_{18}(\S^9)=\{\sigma_9\eta^2_{16},\nu^3_9,\eta_9\varepsilon_{10}\}
\cong(\Z_2)^3$ and $G_{19}(\S^{10})=\{3[\iota_{10},\iota_{10}],
\nu^3_{10},\eta_{10}\varepsilon_{11}\}\cong 3\Z\oplus(\Z_2)^2$.
By Lemma \[lead\], Corollary\[CC\].(3), Proposition \[Gnp0\] and (\[emyu\]), we have determined $G_{n+10}(\S^n)$ for $n\ge 12$.
It is easily seen that $$G_{n+10}(\S^n)= \left\{ \begin{array}{ll}
\{\nu_4\sigma'+E\varepsilon',2E\varepsilon',
\alpha_1(4)\alpha_2(7),&\\
\nu_4\alpha_2(7),\nu_4\alpha'_1(7)\}, & \mbox{if $n=4$};\\
\pi_{15}(\S^5), & \mbox{if $n=5$};\\
\pi^6_{16}\oplus\Z_3\{3\beta_1(6)\}, & \mbox{if $n=6$};\\
\{\sigma_8\nu_{15},\nu_8\sigma_{11},\sigma_8\alpha_1(15)\}, & \mbox{if $n=8$};\\
\{\sigma_9\nu_{16},\beta_1(9)\}, & \mbox{if $n=9$};\\
\pi^{10}_{20}=\{\sigma_{10}\nu_{17},\eta_{10}\mu_{11}\}, & \mbox{if $n=10$};\\
\pi_{21}(\S^{11}), & \mbox{if $n=11$}.\\
\end{array}
\right.$$ Thus, by summing up the above results, we get
\[ToMa2\] The group $G_{n+10}(\S^n)$ is isomorphic to one of the following groups: $\Z_{120}\oplus\Z_6$, $\Z_{72}\oplus\Z_2$, $\Z_{24}\oplus\Z_2$, $\Z_{24}\oplus\Z_8$, $\Z_{24}$, $\Z_4\oplus\Z_2$, $\Z_6\oplus\Z_2$ according as $n=4,5,6,8,9,10,11$. Furthermore, $G_{n+10}(\S^n)$ is isomorphic to the group: $0$ if $n\equiv 0 \ (\bmod\ 4)\ge 12$; $\Z_2$ if $n\equiv 2\ (\bmod\ 4)\ge 14$; $\Z_3$ if $n\equiv 1\ (\bmod\ 4)\ge 13$ and $\Z_6$ if $n\equiv 3\ (\bmod\ 4)\ge 15$.
We recall that $\pi_{n+11}(\S^n;3)=\{\alpha_3(n)\}\cong\Z_3$ for $n=3,4$ and that $\pi_{n+11}(\S^n;3)=\{\alpha'_3(n)\}\cong\Z_9$ for $n\ge 5$, where $3\alpha'_3(n)=\alpha_3(n)$ for $n\ge 5$. By [@Mimura], [@M-M-O], [@M-T], [@T] and (\[zeta6\]), $\sharp[\iota_n,\zeta_n]=1,4,8,1,4,1,8,1$ according as $n=5,6,8,9,10,11,12,13$. We easily obtain that $$G_{n+11}(\S^n)= \left\{ \begin{array}{ll}
\{\nu_4\sigma'\eta_{14},\nu_4\bar{\nu}_7,\nu_4\varepsilon_7,\\
2E\mu',\varepsilon_4\nu_{12},(E\nu')\varepsilon_7\}, & \mbox{if $n=4$};\\
\pi_{16}(\S^5), & \mbox{if $n=5$};\\
\{4\zeta_6,\bar{\nu}_6\nu_{14}\}, & \mbox{if $n=6$};\\
\{\bar{\nu}_8\nu_{16}\}, & \mbox{if $n=8$};\\
\pi_{20}(\S^9), & \mbox{if $n=9$};\\
4\pi_{21}^{10}, & \mbox{if $n=10$};\\
\pi_{22}(\S^{11}), & \mbox{if $n=11$};\\
\{3[\iota_{12},\iota_{12}]\}, & \mbox{if $n=12$}.\\
\end{array}
\right.$$
By abuse of notations, $\zeta_n$ for $n\ge 5$ represents a generator of the direct summands $\Z_8$ of $\pi^n_{n+11}$ and $\Z_{504}$ of $\pi_{n+11}(\S^n)$, respectively. By [@M2], [@Mimura], [@M-M-O], [@T], Corollary \[CC\].(3), Proposition \[Gnp0\] and (\[zeta6\]), we obtain $$\sharp[\iota_n,\zeta_n] =\left\{\begin{array}{ll}
1,&\ \mbox{if} \ n\equiv 1,5,7\ (\bmod\ 8)\ge 5;\\
252,& \ \mbox{if} \ n\equiv 2\ (\bmod\ 4)\ge 6;\\
504,&\ \mbox{if} \ n\equiv 0\ (\bmod\ 4)\ge 8.
\end{array}
\right.$$ Assume that $n\equiv 3\,(\bmod\,8)\ge 19$. Then, by [@B-M], [@Bott] and [@H-M], we obtain $$\pi_{n+10}(SO(n))\cong\left\{\begin{array}{ll}
(\Z_2)^3,&\ \mbox{if} \ n\equiv 3\ (\bmod\ 32)\ge 35;\\
(\Z_2)^2\oplus\Z_4,& \ \mbox{if} \ n\equiv 19\ (\bmod\ 64);\\
(\Z_2)^2\oplus\Z_8,& \ \mbox{if} \ n\equiv 51\ (\bmod\ 128);\\
(\Z_2)^2\oplus\Z_{16},& \ \mbox{if} \ n\equiv 115\ (\bmod\ 128);\\
(\Z_2)^2,&\ \mbox{if} \ n\equiv 11\ (\bmod\ 16)\ge 27
\end{array}
\right.$$ and $$\pi_{n+10}(SO(n+1))\cong\left\{\begin{array}{ll}
(\Z_2)^4\oplus\Z_3,&\ \mbox{if} \ n\equiv 3\ (\bmod\ 32)\ge 35;\\
(\Z_2)^3\oplus\Z_{12},& \ \mbox{if} \ n\equiv 19\ (\bmod\ 64);\\
(\Z_2)^3\oplus\Z_{24},& \ \mbox{if} \ n\equiv 51\ (\bmod\ 64);\\
(\Z_2)^3\oplus\Z_3,&\ \mbox{if} \ n\equiv 11\ (\bmod\ 16)\ge 27.
\end{array}
\right.$$ In the exact sequence $(\mathcal{SO}^n_{n+10})$, we get $p_{n+1}(\R)_\ast([\eta_n]\mu_{n+1})=\eta_n\mu_{n+1}$ and $p_{n+1}(\R)_\ast[\beta_1(n)]
=\beta_1(n)$. So, $p_{n+1}(\R)_\ast$ is a split epimorphism. Whence, by the group structures of $\pi_{n+10}(SO(n+k))$ for $k=0,1$, we obtain $$\Delta\zeta_n=0, \; \mbox{if} \; n\not\equiv 115\ (\bmod\ 128) \;
\mbox{and} \; \Delta\zeta_n\ne 0, \; \mbox{if} \; n\equiv
115\ (\bmod\ 128).$$ So, we have $
[\iota_n,\zeta_n]=0, \; \mbox{if}\; n\equiv 3\,(\bmod\ 8)\ge 19 \;\mbox{and}\;
n\not\equiv 115\ (\bmod\ 128)$. Consequently, by use of Lemma \[lead\] and [@T], the groups $G_{n+11}(\S^n)$ have been determined if $n\ge 13$ except $n\equiv 115\ (\bmod\ 128)$. Thus, by summing up the above results, we get
\[11\] The group $G_{n+11}(\S^n)$ is isomorphic to one of the following groups: $(\Z_2)^6$, $\Z_{504}\oplus(\Z_2)^2$, $\Z_2\oplus\Z_4$, $\Z_2$, $\Z_{504}\oplus\Z_2$, $\Z_2$, $\Z_{504}$, $3\Z$ according as $n=4,5,6,8,9,10,11,12$. Furthermore, $G_{n+11}(\S^n)$ is isomorphic to the group: $\Z_{504}$ if $n\equiv 1,5,7\
(\bmod\ 8)\ge 13$; $\Z_2$ if $n\equiv 2\ (\bmod\ 4)\ge 14$; $0$ if $n\equiv 0\ (\bmod\ 4)\ge 16$ and $\Z_{504}$ if $n\equiv 3\ (\bmod\ 8)\ge 19$ provided $n\not\equiv 115\ (\bmod\ 128)$.
We recall that $\zeta_n\in\{2\iota_n,\eta^3_n,\sigma_{n+3}\}$ for $n\ge 11$. So, by the fact that $2\Delta\iota_n=0$ for $n$ odd, we obtain $$\Delta\zeta_n=-\{\Delta\iota_n,2\iota_{n-1},\eta^3_{n-1}\}\circ\sigma_{n+3
}\ \mbox{for}\ n \ \mbox{odd}\ \mbox{and} \ n\ge 11.$$ We see that $2\{\Delta\iota_n,2\iota_{n-1},\eta^3_{n-1}\}=-\Delta\iota_n
\circ\{2\iota_{n-1},\eta^3_{n-1},2\iota_{n+2}\}=0$. Hence, by the fact that $\pi_{n+3}(SO(n))\cong\Z_{16}$ for $n\equiv 3\ (\bmod\ 8)\ge 11$, we obtain
[*$\Delta\zeta_n\in 8(\pi_{n+3}(SO(n))\circ\sigma_{n+3})$ if $n\equiv 3\ (\bmod\ 8)\ge 11$.*]{}
Finally, we recall $\pi_{22}(\S^{10})=\{[\iota_{10},\nu_{10}]\}\cong\Z_{12}$. By Proposition \[Toda\], $G_{22}(\S^{10})=\pi^{10}_{22}$. It is easily seen, in the light of [@T], Corollary \[CC\].(3) and Proposition \[Gnp0\], that $G_{n+12}(\S^n)=\pi_{n+12}(\S^n)$ unless $n=10$ and $$G_{n+13}(\S^n)= \left\{ \begin{array}{ll}
\{\nu^2_4\sigma_{10},\nu_4\eta_7\mu_8,(E\nu')\eta_7\mu_8,\\
(\alpha_1(4)+\nu_4)\beta_1(7)\}\cong\Z_{24}\oplus(\Z_2)^2, & \mbox{if $n=4$};\\
\{3[\iota_{14},\iota_{14}]\}\cong 3\Z, & \mbox{if $n=14$};\\
\pi^n_{n+13}, & \mbox{if $n$ is even}\\
& \mbox{unless $n=2,4,14$};\\
\pi_{n+13}(\S^n), & \mbox{if $n$ is odd}.
\end{array}
\right.$$ We close the paper with the table of $G_{n+k}(\S^n)$ for $1\le k\le 13$ and $2\le n\le 26$:
$G_{n+k}(\S^n)$ n=2 n=3 n=4 n=5 n=6 n=7 n=8
----------------- ------------ ------------ ------------- ------------- ----------- --------- -------------
k=1 $\infty$ $2$ $0$ $0$ $2$ $2$ $0$
k=2 $2$ $2$ $0$ $2$ $2$ $2$ $0$
k=3 $2$ $12$ $3\infty+2$ $24$ $2$ $24$ $0$
k=4 $12$ $2$ $(2)^2$ $2$ $0$ $0$ $0$
k=5 $2$ $2$ $(2)^2$ $2$ $3\infty$ $0$ $0$
k=6 $2$ $3$ $24+3$ $2$ $0$ $2$ $0$
k=7 $3$ $15$ $0$ $30$ $0$ $120$ $3\infty+2$
k=8 $15$ $2$ $0$ $0$ $24+2$ $(2)^3$ $(2)^2$
k=9 $2$ $(2)^2$ $2$ $(2)^2$ $(2)^3$ $(2)^4$ $(2)^2$
k=10 $(2)^2$ $12+2$ $120+6$ $72+2$ $24+2$ $24+2$ $24+8$
k=11 $12+2$ $84+(2)^2$ $(2)^6$ $504+(2)^2$ $4+2$ $504+2$ $2$
k=12 $84+(2)^2$ $(2)^2$ $(2)^6$ $(2)^3$ $240$ $0$ $0$
k=13 $(2)^2$ $6$ $24+(2)^2$ $6+2$ $2$ $6$ $(2)^2$
$G_{n+k}(\S^n)$ n=9 n=10 n=11 n=12 n=13 n=14 n=15 n=16 n=17
----------------- --------- ----------------- --------- ----------- --------- ----------- --------- ------ -------
k=1 $0$ $0$ $2$ $0$ $0$ $0$ $2$ $0$ $0$
k=2 $0$ $2$ $2$ $0$ $0$ $2$ $2$ $0$ $0$
k=3 $12$ $2$ $12$ $2$ $24$ $2$ $24$ $0$ $12$
k=4 $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$
k=5 $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$
k=6 $0$ $0$ $2$ $2$ $2$ $0$ $2$ $0$ $0$
k=7 $120$ $0$ $240$ $0$ $120$ $0$ $240$ $0$ $120$
k=8 $2$ $2$ $(2)^2$ $0$ $0$ $2$ $(2)^2$ $0$ $0$
k=9 $(2)^3$ $3\infty+(2)^2$ $(2)^3$ $2$ $(2)^2$ $(2)^2$ $(2)^3$ $0$ $2$
k=10 $24$ $4+2$ $6+2$ $0$ $3$ $2$ $6$ $0$ $3$
k=11 $504+2$ $2$ $504$ $3\infty$ $504$ $2$ $504$ $0$ $504$
k=12 $0$ $4$ $2$ $(2)^2$ $2$ $0$ $0$ $0$ $0$
k=13 $6$ $2$ $6+2$ $(2)^2$ $6$ $3\infty$ $3$ $0$ $3$
$G_{n+k}(\S^n)$ n=18 n=19 n=20 n=21 n=22 n=23 n=24 n=25 n=26
----------------- --------- --------- ------ --------- --------- --------- ------ ------- ---------
k=1 $0$ $2$ $0$ $0$ $0$ $2$ $0$ $0$ $0$
k=2 $2$ $2$ $0$ $0$ $2$ $2$ $0$ $0$ $2$
k=3 $2$ $12$ $0$ $12$ $2$ $24$ $0$ $12$ $2$
k=4 $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$
k=5 $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$
k=6 $0$ $0$ $2$ $2$ $0$ $2$ $0$ $0$ $0$
k=7 $0$ $120$ $0$ $120$ $0$ $120$ $0$ $120$ $0$
k=8 $2$ $(2)^2$ $0$ $0$ $2$ $(2)^2$ $0$ $0$ $2$
k=9 $(2)^2$ $(2)^3$ $2$ $(2)^2$ $(2)^2$ $(2)^3$ $0$ $2$ $(2)^2$
k=10 $2$ $6$ $0$ $3$ $2$ $6$ $0$ $3$ $2$
k=11 $2$ $504$ $0$ $504$ $2$ $504$ $0$ $504$ $2$
k=12 $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$
k=13 $0$ $3$ $0$ $3$ $0$ $3$ $0$ $3$ $0$
Like in [@T], an integer $n$ indicates the cyclic group $\Z_n$ of order $n$, the symbol $\infty$ an infinite cyclic group $\Z$, the symbol $+$ the direct sum of groups and $(2)^k$ indicates the direct sum of $k$-copies of $\Z_2$.
[99]{} J.F. Adams, [*Vector fields on spheres*]{}, Ann. of Math. (1962), 603-632. W.D. Barcus and M.G. Barratt, [*On the homotopy classification of the extensions of a fixed map*]{}, Trans. Amer. Math. Soc. (1958), 57-74. M.G. Barratt, [*Note on a formula due to Toda*]{}, J.London Math. Soc. [**36**]{} (1661), 95-96. M.G. Barratt and M.E. Mahowald, [*The metastable homotopy of $O(n)$*]{}, Bull. Amer. Math. Soc. [**70**]{} (1964), 758-760. R. Bott, [*The stable homotopy of the classical groups*]{}, Ann. of Math. [**70**]{} (1959), 313-337. M. Golasiński and D.L. Gonçalves, [*Postnikov towers and Gottlieb groups of orbit spaces*]{}, Pacific J. Math. [**197**]{}, no. 2 (2001), 291-300. D. Gottlieb, [*A certain subgroup of the fundamental group*]{}, Amer. J. of Math. [**87**]{} (1965), 840-856. ————— , [*Evaluation subgroups of homotopy groups,*]{} Amer. J. of Math. [**91**]{} (1969), 729-756. ————— , [*On fibre spaces and the evaluation map*]{}, Ann. of Math. [**87**]{} (1968), 42-56; correction: ibid. [**91**]{} (1970), 640-642. P.J. Hilton, [*A note on the $P$-homomorphism in homotopy groups of spheres*]{}, Proc. Camb. Phil. Soc. [**59**]{} (1955), 230-233. P.J. Hilton and J.H.C. Whitehead, [*Note on the Whitehead product*]{}, Ann. of Math. [**58**]{} (1953), 429-442. C.S. Hoo and M. Mahowald, [*Some homotopy groups of Stiefel manifolds*]{}, Bull. Amer. Math. Soc. [**71**]{} (1965), 661-667. H. Kachi and J. Mukai, [*Some homotopy groups of rotation groups $R_{n}$,*]{} Hiroshima J. Math. [**29**]{} (1999), 327-345. M. Kervaire, [*Some nonstable homotopy groups of Lie groups*]{}, Illinois J. Math. [**4**]{} (1960), 161-169. L. Kristensen and I. Madsen, [*Note on Whitehead products of spheres*]{}, Math. Scand. [**21**]{} (1967), 301-314. G.E. Lang, [*Localizations and evaluation subgroups,*]{} Proc. Amer. Math. Soc. [**5**]{} (1975), 489-494. A. Liulevicius, “The factorization of cyclic reduced powers by secondary cohomology operations", Mem. Amer. Math. Soc. [**42**]{} (1962). M. Mahowald, [*Some Whitehead products in $\S^n$*]{}, Topology [**4**]{} (1965), 17-26. ———– , “The metastable homotopy of $\S^n$", Mem. Amer. Math. Soc. [**72**]{} (1967). ———– , [*A new infinite family in $_2\pi^s_\ast$*]{}, Topology [**16**]{} (1977), 249-256. ———– , [*Private communication*]{}. M. Mimura, [*On the generalized Hopf homomorphism and the higher composition, Part I; Part II. $\pi_{n+i}(\S^n)$ for $i = 21$ and $22$*]{}, J. Math. Kyoto Univ. [**4**]{} (1964), 171-190; [**4**]{} (1965), 301-326. M. Mimura, M. Mori and N. Oda, [*Determination of $2$-components of the $23$ and $24$-stems in homotoy groups of spheres*]{}, Fac. Sci. Kyushu Univ. A [**29**]{} (1975), 1-42. M. Mimura and H. Toda, [*The $(n+20)$-th homotopy groups of $n$-sphere*]{}, J. Math. Kyoto Univ. [**3**]{} (1963), 37-58. Y. Nomura, [*Note on some Whitehead products*]{}, Proc. Japan Acad. [**50**]{} (1974), 48-52. ———– , [*On the desuspension of Whitehead products*]{}, J. London Math. Soc. (2) [**22**]{} (1980), 374-384. N. Oda, [*Hopf invariants in metastable homotoy groups of spheres*]{}, Mem. Fac. Sci. Kyushu Univ. A [**30**]{}(1976), 221-246. ———– , “Unstable homotopy groups of spheres", Bull. Inst. Advanced Research, Fukuoka Univ. no. 44 (1979), 49-152. J.-P. Serre, [*Groupes d’homotopie et classes groupes abéliens*]{}, Ann. of Math. [**58**]{} (1953), 258-294. S. Thomeier, [*Einige Ergebnisse über Homotopiegruppen von Sphären*]{}, Math. Ann. [**164**]{} (1966), 225-250. H. Toda, “Composition methods in homotopy groups of spheres", Ann. of Math. Studies [**49**]{}, Princeton (1962). ———– , [*Order of the identity class of a suspension space*]{}, Ann. of Math. [**78**]{} (1963), 300-325. ———– , [*On iterated suspensions, I; II*]{}, J. Math.Kyoto Univ. [**5**]{}(1965), 87-142; [**5**]{}(1966), 209-250. ———– , “Unstable $3$-primary homotopy groups of spheres", Study of Econoinformatics no. 29, Himeji Dokkyo Univ. (2003). ———– , [*Private communication*]{}. G.W. Whitehead, [*A generalization of the Hopf invariant*]{}, Ann. of Math. [**51**]{}(1950), 192-237. ———– , “Elements of homotopy theory", Springer-Verlag, Berlin (1978).
\
e-mail: marek@mat.uni.torun.pl
[Department of Mathematical Sciences\
Faculty of Science, Shinshu University\
Matsumoto 390-8621, Japan]{}\
e-mail: jmukai0@gipac.shinshu-u.ac.jp
[^1]: 2000 [*Mathematics Subject Classification*]{}.[Primary 55M35, 55Q52; Secondary 57S17]{}. : [EHP sequence, fibration, Gottlieb group, rotation group, Stiefel manifold, Whitehead product.]{}The first author was partially supported by Akio Sat$\bar{\rm o}$, Shinshu University 101501. The second author was partially supported by Grant-in-Aid for Scientific Research (No. 15540067 (c), (2)), Japan Society for the Promotion of Science and the Faculty of Mathematics and Computer Science in Toruń.
|
---
abstract: |
In this article, we review the prospects for the Fermi satellite (formerly known as GLAST) to detect gamma rays from dark matter annihilations in the Central Region of the Milky Way, in particular on the light of the recent astrophysical observations and discoveries of Imaging Atmospheric Cherenkov Telescopes. While the existence of significant backgrounds in this part of the sky limits Fermi’s discovery potential to some degree, this can be mitigated by exploiting the peculiar energy spectrum and angular distribution of the dark matter annihilation signal relative to those of astrophysical backgrounds.\
\
[*Keywords*]{}: Dark Matter, Gamma Rays, Galactic Center
address:
- 'Physics Division, Theory Group, CERN, CH-1211 Geneva 23, Switzerland'
- 'Theoretical Astrophysics, Fermi National Accelerator Laboratory, Batavia, USA'
- 'Department of Astronomy and Astrophysics, The University of Chicago, USA'
author:
- Pasquale Dario Serpico
- Dan Hooper
title: Gamma rays from Dark Matter Annihilation in the Central Region of the Galaxy
---
CERN-PH-TH/2009-025, FERMILAB-PUB-09-045-A
Introduction {#intro}
============
Despite the numerous cosmological and astrophysical indications of the presence of non-baryonic dark matter (DM), the particle nature of this substance remains unknown. If the DM consists of weakly interacting massive particles (WIMPs), an important tool for inferring their properties could be “indirect detection”, i.e. astrophysical observations of the annihilation (or decay) products of DM in our Galaxy or beyond. For WIMPs with masses at or around the electroweak scale, $m_X\sim\mathcal{O}$(0.1-1) TeV, the annihilation products are typically found at GeV-TeV energies, the domain of high-energy astrophysics. Of the different annihilation products, gamma rays and neutrinos have the important advantage of retaining directional information while not suffering energy losses. The very small cross sections of neutrinos, however, make their flux from the region of the Galactic Center very difficult to detect. On the contrary, the gamma-ray spectrum from DM annihilation or decay may be detectable with sufficient statistics, energy resolution, and over an extended angular distribution, to provide a very distinctive set of information related to both the particle identity of the WIMP and its astrophysical distribution. A major challenge is in separating the DM signal from any astrophysical backgrounds, whose energy spectrum and angular distribution are not well known.
A major change in the prospects for DM detection has occurred in the last few years, following the discovery of a bright astrophysical source of TeV gamma rays from the Galactic Center. As a result of this discovery, we now know that DM emission from the Galactic Center will not be detectable in a (quasi) background-free regime, and—unless one turns the attention to other targets—the peculiar spectral shape and angular distribution of the signal must be used to extract it from this and other backgrounds. This topic is the main subject of this review. The remainder of this paper is structured as follows: In Sec. \[instr\] we briefly describe the capabilities of current high energy gamma-ray telescopes. In Sec. \[signal\] we review the features of the DM annihilation signal, both in its energy (Sec. \[espectra\]) and angular dependence (Sec. \[angshape\]). In Sec. \[GC\] we review the prospects for detecting or constraining DM annihilations in the Galactic Center region, while in Sec. \[innerhalo\] we briefly discuss some interesting aspects of the diffuse signal from the inner halo. In Sec. \[conclusions\] we report our conclusions. Appendix A briefly illustrates the qualitative changes to the considerations we develop in the main text when decaying (as opposed to annihilating) DM is considered. Appendix B describes a subtlety regarding the angular shape of the signal arising for a velocity-dependent annihilation cross section.
The Instruments {#instr}
===============
The ability of gamma-ray experiments to identify DM annihilation radiation from the Galactic Center region relies on the effective area, angular and energy resolution of the existing telescopes, as well as the rejection of other (mostly hadronic) cosmic ray events contaminating the gamma-ray sample. Since the mean free path of gamma rays is much shorter than the atmospheric slant depth, direct observations in the GeV region and above can only be done from space—which is the strategy pursued by the LAT detector on the Fermi gamma-ray space telescope (formerly GLAST) [@GLASTurl]—or indirectly by ground-based Imaging Atmospheric Cerenkov Telescopes (IACTs) such as HESS [@HESSurl], MAGIC [@MAGICurl], VERITAS [@VERITASurl] and CANGAROO-III [@CANGAROOIIIurl]. In the latter category, the direction and energy of the primary particle hitting the atmosphere is reconstructed from the Cherenkov emission of the secondary charged particles generated in the atmospheric shower.
These differences lead the two classes of experiments to adopt different strategies in the search for DM. Fermi-LAT is very effective in rejecting hadronic events, and continuously monitors a large fraction of the sky, but has an effective area of only $\sim 1\,$m$^2$, far smaller than that of ground-based telescopes, $\sim 10^4\,$m$^2$. On the other hand, IACTs study small angular fields and have a lower rejection capability, but much greater overall exposure. As a consequence, diffuse gamma-ray signals are better probed by Fermi-LAT. Any unidentified sources detected by Fermi-LAT which lack a low-energy counterpart could be potentially attributed to DM substructure. IACTs would be very effective in providing detailed follow-up observations of such sources.
In addition, the accessible energy range is very different between these two classes of experiments: $\sim$100 MeV to 300 GeV for Fermi-LAT, and above $\sim$100 GeV for ACTs. This difference makes Fermi-LAT most sensitive to DM particles lighter than a few hundreds GeV, while IACTs are better suited for TeV-scale or heavier WIMPs. On the other hand, the Fermi-LAT has poorer angular resolution than IACTs, so it is less accurate in the localization of point-like sources. For both instrument classes, the search for indirect DM signatures is among the top physics priorities. The reach of Fermi-LAT and current and future IACTs has been recently assessed in Ref. [@Baltz:2008wd] and Ref. [@Buckley:2008zc], respectively.
The Dark Matter Signal {#signal}
======================
The differential flux of gamma rays (photons per unit area, time, energy and steradian) produced in DM annihilations[^1] is described by $$\Phi_{\gamma} (E_\gamma,\Omega)= \left[ \frac{\d N_{\gamma}}{\d E_{\gamma}} (E_\gamma)\frac{\sigv}{8\pi m^2_X} \right]\int_{\rm{los}} \rho^2(\ell,\Omega)\, \d \ell,
\label{flux1}$$ where $\sigv$ is the WIMP annihilation cross section multiplied by the relative velocity of the two WIMPs (averaged over the WIMP velocity distribution), $m_X$ is the mass of the WIMP, $\rho$ is the position-dependent DM density, and the integral is performed over the line-of-sight (los) in the direction of the sky, $\Omega$. The gamma-ray spectrum generated per WIMP annihilation is $\d N_{\gamma}/\d E_{\gamma}$, it has units of Energy$^{-1}$ and its integral over energy is equal to 1. If the DM is not its own antiparticle as assumed here, Eq. (\[flux1\]) should be multiplied further by a factor 1/2 (if $X$ and $\bar{X}$ are equally abundant). The factor in square brackets in Eq. (\[flux1\]) depends only on particle physics: in particular, cross section, mass, and the spectrum of gamma rays produced through DM annihilations depends on the nature of the WIMP. The integral over the line-of-sight determines instead the angular dependence of the signal and is controlled by the astrophysical distribution of DM. We shall discuss each of these two terms in the following, assuming that this factorization holds (see \[entanglingEandAng\] for a discussion of possible violations of this hypothesis). In convenient units, Eq. (\[flux1\]) can be recast as: $$\frac{\Phi_{\gamma}(E_{\gamma},\Omega)}{{\rm cm}^{-2} \, {\rm s}^{-1}\, {\rm sr}^{-1}} \approx 2.8 \times 10^{-10}\,J(\Omega)\, \frac{\d N_{\gamma}}{\d E_{\gamma}}(E_\gamma)\frac{\sigv}{\rm pb}\, \left(\frac{100\, \rm{GeV}}{m_{\rm{X}}}\right)^2.
\label{flux2}$$ The dimensionless function $J(\Omega)$ depends only on the DM distribution in the halo and is defined by convention as [@Bergstrom:1997fj] $$J(\Omega) = \frac{1}{8.5 \, \rm{kpc}} \bigg(\frac{1}{0.3 \, \rm{GeV}/\rm{cm}^3}\bigg)^2 \, \int_{\rm{los}} \rho^2(\ell,\Omega) {\rm d} \ell\,.
\label{jpsi}$$ For typical halo models (see Sec. 3.2) this is a function strongly peaked towards the Galactic Center (for an illustration, see for example Fig. A1).
The benchmark value for the cross-section, $\sigv \approx $1 picobarn (or in cgs units $\sim 3 \times 10^{-26} \,\rm{cm}^3/\rm{s}$), is motivated by the fact that a WIMP annihilating with such a cross section during the freeze-out epoch will be generated as a thermal relic with a density similar to the measured DM abundance (for a review, see Ref. [@Bertone:2004pz]). WIMPs constituting the [*cold*]{} DM annihilate in the non-relativistic limit. If annihilations take place largely through $S$-wave processes, then the annihilation cross section of WIMPs in the Galactic halo (i.e. in the low velocity limit) will also be approximately equal to this value, which justifies the benchmark value used in Eq. (\[flux2\]). Yet, it is important to stress that much lower signals are possible (e.g. if $P-$wave annihilation dominates the freeze-out process), as well as significantly enhanced ones in models where the DM is non-thermally produced in the early universe (see e.g. Ref. [@Moroi:1999zb]). Some further considerations and refs. can be found in \[entanglingEandAng\].
It is also worth noting that for S-wave annihilating thermal relics the indirect detection signal has two advantages compared to direct detection via nuclear recoils in underground detectors: i) it is proportional to a relatively large annihilation cross section; ii) it is less dependent from the particle physics details. For example, even under the (possibly unrealistic) assumption that DM annihilates mostly into quarks of the first generation, the natural expectation value for DM-nucleon elastic cross section is at the level of $(m_N/m_X)^2\times 1\,$pb$\approx 10^{-40}\,(m_X/0.1\,{\rm TeV})^2\,$cm$^2$. For DM annihilating predominantly into heavier particles, a further suppression is expected.
Energy Spectra {#espectra}
--------------
\[spectra\]
The gamma-ray spectrum from DM annihilation originates from several different contributions. Typically, the most abundant source of photons is the hadronization and/or decays of unstable particles. For example, neutralinos in the Minimal Supersymmetric Standard Model (MSSM), dominantly annihilate to final states consisting of heavy fermions $b \bar{b}$, $t \bar{t}$, $\tau^+ \tau^-$ (i.e. bottom quarks, top quarks and tau leptons, respectively) or bosons $ZZ$, $W^+ W^-$, $HA$, $hA$, $ZH$, $Zh$, $ZA$, $W^{\pm} H^{\pm}$, where $W^{\pm},Z$ are the gauge bosons mediating the weak interactions and $H$, $h$, $A$ and $H^{\pm}$ are the Higgs bosons of the MSSM [@jungman]. With the exception of the $\tau^{+} \tau^-$ channel, each of these annihilation modes result in a very similar spectrum of gamma rays, dominated eventually by the decay of mesons, especially $\pi^0$, generated in the cascade. In Fig. 1 we show the predicted gamma-ray spectrum per annihilation, for several possible WIMP annihilation modes. The harder spectrum here shown from annihilation into the $\tau^{+}\tau^{-}$ channel is not typical of SUSY models, although it might be a distinctive feature of DM annihilating dominantly to charged leptons pairs; this arises, for example, in Kaluza-Klein DM models (for a review of particle DM models, see e.g. [@Bergstrom:2009ib]).
Although these secondary photons provide the dominant emission, other important channels can exist at the one-loop level. A particularly striking signature would be the mono-energetic photons resulting from final states such as $\gamma \gamma$, $\gamma Z$ or $\gamma h$ (see Ref. [@lines] for a discussion of these processes within the context of supersymmetry). Unfortunately, such processes are expected to produce far fewer events than continuum emission and in typical models can not easily be detected (for a counterexample, see Ref. [@Gustafsson:2007pc]). Bremsstrahlung (with an additional photon appearing in the final state) is automatically present when annihilations produce charged final states, and can dominate the high energy region of the spectrum when those charged annihilation products are much lighter than the WIMPs. This is particularly important when the tree-level processes to a pair of light fermions are disfavored by “selections rules”, but no suppression is present for three body final states [@Birkedal:2005ep; @Bringmann:2007nk]. For more details, see the review by L. Bergstrom in this issue [@Bergstrom:2009ib]. Note that gamma-ray emission from DM annihilating into standard model particles is unavoidable, even in the most extreme case when only neutrino final states are allowed at tree level, due to $W$ and $Z$-strahlung [@Kachelriess:2007aj; @Bell:2008ey].
Angular Shape Of The Signal {#angshape}
---------------------------
The gamma-ray spectrum and angular distribution predicted by Eq. (\[flux1\]) is rather general (see, however, Appendix B), and could be applied to the case of a smoothly distributed Galactic halo, or alternative targets such as dwarf spheriodal galaxies, microhalos/clumps, density spikes around intermediate mass black holes, or the integrated extragalactic diffuse background (described in other contributions to this special issue). While some of these targets have interesting observational prospects, the intensity of these signals is strongly dependent on often unknown cosmological and astrophysical properties, such as the quantity of small-scale structures in DM halos or the population of intermediate mass black holes in the Milky Way. In this article, we will focus on the smooth Galactic halo.
Naively, by inputing the DM density profile inferred from kinematical observations of the Milky Way into Eq. (\[flux1\]), one could obtain an approximate lower bound on the gamma-ray flux from DM annihilation. In principle, this would enable us to translate the observations from ground or space-based gamma-ray telescopes into constraints on the particle physics properties of the WIMP (mass, annihilation cross section and dominant modes). Unfortunately, even the average, smooth distribution of the DM particles in our Galaxy is not well known, especially in the volume within the solar circle. This is due to the fact that the baryonic material dominates the gravitational potential in the inner Galaxy, and lacking a detailed knowledge of its distribution, a reconstruction of the DM distribution “by subtraction” is unfeasible. It is not surprising, then, that very different profiles have been claimed to fit the observations (for example, compare the results of Ref. [@Binney:2001wu] and Ref. [@Klypin:2001xu]).
On general grounds, a class of spherically symmetric, smooth halo distributions can be used to approximately fit both the observed rotation curves of galaxies and the results of numerical simulations of DM halos. A non-trivial angular dependence of the gamma-ray signal results from the off-center position of the Sun within the halo (see e.g. Ref. [@Hooper:2007be] and references therein for a discussion of sub-leading effects determining the angular distribution of the signal). The function $J$ introduced in Eq. (\[flux2\]) and Eq. (\[jpsi\]) then depends only on the angle $\theta$ between the observed direction of the sky and the Galactic Center or, in terms of galactic latitude $b$ and longitude $l$, only on $\cos\theta=\cos b\cos l$. The radial variable, $r$, can be expressed in terms of the relevant quantities $\{\ell,\theta\}$ as $$r(\ell,\theta)=\sqrt{r_\odot^2+\ell^2-2\,r_\odot\,\ell\cos\theta}\,, \label{rspsi}$$ where $r_\odot\approx 8.33\pm 0.35\,$kpc [@Gillessen:2008qv] is the distance of the Solar System from the Galactic Center. Typically, one considers a distribution of the form $$\rho(r)=\left(\frac{r_s}{r}\right)^\gamma
\frac{\rho_0}{[1+(r/r_s)^\alpha]^{(\beta-\gamma)/\alpha}},\label{prof}$$ where $\rho_0$ is a normalization constant and $r_s$ is a characteristic radius below which the profile scales as $r^{-\gamma}$. A very well known, universal profile of this class fit to DM-only ([*i.e.*]{} neglecting baryons) N-body simulations has been proposed by Navarro, Frenk and White (NFW) [@NFW], corresponding to the choice $\{\alpha,\beta,\gamma\}=\{1.0,3,1.0\}$. Steeper or softer profiles have also been extensively discussed, such as that proposed by Moore et al. [@Moore] and Kravtsov et al. [@Kravtsov:1997dp], respectively. While simulations (and data as well) typically agree on the shape of profile in the outskirts of the halos, a disagreement clearly exists concerning the inner slope, $\gamma$. More recent simulations [@Power:2002sw; @Navarro:2003ew; @Reed:2003hp; @Merritt:2005xc; @Navarro:2008kc] suggest that halo density profiles are better represented by a function with a continuously-varying slope, as the one proposed by Einasto [@Einasto] $$\rho(r)=\rho_{-2}\,e^{-\frac{2}{\alpha}\left[\left(\frac{r}{r_{-2}}\right)^\alpha-1\right]}\,,\label{einasto}$$ with clear hints for non-universality in the form of halo-to-halo variations in the quantity $\alpha$. The quantity $r_{-2}\simeq 25\,$kpc denotes the radius at which the logarithmic slope of the profile, $\d \log\rho/\d \log r$, assumes the value $-2$; the other free parameters are $\alpha$ and the overall normalization, here chosen as the density $\rho_{-2}\equiv \rho(r_{-2})$. At $r \gsim 1$ kpc, these newly proposed fitting formulae provide only marginal improvement with respect to the more traditional ones. At smaller radii, however, these recent results lead us to expect the inner slope of DM halos to be shallower that that predicted by Moore [*et al.*]{}, and probably shallower than NFW, as well.
\[TableII\]
-------------------- ---------- --------- ---------- ---------------------- ---------
Model $\alpha$ $\beta$ $\gamma$ $\rho_\odot$ $r_s$
\[GeV$\,$cm$^{-3}$\] \[kpc\]
Moore [*et al.*]{} 1.5 3 1.5 0.27 28
NFW 1.0 3 1.0 0.30 20
Kravtsov 2.0 3 0.4 0.37 10
-------------------- ---------- --------- ---------- ---------------------- ---------
: Parameters describing some common halo profiles of the form described by Eq. (\[prof\]), where $\rho_\odot$ is the DM density at the Solar distance from the GC. See text for more details.
There are a number of astrophysical processes that may potentially modify the DM distribution, none of which are taken into account in the above-mentioned profiles. It is very difficult to reliably account for these effects in simulations, and only a few results are available (see e.g. Refs. [@Gnedin04; @Gustafsson:2006gr; @Read:2009iv]). Qualitatively, since baryons can cool and contract, one expects them to steepen the gravitational potential in the central regions galaxies and, as a result, enhance the DM density [@ac]. On the other hand, other feedback or frictional effects have been proposed that could reduce the DM density in the inner halos and bring the prediction closer to observations (for a recent review, see Ref. [@Sellwood:2008bd]). For some galaxies, it has actually been argued that flat-cored profiles fit the observations better than cusped profiles.
Even greater uncertainties exist concerning the DM distribution at the very center of the Milky Way, in the region immediately surrounding the central supermassive black hole. Adiabatic accretion may lead to the formation of a spike in the DM distribution, resulting in a very high DM annihilation rate in the innermost parsecs of the galaxy [@spike]. Mergers as well as scattering on the dense stellar cusp around the central black hole may potentially destroy density enhancements, however. In general, these effects only affect regions too close to the Galactic Center to be resolved angularly by present detectors, leaving only the energy spectrum to be used for separating the DM signal from the background. Yet, such a spike might lead to a measurable gamma-ray flux from the innermost angular bin, even in presence of relatively large astrophysical backgrounds. For more details, we direct the reader to the references cited in Ref. [@Bertone:2004pz] or Ref. [@Fornasa:2007nr].
Given these considerably uncertainties, we have chosen to use three illustrative choices for the DM halo profiles: the Moore [*et al.*]{}, NFW, and Kravtsov profiles (with the parameters given in Table I). In each case, as in Ref. [@Yuksel:2007ac], the normalization has been chosen so that the mass contained within the solar circle provides the appropriate DM contribution to the local rotational curves. In all of the cases we have discussed, even for the most conservative cored profile, the DM signal peaks at the Galactic Center. In presence of an isotropic (or even vanishing) astrophysical background, the Galactic Center region thus becomes the natural location to look for a DM signal. Unfortunately, there is also a higher background density as we look toward the inner region of our Galaxy. Still, as a first step, one can consider the detection prospects for this region, a task we address in Sec. \[GC\], before turning to more general arguments regarding the optimal regions of the sky for DM searches in Sec. \[innerhalo\].
The Galactic Center {#GC}
===================
The Galactic Center has long been considered to be among the most promising targets for the detection of DM annihilation, particularly if the halo profile of the Milky Way is cusped in its inner volume [@Bergstrom:1997fj; @gchist]. This has been complicated, however, by the recent discovery of astrophysical gamma-ray sources from the Galactic Center. Following an earlier claim by the WHIPPLE IACT [@whipple], very high-energy gamma rays from the Galactic Center have been detected by HESS [@hess], MAGIC [@magic], and CANGAROO-II [@cangaroo]. This source is consistent with point-like emission and is located at $l = 359^\circ 56^\prime 41.1^{\prime\prime}\pm 6.4^{\prime\prime}$ (stat), $b = -0^\circ2^{\prime}39.2^{\prime\prime}\pm 5.9^{\prime\prime}$ (stat) with a systematic pointing error of 28$^{\prime\prime}$ [@van; @Eldik:2007yi], coincident with the position of Sgr A$^{\star}$, the black hole constituting the dynamical center of the Milky Way. The spectrum of this source is well described by a power-law with a spectral index of $\alpha=2.25 \pm 0.04 (\rm{stat}) \pm 0.10 (\rm{syst})$ over the range of approximately 160 GeV to 20 TeV. Although speculations were initially made that this source could be the product of annihilations of very heavy ($\sim$10 to 50 TeV) DM particles [@actdark], this interpretation is disfavored by the power-law form of the observed spectrum and the wide energy range over which it extends (see Fig. \[fig:edis\]).
![\[fig:edis\] Spectral energy density of the Galactic Center source as measured by HESS in 2004 (full points) and 2003 [@hess] (open points). Upper limits are 95% CL. The shaded area shows the power-law fit $\mathrm{d}N/\mathrm{d}E \sim E^{-\Gamma}$, with $\Gamma=2.25\pm0.04\,$(stat.)$\pm 0.10\,$(syst.). The dashed line illustrates typical spectra of phenomenological MSSM DM annihilation for best fit neutralino masses of 14 TeV. The dotted line shows the distribution predicted for Kaluza-Klein DM with a mass of 5 TeV. The solid line gives the spectrum of a 10 TeV DM particle annihilating into $\tau^+\tau^-$ (30%) and $b\bar{b}$ (70%). From Ref. [@Aharonian:2006wh].](fig2.eps){width=".6\textwidth"}
The source of these gamma rays is, instead, likely an astrophysical accelerator associated with our Galaxy’s central supermassive black hole [@hessastro]. In recent analyses, this source has been treated as a background for DM searches [@Zaharijas:2006qb; @Hooper:2007gi]. Given the presence of this background, the prospects for detecting DM annihilation products from the Galactic Center appear considerably less promising than they had a few years ago.
The main challenge involved in DM searches with Fermi will be to distinguish the signal from this and other backgrounds. This is made particularly difficult by our ignorance regarding the nature of these backgrounds. The Galactic Center is indeed a complex region of the sky at all wavelengths, the gamma-ray window being no exception [@vanEldik:2008ye].
An attempt to use angular and spectral information to separate DM annihilation products from these backgrounds was performed in Ref. [@Dodelson:2007gd], studying a $2^\circ\times 2^\circ$ region arond the Galactic Center (see also the analysis of the Fermi DM team in Ref. [@Baltz:2008wd]). This study considered two known point-source backgrounds: a yet unidentified source detected by EGRET approximately 0.2$^{\circ}$ away from the dynamical center of our galaxy [@dingus; @pohl], and the IACT source discussed above. In addtion, a diffuse spectrum with a free power-law index was also included. More in detail, we describe the spectrum of the source revealed by IACTs at the Galactic Center as a power-law given by: $$\Phi^{\rm ACT} = 1.0 \times 10^{-8} \left({ {E_{\gamma}}\over{{\rm GeV}}}\right)^{-2.25} {\rm GeV}^{-1} \, {\rm cm}^{-2} \, {\rm s}^{-1}.$$ The flux from the EGRET source slightly off-set is instead modeled as: $$\Phi^{\rm EG} = 2.2 \times 10^{-7} \,\left({ {E_{\gamma}}\over{{\rm GeV}}}\right)^{-2.2} e^{-\frac{E_{\gamma}}{30 \, {\rm GeV}}}\, {\rm GeV}^{-1} \, {\rm cm}^{-2} \, {\rm s}^{-1} \,,$$ where the exact value of the cutoff energy (here 30 GeV) is somewhat arbitrary, but reflects the fact that this source has not been observed yet by IACTs at $E\gsim 100\,$GeV. Finally, we allow for a diffuse/unresolved flux with spectrum $$\Phi^{\rm diff}(A,\alpha) = A \left({ {E_{\gamma}}\over{{\rm GeV}}}\right)^{-\alpha} {\rm GeV}^{-1} \, {\rm cm}^{-2} \, {\rm s}^{-1} \, {\rm sr}^{-1}\,,$$ where $\alpha$ is allowed to vary between 1.5 and 3.0. We adopt an overall normalization, $A$, such that the integrated flux of the diffuse background between 1 GeV and 300 GeV in a $2^\circ\times 2^\circ$ field of view around the Galactic Center is equal to $10^{-4} \, {\rm cm}^{-2} \, {\rm s}^{-1} \, {\rm sr}^{-1}$. We do not, however, assume that this normalization is known in our analysis, leaving open the possibility that some of the diffuse gamma rays observed are the product of dark matter annihilations.
A multi-parameter $\chi^2$ analysis of the simulated sky against models including a contribution from DM annihilation radiation yields the projected exclusion limits at the 95% confidence level shown in Fig. \[limitnfw\] (for ten years of collection time by Fermi-LAT). The simulated sky used to produce this figure contains the two resolved sources and the diffuse background but no dark matter, and to derive the exclusion limits we compare the signal thus obtained to a model which includes both the backgrounds and a signal from dark matter. The left and right panels refer to the cases of an NFW profile and Moore [*et al.*]{} profile (replaced by a flat core within $10^{-2}$ pc of the Galactic Center to avoid a divergency), respectively. The WIMP is always assumed to annihilate dominantly to $b \bar{b}$. This assumption is not particularly restrictive, since for many annihilation modes the spectrum would look very similar (see Fig. 1). Apart for some cases, as possibly Kaluza Klein DM, the gamma-ray spectrum alone does not help much in discriminating among several DM candidates, see e.g. [@Hooper:2006xe] for details.
To give a feeling for the numbers involved, the integral of the function $J$ over the considered region leads to a factor $\sim 1.3$ for the NFW model; in this model, for the benchmark 100 GeV WIMP with $\sigv \sim 3 \times 10^{-26}$ cm$^3$/s one would expect less than 3000 events from DM above 1 GeV, to be compared with more than 2.3$\times 10^5$ background photons. The background is almost evenly split between the EGRET point source and the diffuse flux; the IACT source contributes less than 7000 events, but it is located at the GC and, differently from the EGRET source, is not cut-off at high energy, so it is important to include it especially for high WIMP masses. The solid line in each frame represents the limit found if the diffuse background is assumed to be distributed isotropically, while the dashed line represents the conservative limit obtained if the diffuse background has the same angular distribution as the DM signal ([*i.e.*]{} the case in which angular information is not useful in disentangling the signal from the diffuse background). For values of $\sigv$ below the corresponding lines, a pure background model is expected to be consistent with the data. The fact that the limits are significantly stronger in the uniform background case is the manifestation of the improved sensitivity which can be achieved by an analysis including both energy and angular information. For comparison, in Fig. \[limitnfw\] it is also shown the region already excluded by EGRET [@dingus] (above the dotted line) and the mass and cross section of neutralino models found in a random scan over supersymmetric parameters, as calculated using DarkSUSY [@darksusy]. As expected, many of the models cluster around $\sigv \sim 3 \times 10^{-26}$ cm$^3$/s, the value required of a thermal relic annihilating via an $S$-wave amplitude. Each point shown represents a model which respects all direct collider constraint and generates a thermal DM abundance consistent with the observed DM density. In the scan the SUSY parameters varied were $M_2$, $|\mu|$ and $m_{\tilde{q}}$ up to 2 TeV, $m_A$ and $m_{\tilde{l}}$ up to 1 TeV and $\tan \beta$ up to 60. Also, the gaugino masses were assumed to evolve to a single unified scale, such that $M_1 \approx 0.5 M_2$, $M_3 \approx 2.7 M_2$.
Should gamma rays be identified as having been produced in DM annihilations, such observations could then be used to measure the characteristics of the DM particle, including its mass, annihilation cross section and spatial distribution. Such determinations are an important step toward identifying the particle nature of DM. A calculation similar to the one leading to the results of Fig. \[limitnfw\] can be performed, this time including in the template a contribution from a fiducial model of DM annihilation, and asking the accuracy by which a reconstruction of its input properties is possible. In Fig. \[ellipsenfw\], the ability of Fermi-LAT to determine the WIMP mass and annihilation cross section for WIMPs distributed with an NFW halo profile is illustrated, for the case of an annihilation cross section of $3 \times 10^{-26} \,\rm{cm}^3/\rm{s}$ and a mass of 100 GeV. In each frame the projected 2 and 3$\,\sigma$ constraints on the input parameters are reported, assuming an isotropic diffuse background (in addition to background point sources). In the left frame, we treat the shape of the halo profile (NFW) as if it is known in advance. Of course, this is not a realistic assumption, and a less accurate determination of the WIMP mass must be expected in a more realistic treatment. In the right panel of Fig. \[ellipsenfw\] we report the results obtained by marginalizing over the inner slope of the halo profile, $\gamma$. In the absence of an assumption on the inner halo slope, the constraint on the DM mass worsens by a factor $\sim $2.
If the spectrum and angular distribution of gamma rays from DM annihilations in the Galactic Center region are sufficiently well measured, it will also be possible to measure the underlying DM distribution. Fig. \[ellipseslope\] shows the results from the right frame of Fig. \[ellipsenfw\] in the $\{m_X,\,\gamma\}$ plane, marginalized over the annihilation cross section. In the above benchmark model, the inner slope of the halo profile can be determined at approximately the $\sim 10\%$ level.
A few caveats are in order: We are possibly oversimplifying the spectral shape of the background, since we are extrapolating the known point-like source properties from lower and higher energies. Also, the bounds are slightly optimistic, in the sense that we are considering the longest plausible lifetime of Fermi (ten years) and that further effects degrading the angular resolution or accounting for dead-time may loosen the constraints by up to a factor$\sim\mathcal{O}$(2). Finally, we have included only the HESS and EGRET sources (in addition to the diffuse background) in our analysis, but it is likely that other astrophysical point sources with different spectra will be discovered by Fermi-LAT. The above estimates should be quite realistic as long as a limited number of discrete sources will “contaminate” the inner Galactic Center angular bins, so that they could be removed effectively by the Fermi team. On the other hand, it is worth taking a lesson from the “sudden” discovery of an astrophysical source just at the Galactic Center: In the worst case, the Galactic center region might so crowded with (yet unknown) sources that the separation of background and signal might be degraded with respect to expectations based on present knowledge. Still, it is worth mentioning that the bulk of the statistical significance of the dark matter annihilation signal typically does not come from the inner $0.1^{\circ}$ around the Galactic Center (where the IACT source dominates), rather from the surrounding angular region. It is then interesting to see what are the perspectives to detect a DM signal from a more extended region, which we address in the following.
The Inner Halo {#innerhalo}
==============
The emission of radiation per unit solid angle from DM annihilation is expected to be maximized at the Galactic Center. Yet, geometric factors and the presence of point-like and diffuse backgrounds make the choice of the optimal window size a non-trivial problem [@Baltz:2008wd; @Buckley:2008zc; @Bergstrom:1997fj; @Stoehr:2003hf; @Evans:2003sc; @Serpico:2008ga]. Unless the DM halo is very cuspy towards the Galactic Center (say, cuspier than NFW), the optimal strategy for DM searches is never to focus on the inner sub-degree around the Galactic Center. Rather, a window size up to $\sim 50^\circ$ or more is preferred. The optimal shape of the window depends on the angular distribution of the signal and backgrounds, but also on the details of the analysis (like energy cuts, astrophysical foreground removal, etc.). It could be optimized from morphological studies of the low-energy emission measured by Fermi, but a circular annulus around the Galactic Center or a “rectangular” window in Galactic coordinates—with an inner rectangular mask— generally work fairly well. Here we illustrate this issue within a simplified model: Based on EGRET data, in the diffuse background one can identify an isotropic, extragalactic component $\propto E^{-2.1}$ and a Galactic component scaling as $\sim E^{-2.7}$, which is however dominant at GeV energies and peaks towards the Galactic Plane (see [@Bergstrom:1997fj; @Serpico:2008ga] for details). The energy spectrum of the DM signal thus mostly enters the game in determining which one of the two spectra (of different angular shape) dominates the background: the harder the DM spectrum, the closer the background is to an isotropic emission, the smaller the optimal angular window. Still, even in the quite extreme[^2] case of $70\,$GeV monochromatic lines, the optimal angular window is very extended. This is illustrated in Fig. \[Fig2\] , where we plot the relative signal-to-noise ($S/N$) as a function of the maximum galactic latitude, $b_{\rm max}$, for a Galactic region $0.4^\circ<|b|<b_{\rm max}$, $0^\circ<|l|<l_{\rm max}=b_{\rm max}$. The solid lines refer to an hypothetical gamma-line emission for the benchmark “model I” in the inert Higgs doublet scenario (see Ref. [@Gustafsson:2007pc] for details) for three different halo profiles. The dot-dashed line refers to the continuum spectrum of a 100 GeV neutralino annihilating into $W^{+}W^{-}$ with an NFW distribution, where indeed we see that the optimal window is even more extended than for the line signal: this is due to the fact that the background at the typical energies of the continuum DM photons, $ E \lsim m_X/10$, is mostly dominated by the Galactic unresolved background, which peaks towards the Galactic Plane/Center.
To give a feeling of the quantitative difference with respect to the GC case treated in the previous section, let us note that for our usual benchmark model ($m_X=100\,$GeV, $\langle\sigma v\rangle=1\,$pb with $b\bar{b}$ final state), a “naive” count statistics above 1 GeV (including all backgrounds) would lead to a $S/N~\sim 6$ for the $2^\circ\times 2^\circ$ window considered in Sec. 4, while for the region considered above and $b_{\rm max}=l_{\rm max}=25^\circ$ one would find almost 40000 events from DM vs. about seven millions background events, resulting in a $S/N>14$. Of course, this analysis does not take into account angular and energy cuts that (as shown for the GC case above) do improve the diagnostic power, as well as other possible systematics. Yet, the importance of this diffuse signal cannot be underestimated (and it is actually confirmed by other analyses, see Sec. 4.2 in Ref. \[6\]), especially in the case no signal is revealed from the GC, which may be due to a inner halo profile more cored than NFW. The extended signal has indeed a milder dependence on the profile. Also, from Fig. \[Fig2\] it is clear that—at least for the simple $S/N$ estimator—it is only important to adopt the right angular cut within a factor of two or so in order not to degrade the sensitivity by more than$\sim 20\%$. But “blindly” focusing on a too narrow window (degree scale) around the GC might be overly penalizing; especially for relatively cored profiles, the loss in sensitivity may reach a factor of two or more.
Besides improving the prospects for detection, other advantages in focusing on an extended region around the Galactic Center include:
- enabling an empirical determination of the dark matter profile slope outside of the region dominated by the gravitational potential of the central supermassive black hole.
- obtaining independent evidence that an “excess” signal with respect to backgrounds is the product of DM annihilation, by comparing the emission from many angular regions. In contrast, if the spectrum is found to vary with location, it is most likely the product of astrophysical backgrounds.
- constraining the quantity of dark matter substructure in the halo, by observing the angular distribution of the emission. The shape of the smooth, unresolved inner profile [@Hooper:2007be] could be studied, in addition to an anisotropy/multipole analysis [@SiegalGaskins:2008ge; @Lee:2008fm; @Fornasa:2009qh]. This, in turn, may have important implications for DM cosmology; see the review by T. Bringmann in this issue.
Conclusions
===========
The most challenging task for indirect dark matter searches is not to detect a few events, but to confidently identify those events as the products of dark matter annihilations. In particular, any signal must be separated from astrophysical backgrounds if it is to be reliably claimed to be a detection of dark matter. This is certainly true in the case of gamma-ray telescopes hoping to observe dark matter annihilations in the region of the Galactic Center (GC). The discovery of a bright TeV source of astrophysical origin at the GC by IACTs has changed the prospects for such searches considerably. One strategy in light of this is to focus on different targets with lesser astrophysical background (such as dwarf galaxies). Another is to search for a signal in the “noisy” GC region, by taking advantage of the peculiar spectral and angular properties expected from dark matter annihilation products.
In this paper, we have reviewed the latter strategy, reporting on recent studies of the GC and inner Galaxy. Given the characteristics of this search, the LAT instrument on board of the Fermi satellite is better suited than existing IACTs. Fermi-LAT instrument will detect a number of astrophysical sources in the region of the sky around the GC, including the point sources previously identified by HESS and EGRET, and perhaps others. A diffuse gamma-ray background will also likely be present. Although predictions of Fermi’s sensitivity are unavoidably limited by our incomplete knowledge of these backgrounds, we have shown that the spectral and angular differences between the signal and backgrounds should be distinctive enough to allow one to separate signal from background over a significant region of the parameter space, at least for a sufficiently cusped dark matter profile (NFW-like or steeper).
In the optimistic case where dark matter annihilation products are identified by Fermi, then it may also be possible to measure or constrain the properties of the dark matter, including its mass, annihilation cross section, and spatial distribution. It is unlikely that Fermi will determine the WIMP’s mass with high precision, however. For example, for the case of a 100 GeV WIMP with an annihilation cross section of $3 \times 10^{-26}$ cm$^3$/s, distributed with an NFW halo profile, the mass could be determined to lie within approximately 50-300 GeV. In the same benchmark model, the inner slope of the dark matter halo profile could be determined to $\sim 10\%$ precision. The combination of several indirect detection channels will be crucial to both confirm such a detection, and to best constrain the WIMP’s properties. On the other hand, it is not excluded that Fermi will lead to a radical revision of the present gamma-ray picture of the GC, revealing a more complicated zoo of astrophysical accelerators than envisaged in the present estimates. In the case where either the DM signal from the GC is too low or the the background is too large/complex, a DM discovery in gamma rays is still possible by looking at the emission from an extended region in the inner halo with Fermi, or from other dark matter substructures with both Fermi-LAT and IACTs. In particular, the morphology and the spectral properties of the unresolved Galactic background at $E\lsim\,$GeV will be useful to optimize the angular and energy-cut templates for searches of the DM emission from an annulus of several tens of degrees around the GC.
Acknowledgments {#acknowledgments .unnumbered}
===============
PS would like to thank the Galileo Galilei Institute for Theoretical Physics for the hospitality and the INFN for partial support during the completion of this work. DH is supported in part by the Fermi Research Alliance, LLC under Contract No. DE-AC02-07CH11359 with the US Department of Energy and by NASA grant NNX08AH34G. Finally, we thank Sergio Palomares Ruiz for noticing a typo in the v2 of the manuscript.
The Case of Decaying Dark Matter {#DDM}
================================
In the case of decaying dark matter, Eq. (\[flux1\]) is modified to $$\Phi_\gamma= \frac{\d N_{\gamma}}{\d E_{\gamma}} \frac{\Gamma}{4\pi\,m_X} \int_{\rm{los}} \rho(\ell,\Omega)\d \ell,
\label{fluxdec}$$ where $\Gamma$ is the decay width (inverse lifetime) and the spectrum now refers to the photons generated in the decay process. Unlike with the cross section in the case of annihilating dark matter, one does not have any strong theoretical motivation for considering any particular lifetime for an unstable DM particle. In any case, arguments have been put forward justifying the typical range of the lifetimes needed for significant signatures in astrophysics with $\sim$TeV mass particles and GUT-scale physics mediating the process (in analogy with the expected proton decay in GUTs), see e.g. [@Arvanitaki:2008hq]. From the phenomenological point of view, there are a couple of points worth mentioning regarding decaying DM candidates:
- The DM distribution and the role of substructures in particular is of little importance in determining the level of the signal.
- The angular distribution of the gamma-ray signal is very distinctive, and much flatter than the corresponding annihilation signal, as illustrated for a NFW profile in Fig. (\[fig:angular\]).
Should gamma rays be detected from DM, a comparison between the emission in the inner Galaxy and the emission at high latitude would immediately reveal the nature of the particle physics process (annihilation or decay) responsible for the emission [@Bertone:2007aw]. Notice that this information is very difficult to extract with other cosmic ray probes.
A Comment on the Factorization Assumption in Eq. (\[flux1\]) {#entanglingEandAng}
============================================================
Although the factorization between the particle physics term and astrophysical term in Eq. (\[flux1\]) is a useful approximation and valid in most practical cases, there are exceptions. More correctly, one should write $$\Phi_{\gamma} (E_\gamma,\Omega)=\frac{\d N_{\gamma}}{\d E_{\gamma}} (E_\gamma)\frac{1}{8\pi m^2_X} \int_{\rm{los}}\left[\int\sigma (v_{r})\,v_{r}\,u({\bf v}_r){\rm d}^3{\bf v}_r\right] \rho^2(\ell,\Omega)\, \d \ell,
\label{flux3}$$ where ${\bf v}_r$ is the relative velocity between the two particles (with $v_r\equiv |{\bf v}_r|$) and $u({\bf v}_r)$ its distribution function (not necessarily isotropic), whose integral over ${\rm d}^3{\bf v}_r$ is normalized to unity. The factorization assumed in Eq. (\[flux1\]) holds only if the integral in square brackets—which is nothing but $\sigv$—is independent of position. A sufficient condition for this is that $\sigma (v_{r})\,v_{r}$ is velocity-independent. In general, the integral depends on the kinematical structure of the halo via the position-dependent velocity dispersion, anisotropy, etc. Then, both the astrophysical distribution of the DM and the particle physics contribute in determining the angular shape of the signal.
One case in which the factorization is not valid can be found when the WIMP annihilations mainly through a $P$-wave process, such that $\sigma (v_{r})\,v_{r}\propto v_r^2$ [@Amin:2007ir]. This is a largely academic case, however, since whenever $P$-wave annihilation is dominant the gamma-ray signal is expected to be suppressed. More interesting is the case in which non-perturbative processes lead to large “Sommerfeld enhancements” to the annihilation cross section at low velocities [@Hisano:2004ds]. This effect can be thought of as the distortion of the wave-function due to a relatively long-range attraction between the WIMPs. In this case, a further steepening of the signal towards the Galactic Center is expected, which in turn should ease the detection of gamma rays from the inner Galaxy [@Robertson:2009bh].
References {#references .unnumbered}
==========
[00]{}
`http://www-glast.slac.stanford.edu/` `http://www.mpi-hd.mpg.de/hfm/HESS/HESS.html` `http://www.magic.iac.es/` `http://veritas.sao.arizona.edu/` `http://icrhp9.icrr.u-tokyo.ac.jp/`
E. A. Baltz [*et al.*]{}, JCAP [**0807**]{}, 013 (2008) \[arXiv:0806.2911 \[astro-ph\]\].
J. Buckley [*et al.*]{}, arXiv:0812.0795 \[astro-ph\]. L. Bergstrom, P. Ullio and J. H. Buckley, Astropart. Phys. [**9**]{}, 137 (1998) \[arXiv:astro-ph/9712318\]. G. Bertone, D. Hooper and J. Silk, Phys. Rept. [**405**]{}, 279 (2005) \[arXiv:hep-ph/0404175\]. T. Moroi and L. Randall, Nucl. Phys. B [**570**]{}, 455 (2000) \[arXiv:hep-ph/9906527\].
G. Jungman, M. Kamionkowski and K. Griest, Phys. Rept. [**267**]{}, 195 (1996) \[hep-ph/9506380\]. S. Dodelson, D. Hooper and P. D. Serpico, Phys. Rev. D [**77**]{}, 063512 (2008) \[arXiv:0711.4621 \[astro-ph\]\].
L. Bergstrom, arXiv:0903.4849 \[hep-ph\].
L. Bergstrom and P. Ullio, Nucl. Phys. B [**504**]{}, 27 (1997) \[hep-ph/9706232\]; P. Ullio and L. Bergstrom, Phys. Rev. D [**57**]{}, 1962 (1998) \[hep-ph/9707333\]. M. Gustafsson, E. Lundstrom, L. Bergstrom and J. Edsjo, Phys. Rev. Lett. [**99**]{}, 041301 (2007) \[arXiv:astro-ph/0703512\].
A. Birkedal, K. T. Matchev, M. Perelstein and A. Spray, arXiv:hep-ph/0507194. T. Bringmann, L. Bergstrom and J. Edsjo, JHEP [**0801**]{}, 049 (2008) \[arXiv:0710.3169 \[hep-ph\]\].
M. Kachelriess and P. D. Serpico, Phys. Rev. D [**76**]{}, 063516 (2007) \[arXiv:0707.0209 \[hep-ph\]\].
N. F. Bell, J. B. Dent, T. D. Jacques and T. J. Weiler, Phys. Rev. D [**78**]{}, 083540 (2008) \[arXiv:0805.3423 \[hep-ph\]\].
J. J. Binney and N. W. Evans, Mon. Not. Roy. Astron. Soc. [**327**]{}, L27 (2001) \[arXiv:astro-ph/0108505\]. A. Klypin, H. Zhao and R. S. Somerville, Astrophys. J. [**573**]{}, 597 (2002) \[arXiv:astro-ph/0110390\]. D. Hooper and P. D. Serpico, JCAP [**0706**]{}, 013 (2007) \[astro-ph/0702328\]. S. Gillessen, F. Eisenhauer, S. Trippe, T. Alexander, R. Genzel, F. Martins and T. Ott, arXiv:0810.4674 \[astro-ph\].
J. F. Navarro [*et al.*]{}, Astrophys. J. 462 563 (1996).
B. Moore [*et al.*]{}, Phys. Rev. D [**64**]{} 063508 (2001).
A. V. Kravtsov, A. A. Klypin, J. S. Bullock and J. R. Primack, Astrophys. J. [**502**]{}, 48 (1998) \[astro-ph/9708176\]. C. Power [*et al.*]{}, Mon. Not. Roy. Astron. Soc. [**338**]{}, 14 (2003) \[astro-ph/0201544\]. J. F. Navarro [*et al.*]{}, Mon. Not. Roy. Astron. Soc. [**349**]{}, 1039 (2004) \[astro-ph/0311231\]. D. Reed [*et al.*]{}, Mon. Not. Roy. Astron. Soc. [**357**]{}, 82 (2005) \[astro-ph/0312544\].
D. Merritt, J. F. Navarro, A. Ludlow and A. Jenkins, Astrophys. J. [**624**]{}, L85 (2005) \[astro-ph/0502515\]. J. F. Navarro [*et al.*]{}, arXiv:0810.1522 \[astro-ph\]. J. Einasto, Trudy Inst. Astroz. Alma-Ata, 51, 87 (1965)
O. Y. Gnedin, A. V. Kravtsov, A. A. Klypin and D. Nagai, Astrophys. J. [**616**]{}, 16 (2004) \[astro-ph/0406247\]; M. Gustafsson, M. Fairbairn and J. Sommer-Larsen, Phys. Rev. D [**74**]{}, 123522 (2006) \[arXiv:astro-ph/0608634\]. J. I. Read, L. Mayer, A. M. Brooks, F. Governato and G. Lake, arXiv:0902.0009 \[astro-ph.GA\]. F. Prada, A. Klypin, J. Flix, M. Martinez and E. Simonneau, Phys. Rev. Lett. [**93**]{}, 241301 (2004) \[arXiv:astro-ph/0401512\]. G. Bertone and D. Merritt, Mod. Phys. Lett. A [**20**]{}, 1021 (2005) \[astro-ph/0504422\]; G. Bertone and D. Merritt, Phys. Rev. D [**72**]{}, 103502 (2005) \[astro-ph/0501555\]. J. A. Sellwood, arXiv:0807.1973 \[astro-ph\]. P. Gondolo and J. Silk, Phys. Rev. Lett. [**83**]{}, 1719 (1999) \[astro-ph/9906391\]; P. Ullio, H. Zhao and M. Kamionkowski, Phys. Rev. D [**64**]{}, 043504 (2001) \[astro-ph/0101481\]; G. Bertone, G. Sigl and J. Silk, Mon. Not. Roy. Astron. Soc. [**337**]{}, 98 (2002) \[astro-ph/0203488\].
M. Fornasa and G. Bertone, Int. J. Mod. Phys. D [**17**]{}, 1125 (2008) \[arXiv:0711.3148 \[astro-ph\]\].
H. Yuksel, S. Horiuchi, J. F. Beacom and S. Ando, Phys. Rev. D [**76**]{}, 123506 (2007) \[arXiv:0707.0196 \[astro-ph\]\]. L. Bergstrom, J. Edsjo and P. Ullio, Phys. Rev. Lett. [**87**]{}, 251301 (2001) \[astro-ph/0105048\]; V. Berezinsky, A. Bottino and G. Mignola, Phys. Lett. B [**325**]{}, 136 (1994) \[hep-ph/9402215\]; A. Cesarini, F. Fucito, A. Lionetto, A. Morselli and P. Ullio, Astropart. Phys. [**21**]{}, 267 (2004) \[astro-ph/0305075\]; P. Ullio, L. Bergstrom, J. Edsjo and C. G. Lacey, Phys. Rev. D [**66**]{}, 123502 (2002) \[astro-ph/0207125\].
K. Kosack [*et al.*]{} \[The VERITAS Collaboration\], Astrophys. J. [**608**]{}, L97 (2004) \[astro-ph/0403422\]. F. Aharonian [*et al.*]{} \[The HESS Collaboration\], Astron. Astrophys. [**425**]{}, L13 (2004) \[astro-ph/0408145\]. J. Albert [*et al.*]{} \[MAGIC Collaboration\], Astrophys. J. [**638**]{}, L101 (2006) \[astro-ph/0512469\].
K. Tsuchiya [*et al.*]{} \[CANGAROO-II Collaboration\], Astrophys. J. [**606**]{}, L115 (2004) \[astro-ph/0403592\].
C. van Eldik, O. Bolz, I. Braun, G. Hermann, J. Hinton and W. Hofmann \[for the HESS Collaboration\], arXiv:0709.3729 \[astro-ph\]. D. Hooper and J. March-Russell, Phys. Lett. B [**608**]{}, 17 (2005) \[hep-ph/0412048\]; D. Hooper, I. de la Calle Perez, J. Silk, F. Ferrer and S. Sarkar, JCAP [**0409**]{}, 002 (2004) \[astro-ph/0404205\]; S. Profumo, Phys. Rev. D [**72**]{}, 103521 (2005) \[astro-ph/0508628\]; L. Bergstrom, T. Bringmann, M. Eriksson and M. Gustafsson, Phys. Rev. Lett. [**94**]{}, 131301 (2005) \[astro-ph/0410359\].
F. Aharonian [*et al.*]{} \[H.E.S.S. Collaboration\], Phys. Rev. Lett. [**97**]{}, 221102 (2006) \[Erratum-ibid. [**97**]{}, 249901 (2006)\] \[arXiv:astro-ph/0610509\].
F. Aharonian and A. Neronov, Astrophys. J. [**619**]{}, 306 (2005) \[astro-ph/0408303\]; astro-ph/0503354; AIP Conf. Proc. [**745**]{}, 409 (2005); A. Atoyan and C. D. Dermer, Astrophys. J. [**617**]{}, L123 (2004) \[astro-ph/0410243\]. G. Zaharijas and D. Hooper, Phys. Rev. D [**73**]{}, 103501 (2006) \[arXiv:astro-ph/0603540\].
D. Hooper, G. Zaharijas, D. P. Finkbeiner and G. Dobler, Phys. Rev. D [**77**]{}, 043511 (2008) \[arXiv:0709.3114 \[astro-ph\]\].
C. van Eldik, arXiv:0811.0931 \[astro-ph\]. D. Hooper and B. L. Dingus, Phys. Rev. D [**70**]{}, 113007 (2004) \[astro-ph/0210617\]; D. Hooper and B. Dingus, Proc. of the 34th COSPAR Scientific Assembly, Houston, Texas (2002), astro-ph/0212509. M. Pohl, Astrophys. J. [**626**]{}, 174 (2005).
D. Hooper and G. Zaharijas, Phys. Rev. D [**75**]{}, 035010 (2007) \[arXiv:hep-ph/0612137\].
P. Gondolo, J. Edsjo, P. Ullio, L. Bergstrom, M. Schelke and E. A. Baltz, JCAP [**0407**]{}, 008 (2004) \[astro-ph/0406204\]. F. Stoehr, S. D. M. White, V. Springel, G. Tormen and N. Yoshida, Mon. Not. Roy. Astron. Soc. [**345**]{}, 1313 (2003) \[astro-ph/0307026\]. N. W. Evans, F. Ferrer and S. Sarkar, Phys. Rev. D [**69**]{}, 123501 (2004) \[astro-ph/0311145\];
P. D. Serpico and G. Zaharijas, Astropart. Phys. [**29**]{}, 380 (2008) \[arXiv:0802.3245 \[astro-ph\]\]. J. M. Siegal-Gaskins, JCAP [**0810**]{}, 040 (2008) \[arXiv:0807.1328 \[astro-ph\]\].
S. K. Lee, S. Ando and M. Kamionkowski, arXiv:0810.1284 \[astro-ph\]. M. Fornasa, L. Pieri, G. Bertone and E. Branchini, arXiv:0901.2921 \[astro-ph\]. A. Arvanitaki, S. Dimopoulos, S. Dubovsky, P. W. Graham, R. Harnik and S. Rajendran, arXiv:0812.2075 \[hep-ph\]. G. Bertone, W. Buchmuller, L. Covi and A. Ibarra, JCAP [**0711**]{}, 003 (2007) \[arXiv:0709.2299\]. M. A. Amin and T. Wizansky, Phys. Rev. D [**77**]{}, 123510 (2008) \[arXiv:0710.5517 \[astro-ph\]\]. J. Hisano, S. Matsumoto, M. M. Nojiri and O. Saito, Phys. Rev. D [**71**]{}, 063528 (2005) \[arXiv:hep-ph/0412403\].
B. Robertson and A. Zentner, arXiv:0902.0362 \[astro-ph.CO\].
[^1]: See Appendix A for the case of DM decay.
[^2]: Even for heavy dark matter particles of TeV mass, the continuum photon spectrum peaks at or below 30 GeV, explaining in which sense the line emission considered here is “extreme”.
|
---
author:
- David Zwicker
bibliography:
- 'bibdesk.bib'
title:
- 'Supporting Information: Olfactory coding with global inhibition'
- 'Supporting Information: Normalized neural representations of natural odors'
---
Statistics of normalized concentrations and excitations
=======================================================
Let $p_i$ be the probability that ligand $i$ is present in an odor. If it is present, its concentration $c_i$ is drawn from a log-normal distribution with mean $\mu_i$ and standard deviation $\sigma_i$, while $c_i=0$ if the ligand is not present. Hence,
$$\begin{aligned}
{\langle c_i \rangle} &= p_i\mu_i
\\
\operatorname{var}(c_i) &= (p_i - p_i^2) \mu_i^2 + p_i\sigma_i^2
\;,\end{aligned}$$
while the covariances $\operatorname{cov}(c_i, c_j) = {\langle c_ic_j \rangle} - {\langle c_i \rangle}{\langle c_j \rangle}$ vanish for $i\neq j$ since the ligands are independent. The statistics of the total concentration ${c_{\rm tot}}= \sum_i c_i$ read $$\begin{aligned}
{\langle {c_{\rm tot}}\rangle} &= {\sum_{i=1}^{{{N_{\rm L}}}}}{\langle c_i \rangle}
& \text{and} &&
\operatorname{var}({c_{\rm tot}}) &= {\sum_{i=1}^{{{N_{\rm L}}}}}\operatorname{var}(c_i)
\;.
\label{eqn:ctot_stats}\end{aligned}$$ The excitations $e_n$ are given by $e_n = \sum_i S_{ni} c_i$, where the sensitivities $S_{ni}$ are log-normally distributed with mean ${\langle S_{ni} \rangle} = \bar S$ and variance $\operatorname{var}(S_{ni}) = \bar S^2(e^{\lambda^2} - 1)$. Hence,
$$\begin{aligned}
{\langle e_n \rangle} &= \bar S {\langle {c_{\rm tot}}\rangle}
\\
\operatorname{var}(e_n) &= \bar S^2 \operatorname{var}({c_{\rm tot}})
+ \operatorname{var}(S_{ni}) {\sum_{i=1}^{{{N_{\rm L}}}}}{\langle c_i^2 \rangle}
\;,\end{aligned}$$
where ${\langle c_i^2 \rangle} = p_i(\mu_i^2 + \sigma_i^2)$ and $\operatorname{cov}(e_n, e_m)=0$ for $n\neq m$.
We next determine the statistics of the normalized concentrations $\hat c_i = c_i/{c_{\rm tot}}$. For simplicity, we consider large odors, $\sum_i p_i \gg 1$, where ${c_{\rm tot}}$ can be considered as an independent random variable. Since ${c_{\rm tot}}$ is the sum of (a variable) number of log-normally distributed random variables, its distribution can be approximated by another log-normal distribution [@Wu2005], which we parameterize by its mean $\mu_{\rm tot}$ and variance $\sigma_{\rm tot}^2$. We consider the simple approximation where these parameters are directly given by [@Fenton1960]. This choice approximates the tail of the distribution well, but leads to errors in the vicinity of the mean [@Wu2005].
Since both ${c_{\rm tot}}$ and $c_i$ are log-normally distributed when ligand $i$ is present in an odor ($c_i>0$), $\hat c_i$ is also log-normally distributed in this case and
$$\begin{aligned}
{\langle \hat c_i \rangle}_{c_i > 0} &= \frac{\mu_i}{\mu_{\rm tot}} \chi
\\
\operatorname{var}(\hat c_i)_{c_i > 0} &=
\frac{\mu_i^2\chi^2}{\mu_{\rm tot}^2}
\left(\frac{\sigma_i^2}{\mu_i^2} \chi
+ \chi - 1
\right)
\;,\end{aligned}$$
where $\chi = 1 + \sigma_{\rm tot}^2 \mu_{\rm tot}^{-2}$. Since $\hat c_i=0$ with probability $1-p_i$, the statistics of $\hat c_i$ read
\[eqn:norm\_conc\_stats\] $$\begin{aligned}
{\langle \hat c_i \rangle} &= \frac{p_i\mu_i}{\mu_{\rm tot}} \chi
\label{eqn:norm_conc_mean}
\\
\operatorname{var}(\hat c_i) &=
\frac{p_i\mu_i^2\chi^2}{\mu_{\rm tot}^2}\left(
\frac{\sigma_i^2}{\mu_i^2}\chi
+ \chi - p_i
\right)
\;.\end{aligned}$$
Note that the covariance $\operatorname{cov}(\hat c_i, \hat c_j)$ does not vanish since the $\hat c_i$ are not independent. In particular, $\operatorname{var}(\sum_i \hat c_i)=0$, since $\sum_i \hat c_i = 1$ by definition. This condition is only consistent with ${\mbox{Eq.\hspace{0.25em}{\ref{eqn:norm_conc_mean}}}}$ if $\chi \approx 1$, which implies that ${c_{\rm tot}}$ must not vary much, $\frac{\sigma_{\rm tot}}{\mu_{\rm tot}} \ll 1$. Using $\chi = 1$, the statistics of the normalized excitations $\hat e_n = \bar S^{-1} \sum_i S_{ni} \hat c_i$ read
$$\begin{aligned}
{\langle \hat e_n \rangle} &= 1 \\
\operatorname{var}(\hat e_n) &=
\frac{\operatorname{var}(S_{ni})}{\bar S^2} \Bigl\langle\sum_i \hat c_i^2 \Bigr\rangle
\;,\end{aligned}$$
where ${\langle \sum_i \hat c_i^2 \rangle} \approx \sum_i{\langle \hat c_i^2 \rangle}$ with ${\langle \hat c_i^2 \rangle} = {\langle \hat c_i \rangle}^2 + \operatorname{var}(\hat c_i)$ and the statistics given in . In the simple case where all ligands are drawn from the same distribution ($p_i = p$, $\mu_i = \mu$, $\sigma_i=\sigma$), we obtain $$\begin{aligned}
{\langle \hat c_i \rangle} &\approx \frac1{{N_{\rm L}}}\;,
&
\operatorname{var}(\hat c_i) &\approx
\frac{1-p+\frac{\sigma^2}{\mu^2}}{s{{N_{\rm L}}}}
\;,\end{aligned}$$ and ${\langle \hat c_i^2 \rangle} \approx \frac{1}{s{{N_{\rm L}}}}(\frac{\sigma^2}{\mu^2} + 1)$, such that $$\begin{aligned}
\operatorname{var}(\hat e_n) & \approx
\frac{1}{s}
\left( 1 + \frac{\sigma^2}{\mu^2} \right)
\frac{\operatorname{var}(S_{ni})}{\bar S^2}
\;,\end{aligned}$$ which is equivalent to Eq. [**5**]{} in the main text.
Numerical simulations
=====================
We numerically calculated ensemble averages over odors ${\boldsymbol}c$ and sensitivity matrices $S_{ni}$. Here, we first choose $S_{ni}$ by drawing all entries independently from a log-normal distribution with mean $\bar S=1$ and variance $\operatorname{var}(S_{ni}) = e^{\lambda^2} - 1$. We then draw an odor ${\boldsymbol}c$ using the following procedure: First, we determine which of the ${{N_{\rm L}}}$ ligands are present according to their probabilities $p_i$. Second, we draw the concentrations $c_i$ for each ligand $i$ that is present from a log-normal distribution with mean $\mu_i$ and standard deviation $\sigma_i$. We then use Eqs. [**1**]{}–[**3**]{} given in the main text to map the odor ${\boldsymbol}c$ to a binary activity vector ${\boldsymbol}a$, from which we can for instance calculate the number of active channels. We obtain ensemble averages of such quantities by repeating these steps $10^5$ times. This allows us to calculate the mean activities ${\langle a_n \rangle}$, the covariances $\operatorname{cov}(a_n, a_m)$, and the Pearson correlation coefficient $\rho$, which is defined as $$\begin{aligned}
\rho = \frac{1}{{{N_{\rm R}}}^2 - {{N_{\rm R}}}} \sum_{n \neq m}
\frac{\operatorname{cov}(a_n, a_m)}{\bigl[\operatorname{var}(a_n)\operatorname{var}(a_m)\bigr]^{\frac12}}
\;.
\label{eqn:pearson}\end{aligned}$$ [Fig.\[fig:activity\_width\]]{} shows these quantities as a function of the width $\lambda$ of the sensitivity distribution. We also estimate $P({\boldsymbol}a)$ from an ensemble average to calculate the information $I$ from its definition given in Eq. [**4**]{} in the main text.
![ Influence of the width $\lambda$ of the sensitivity distribution on the statistics of the odor representations. (A) Expected channel activity ${\langle a_n \rangle}$ as a function of $\lambda$ for several inhibition strengths $\alpha$. Intermediated values, $\lambda \approx 1$, lead to larger activities. (B) Mean Pearson correlation coefficient ${\langle \rho \rangle}$ calculated from an ensemble average of as a function of $\lambda$ for several $\alpha$. For small $\lambda$, ${\langle a_n \rangle}$ was too small to estimate $\rho$ reliably. (A–B) Results are shown for small ($\frac\sigma\mu=1$, solid lines) and large ($\frac\sigma\mu=10$, dashed lines) concentration variability. Remaining parameters are ${{N_{\rm R}}}=32$, ${{N_{\rm L}}}=256$, and $p_i=0.1$. \[fig:activity\_width\] ](FigS1_ActivityWidth){width="\columnwidth"}
Approximate channel activity
============================
We estimate the expected activity ${\langle a_n \rangle}$ by the probability that the normalized excitations $\hat e_n$ exceed the expected normalized threshold $\alpha$. Since both the sensitivities $S_{ni}$ and the normalized concentrations $\hat c_i$ are approximately log-normally distributed, $\hat e_n$ can also be approximated by a log-normal distribution [@Fenton1960]. The associated probability distribution function reads $$\begin{aligned}
f(\hat e_n) &=
\frac{1}{\sqrt{2 \pi } S_n \hat e_n}
\exp\left[-\frac{\bigl(M_n - \ln (\hat e_n)\bigr)^2}{2 S_n^2}\right]
\label{eqn:lognorm_pdf}\end{aligned}$$ and the cumulative distribution function is $$\begin{aligned}
F(\hat e_n) &=\frac12 \operatorname{erfc}\left[
\frac{M_n - \ln(\hat e_n)}{\sqrt 2 S_n}
\right]
\label{eqn:lognorm_cdf}
\;.\end{aligned}$$ The parameters $M_n$ and $S_n$ can be determined from the mean and variance
\[eqn:normalized\_excitations\_statistics\] $$\begin{aligned}
{\langle \hat e_n \rangle} &= \exp\left(M_n + \frac{S_n^2}{2}\right)
\\
\operatorname{var}(\hat e_n) &= e^{2M_n + S_n^2} \left(e^{S_n^2} -1\right)
\;.\end{aligned}$$
Solving these equations for $M_n$ and $S_n$, we obtain $$\begin{aligned}
M_n &= \ln{\langle \hat e_n \rangle} - \zeta
& \text{and} &&
S_n &= \sqrt{2\zeta}
\;,
\label{eqn:lognorm_parameter}\end{aligned}$$ where $\zeta = \frac12 \ln(1 + \operatorname{var}(\hat e_n){\langle \hat e_n \rangle}^{-2})$. Eq. [**6**]{} of the main text follows from this and Eq. [**5**]{}. For small ${\langle a_n \rangle}$ we have $$\begin{aligned}
{\langle a_n \rangle} \approx
\frac{2 \sqrt{\zeta/\pi } }{\ln (\alpha )+\zeta}
\exp\left[-\frac{(\ln (\alpha )+\zeta )^2}{4 \zeta }\right]
\;,\end{aligned}$$ which follows from $\operatorname{erfc}(x) \approx e^{-x^2}/(x\sqrt\pi)$, valid for $x \gg 1$. For small $\zeta$, we obtain the approximate scaling $\ln{\langle a_n \rangle} \sim -(\ln \alpha)^2/(4\zeta)$, where $\zeta \sim s^{-1}$ for $s \gg 1$.
Odor discriminability
=====================
We quantify the discriminability of two odors by the Hamming distance $d$ of their respective representations ${\boldsymbol}a$ for several different cases:
#### Uncorrelated odors
The expected distance ${\langle d \rangle}$ between the activity patterns ${\boldsymbol}a^{(1)}$ and ${\boldsymbol}a^{(2)}$ of two independent odors is $$\begin{aligned}
{\langle d \rangle} &= {{N_{\rm R}}}\left({\langle a_n^{(1)} \rangle} + {\langle a_n^{(2)} \rangle} - 2{\langle a_n^{(1)} \rangle}{\langle a_n^{(2)} \rangle}\right)
\label{eqn:discriminability_independent}
\;,\end{aligned}$$ where ${\langle a_n^{(1)} \rangle}$ and ${\langle a_n^{(2)} \rangle}$ denote the expected activities of the two odors, averaged over sensitivity matrices, and we neglect correlations $\operatorname{cov}(a_n, a_m)$ for simplicity.
#### Adding target to background
We calculate the expected change ${\langle d \rangle}$ of the representation when a target odor ${\boldsymbol}c^{\rm t}$ is added to a background odor ${\boldsymbol}c^{\rm b}$. Because the odor concentrations are specified, we consider the actual excitations $e_n$ instead of the normalized quantities $\hat e_n$. Taking an ensemble average over sensitivity matrices, the excitations associated with the two odors are characterized by probability distribution functions $f_E^{\rm t}(e^{\rm t})$ and $f_E^{\rm b}(e^{\rm b})$ for the target and the background, respectively. We here consider log-normally distributed $e_n$, which are parameterized by their mean and variance, $$\begin{aligned}
{\langle e_n \rangle} &= \bar S {\sum_{i=1}^{{{N_{\rm L}}}}}c_i
&
\operatorname{var}(e_n) &= \operatorname{var}(S_{ni}) {\sum_{i=1}^{{{N_{\rm L}}}}}c_i^2
\;,\end{aligned}$$ where $\operatorname{var}(S_{ni}) = \bar S^2(e^{\lambda^2} - 1)$. When the target is added to the background, the expected threshold ${\langle \gamma \rangle}$ increases from $\gamma^{\rm b}= \alpha {\langle e^{\rm b} \rangle}$ to $\gamma^{\rm s} = \alpha ({\langle e^{\rm b} \rangle} + {\langle e^{\rm t} \rangle})$, where ${\langle e^\kappa \rangle}$ denotes the mean excitation ${\langle e^\kappa \rangle} = \int \! z \, f_E^\kappa(z) \, {\text{d}}z$ for $\kappa={\rm t}, {\rm b}$. This increase in the threshold can deactivate a channel if it was previously active, if its excitation was larger than the threshold associated with the background, $e^{\rm b} > \gamma^{\rm b}$. For such $e^{\rm b}$, the probability that the receptor gets deactivated by adding the target is $P(e^{\rm b} + e^{\rm t} < \gamma^{\rm s} | e^{\rm b})$. Integrating over all possible $e^{\rm b}$, we thus get the probability $p_{\rm off}$ that a channel becomes inactive, $$\begin{aligned}
p_{\rm off} &= \int_{\gamma^{\rm b}}^{\infty}
\! P\left(e^{\rm b} + e^{\rm t} < \gamma^{\rm s} \,\middle | \, e^{\rm b} \right) f^{\rm b}_E\bigl(e^{\rm b}\bigr) \, {\text{d}}e^{\rm b}
\notag\\
&= \int_{\gamma^{\rm b}}^{\infty}
F^{\rm t}_E\bigl(\gamma^{\rm s} - e^{\rm b}\bigr) f^{\rm b}_E\bigl(e^{\rm b}\bigr) \, {\text{d}}e^{\rm b}
\;,\end{aligned}$$ where $F^{\rm t}_E(e^{\rm t})$ is the cumulative distribution function associated with $f^{\rm t}_E(e^{\rm t})$. Conversely, a channel becomes active when the additional excitation by the target odor brings it above the threshold $\gamma^{\rm s}$. The associated probability $p_{\rm on}$ reads $$\begin{aligned}
p_{\rm on} &= \int_0^{\gamma^{\rm b}}
\left[1 - F^{\rm t}_E\bigl(\gamma^{\rm s} - e^{\rm b}\bigr)\right] f^{\rm b}_E\bigl(e^{\rm b}\bigr) \, {\text{d}}e^{\rm b}
\;.\end{aligned}$$ Taken together, the expected number ${\langle d \rangle}$ of channels that change their state reads $$\begin{aligned}
{\langle d \rangle} &= {{N_{\rm R}}}\cdot \bigl(p_{\rm on} + p_{\rm off} \bigr)
\label{eqn:mixture_hamming}
\;.\end{aligned}$$ There are three simple limits that we can solve analytically: If there is no target, ${\langle e^{\rm t} \rangle}=0$, the activation pattern does not change and we have ${\langle d \rangle}=0$. In the opposing limit of a dominant target, ${\langle e^{\rm t} \rangle} \rightarrow \infty$, the activation patterns are independent and we recover the distance ${\langle d \rangle}_{\rm max}$ for uncorrelated odors, which is given by . Lastly, in the case where the target and the background are identically distributed, ${\langle e^{\rm b} \rangle} = {\langle e^{\rm t} \rangle}$ and $\operatorname{var}(e^{\rm b})=\operatorname{var}(e^{\rm t})$, we have ${\langle d \rangle}=\frac12 {\langle d \rangle}_{\rm max}$.
#### Discriminating two odors of equal size
We consider the simple case of two odors that each contain $s$ ligands at equal concentration, sharing $s_{\rm b}$ of them, such that the expected threshold ${\langle \gamma \rangle}$ is the same for both odors. Similar to the derivation above, we here calculate the probability $p$ that a channel is active for one odor, but not for the other. The $s_{\rm b}$ ligands that are present in both odors cause a baseline excitation $e^{\rm b}$, which is distributed according to $f^{\rm b}_E(e^{\rm b})$. A channel is inactive for an odor with probability $F^{\rm d}_E({\langle \gamma \rangle} - e^{\rm b})$, where $F^{\rm d}_E(e^{\rm d})$ is the cumulative distribution function of the excitation caused by the $s_{\rm d} = s - s_{\rm b}$ different ligands. Hence, $$\begin{aligned}
p &=
2\int_0^{{\langle \gamma \rangle}}
F^{\rm d}_E(z)
\bigl[1 - F^{\rm d}_E(z)\bigr]
f^{\rm b}_E(e^{\rm b})
{\text{d}}e^{\rm b}
\;,\end{aligned}$$ where $z = {\langle \gamma \rangle} - e^{\rm b}$. Note that the upper bound of the integral is ${\langle \gamma \rangle}$ since channels will be active for both odors if $e^{\rm b} \ge {\langle \gamma \rangle}$. The associated Hamming distance ${\langle d \rangle}$ between the two odors is then given by ${\langle d \rangle}=p {{N_{\rm R}}}$.
Receptor binding model
======================
We consider a simple model where receptors $R_n$ get activated when they bind ligands $L_i$. This binding is described by the chemical reaction , where $R_nL_i$ is the receptor-ligand complex. In equilibrium, the concentrations denoted by square brackets obey $[R_nL_i] = K_{ni} \cdot [R_n][L_i]$, where $K_{ni}$ is the binding constant of the reaction. Hence, $$\begin{aligned}
[R_nL_i] &= \frac{c^{\rm rec}_n K_{ni} c_i}{1 + \sum_i K_{ni} c_i}
\label{eqn:binding}
\;,\end{aligned}$$ where we consider the case where multiple ligands compete for the same receptor. Here, $c_i = [L_i]$ is the concentration of free ligands and $c^{\rm rec}_n=[R_n] + \sum_i [R_nL_i]$ denotes the fixed concentration of receptors, which is related to the copy number of receptors of type $n$. We consider a simple receptor model where the excitation is proportional to the concentration of the bound ligands, such that the excitation accumulated in glomerulus $n$ reads $$\begin{aligned}
e_n &= \beta_n \frac{N^{\rm rec}_n}{c^{\rm rec}_n} {\sum_{i=1}^{{{N_{\rm L}}}}}[R_nL_i]
= \beta_n N^{\rm rec}_n \frac{\sum_i K_{ni} c_i}{1 + \sum_i K_{ni} c_i}
\label{eqn:receptor_occupancy}
\;.\end{aligned}$$ Here, $N^{\rm rec}_n$ is the copy number of receptors of type $n$ and $\beta_n$ characterizes their excitability, which could for instance be modified by point mutations [@Yu2015]. Defining $S_{ni} = \beta_n N^{\rm rec}_n K_{ni}$, we recover Eq. [**1**]{} of the main text in the limit of small concentrations, $\sum_i K_{ni} c_i \ll 1$. The sensitivities are thus proportional to the copy number $N^{\rm rec}_n$ and the biochemical details encoded in $\beta_nK_{ni}$.
|
---
abstract: 'In a previous paper, the author compute the dimension of Hochschild cohomology groups of Jacobian algebras from (unpunctured) triangulated surfaces, and gave a geometric interpretation of those numbers in terms of the number of internal triangles, the number of vertices and the existence of certain kind of boundaries. The aim of this note is computing the cyclic (co)homology and the Hochschild homology of the same family of algebras and giving an interpretation of those dimensions through elements of the triangulated surface.'
address: 'Instituto de Matemáticas, UNAM Área de la Investigación Científica, Circuito exterior, Ciudad Universitaria, CDMX, 04510, México.'
author:
- 'Yadira Valdivieso-Díaz'
title: 'A note on (co)homologies of algebras from unpunctured surfaces'
---
Introduction
============
A *surface with marked points*, or simply a *surface*, is a pair $(S,M)$, where $S$ is a compact connected Riemann surface with (possibly empty) boundary, and $M$ is a non-empty finite subset of $S$ containing at least one point from each connected component of the boundary of $S$. We said that $(S,M)$ is an *unpunctured surface* if $M$ is contained in the boundary of $S$. We define a *triangulation* as a maximal collection of non-crossing arcs with endpoints in $M$.
Given an *(tagged) triangulation* ${\mathbb{T}}$, it is possible to construct a finite dimensional algebra $A_{\mathbb{T}}$, which turns out to be gentle if $(S,M)$ is an unpunctured surface (see [@LF09; @ABCJP10; @Lad12; @TV15]), and in particular quadratic monomial. In this note, we compute three different (co)homologies of those gentle algebras coming from unpunctured surfaces, namely: *Hochschild homology* and *cyclic homology and cohomology*, and we show that there is a combinatorial interpretation of those (co)homologies through the elements of the surface and the triangulation.
Given an associative algebra $A$ over a field $\mathbf k$ and $M$ an $A$-bimodule, we define the *Hochschild cohomology* of $A$, with coefficients in $M$, as the graded vector space ${\operatorname{HH}}^*(A,M)= \operatorname{Ext}^*_{A\otimes_{\mathbf k} A^{\operatorname{op}}}(A,M)$, where $A^{\operatorname{op}}$ is the algebra $A$ with the opposite multiplication. The original definition was introduced by Hochschild in [@Hoc45] using a resolution of $A$ as bimodule. Later, Cartan and Eilenberg in [@CE56 Chapter 9] extended it to algebras over more general rings, and also dualized it, giving the definition of *Hochschild homology*.
The *cyclic cohomology* can be defined in several ways, but the original definition was given by Connes in [@Con81], as a variation of the de Rham homology in spaces with bad behaviour. He used a sub-complex of the Hochschild complex when $M=\operatorname{Hom}_{\mathbf k}(A, {\mathbf k})$, called *cyclic complex*. In this note, we use the definition of cyclic (co)homology given in [@Lod92 Theorem 4.1.13], for algebras over rings of characteristic zero. As in the Hochschild cohomology case, the dual definition of cyclic cohomology was given later by several authors: Loday, Kassel, Quillen and Tsygan.
To compute those (co)homologies, we use the computation of the Hochschild homology of quadratic monomial algebras given by Skölderberg in [@Sko99], and two results of Loday [@Lod92 Theorem 4.1.13, Section 2.4.8], which relate the Hochschild homology and the cyclic homology by one hand, and the last one with the cyclic cohomology by other hand.
The main result of this note is the following:
Let $(S,M, {\mathbb{T}})$ be a triangulated surface and $A_{\mathbb{T}}$ be the algebra associated to $(S,M,{\mathbb{T}})$. Then,
1. ${\operatorname{HH}}_n(A_{\mathbb{T}})\simeq \begin{cases}
\mathbf{k}(Q_{\mathbb{T}})_0 & \textrm{if $n=0$}\\
(A_{\mathbb{T}}^!/[A_{\mathbb{T}}^!, A_{\mathbb{T}}^!])_{n} & \textrm{if $n\equiv 3\pmod{6}$}\\
(A_{\mathbb{T}}^!/[A_{\mathbb{T}}^!, A_{\mathbb{T}}^!])_{n+1} & \textrm{if $n\equiv 2\pmod{6}$}\\
0 & \textrm{otherwise}
\end{cases}$
2. ${\operatorname{HC}}_{n}(A_{\mathbb{T}})\simeq \begin{cases}
\mathbf{k} (Q_{\mathbb{T}})_0 & \textrm{if $n=0$}\\
\mathbf k (Q_{\mathbb{T}})_0 \oplus (A_{\mathbb{T}}^!/[A_{\mathbb{T}}^!, A_{\mathbb{T}}^!])_{n+1} & \textrm{if $n\equiv 2\pmod{6}$}\\
0 & \textrm{otherwise}
\end{cases}
$
Moreover, the dimension of the quotients $(A_{\mathbb{T}}^!/[A_{\mathbb{T}}^!, A_{\mathbb{T}}^!])_{k}$ appearing in ${\operatorname{HH}}_n(A_{\mathbb{T}})$ and ${\operatorname{HC}}_n(A_{\mathbb{T}})$, are equal to the number of internal triangles of ${\mathbb{T}}$.
As consequence of the main Theorem, it follows that the dimension of the cyclic cohomology is computed as follows.
Let $A_{\mathbb{T}}$ be the algebra associated to the triangulated surface $(S, M, {\mathbb{T}})$ and denote by ${\operatorname{int}({\mathbb{T}})}$ the set of internal triangles of ${\mathbb{T}}$. Then $$\operatorname{dim}_{\mathbf k}({\operatorname{HC}}^{n}(A_{\mathbb{T}}))= \begin{cases}
\mid (Q_{\mathbb{T}})_0\mid & \textrm{if $n=0$}\\
\mid (Q_{\mathbb{T}})_0\mid + \mid {\operatorname{int}({\mathbb{T}})}\mid & \textrm{if $n\equiv 2\pmod{6}$}\\
0 & \textrm{otherwise}
\end{cases}$$
According to the previous results, the dimensions of the cyclic (co)homology and Hochschild homology of the algebra $A_{\mathbb{T}}$ depend on $\mid{\operatorname{int}({\mathbb{T}})}\mid $ and the number of vertices of the quiver $Q_{\mathbb{T}}$, then it is easy to observe those (co)homologies are not invariant under *flips of arcs*, and therefore they are not invariant under *mutation of quivers with potentials*. See [@FST08] for definition of flips of arcs in triangulations, [@DWZ08] for definitions of mutations of quivers with potentials and [@LF09] for details about the relation between algebras from surfaces and quivers with potentials and its mutations.
Results
=======
In the first part of this section, we recall some definitions and notations of path algebras and surfaces with marked points and we include the computation of the Hochschild homology given by Sköldberg in [@Sko99] for completeness.
Let $Q=(Q_0, Q_1)$ be a finite quiver with a set of vertices $Q_0$ and a set of arrows$Q_1$. We denote the source and the target of an arrow $a\in Q_1$ by $s(a)$ and $t(a)$, respectively. A *path* $w$ of length $l$ is a sequence of $l$ arrows $a_1\cdots a_l$, such that $t(a_k)=s(a_{k+1})$ for every $k=1, \dots, l-1$, we say that its source $s(w)$ is $s(\alpha_1)$ and its target $t(w)$ is $t(\alpha_l)$. We denote by ${\lvert w\rvert}$ the length of the path $w$. We write $[e_i\mid a_1\cdots a_l\mid e_j]$ instead of $a_1 \cdots a_l$ to emphasize that the source of the path $a_1 \cdots a_l$ is $e_i$ and its target is $e_j$.
Let $\mathbf k$ be an algebraically closed field. The path algebra $kQ$ is the $\mathbf k$-vector space with basis the set of paths in $Q$ and the product of the basis elements is given by the concatenations of the sequences of arrows of the paths $w$ and $w'$ if they form a path and zero otherwise. Let $F$ be the two-sided ideal of $kQ$ generated by the arrows of $Q$. A two-sided ideal $I$ is said to be admissible if there exists an integer $m_0\geq 2$ such that $F^{m_0}\subseteq I \subseteq F^2$ and its elements are called *relations*. The pair $(Q,I)$ is called a *bounded quiver*.
The quotient algebra $kQ/I$ is said a *quadratic monomial algebra* if the admissible ideal $I$ is generated by paths of length 2. We associate its Koszul dual $A^!$ which is the quotient algebra $A^!={\mathbf{k}Q}/J$ where $J$ is generated by all paths $w$ of length 2 such that $w\notin I$.
In the next few paragraphs, we give a construction of a quadratic monomial algebra from an unpunctured surface.
Let $(S,M)$ be an unpunctured surface. An *arc* $\tau$ in $(S,M)$ is a not self-crossing curve in $S$ with endpoints in $M$ and not isotopic to a point or to a boundary segment.
For any two arcs $\tau$ and $\tau'$ in $S$, let $e(\tau, \tau')$ be the minimal number of crossings of $\tau$ and $\tau'$, that is, $e(\tau, \tau')$ is the minimum of numbers of crossings of curves $\sigma$ and $\sigma'$, where $\sigma$ is isotopic to $\tau$ and $\sigma$ is isotopic to $\tau'$. Two arcs $\tau$ and $\tau'$ are called *non-crossing* if $e(\tau,\tau')=0$. A *triangulation* ${\mathbb{T}}$ is a maximal collection of non-crossing arcs. The arcs of a triangulation ${\mathbb{T}}$ cut the surface into *triangles*. A triangle $\triangle$ in ${\mathbb{T}}$ is called an *internal triangle* if none of its sides is a boundary segment. We refer to the triple $(S,M,{\mathbb{T}})$ as a *triangulated surface*.
If ${\mathbb{T}}=\{\tau_1, \cdots\tau_m\}$ is a triangulation of an unpunctured surface $(S,M)$, we define a quiver $Q_{\mathbb{T}}$ as follows: $Q_{\mathbb{T}}$ has $m$ vertices, one for each arc in ${\mathbb{T}}$. We will denote the vertex corresponding to $\tau_i$ by $e_i$ (or $i$ if there is no ambiguity). The number of arrows from $i$ to $j$ is the number of triangles $\triangle$ in ${\mathbb{T}}$ such that the arcs $\tau_i, \tau_j$ form two sides of $\triangle$, with $\tau_j$ following $\tau_i$ when going around the triangle $\triangle$ in the counter-clockwise orientation. Note that the interior triangles in ${\mathbb{T}}$ correspond to oriented 3-cycles in $Q_{{\mathbb{T}}}$.
Following [@ABCJP10; @LF09], in the unpunctured case, the algebra $A_{\mathbb{T}}$ is the quotient of the path algebra of the quiver $Q_{\mathbb{T}}$ by the two-sided ideal generated by the subpaths of length two of each oriented 3-cycle in $Q_{\mathbb{T}}$, then $A_{\mathbb{T}}$ is a quadratic monomial algebra. It is easy to see that $A_{\mathbb{T}}$ is also a gentle algebra.
Since $A_{\mathbb{T}}$ is a quadratic monomial algebra, for any triangulated surface $(S,M,{\mathbb{T}})$, we use the computations of Sköldberg in [@Sko99 Corollary 1]to compute the Hochschild homology of $A_{\mathbb{T}}$. Then, as we mention before, we use [@Lod92 Theorem 4.1.13 and Section 2.4.8] to compute the dimension of the cyclic homology and cohomology. Before give the result of Sköldberg we need to introduce some definitions and notations.
Let $A$ be a quadratic monomial algebra, observe that the algebra $A= \amalg_{n\in \mathbb N} A_n$ ( $A^!=\amalg_{n\in \mathbb N}A^!_{n})$ is $\mathbb N$-graded, where $A_n$ ($A_n^!$) is the $\mathbf k$ vector space generated by all paths of length $n$ of $A$(and $A^!$ respectively). This $\mathbb N$-graded is called the *internal grading*. For a $\mathbb N$-graded vector space $V$, we define $V_{\geq i}$ by $\amalg_{i\geq n}V_i$ for a natural $n$.
We give both $A$ and its dual $A^!$ an other $\mathbb N$-graded, called *homological grading*, by assigning to a basis element $a_1 \cdots a_l$ of $A$ homological degree $0$, and to a basis element $b_1\cdots b_n$ of $A^!$ homological degree $n$. For two elements $x, y$ homogeneous with respect to the homological grading, in a bi-graded algebra, we define their graded commutator $[x, y]=xy-(-1)^{\operatorname{homdeg}(x)\operatorname{homdeg}(y)}yx$, where $\operatorname{homdeg}(y)$ is the homological degree, and we extend this definition bi-linearly to any pair of elements. If $A$ is a bi-graded algebra we define the vector space of commutators $[A,A]$ as
$$[A,A]=\operatorname{span}_{\mathbf k}\{[x,y]\mid x, y\in A\}.$$
Denote by $\mathcal C_m$ the set of cycles $Q$ of length $m$. Observe that the cyclic group $C_n=\langle g \rangle$ acts on $\mathcal C_n$ by the action $g a_1 \cdots a_n=a_n a_1 \cdots a_{n-1}$. We say that two cycles $w$ and $w'$ are *cyclically equivalent* if $w$ and $w'$ are in the same orbit. Moreover, we say that any two cycles $w, w'$ are $[A,A]$-equivalent if $w$ and $w'$ are not elements of $[A,A]$ and they are cyclically equivalent, we denote this situation in the follow way $w\equiv w' \pmod{[A,A]}$.
According to Sköldberg [@Sko99 Corollary 1] the Hochschild homology of a quadratic monomial algebra is computed as follows.
\[computations\] The Hochschild homology of any quadratic monomial algebra $A= {\mathbf{k}Q}/I$ is given by
$$HH_{n}(A)\simeq \begin{cases}
\mathbf k Q_0 \oplus (A/[A,A])_{\geq 1} & \textrm{if $n=0$}\\
(A^!/[A^!, A^!])_{2} \oplus (A/[A,A])_{\geq 1} & \textrm{if $n=1$}\\
(A^!/[A^!, A^!])_{n} \oplus (A^!/[A^!, A^!])_{n+1} & \textrm{if $n\geq 2$}
\end{cases}$$
The computation of the cyclic homology of a quadratic monomial algebra over a field, not necessarily of characteristic zero, was given by Sköldberg in [@Sko01]. However, as consequence of Theorem \[computations\] and the short exact sequence
$$0\longrightarrow \widetilde{{\operatorname{HC}}}_{n-1}\longrightarrow \widetilde{{\operatorname{HH}}}_{n}\longrightarrow \widetilde{{\operatorname{HC}}}_{n}\longrightarrow 0,$$ where $\widetilde{{\operatorname{HH}}}_{n}(A)$ is the quotient ${\operatorname{HH}}_n(A)/{\operatorname{HH}}_n(A_0)$ and $\widetilde{{\operatorname{HC}}}_{n}(A)$ is the quotient ${\operatorname{HC}}_n(A)/{\operatorname{HC}}_n(A_0)$, see [@Lod92 Theorem 4.1.13], it is possible to compute the cyclic homology of a quadratic monomial algebra using an inductive argument, as follows.
[@Sko01 Theorem 4]\[computations2\] Let $A$ be a quadratic monomial algebra, where $\mathbf k$ is a field of characteristic $0$, the cyclic homology of $A$ is given by $$HC_{n}(A)\simeq \begin{cases}
\mathbf{k} Q_0 \oplus (A/[A,A])_{\geq 1} & \textrm{if $n=0$}\\
\mathbf k Q_0 \oplus (A^!/[A^!, A^!])_{n+1} & \textrm{if $n\equiv 0\pmod{2}$ and $n\neq 0$}\\
(A^!/[A^!, A^!])_{n+1} & \textrm{otherwise}
\end{cases}$$
As shown in Theorem \[computations\] and Theorem \[computations2\], to compute the cyclic and Hochschild homology of any algebra $A$, we need to compute the groups $A/[A,A]$ and $A^!/[A^!, A^!]$. In the following lemmas we show that those groups, for algebras coming from triangulated surfaces, are related to the internal triangles of the triangulation. In order to do that, we need to introduce some notation. Denote by $\operatorname{Int}({\mathbb{T}})=\{\Delta_1, \cdots, \Delta_t\}$ the set of internal triangles of the triangulated surface $(S,M, {\mathbb{T}})$ and by $Q(\Delta_i)$ the subquiver of $Q_{\mathbb{T}}$ associated to the internal triangle $\Delta_i$ for each $i=1, \dots, t$. By construction of $Q_{\mathbb{T}}$, the quiver $Q(\Delta_i)$ is a 3-cycle.
\[paths\] By definition $[A, A]$ is generated by the elements $[x,y]$ such that $x, y\in A$. In particular, if $w=[e_i\mid a_1a_2\dots a_l\mid e_j]$ is a path such that $e_i\neq e_j$, we have that $w= [w, e_j]$ . Then any path which is not a cycle is an element of $[A,A]$.
\[quotient1\] Let $(S,M, {\mathbb{T}})$ be a triangulated surface and $A_{\mathbb{T}}$ be the algebra associated to $(S,M, {\mathbb{T}})$. Then the quotient $(A_{\mathbb{T}}/[A_{\mathbb{T}},A_{\mathbb{T}}])_{\geq 1}$ in trivial.
Since any path which is not a cycle is an element of $[A_{{\mathbb{T}}},A_{\mathbb{T}}]$, it is enough to show that any cycle , non-zero in $A_{\mathbb{T}}$ of positive length, is an element of the commutator $[A_{\mathbb{T}},A_{\mathbb{T}}]$.
Suppose $w=[e_i\mid a_1a_2\dots a_l\mid e_i]$ is a cycle. Since $Q_{\mathbb{T}}$ has no loops, we have that $\mid w \mid\geq 2$. Moreover, the algebra $A_{\mathbb{T}}$ is a finite dimensional quadratic monomial algebra, then $a_l a_i$ is an element of the ideal $I_{\mathbb{T}}$, therefore $w=[a_1\cdots a_{l-1}, a_l]$, as we claim.
\[quotient2\] Let $(S, M, {\mathbb{T}})$ be a triangulated surface and $A_{\mathbb{T}}$ be the algebra associated to $(S,M, {\mathbb{T}})$ and $n\neq 3(2t+1)$ for any $t\in\mathbb Z^{\geq 0}$. Then the quotient $(A_{\mathbb{T}}^!/[A_{\mathbb{T}}^!, A_{\mathbb{T}}^!])_n$ is trivial. Moreover, if $n=3(2t+1)$ for some $t\in\mathbb Z^{\geq 0}$, then $\operatorname{dim}_{\mathbf k}(A_{\mathbb{T}}^!/[A_{\mathbb{T}}^!, A_{\mathbb{T}}^!])_n=\mid int({\mathbb{T}})\mid$
By definition $A^!$ is the path algebra $kQ_{\mathbb{T}}/J$, where $J$ is the ideal generated by the paths $m$ of length 2 such that $m\notin I$. Before give a basis for $A^!/[A^!, A^!]$, we first give the generators of $J$ and the non-zero paths in $A^!$.
Recall any path $w=[e_i\mid a_1a_2\dots a_l\mid e_j]$, non-zero in $A$, is coming from arcs attached to a marked point $x$ as in Figure \[path\], and each arrow is opposite to $x$. Denote by $w_x$ the maximal non-zero path coming from the arcs attached to the marked point $x$. Then $J$ is generated by the subpaths of length $2$ of each maximals non-zero paths, therefore element of the basis of $A^!$ is a sequence of consecutive arrows of a $3$-cycle $Q(\Delta)$ associated to an internal triangle $\Delta$ of ${\mathbb{T}}$ or an arrow in $Q_{\mathbb{T}}$.
(0,0) ellipse (0.5cm and 0.8cm); (0,-0.8) node\[above\][$x$]{}circle (1.5pt); ; (0,-0.8) to node\[midway\][$\tau_j$]{} (1.7,-2) (0,-0.8) to (-0.3,-2) (0,-0.8) to node[$\tau_i$]{} (-1.7,-2); at (0.45, -1.8);
Since the length of any cycle in $A^!$ is always a multiple of $3$ and any path which is not a cycle is element of $[A^!, A^!]$, it is clear that $[A^!, A^!]\bigcap A^!_n=A^!_n$ for any $n$ which is not multiple of 3.
Now, suppose $n$ is multiple of 3 and even. We claim that $[A^!, A^!]\bigcap A^!_n=A^!_n$. Let $w$ be path of length $n$. If $w$ is not a cycle, then $w$ is an element of $[A^!, A^!]\bigcap A_n$, see Remark \[paths\]. Suppose $w$ is a cycle, then $w=(a_1a_2a_3)^t$ for some $t\in\mathbb N$ such that $n=3t$. Therefore $[a_1a_2a_3, (a_1a_2a_3)^{t-1}]=(a_1a_2a_3)^t-(-1)^{(3)(3t-3)}(a_1a_2a_3)^t$, observe that $(-1)^{(3)(3t-3)}=-1$, then $[a_1a_2a_3, (a_1a_2a_3)^{t-1}]=2(a_1a_2a_3)^t$, hence $[A^!, A^!]\bigcap A^!_n=A^!_n$.
Then, in both cases, when $n$ is not multiple of $3$ or $n$ is multiple of $3$ and even, we have that $(A^!/[A^!, A^!])_n=0$.
Finally, suppose $n$ is multiple of 3 and odd. By definition $$[w_1, w_2]=w_1w_2 - (-1)^{\mid w_1\mid \mid w_2\mid}w_2w_1$$ is a generator of $[A^!, A^!]\bigcap A^!_n$, if $w_1$ and $w_2$ are paths such that the length $\mid w_1w_2\mid=n$ or $\mid w_2w_1\mid =n$. Moreover, since $n$ is odd and $n= \mid w_1 \mid +\mid w_2 \mid$, we have that $(-1)^{\mid w_1\mid \mid w_2\mid}=1$, then $$[w_1, w_2]=w_1w_2 - w_2w_1.$$
Let $g$ be a generator of the cyclic group $C_n$. We claim that any generator of $[A^!, A^!]\bigcap A^!_n$ is an element of the form:
- $(c_1c_2c-3)^t -g^r(c_1c_2c_3)^t$ where $c_1c_2c_3$ is a 3-cycle coming from an internal triangle $\Delta_k$ of ${\mathbb{T}}$ and $r\in \{1, 2, \dots, n-1\}$ or
- a path $w'$ which is not a cycle, of length $n$.
Let $w_1=a_1a_2\dots a_{l_1}$ and $w_2=b_1b_2\dots b_{l_2}$ paths such that the length $\mid w_1w_2\mid=n$ or $\mid w_2w_1\mid=n$. Suppose $[w_1,w_2]$ is not a path, then $w_1w_2$ and $w_2w_1$ are non-zero cycles in $A^!$, and therefore both of them are sequences of consecutive arrows of the same $3$-cycle $Q(\Delta)$ associated to an internal triangle $\Delta$ of ${\mathbb{T}}$, hence $w_2w_1= g^r w_1w_2$ for some $r\in \{1,2\dots, n-1\}$ and $w_1w_2=(c_1c_2c_3)^t$, for some $c_1c_2c_3$ is a 3-cycle coming from an internal triangle $\Delta_k$ of ${\mathbb{T}}$, as we claim.
And, as consequence, $\operatorname{dim}_{\mathbf k}((A^!/[A^!, A^!])_n)= \mid {\operatorname{int}({\mathbb{T}})}\mid$.
Let $A_{\mathbb{T}}$ be the algebra associated to the triangulated surface $(S,M,{\mathbb{T}})$.
We first compute the cyclic homology groups ${\operatorname{HC}}_n(A_{\mathbb{T}})$. Since $(A_{\mathbb{T}}/[A_{\mathbb{T}},A_{\mathbb{T}}])_{\geq 1}$ is trivial by Lemma \[quotient1\], we have that ${\operatorname{HC}}_{0}(A_{\mathbb{T}})=\mathbf{k}Q_0$. Let $n\geq 1$, by Lemma \[quotient2\] we have that the quotient $(A_{\mathbb{T}}^!/[A_{\mathbb{T}}^!, A_{\mathbb{T}}^!])_{n+1}$ is also trivial for any $n+1\neq 3(2k+1)$, then by Theorem \[computations2\] we have that the $n$-cyclic homology group of $A_{\mathbb{T}}$ is given by $${\operatorname{HC}}_{n}(A_{\mathbb{T}})\simeq \begin{cases}
\mathbf k Q_0 & \textrm{if $n=0$}\\
\mathbf k Q_0 \oplus (A_{\mathbb{T}}^!/[A_{\mathbb{T}}^!, A_{\mathbb{T}}^!])_{n+1} & \textrm{if $n\equiv 2\pmod{6}$}\\
0 & \textrm{otherwise}
\end{cases}$$
Similarly by Theorem \[computations\] and also Lemmas \[quotient1\] and \[quotient2\], we have that the Hochschild homology groups of $A_{\mathbb{T}}$ is given by
$${\operatorname{HH}}_n(A_{\mathbb{T}})\simeq \begin{cases}
\mathbf{k}(Q_{\mathbb{T}})_0 & \textrm{if $n=0$}\\
(A_{\mathbb{T}}^!/[A_{\mathbb{T}}^!, A_{\mathbb{T}}^!])_{n} & \textrm{if $n\equiv 3\pmod{6}$}\\
(A_{\mathbb{T}}^!/[A_{\mathbb{T}}^!, A_{\mathbb{T}}^!])_{n+1} & \textrm{if $n\equiv 2\pmod{6}$}\\
0 & \textrm{otherwise}
\end{cases}$$
It follows from Loday [@Lod92 Section 2.4.8], that for any finite dimensional algebra $A$ with unit the $n$-cyclic homology group ${\operatorname{HC}}^n(A)$ of $A$ and the $n$-cyclic cohomology group $\operatorname{Hom}_k({\operatorname{HC}}_n(A), k)$ of $A$ are isomorphic. Since $A_{\mathbb{T}}$ is a finite dimensional algebra for any triangulated surface $(S,M,{\mathbb{T}})$, the Corollary follows from the main Theorem.
To conclude this note, we compute the dimension of the cyclic (co)homologies groups and dimension of the Hochschild homology groups of two algebras from surfaces, which are closed related.
Consider the triangulated surface $(S,M,{\mathbb{T}})$ of the Figure \[3boundaries\], where $S$ is a surface of genus zero with 3 boundaries components and four marked points.
Observe that the quiver $Q_{\mathbb{T}}$ has 7 vertices and there are 3 internal triangles: $\triangle_1(\tau_1, \tau_5, \tau_7)$, $\triangle_2(\tau_1,\tau_6,\tau_2)$ and $\triangle_3(\tau_6,\tau_5,\tau_4)$. Then, according to our main Result, the dimension of the Hochschild homology groups ${\operatorname{HH}}_n(A_{\mathbb{T}})$ are computed as follows:
$$\operatorname{dim}_{\mathbf k}{\operatorname{HH}}_n(A_{\mathbb{T}})= \begin{cases}
7 & \textrm{if $n=0$}\\
3 & \textrm{if $n\equiv 3\pmod{6}$ or $n\equiv 2\pmod{6}$ }\\
0 & \textrm{otherwise}
\end{cases}$$
Since $\operatorname{dim}_{\mathbf k}({\operatorname{HC}}^n(A_{\mathbb{T}}))=\operatorname{dim}_{\mathbf k}({\operatorname{HC}}_n(A_{\mathbb{T}}))$ by Corollary for any $n\in \mathbb Z^{\geq 0}$, the dimension of the cyclic (co)homology groups ${\operatorname{HC}}_n(A_{\mathbb{T}})$ and ${\operatorname{HC}}^n(A_{\mathbb{T}})$ are computed as follows:
$$\operatorname{dim}_{\mathbf k}({\operatorname{HC}}^n(A_{\mathbb{T}}))=\operatorname{dim}_{\mathbf k}({\operatorname{HC}}_n(A_{\mathbb{T}}))= \begin{cases}
7 & \textrm{if $n=0$}\\
10 & \textrm{if $n\equiv 2\pmod{6}$}\\
0 & \textrm{otherwise}
\end{cases}$$
Finally, consider the triangulated surface $(S,M, {\mathbb{T}}')$ of Figure \[3boundaries2\], which is obtained by removing the arc $\tau_7$ of the triangulation ${\mathbb{T}}$ and replacing it by $\tau_7'$, that is, a *flip of arc $\tau_7$*.
In this case, the quiver $Q_{\mathbb{T}}'$ has also 7 vertices, which is actually an invariant of $(S,M)$, but there are only 2 internal triangles: $\Delta_2(\tau_1,\tau_6,\tau_2)$ and $\Delta_3(\tau_6, \tau_5, \tau_4)$ , then:
$$\operatorname{dim}_{\mathbf k}{\operatorname{HH}}_n(A_{\mathbb{T}}')= \begin{cases}
2 & \textrm{if $n\equiv 3\pmod{6}$ or $n\equiv 2\pmod{6}$ }\\
\operatorname{dim}_{\mathbf k}{\operatorname{HH}}_n(A_{\mathbb{T}}) & \textrm{otherwise}
\end{cases}$$ and $$\operatorname{dim}_{\mathbf k}({\operatorname{HC}}^n(A_{\mathbb{T}}'))=\operatorname{dim}_{\mathbf k}({\operatorname{HC}}_n(A_{\mathbb{T}}'))= \begin{cases}
9 & \textrm{if $n\equiv 2\pmod{6}$}\\
\operatorname{dim}_{\mathbf k}({\operatorname{HC}}^n(A_{\mathbb{T}})) & \textrm{otherwise}
\end{cases}$$
Therefore, the (co)homologies computed in this note are not invariant under *flips* and *mutations of quivers with potentials*.
[10]{}
I. Assem, T. Br[ü]{}stle, G. Charbonneau-Jodoin, and P.-G. Plamondon. Gentle algebras arising from surface triangulations. , 4(2):201–229, 2010.
H. Cartan and S. Eilenberg. . Princeton University Press, Princeton, N. J., 1956.
A. Connes. Spectral sequence and homology of currents for operator algebras. , 41(81):27–9, 1981.
H. Derksen, J. Weyman, and A. Zelevinsky. Quivers with potentials and their representations. [I]{}. [M]{}utations. , 14(1):59–119, 2008.
S. Fomin, M. Shapiro, and D. Thurston. Cluster algebras and triangulated surfaces. [I]{}. [C]{}luster complexes. , 201(1):83–146, 2008.
G. Hochschild. On the cohomology groups of an associative algebra. , 46:58–67, 1945.
D. Labardini-Fragoso. Quivers with potentials associated to triangulated surfaces. , 98(3):797–839, 2009.
S. Ladkani. On <span style="font-variant:small-caps;">J</span>acobian algebras from closed surfaces. arXiv:1207.3778.
J.-L. Loday. , volume 301 of [*Grundlehren der Mathematischen Wissenschaften \[Fundamental Principles of Mathematical Sciences\]*]{}. Springer-Verlag, Berlin, 1992. Appendix E by Mar[í]{}a O. Ronco.
E. Sk[ö]{}ldberg. The [H]{}ochschild homology of truncated and quadratic monomial algebras. , 59(1):76–86, 1999.
E. Sk[ö]{}ldberg. Cyclic homology of quadratic monomial algebras. , 156(2-3):345–356, 2001.
S. Trepode and Y. Valdivieso-D[í]{}az. On finite dimensional [J]{}acobian algebras. , pages 1–14, 2015.
|
---
abstract: 'We consider a family of stochastic 2D Euler equations in vorticity form on the torus, with transport type noises and $L^2$-initial data. Under a suitable scaling of the noises, we show that the solutions converge weakly to that of the deterministic 2D Navier–Stokes equations. Consequently, we deduce that the weak solutions of the stochastic 2D Euler equations are approximately unique and “weakly quenched exponential mixing”.'
author:
- 'Franco Flandoli[^1] Lucio Galeati[^2] Dejun Luo[^3]'
title: 'Scaling limit of stochastic 2D Euler equations with transport noises to the deterministic Navier–Stokes equations'
---
Introduction
============
Let $\T^2=\R^2/\Z^2$ be the 2D torus and $\Z^2_0= \Z^2\setminus \{0\}$ the nonzero lattice points. Define $$\sigma_k(x)= \frac{k^\perp}{|k|} e_k(x),\quad x\in \T^2,\ k\in \Z^2_0,$$ where $k^\perp= (k_2, -k_1)$ and $\{e_k \}_{k\in \Z^2_0}$ is the usual trigonometrical basis of $L^2(\T^2)$, see the beginning of Section 2. Then $\{\sigma_k\}_{k\in \Z^2_0}$ is a complete orthonormal basis of the space of square integrable, divergence free vector fields on $\T^2$ with zero mean.
In a previous work [@FlaLuo-2], the first and the third named authors studied the vorticity form of the stochastic 2D Euler equations with transport type noise: $$\label{approx-eq}
\d \xi^N + u^N\cdot\nabla\xi^N\,\d t= 2\sqrt{\nu}\,\eps_N \sum_{|k|\leq N} \frac1{|k|} \sigma_{k} \cdot \nabla\xi^N \circ \d W^{k},$$ where $\nu>0$ is a constant, $\{\eps_N\}_{N\geq 1}$ a sequence of positive numbers and $\{W^k\}_{k\in \Z^2_0}$ is a family of independent standard Brownian motions on some filtered probability space $(\Omega, \mathcal F, \mathcal F_t, \P)$. In the above equation, $u^N= (u^N_1, u^N_2)$ is the velocity field and $\xi^N= \nabla^\perp \cdot u^N= \partial_2 u^N_1 - \partial_1 u^N_2$ is the vorticity; conversely, $u^N =K\ast \xi^N$ where $K$ is the Biot–Savart kernel. The equation has the enstrophy measure $\mu$ on $\T^2$ as the invariant measure, which is supported on $H^{-1-}(\T^2) = \bigcap_{s<-1} H^s(\T^2)$, $H^s(\T^2)$ being the usual Sobolev space on $\T^2$. For any fixed $N\geq 1$, it is known that admits a stationary solution $\xi^N$ with paths in $C\big([0,T], H^{-1-}(\T^2)\big)$ (taking $\rho_0\equiv 1$ in [@FlaLuo-1 Theorem 1.3]). We choose the parameter $\eps_N$ in such a way that it compensates the coefficient appearing in the Itô–Stratonovich correction term. More precisely, let $$\eps_N= \bigg(\sum_{|k|\leq N} \frac1{|k|^2} \bigg)^{-1/2} \sim (\log N)^{-1/2},$$ then, in the Itô formulation, becomes $$\d \xi^N + u^N\cdot\nabla\xi^N\,\d t= 2\sqrt{\nu}\,\eps_N \sum_{|k|\leq N} \frac1{|k|} \sigma_{k} \cdot \nabla\xi^N \, \d W^{k} + \nu\Delta \xi^N\,\d t.$$ It was proved in [@FlaLuo-2] that the stationary solutions $\xi^N$ of converge to the unique-in-law stationary solution of the stochastic 2D Navier–Stokes equations driven by additive space-time white noise $$\label{NS-vort}
\d\xi +u\cdot\nabla\xi\,\d t=\nu\Delta\xi \,\d t+ \sqrt{2\nu } \, \nabla^{\perp} \cdot\d W.$$ Here, $W= \sum_{k\in \Z^2_0} \sigma_{k} W^{k}$ is a cylindrical Brownian motion in the Hilbert space of divergence free vector fields on $\mathbb T^2$. The equation , in the velocity form, has been studied intensively in the past three decades, see for instance [@AC; @AF; @DaPD]. In particular, it was shown in [@DaPD] that has a pathwise unique strong solution for $\mu$ almost every initial data in Besov spaces of negative order. As a consequence of the Yamada–Watanabe type theorem (see e.g. [@Kur]), the stationary solutions of are unique in law.
On the other hand, the second named author considered in [@Gal] a similar scaling limit for a sequence of stochastic transport linear equations, but in a different regime, namely for function-valued solutions of suitable regularity. To state the result we introduce the notation $\ell^p\ (p\in [1,\infty])$ which are the usual spaces of real sequences indexed by $\Z^2_0$ and denote the norm by $\|\cdot\|_{\ell^p}$. Let $\theta^N_\cdot \in \ell^2 (N\geq 1)$ be a sequence verifying, for all $N\in \N$, $$\label{theta}
\theta^N_k = \theta^N_j \quad \mbox{whenever } |k|= |j|$$ and $$\label{theta-N}
\lim_{N\to\infty} \frac{\|\theta^N_\cdot\|_{\ell^\infty}} {\|\theta^N_\cdot\|_{\ell^2}} =0.$$ The main result of [@Gal] asserts that, if $$\label{eps-N}
\eps_N = \frac{2\sqrt{\nu}} {\|\theta^N_\cdot\|_{\ell^2}},$$ then the solutions $\xi^N$ of the sequence of stochastic transport linear equations ($b$ is a vector field on $\T^2$) $$\d \xi^N = b\cdot \nabla \xi^N\,\d t + \eps_N \sum_{k\in \Z^2_0} \theta^N_k \sigma_k \cdot \nabla \xi^N \circ \d W^k$$ converge to the solution of the parabolic Cauchy problem $$\partial_t \xi= \nu \Delta \xi+ b\cdot \nabla \xi, \quad \xi(0)= \xi_0.$$ For $\xi_0\in L^2(\T^2)$, this last equation admits a unique weak solution in $L^2\big(0,T; L^2(\T^2) \big)$ under mild assumptions on $b$, see for instance [@Gal Lemma 3.3].
Motivated by the above discussions, we consider, in the regime of regular solutions (compared to the white noise solutions considered in [@FlaLuo-2]), the stochastic 2D Euler equations $$\label{SEE-vort}
\d \xi^N + u^N\cdot\nabla\xi^N\,\d t= \eps_N \sum_{k\in \Z^2_0} \theta^N_k \sigma_{k} \cdot \nabla\xi^N \circ \d W^{k},$$ where $\big\{ \theta^N_\cdot \big\}_{N\geq 1}$ satisfies and , and $\eps_N$ is defined as in . We assume $\xi^N_0 = \xi_0\in L^2(\T^2)$ with zero mean. Then one can show that the equation admits a solution $\xi^N$ (weak in both analytic and probabilistic sense), satisfying $$\sup_{t\in [0,T]} \int_{\T^2} |\xi^N(t,x)|^2 \,\d x <+\infty.$$ We will prove that such more regular solutions of equation converge to the unique solution of the *deterministic* 2D Navier–Stokes equations $$\label{NSE}
\partial_{t}\xi +u\cdot\nabla\xi=\nu\Delta\xi, \quad \xi(0)=\xi_0 .$$ According to the classical theory of 2D Navier–Stokes equations (see [@Teman Theorem 3.2] for the velocity form), the above equation has a unique solution.
A direct consequence of the above scaling limit is that the transport type noises considered here regularize the 2D Euler equations asymptotically. More precisely, it is well known that the 2D Euler equations has a unique solution if the initial data $\xi_0$ belongs to $L^\infty(\T^2)$, while the uniqueness of solutions remains an open problem in the case $\xi_0\in L^p(\T^2)$ for $p<\infty$. Although we cannot prove that the stochastic 2D Euler equation has a unique solution for $L^2(\T^2)$-initial data, the above result shows that, in the limit, we get the uniquely solvable 2D Navier–Stokes equation . As a result, the distances between the laws of weak solutions of tend to zero as $N\to \infty$. We call such a property the approximate weak uniqueness, see Section \[subsec-uniqueness\] for more details.
Our main result is the convergence to deterministic Navier–Stokes equations; however, tuning parameters in the right way we may construct sequences converging to deterministic Euler equation. More precisely, given any viscosity solution of 2D Euler, we can find a suitable sequence converging to it, see Section \[sec 6.2\] for more details. We do not know the converse, namely if every limiting measure constructed in this way is a superposition of viscosity solutions, but our result makes this conjecture plausible. It is very important to identify selection criteria, for instance by viscosity, by noise or by additional physical requirements, in view of the multiplicity of solutions found recently by the method of convex integration [@DeLellSze]; although our result is not conclusive, it makes plausible that the zero-noise limit selects viscosity solutions. Notice that this is different from what happens for certain examples of linear transport equations [@AttFla].
Our result also has interesting implications related to the mixing behavior of incompressible flows, a phenomenon which recently attracted a lot of attention, see for instance [@ACM14; @ACM19; @YZ] and the references therein. In [@ACM19], Alberti et al. considered the solutions to the continuity equation $$\label{CE}
\partial_t \rho_t + \div(\rho_t u) =0$$ and estimated the “mixedness” of $\rho_t$ as $t\to \infty$ in terms of the negative Sobolev norm $\|\rho_t\|_{\dot H^{-1}}$. Here $\dot H^{s}(\T^2)\, (s\in\R)$ denote the homogeneous Sobolev spaces. They constructed a bounded and divergence free vector field $u\in C^\infty \big([0,\infty) \times \R^d, \R^d\big)$ and a bounded solution $\rho\in C^\infty \big([0,\infty) \times \R^d \big)$ to such that, for any $0<s<2$, it holds $$\|\rho_t\|_{\dot H^{-s}} \leq C_s\, e^{-c s t}, \quad t\geq 0,$$ where $C_s>0$ and $c>0$ are constants. Such exponential mixing result is in fact optimal, taking into account the lower bounds on functional mixing scale proved in [@IKX; @Seis]. Using our limit result and the exponential decay of the energy and the enstrophy of the solution to the Navier–Stokes equation , we can prove that the solutions to the stochastic 2D Euler equations satisfy the “weakly quenched exponential mixing” property, see Section \[subsec-mixing\] for more precise statements.
However, the decay in $\dot{H}^{-s}$-norms does not extend to the $L^{2}$-norm and our result does not imply anomalous dissipation of enstrophy. This is a difficult open question, which is discussed in Section \[subsec-anomalous\].
This paper is organized as follows. In Section 2, we introduce some notations and state the main results, including the existence of weak solutions to the stochastic 2D Euler equations and the scaling limit to the deterministic 2D Navier–Stokes equation , as well as a finite dimensional convergence result. The proofs of these results are provided in Sections 3 to 5. In the last sections, we discuss the consequences of the scaling limit in more detail.
Functional settings and main results {#sec2}
====================================
In this section, we give some more notations for functional spaces and state the main results of the paper. Let $C^\infty(\T^2)$ be the space of smooth function on $\T^2$. We write $\<\cdot , \cdot \>$ and $\|\cdot \|_{L^2}$ for the inner product and the norm in $L^2(\T^2)$. Recall also the Sobolev spaces $H^s(\T^2),\, s\in \R$. Denote by $$e_k(x) = \sqrt{2} \begin{cases}
\cos(2\pi k\cdot x), & k\in \Z^2_+, \\
\sin(2\pi k\cdot x), & k\in \Z^2_-,
\end{cases} \quad x\in \T^2,$$ where $\Z^2_+= \{k\in \Z^2_0: (k_1>0) \mbox{ or } (k_1=0,\, k_2>0)\}$ and $\Z^2_-= -\Z^2_+$. To save notations, we shall write the vector valued spaces $L^2(\T^2, \R^2)$ and $H^s(\T^2, \R^2)$ simply as $L^2(\T^2)$ and $H^s(\T^2)$. We denote by $H$ (resp. $V$) the subspace of $L^2(\T^2)$ (resp. $H^1(\T^2)$) of functions with zero mean. Moreover, we assume $\{W^k\}_{k\in \Z^2_0}$ is a family of independent $\mathcal F_t$-Brownian motions on the filtered probability space $(\Omega, \mathcal F, \mathcal F_t, \P)$.
First, we fix $\theta_\cdot \in \ell^2$ verifying and consider the following stochastic 2D Euler equation in vorticity form: $$\label{SEE-vort-1}
\d \xi + u\cdot \nabla\xi\,\d t= \eps \sum_{k\in \Z^2_0} \theta_k \sigma_{k} \cdot \nabla\xi \circ \d W^{k}, \quad \xi(0)= \xi_0\in H,$$ with $$\label{epsilon}
\eps =\frac{2\sqrt{\nu}}{\|\theta_\cdot\|_{\ell^2}} .$$ Using , it is not difficult to prove the simple equality (cf. [@FlaLuo-2 Lemma 2.6]) $$\label{useful-equality}
\sum_{k\in \Z^2_0} \theta_k^2\, \sigma_k(x)\otimes \sigma_k(x) \equiv \frac12 \|\theta_\cdot\|^2_{\ell^2} I_2,\quad x\in\mathbb{T}^2,$$ where $I_2$ is the $2\times 2$ identity matrix. From this we deduce the Itô formulation of : $$\label{SEE-vort-2}
\d \xi + u\cdot \nabla\xi\,\d t= \nu \Delta\xi\,\d t + \eps \sum_{k\in \Z^2_0} \theta_k \sigma_{k} \cdot \nabla\xi\, \d W^{k}.$$ This equation is understood as follows: for any $\phi \in C^\infty(\T^2)$, it holds $\P$-a.s. for all $t\in [0,T]$, $$\label{SEE-formulation}
\<\xi_t,\phi\> = \<\xi_0,\phi\> + \int_0^t \<\xi_s, u_s\cdot \nabla\phi\>\,\d s + \nu \int_0^t \<\xi_s, \Delta\phi\>\,\d s - \eps \sum_{k\in \Z^2_0} \theta_k \int_0^t \<\xi_s, \sigma_k\cdot \nabla\phi\>\,\d W^k_s.$$ Recall that $u$ is related to $\xi$ via the Biot–Savart kernel $K$ on $\T^2$: $u=K\ast \xi$. Thus if $\xi\in L^2 \big(\Omega,L^2 (0,T; H) \big)$, then $u\in L^2 \big(\Omega,L^2 (0,T; V) \big)$. Under this condition, if $\xi$ (and also $u$) is $\mathcal F_t$-progressively measurable, it is clear that all the terms in the above equation makes sense. For instance, the stochastic integral is a square integrable martingale since $$\aligned
\E \Bigg(\sum_{k\in \Z^2_0} \theta_k^2 \int_0^t \< \xi_s, \sigma_k\cdot \nabla\phi\>^2\,\d s \Bigg) &
\leq \|\theta_\cdot\|_{\ell^\infty}^2 \E \Bigg( \int_0^T \sum_{k\in \Z^2_0}\< \xi_s\nabla\phi, \sigma_k \>^2\,\d s\Bigg)\\
&\leq \|\theta_\cdot\|_{\ell^\infty}^2 \E \bigg( \int_0^T \Vert\xi_s\nabla\phi \Vert_{L^2}^2\,\d s\bigg)\\
&\leq \|\theta_\cdot\|_{\ell^\infty}^2 \|\nabla\phi\|_\infty^2 \E \int_0^T \|\xi_s\|_{L^2}^2 \,\d s <+\infty,
\endaligned$$ where we used the fact that $\{\sigma_k\}_{k\in\mathbb{Z}_0^2}$ form an (incomplete) orthonormal system in $L^2(\T^2,\R^2)$. From this result we can give the definition of solutions to .
\[SEE-def\] We say that has a weak solution if there exist a filtered probability space $\big( \Omega, \mathcal F, \mathcal F_t, \P\big)$, a sequence of independent $\mathcal F_t$-Brownian motions $\{W^k\}_{k\in \Z^2_0}$ and an $\mathcal F_t$-progressively measurable process $\xi\in L^2 \big(\Omega,L^2 (0,T; H) \big)$ with $\mathbb{P}$-a.s. weakly continuous trajectories such that for any $\phi \in C^\infty(\T^2) $, the equality holds $\P$-a.s. for all $t\in [0,T]$.
Note that the solution is weak in both the probabilistic and the analytic sense. Our first result is the existence of solutions to .
\[thm-existence\] For any $\xi_0\in H$, there exists at least one weak solution to , satisfying $$\label{energy estimate thm-existence}
\sup_{t\in [0,T]} \|\xi_t \|_{L^2} \leq \|\xi_0 \|_{L^2}\quad \mathbb{P}\text{-a.s.}$$
Next, we take a sequence $\theta^N \in \ell^2$, satisfying and , and consider the stochastic 2D Euler equations . Similarly to the above discussions, is understood as follows: for any $\phi\in C^\infty(\T^2)$ and $t\in [0,T]$, $$\label{SEE-approx-formulation}
\big\<\xi^N_t,\phi \big\> = \<\xi_0,\phi\> + \int_0^t \big\<\xi^N_s, u^N_s\cdot \nabla\phi \big\>\,\d s + \nu \int_0^t \big\<\xi^N_s, \Delta\phi \big\>\,\d s - \eps_N \sum_{k\in \Z^2_0} \theta^N_k \int_0^t \big\<\xi^N_s, \sigma_k\cdot \nabla\phi \big\>\,\d W^k_s.$$ We remark that Theorem \[thm-existence\] only provides us with weak solutions, thus the processes $\xi^N_\cdot$ might be defined on different probability spaces. The relevant notion of convergence of these processes is the weak convergence of their laws. Here is the main result of this paper.
\[thm-main\] Let $Q^N$ be the law of $\xi^N_\cdot$, $N\geq 1$. Then the family $\big\{ Q^N \big\}_{N\geq 1}$ is tight in $C([0,T];H^-)$ and it converges weakly to $\delta_{\xi_\cdot}$, where $\xi_\cdot$ is the unique solution of the 2D Navier–Stokes equations .
Theorem \[thm-main\] also implies convergence of the associated advected passive scalars, see Corollary \[corollary sec4\] for the precise statement.
\[rem-theta\] If $\xi_0 \in L^\infty(\T^2)$, then under slightly stronger conditions on $\theta_\cdot$ (e.g. assume $\theta_k \sim |k|^{-2-\delta}$ for some $\delta>0$), the equation has a unique solution in $L^\infty \big( [0,T] \times \T^2 \big)$, see for instance [@BrFlMa Theorem 2.10]. Note that in the approximating equations , we can take $\theta^N\in \ell^2$ such that there are only finitely many $k$ for which $\theta^N_k\neq 0$, and at the same time satisfying , for instance, $\theta^N_k = {\bf 1}_{\{|k|\leq N \}}$. Therefore, if we approximate $\xi_0\in L^2(\T^2)$ by a sequence of bounded functions $\xi^N_0 \in L^\infty(\T^2)$, then the approximating sequence $\xi^N_\cdot\, (N\geq 1)$ are unique solutions of the equations . Moreover in this case we can consider the sequence $\xi^N_\cdot$ to be defined on the same probability space $(\Omega, \mathcal F, \P)$, again by the results in [@BrFlMa]; thus convergence in law to a deterministic limit implies also convergence in probability. The energy bound then also implies convergence in $L^p(\Omega, \mathbb{P})$, for any $p>\infty$.
In Sections \[sec3\] and \[sec4\], we prove Theorems \[thm-existence\] and \[thm-main\] respectively. Then in Section \[sec5\] we show that the same result can be achieved, under the same scaling, already working with finite dimensional approximations of Galerkin type. More precisely, denoting by $\Pi_N$ the orthogonal projection of $L^2(\T^2)$ into $H_N=\text{span}\{e_k: k\in\mathbb{Z}^2_0, \vert k\vert\leq N \}$, we consider for each $N$ the solution $\tilde\xi^N$ of the SDE: $$\label{SDE sec2}
\d \tilde\xi^N = -\Pi_N\big( \big(K\ast \tilde\xi^N \big)\cdot\nabla\tilde\xi^N \big)\,\d t + \varepsilon_N \sum_{k\in\mathbb{Z}^2_0}\theta^N_k \Pi_N\big( \sigma_k\cdot\nabla \tilde\xi^N \big)\circ\d W^k, \quad \tilde\xi^N_0=\Pi_N \xi_0.$$ The variables $\big\{\tilde\xi^N \big\}_{N\in \N}$ are defined on the same probability space with respect to the same Brownian motions $\{W^k \}_{k\in\mathbb{Z}^2_0}$. In this case we can prove the following
\[thm sec5\] Suppose the sequence $\big\{\theta^N \big\}_N\subset \ell^2$ satisfies and the additional condition $$\label{condition thm5}
\lim_{N\to\infty} \big\Vert \theta^N_\cdot \big\Vert_{\ell^2}^{-2} \sum_{k: \vert k-j\vert>N} \big(\theta^N_k \big)^2 =0\quad \forall j\in\mathbb{Z}_0^2.$$ Then the sequence $\big\{ \tilde\xi^N \big\}_{N\geq 1}$ converges in probability to $\xi_\cdot$, where $\xi_\cdot$ is the unique solution of Navier–Stokes equation with initial data $\xi_0$.
It can be checked for instance that condition is satisfied for $\big\{\theta^N_\cdot \big\}_N$ given by $\theta^N_k = \vert k\vert^{-\alpha}\,{\bf 1}_{\{\vert k\vert\leq N \}}$, for any $\alpha\in [0,1]$.
Existence of solutions to {#sec3}
==========================
In this section we give a proof of Theorem \[thm-existence\] by using the Galerkin approximation and the compactness method.
To use the method of Galerkin approximation, we introduce some notations. For $N\geq 1$, let $H_N={\rm span}\{e_k: k\in \Z^2_0, |k|\leq N\}$ which is a finite dimensional subspace of $H$. Denote by $\Pi_N: H\to H_N$ the orthogonal projection: $\Pi_N \xi= \sum_{|k|\leq N} \<\xi, e_k\> e_k$. $\Pi_N$ can also act on vector valued functions. Let $$b_N(\xi) = \Pi_N\big( (K\ast \Pi_N\xi) \cdot \nabla(\Pi_N\xi) \big), \quad G_N^k(\xi)= \Pi_N\big( \sigma_k \cdot \nabla(\Pi_N\xi) \big),\quad k\in \Z^2_0.$$ Note that, for fixed $N$, there are only finitely many $k\in \Z^2_0$ such that $G_N^k$ is not zero. We shall view $b_N$ and $G_N^k$ as vector fields on $H_N$ whose generic element is denoted by $\xi_N$. These vector fields have the following useful properties: $$\label{properties}
\big\< b_N(\xi_N), \xi_N \big\> = \big\< G_N^k(\xi_N), \xi_N \big\>=0\quad \mbox{for all } \xi_N\in H_N,$$ which can be proved easily from the definitions of $b_N$ and $G_N^k$, and the integration by parts formula. Consider the finite dimensional version of on $H_N$: $$\label{SDE}
\d\xi_N(t)= -b_N(\xi_N(t))\,\d t + \nu \Delta \xi_N(t)\,\d t + \eps \sum_{k\in \Z^2_0} \theta_k G_N^k(\xi_N(t)) \, \d W^{k}_t, \quad \xi_N(0)= \Pi_N\xi_0,$$ where $\xi_0\in H$ is the initial condition in Theorem \[thm-existence\]. We remark that the sum over $k$ is a finite sum. Its generator is $$\L_N \varphi(\xi_N) = \< - b_N(\xi_N) +\nu \Delta \xi_N, \nabla_N \varphi(\xi_N)\> + \frac{\eps^2}2 \sum_{k\in \Z^2_0} \theta_k^2 \, {\rm Tr}\big[ \big(G_N^k \otimes G_N^k \big) \nabla_N^2 \varphi \big](\xi_N)$$ for any $\varphi\in C_b^2(H_N)$.
\[energy-estimate\] The equation has a unique strong solution $\xi_N(t)$ satisfying $$\sup_{t\in [0,T]} \|\xi_N(t) \|_{L^2} \leq \|\xi_N(0) \|_{L^2} \quad \P \mbox{\rm-}a.s.$$
The vector fields $b_N$ and $G^k_N$ are respectively quadratic and linear on the finite dimensional space $H_N$, therefore they are smooth. By the standard SDE theory, local existence and uniqueness of strong solutions to holds for any initial data. By the Itô formula, $$\label{energy-estimate-1}
\aligned
\d\|\xi_N(t) \|_{L^2}^2=&\ -2 \<\xi_N(t), b_N(\xi_N(t))\>\,\d t +2\nu \<\xi_N(t), \Delta \xi_N(t) \> \,\d t\\
&\ + 2\,\eps \sum_{k\in \Z^2_0} \theta_k \big\<\xi_N(t), G_N^k(\xi_N(t)) \big\>\, \d W^{k}_t+ \eps^2 \sum_{k\in \Z^2_0} \theta_k^2 \big\| G_N^k(\xi_N(t))\big\|_{L^2}^2\,\d t.
\endaligned$$ The first and the third terms on the right hand side vanish due to . Moreover, noting that $\Pi_N:H \to H_N$ is an orthogonal projection, $$\big\| G_N^k(\xi_N(t))\big\|_{L^2} = \big\| \Pi_N( \sigma_k \cdot \nabla \xi_N(t))\big\|_{L^2} \leq \| \sigma_k \cdot \nabla \xi_N(t)\|_{L^2}.$$ Therefore, $$\eps^2 \sum_{k\in \Z^2_0} \theta_k^2 \big\| G_N^k(\xi_N(t))\big\|_{L^2}^2 \leq \eps^2 \sum_{k\in \Z^2_0} \theta_k^2 \int_{\T^2} (\sigma_k \cdot \nabla \xi_N(t))^2 \,\d x = 2\nu \| \nabla \xi_N(t) \|_{L^2}^2,$$ where the last equality is due to and . Combining these results with we obtain $\d \|\xi_N(t) \|_{L^2}^2 \leq 0$, which implies the desired inequality and also the global existence of solution to .
Lemma \[energy-estimate\] shows that $\{\xi_N(\cdot)\}_{N\geq 1}$ is bounded in $L^p \big(\Omega,L^p(0,T; H) \big)$ for any $p>2$: $$\label{key-bound}
\E \int_0^T \|\xi_N(t) \|_{L^2}^p \,\d t \leq T\|\xi_N(0) \|_{L^2}^p \leq T\|\xi_0 \|_{L^2}^p.$$ Thus we can find a weakly convergent subsequence. Denote by $u_N = K\ast \xi_N,\, N\geq 1$; then $\{u_N(\cdot)\}_{N\geq 1}$ is bounded in $L^2 \big(\Omega, L^2(0,T; V) \big)$. In order to pass to the limit in the nonlinear term, we need $u_N$ to be strongly convergent in $L^2 \big(\Omega,L^2(0,T; H) \big)$. In fact, we will show that the laws $\eta_N$ of $u_N(\cdot)$ are tight in $C\big([0,T], H^{1-}(\T^2) \big)$. To this end we first recall the compactness result by J. Simon [@Simon Corollary 9, p.90].
Take any $\delta\in (0,1)$ small enough and $\beta> 4$ (this choice is due to computations below). We have the compact inclusions $$V= H^1 \subset H^{1-\delta} \subset H^{-\beta},$$ and there exists $C>0$ such that $$\|f\|_H \leq C \|f\|_V^{1-\kappa} \|f\|_{H^{-\beta}}^{\kappa}, \quad f\in V,$$ where $\kappa = \delta/(1+\beta)$. Recall that, for $\alpha\in (0,1)$, $p>1$ and a normed linear space $(Y, \|\cdot \|_Y)$, the fractional Sobolev space $W^{\alpha, p}(0,T; Y)$ is defined as those functions $f\in L^p(0,T; Y)$ such that $$\int_0^T\! \int_0^T \frac{\|f(t)-f(s) \|_Y^p}{|t-s|^{1+ \alpha p}}\,\d t\d s< +\infty.$$ The next result follows from [@Simon Corollary 9, p.90].
\[thm-simon\] Let $\beta>4$ be given. If $p> 12 (1+\beta -\delta)/\delta$, then $$L^p(0,T; V) \cap W^{1/3, 4} \big(0,T; H^{-\beta} \big) \subset C\big( [0,T]; H^{1-\delta} \big)$$ with compact inclusion.
If we can prove that $\{\eta_N \}_{N\in \N}$ are tight on $C\big( [0,T]; H^{1-\delta} \big)$ for any $\delta\in (0,1)$, then the tightness of $\{\eta_N \}_{N\in \N}$ on $C\big([0,T], H^{1-}(\T^2) \big)$ follows immediately.
To show the tightness of $\{\eta_N\}_{N\geq 1}$ on $ C\big( [0,T]; H^{1-\delta} \big)$, by Theorem \[thm-simon\], it is sufficient to prove, for each $N\geq 1$, $$\label{tightness-estimate-1}
\E \int_0^T \|u_N(t)\|_V^p\, \d t + \E \int_0^T\! \int_0^T \frac{\|u_N(t) - u_N(s)\|_{H^{-\beta}}^4}{|t-s|^{7/3}}\,\d t\d s \leq C.$$ By , we immediately get the uniform boundedness of $\{u_N(\cdot)\}_{N\geq 1}$ in $L^p \big(\Omega, L^p(0,T; V) \big)$. It remains to estimate the second expected value.
\[lem-estimate\] There is a constant $C>0$ such that for any $N\geq 1$ and $0\leq s<t\leq T$, $$\E \big(\<\xi_N(t) - \xi_N(s), e_k\>^4 \big) \leq C |k|^8 |t-s|^2 \quad \mbox{for all } k\in \Z^2_0.$$
It is enough to consider $|k|\leq N$. By , we have $$\label{lem-estimate-1}
\aligned
\<\xi_N(t) - \xi_N(s), e_k\> =&\, \int_s^t \<\xi_N(r), u_N(r)\cdot \nabla e_k\>\,\d r + \nu \int_s^t \<\xi_N(r), \Delta e_k\>\,\d r \\
& - \eps \sum_{l\in \Z^2_0} \theta_l \int_s^t \<\xi_N(r), \sigma_l\cdot \nabla e_k\>\,\d W^l_r.
\endaligned$$ Using the Hölder inequality and Lemma \[energy-estimate\], we obtain $$\aligned
\E \bigg( \Big| \int_s^t \<\xi_N(r), u_N(r)\cdot \nabla e_k\>\,\d r \Big|^4 \bigg) &\leq |t-s|^3\, \E \int_s^t \<\xi_N(r), u_N(r)\cdot \nabla e_k\>^4\,\d r \\
& \leq |t-s|^3\, \E \int_s^t \|\xi_N(r)\|_{L^2}^4 \| u_N(r) \|_{L^2}^4 \|\nabla e_k\|_\infty^4 \,\d r \\
&\leq C \|\xi_0\|_{L^2}^8 |k|^4 |t-s|^4,
\endaligned$$ where the last step is due to the fact $\nabla e_k = 2\pi k e_{-k}$. In the same way, since $\Delta e_k=- 4\pi^2\vert k\vert^2\,e_k$, $$\E \bigg( \Big| \int_s^t \<\xi_N(r), \Delta e_k\>\,\d r \Big|^4 \bigg) \leq C \|\xi_0\|_{L^2}^4 |k|^8 |t-s|^4.$$ Next, by Burkholder’s inequality, $$\aligned
\E \bigg( \Big| \eps \sum_{l\in \Z^2_0} \theta_l \int_s^t \<\xi_N(r), \sigma_l\cdot \nabla e_k\>\,\d W^l_r \Big|^4 \bigg) &\leq C \eps^4\, \E \bigg( \Big| \sum_{l\in \Z^2_0} \theta_l^2 \int_s^t \<\xi_N(r), \sigma_l\cdot \nabla e_k\>^2\,\d r \Big|^2 \bigg) .
\endaligned$$ We have $$\aligned
\sum_{l\in \Z^2_0} \theta_l^2 \<\xi_N(r), \sigma_l\cdot \nabla e_k\>^2 &\leq \|\theta \|_{\ell^\infty}^2 \sum_{l\in \Z^2_0} \<\xi_N(r)\nabla e_k, \sigma_l\>^2 \\
&\leq \|\theta \|_{\ell^\infty}^2 \|\xi_N(r)\nabla e_k \|_{L^2}^2 \leq C\|\theta \|_{\ell^\infty}^2 |k|^2 \|\xi_0\|_{L^2}^2,
\endaligned$$ where we have used the fact that $\{\sigma_l\}_{l\in \Z^2_0}$ is an orthonormal family. Therefore, $$\E \bigg( \Big| \eps \sum_{l\in \Z^2_0} \theta_l \int_s^t \<\xi_N(r), \sigma_l\cdot \nabla e_k\>\,\d W^l_r \Big|^4 \bigg) \leq C\eps^4 \|\theta \|_{\ell^\infty}^4 |k|^4 \|\xi_0\|_{L^2}^4 |t-s|^2 \leq C' |k|^4 |t-s|^2.$$ Combining the above estimates with we finally get the desired inequality.
Using the above estimate and Cauchy’s inequality, $$\aligned
\E \big[\|\xi_N(t) - \xi_N(s)\|_{H^{-\beta-1}}^4 \big] &= \E \Bigg[\sum_{k\in \Z^2_0 } \frac{\<\xi_N(t) - \xi_N(s), e_k\>^2 } {|k|^{2(\beta +1)}} \Bigg]^2\\
&\leq \Bigg[\sum_{k\in \Z^2_0 } \frac{1 } {|k|^{2(\beta +1)}} \Bigg] \Bigg[\sum_{k\in \Z^2_0 } \frac{\E \big( \<\xi_N(t) - \xi_N(s), e_k\>^4 \big) } {|k|^{2(\beta +1)}} \Bigg] \\
&\leq C|t-s|^2 \sum_{k\in \Z^2_0 } \frac1{|k|^{2(\beta +1) -8}} \leq C'|t-s|^2,
\endaligned$$ since $\beta>4$. Consequently, $$\E \big[\|u_N(t) - u_N(s)\|_{H^{-\beta}}^4 \big] \leq C'|t-s|^2,$$ which implies $$\E \int_0^T\! \int_0^T \frac{\|u_N(t) - u_N(s)\|_{H^{-\beta}}^4}{|t-s|^{7/3}}\,\d t\d s \leq C .$$ Thus we have proved and we obtain the tightness of $\{\eta_N\}_{N\geq 1}$ on $ C\big( [0,T]; H^{1-} \big)$. Equivalently, we have proved the tightness of the laws $\bar\eta_N$ of $\xi_N\, (N\geq 1)$ on $\mathcal X:= C\big( [0,T]; H^{-} \big)$.
Since we are dealing with the SDEs , we need to consider $\bar\eta_N$ together with the laws of Brownian motions $\big\{ (W^k_t)_{0\leq t\leq T}: k\in \Z^2_0 \big\}$. To this end, we endow $\R^{\Z^2_0}$ with the metric $$d_\infty(a,b)= \sum_{k\in \Z^2_0} \frac{|a_k-b_k| \wedge 1}{2^{|k|}}, \quad a,b \in \R^{\Z^2_0}.$$ Then $\big( \R^{\Z^2_0}, d_\infty(\cdot, \cdot) \big)$ is separable and complete (see [@Billingsley Example 1.2, p.9]). The distance in $\mathcal Y:= C\big([0,T], \R^{\Z^2_0} \big)$ is given by $$d_{\mathcal Y}(w,\hat w) = \sup_{t\in [0,T]} d_\infty(w(t), \hat w(t)),\quad w, \hat w \in \mathcal Y,$$ which makes $\mathcal Y$ a Polish space. Denote by $\mathcal W$ the law on $\mathcal Y$ of the sequence of independent Brownian motions $\big\{ (W^k_t)_{0\leq t\leq T}: k\in \Z^2_0 \big\}$.
To simplify the notations, we write $W_\cdot= (W_t)_{0\leq t\leq T}$ for the whole sequence of processes $\big\{ (W^k_t)_{0\leq t\leq T}: k\in \Z^2_0 \big\}$ in $\mathcal Y$. For any $N\geq 1$, denote by $P_N$ the joint law of $(\xi_N(\cdot), W_\cdot )$ on $$\mathcal X \times \mathcal Y = C\big( [0,T]; H^{-} \big)\times C\big([0,T], \R^{\Z^2_0} \big).$$ Since the marginal laws $\{ \bar\eta_N \}_{N\in \N}$ and $\{\mathcal W\}$ are respectively tight on $\mathcal X$ and $\mathcal Y$, we conclude that $\{ P_N \}_{N\in \N}$ is tight on $\mathcal X \times \mathcal Y$. The Prohorov theorem (see [@Billingsley Theorem 5.1, p.59]) implies that there exists a subsequence $\{N_i\}_{i\in \N}$ such that $P_{N_i}$ converge weakly as $i\to \infty$ to some probability measure $P$ on $\mathcal X \times \mathcal Y$. By Skorokhod’s representation theorem ([@Billingsley Theorem 6.7, p.70]), there exist a probability space $\big(\tilde\Omega, \tilde{\mathcal F}, \tilde \P \big)$, and stochastic processes $\big(\tilde \xi_{N_i}(\cdot), \tilde W^{N_i}_\cdot \big)_{i\in \N}$ and $\big(\tilde \xi(\cdot), \tilde W_\cdot \big)$ on this space with the corresponding laws $P_{N_i}$ and $P$ respectively, such that $\big(\tilde \xi_{N_i}(\cdot), \tilde W^{N_i}_\cdot \big)$ converge $\tilde\P$-a.s. in $\mathcal X\times \mathcal Y$ to the limit $\big(\tilde \xi(\cdot), \tilde W_\cdot \big)$. We are going to prove that $\big(\tilde \xi(\cdot), \tilde W_\cdot \big)$ is a weak solution to the equation .
Denote by $\tilde u_{N_i} = K\ast \tilde \xi_{N_i}$ and $\tilde u = K\ast \tilde \xi$ which are the velocity fields defined on the new probability space $\big(\tilde\Omega, \tilde{\mathcal F}, \tilde \P \big)$. By the above discussions, we know that $$\label{strong-convergence}
\tilde\P \mbox{-a.s.}, \quad \tilde \xi_{N_i}(\cdot) \mbox{ converge strongly to } \tilde \xi(\cdot) \mbox{ in } C([0,T]; H^{-}),$$ which implies that $$\tilde\P \mbox{-a.s.}, \quad \tilde u_{N_i}(\cdot) \mbox{ converge strongly to } \tilde u(\cdot) \mbox{ in } C([0,T]; H^{1-}).$$ The new processes $\tilde \xi_{N_i}(\cdot)$ (resp. $\tilde u_{N_i}(\cdot)$) have the same law with $\xi_{N_i}(\cdot)$ (resp. $u_{N_i}(\cdot)$), and thus by Lemma \[energy-estimate\], we have $$\label{sect-3.1}
\sup_{t\in [0,T]} \big\Vert \nabla^\perp\cdot \tilde u_{N_i}(t) \big\Vert_{L^2}
= \sup_{t\in [0,T]} \big\Vert \tilde\xi_{N_i}(t) \big\Vert_{L^2}
\leq \Vert \xi_0 \Vert_{L^2}\quad \tilde\P \mbox{-a.s.}$$
\[lem-bddness\] The process $\tilde\xi$ has $\tilde{\mathbb{P}}$-a.s. weakly continuous trajectories in $L^2$ and satisfies $$\label{estim-final}
\sup_{t\in [0,T]} \Vert \tilde\xi(t)\Vert_{L^2}\leq \Vert \xi_0\Vert_{L^2}\quad \tilde{\mathbb{P}}\text{-a.s.}$$
Thanks to , there exists a set $\Gamma\subset \tilde{\Omega}$ of full measure such that, for every $\omega\in \Gamma$, holds and $$\label{estim lem-bddness}
\sup_{i\geq 1} \sup_{t\in [0,T]} \Vert \tilde\xi_{N_i}(\omega, t)\Vert_{L^2} \leq \Vert \xi_0\Vert_{L^2}.$$ Let us fix $\omega\in \Gamma$. Then by the sequence $\{\tilde\xi_{N_i} (\omega,\cdot)\}_{i\geq 1}$ is bounded in $L^\infty(0,T;L^2)$ and so we can extract a subsequence (not relabelled for simplicity) which is weak-$\ast$ convergent. But weak-$\ast$ convergence in $L^\infty(0,T;L^2)$ implies weak-$\ast$ convergence in $L^\infty(0,T;H^{-})$, which implies by that the limit is necessarily $\tilde\xi$; therefore by properties of weak-$\ast$ convergence $$\nonumber
\Vert \tilde\xi(\omega,\cdot)\Vert_{L^\infty(0,T;L^2)}\leq \liminf_N \Vert \tilde\xi_N(\omega,\cdot)\Vert_{L^\infty(0,T;L^2)}\leq \Vert \xi_0\Vert_{L^2}.$$ In particular, there exists a subset $S_\omega\subset [0,T]$ of full Lebesgue measure (thus dense) such that $\Vert \tilde\xi(\omega,s) \Vert_{L^2}\leq \Vert\xi_0\Vert_{L^2}$ for every $s\in S_\omega$. Now let $t\in [0,T]\setminus S_\omega$ and consider a sequence $t_n\to t$, $t_n\in S_\omega$. Then the sequence $\{\tilde\xi(\omega, t_n)\}_n$ is uniformly bounded in $L^2$ and we can therefore extract a weakly convergent subsequence; but $\tilde\xi(\omega,\cdot)\in C([0,T];H^{-})$, therefore $\tilde\xi(\omega,t_n)\to \tilde\xi(\omega, t)$ in $H^{-}$ and so the weak limit must be $\tilde\xi(\omega,t)$. By properties of weak convergence we have $$\nonumber
\Vert \tilde\xi(\omega,t)\Vert_{L^2}\leq \liminf_n \Vert \tilde\xi(\omega,t_n)\Vert_{L^2}\leq \Vert \xi_0\Vert_{L^2}.$$ As the reasoning holds for any $t\in [0,T]\setminus S_\omega$, for any $\omega\in\Gamma$, we have obtained $$\nonumber
\sup_{t\in [0,T]} \Vert \tilde\xi(\omega,t)\Vert_{L^2}\leq \Vert \xi_0\Vert_{L^2}\quad \forall\,\omega\in \Gamma,$$ namely . It remains to show that, for every $\omega\in\Gamma$, $t\mapsto \tilde\xi(\omega,t)$ is weakly continuous in $L^2$. Let $t_n\to t$, then by the sequence $\{\tilde\xi(\omega,t_n)\}_n$ is bounded in $L^2$ and so it admits a weakly convergent subsequence. But $\tilde\xi(\omega,\cdot)\in C([0,T];H^{-})$, therefore the weak limit is necessarily $\tilde\xi (\omega,t)$; as the reasoning holds for any subsequence of $\{\tilde\xi(\omega,t_n)\}_n$, we deduce that the whole sequence is weakly converging to $\tilde\xi(\omega,t)$.
Finally we can give the
The processes $\big(\tilde \xi_{N_i}(\cdot), \tilde W^{N_i}_\cdot \big)$ on the new probability space $\big(\tilde\Omega, \tilde{\mathcal F}, \tilde \P \big)$ have the same laws with that of $(\xi_{N_i}(\cdot), W_\cdot )$, which satisfy the equation with $N$ replaced by $N_i$. Some classical arguments show that the stochastic integrals involved below make sense, see e.g. [@Krylov Section 2.6, p.89]. Therefore, for any $\phi\in C^\infty(\T^2)$, one has, $\tilde \P$-a.s for all $t\in [0,T]$, $$\label{proof-0}
\aligned
\big\< \tilde\xi_{N_i}(t),\phi \big\> =&\, \big\< \xi_{N_i}(0),\phi \big\> +\int_0^t \big\< \tilde\xi_{N_i}(s), \tilde u_{N_i}(s)\cdot \nabla \phi \big\>\,\d s + \nu \int_0^t \big\<\tilde\xi_{N_i}(s), \Delta \phi \big\>\,\d s \\
& - \eps \sum_{k\in \Z^2_0} \theta_k \int_0^t \big\<\tilde\xi_{N_i}(s), \sigma_k\cdot \nabla \phi \big\>\,\d\tilde W^{N_i,k}_s.
\endaligned$$ We regard all the quantities as real valued stochastic processes. From the above discussions, we can prove that, as $i\to \infty$, all the terms of the first line converge in $L^1\big( \tilde\Omega, C([0,T], \R)\big)$ to the corresponding ones. Indeed, considering $\<\cdot, \cdot\>$ as the duality between distributions and smooth functions, then implies that, $\tilde\P$-a.s., $\big\< \tilde\xi_{N_i}(\cdot),\phi \big\>$ converge in $C([0,T], \R)$ to $\big\< \tilde\xi(\cdot),\phi \big\>$. Moreover, by , $$\big| \big\< \tilde\xi_{N_i}(t),\phi \big\>\big| \leq \|\xi_0\|_{L^2} \|\phi\|_{L^2} \quad \tilde\P \mbox{-a.s. for all } t\in [0,T].$$ Thus the dominated convergence theorem implies the desired result. For the nonlinear term, we have $$\aligned
&\ \E_{\tilde \P} \bigg[\sup_{t\in [0,T]}\bigg| \int_0^t \big\< \tilde\xi_{N_i}(s), \tilde u_{N_i}(s)\cdot \nabla \phi \big\>\,\d s - \int_0^t \big\< \tilde\xi(s), \tilde u(s)\cdot \nabla \phi \big\>\,\d s \bigg| \bigg] \\
\leq &\ \E_{\tilde \P} \bigg[\sup_{t\in [0,T]}\bigg| \int_0^t \big\< \tilde\xi_{N_i}(s), \tilde u_{N_i}(s)\cdot \nabla \phi \big\>\,\d s - \int_0^t \big\< \tilde\xi_{N_i}(s), \tilde u(s)\cdot \nabla \phi \big\>\,\d s \bigg| \bigg]\\
&\, + \E_{\tilde \P} \bigg[ \sup_{t\in [0,T]}\bigg| \int_0^t \big\< \tilde\xi_{N_i}(s), \tilde u(s)\cdot \nabla \phi \big\>\,\d s - \int_0^t \big\< \tilde\xi(s), \tilde u(s)\cdot \nabla \phi \big\>\,\d s \bigg| \bigg].
\endaligned$$ Thanks to and , and the strong convergence of $\tilde u_{N_i}$ to $\tilde u$ in $L^2\big(\tilde\Omega, L^2(0,T; H) \big)$, the first term on the right hand side vanishes as $i\to \infty$. For the second term, by , the quantity in the square bracket tends to 0 $\tilde\P$-a.s., which together with the bounds and , the dominated convergence theorem leads to the desired result.
It remains to show the convergence of the stochastic integrals. Fix any $M\in \N$; we have $$\label{proof-1}
\aligned
&\ \E_{\tilde \P} \bigg[\sup_{t\in [0,T]}\bigg| \sum_{k\in \Z^2_0} \theta_k \int_0^t \big\<\tilde\xi_{N_i}(s), \sigma_k\cdot \nabla \phi \big\>\,\d \tilde W^{N_i,k}_s - \sum_{k\in \Z^2_0} \theta_k \int_0^t \big\<\tilde\xi(s), \sigma_k\cdot \nabla \phi \big\>\,\d\tilde W^{k}_s \bigg| \bigg] \\
\leq &\ \E_{\tilde \P} \bigg[ \sup_{t\in [0,T]}\bigg| \sum_{|k|\leq M} \theta_k \bigg(\int_0^t \big\<\tilde\xi_{N_i}(s), \sigma_k\cdot \nabla \phi \big\>\,\d \tilde W^{N_i,k}_s - \int_0^t \big\<\tilde\xi(s), \sigma_k\cdot \nabla \phi \big\>\,\d\tilde W^{k}_s \bigg) \bigg| \bigg] \\
&\, + \E_{\tilde \P} \bigg[ \sup_{t\in [0,T]}\bigg| \sum_{|k|> M} \theta_k \int_0^t \big\<\tilde\xi_{N_i}(s), \sigma_k\cdot \nabla \phi \big\>\,\d \tilde W^{N_i,k}_s \bigg| \bigg] \\
&\, + \E_{\tilde \P} \bigg[ \sup_{t\in [0,T]}\bigg| \sum_{|k|> M} \theta_k \int_0^t \big\<\tilde\xi(s), \sigma_k\cdot \nabla \phi \big\>\,\d \tilde W^{k}_s \bigg| \bigg].
\endaligned$$ We denote the three expectations on the right hand side by $J_{N_i}^{(n)},\, n=1,2,3$. First, $$\aligned
\big| J_{N_i}^{(2)} \big| &\leq C\, \E_{\tilde \P} \bigg[ \bigg( \sum_{|k|> M} \theta_k^2 \int_0^T \big\<\tilde\xi_{N_i}(s), \sigma_k\cdot \nabla \phi \big\>^2\,\d s \bigg)^{1/2} \bigg] \\
&\leq C\|\theta\|_{\ell^\infty_{>M}}\, \E_{\tilde \P} \bigg[ \bigg( \sum_{|k|> M} \int_0^T \big\<\tilde\xi_{N_i}(s)\nabla \phi , \sigma_k \big\>^2 \,\d s \bigg)^{1/2} \bigg] \\
&\leq C\|\theta\|_{\ell^\infty_{>M}}\, T^{1/2} \|\xi_0\|_{L^2} \|\nabla\phi \|_\infty,
\endaligned$$ where $\|\theta\|_{\ell^\infty_{>M}} = \sup_{|k|>M} |\theta_k|$ tends to 0 as $M\to \infty$. Similar estimate holds for $J_{N_i}^{(3)}$ by Lemma \[lem-bddness\].
Finally, we deal with $J_{N_i}^{(1)}$ for which we need Skorohod’s result for convergence of stochastic integrals, see for instance [@GyMa Lemma 5.2] and [@Luo Lemma 3.2] for a slightly more general version. By the discussions above Lemma \[lem-bddness\], we known that as $i\to \infty$, $\tilde\P$-a.s. for all $s\in [0,T]$, $\big\<\tilde\xi_{N_i}(s), \sigma_k\cdot \nabla \phi \big\> \to \big\<\tilde\xi(s), \sigma_k\cdot \nabla \phi \big\>$ and $\tilde W^{N_i,k}_s \to \tilde W^{k}_s$. Since there are only finitely many stochastic integrals, by [@Luo Lemma 3.2], it is sufficient to show that, for any $|k|\leq M$, $$\bigg(\E_{\tilde \P} \int_0^T \big\<\tilde\xi(s), \sigma_k\cdot \nabla \phi \big\>^4 \,\d s \bigg) \bigvee \bigg(\sup_{i\geq 1} \E_{\tilde \P} \int_0^T \big\<\tilde\xi_{N_i}(s), \sigma_k\cdot \nabla \phi \big\>^4 \,\d s \bigg)<+\infty.$$ Indeed, by Lemma \[lem-bddness\], $$\E_{\tilde \P} \int_0^T \big\<\tilde\xi(s), \sigma_k\cdot \nabla \phi \big\>^4 \,\d s \leq \E_{\tilde \P} \int_0^T \big\|\tilde\xi(s) \big\|_{L^2}^4 \big\|\sigma_k\cdot \nabla \phi \big\|_{L^2}^4 \,\d s \leq T \|\xi_0\|_{L^2}^4 \|\nabla\phi\|_\infty^4.$$ Analogous uniform estimate holds for the second part. Therefore we obtain $\lim_{i\to \infty} J_{N_i}^{(1)} =0$. First letting $i\to \infty$ and then $M\to \infty$ in , we have proved the convergence of stochastic integrals.
Therefore, letting $i\to \infty$ in , we obtain, $\tilde\P$-a.s. for all $t\in [0,T]$, $$\aligned
\big\< \tilde\xi(t),\phi \big\> =&\, \big\< \xi(0),\phi \big\> +\int_0^t \big\< \tilde\xi(s), \tilde u(s)\cdot \nabla \phi \big\>\,\d s + \nu \int_0^t \big\<\tilde\xi(s), \Delta \phi \big\>\,\d s \\
& - \eps \sum_{k\in \Z^2_0} \theta_k \int_0^t \big\<\tilde\xi(s), \sigma_k\cdot \nabla \phi \big\>\,\d\tilde W^{k}_s.
\endaligned$$ This completes the proof.
Convergence to 2D Navier–Stokes equations {#sec4}
=========================================
In this section we show that the solutions to converge weakly to the unique solution of the deterministic 2D Navier–Stokes equations.
Let us briefly recall the setting: we fix $\xi_0\in L^2$ and $\nu>0$, we consider a sequence $\big\{\theta^N_\cdot \big\}_{N\geq 1}$ satisfying and , and define $\varepsilon_N$ by . For each $N$, we consider a weak solution $\xi^N$ of with initial data $\xi_0$ satisfying , whose existence is granted by Theorem \[thm-existence\]. Since we are dealing with weak solutions, the processes $\xi^N$ might be defined on different probability space; however, for the sake of simplicity, in the following we do not distinguish the notation $\E$, $\P$, $\Omega$, etc.
Let us immediately remark that conditions and together imply $$\lim_{N\to\infty} \varepsilon_N \big\Vert \theta^N_\cdot \big\Vert_{\ell^\infty} =0,$$ therefore the sequence $\big\{\varepsilon_N \big\Vert \theta^N_\cdot \big\Vert_{\ell^\infty} \big\}_{N\geq 1}$ is bounded by a suitable constant.
Let $Q^N$ denote the law of $\xi^N$; we are going to show that the sequence $\big\{Q^N \big\}_{N\geq 1}$ is tight in $L^2\big(0,T; H^{-1} \big)$. To this end, let us recall the following Aubin–Lions theorem (see [@Lions Theorem 5.2, p.61]).
Let $\alpha\in (0,1/2)$ and $\beta>0$. Then $$\nonumber
L^2\big(0,T;L^2 \big)\cap W^{\alpha,2} \big(0,T;H^{-1-\beta} \big)\subset L^2\big(0,T; H^{-1} \big)$$ with compact inclusion.
To show the tightness of $\big\{Q^N \big\}_{N\geq 1}$ in $L^2\big(0,T;H^{-1} \big)$, by the Aubin–Lions theorem and the estimate , it is enough to show that $$\nonumber
\sup_{N\geq 1} \mathbb{E}\int_0^T\! \int_0^T \frac{\big\Vert \xi^N_t-\xi^N_s \big\Vert_{H^{-1-\beta}}^2}{\vert t-s\vert^{1+2\alpha}}\,\d t\d s <\infty.$$ To this aim, it suffices to obtain estimates similar to those of Lemma \[lem-estimate\], taking care that all the constants involved do not depend on $\theta^N_\cdot$ nor $\varepsilon_N$.
\[lem-estimate 2\] There is a constant $C>0$ such that for any $N\geq 1$, $0\leq s<t \leq T$, $$\E \big( \big\<\xi^N_t - \xi^N_s, e_k \big\>^2 \big) \leq C |k|^4 |t-s| \quad \mbox{for all } k\in \Z^2_0.$$
For any fixed $k$, since $\xi^N$ is a solution of , it holds $$\nonumber\begin{split}
\big\<\xi^N_t-\xi^N_s,e_k \big\> = \int_s^t \big\<\xi^N_r, u^N_r\cdot \nabla e_k \big\>\,\d r + \nu \int_s^t \big\<\xi^N_r, \Delta e_k \big\> \,\d r - \eps_N \sum_{l\in \Z^2_0} \theta^N_l \int_s^t \big\<\xi^N_r, \sigma_l\cdot \nabla e_k \big\>\,\d W^l_r
\end{split}$$ and therefore $$\nonumber\begin{split}
\mathbb{E}\big( \big\<\xi^N_t-\xi^N_s,e_k \big\>^2\big)
\leq 3\bigg( &\mathbb{E}\,\Big\vert\int_s^t \big\<\xi^N_r, u^N_r\cdot \nabla e_k \big\>\,\d r\Big\vert^2
+ \nu^2\, \mathbb{E}\,\Big\vert\int_s^t \big\<\xi^N_r, \Delta e_k \big\>\,\d r\Big\vert^2
\\ &+ \eps_N^2\, \mathbb{E}\,\Big\vert \sum_{l\in \Z^2_0} \theta^N_l \int_s^t \big\<\xi^N_r, \sigma_l\cdot \nabla e_k \big\>\,\d W^l_r\Big\vert^2\,\bigg).
\end{split}$$ Since $\Vert\nabla e_k\Vert_{L^\infty} = 2\sqrt{2}\, \pi\vert k \vert$, the first term on the right hand side can be estimated by $$\nonumber\begin{split}
\mathbb{E}\,\Big\vert\int_s^t \big\<\xi^N_r, u^N_r\cdot \nabla e_k \big\>\,\d r\Big\vert^2
& \leq \vert t-s\vert \int_s^t \mathbb{E}\big( \big\<\xi^N_r, u^N_r\cdot\nabla e_k \big\>^2\big)\,\d r\\
& \leq 8\pi^2\vert k\vert^2\,\vert t-s\vert \int_s^t \mathbb{E}\big( \big\Vert \xi_r^N \big\Vert^2_{L^2}\, \big\Vert u_r^N \big\Vert^2_{L^2}\big)\, \d r\\
& \leq 8\pi^2 \vert k\vert^2 \Vert\xi_0\Vert_{L^2}^4 \vert t-s\vert^2.
\end{split}$$ Using the fact that $\Delta e_k=-4\pi^2\vert k\vert^2 e_k$, the second term can be similarly estimated by $$\nonumber
\mathbb{E}\,\Big\vert\int_s^t \big\<\xi^N_r, \Delta e_k \big\>\,\d r\Big\vert^2
\leq 32 \pi^4\vert k\vert^4 \Vert\xi_0\Vert_{L^2}^2 \vert t-s\vert^2.$$ Finally, for the last term, by the Itô isometry, $$\nonumber\begin{split}
\varepsilon_N^2\, \mathbb{E}\,\Big\vert \sum_{k\in \Z^2_0} \theta^N_k \int_s^t \big\<\xi^N_r, \sigma_k\cdot \nabla e_k \big\>\,\d W^k_r\Big\vert^2
& = \varepsilon_N^2\, \mathbb{E}\, \sum_{k\in \Z^2_0} \big(\theta^N_k \big)^2 \int_s^t \big\<\xi^N_r, \sigma_k\cdot \nabla e_k \big\>^2\, \d r\\
& \leq \varepsilon_N^2\, \big\Vert \theta^N \big\Vert_{\ell^\infty}^2 \int_s^t\mathbb{E} \sum_{k\in \Z^2_0} \big\<\xi^N_r \,\nabla e_k, \sigma_k \big\>^2 \, \d r\\
& \leq C \int_s^t \mathbb{E}\big(\Vert \xi_r^N\,\nabla e_k\Vert_{L^2}^2\big)\, \d r\\
&\leq 8\pi^2 \vert k\vert^2 C\, \Vert \xi_0\Vert_{L^2}^2 \vert t-s\vert.
\end{split}$$ Combining all the estimates we obtain the conclusion.
From Lemma \[lem-estimate 2\] we deduce the following:
\[lem tightness sec 4\] The family $\big\{Q^N \big\}_{N\geq 1}$ is tight in $L^2\big(0,T;H^{-1} \big)$.
It holds $$\nonumber
\mathbb{E}\big( \big\Vert \xi^N_t-\xi^N_s \big\Vert_{H^{-\beta-1}}^2\big)
= \sum_{k\in\mathbb{Z}^2_0} \frac1{|k|^{2\beta +2}}\, \mathbb{E}\big(\big\langle \xi^N_t-\xi^N_s,e_k \big\rangle^2 \big)
\leq C\vert t-s\vert \sum_{k\in\mathbb{Z}^2_0} \frac1{|k|^{2\beta -2}}
\leq C'\vert t-s\vert$$ for $\beta> 2$; for such choice of $\beta$, $$\nonumber
\mathbb{E}\int_0^T\! \int_0^T \frac{\big\Vert \xi^N_t-\xi^N_s \big\Vert_{H^{-\beta-1}}^2}{\vert t-s\vert^{1+2\alpha}}\,\d t\d s
\leq C' \int_0^T\! \int_0^T \frac{\d t\d s}{|t-s|^{2\alpha}} <\infty$$ for any $\alpha<1/2$. Therefore Aubin–Lions theorem can be applied and the conclusion follows.
Sharper estimates allow us to show that, similarly to Section \[sec3\], the sequence $\big\{Q^N \big\}_{N\geq 1}$ is tight in $C([0,T];H^-)$. This fact will be used in Section \[sec-consequences\]. We omit the proof here since it is the same as those in Section \[sec3\].
By estimate we know that, for all $N$, almost every realization of $\xi^N$ satisfies $$\nonumber
\int_0^T \big\Vert \xi^N_r \big\Vert_{L^2}^2\,\d r\leq T\Vert\xi_0\Vert_{L^2}^2.$$ In particular, if we fix a radius $R\geq \sqrt{T}\Vert \xi_0\Vert_{L^2}$ and consider the space $$\label{weak-space}
L^2_{R,w} = \big\{ f\in L^2(0,T;H) : \Vert f\Vert_{L^2(0,T;H)}\leq R \big\}$$ endowed with the weak topology, then it is a metrizable, compact space (see for instance [@Bre]); we can regard $\big\{ \xi^N \big\}_{N\geq 1}$ as random variables taking values in $L^2_{R,w}$ and so by compactness their laws form a tight sequence in such space. Together with Lemma \[lem tightness sec 4\] this implies tightness of $\big\{Q^N \big\}_{N\geq 1}$ in $L^2\big(0,T;H^{-1} \big)\cap L^2_{R,w}$.
Before giving the proof of the second part of Theorem \[thm-main\], we need the following lemma.
\[lem continuity sec 4\] For any $\phi\in C^\infty(\mathbb{T}^2)$, consider the map $$\nonumber
F_\phi (f)_{\cdot} = \langle f_\cdot,\phi\rangle - \langle \xi_0,\phi\rangle -\int_0^\cdot \langle (K\ast f_s)\cdot\nabla\phi, f_s\rangle\,\d s - \nu \int_0^\cdot \<f_s, \Delta\phi\>\,\d s.$$ Then $F_\phi$ is a continuous bounded map from $L^2\big(0,T;H^{-1} \big)\cap L^2_{R,w}$ into $L^2(0,T;\mathbb{R})$.
Let us show boundedness first. We have $$\nonumber\begin{split}
\vert F_\phi(f)_t\vert
& \leq \Vert f_t\Vert_{L^2}\,\Vert \phi\Vert_{L^2} + \Vert \xi_0\Vert_{L^2}\,\Vert \phi\Vert_{L^2} + \int_0^t \vert \langle (K\ast f_s)\cdot\nabla\phi, f_s\rangle\vert\,\d s + \nu \int_0^t |\<f_s, \Delta\phi\>| \,\d s \\
& \leq \Vert f_t\Vert_{L^2}\,\Vert \phi\Vert_{L^2} + \Vert \xi_0\Vert_{L^2}\,\Vert \phi\Vert_{L^2} + \Vert \nabla\phi\Vert_{L^\infty} \int_0^T \Vert f_s\Vert_{L^2}^2\, \d s + \nu \|\Delta\phi\|_\infty \int_0^T \|f_s\|_{L^2} \,\d s\\
& \leq \Vert \phi\Vert_{C^2} (\Vert f_t\Vert_{L^2} + \Vert \xi_0\Vert_{L^2} + C_{R,T}),
\end{split}$$ where we used the fact that $f\in L^2_{w,R}$, and $C_{R,T}$ is a constant depending on $R$ and $T$. Therefore $$\Vert F_\phi(f)\Vert_{L^2(0,T;\mathbb{R})}
\leq \Vert \phi\Vert_{C^2} \big(\Vert f\Vert_{L^2(0,T;L^2)} + \Vert \xi_0\Vert_{L^2} + C_{R,T} \big)
\leq \Vert \phi\Vert_{C^2} \big(\Vert \xi_0\Vert_{L^2} + C'_{R,T} \big).$$ Regarding continuity: let $f^n$ be a sequence converging to $f$ in $L^2\big(0,T;H^{-1} \big)\cap L^2_{R,w}$, namely $f^n \to f$ strongly in $L^2\big(0,T;H^{-1} \big)$ and weakly in $L^2\big(0,T;L^2 \big)$. Strong convergence in $L^2\big(0,T;H^{-1} \big)$ implies convergence of $\langle f^n,\phi\rangle$ to $\langle f,\phi\rangle$ in $L^2(0,T)$, similarly for $\int_0^\cdot \langle f^n,\Delta\phi\rangle\,\d s$ to $\int_0^\cdot \langle f,\Delta\phi\rangle\,\d s$; so we only need to check convergence of the nonlinear term. By properties of the Biot–Savart kernel, $K\ast f^n\to K\ast f$ strongly in $L^2\big(0,T;L^2 \big)$; combining the strong convergence of $K\ast f^n$ and the weak convergence of $f^n$ we obtain, that for any $t\in (0,T)$, $$\nonumber
\int_0^t \big\langle (K\ast f^n_s)\cdot\nabla\phi, f^n_s \big\rangle\,\d s \to \int_0^t \big\langle (K\ast f_s)\cdot\nabla\phi, f_s \big\rangle \,\d s.$$ Therefore pointwise convergence holds; the previous estimates also show uniform boundedness of the integral processes, therefore by dominated convergence we obtain the conclusion.
Finally we can complete the
The fact that $\xi^N$ are solutions of may be formulated as follows: for every $\phi\in C^\infty(\mathbb{T}^2)$, the equality $F_\phi\big(\xi^N \big)=M^N_\phi$ holds, where $F_\phi$ is defined as in Lemma \[lem continuity sec 4\] and $M^N_\phi$ is the process given by $$\nonumber
M^N_\phi = -\varepsilon_N\sum_{k\in\mathbb{Z}^2_0} \theta^N_k \int_0^\cdot \big\< \xi^N_s,\sigma_k\cdot\nabla\phi \big\>\,\d W^k_s.$$ The sequence $\big\{Q^N \big\}_{N\geq 1}$ is tight in $L^2\big(0,T;H^{-1} \big)\cap L^2_{R,w}$, therefore by Prohorov theorem we can extract a subsequence (not relabelled for simplicity) which is weakly converging to the law $Q$ of some $L^2\big(0,T;H^{-1} \big)\cap L^2_{R,w}$-valued random variable $\xi$. By Lemma \[lem continuity sec 4\], $F_\phi$ is a continuous and bounded map, therefore by properties of convergence in law $F_\phi(\xi^N)$ are also converging in distribution to $F_\phi(\xi)$; in particular this implies that $M^N_\phi$ are also converging to some limit. On the other side, by Itô’s isometry we have $$\nonumber\begin{split}
\mathbb{E}\int_0^T \big\vert M^N_\phi(t) \big\vert^2\,\d t
& = \varepsilon_N^2\int_0^T \mathbb{E}\int_0^t \sum_{k\in\mathbb{Z}_0^2} \big(\theta_k^N \big)^2 \big\<\xi^N_s,\sigma_k\cdot \nabla\phi \big\>^2\, \d s\, \d t\\
& \leq T \varepsilon_N^2 \big\Vert \theta^N_\cdot \big\Vert_{\ell^\infty}^2 \mathbb{E}\int_0^T\sum_{k\in\mathbb{Z}_0^2} \big\< \xi^N_s\,\nabla\phi, \sigma_k \big\>^2\,\d s\\
& \leq T \varepsilon_N^2 \big\Vert \theta^N_\cdot \big\Vert_{\ell^\infty}^2 \int_0^T \mathbb{E}\Big( \big\Vert \xi_s^N\,\nabla\phi \big\Vert_{L^2}^2 \Big)\,\d s\\
& \leq T^2\Vert \xi_0\Vert^2_{L^2}\,\Vert\nabla\phi\Vert_{L^\infty}^2\,\varepsilon_N^2 \big\Vert \theta^N_\cdot \big\Vert_{\ell^\infty}^2 \to 0 \quad \text{ as }N\to\infty
\end{split}$$ which implies that $M^N_\phi$ is converging in law to 0; therefore $F_\phi(\xi)=0$, up to a $Q$-negligible set. Given a countable dense set $\{\phi_n\}_n$, we can deduce that the support of $Q$ satisfies $F_{\phi_n}(\xi)=0$ for all $n$. This, together with its $L^2$-boundedness, implies that $F_\phi(\xi)=0$ for all $\phi$. Namely, the support of $Q$ is made of solutions of the deterministic 2D Navier–Stokes equation starting at $\xi_0$; therefore by uniqueness $Q$ is given by $\delta_\xi$, where $\xi$ is such unique solution. As the reasoning applies to any subsequence of $\big\{Q^N \big\}_{N\geq 1}$, we deduce convergence in law of the whole sequence to $\delta_\xi$.
As a consequence of Theorem \[thm-main\] we deduce convergence of the passive scalars advected by $u^N$ to those advected by $u$, where as usual $u^N$ and $u$ denote the velocity fields associated to $\xi^N$ and $\xi$. To state the result, we assume for simplicity the sequence $u^N$ to be defined on the same filtered probability space $(\Omega,\mathcal{F},\mathcal{F}_t,\mathbb{P})$ and such that $u^N(\omega)\to u$ in $L^2\big(0,T;L^2 \big)$ for every $\omega\in \Gamma$, a set of full probability; this comes without loss of generality by applying Skorokhod’s theorem. For a given $\rho_0\in L^p(\mathbb{T}^2)$, $p\in (1,\infty)$, we denote by $\rho^N$ the passive scalar advected by $u^N$ with initial configuration $\rho_0$, i.e. the solution of $$\label{passive scalar eq sec 4}
\begin{cases}
\partial_t \rho^N + u^N\cdot\nabla\rho^N = 0,\\ \rho^N(0)=\rho_0;
\end{cases}$$ similarly for $\rho$ and $u$. By , we can take $\Gamma$ such that $\sup_{N\geq 1} \big\Vert u^N(\omega) \big\Vert_{L^\infty(0,T; H^1)}\leq \Vert \xi_0\Vert_{L^2}$ for every $\omega\in \Gamma$ and thus, by the DiPerna–Lions theory, equation admits a unique weak solution, which belongs to $C([0,T];L^p)$; similarly for $\rho$. We have the following
\[corollary sec4\] For any $\omega\in \Gamma$, any $p\in(1,+\infty)$ and any $\rho_0\in L^p$, $\rho^N(\omega)\to \rho$ in $C([0,T];L^p)$.
It follows immediately from [@DiPLio Theorem II.5, p. 527].
Convergence of finite dimensional approximations {#sec5}
================================================
The setting of this section is the same as Section \[sec4\] in terms of $\xi_0$, $\nu$, $\big\{\theta^N_\cdot \big\}_N$ and $\varepsilon_N$. However, for any $N$ we now consider $\xi^N$ to be an $H_N$-valued solution of the following SDE:
$$\label{SDE-sec5}
\d \xi^N = -b_N\big(\xi^N \big)\,\d t + \varepsilon_N \sum_{k\in\mathbb{Z}^2_0}\theta^N_k G_N^k\big( \xi^N\big)\circ\d W^k, \quad \xi^N_0=\Pi_N \xi_0,$$
where the vector fields $b_N$ and $ G_N^k$ are defined at the beginning of Section \[sec3\]. Recall that $G_N^k\big( \xi^N\big)=0$ whenever $\vert k\vert>2N$, thus the series appearing on the right hand side is finite. We are interested in determining conditions on $\big\{ \theta^N_\cdot \big\}_N$ under which $\xi^N$ converge in law to the unique solution of . Different finite dimensional schemes, like , can also be considered; here we use in order to show that the method is fairly robust and does not depend directly on the nature of the system, being dissipative while being conservative. The additional difficulty with respect to the previous sections is that the Itô–Stratonovich corrector is not exactly $\nu\Delta\xi$, but is dependent of the finite-dimensional approximation, therefore we need to take care of its convergence in the limit.
Equation admits a unique strong solution $\xi^N$, satisfying $$\label{energy conservation sec5}
\mathbb{P}\big( \big\Vert \xi^N_t \big\Vert_{L^2} = \big\Vert \xi^N_0 \big\Vert_{L^2} \mbox{ for all } t\in [0,T]\big)=1.$$
All vector fields in are smooth and $H_N$ is finite dimensional, so local existence and uniqueness follows. By Stratonovich chain rule, $$\nonumber
\d \Big( \frac{1}{2}\Vert \xi^N\Vert^2_{L^2}\Big)
= \langle \xi^N,\circ\, \d \xi^N\rangle
= -\big\langle b_N\big(\xi^N \big),\xi^N \big\rangle\,\d t + \varepsilon_N\sum_{k\in\mathbb{Z}^2_0} \theta_k^N \big\langle G_N^k\big( \xi^N\big) ,\xi^N \big\rangle\circ \d W^k = 0$$ where the last equality follows from . This shows that $\Vert\cdot\Vert_{L^2}$ is invariant and implies global existence as well as the last statement.
Before deriving the corresponding weak Itô formulation of equation , we need to introduce some notation. Recall that $\Pi_N$ is the orthogonal projection on $H_N$; with a slight abuse we identify it with the associated convolution kernel: $\Pi_N\xi=\Pi_N\ast \xi$. We denote the scalar product between matrices by $A:B=\text{Tr}(A^T B)$. For fixed $N$, let us define $$\label{covariance-matrix}
A_N(x,y):= \sum_{k\in\mathbb{Z}^2_0} \big(\theta^N_k \big)^2\,\sigma_k(x)\otimes\sigma_k(y),$$ which is the covariance operator associated to the noise $$\nonumber
W_N(t,x)=\sum_{k\in\mathbb{Z}^2_0} \theta^N_k\,\sigma_k(x)\,W^k(t).$$ It is easy to check that $A_N$ is homogeneous and it holds $$\nonumber
A_N(x,y)=A_N(x-y)= \sqrt{2} \sum_{k\in\mathbb{Z}^2_+} \big(\theta_k^N \big)^2\frac{k^\perp\otimes k^\perp}{\vert k\vert^2}\, e_k(x-y);$$ in particular, identity can be rewritten as $$\label{useful-equal}
A_N(x, x)=A_N(0)=\frac{1}{2} \big\Vert \theta^N_\cdot \big\Vert_{\ell^2}^2 I_2.$$ Moreover, $A_N$ has Fourier transform given by $$\nonumber
\hat{A}_N(k) = \sqrt{2}\, \big(\theta_k^N \big)^2 \frac{k^\perp\otimes k^\perp}{\vert k\vert^2} {\bf 1}_{\{k\in \Z^2_+\}} ,$$ which implies $$\nonumber
\big\Vert \hat{A}_N \big\Vert_{\ell^1} = \sqrt{2} \sum_{k\in\mathbb{Z}^2_+} \big(\theta^N_k \big)^2 = \frac{\sqrt{2}}2 \big\Vert \theta^N_\cdot \big\Vert_{\ell^2}^2.$$ We can now prove the following
\[fundamental lemma sec5\] $\xi^N$ is a solution of if and only if $\xi^N(0)=\Pi_N\xi_0$ and for any $\phi\in H_N$, $$\nonumber\begin{split}
\d\langle\xi^N,\phi\rangle
= \big\langle \big(K\ast\xi^N \big)\cdot\nabla\phi,\xi^N \big\rangle\,\d t -\varepsilon_N\sum_{k\in\mathbb{Z}^2_0} \theta_k^N \big\langle \sigma_k\cdot\nabla\phi,\xi^N \big\rangle\,\d W^k
+ \frac{\varepsilon_N^2}{2} \big\langle C_N\phi,\xi^N \big\rangle\,\d t,
\end{split}$$ where the operator $C_N$ is given by $$\nonumber
C_N\phi(x) = (\Pi_N A_N)\ast \nabla^2\phi\, (x) = \int_{\mathbb{T}^2} \Pi_N(x-y)A_N(x-y): \nabla^2\phi(y)\,\d y$$ and satisfies $$\nonumber
\Vert C_N\phi\Vert_{L^2} \leq \Vert\theta^N_\cdot\Vert_{\ell^2}^2\,\Vert \nabla^2\phi\Vert_{L^2}.$$
It is clear that $\xi^N$ is a solution of if and only if, for any $\phi\in H_N$, it holds $$\nonumber
\d \big\langle\xi^N,\phi \big\rangle
= -\big\langle b_N\big(\xi^N \big),\phi \big\rangle\,\d t +\varepsilon_N\sum_{k\in\mathbb{Z}^2_0} \theta_k^N \big\langle G_N^k \big(\xi^N \big),\phi \big\rangle\circ\d W^k.$$ Integration by parts and properties of the Stratonovich integral then yield $$\nonumber\begin{split}
\d\big\langle \xi^N,\phi \big\rangle
& = \big\langle \big(K\ast\xi^N \big)\cdot\nabla\phi,\xi^N \big\rangle\,\d t -\varepsilon_N\sum_{k\in\mathbb{Z}^2_0} \theta^N_k \big\langle \Pi_N(\sigma_k\cdot\nabla\phi),\xi^N \big\rangle\circ\d W^k\\
& = \big\langle \big(K\ast\xi^N \big)\cdot\nabla\phi,\xi^N \big\rangle\,\d t -\varepsilon_N\sum_{k\in\mathbb{Z}^2_0} \theta^N_k \big\langle \sigma_k\cdot\nabla\phi,\xi^N \big\rangle\,\d W^k + \frac{\varepsilon_N^2}{2} \big\langle C_N\phi,\xi^N \big\rangle\,\d t,
\end{split}$$ where $C_N$ is given by $$\nonumber
C_N\phi = \sum_{k\in\mathbb{Z}^2_0} \big(\theta_k^N \big)^2\, \sigma_k\cdot\nabla(\Pi_N(\sigma_k\cdot\nabla\phi)).$$ Recall that for fixed $N$, the sum over $k$ has a finite amount of non zero terms, so all the above calculations (and the following) are rigorous. It remains to compute $C_N$ explicitly; using the fact that $\Pi_N$ and $\nabla$ commute, we have $$\nonumber\begin{split}
C_N\phi(x)
& = \sum_{k\in\mathbb{Z}^2_0} \big(\theta_k^N \big)^2\, \sigma_k(x)\cdot\Pi_N[\nabla(\sigma_k\cdot\nabla\phi))](x)\\
& = \sum_{k\in\mathbb{Z}^2_0} \big(\theta_k^N \big)^2\, \int_{\mathbb{T}^2}\Pi_N(x-y)\,\sigma_k(x)\cdot\nabla(\sigma_k\cdot\nabla\phi)(y)\,\d y.
\end{split}$$ Note that $\sigma_k(x)\cdot\nabla\sigma_k(y)=0$ for all $k$, $x$ and $y$, thus by , $$\nonumber\begin{split}
C_N\phi(x)
& = \sum_{k\in\mathbb{Z}^2_0} \big(\theta_k^N \big)^2\, \int_{\mathbb{T}^2}\Pi_N(x-y)\,\sigma_k(x)\otimes \sigma_k(y) : \nabla^2\phi(y)\,\d y\\
& = \int_{\mathbb{T}^2} \Pi_N(x-y)A_N(x-y) : \nabla^2\phi(y)\,\d y.
\end{split}$$ Finally, by Parseval identity and Young inequality we have $$\nonumber\begin{split}
\Vert C_N\phi\Vert_{L^2}
& = \big\Vert (\Pi_N A_N)\ast \nabla^2\phi \big\Vert_{L^2}
= \big\Vert \big(\hat{\Pi}_N\ast\hat{A}_N \big) \widehat{\nabla^2\phi} \big\Vert_{\ell^2}
\leq \big\Vert \hat{\Pi}_N\ast\hat{A}_N \big\Vert_{\ell^\infty}\, \big\Vert \widehat{\nabla^2\phi} \big\Vert_{\ell^2}\\
& \leq \big\Vert \hat{\Pi}_N \big\Vert_{\ell^\infty} \big\Vert \hat{A}_N \big\Vert_{\ell^1} \big\Vert \nabla^2\phi \big\Vert_{L^2}
\leq \big\Vert \theta^N_\cdot \big\Vert_{\ell^2}^2 \big\Vert \nabla^2\phi \big\Vert_{L^2}.
\end{split}$$ We have only shown that the Stratonovich formulation implies the corresponding Itô one, but all calculations done backwards provide the converse implication.
It follows from Lemma \[fundamental lemma sec5\] and our choice of $\varepsilon_N$ that for any $\phi\in H_N$ it holds $$\label{estimate approx. corrector sec5}
\frac{\varepsilon_N^2}{2} \Vert C_N\phi\Vert_{L^2} \leq 2\nu \big\Vert \nabla^2\phi \big\Vert_{L^2}.$$ This allows to control the correctors $C_N$ when taking the limit as $N\to\infty$. Let $Q^N$ denote the law of $\xi^N$, then we can prove the following:
The family $\big\{ Q^N \big\}_N$ is tight in $C\big([0,T];H^-(\mathbb{T}^2) \big).$
We only sketch the proof briefly since most of the calculations are identical to those of Section \[sec3\]. Indeed by the energy equality and Theorem \[thm-simon\], we only need to show that there exists a constant $C$ such that, for any $N\geq 1$, $$\nonumber
\mathbb{E}\big( \big\langle \xi^N_t-\xi^N_s,e_k \big\rangle^4 \big) \leq C\vert k\vert^8\vert t-s\vert^2\quad \text{for all } k\in\mathbb{Z}^2_0;$$ again, we only need to show the estimate for $\vert k\vert\leq N$ and by Lemma \[fundamental lemma sec5\] it holds $$\nonumber\begin{split}
\big\langle \xi^N_t-\xi^N_s,e_k \big\rangle
= & \int_s^t \big\langle \big(K\ast\xi_r^N \big)\cdot\nabla e_k,\xi_r^N \big\rangle\,\d r + \frac{\varepsilon_N^2}{2}\int_s^t \big\langle C_N e_k,\xi^N_r \big\rangle\,\d r\\
& -\varepsilon_N\sum_{k\in\mathbb{Z}^2_0} \theta_k^N\int_s^t \big\langle\sigma_k\cdot\nabla e_k,\xi_r^N \big\rangle\,\d W^k_r.
\end{split}$$ The first and the last term on the right hand side can be estimated similarly to Lemma \[lem-estimate\] using respectively the Hölder and Burkholder inequality. For the term involving the corrector $C_N$, thanks to the energy identity and estimate , we have $$\nonumber\begin{split}
\Big\vert \frac{\varepsilon_N^2}{2}\int_s^t \big\langle C_Ne_k,\xi_r^N \big\rangle\,\d r\Big\vert
& \leq \frac{\varepsilon_N^2}{2} \Vert C_Ne_k\Vert_{L^2}\int_s^t \big\Vert \xi_r^N \big\Vert_{L^2}\,\d r\\
& \leq 2\nu \Vert \nabla^2 e_k\Vert_{L^2} \vert t-s\vert\, \Vert \xi_0\Vert_{L^2}
\leq C\vert k\vert^2 \vert t-s\vert,
\end{split}$$ which implies the conclusion.
We are now ready to complete the
We only sketch the proof, highlighting the passages which require to be handled differently from the previous sections. Observe first of all that $\big\{\xi^N \big\}_{N\geq 1}$ is a sequence of variables all defined on the same probability space, therefore convergence in probability to a deterministic limit is equivalent to convergence in law to it. As the sequence $\big\{Q^N \big\}_{N\geq 1}$ is tight, it suffices to show that any weakly convergent subsequence we extract converges to $\delta_{\xi_\cdot}$, $\xi$ being the unique solution of . Assume we have extracted a (not relabelled) subsequence $\xi^N$ whose laws $Q^N$ are converging in the topology of $C\big([0,T];H^-(\mathbb{T}^2) \big)\cap L^2_{R,w}$ to the law $Q$ of a random variable $\tilde\xi$. Then $\Pi_N \xi_0 \to \xi_0$ in $L^2$ and the convergence of the nonlinear term and the stochastic integral can be treated in the same way as in Section \[sec4\]. The only term which requires a different analysis is the convergence in a suitable sense of the corrector $\varepsilon_N^2 C_N/2$ to $\nu\Delta$. In particular, given a countable dense set $\{\phi_n\}_n$, it suffices to show that, for all $n$, $$\label{eq main proof sec5}
\frac{\varepsilon_N^2}{2} C_N\phi_n \to \nu\Delta\phi_n\quad \text{in } L^2,$$ as this implies that, for all $n$, $$\nonumber
\frac{\varepsilon_N^2}{2}\int_0^\cdot \big\langle C_N \phi_n,\xi^N_s \big\rangle\,\d s \to \nu \int_0^\cdot \big\langle \Delta \phi_n, \xi^N_s \big\rangle\,\d s\quad \text{in law}.$$ Let $\Pi_N^\perp$ denote the orthogonal projection on $H_N^\perp$, which, with a slight abuse of notation, is identified with the associated convolution kernel. In this way, $\Pi_N+\Pi_N^\perp=I$ in the sense of linear operators on $L^2$ and $\Pi_N+\Pi_N^\perp = \delta $ in the sense of convolution with a distribution. Then for any fixed $N$ and any $\phi$ smooth, by , it holds $$\nonumber\begin{split}
\nu\Delta\phi(x)
& = \frac{\varepsilon_N^2}{2} \int_{\mathbb{T}^2} A_N(x-y): \nabla^2\phi(y)\, \delta(\d y)\\
& = \frac{\varepsilon_N^2}{2} \bigg[ \int_{\mathbb{T}^2} \Pi_N(x-y)A_N(x-y): \nabla^2\phi(y)\,\d y
+ \int_{\mathbb{T}^2} \Pi^\perp_N(x-y)A_N(x-y): \nabla^2\phi(y)\,\d y \bigg] \\
& =: \frac{\varepsilon_N^2}{2} C_N\phi + \frac{\varepsilon_N^2}{2} C_N^\perp \phi.
\end{split}$$ Assertion then is equivalent to showing that, for all $n$, $\varepsilon_N^2 C_N^\perp \phi_n\to 0$ as $N\to\infty$. We can take the collection $\{\phi_n\}$ to be finite linear combinations of $e^{-i 2\pi j\cdot x},\, j\in \mathbb{Z}_0^2$. In this case, it is enough to prove that, for any $j\in \Z_0^2$, $\varepsilon_N^2 C_N^\perp e^{-i 2\pi j\cdot x} \to 0$ as $N\to \infty$. We have $$\nonumber\begin{split}
\varepsilon_N^2 \big\Vert C_N^\perp e^{-i 2\pi j\cdot x} \big\Vert_{L^2}
= 4\pi^2 \varepsilon_N^2 \Big\vert \widehat{A_N \Pi_N^\perp}(j) : (j\otimes j)\Big\vert
\leq K|j|^2 \big\Vert \theta^N_\cdot \big\Vert_{\ell^2}^{-2} \sum_{k:\vert k-j\vert>N} \big(\theta_k^N \big)^2.
\end{split}$$ This shows that, under condition , claim holds and the conclusion follows.
\[rem end sec5\] In this case the tightness of $\big\{Q^N \big\}_N$ in $C([0,T];H^-)$ is optimal, in the sense that it is not possible to prove tightness in $C\big([0,T]; L^2\big)$. Indeed, if this were true, since the sequence $\xi^N$ satisfies , the same should hold for the limit $\xi$, namely $\Vert\xi_t\Vert_{L^2}$ being constant; but we know that $\xi$ is a solution of Navier–Stokes equation, which is dissipative.
Consequences of the scaling limit {#sec-consequences}
=================================
In this section we discuss some implications of our scaling limit on the stochastic 2D Euler equations , including the approximate weak uniqueness, the existence of recovery sequences for Euler equations and a “weak quenched mixing property” of the weak solutions. We also give a discussion on possible dissipation of enstrophy in Section \[subsec-anomalous\].
Approximate uniqueness {#subsec-uniqueness}
----------------------
Uniqueness of solutions for 2D Euler equations when vorticity is in $L^{2}$ is a famous open problem. In view of certain regularization by noise results, where uniqueness is restored by a suitable noise, it is natural to ask whether a suitable noise may provide uniqueness, at least in law, for the solution of the corresponding stochastic 2D Euler equations with vorticity in $L^{2}$. We cannot prove such a strong result but we identify a new kind of property which we may call “approximate uniqueness” in law. The precise statement is given in Corollary \[corollary approx uniq\] below; roughly speaking it claims that all different solutions of a suitable stochastic 2D Euler equations, with a given initial vorticity in $L^{2}$, are very close to each other in law; for any degree of closedness we find a noise with such property.
On the family of all Borel probability measures on $C\left( \left[
0,T\right] ;H^{-}\right) $, let $d\left( \cdot,\cdot\right) $ be a distance that metrizes weak convergence.
For every $N$, let $\mathcal{C}_{N}$ be the class of all weak solutions of equation satisfying and let $\mathcal{C}=
\bigcup_{N\in\mathbb{N}} \, \mathcal{C}_{N}$. We denote by $Q_{N}$ the elements of $\mathcal{C}_{N}$ and generically by $Q$ those of $\mathcal{C}$, interpreting weak solutions as measures on the path space $C\left( \left[ 0,T\right] ;H^{-}\right) $.
\[definition approx uniq\] The family of weak solutions $\left\{ Q;Q\in\mathcal{C}\right\} $ is said to converge to a probability measure $\mu$ on $C\left( \left[ 0,T\right] ;H^{-}\right)
$ if, for every $\epsilon>0$, there is $N_{0}\in\mathbb{N}$ such that for all $N\geq N_{0}$, it holds $d\left( Q_{N},\mu\right) <\epsilon$ for all $Q_{N}\in\mathcal{C}_{N}$.
\[thm approx uniq\] Given $\xi_{0}\in L^{2}$, the family of weak solutions $\left\{
Q;Q\in\mathcal{C}\right\} $ converges to $\delta_{\xi}$ on $C\left( \left[
0,T\right] ;H^{-}\right) $, where $\xi$ is the unique solution of the deterministic Navier–Stokes equations .
We argue by contradiction. Assume there is $\epsilon>0$ such that for every $k\in\mathbb{N}$ there exist $N_{k}\geq k$ and $Q_{N_{k}}\in\mathcal{C}_{N_{k}}$ with the property $d\left( Q_{N_{k}},\delta_{\xi}\right)
\geq\epsilon$. The family $\left\{ Q_{N_{k}}\right\} _{k\in\mathbb{N}}$ is tight on $C([0,T], H^-)$ (for reasons similar to those proved above for a generic sequence of the form $\{ Q_{n} \} _{n\in\mathbb{N}}$). Hence it has a subsequence $\big\{ Q_{N_{k_{l}}}\big\} _{l\in\mathbb{N}}$, where we may assume $\left\{ N_{k_{l}}\right\} $ increasing, which converges weakly, thus to $\delta_{\xi}$ by the argument developed above. This is in contradiction with $d\left( Q_{N_{k}},\delta_{\xi}\right) \geq\epsilon$ for every $k\in\mathbb{N}$.
\[corollary approx uniq\] For every $\epsilon>0$, there is $N_{0}\in\mathbb{N}$ such that for all $N\geq N_{0}$, we have $d\left( Q_{N},Q_{N}^{\prime}\right) <\epsilon$ for all $Q_{N},Q_{N}^{\prime}\in
\mathcal{C}_{N}$.
It follows from the previous theorem by triangle inequality.
\[remark wassertein dist\] If we denote by $d_p$ the $p$-th Wasserstein distance for Borel probability measures on $C([0,T];H^-)$, then Theorem \[thm-main\] implies convergence of $Q_N$ to $\delta_{\xi}$ in the $p$-th Wasserstein distance, for any $p<\infty$. To see this, we can consider by Skorokhod Theorem a sequence $\tilde{\xi}^N$ distributed as $Q_N$, converging $\tilde{\mathbb{P}}$-a.s. to $\xi$ and satisfying the energy bound ; by dominated convergence this implies $$\lim_{N\to\infty} \tilde{\mathbb{E}}\Big[ \big\Vert \tilde\xi^N-\xi \big\Vert_{C([0,T];H^{-\delta})}^p\Big]\to 0$$ for any $\delta>0$ and $p<\infty$. In particular it is easy to see that Definition \[definition approx uniq\], Theorem \[thm approx uniq\] and Corollary \[corollary approx uniq\] still hold if we replaced $d$ by $d_p$.
Recovery sequences for Euler equations {#sec 6.2}
--------------------------------------
We are now going to show that, given any viscosity solution $\xi$ of Euler equations, we can find a suitable sequence $\xi^N$ of solutions of such that their laws $Q_N$ converge to $\delta_\xi$. This may be seen as a result of existence of recovery sequences, in a nice parallelism with the theory of $\Gamma$-convergence; we stress however that no variational problems are involved in our setting and this is merely an analogy. This result may help understanding the structure of viscosity solutions of Euler equations, deducing their properties from those of the sequence $\big\{ \xi^N \big\}_{N\in \N}$.
We consider a fixed sequence $\theta^N\in \ell^2$ satisfying the usual conditions and a fixed initial data $\xi_0\in L^2$. However we now allow the parameter $\nu$ to vary on $(0,+\infty)$; for fixed $\nu$, $\varepsilon_N$ depends on $\nu$ and $\theta^N$ in the usual way. We denote by $\xi^\nu$ the unique solution of Navier–Stokes with initial data $\xi_0$ and coefficient $\nu$; as in the previous section, we identify any solution of satisfying with a Borel probability measure on $C([0,T];H^-)$ and we denote by $d(\cdot,\cdot)$ the distance which metrizes weak convergence. We denote by $\mathcal{C}_{N,\nu}$ the set of laws of weak solutions of satisfying , with initial data $\xi_0$ and with respect to the parameters $\theta^N$, $\nu$; a generic element of $\mathcal{C}_{N,\nu}$ is denoted by $Q_{N,\nu}$.
We define $\mathcal{H}$ to be the set of viscosity solutions of Euler equations with initial data $\xi_0$, namely $\xi\in \mathcal{H}$ if there exists a sequence $\nu_n\to 0$ such that $\xi^{\nu_n}\to \xi$ in $C([0,T];H^-)$; if uniqueness of viscosity solutions of Euler were true, than $\mathcal{H}$ would consist of a singleton.
\[cor sec 6.2\] For any $\xi\in \mathcal{H}$ there exist sequences $\nu_i\downarrow 0$, $N_i\uparrow\infty$ such that $$\lim_{i\to\infty} d(Q_{N_i,\nu_i},\delta_\xi) = 0.$$
Since $\xi\in\mathcal{H}$, there exists a sequence $\nu_i\downarrow 0$ such that $\xi^{\nu_i}\to \xi$. By Theorem \[thm approx uniq\], for fixed $\nu_i$, we can find $N_i$ and an element $Q_{N_i,\nu_i}$ such that $d(Q_{N_i,\nu_i} , \delta_{\xi^{\nu_i}})\leq 1/i$; moreover, since we can construct the sequence inductively, we can always take $N_{i+1}\geq N_i$. Then by the triangle inequality, $$d(Q_{N_i,\nu_i},\delta_\xi)\leq d(Q_{N_i,\nu_i},\delta_{\xi^{\nu_i}}) + d(\delta_{\xi^{\nu_i}},\delta_{\xi}) \leq \frac{1}{i} + \Vert \xi^{\nu_i} -\xi\Vert_{C([0,T];H^-)}$$ and the conclusion follows.
Similarly to Remark \[remark wassertein dist\], the result still holds if we work with the $p$-th Wasserstein distance $d_p$ instead of $d$, for any $p<\infty$.
Next, we consider two sequences $\nu_i\to 0$ and $N_i\to\infty$, and for any $i$ an element $Q_{N_i,\nu_i}\in \mathcal{C}_{N_i,\nu_i}$. Using the same arguments in the previous sections, tightness of $\{Q_{N_i,\nu_i}\}_i$ in $C([0,T];H^-)$ can be shown; by Prohorov theorem we can therefore extract a subsequence which is weakly convergent to some probability law $Q$. Then, repeating the arguments in Section \[sec4\] and observing that this time also the corrector $\nu_i\Delta$ is infinitesimal, we find that almost every realization of $Q$ is a weak solution of deterministic Euler equations with initial data $\xi_0$. Since uniqueness in this case is not known, we cannot conclude that $Q$ is of the form $\delta_\xi$; rather it is a probability distribution on the weak solutions of Euler equation starting at $\xi_0$ – a *superposition solution*. Observe that in the above argument in principle we did not need to vary $N$: convergence of a subsequence to a superposition solution of deterministic Euler equations also holds if we considered a sequence $Q_{N,\nu_i}\in\mathcal{C}_{N,\nu_i}$ with $N$ fixed. However, the scaling limits we have obtained suggest that varying $N$ should allow to deduce non trivial properties in the limit which are not necessarily present for $N$ fixed; in particular, Corollary \[cor sec 6.2\] leads us to the following conjecture.
For any weakly convergent sequence $\{Q_{N_i,\nu_i}
\}_i$, the limit $Q$ is a probability measure supported on $\mathcal{H}$, the set of viscosity solutions of Euler equations starting at $\xi_0$.
Weakly quenched exponential mixing properties {#subsec-mixing}
---------------------------------------------
The multiplicative transport noise in Stratonovich form used above to perturb 2D Euler equations is formally vorticity-conservative but not formally energy-conservative. In general, the energy budget is not clear, namely we cannot say whether such noise increases or decreases the energy. Due to our convergence result to the Navier–Stokes equations, however, we can state an energy-dissipation result, in the precise form of Corollary \[Corollary energy dissipation\] below.
To avoid misunderstandings, we are not claiming that this noise produces an anomalous dissipation. Such property means a true dissipation when the equation is formally energy-conservative. Our noise is not formally energy-conservative. Thus the only relevant information of Corollary \[Corollary energy dissipation\] below is to clarify in which direction energy goes.
On the torus $\T^2$, for the unique solution $\xi_{t}\in C\left( [0,T];L^{2}\right) $ of the deterministic Navier–Stokes equations with initial condition $\xi_{0}\in L^{2}$, we have $$\begin{aligned}
\frac{\d}{\d t} \left\Vert \xi_{t}\right\Vert _{L^{2}}^{2}+\alpha\left\Vert
\xi_{t}\right\Vert _{L^{2}}^{2} & \leq0 , \\
\frac{\d}{\d t} \left\Vert u_{t}\right\Vert _{L^{2}}^{2}+\alpha\left\Vert
u_{t}\right\Vert _{L^{2}}^{2} & \leq0\end{aligned}$$ where $\alpha= 8\nu \pi^2$, as a consequence of the inequality $\left\langle -\Delta
f,f\right\rangle \geq 4\pi^2 \left\Vert f\right\Vert _{L^{2}}^{2}$ for smooth $f$. It follows that $$\begin{aligned}
\left\Vert \xi_{t}\right\Vert _{L^{2}}^{2} & \leq e^{-\alpha t}\left\Vert
\xi_{0}\right\Vert _{L^{2}}^{2}, \\
\left\Vert u_{t}\right\Vert _{L^{2}}^{2} & \leq e^{-\alpha t}\left\Vert
u_{0}\right\Vert _{L^{2}}^{2}.\end{aligned}$$
For every $\xi_{\cdot}\in C\left( \left[ 0,T\right] ;H^- \right) $, we call *energy profile* the real valued continuous function $$e(t) :=\frac{1}{2}\left\Vert K\ast\xi_{t}\right\Vert _{L^{2}}^{2}, \quad t\in [0,T] .$$
The map $\xi_{\cdot}\mapsto e\left( \cdot\right) $ from $C\left( \left[
0,T\right] ;H^-\right) $ to $C\left( \left[ 0,T\right] ;\mathbb{R}\right) $ is continuous. Given $\xi_{0}\in L^{2}$, the energy profile of the unique solution $\xi$ of the deterministic Navier–Stokes equations satisfies $e\left( t\right) \leq e^{-\alpha t}e\left( 0\right) $. Concerning solutions of the stochastic 2D Euler equations, always with initial condition $\xi_{0}\in L^{2}$, since their trajectories are of class $C\left(
\left[ 0,T\right] ;H^{-}\right) $, the energy profile is well defined also for them, being in this case a real-valued continuous stochastic process.
\[cor-weak-converg\] Let $\xi^{N}$ be solutions of the stochastic Euler equations, converging in law to $\xi$; call $e_{N}$ and $e$ the corresponding energy profiles. Then $e_{N}$ converges in law to $e$ on $C([0,T] ; \mathbb{R}) $.
It follows from the convergence in law of $\xi^{N}$ to $\xi$ on $C([0,T]; H^{-})$, and the stability of convergence in law by composition with continuous functions.
\[Corollary energy dissipation\] For every $\epsilon>0$, $$\lim_{N\rightarrow\infty}\mathbb{P}\left( e_{N}(t) \leq
e^{-\alpha t} (e(0) +\epsilon) \mbox{ for all } t\in [0,T] \right) =1.$$
Given $\epsilon>0$, by Corollary \[cor-weak-converg\], $$\lim_{N\rightarrow\infty} \mathbb{P}\left( \|e_{N}(\cdot) - e(\cdot)\|_{C([0,T] ; \mathbb{R})} \leq
\epsilon\right) =1.$$ Since $e(t) \leq e^{-\alpha t} e(0)$ for all $t\in [0,T]$, we have $$\label{Corollary energy dissipation-1}
\lim_{N\rightarrow\infty} \mathbb{P}\big( e_{N}(t) \leq e^{-\alpha t} e(0) + \epsilon \mbox{ for all } t\in [0,T] \big) =1.$$ Note that $e^{-\alpha t} e(0) + e^{-\alpha T}\epsilon \leq e^{-\alpha t} (e(0) +\epsilon)$ for all $t\in [0,T]$, replacing $\epsilon$ by $e^{-\alpha T}\epsilon$ in gives us the result.
We cannot state a similar result for the enstrophy profile$$i( t) := \Vert \xi_{t}\Vert_{L^{2}}^{2},$$ even if it is well defined for both $\xi^{N}$ and $\xi$. Indeed, $\xi\in C\left(
\left[ 0,T\right] ;L^{2}\right) $, hence $i\left( \cdot\right) \in
C\left( \left[ 0,T\right] ;\mathbb{R}\right) $, but we only know that $\xi^{N}\in L^{\infty}\left( \left[ 0,T\right] ;L^{2}\right) $ and that $\xi^{N}$ converges in law to $\xi$ in the strong topology of $C\left(
\left[ 0,T\right] ;H^{-}\right) $. Thus we cannot say that enstrophy is dissipated (in a probabilistic sense). If true, this would be a result of anomalous enstrophy dissipation, because formally the enstrophy is conserved by the stochastic dynamics.
However, for the solution to the deterministic 2D Navier–Stokes equation, we have$$\begin{aligned}
\left\Vert \xi_{t}\right\Vert _{H^{-\delta}}^{2} & \leq \left\Vert \xi
_{t}\right\Vert _{L^2}^{2} \leq e^{-\alpha t}\left\Vert \xi_{0}\right\Vert _{L^2}^{2}$$ and the convergence in law of $\xi^N$ to $\xi$ in $C\left([0,T] ;H^{-}\right)
$. Repeating the argument above gives us the asymptotically exponential decay of vorticity in negative Sobolev norms.
\[prop-mixing-in-probab\] For every $\epsilon,\delta>0$, $$\lim_{N\rightarrow\infty}\mathbb{P}\left( \left\Vert \xi_{t}^{N}\right\Vert
_{H^{-\delta}}^{2}\leq e^{-\alpha t} \big(\Vert \xi_{0}\Vert _{L^{2}}^2 +\epsilon \big) \mbox{ for all } t\in [0,T] \right) =1.$$
In the rest of this subsection, to avoid technical problems (cf. Remark \[rem-theta\]), we take $\theta^N_k = {\bf 1}_{\{|k|\leq N\}},\, k\in \Z^2_0$. Denote by $L^\infty_0 = L^\infty_0(\T^2)$ the space of functions in $L^\infty (\T^2)$ with zero mean. Then for any $\xi_0 \in L^\infty_0$, by [@Bre Theorem 2.10], the following stochastic Euler equation on $\T^2$ $$\d \xi^N + u^N\cdot\nabla\xi^N\,\d t= \eps_N \sum_{|k|\leq N} \theta^N_k \sigma_{k} \cdot \nabla\xi^N \circ \d W^{k}, \quad \xi^N|_{t=0} = \xi_0$$ admits a unique solution $\xi^{N,\xi_0}$ in $L^\infty_0$; moreover, [@Bre Theorem 2.14] implies that the equation of characteristics $$\d X_t= u_t^N(X_t)\,\d t + \eps_N \sum_{|k|\leq N} \theta^N_k \sigma_{k}(X_t)\circ \d W^k_t$$ generates a stochastic flow $\varphi^{N, \xi_0}_t$ of homeomorphisms on $\T^2$, such that, $\P$-a.s. for all $t\geq 0$ and $x\in \T^2$, it holds $$\label{repres-flow}
\xi^{N,\xi_0}_t(\omega, x) = \xi_0\big(\varphi^{N, \xi_0}_{-t} (\omega, x)\big),$$ where $\varphi^{N, \xi_0}_{-t} (\omega, \cdot)$ is the inverse map of $\varphi^{N, \xi_0}_{t} (\omega, \cdot)$. Moreover, $\P$-a.s., the Lebesgue measure on $\T^2$ is invariant under the stochastic flow $\varphi^{N, \xi_0}_t$ for all $t\geq 0$. The above formula implies that, $\P$-a.s., the norms $\big\| \xi^{N,\xi_0}_t \big\|_{L^p}\ (p>1)$ are preserved. In particular, taking $p=2$ and $\delta>0$, by the interpolation inequality, $\P$-a.s., $$\|\xi_0\|_{L^2}^2 = \big\| \xi^{N,\xi_0}_t \big\|_{L^2}^2 \leq \big\| \xi^{N,\xi_0}_t \big\|_{H^\delta} \big\| \xi^{N,\xi_0}_t \big\|_{H^{-\delta}}\quad \mbox{for all } t>0.$$ Combining this with Proposition \[prop-mixing-in-probab\], we obtain the asymptotically exponential increase of vorticity in positive Sobolev norms.
Given $\xi_0\in L^\infty_0$, for any $\delta>0$ and $T>0$, $$\lim_{N\to \infty} \P \Big( \big\| \xi^{N,\xi_0}_t \big\|_{H^\delta} \geq \frac12 e^{\alpha t/2} \|\xi_0\|_{L^2} \mbox{ for all } t\in [0,T] \Big) =1.$$
Next we will deduce a result on the weakly quenched mixing behavior of the stochastic flows $\varphi^{N, \xi_0}_t$.
Let $\xi _{0}\in L^\infty_0$. There exists a null set $\mathcal N\subset \Omega $ such that for all $\omega \in \mathcal N^{c}$, for all $N\in \N$, for every $ f\in H^\delta$ and all $t\geq 0$, we have $$\begin{aligned}
\left\vert \int_{\mathbb{T}^{2}}f\left( \varphi _{t}^{N, \xi _{0}}( \omega
,x) \right) \xi _{0}( x) \,\d x
\right\vert &\leq &\big\Vert \xi _{t}^{N, \xi _{0}}( \omega) \big\Vert
_{H^{-\delta}} \Vert f \Vert_{H^{\delta}}.\end{aligned}$$
For any $N\in \N$, there exists a null set $\mathcal N_{N} \subset \Omega$ such that for all $\omega \in \mathcal N_{N}^{c}$, for all $t\geq 0$, the formula holds and the Lebesgue measure is invariant under the map $\varphi _{t}^{N, \xi _{0}}(\omega, \cdot)$. For every $f\in H^\delta$, we have $$\aligned
\left\vert \int_{\mathbb{T}^{2}}f\left( \varphi _{t}^{N, \xi _{0}}( \omega,x) \right) \xi _{0}( x) \,\d x \right\vert
& =\left\vert \int_{\mathbb{T}^{2}}\xi _{t}^{N, \xi _{0}}( \omega ,x) f( x) \,\d x \right\vert \\
&\leq \big\Vert \xi _{t}^{N, \xi _{0}}( \omega) \big\Vert_{H^{-\delta}} \Vert f\Vert_{H^{\delta}}.
\endaligned$$ Now it is clear that the assertion holds.
The above result plus Proposition \[prop-mixing-in-probab\] gives us the weakly quenched exponential mixing property of the stochastic flows $\varphi^{N, \xi_0}_t$.
Under the previous notations, for every $\xi _{0}\in L_0^\infty, f\in H^\delta$, for every $\epsilon >0$, $$\lim_{N\rightarrow \infty } \P\left( \left\vert \int_{\mathbb{T}^{2}}f\left(
\varphi _{t}^{N, \xi _{0}} ( x) \right) \xi _{0}( x)
\,\d x \right\vert \leq e^{-\alpha t/2} (\Vert \xi_{0} \Vert_{L^2}+\epsilon) \|f \|_{H^\delta} \mbox{ for all } t\in [0,T]
\right) =1.$$
Further discussions on anomalous dissipation of enstrophy {#subsec-anomalous}
---------------------------------------------------------
As already pointed out earlier, the fact that the vorticity processes $\xi^N$ converge to a limit $\xi$ which is explicitly dissipating suggests that a partial dissipation should already take place at the level of $\xi^N$; the problem is that we only have convergence in $C([0,T];H^-)$ and not in $C([0,T];L^2)$, which does not allow to conclude.
The problem is not only technical: examples of processes $\xi^N$ which preserve vorticity almost surely but are converging in $C([0,T];H^-)$ to the solution $\xi$ of deterministic Navier–Stokes equations can be indeed found. One example is given by the processes from Section \[sec5\], as pointed out in Remark \[rem end sec5\].
Another example is the following: let $\xi_0\in C^\infty(\mathbb{T}^2)$ and the sequence $\theta^N$ be taken as in the last subsection, that is, for each fixed $N$, only a finite number of $\theta^N_k$ are non zero. Then the solution $\xi^N$ will preserve spatial regularity over time, for instance because $\Vert \xi^N\Vert_{L^\infty}$ can be controlled uniformly and Beale–Kato–Majda criterion can be applied, see [@CrFlHo]. This implies that the formal computation on vorticity invariance is actually rigorous and so $\big\Vert \xi^N_t \big\Vert_{L^2}=\Vert\xi_0\Vert_{L^2}$ for all $t>0$.
The above examples show that our scaling limit does not a priori give any information on whether anomalous dissipation will take place. It definitely does not take place for all solutions, but it might at least for *some* of them. Before proceeding further, let us give a rigorous definition.
Let $\xi_\cdot$ be a weak solution of satisfying . We say that anomalous dissipation of enstrophy takes place with positive probability if, for some $t\in [0,T]$, it holds $$\label{condition def anomalous dissipation}
\mathbb{P}\big(\Vert \xi_t\Vert_{L^2}<\Vert \xi_0\Vert_{L^2} \big)>0.$$
Since $\Vert\xi_t\Vert\leq \Vert \xi_0\Vert_{L^2}$ with probability one, condition is equivalent to requiring that, for some $t\in [0,T]$, $\mathbb{E}(\Vert \xi_t\Vert_{L^2}) <\mathbb{E}(\Vert \xi_0\Vert_{L^2})$. Such a quantity might be easier to handle because it is possible that $\xi$ does not have trajectories in $C([0,T];L^2)$, yet the map $t\mapsto \mathbb{E}(\Vert \xi_t\Vert_{L^2})$ is continuous as an effect of the averaging. However, condition is not equivalent to $$\nonumber
\mathbb{P}\big( \Vert \xi_t\Vert_{L^2}<\Vert \xi_0\Vert_{L^2} \text{ for some } t\in [0,T]\big)>0;$$ while the latter seems a more natural definition of anomalous dissipation, the fact that it involves evaluation on an uncountable set $[0,T]$ for a process $\xi_t$ with possibly not continuous trajectories in $L^2$ (not even right/left continuous) makes it very difficult to be handled.
The occurrence of anomalous dissipation might rely on the kind of noise we use. Here we restrict to the case of a noise constructed from $\theta\in\ell^2$ and $\{\sigma_k\}_{k\in \Z_0^2}$ as before, but observe that this is a very specific choice: it is an isotropic, divergence-free noise whose covariance operator is a Fourier multiplier; this leaves open the question whether other choices of noise might be better suited for obtaining an anomalous dissipation effect. In any case it would be interesting to give an answer to the following:
\[problem1\] Do there exist an initial data $\xi_0\in L^2$, a family of coefficients $\theta\in\ell^2$ and an associated solution $\xi$ which displays anomalous dissipation of enstrophy?
A different question, in the case of a positive answer for Problem \[problem1\], is related to anomalous dissipation occurring for *all* initial data.
\[problem4\] Does there exist a family of coefficients $\theta$ such that any solution of satisfying , for any initial data $\xi_0\in L^2$, displays anomalous dissipation of enstrophy with positive probability?
Clearly, if a positive answer to Problem \[problem4\] could be given, then the previous examples would show that $\theta$ cannot consist of all but a finite number of $\theta_k$ being $0$; more refined arguments show that in general $\theta_k$ cannot decay too fast as $k\to\infty$. On the other hand, condition $\theta\in\ell^2$, which is required for the equation to be meaningful, implies that such decay cannot be too slow either. It would be interesting to explore the case of $\theta_k$ decaying “almost as slowly as possible”, for instance taking $$\theta_k \sim \frac{1}{\vert k\vert \log\vert k\vert}.$$ Observe however that dealing with such a choice of $\theta$ is highly non trivial: uniqueness of solutions of for such $\theta$, even in the case of smooth initial data, is not known.
**Acknowledgements.** The third named author is grateful to the financial supports of the grant “Stochastic models with spatial structure” from the Scuola Normale Superiore di Pisa, the National Natural Science Foundation of China (Nos. 11571347, 11688101), and the Youth Innovation Promotion Association, CAS (2017003).
[9]{} G. Alberti, G. Crippa, A. L. Mazzucato, Exponential self-similar mixing and loss of regularity for continuity equations. *C. R. Math. Acad. Sci. Paris* **352** (2014), no. 11, 901–906.
G. Alberti, G. Crippa, A. L. Mazzucato, Exponential self-similar mixing by incompressible flows. *J. Amer. Math. Soc.* **32** (2019), no. 2, 445–490.
S. Albeverio, A. B. Cruzeiro, Global flows with invariant (Gibbs) measures for Euler and Navier–Stokes two-dimensional fluids. *Comm. Math. Phys.* **129** (1990), 431–444.
S. Albeverio, B. Ferrario, Uniqueness of solutions of the stochastic Navier–Stokes equation with invariant measure given by the enstrophy. *Ann. Probab.* **32** (2004), 1632–1649.
S. Attanasio, F. Flandoli, Zero-noise solutions of linear transport equations without uniqueness: an example, *C. R. Math. Acad. Sci. Paris* **347** (2009), no. 13–14, 753–756.
P. Billingsley, Convergence of Probability Measures. Second edition. Wiley Series in Probability and Statistics: Probability and Statistics. A Wiley-Interscience Publication. *John Wiley & Sons, Inc., New York,* 1999.
H. Brezis, Functional analysis, Sobolev spaces and partial differential equations. *Springer Science and Business Media* (2010).
Z. Brzeźniak, F. Flandoli, M. Maurelli, Existence and uniqueness for stochastic 2D Euler flows with bounded vorticity. *Arch. Ration. Mech. Anal.* **221** (2016), no. 1, 107–142.
D. Crisan, F. Flandoli, D.D. Holm, Solution properties of a 3D stochastic Euler fluid equation. *J. Nonlinear Sci.* **29** (2019), no. 3, 813–870.
G. Da Prato, A. Debussche, Two-Dimensional Navier–Stokes Equations Driven by a Space–Time White Noise. *J. Funct. Anal.* **196** (2002), 180–210.
G. Da Prato, J. Zabczyk, Stochastic equations in infinite dimensions, *Cambridge university press* (2014).
C. De Lellis, L. Székelyhidi, The Euler equations as a differential inclusion. *Ann. of Math.* (2) **170** (2009), no. 3, 1417–1436.
R.J. DiPerna, P.L. Lions, Ordinary differential equations, transport theory and Sobolev spaces. *Invent. Math.* **98** (1989), no. 3, 511–547.
F. Flandoli, D. Luo, $\rho$-white noise solutions to 2D stochastic Euler equations. *Probab. Theory Relat. Fields* (2019), https://doi.org/10.1007/s00440-019-00902-8.
F. Flandoli, D. Luo, Convergence of transport noise to Ornstein-Uhlenbeck for 2D Euler equations under the enstrophy measure. *Ann. Probab.* (2019), accepted.
L. Galeati, On the convergence of stochastic transport equations to a deterministic parabolic one. arXiv:1902.06960.
I. Gyöngy, T. Martinez, On stochastic differential equations with locally unbounded drift. *Czechoslovak Mathematical Journal* **51** (2001) 763–783.
G. Iyer, A. Kiselev, X. Xu, Lower bounds on the mix norm of passive scalars advected by incompressible enstrophy-constrained flows. *Nonlinearity* **27** (5) (2014) 973–985.
N.V. Krylov, Controlled Diffusion Processes. Translated from the Russian by A. B. Aries.Applications of Mathematics, vol. 14. Springer, New York (1980).
T.G. Kurtz, The Yamada–Watanabe–Engelbert theorem for general stochastic equations and inequalities. *Electron. J. Probab.* **12** (2007), 951–965.
J.L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires. (French) Dunod; Gauthier–Villars, Paris, 1969.
D. Luo, Absolute continuity under flows generated by SDE with measurable drift coefficients. *Stochastic Process. Appl.* **121** (2011), no. 10, 2393–2415.
C. Seis, Maximal mixing by incompressible fluid flows. *Nonlinearity* **26** (2013), no. 12, 3279–3289.
J. Simon, Compact sets in the space $L^p(0,T; B)$. *Ann. Mat. Pura Appl.* **146** (1987), 65–96.
R. Temam, Navier–Stokes equations and nonlinear functional analysis. Second edition. CBMS–NSF Regional Conference Series in Applied Mathematics, 66. *Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA*, 1995.
Y. Yao and A. Zlatoš, Mixing and un-mixing by incompressible flows. *J. Eur. Math. Soc. (JEMS)* **19** (2017), no. 7, 1911–1948.
[^1]: Email: franco.flandoli@sns.it. Scuola Normale Superiore of Pisa, Italy.
[^2]: Email: lucio.galeati@iam.uni-bonn.de. Institute of Applied Mathematics, University of Bonn, Germany.
[^3]: Email: luodj@amss.ac.cn. RCSDS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China, and School of Mathematical Sciences, University of the Chinese Academy of Sciences, Beijing 100049, China.
|
---
abstract: 'This paper aims to explore the temporal-spatial spreading and asymptotic behaviors of West Nile virus (WNv) by a reaction-advection-diffusion system with free boundaries, especially considering the impact of advection term on the extinction and persistence of West Nile virus. We define the spatial-temporal risk index $R^{F}_{0}(t)$ with the advection rate ($\mu$) and the general basic disease reproduction number $R^D_0$ to get the vanishing-spreading dichotomy regimes of West Nile virus. We show that there exists a threshold value $\mu^{*}$ of the advection rate, and obtain the threshold results of $\mu^*$. When the spreading occurs, we investigate the asymptotic dynamical behaviors of the solution in the long run and first give a sharper estimate that the asymptotic spreading speed of the leftward front is less than the rightward front in the case of $0<\mu<\mu^*$. At last, we give some numerical simulations to identify the significant effects of the advection.'
---
Chengcheng Cheng
Zuohuan Zheng$^*$
Introduction {#s1}
============
Infectious disease has become an essential factor which threatens people’s life nowadays. West Nile virus (WNv), originated from Africa, is one of the fatal mosquito-borne contagious diseases that has widely spread all over the north America since it broke out in 1999 for the first time [@koar2003west; @nash2001out]. In 1999–2001, West Nile virus lead to 149 cases of clinical neurologic disease in humans and 11932 deaths in birds in the United States [@koma2003exper]. Since 2008, many central European countries were invaded by West Nile virus which resulted in several hundreds of human and animal neuroinvasive cases [@rudo2017west]. In the 77 years since West Nile virus was discovered, this virus has spread to a large region of the earth and is considered one of the most important causative agent of viral encephalitis all over the world [@chancey2015global]. Therefore, it is urgent to understand how the disease can spread spatially to large region to cause large-scale epidemic and investigate the vanishing-spreading dichotomy regimes of West Nile virus.
In order to investigate the spreading dynamics of WNv, Lewis et al. [@lewis2006traveling] first discussed a reaction-diffusion sysytem of WNv with diffusion terms describing the movement of birds and mosquitoes. Then Maidana and Yang [@maidana2009spatial] proposed a reaction-advection-diffusion equation to study the vanishing and spreading of WNv across America. Recently, Li et al. [@Li2019vertical] formulated and analyzed a periodic delay differential equation of WNv model with vertical transmission. There are also other studies about WNv, such as, Wonham et al. [@wohn2006transmission], Hartley et al. [@hartley2012effect] and references therein.
The free boundary problems associated with the ecological models have attracted considerable research interests in the past, and several results have been applied to lots of fields, such as [@vuik1985numerical; @yi2004one; @amadori2005singular; @chen2000free; @chen2003free; @tao2005elliptic]. There are also a lot of reaction-diffusion biological models to use free boundary for studying. Du and Lin [@du2010spreading] first investigated a diffusive logistic equation with a free boundary in one dimension space. Wang [@wang2014free] studied the asymptotic behaviors about some free boundary problems for the Lotka-Volterra model of two species. Tarboush et al.[@tarboush2017spreading] considered a WNv problem with a coupled system, which described the diffusion of birds by a partial differential equation and the movement of mosquitoes by an ordinary differential equation. Lin and Zhu [@lin2017spatial] put forward a reaction-diffusion system with moving fronts to investigate the spreading dynamics of WNv between mosquitoes and host birds across North America.
Although there have been many works to investigate the propagation of WNv, advection terms have not received much attention in studying the spreading and vanishing of WNv. However, advection, especially the bird advection, plays an important role in spreading of WNv. For instance, in order to investigate the spreading of WNv in North America, it was observed in [@maidana2009spatial] that WNv appeared for the first time in New York city in 1999. In 2004, WNv was detected among birds in California. It has been spread across almost the whole America continent since it broke out in America. Thus, the advection movements of birds and diffusion lead to the biological invasion of WNv from the east to the western coast of the USA. Therefore, it is much worthwhile to take into consideration the advection movement in modeling West Nile virus.
Considering the previous preliminaries, in order to more explicitly describe the spreading and vanishing of WNv, we are planning to study a reaction-advection-diffusion epidemic model with free boundaries to describe the spreading and vanishing of WNv on the basis of [@maidana2009spatial]. Since the movements of the birds and mosquitoes change with time, we assume that the habitats of the birds and mosquitoes have moving boundaries. And the impact of the advection movement on the asymptotic spreading speeds of the double fronts is mainly investigated when spreading happens. Since the effect of advection rate on the mosquitoes is small enough, so we only consider the impact of advection movement on the birds.
For simplicity, let $$\label{1.2}
\begin{aligned}
&a_{1}:=\frac{\alpha_{1} \beta}{N_{1}}, a_{2}:=\frac{\alpha_{2} \beta}{N_{1}}. \end{aligned}$$ Here what $\alpha_{1},\alpha_{2},\beta$ represent will be explained later. Now we are going to discuss the following simplied reaction-advection-diffusion system with free boundaries of WNv between mosquitoes and birds $$\label{1.3}
\left\{\begin{array}{ll}{U_{t}=D_{1} U_{x x}-\mu U_{x}+a_{1}\left(N_{1}-U\right) V-\gamma U,} & {g(t)<x<h(t), \enspace t>0,} \\ {V_{t}=D_{2} V_{x x}+a_{2}\left(N_{2}-V\right) U-d V,} & {g(t)<x<h(t), \enspace t>0,} \\ {U(x, t)=V(x, t)=0,} & {x=h(t) \text { or } x=g(t),\enspace t>0,} \\ {h(0)=h_{0},\enspace h^{\prime}(t)=-\nu U_{x}(h(t), t),} & {t>0,} \\ {g(0)=-h_{0}, \enspace g^{\prime}(t)=-\nu U_{x}(g(t), t),} & {t>0,} \\ {U(x, 0)=U_{0}(x), \quad V(x, 0)=V_{0}(x),} & {-h_{0} \leq x \leq h_{0},}\end{array}\right.$$ where $U(x,t)$ and $V(x, t)$ represent the spatial infected densities of birds and mosquitoes at location $x$ and time $t$, respectively; $N_1$ and $N_2$ are the total carrying capacities of the birds and mosquitoes ; $D_{1}$ and $D_{2}$ represent the diffusion rates of the birds and mosquitoes and $D_2\ll D_1$; $\mu $ is the advection rate caused by the wind on the birds; $\alpha_{1}$ and $\alpha_{2}$ represent the WNv transmission probabilities per bite to birds and mosquitoes; $\beta$ is the biting rate of mosquitoes to birds; $\gamma$ is the recovery rate of birds from infection; $d$ is the death rate of the mosquitoes. The moving region $(g(t),h(t))$ represents the infected habitat of WNv. Suppose that the free boundaries satisfy Stefan conditions: $$g^{\prime}(t)=-\nu U_{x}(g(t), t)$$ and $$h^{\prime}(t)=-\nu U_{x}(h(t), t),$$ where $\nu$ is a positive constant which represents the boundary expanding capacity. Furthermore, we assume that the initial conditions $U_{0}$ and $V_{0}$ satisfy $$\label{1.4}
\left\{\begin{array}{ll}{U_{0} \in C^{2}\left[-h_{0}, h_{0}\right],} & {U_{0}(\pm h_{0})=0, \enspace 0<U_{0}(x) \leq N_{1} \text { in }\left(-h_{0}, h_{0}\right),} \\ {V_{0} \in C^{2}\left[-h_{0}, h_{0}\right],} & {V_{0}(\pm h_{0})=0,\enspace 0<V_{0}(x) \leq N_{2} \text { in }\left(-h_{0}, h_{0}\right).}\end{array}\right.$$ Considering small advection rate and high risk at infinity, in this paper, we make the following hypothesis $$(H) \quad\quad\quad {a_{1}a_{2}N_{1}N_{2}}> d\gamma, |\mu|<\mu^{*},$$ where $\mu^*:=2\sqrt{D_{1}(\dfrac{a_{1}a_{2}N_{1}N_{2}}{d}-\gamma)},$ which is defined as a threshold value of the advection (see to Section \[s3\]). What is more, as Guo and Wu [@guo2012free], we define $\lim\limits_{t\rightarrow +\infty}\limits\frac{-g(t)}{t}$ and $\lim\limits_{t\rightarrow +\infty}\limits\frac{h(t)}{t}$ as the leftward and rightward asymptotic spreading speeds, respectively.
As is discussed before, the main goal of this paper is to explore the propagation of WNv and investigate the effect of the advection movement on the asymptotic behaviors by a more general epidemic system (\[1.3\]). In view of the bird avection movement ($\mu$), the diffusion of birds and mosquitoes ($D_1,D_2$) and moving infected regions ($(g(t),h(t))$), this reaction-advection-diffusion model is more coincident with the laws of WNv than [@lewis2006traveling] and [@lin2017spatial]. By introducing the spatial-temporal risk index $R^F_{0}(t)$ with respect to advection and time as a threshold condition, the vanishing-spreading dichotomy regimes of WNv are obtained (see to Theorem \[t54\]). It is worthy of note that spreading speed is an important factor to influence the frontier propagation rate of WNv, while many previous studies can not calculate the spreading speed of the epidemic. Our main result is that we give an estimate that the asymptotic spreading speed of the leftward front is less than the rightward front for $0<\mu<\mu^*$ when the spreading occurs (see to Theorem \[t56\]).
This paper is organized as follows: in section 2, we prove the existence and uniqueness of the solution for system (\[1.3\]) by contraction mapping theorem, standard $L^p$ estimate and Sobolev embedding theorem. In section 3, we introduce spatial-temporal risk index $R^F_{0}(t)$ and the general basic reproduction number $R^D_0$. In section 4 and 5, we discuss the vanishing-spreading dichotomy regimes of WNv by applying $R^F_{0}(t)$ and $R^D_0$. In section 6, we mainly give an explicit estimate of the asymptotic spreading speeds about the leftward front and the rightward front compared with the corresponding reaction-diffusion model without advection. In section 7, several numerical simulations are given to illustrate our analytic results. At last, we sum up this paper by a brief discussion.
Existence and uniqueness {#s2}
========================
In this section, we first give the basic results about the local existence and uniqueness for the problem (\[1.3\]) with the initial conditions (\[1.4\]).
\[t21\] For any given $(U_{0},V_{0})$ satisfying (\[1.4\]) and any $\alpha \in (0,1)$, there exists $T>0$ such that the system (\[1.3\]) admits a unique solution $$\label{2.1}
\begin{aligned}
&(U,V;g,h)\in ({C^{1+\alpha,(1+\alpha)/2}}(\overline{D_{T}}))^{2}\times (C^{1+\alpha/2}([0,T]))^{2}
\end{aligned}$$ Further, $$\label{2.2}
||U||_{{C^{1+\alpha,(1+\alpha)/2}}(\overline{D_{T}})}+||V||_{{C^{1+\alpha,(1+\alpha)/2}}(\overline{D_{T}})}
+||g||_{C^{1+\alpha/2}([0,T])}+||h||_{C^{1+\alpha/2}([0,T])}\leq C_{1}.$$ Where $$\label{2.3}
D_{T}=\{(x,t)\in \mathbb{R}^{2}:x\in(g(t),h(t)),t\in(0,T]\},$$ positive constants $T, C_{1}$ depend only on $ ||U_0||_{C^{2}([-h_0,h_0])}$, $||V_0||_{C^2([-h_0,h_0])}$, $h_0$ and $ \alpha$.
Now we only provide a simple sketch to prove this theorem. First, straighten the free boundary; then, use the contraction mapping theorem, standard $L^p$ estimate and Sobolev embedding theorem to get the local existence and uniqueness of $(U, V; g, h)$; finally, apply the Schauder estimates to obtain the regularity of the solution. The detailed proof can refer to Theorem 2.1 in [@chen2000free], Lemma 2.1 in [@guo2012free] or Theorem 2.1 Wang and Zhao [@wang2017free].
In order to prove the boundness of the local solution, we need to use the following Comparison Principle to eatimate $U(x,t)$, $V(x,t)$ and the free boundaries $x=g(t)$, $x=h(t)$. The proof is similar to Lemma 3.5 in [@du2010spreading], so we omit it here.
\[l22\] Assume that $T\in(0,+\infty)$, $\overline{h}(t), \overline{g}(t)\in C^{1}([0,T])$, $\overline{U},\overline{V}\in C(\overline{D_{T}^{*}})\bigcap C^{2,1}(D_{T}^{*})$, and $$\label{2.4}
\left\{\begin{array}{ll}{\overline{U}_{t}-D_{1} \overline{U}_{x x} \geq -\mu\overline{U}_{x}+a_{1}\left(N_{1}-\overline{U}\right) \overline{V}-\gamma \overline{U},} & {\overline{g}(t)<x<\overline{h}(t), \enspace 0<t<T,} \\ {\overline{V}_{t}-D_{2} \overline{V}_{x x} \geq a_{2}\left(N_{2}-\overline{V}\right) \overline{U}-d \overline{V},} & {\overline{g}(t)<x<\overline{h}(t), \enspace 0<t<T,} \\ {\overline{U}(0, t) \geq U(0, t), \enspace \overline{V}(0, t) \geq V(0, t),} & {0<t<T,} \\ {\overline{U}(x, t)= 0,\overline{V}(x, t)= 0,} & {x=\overline{g}(t)\enspace or\enspace \overline{h}(t), \enspace 0<t<T,} \\ {\overline{h}^{\prime}(t) \geq-\nu \overline{U}_{x}(\overline{h}(t), t),\overline{g}^{\prime}(t) \leq-\nu \overline{U}_{x}(\overline{g}(t), t),} & {\enspace 0<t<T,} \\\overline{U}(x, 0)\geq {U_{0}(x), \enspace\overline{V}(x, 0)\geq V_{0}(x),} & {-h_{0} \leq x \leq h_{0},}\end{array}\right.$$ then the solution $(U,V;g,h)$ of (\[1.3\]) satisfies $$\label{2.5}
\begin{aligned}
&\overline{ U}(x,t)\geq {U}(x,t), \enspace \overline{V}(x,t)\geq {V}(x,t),
\\ & \overline{h}(t)\geq {h}(t),\enspace {g}(t)\geq \overline{ g}(t), for\enspace g(t)\leq x\leq h(t),t\in(0,T],
\end{aligned}$$ where $D_{T}^{*}=\{(x,t)\in\mathbb{R}^{2}: x\in(\overline g(t),\overline{h}(t)),t\in(0,T]\}$.
\[r23\] Assume that $T\in(0,+\infty)$, $\underline{h}(t), \underline{g}(t)\in C^{1}([0,T])$, $\underline{U}, \underline{V}\in C(\overline{D_{T}^{**}})\bigcap
C^{2,1}(D_{T}^{**})$, if the reverse inequalities of (\[2.4\]) are satisfied, then $$\label{2.6}
\begin{aligned}
&\underline{U}(x,t)\leq U(x,t),\enspace \underline{V}\leq V(x,t),
\\&\underline{h}(t)\leq h(t),\enspace g(t)\leq\underline{g}(t),for\enspace \underline{g}(t)\leq x\leq\underline{h}(t),t\in(0,T],
\end{aligned}$$ where $D_{T}^{**}=\{(x,t)\in\mathbb{R}^{2}: x\in(\underline{g}(t),\underline{h}(t)),t\in(0,T]\}$.
\[r24\] If $(\overline{U},\overline{V};\overline{g},\overline{h})$ and $(\underline{U},\underline{V};\underline{g},\underline{h})$ satisfy the above conditions, then they are called the upper and lower solution of (\[1.3\]), respectively.
The following Lemma gives some estimates of $U(x,t)$ and $V(x,t)$.
\[l25\] Assume that $T\in(0,+\infty)$. Let $(U,V;g,h)$ be a solution of (\[1.3\]) for $t\in(0,T]$, then there exist positive constants $C_{2},C_{3}$ independent of $T$ such that $$\label{2.7}
\begin{aligned}
&0<U(x,t)\leq C_{2}, for\enspace g(t)< x< h(t), 0< t\leq T.
\\&0<V(x,t)\leq C_{3}, for\enspace g(t)< x< h(t), 0< t\leq T.
\end{aligned}$$ Indeed, we can take $C_2=N_1,C_3=N_2$.
Since $g(t), h(t)$ are fixed, by the strong maximum principle, we can get $$U(x,t)>0,V(x,t)>0 \enspace for\enspace(x,t)\enspace \in (g(t),h(t))\times (0,T].$$ For the problem (\[1.3\]), consider the following system $$\label{2.8}
\left\{\begin{array}{ll}{u^{\prime}=a_{1}(N_{1}-u)v-\gamma u,} & { 0<t\leq T,} \\{v^{\prime}=a_{2}(N_{2}-v)u-d v,} & {0<t\leq T,} \\ {u(0)=\sup\limits_{x\in[-h_{0},h_{0}]}U_{0}(x),}\\{v(0)=\sup\limits_{x\in[-h_{0},h_{0}]}V_{0}(x).} \end{array}\right.$$ Then $$u(h(t),t)>U(h(t),t)=0, v(h(t),t)>V(h(t),t)=0$$ for $t\in(0,T].$ And $$u|_{t=0}\geq U_0(x), v|_{t=0}\geq V_0(x)$$ for $x\in [-h_{0},h_{0}].$ Hence, the solution $(u,v)$ of equation (\[2.8\]) is an upper solution of the system (\[1.3\]). Apply the upper and lower solutions theorem (Theorem 2.1 in [@pao1995reaction]) and Lemma \[l22\], we get $$\label{2.9}
{u}\geq U(x,t), {v}\geq V(x,t),for\enspace (x,t)\in[g(t),h(t)]\times[0,T].$$ Since the solution of the ordinary differential equation (\[2.8\]) satisfies $$\label{2.10}
\begin{aligned}
&{u}\leq \max\limits\{\sup\limits_{x\in[-h_{0},h_{0}]}U_{0}(x),N_{1}\}=N_1,for\enspace (x,t)\in[g(t),h(t)]\times[0,T].
\\&{v}\leq \max\limits\{\sup\limits_{x\in[-h_{0},h_{0}]}V_{0}(x),N_{2}\}=N_2,for\enspace (x,t)\in[g(t),h(t)]\times[0,T].
\end{aligned}$$ Therefore, $$\label{2.11}
\begin{aligned}
&{U}\leq \max\limits\{\sup\limits_{x\in[-h_{0},h_{0}]}U_{0}(x),N_{1}\}=N_1,for\enspace (x,t)\in[g(t),h(t)]\times[0,T].
\\&{V}\leq \max\limits\{\sup\limits_{x\in[-h_{0},h_{0}]}V_{0}(x),N_{2}\}=N_2,for\enspace (x,t)\in[g(t),h(t)]\times[0,T].
\end{aligned}$$ Similarly, considering the following system $$\label{2.12}
\left\{\begin{array}{ll}{\underline{u}^{\prime}=a_{1}(N_{1}-\underline{u})\underline{v}-\gamma \underline{u},} & { 0<t\leq T,} \\{\underline{v}^{\prime}=a_{2}(N_{2}-\underline{v})\underline{u}-d\underline{v},} & {0<t\leq T,} \\ {\underline{u}(0) =\inf\limits_{x\in[-h_{0},h_{0}]}U_{0}(x),} & {t>0,} \\{\underline{v}(0) =\inf\limits_{x\in[-h_{0},h_{0}]}V_{0}(x),} & {t>0,} \end{array}\right.$$ then $(\underline{u},\underline{v})$ is the lower solution of (\[1.3\]). Moreover, $$\label{2.13}
0<\underline{u}\leq U(x,t),0<\underline{v}\leq V(x,t), for\enspace (x,t)\in(g(t),h(t))\times(0,T].$$ Therefore, take $$C_{2}=\max\{\sup\limits_{x\in[-h_{0},h_{0}]}U_{0}(x),N_{1}\}=N_1,C_{3}=\max\{\sup\limits_{x\in[-h_{0},h_{0}]}V_{0}(x),N_{2}\}=N_2.$$ Then $C_{2},C_{3}$ satisfy (\[2.7\]).
The following lemma gives an estimate about the upper bound and lower bound of the asymptotic spreading speeds for the leftward front and the rightward front, which is similar to Lemma 2.2 in [@du2010spreading] or Lemma 3.3 in [@tian2017free].
\[l26\] Assume that $T\in(0,+\infty)$. Let $(U,V;g,h)$ be a solution of (\[1.3\]) for $t\in(0,T]$, then there exists a positive constant $C$ independent of $T$ such that $$0<-g^{\prime}(t), h^{\prime}(t)\leq C, for\enspace 0<t\leq T.$$
The following theorem shows that the solution of (\[1.3\]) can be extend to $[0,\infty)$.
\[t26\] For any given $(U_{0},V_{0})$ satisfying the initial conditions, the solution of free boundary problem (\[1.3\]) exists and is unique for any $t\in[0,\infty)$.
For fixed $T_0>0$, the uniqueness and local existence of the solution to the problem (\[1.3\]) can be obtained following from Theorem \[t21\]. Now we show the existence of the global solution. If $[0,T_{0}]$ is the maximal existence interval of the solution, we will show $T_{0}=+\infty$. Assuming that $T_{0}<+\infty$, for the fixed $\delta\in(0,T_{0})$, by the $L^{p}$ estimates, the Sobolev’s embedding theorem and the Lölder estimates about parabolic equations, there exists $C_{4}=C_{4}(\delta,T,C_1,C_{2},C_{3})>0$ such that $$\|U(x,t)\|_{C^{2}(g(t),h(t))},\|V(x,t)\|_{C^{2}(g(t),h(t))} \leq C_{4}, for \enspace t\in[\delta,T_{0}].$$ Then there exists a small $\varepsilon>0$ such that the solution of (\[1.3\]) with initial time $T_{0}-\varepsilon/2$ can be extended to the time $T_{0}+\varepsilon/2$ uniquely by Theorem \[t21\] and Zorn’s lemma. This is contradict to the choice of $T_{0}$. Hence, we have completed the proof.
Basic reproduction number {#s3}
=========================
In this section, we will define and study the basic reproduction number for the system(\[1.3\]). According to L[ó]{}pez-G[ó]{}mez [@caudevilla2008asymptotic], there exist a principal eigenvalue and corresponding unique eigenfunction $(\phi,\psi)$ (subject to a multiplicative positive constant) satisfying the following eigenvalue problem $$\label{3.1}
\left\{\begin{array}{ll}{-D_{1} \phi_{x x}=-\mu\phi_{x}+\frac{a_{1} N_{1} }{R_{0}^{D}}\psi-\gamma \phi,} & {x\in (-h_{0},h_{0}),} \\ {-D_{2} \psi_{x x}=\frac{a_{2} N_{2} }{R_{0}^{D}}\phi-d \psi,} & {x\in (-h_{0},h_{0}), } \\ {\phi(x)=\psi(x)=0,} & {x=\pm h_{0}.} \end{array}\right.$$ By applying a similar way as discussed by Diekmann et al. [@diekmann1990definition], Allen et al. [@Allen2008Asymptotic] or Zhao [@zhao2003dynamical] to calculate the principal eigenvalue of (\[3.1\]), we define the unique positive principal eigenvalue $R^D_0$ with Dirichlet boundary condition and the advection rate for problem (\[3.1\]) as the general basic reproduction number: $$\label{3.2}
\begin{aligned}
&R_{0}^{D}:=R_{0}^{D}((-h_0,h_0),\mu,D_1,D_2)\\&=\sqrt{\dfrac{a_{1}a_{2}N_{1}N_{2}}{[{D_{1}(\dfrac{\pi}{2h_{0}})^2}+\dfrac{\mu^{2}}{4D_{1}}+\gamma][D_{2}(\dfrac{\pi}{2h_{0}})^{2}+d]}}.
\end{aligned}$$
Applying the similar variational methods from Cantrell and Cosner [@cantrell2004spatial] or Lemma 2.3 in [@Huang2010Dynamics], the following result of $R^{D}_{0}$ holds.
\[l31\] $1-R^{D}_{0}$ and $\lambda_{0}$ have the same sign, where $\lambda_{0}$ is the principal eigenvalue of the following problem $$\label{3.3}
\left\{\begin{array}{ll}{-D_{1} \phi_{x x}=-\mu\phi_{x}+a_{1} N_{1} \psi-\gamma \phi+\lambda_0\phi,} & {x\in (-h_0,h_0),} \\ {-D_{2} \psi_{x x}=a_{2} N_{2} \phi-d \psi+\lambda_0\psi,} & {x\in (-h_0,h_0), } \\ {\phi(x)=\psi(x)=0,} & {x=\pm h_0 .} \end{array}\right.$$ where $(\phi(x),\psi(x))>0$ for $x\in (-h_0,h_0)$, $\phi^\prime(-h_0)>0,\psi^\prime(-h_0)>0$ and $\phi^\prime(h_0)<0,\psi^\prime(h_0)<0.$
Since the boundary $(g(t),h(t))$ changes with time, we introduce the spatial-temporal risk index with the advection rate and time as the basic reproduction number in epidemiology $$\label{3.4}
\begin{aligned}
&R^{F}_{0}(t):=R^D_{0}((g(t),h(t)),\mu,D_{1},D_{2})
\\&=\sqrt{\dfrac{a_{1}a_{2}N_{1}N_{2}}{[{D_{1}(\dfrac{\pi}{h(t)-g(t)})^2}+\dfrac{\mu^{2}}{4D_{1}}+\gamma][D_{2}(\dfrac{\pi}{h(t)-g(t)})^{2}+d]}}.
\end{aligned}$$
Following from the definition of $R^{F}_{0}(t)$, we can easily get
\[p32\]The following properties of $R^{F}_{0}(t)$ hold.\
(1) $R^{F}_{0}(t)$ is a positive and monotonically decreasing function of $\mu: R^{F}_{0}(t)\rightarrow 0$ as $\mu \rightarrow+\infty;$\
(2) If $\mu\neq 0$, then $R^{F}_{0}(t)\rightarrow $ 0 as $D_{1}\rightarrow 0$ and $D_{2}\rightarrow 0$ or $D_1 \rightarrow \infty;$\
(3) $R^{F}_{0}(t)$ is strictly monotonically increasing function of $t:$ when $\mu\neq 0,$ if $h(t)-g(t)\rightarrow +\infty$ as $t\rightarrow +\infty$, then $R^{F}_{0}(t)\rightarrow R_0(\mu) $ as $t\rightarrow +\infty$, where $R_{0}(\mu)=\sqrt{\dfrac{a_{1}a_{2}N_{1}N_{2}}{(\dfrac{\mu^{2}}{4D_{1}}+\gamma)d}}.$
Since the left boundary $x=g(t) $ is monotonically decreasing and the right boundary $x=h(t)$ is monotonically increasing, there exist $g_\infty\in[-\infty,0)$ and $h_\infty\in(0,\infty]$ such that $g_\infty=\lim\limits_{t\rightarrow +\infty}g(t)$ and $h_\infty=\lim\limits_{t\rightarrow +\infty}h(t).$ Moreover, when $\mu=0$, we suppose that the habitat at far distance is in high risk, that is, $$R^D_0((-\infty,0),0,D_1,D_2)>1 \enspace and \enspace R^D_0((0,\infty),0,D_1,D_2)>1,$$ equivalently, ${a_{1}a_{2}N_{1}N_{2}}> d\gamma.$
In view of the above properties of $R^F_0(t)$, there exists a threshold value $$\mu^{*}:=2\sqrt{D_{1}(\dfrac{a_{1}a_{2}N_{1}N_{2}}{d}-\gamma)}.$$ If $|\mu|<\mu^{*}$, then there is a $t_0\geq0$ such that $$R^{F}_{0}(t_0)=R^D_{0}((g(t_0),h(t_0)),
\mu,D_1,D_2)\geq1$$ and $$R^F_0(\infty):=R^D_{0}((g_\infty,h_\infty),
\mu,D_1,D_2)>1$$ under the assumption of $h_\infty-g_\infty=\infty$; if $|\mu|>\mu^{*}$, then $$R^{F}_{0}(t)=R^D_{0}((g(t),h(t)),\mu,D_{1},D_{2})< 1$$ for any $t\geq 0$.
Considering the above arguments about high-risk habitat at far distance and small advection, we make the assumption of $(H)$.
The vanishing regime of WNv {#s4}
===========================
Now we will introduce the definitions of vanishing and spreading from [@ge2015sis].
The disease is **vanishing** if $h_{\infty}-g_{\infty}<\infty$ and $$\lim\limits_{t\rightarrow +\infty}(||U(\cdot,t)||_{C(g(t),h(t))}+||V(\cdot,t)||_{C(g(t),h(t))})=0;$$ The disease is **spreading** if $h_{\infty}-g_{\infty}=\infty$ and $$\lim\limits_{t\rightarrow +\infty}\sup\limits (||U(\cdot,t)||_{C(g(t),h(t))}+ ||V(\cdot,t)||_{C(g(t),h(t))})>0.$$
The following theorem gives the relationship about the spreading boundaries with the densities of birds and mosquitoes. We show that if the infected region of WNv is bounded, then the densities of the birds and the mosquitoes will decay to 0 and the asymptotic spreading speeds of the double boundaries will decay to 0, that is, the diseases will be extinct, which is coincident with the biological reality.
\[t42\] Let $(U,V;g,h)$ be a solution of (\[1.3\]), if $ h_{\infty}-g_{\infty}<\infty$, then $$\label{4.1}
\lim\limits_{t\rightarrow +\infty}||U(\cdot,t)||_{C(g(t),h(t))}=\lim\limits_{t\rightarrow +\infty}||V(\cdot,t)||_{C(g(t),h(t))}=0$$ and $$\label{4.2}
\lim\limits_{t\rightarrow +\infty}g^{\prime}(t)=\lim\limits_{t\rightarrow +\infty}h^{\prime}(t)=0.$$
We will prove this theorem by two steps.\
**Step 1.** we will show $$\label{4.3}
||U||_{{C^{1+\alpha,(1+\alpha)/2}}((g(t),h(t))\times[1,\infty))}+||V||_{{C^{1+\alpha,(1+\alpha)/2}}((g(t),h(t))\times[1,\infty))}\leq \tilde{C},$$ and $$\label{4.4}
||g||_{C^{1+\alpha,(1+\alpha)/2}([1,\infty))}+||h||_{C^{1+\alpha,(1+\alpha)/2}([1,\infty))}\leq \tilde{C}$$ for any $\alpha\in(0,1)$, where $\tilde{C}=\tilde{C}(\alpha,h_{0},||U_{0}||_{C^{2}([-h_{0},h_{0}])},||V_{0}||_{C^{2}([-h_{0},h_{0}])},g_{\infty},h_{\infty})>0$.
Now we straighthen the free boundaries by making a transformation motivated by Wang[@wang2014some; @wang2016diffusive] $$\label{4.5}
y=\frac{2x}{h(t)-g(t)}-\frac{h(t)+g(t)}{h(t)-g(t)}.$$ then the boundary $x=g(t)$ changes into $y=-1$ and $x=h(t)$ changes into $y=1$. A straightforward calculation gives $$\label{4.6}
\begin{aligned}
&\frac{\partial y}{\partial x}=\frac{2}{h(t)-g(t)}:=\sqrt{A(g(t),h(t),y)}, \frac{\partial^2 y}{\partial x^2}=0,
\\&\frac{\partial y}{\partial t}=-\frac{y({h^\prime(t)}-{g^\prime(t)})+({h^\prime(t)}+{g^\prime(t)})}{h(t)-g(t)}
\\&\enspace\enspace:=B(g(t),{g^\prime(t),h(t),{h^\prime(t)},y)}.
\end{aligned}$$ Let $W(y,t)=U(x,t), Z(y,t)=V(x,t)$, then $W(y,t)$ satisfies $$\label{4.7}
\left\{\begin{array}{ll}{W_{t}-D_{1}AW_{yy}+(\mu \sqrt{A}+B)W_{y}=f(W,Z),} & {y\in (-1,1),t>0,} \\ {W(\pm 1,t)=0 ,} & {t>0, } \\ {W(y,0)=U_{0}( h_0 y),} & {y\in(-1,1).} \end{array}\right.$$ For any integer $n\geq 0,$ define $$W^{n}(y,t)=W(y,t+n), Z^{n}(y,t)=Z(y,t+n),$$ then (\[4.7\]) becomes $$\label{4.8}
\left\{\begin{array}{ll}{W^{n}_{t}-D_{1}A^{n} W^{n}_{yy}+(\mu \sqrt{A^{n}}+B^{n})W^{n}_{y}=f(W^n,Z^n),} & {y\in [-1,1],t\in(0,3],} \\ {W^n(\pm 1,t)=0 ,} & {t\in (0,3], } \\ {W^n(y,0)=U(\frac{y(h(n)-g(n))+h(n)+g(n)}{2},n),} & {y\in[-1,1],} \end{array}\right.$$ where $A^n=A(t+n), B^n=B(t+n)$. We can see that $W^n,Z^n,A^n $ and $B^n$ are uniformly bounded on $n$ according to Theorem \[t21\] and Lemma \[l25\]. Moreover, $$\label{4.9}
\max _{0 \leq t_{1}<t_{2} \leq 3,\left|t_{1}-t_{2}\right| \leq \tau}\left|A^{n}\left(t_{1}\right)-A^{n}\left(t_{2}\right)\right| \leq \frac{8\left(h^{n}(t)-g^{n}(t)\right)^{\prime}}{\left(h^{n}(t)-g^{n}(t)\right)^{3}} \leq \frac{2 C_{1} \tau}{h_{0}^{3}} \rightarrow 0 , as \enspace \tau \rightarrow 0.$$ where $h^{n}(t)=h(t+n)$ and $g^{n}(t)=g(t+n)$. When $ h_{\infty}-g_{\infty}<\infty,$ we obtain $A^{n} \geq \frac{4}{\left(h_{\infty}-g_{\infty}\right)^{2}}$ for any $n\geq 0$ and $t\in (0,3]$.
Taking $p\gg1$, by applying the interior $L^{P}$ eatimate, there exists a postive constant $\overline{C}$ independent of $n$ such that $||U^n||_{W^{2,1}_{p}([-1,1]\times [1,3])}\leq \overline{C}$ for any $n>0$. Therefore, by Sobolev’s embedding theorem, $$||U^n||_{C^{1+\alpha,(1+\alpha)/2}([-1,1]\times [1,3])} \leq \overline{C}.$$ Moreover, $||U||_{C^{1+\alpha,(1+\alpha)/2}([-1,1]\times [n+1,n+3])}\leq \overline{C}$. Similarly, we can get $$||V||_{C^{1+\alpha,(1+\alpha)/2}([-1,1]\times [n+1,n+3])}\leq \overline{C_1}$$ for some $\overline{C_1}>0.$
Since $$g^{\prime}(t)=-\nu U_{x}(g(t),t),\enspace U_{x}(g(t),t)=\frac{2}{h(t)-g(t)}W_{y}(-1,t),$$ $$h^{\prime}(t)=-\nu U_{x}(h(t),t),\enspace U_{x}(h(t),t)=\frac{2}{h(t)-g(t)}W_{y}(1,t).$$ in $[-1,1]\times [n+1,n+3],$ and $g^{\prime}(t)$ and $h^{\prime}(t)$ are bounded, then $$\label{4.10}
||g||_{C^{1+\alpha/2}([n+1,n+3])}+||h||_{C^{1+\alpha/2}([n+1,n+3])}\leq \hat{C},$$ for some $\hat{C}>0$. Since the rectangles $[-1,1]\times [n+1,n+3]$ overlap and $\overline{C}$, $\overline{C_1}$ and $\hat{C}$ are independent on $n$, take $\tilde{C}=\overline{C}+\overline{C_1}+\hat{C}$, so (\[4.3\]) and (\[4.4\]) hold. Moreover, since $h_\infty-g_\infty<\infty$, we can get $g^{\prime}(t)\rightarrow 0$ and $h^{\prime}(t)\rightarrow 0$ as $t\rightarrow \infty$. Thus (\[4.2\]) has been proved.
**Step 2.** we will show $ \lim\limits_{t\rightarrow +\infty}||U(\cdot,t)||_{C(g(t),h(t))}=0.$
On the contrary, we assume that $$\lim\limits_{t\rightarrow +\infty}\sup\limits||U(\cdot,t)||_{C(g(t),h(t))}=\theta>0.$$ Then there is a sequence$\{(x_{k},t_{k})\} $ in $(g(t),h(t))\times (0,\infty)$ such that $U(x_{k},t_{k})\geq \frac{\theta}{2}$ for any $ k \in N$ and $t_{k}\rightarrow \infty $ as $k\rightarrow \infty.$ Since $$-\infty <g_{\infty}<g_{t}<x_{k}<h_{t}<h_{\infty}<\infty,$$ then there exists a subsequence $\{x_{k_{n}}\}$ of $\{x_{k}\}$ such that $x_{k_{n}} \rightarrow x_{0}\in(g_{\infty},h_{\infty})$ as $n\rightarrow \infty.$ Without loss of generality, we assume that $x_{k}\rightarrow x_{0}$ as $k\rightarrow \infty.$
Let $$U_{k}(x,t)=U(x,t_{k}+t), V_{k}(x,t)=V(x,t_{k}+t),$$ for $(x,t)\in [g(t_{k}+t),h(t_{k}+t)]\times (-t_{k},\infty).$ In view of (\[2.2\]) and (\[4.3\]), there exists a subsequence $\{(U_{k_{n}},V_{k_{n}})\}$ of $\{(U_{k},V_{k})\}$ such that $(U_{k_{n}},V_{k_{n}})\rightarrow (\tilde{U},\tilde{V}) \enspace as\enspace n\rightarrow \infty.$ And $(\tilde{U},\tilde{V})$ satisfies $$\label{4.11}
\left\{\begin{array}{ll}{\tilde{U}_{t}=D_{1} \tilde{U}_{x x}-\mu \tilde{U}_{x}+a_{1}\left(N_{1}-\tilde{U}\right) \tilde{V}-\gamma \tilde{U},} & {g_{\infty}<x<h_{\infty}, \enspace t\in (-\infty,\infty),} \\ {\tilde{V}_{t}=D_{2} \tilde{V}_{x x}+a_{2}\left(N_{2}-\tilde{V}\right) \tilde{U}-d \tilde{V},} & {g_{\infty}<x<h_{\infty}, \enspace t \in (-\infty,\infty),} \end{array}\right.$$ with $\tilde{U}(h_{\infty},t)=0$ for $t\in (-\infty,\infty).$ Note that $\tilde{U}(x_{0},0)\geq \frac{\theta}{2}$, then by strong maximum principle, we get $\tilde{U}>0$ in $(g_{\infty},h_{\infty})\times(-\infty,\infty)$. Since $$\tilde{U}_{t}-D_{1} \tilde{U}_{x x}+\mu \tilde{U}_{x}+({a_{1}N_{2}+\gamma})\tilde{U}\geq 0,$$ applying Hopf Lemma at the point $(h_{\infty},0)$, we can get $\tilde{U}_{x}(h_{\infty},0)<0.$ It implies that there exists a $\delta_0>0$ such that $$\label{4.12}
U_{x}(h(t_{k_{n}}),t_{k_{n}})=(U_{k_{n}})_{x}(h(t_{k_{n}}),0)\leq-\delta_0<0, for \enspace n\gg 1,$$ so $h^\prime(t_{k_{n}})\geq \nu \delta_0>0$ as $ n$ sufficiently large. Since $h(t)$ is bounded, then $h^{\prime}(t)\rightarrow 0$ as $t\rightarrow \infty$, so $h^\prime(t_{k_{n}})\rightarrow 0$ as $n\rightarrow \infty $, which is a contradiction. Therefore, $ \lim\limits_{t\rightarrow +\infty}||U(\cdot,t)||_{C(g(t),h(t))}=0.$
For any given $\epsilon >0$, there exists a $T>0$ such that $U(x,t)\leq \epsilon$ for $x\in[g(t),h(t)]$ and $t>T$. Therefore, $V_{t}-D_{2} V_{x x}\leq a_{2}N_{2}\epsilon-\gamma V.$ By Comparison Principle, we get $$\lim\limits_{t\rightarrow +\infty}\sup\limits||V(\cdot,t)||_{C(g(t),h(t))}\leq\frac{a_{2}N_{2}}{\gamma}\epsilon.$$ Since $\epsilon$ is arbitrary, $\lim\limits_{t\rightarrow +\infty}||V(\cdot,t)||_{C(g(t),h(t))}=0$ holds.
Moreover, we have the following result.
\[t43\] If $ h_{\infty}-g_{\infty}<\infty$, then $R^D_{0}((g_{\infty},h_{\infty}),\mu,D_{1},D_{2})=R^D_{0}((g_{\infty},h_{\infty}),\mu,\gamma,d)\leq 1.$
We prove this theorem by contradiction. Assume that $R^D_{0}((g_{\infty},h_{\infty}),\mu,D_{1},D_{2})> 1$, then there exists $T\gg 1$ such that $R^D_{0}((g(T),h(T)),\mu,\gamma,d)>1$. For small $\varepsilon>0,$ according to the continuity of $R^D_{0}((g(T),h(T)),\mu,\gamma,d)$ in $\gamma$ and $d$, we get $$R^D_{0}((g(T),h(T)),\mu,\gamma+\varepsilon,d+\varepsilon)>1$$ with $\varepsilon$ dependent on $T$.
Let $(H(x,t),M(x,t))$ be the solution of $$\label{4.13}
\left\{\begin{array}{ll}{H_{t}=D_{1} H_{x x}-\mu H_{x}+a_{1}\left(N_{1}-H\right) M-(\gamma+\varepsilon) H,} & {g(T)<x<h(T), \enspace t>T,} \\ {M_{t}=D_{2} M_{x x}+a_{2}\left(N_{2}-M\right) H-(d+\varepsilon) M,} & {g(T)<x<h(T), \enspace t>T,} \\ {H(g(T), t)=H(h(T), t)=0,} & { t>T,}\\ {M(g(T), t)=M(h(T), t)=0,} & { t>T,} \\ {H(x,T)=U(x,T),} & {g(T)\leq x\leq h(T),} \\ {M(x,T)=V(x,T),} & {g(T)\leq x\leq h(T).} \end{array}\right.$$ By maximal principle, it follows that $U(x,t)\geq e^{\varepsilon(t-T)}H(x,t)$ and $V(x,t)\geq e^{\varepsilon(t-T)}M(x,t)$ in $[g(T),h(T)]\times [T,\infty).$ Moreover, in view of $R^D_{0}((g(t),h(t)),\mu,\gamma+\varepsilon,d+\varepsilon)>1$, by using the upper and lower solution meothod with monotone iterations in [@pao2012nonlinear] (also see to [@tian2018advection]), we can get $$\lim\limits_{t\rightarrow +\infty}H(x,t)=\tilde{H}(x), \lim\limits_{t\rightarrow +\infty}M(x,t)=\tilde{M}(x)$$ uniformly on $[g(T),h(T)]$, where $(\tilde{H}(x),\tilde{M}(x))$ is the positive steady solution of (\[4.13\]) and satisfies $$\label{4.14}
\left\{\begin{array}{ll}{-D_{1} \tilde{H}^{\prime\prime}+\mu \tilde{H}^{\prime}=a_{1}\left(N_{1}-\tilde{H}\right) \tilde{M}-(\gamma+\varepsilon) \tilde{H},} & {g(T)<x<h(T), } \\ {-D_{2}\tilde{M}^{\prime\prime}=a_{2}\left(N_{2}-\tilde{M}\right) \tilde{H}-(d+\varepsilon) \tilde{M},} & {g(T)<x<h(T),} \\ {\tilde{H}(g(T))=\tilde{H}(h(T))=0,}\\{\tilde{M}(g(T))=\tilde{M}(h(T))=0.} \end{array}\right.$$ Therefore, $$\lim\limits_{t\rightarrow +\infty}H(0,t)=\tilde{H}(0), \lim\limits_{t\rightarrow +\infty}M(x,0)=\tilde{M}(0),$$ it imples that $$\label{4.145}
\begin{aligned}
U(0,t)\geq e^{\varepsilon(t-T)}\tilde{H}(0)>0, V(0,t)\geq e^{\varepsilon(t-T)}\tilde{M}(0)>0
\end{aligned}$$ in $[T,\infty)$. Since $h_\infty-g_\infty<\infty,$ by Theorem \[t42\], we get $U(0,t)\rightarrow 0$, $V(0,t)\rightarrow 0$ as $t\rightarrow \infty$, which is contradict to (\[4.145\]). Hence, the proof is completed.
The following result is an ordinary corollary of the above theorem, which is similar to the argument in [@ge2015sis]. It implies that $h_\infty$ and $g_\infty$ will be finite or infinite simultaneously under the assumption of $(H)$.
\[t44\] Assume that (H) holds, if $h_\infty<\infty$ or $-g_\infty<\infty$, then $h_\infty-g_\infty<\infty$. Moreover, we get $R^D_{0}((g_\infty,h_\infty),\mu,D_{1},D_{2})\leq 1.$
Next we will give some suffient conditions for vanishing of the virus.
\[t45\] If $R^F_{0}(0)=R^D_{0}((-h_{0},h_{0}),\mu,D_{1},D_{2})< 1,$ then $h_{\infty}-g_{\infty}<\infty,$ and $$\label{4.15}
\lim\limits_{t\rightarrow +\infty}||U(\cdot,t)||_{C(g(t),h(t))}=\lim\limits_{t\rightarrow +\infty}||V(\cdot,t)||_{C(g(t),h(t))}=0,$$ if given $||U_{0}||_{L_{\infty}}$ and $||V_{0}||_{L_{\infty}}$ are so small.
Following from the Lemma \[l31\], if $R^F_{0}(0)=R^D_{0}((-h_{0},h_{0}),\mu,D_{1},D_{2})< 1$, then there exists $\lambda_{0}>0$ satisfying $$\label{4.16}
\left\{\begin{array}{ll}{-D_{1} \phi_{x x}=-\mu\phi_{x}+a_{1} N_{1} \psi-\gamma \phi+\lambda_{0}\psi,} & {x\in (-h_{0},h_{0}),} \\ {-D_{2} \psi_{x x}=a_{2} N_{2} \phi-d \psi+\lambda_{0}\psi,} & {x\in (-h_{0},h_{0}), } \\ {\phi(x)=\psi(x)=0,} & {x=\pm h_{0},} \end{array}\right.$$ where $(\phi,\psi)>0$ in $(-h_0,h_0)$.
Next, we will prove two claims.\
**Claim 1.** There exists $L\gg 1$ such that $$\label{4.17}
x\phi^{\prime}(x)<L\phi(x) , x\psi^{\prime}(x)<L\psi(x)$$ for $x \in[-h_{0},h_{0}].$
Now we first discuss the case of $\phi(x)$. In fact, since $\phi^{\prime}(-h_{0})>0$ and $\phi^{\prime}(h_{0})<0$,we denote that $x_{1}$ and $x_{2}$ are the first and last critial point from $-h_{0}$ to $h_{0}$, then $\phi^{\prime}(x_{1})=0, \phi^{\prime}(x_{2})=0$ and $-h_{0}< x_{1}\leq x_{2}<h_{0}.$ Therefore, there exists a $L_1>0$ such that $x\phi^{\prime}(x)<L_1\phi(x)$ holds for $x\in[-h_{0},x_{1}]$ or $x\in[x_{2},h_{0}]$.
Since $\phi(x)>0$, then $$x\phi^{\prime}(x)\leq x||\phi^{\prime}||_{L^{\infty}([x_{1},x_{2}])}\leq L_2 \min\limits_{[x_{1},x_{2}]}\phi(x)\leq L_2\phi(x)$$ for $x\in[x_{1},x_{2}]$, where $L_2\geq \dfrac{h_{0}||\phi^{\prime}||_{L^{\infty}([x_{1},x_{2}])}}{\min\limits_{[x_{1},x_{2}]}\phi(x)}.$ Take $L=\max\{L_1,L_2\}$, then $L$ satisfies the requirement of (\[4.17\]). Similarly, we can take $L\gg 1$, such that $x\psi^{\prime}(x)<L\psi(x)$ for $x\in[-h_0,h_0].$
**Claim 2.** There exists $\tilde{L}>0$ such that $$\label{4.19}
\frac{1}{\tilde{L}}\leq \dfrac{\phi(x)}{\psi(x)}\leq \tilde{L}, \enspace for \enspace x\in[-h_{0},h_{0}].$$
Indeed, since $\phi^{\prime}(h_{0})<0$ and $\psi^{\prime}(h_{0})<0$, then there exists small $\sigma_{1}>0$ such that $\phi^{\prime}(x)<\frac{\phi^{\prime}(h_{0})}{2}<0$ and $\psi^{\prime}(x)<\frac{\psi^{\prime}(h_{0})}{2}<0$ for any $x\in [h_{0}-\sigma_{1},h_{0}]$. Let $\tilde{L}_{1}=\max\limits_{[h_{0}-\sigma_{1},h_{0}]}\left\{\dfrac{\phi^{\prime}(x)}{\psi^{\prime}(x)}\right\},$ then $\dfrac{\phi^{\prime}(x)}{\psi^{\prime}(x)}\leq \tilde{L}_1,$ by Cauchy mean value theorem, it follows $\dfrac{\phi(x)}{\psi(x)}=\dfrac{\phi(x)-\phi(h_{0})}{\psi(x)-\psi(h_{0})}
=\dfrac{\phi^{\prime}(\hat{x})}{\psi^{\prime}(\hat{x})}\leq \tilde{L}_1$ for any $x\in[h_{0}-\sigma_{1},h_{0}]$ and some $\hat{x}\in [h_{0}-\sigma_{1},h_{0}].$ Let $\tilde{L}_2=\max\limits_{[h_{0}-\sigma_{1},h_{0}]}\left\{\dfrac{\psi^{\prime}(x)}{\phi^{\prime}(x)}\right\},$ then $\dfrac{\psi^{\prime}(x)}{\phi^{\prime}(x)}\leq \tilde{L}_2.$ Similarly, we get $\dfrac{\psi(x)}{\phi(x)}\leq \tilde{L}_{2}$ for any $x\in[h_{0}-\sigma_{1},h_{0}].$ Take $\tilde{L}_{3}=\max\limits\left\{\tilde{L}_{1},\tilde{L}_{2}\right\},$ then $\frac{1}{\tilde{L}_{3}}\leq \dfrac{\phi(x)}{\psi(x)}\leq \tilde{L}_{3}$ for $x\in[h_{0}-\sigma_{1},h_{0}].$ Since $\phi^{\prime}(-h_{0})>0$ and $\psi^{\prime}(-h_{0})>0$, then there exists small $\sigma_{2}>0$ such that $\phi^{\prime}(x)>\frac{\phi^{\prime}(-h_{0})}{2}>0$ and $\psi^{\prime}(x)>\frac{\psi^{\prime}(-h_{0})}{2}>0$ for any $x\in [-h_{0},-h_{0}+\sigma_{2}]$. Thus, there exists $\tilde{L}_{4}$ such that $\frac{1}{\tilde{L}_{4}}\leq \dfrac{\phi(x)}{\psi(x)}\leq \tilde{L}_{4}$ for $x\in [-h_{0},-h_{0}+\sigma_{2}]$. Since $\phi(x)>0, \psi(x)>0$, then there exists $\tilde{L}_{5}>0$ such that $\frac{1}{\tilde{L}_{5}}\leq \dfrac{\phi(x)}{\psi(x)}\leq \tilde{L}_{5}$ for $x\in [-h_{0}+\sigma_{2},h_{0}-\sigma_{1}]$. Therefore, let $\tilde{L}=\max\limits\{\tilde{L}_{3},\tilde{L}_{4},\tilde{L}_{5}\}$, then (\[4.19\]) holds.
Let $$\label{4.20}
\begin{aligned}
&\vartheta(t)=h_{0}(1+\delta-\frac{\delta}{2}e^{-\delta t}),
\\&\overline{U}(x,t)=a_0 e^{-\delta t}\phi\left(\frac{x h_{0}}{\vartheta(t)}\right)e^{{\frac{\mu}{2D_{1}}}\left(1-\frac{h_{0}}{\vartheta(t)}\right)x},
\\&\overline{V}(x,t)=a_0 e^{-\delta t}\psi\left(\frac{x h_{0}}{\vartheta(t)}\right),
\end{aligned}$$ for any $x\in(-\vartheta(t),\vartheta(t))$ and $t\geq0,$ where $a_0>0$ and $0<\delta\ll 1$ such that $$\label{4.21}
\begin{aligned}
&-\delta-\frac{L h^2_{0}}{\vartheta^2(t)} \frac{\delta^2}{2}-\dfrac{\mu h^2_{0}}{4D_{1}}\dfrac{\delta^2}{\vartheta(t)}+\frac{\mu^2}{4D_{1}}\left(1-\dfrac{h^2_{0}}{\vartheta^2(t)}\right)+\gamma\left(1-\dfrac{h^2_{0}}{\vartheta^2(t)}\right)\\&+\lambda_{0}\dfrac{h^2_{0}}{\vartheta^2(t)}
+a_{1}N_{1}\tilde{L}\left(\dfrac{h^2_{0}}{\vartheta^2(t)}-\frac{\delta}{1+\frac{\delta}{2}}\right)\geq 0,
\end{aligned}$$ and $$\label{4.22}
\begin{aligned}
&-\delta -\dfrac{L h^2_{0}}{\vartheta^2(t)} \frac{\delta^2}{2} +a_{2}N_{2}\tilde{L}\left(\dfrac{h^2_{0}}{\vartheta^2(t)}-1\right)+d\left(1-\dfrac{h^2_{0}}{\vartheta^2(t)}\right)+\lambda_{0}\dfrac{h^2_{0}}{\vartheta^2(t)}\geq 0.
\end{aligned}$$ Further, direct calculation gives $$\label{4.23}
\begin{aligned}
&\overline{U}_{t}-D_{1}\overline{U}_{xx}+\mu\overline{U}_{x}-a_{1}(N_{1}-\overline{U})\overline{V}+\gamma\overline{U}
\\&\geq \overline{U}_{t}-D_{1}\overline{U}_{xx}+\mu\overline{U}_{x}
-a_{1}N_{1}\overline{V}+\gamma\overline{U}
\\&=-\delta \overline{U}-\frac{xh^2_{0}}{\vartheta^2(t)} \frac{\delta^2}{2} \dfrac{\phi^{\prime}}{\phi}\overline{U}+\dfrac{\mu h^2_{0}x}{4D_{1}}\dfrac{\delta^2}{\vartheta^2(t)}\overline{U}+\frac{\mu^2}{4D_{1}}\left(1-\dfrac{h^2_{0}}{\vartheta^2(t)}\right)\overline{ U}
\\&+a_{1}N_{1}\overline{V}\left(\dfrac{h^2_{0}}{\vartheta^2(t)}e^{{\frac{\mu}{2D_{1}}}\left(1-\frac{h_{0}}{\vartheta(t)}\right)x}-1\right)+\gamma\left(1-\dfrac{h^2_{0}}{\vartheta^2(t)}\right)\overline{V}+\lambda_{0}\dfrac{h^2_{0}}{\vartheta^2(t)}\overline{U}
\\&\geq\overline{U}\left(-\delta-\frac{L h^2_{0}}{\vartheta^2(t)} \frac{\delta^2}{2}-\dfrac{\mu h^2 _{0}}{4D_{1}}\dfrac{\delta^2}{\vartheta(t)}+\frac{\mu^2}{4D_{1}}\left(1-\dfrac{h^2_{0}}{\vartheta^2(t)}\right)\right)
\\&+\overline{U}\left(\gamma\left(1-\dfrac{h^2_{0}}{\vartheta^2(t)}\right)+\lambda_{0}\dfrac{h^2_{0}}{\vartheta^2(t)}+a_{1}N_{1}\tilde{L}\left(\dfrac{h^2_{0}}{\vartheta^2(t)}-\frac{\delta}{1+\frac{\delta}{2}}\right)\right)
\\&\geq 0
\end{aligned}$$ and $$\label{4.24}
\begin{aligned}
&\overline{V}_{t}-D_{2}\overline{V}_{xx}-a_{2}(N_{2}-\overline{V})\overline{U}+d\overline{V}
\\&\geq\overline{V}_{t}-D_{2}\overline{V}_{xx}-a_{2}N_{2}\overline{U}+d\overline{V}
\\&=-\delta \overline{V}-\dfrac{x h^2_{0}}{\vartheta^2(t)} \frac{\delta^2}{2} \dfrac{\psi^{\prime}}{\psi}\overline{V}+a_{2}N_{2}\dfrac{\phi}{\psi}\overline{V} \left(\dfrac{h^2_{0}}{\vartheta^2(t)}-1\right)
\\&+d\overline{V}\left(1-\dfrac{h^2_{0}}{\vartheta^2(t)}\right)+\lambda_{0}\overline{V}\dfrac{h^2_{0}}{\vartheta^2(t)}
\\&\geq\overline{V}\left(-\delta -\dfrac{L h^2_{0}}{\vartheta^2(t)} \frac{\delta^2}{2} +a_{2}N_{2}\tilde{L}\left(\dfrac{h^2_{0}}{\vartheta^2(t)}-1\right)+d\left(1-\dfrac{h^2_{0}}{\vartheta^2(t)}\right)+\lambda_{0}\dfrac{h^2_{0}}{\vartheta^2(t)}\right)
\\&\geq0
\end{aligned}$$ for any $x\in(-\vartheta(t),\vartheta(t))$ and $t\geq0.$
Take $a_0=\frac{\delta^2h_{0}}{2\nu e^{{\frac{\mu}{2D_{1}}}h_{0}\delta}}\min\limits\left\{\frac{-1}{\phi^{\prime}(h_{0})},\frac{1}{\phi^{\prime}(-h_{0})} \right\}$, then
$$\label{4.25}
\left\{\begin{array}{ll}{\overline{U}_{t}-D_{1} \overline{U}_{x x} \geq -\mu\overline{U}_{x}+a_{1}\left(N_{1}-\overline{U}\right) \overline{V}-\gamma \overline{U},} & {-\vartheta(t)<x<\vartheta(t), \quad t\geq0,} \\ {\overline{V}_{t}-D_{2} \overline{V}_{x x} \geq a_{2}\left(N_{2}-\overline{V}\right) \overline{U}-d \overline{V},} & {-\vartheta(t)<x<\vartheta(t), \enspace t\geq0,} \\ {\overline{U}(0, t) \geq U(0, t), \quad \overline{V}(0, t) \geq V(0, t),} & {t\geq0,} \\ {\overline{U}(x, t)\geq 0,\overline{V}(x, t)\geq 0,} & {x= \pm \vartheta(t), \quad t\geq0,} \\ {\vartheta^{\prime}(t) \geq-\nu \overline{U}_{x}(\vartheta(t), t),-\vartheta^{\prime}(t) \leq-\nu \overline{U}_{x}(-\vartheta(t), t),} & {\enspace t\geq 0.} \end{array}\right.$$
If $$\label{4.26}
\begin{aligned}
&||U_{0}||_{L_{\infty}}\leq a_0 \min\limits_{x\in [-h_{0},h_{0}]}\phi(\frac{x}{1+\delta})e^{{\frac{\mu}{2D_{1}}}(\frac{\delta}{1+\delta})x},
\\&||V_{0}||_{L_{\infty}}\leq a_0 \min\limits_{x\in [-h_{0},h_{0}]}\psi(\frac{x}{1+\delta}),
\end{aligned}$$ then $$\label{4.27}
\begin{aligned}
&\overline{U}(x, 0)=a_0\phi(\frac{x}{1+\delta})e^{{\frac{\mu}{2D_{1}}}(\frac{\delta}{1+\delta})x}
\geq {U_{0}(x)},
\\&\enspace\overline{V}(x, 0)=a_0\psi(\frac{x}{1+\delta})\geq V_{0}(x).
\end{aligned}$$ Therefore, $(\overline{ U}(x,t),\overline{V}(x,t);-\vartheta(t),\vartheta(t))$ is an upper solution of (\[1.3\]). Hence, $h(t)\leq \vartheta(t), g(t)\geq -\vartheta(t)$, then $h_{\infty}-g_{\infty}\leq \lim\limits_{t\rightarrow +\infty}2\vartheta(t)\leq 2h_{0}(1+\delta).$ By Theorem \[t42\], we can get $\lim\limits_{t\rightarrow +\infty}||U(\cdot,t)||_{C(g(t),h(t))}=\lim\limits_{t\rightarrow +\infty}||V(\cdot,t)||_{C(g(t),h(t))}=0.$
\[c46\] When given initial data $U_0(x)$ and $V_0(x)$ are small enough, if $\mu=0,$ then $R^D_{0}((-h_{0},h_{0}),$ $0, D_{1},D_{2})=1,$ the disease is spreading according to the arrguments in [@lin2017spatial]. However, if $\mu\neq 0,$ then $R^D_{0}((-h_{0},h_{0}),\mu,D_{1},D_{2})<1,$ the disease is vanishing by Theorem \[t45\].
In fact, even if $U_0(x)$ and $V_0(x)$ are not small, the disease will be extinct when the expanding capability $\nu$ is sufficiently small. Detailed proof can refer to Lemma 3.8 in [@du2010spreading].
\[t47\] If $R^F_0(0)=R^D_{0}((-h_{0},h_{0}),\mu, D_{1},D_{2})<1,$ then there exists a small $\nu^*>0$ depending on $U_0$ and $V_0$ such that $h_\infty-g_\infty <\infty$ and $\lim\limits_{t\rightarrow +\infty}||U(\cdot,t)||_{C(g(t),h(t))}=\lim\limits_{t\rightarrow +\infty}||V(\cdot,t)||_{C(g(t),h(t))}=0$ when $\nu<\nu^*$.
**The spreading regime of WNv** {#s5}
===============================
Next, we will discuss the spreading conditions of the disease and investigate the impact of $R^F_0(t)$ on the infected habitats and the densities of mosquitoes. It implies that the spreading will occur when $R^F_0(0)\geq1$.
\[t51\] If $R^F_{0}({0})=R^D_0((-h_0,h_0),\mu,D_1,D_2)\geq 1$, then $ h_{\infty}-g_{\infty}=\infty$ and $$\lim\limits_{t\rightarrow +\infty}inf||U(\cdot,t)||_{C(g(t),h(t))}>0, \lim\limits_{t\rightarrow +\infty}inf||V(\cdot,t)||_{C(g(t),h(t))}>0.$$ It means that the disease will spread.
In the case of $R^F_{0}({0})=R^D_0((-h_0,h_0),\mu,D_1,D_2)>1$, by Lemma \[l31\], there exists a principal eigenvalue $\lambda_0<0$ with positive eigenvalue function $(\phi(x),\psi(x))$ for the following problem $$\label{5.01}
\left\{\begin{array}{ll}{-D_{1} \phi_{x x}=-\mu\phi_{x}+a_{1} N_{1} \psi-\gamma \phi+\lambda_0\phi,} & {x\in (-h_0,h_0),} \\ {-D_{2} \psi_{x x}=a_{2} N_{2} \phi-d \psi+\lambda_0\psi,} & {x\in (-h_0,h_0), } \\ {\phi(x)=\psi(x)=0,} & { x=\pm h_0 .} \end{array}\right.$$ Let us construct a lower solution to system (\[1.3\]). Set $$\underline U(x,t)=\upsilon \phi(x), \underline V(x,t)=\upsilon \psi(x),$$ where $x\in[-h_0,h_0],t\geq 0$ and $0<\upsilon\ll1 $ .
Simple computation gives $$\label{5.02}
\begin{aligned}
&\underline{U}_{t}-D_{1} \underline{U} _{x x}+\mu \underline{U}_x-a_1(N_1 - {\underline U}){\underline {V}}+\gamma \underline U
\\&=\upsilon (-D_1\phi_{xx}+\mu \phi_x-a_1N_1\psi+\upsilon a_1 \phi\psi+\gamma \phi)
\\&= \upsilon(\lambda_{0}\phi+\upsilon a_1 \phi\psi)
\end{aligned}$$ and $$\label{5.03}
\begin{aligned}
&\underline{V}_{t}-D_{2} \underline{V}_{x x}-a_2(N_2-\underline {V}) \underline {U}+d \underline V
\\&=\upsilon (-D_2\psi_{xx}-a_2N_2\phi+\upsilon a_2 \phi\psi+d \phi)
\\&= \upsilon(\lambda_{0}\psi+\upsilon a_2 \phi\psi).
\end{aligned}$$ Since $\lambda_{0}<0$, recalling that $\phi^\prime(-h_0), \psi^\prime(-h_0)>0$ and $\phi^\prime(h_0), \psi^\prime(h_0)<0$, take $\upsilon$ sufficiently small such that $$\label{5.04}
\left\{\begin{array}{ll}{\underline{U}_{t}-D_{1} \underline{U}_{x x} \leq -\mu\underline{U}_{x}+a_{1}\left(N_{1}-\underline{U}\right) \underline{V}-\gamma \underline{U},} & {-h_0<x<h_0, \quad t>0,} \\ {\underline{V}_{t}-D_{2} \underline{V}_{x x} \leq a_{2}\left(N_{2}-\underline{V}\right) \underline{U}-d \underline{V},} & {-h_0<x<h_0, \enspace t>0,} \\ {\underline{U}(x, t)\leq 0,\underline{V}(x, t)= 0,} & {x= \pm h_0, \quad t>0,} \\{0=(-h_0)^\prime\geq -\nu \underline U_x(-h_0,t),} & {t>0,}\\{0=(h_0)^\prime\leq -\nu \underline U_x(h_0,t),} & {t>0,}\\ {\underline{U}(0, t) \leq U(0, t), \quad \underline{V}(0, t) \leq V(0, t),} & {t>0.} \end{array}\right.$$ Thus, by Lemma \[l22\], $U(x,t)\geq\underline U(x,t)$ and $V(x,t)\geq \underline V(x,t)$ for $x\in[-h_0,h_0]$ and $t\geq 0.$ It implies that $\lim\limits_{t\rightarrow +\infty}inf||U(\cdot,t)||_{C(g(t),h(t))}\geq\upsilon \phi(0)>0$ and $\lim\limits_{t\rightarrow +\infty}inf||V(\cdot,t)||_{C(g(t),h(t))}\geq\upsilon \psi(0)>0.$ Moreover, by Theorem \[t42\], we get $h_\infty-g_\infty=\infty.$
For $R^F_0(0)=R^D_0((-h_0,h_0),\mu,D_1,D_2)=1$, by Property \[p32\], then $R^F_0(t_0)>R^F_0(0)=1$ for any $t_0>0$. Take the initial time from 0 to $t_0(>0)$ and repeat the above procedures, we can get $h_\infty-g_\infty=\infty.$
\[r51\] Assume that $(H)$ holds. $R^F_{0}(t_{0})\geq 1$ for some $t_{0}\geq 0$ if and only if the disease will spread. Indeed, if $R^F_0(t)<0$ for any $t\geq 0$, then $h_\infty-g_\infty<\infty$, that is, the disease will be extinct.
The following theorem can be obtained by constructing upper and lower sulution and analysis about the steady state of (\[1.3\]) by Poincar[é]{}-Bendixson theorem and Proposition 2.1 in [@lewis2006traveling].
\[t52\] When spreading occurs, the problem (\[1.3\]) admits a unique coexistent steady state $E^{*}=(U^{*},V^{*})$, where $(U^{*},V^{*})$ is the unique globally asymptoyic stable endemic equilibrium of the following equation $$\label{5.1}
\left\{\begin{array}{ll}{\frac{du}{dt}=a_{1}(N_{1}-u)v-\gamma u,} & {t>0,} \\{\frac{dv}{dt}=a_{2}(N_{2}-v)u-d v,} & {t>0,} \end{array}\right.$$ where $$U^{*}=\dfrac{a_1 a_2 N_1 N_2-\gamma d}{a_1 a_2 N_2 +a_2 \gamma },\enspace V^{*}=\dfrac{a_1 a_2 N_1 N_2-\gamma d}{a_1 a_2 N_1 + a_1 d}.$$
The following theorem implies the disease will spread when the expanding capability is sufficiently large, which is silimar to the arguments of [@tarboush2017spreading].
\[t53\] Assume that $(H)$ holds. If $R^F_0(0)<1,$ then $h_\infty-g_\infty=\infty$ when $\nu$ is large enough.
In view of the above arguments, we can give the vanishing-spreading dichotomy regines of WNv.
\[t54\] Assume that $(H)$ holds. Let $(U(x,t),V(x,t);g(t),h(t))$ be the solution of system (\[1.3\]), then the vanishing-spreading dichototmy regines hold:\
(1)Vanishing: $h_\infty-g_\infty<\infty$ and $\lim\limits_{t\rightarrow +\infty}(||U(\cdot,t)||_{C(g(t),h(t))}+||V(\cdot,t)||_{C(g(t),h(t))})=0$;\
(2)Spreading: $h_\infty-g_\infty=\infty$ and $\lim\limits_{t\rightarrow +\infty}U(x,t)=U^*,
\lim\limits_{t\rightarrow +\infty}V(x,t)=V^*$ uniformly for $x$ in any compact subset of $\mathbb{R}.$
Asymptotic spreading speed {#s6}
==========================
In order to prevent the WNv from dispersing, it is essential to investigate the asymptotic spreading speed of the infected boundary. In Section 7 of [@lin2017spatial], Lin and Zhu compared the definition of the minnimal wave speed, the spreading speed with the asymptotic spreading speed. Now, we aim to give the estimates of the asymptotic spreading speeds about the leftward front and rightward front for system (\[1.3\]). For this purpose, we first recall a lemma.
\[l55\] Assume that $a_{1}a_{2}N_{1}N_{2}>\gamma d$, then there exists a constant $c^{*}>0$ such that for every $c\in [0,c^{*}),$ the system $$\label{5.2}
\left\{\begin{array}{ll}
{D_{1}u^{\prime\prime}-cu^{\prime}+a_{1}(N_{1}-u)v-\gamma u=0,} & { 0<s<\infty,} \\{D_{2}v^{\prime\prime}-cv^{\prime}+a_{2}(N_{2}-v)u-d v=0,} & {0<s<\infty,} \\ {(u(0),v(0) =(0,0),(u(\infty),v(\infty)) =(U^{*},V^{*})}
\end{array}\right.$$ admits a strictly increasing solution $(u_c,u_c)\in C^2(\mathbb{R^+})\times C^2(\mathbb{R^+})$ and there exists unique $c_\nu\in (0,c^*)$ such that $\nu u_{c_\nu}^{\prime}(0)=c_\nu$ for any $\nu>0.$
When the spreading happens, considering the small advection, we will give a sharper estimate for different asymptotic spreading speeds of the leftward and rightward fronts of (\[1.3\]). This is a main contribution of this paper in studying WNv.
\[t56\] Assume that $0<\mu<\mu^{*}$ and $a_1a_2N_1N_2>d\gamma$. Let $(U,V;g,h)$ be a solution of (\[1.3\]) with $h_{\infty}-g_{\infty}=\infty,$ the asymptotic spreading speeds of the leftward front and rightward front satisfy $$\label{5.3}
\lim\limits_{t\rightarrow +\infty}\sup\limits\frac{-g(t)}{t}\leq c_{\nu}\leq \lim\limits_{t\rightarrow +\infty}\inf\limits\frac{h(t)}{t}.$$ where $c_{\nu}$ is the asymptotic spreading speed of the problem (\[1.3\]) without the advection term.
We will divide the proof of this theorem into two steps.
**Step 1.** we will show $$\label{5.4}
\lim\limits_{t\rightarrow +\infty}\inf\limits\frac{h(t)}{t}\geq c_{\nu}.$$ For the small $\omega >0,$ construct the following auxiliary system motivated by Wang et al. [@wang2017asymptotic] $$\label{5.5}
\left\{\begin{array}{ll}
{D_{1}u_{\omega}^{\prime\prime}-c_{\omega}u_{\omega}^{\prime}+a_{1}(N_{1}-u_{\omega})v_{\omega}-(\gamma+2\omega) u_{\omega}=0,} & { 0<s<\infty,} \\{D_{2}v_{\omega}^{\prime\prime}-c_{\omega}v_{\omega}^{\prime}+a_{2}(N_{2}-v_{\omega})u_{\omega}-(d+2\omega) v_{\omega}=0,} & {0<s<\infty,} \\ {(u_{\omega}(0),v_{\omega}(0) =(0,0),(u_{\omega}(\infty),v_{\omega}(\infty)) =(u_{-2\omega}^{*},v_{-2\omega}^{*}),}
\end{array}\right.$$ where $$u_{-2\omega}^{*}=\dfrac{a_1 a_2 N_1 N_2-(\gamma+2\omega)(d+2\omega)}{a_1 a_2 N_2 +(\gamma+2\omega)a_2}, v_{-2\omega}^{*}=\dfrac{a_1 a_2 N_1 N_2-(\gamma+2\omega)(d+2\omega)}{a_1 a_2 N_1 +(d+2\omega)a_1}.$$ According to Lemma \[l55\], there exists unique $c_{\omega}=c(\nu,\omega)>0$ such that (\[5.5\]) has a unique strictly increasing solution $(u_\omega,v_\omega)$ satisfying $$\nu u_{\omega}^{\prime}(0)=c_\omega, \lim\limits_{\omega\rightarrow 0^{+}}c_\omega=c_\nu.$$ Since $h_\infty-g_\infty=\infty$, then $$(U(x,t),V(x,t))\rightarrow (U^{*},V^*)=\left(\dfrac{a_1 a_2 N_1 N_2-\gamma d}{a_1 a_2 N_2 +\gamma a_2},\dfrac{a_1 a_2 N_1 N_2-\gamma d}{a_1 a_2 N_1 +d a_1}\right)\enspace as\enspace t\rightarrow \infty$$ uniformly for $x$ in any compact subset of $\mathbb{R}$. Hence, there exist large $T_1>0$ and $L\in (0,h(T_1))$ such that $$h(t)>L, \enspace(U(x,t),V(x,t))\geq (u_{-\omega}^{*},v_{-\omega}^{*})>(u_{-2\omega}^{*},v_{-2\omega}^{*})$$ for $x\in[0,L]$ and $t\geq T_1,$ where $$u_{-\omega}^{*}=\dfrac{a_1 a_2 N_1 N_2-(\gamma+\omega)(d+\omega)}{a_1 a_2 N_2 +(\gamma+\omega)a_2}, \enspace v_{-\omega}^{*}=\dfrac{a_1 a_2 N_1 N_2-(\gamma+\omega)(d+\omega)}{a_1 a_2 N_1 +(d+\omega)a_1}.$$ Let $$\label{5.6}
\begin{aligned}
&\underline{h}(t)=c_\omega(t-T_1)+L,t\geq T_1,
\\&\underline{U}(x,t)=u_\omega(\underline{h}(t)-x),0\leq x\leq\underline{h}(t),t\geq T_1,
\\&\underline{V}(x,t)=v_\omega(\underline{h}(t)-x),0\leq x\leq\underline{h}(t),t\geq T_1.
\end{aligned}$$ Then $\underline{h}(T_1)=L<h(T_1)$ and $(\underline{U}(x,T_1),\underline{V}(x,T_1))=(u_\omega(\underline{h}(T_1)-x),v_\omega(\underline{h}(T_1)-x))
\leq (u_{-\omega}^{*},v_{-\omega}^{*})\leq ({U}(x,T_1),{V}(x,T_1))$ for $x\in[0,\underline{h}(T_1)]$. And we can see $$(\underline{U}(x,t),\underline{V}(x,t))=(u_\omega(\underline{h}(t)-x),v_\omega(\underline{h}(t)-x))\leq (u_{-\omega}^{*},v_{-\omega}^{*})\leq ({U}(x,t),{V}(x,t))$$ $$(\underline{U}(\underline{h}(t),t),\underline{V}(\underline{h}(t),t))=(0,0),\enspace \underline{h}^\prime(t)=c_\omega=-\nu \underline{U}_{x}(\underline{h}(t),t)$$ for $t\geq T_1.$ Moreover, since $u_\omega ^\prime>0, 0< \mu<\mu^{*},$ we can get $$\label{5.7}
\begin{aligned}
&\underline{U}_{t}-D_{1} \underline{U}_{x x}=c_\omega u_\omega ^\prime-D_1 u_{\omega}^{\prime\prime}
\\&=a_1 (N_1 -u_\omega) v_\omega -(\gamma+2\omega)u_\omega
\\&\leq a_1 (N_1 -u_\omega) v_\omega -\gamma u_\omega+\mu u_{\omega} ^\prime
\\&= a_{1}\left(N_{1}-\underline{U}\right) \underline{V}-\gamma \underline{U} -\mu\underline{U}_{x}
\end{aligned}$$ and $$\label{5.8}
\begin{aligned}
&\underline{V}_{t}-D_{2} \underline{V}_{x x}=c_\omega v_\omega ^\prime-D_2 v_{\omega}^{\prime\prime}
\\&=a_{2}(N_{2}-v_{\omega})u_{\omega}-(d+2\omega) v_{\omega}
\\&\leq a_{2}(N_{2}-v_{\omega})u_{\omega}-d v_{\omega}
\\&= a_{2}\left(N_{2}-\underline{V}\right) \underline{U}-d \underline{V}.
\end{aligned}$$ Therefore, in view of the Comprison Principle, we can get $h(t)\geq \underline{h}(t)\enspace for\enspace t\geq T_1$ and $$\label{5.9}
\lim\limits_{t\rightarrow +\infty}\inf\limits\frac{h(t)}{t}\geq c_{\omega}.$$ It follows that $\lim\limits_{t\rightarrow +\infty}\inf\limits\frac{h(t)}{t}\geq c_{\nu}$ as $\omega\rightarrow 0.$
**Step 2.** we will show $$\label{5.10}
\lim\limits_{t\rightarrow +\infty}\sup\limits\frac{-g(t)}{t}\leq c_{\nu}.$$ Considering the following ODE system $$\label{5.11}
\left\{\begin{array}{ll}{u^{\prime}=a_{1}(N_{1}-u)v-\gamma u,} & { t>0,} \\{v^{\prime}=a_{2}(N_{2}-v)u-d v,} & {t>0,} \\ {u(0)=\sup\limits_{x\in[-h_{0},h_{0}]}U_{0}(x),}\\{v(0)=\sup\limits_{x\in[-h_{0},h_{0}]}V_{0}(x),} \end{array}\right.$$ by Comprison Principle, we can obtain $$U(x,t),V(x,t)\leq(u(t),v(t)), for \enspace x\in[g(t),h(t)], t>0.$$ By Theorem \[t52\], $(u(t),v(t))\rightarrow (U^*,V^*)$ as $t\rightarrow \infty.$ Therefore, we get $$\label{5.12}
\lim\limits_{t\rightarrow +\infty}\sup\limits\max_{x\in[g(t),h(t)]}U(x,t)\leq U^*, \lim\limits_{t\rightarrow +\infty}\sup\limits\max_{x\in[g(t),h(t)]}V(x,t)\leq V^*.$$ Moreover, for any given small $\sigma>0,$ there exists $T_2>0$ such that $$(U(x,t),V(x,t)\leq( U^*,V^*)\leq (u^*_{\sigma},v^*_{\sigma})$$ for $t\geq T_2$ and $x\in [g(t),h(t)]$, where $$u_{\sigma}^{*}=\dfrac{a_1 a_2 N_1 N_2-(\gamma-\sigma)(d-\sigma)}{a_1 a_2 N_2 +(\gamma-\sigma)a_2}, v_{\sigma}^{*}=\dfrac{a_1 a_2 N_1 N_2-(\gamma-\sigma)(d-\sigma)}{a_1 a_2 N_1 +(d-\sigma)a_1}.$$ For the fixed $\sigma>0,$ construct the following auxiliary system $$\label{5.13}
\left\{\begin{array}{ll}
{D_{1}u_{\sigma}^{\prime\prime}-c_{\sigma}u_{\sigma}^{\prime}+a_{1}(N_{1}-u_{\sigma})v_{\sigma}-(\gamma-2\sigma) u_{\sigma}=0,} & { 0<s<\infty,} \\{D_{2}v_{\sigma}^{\prime\prime}-c_{\sigma}v_{\sigma}^{\prime}+a_{2}(N_{2}-v_{\sigma})u_{\sigma}-(d-2\sigma) v_{\sigma}=0,} & {0<s<\infty,} \\ {(u_{\sigma}(0),v_{\sigma}(0) =(0,0),(u_{\sigma}(\infty),v_{\sigma}(\infty)) =(u_{2\sigma}^{*},v_{2\sigma}^{*}),}
\end{array}\right.$$ where $$u_{2\sigma}^{*}=\dfrac{a_1 a_2 N_1 N_2-(\gamma-2\sigma)(d-2\sigma)}{a_1 a_2 N_2 +(\gamma-2\sigma)a_2}, v_{2\sigma}^{*}=\dfrac{a_1 a_2 N_1 N_2-(\gamma-2\sigma)(d-2\sigma)}{a_1 a_2 N_1 +(d-2\sigma)a_1}.$$ According to Lemma \[l55\], there exists unique $c_{\sigma}=c(\nu,\sigma)>0$ such that (\[5.13\]) has a unique strictly increasing solution $(u_\sigma,v_\sigma)$ satisfying $$\nu u_{\sigma}^{\prime}(0)=c_\sigma, \lim\limits_{\sigma\rightarrow 0^{+}}c_\sigma=c_\nu.$$ Since $(u_\sigma(\infty),v_\sigma(\infty))=(u_{2\sigma}^{*},v_{2\sigma}^{*})$, then there exists a $S_0\gg1$ such that $$(u^*_{\sigma},v^*_{\sigma})<(u_\sigma(S_0),v_\sigma(S_0)).$$ Let $$\label{5.14}
\begin{split}
&\overline g(t)=-c_\sigma(t-T_2)-S_0+g(T_2),\enspace t\geq T_2
\\&\overline{U}(x,t)=u_\sigma(x-\overline g(t)),\enspace \overline g(t)\leq x\leq 0,\enspace t\geq T_2,
\\&\overline{V}(x,t)=v_\sigma(x-\overline g(t)),\enspace \overline g(t)\leq x\leq 0,\enspace t\geq T_2.
\end{split}$$ Therefore, $$\overline g(T_2)=-S_0+g(T_2)<g(T_2),\enspace
\overline g^{\prime}(t)=-c_\sigma=-\nu u_\sigma^{\prime}(0)=-\nu\overline{U}_x(\overline g(t),t)$$ for $t\geq T_2,$ and $$\overline{U}(x,T_2)=u_\sigma(x+S_0-g(T_2))\geq u_\sigma(S_0)\geq U(x,T_2),$$ $$\overline{V}(x,T_2)=v_\sigma(x+S_0-g(T_2))\geq v_\sigma(S_0)\geq V(x,T_2)$$ for $x\in[\overline g(T_2), 0].$ Moreover, $$\overline U(\overline g(t),t)=u_\sigma(0)=0,\enspace \overline V(\overline g(t),t)=v_\sigma(0)=0,$$ $$\overline U(0,t)=u_\sigma(-\overline g(t))=u_\sigma(c_\sigma(t-T_2)+S_0-g(T_2))\geq u_\sigma(S_0)\geq U(0,t),$$ $$\overline V(0,t)=v_\sigma(-\overline g(t))=v_\sigma(c_\sigma(t-T_2)+S_0-g(T_2))\geq v_\sigma(S_0)\geq V(0,t)$$ for $t\geq T_2.$ Since $u_\sigma^{\prime}>0$ and $0<\mu<\mu^*,$ then $$\label{5.15}
\begin{aligned}
&\overline{U}_{t}-D_{1} \overline{U}_{x x}=c_\sigma u_\sigma ^\prime-D_1 u_{\sigma}^{\prime\prime}
\\&=a_1 (N_1 -u_\sigma) v_\sigma -(\gamma-2\sigma)u_\sigma
\\&\geq a_1 (N_1 -u_\sigma) v_\sigma -\gamma u_\sigma-\mu u_{\sigma} ^\prime
\\&= a_{1}\left(N_{1}-\overline{U}\right) \overline{V}-\gamma \overline{U} -\mu\overline{U}_{x}
\end{aligned}$$ and $$\label{5.16}
\begin{aligned}
&\overline{V}_{t}-D_{2} \overline{V}_{x x}=c_\sigma v_\sigma ^\prime-D_2 v_{\sigma}^{\prime\prime}
\\&=a_{2}(N_{2}-v_{\sigma})u_{\sigma}-(d-2\sigma) v_{\sigma}
\\&\geq a_{2}(N_{2}-v_{\sigma})u_{\sigma}-d v_{\sigma}
\\&= a_{2}\left(N_{2}-\overline{V}\right) \overline{U}-d \overline{V}.
\end{aligned}$$ Therefore, in view of Comparison Principle, we can get $g(t)\geq \overline{g}(t),\enspace t\geq T_2$ and $$\label{5.17}
\lim\limits_{t\rightarrow +\infty}\sup\limits\frac{-g(t)}{t}\leq c_{\sigma}.$$ It follows that $\lim\limits_{t\rightarrow +\infty}\sup\limits\frac{-g(t)}{t}\leq c_{\nu}$ as $\sigma\rightarrow 0.$ Therefore, our proof is completed.
\[r416\] When the disease is spreading, if the small advection rate $0<\mu<\mu^*$, the asymptotic spreading speed of the leftward front is less than the rightward front. Similarly, if $-\mu^*<\mu<0,$ the asymptotic spreading speed of the leftward front is more than the rightward front. This fact implies that the advection term plays an important role in influencing WNv progapation speed.
Numerical simulations {#s7}
=====================
In this section, we will provide some numerical simulations of our results by applying the Newton-Raphson method and a similar method as Razvan and Gabriel[@stef2008numerical] for the free boundary problem to investigate the impact of advection term on the transmission of West Nile virus.
Take some parameter values of (\[1.3\]) from [@lewis2006traveling]: $$\label{6.1}
\begin{aligned}
&\dfrac{N_2}{N_1}=20,\alpha_1=0.88,\alpha_2=0.16.
\end{aligned}$$ Let $D_1=6,D_2=1,h_0=15,\gamma=0.6,$ $$\label{6.2}
\begin{aligned}
U_0(x) =
\begin{cases}
0.1*cos(\frac{\pi x}{2h_0}), & x\in[-h_0,h_0] \\
0, & x\notin[-h_0,h_0]
\end{cases},
V_0(x) =
\begin{cases}
2* cos(\frac{\pi x}{2h_0}), & x\in[-h_0,h_0] \\
0, & x\notin[-h_0,h_0]
\end{cases}
\end{aligned}$$
Advection affects the basic reproduction number
-----------------------------------------------
\
First, we will study the impact of advection on the basic reproduction number. Fix $$\nu=2, \beta=0.3, a_1=0.88\times 0.3,a_2=0.16\times 0.3,d=0.3.$$ Take $\mu=0$ and $\mu=3$, respectively, then $$R_0:=\dfrac{a_1a_2N_1N_2}{\gamma d}=1.408>1, R^F_0(0,0):=R^D_{0}((-h_{0},h_{0}),0, D_{1},D_{2})=1.2241>1,$$ $$R^F_0(0,3):=R^D_{0}((-h_{0},h_{0}),3, D_{1},D_{2})=0.7831<1.$$ We choose small $U_0(x)$ and $V_0(x)$ as (\[6.2\]), it can be seen that the density of mosquitoes tends to a positive steady state from Fig.1 (a) with $R^F_0(0,0)>1$, that is, the disease will spread and the infected region will expand to the whole habitat; while the density of the infected mosquitoes diseases to 0 as $t\rightarrow \infty$ from Fig.1 (b) with $R^F_0(0,3)<0$, it implies that the disease will vanish and the infected habitat will limit to a bounded region. This comparison identifies the significient impact of advection term.
[cc]{}
![$R_0>R^F_0(0,0)>1>R^F_0(0,3)=0.7831$ []{data-label="fig:compare1"}](fig1-a.jpg "fig:"){width="150.00000%"} \[fig:visual\_smap\_o\]
![$R_0>R^F_0(0,0)>1>R^F_0(0,3)=0.7831$ []{data-label="fig:compare1"}](fig1-b.jpg){width="150.00000%"}
Advection rate affects the spreading of boundary
------------------------------------------------
\
Next, we will choose different advection intensities to study how they affect the spreading speed of boundaries. Fix $$\nu=4, \beta=0.5, a_1=0.88\times 0.5,a_2=0.16\times 0.5,d=0.029,$$ then $$\mu^{*}=2\sqrt{D_{1}(\dfrac{a_{1}a_{2}N_{1}N_{2}}{\gamma}-d)}\approx5.24.$$ Take $\mu=0$ and $\mu=2$, respectively, it is easy to see that the boundaries $x=g(t)$ and $x=h(t)$ expand differently with respect to $\mu$: the spreading speed of the right boundary is faster than the left boundary from Fig.2 (c) and Fig.2 (d) when $0<\mu<\mu^*$. Moreover, when spreading happens, the densities of birds and mosquitoes tend to a steady state in the long run and the infected habitat will expand to the whole area.
Discussions {#s8}
===========
The invasions of mosquitoes with West Nile virus, dengue fever virus and Zika virus or other virus have lead to many epidemic diseases risking people’s health. Understanding the spatial dispersal and dynamics of these virus plays a significant part in preventing and controlling infectious epidemics. In this paper, the dynamical behavior of a reaction-advection-diffusion WNv model with moving boundary conditions $x = g(t)$ and $x = h(t)$ about birds and mosquitoes is investigated by system (\[1.3\]), the description of which is more compatible with the biological reality.
Firstly, we introduce the spatial-temporal risk index $R^F_0(t)$ dependent on advection rate $\mu $ and spreading region $(g(t),h(t)$ as a threshold value to determine whether the disease will spread. According to its definition, the advection rate can influence the values of $R^F_0(t)$ significantly (see to Section \[s3\]). Next, we obtain some asymptotic properties of the temporal-spatial spreading of West Nile virus. Moreover, we get the sufficient and necessary conditions for spreading or vanishing of WNv. If $R^F_0(0)<1$ and initial data $||U_{0}||_{L_{\infty}},||V_{0}||_{L_{\infty}}$ are sufficiently small or $R^F_0(t)<1$ for any $t\geq0$, the epidemic will vanish and $\lim\limits_{t\rightarrow +\infty}(||U(\cdot,t)||_{C(g(t),h(t))}+||V(\cdot,t)||_{C(g(t),h(t))})=0$. Under the assumption of $(H)$, $R^F_0(t_0)\geq 1$ for some $t_0\geq 0$ if and only if the disease will spread and the system (\[1.3\]) admits a unique stable positive equilibrium $(U^*,V^*)$ when the spreading occurs.
[cc]{}
![$\nu=4, a_1=0.88\times 0.5, a_2=0.16\times 0.5$, $\mu^{*}\approx5.24$ []{data-label="fig:compare1"}](fig2-c.jpg "fig:"){width="130.00000%"} \[fig:visual\_smap\_o\]
![$\nu=4, a_1=0.88\times 0.5, a_2=0.16\times 0.5$, $\mu^{*}\approx5.24$ []{data-label="fig:compare1"}](fig2-d.jpg){width="130.00000%"}
Assume that the habitat at far distance is in high risk, we mainly consider the effect of the small advection movement on the double spreading fronts of WNv. On the one hand, when the advection term $\mu$ becomes larger from $0$ to $\mu^{*}$, then disease becomes more difficult to spread. On the other hand, when the spreading occurs and $0< \mu<\mu^{*},$ we prove that the asymptotic spreading speed of the leftward is less than the rightward front:$ \lim\limits_{t\rightarrow +\infty}\sup\limits\frac{-g(t)}{t}\leq c_{\nu}\leq \lim\limits_{t\rightarrow +\infty}\inf\limits\frac{h(t)}{t}.$ At last, we give some numerical simulations to investigate the impact of advection movement on the basic reproduction number and the spreading of boundary (see to Section \[s7\]). These simulation results are coincidate with the arguments by Maidana and Yang in [@maidana2009spatial].
Therefore, people can take measures to prevent diseases from dispersing according to our analysis and simulations. Furthermore, our reaction-advection-diffusion model of West Nile virus with double free boundaries can also be applied to study other cooperative systems of mosquito-borne diseases. In the future, the hign-dimensional system with free boundaries of WNv model will be explored. Moreover, the temperature, humidity and migration of birds with seasonality will be taken into our consideration to study the spreading of WNv.
[99]{}
Komar N. West Nile virus: epidemiology and ecology in North America. Advances in virus research, 2003, 61: 185–234
Nash D, Mostashari F, Fine A, et al. The Outbreak of West Nile Virus Infection in the New York City Area in 1999. New England Journal of Medicine, 2001, 344: 1807–1814.
Komar N, Langevin S, Hinten S, et al. Experimental infection of North American birds with the New York 1999 strain of West Nile virus. Emerging infectious diseases, 2003, 9: 311–322
Rudolf I, Bet[á]{}[š]{}ov[á]{} L, Bla[ž]{}ejov[á]{} H, et al. West Nile virus in overwintering mosquitoes, central Europe. Parasites [&]{} vectors, 2017, 10: 452
Chancey C, Grinev A, Volkova E, et al. The global ecology and epidemiology of West Nile virus. BioMed Research International, 2015, 2015
Lewis M, Renc[ł]{}awowicz J, Van den Driessche P. Traveling waves and spread rates for a West Nile virus model. Bulletin of Mathematical Biology, 2006, 68: 3–23
Maidana N A, Yang H M. Spatial spreading of West Nile Virus described by traveling waves. Journal of Theoretical Biology, 2009, 258: 403–417
Li F, Liu J, Zhao X Q. A West Nile Virus Model with Vertical Transmission and Periodic Time Delays. Journal of Nonlinear Science, 2019: 1–38
Wonham M J, Lewis M A, Renc[ł]{}awowicz J, et al. Transmission assumptions generate conflicting predictions in host-vector disease models: a case study in West Nile virus. Ecology letters, 2006, 9: 706–725
Hartley D M, Barker C M, Le Menach A, et al. Effects of temperature on emergence and seasonality of West Nile virus in California. The American journal of tropical medicine and hygiene, 2012, 86: 884–894
Vuik C, Cuvelier C. Numerical solution of an etching problem. Journal of Computational Physics, 1985, 59: 247–263
Yi F. One dimensional combustion free boundary problem. Glasgow Mathematical Journal, 2004, 46: 63–75
AMADORI A L, V[Á]{}ZQUEZ J L. Singular free boundary problem from image processing, Mathematical Models and Methods in Applied Sciences, 2005, 15: 689–715
Chen X, Friedman A. A free boundary problem arising in a model of wound healing. SIAM Journal on Mathematical Analysis, 2000, 32: 778–800
Chen X, Friedman A. A free boundary problem for an elliptic-hyperbolic system: an application to tumor growth. SIAM Journal on Mathematical Analysis 2003, 35: 974–986
Tao Y, Chen M. An elliptic-hyperbolic free boundary problem modelling cancer therapy, Nonlinearity, 2005, 19: 419–440
Du Y, Lin Z. Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM Journal on Mathematical Analysis, 2010, 42: 377–405
Wang M. On some free boundary problems of the prey-predator model. Journal of Differential Equations, 2014, 256: 3365–3394
Tarboush A K, Lin Z G, Zhang M Y. Spreading and vanishing in a West Nile virus model with expanding fronts. Science China Mathematics, 2017, 60: 841–860
Lin Z, Zhu H. Spatial spreading model and dynamics of West Nile virus in birds and mosquitoes with free boundary. Journal of Mathematical Biology, 2017, 75: 1381–1409
Guo J S, Wu C H. On a free boundary problem for a two-species weak competition system. Journal of Dynamics and Differential Equations, 2012, 24: 873–895
Wang M, Zhao J. A free boundary problem for the predator-prey model with double free boundaries. Journal of Dynamics and Differential Equations, 2017, 29: 957–979
Pao C V. Reaction diffusion equations with nonlocal boundary and nonlocal initial conditions, Journal of Mathematical Analysis and Applications, 1995, 195: 702–718
Tian C, Ruan S. A free boundary problem for *Aedes aegypti* mosquito invasion. Applied Mathematical Modelling, 2017, 46: 203–217
Caudevilla P [Á]{}, L[ó]{}pez-G[ó]{}mez J. Asymptotic behaviour of principal eigenvalues for a class of cooperative systems. Journal of Differential Equations, 2008, 244: 1093–1113
Diekmann O, Heesterbeek J A P, Metz J A J. On the definition and the computation of the basic reproduction ratio [$R_0$]{} in models for infectious diseases in heterogeneous populations. Journal of Mathematical Biology 1990, 28: 365–382
Allen L J S, Bolker B M, Lou Y, et al. Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model. Discrete and Continuous Dynamical Systems, 2008, 21: 1–20
Zhao X Q, Dynamical systems in population biology. Springer Berlin, 2003
Cantrell R S, Cosner C. Spatial ecology via reaction-diffusion equations. John Wiley [&]{} Sons, 2004
Huang W, Han M, Liu K. Dynamics of an SIS reaction-diffusion epidemic model for disease transmission. Mathematical Biosciences and Engineering, 2010, 7: 51–66
Ge J, Kim K I, Lin Z, et al. A SIS reaction-diffusion-advection model in a low-risk and high-risk domain. Journal of Differential Equations, 2015, 259: 5486-5509
Wang M. On some free boundary problems of the prey–predator model. Journal of Differential Equations, 2014, 256: 3365–3394
Wang M. A diffusive logistic equation with a free boundary and sign-changing coefficient in time-periodic environment. Journal of Functional Analysis, 2016, 270: 483–508
Pao C V. Nonlinear parabolic and elliptic equations. Springer Science [&]{} Business Media, 2012
Tian C, Ruan S. On an advection-reaction-diffusion competition system with double free boundaries modeling invasion and competition of *Aedes Albopictus* and *Aedes Aegypti* mosquitoes. Journal of Differential Equations, 2018, 265: 4016–4051
Wang Z, Nie H, Du Y. Spreading speed for a West Nile virus model with free boundary. Journal of Mathematical Biology, 2019, 79: 1–34
Wang Z, Nie H, Du Y. Asymptotic spreading speed for the weak competition system with a free boundary. arXiv preprint arXiv:1710.05485, 2017
tef[ă]{}nescu R, Dimitriu G. Numerical approximation of a free boundary problem for a predator-prey model. International Conference on Numerical Analysis and Its Applications, 2008, 5434: 548–555
|
---
abstract: 'Commensurate and incommensurate antiferromagnetic fluctuations in the two-dimensional repulsive $t-t''$-Hubbard model are investigated using functional renormalization group equations. For a sufficient deviation from half filling we establish the existence of local incommensurate order below a pseudocritical temperature $T_{pc}$. Fluctuations not accounted for in the mean field approximation are important—they lower $T_{pc}$ by a factor $\approx2.5$.'
author:
- 'H. C. Krahl'
- 'S. Friederich'
- 'C. Wetterich'
title: 'Incommensurate antiferromagnetic fluctuations in the two-dimensional Hubbard model'
---
The two-dimensional Hubbard model [@hubbard; @kanamori; @gutzwiller] has attracted much interest in the past two decades because it is a candidate model for the $\rm{CuO}_2$-planes in the high $T_c$-cuprates and may exhibit d-wave superconducting order [@anderson; @scalapino] at finite chemical potential. The model shows other interesting order structures such as incommensurate antiferromagnetism which appears close to half filling.
We focus on repulsive interactions $U$ and not too large next-to-nearest neighbour hopping $t'$, where the model is an antiferromagnet at half filling. Not so far away from half-filling a more complicated form of antiferromagnetism, namely incommensurate antiferromagnetism is suggested by mean field computations and numerical studies for finite systems [@zaanen; @machida; @schulz; @kato; @benard; @chubukov; @mancini; @kaga; @moreo; @bulut]. Incommensurate antiferromagnetism is related to the existence of spiral magnetic states which occur at large values of $U$ [@sarker; @zhou; @wiese]. Experimentally, incommensurate antiferromagnetism manifests itself in the peak structure of the magnetic structure factor which is accessible via neutron-scattering. It has been observed for a variety of high $T_c$-cuprates, for experimental and numerical results see [@birgeneau; @cheong; @sternlieb; @yuan; @lorenzanaseibold; @fujita; @seiboldlorenzana].
In the temperature region where local incommensurate antiferromagnetic order supposedly sets in, the effective interaction between the electrons is large such that perturbative methods are not reliable. Collective fluctuations of electron-hole pairs in the antiferromagnetic channel play an important role. Since they are omitted in a mean field treatment one may doubt whether the mean field results for incommensurate antiferromagnetism are reliable. For these reasons we investigate the issue of incommensurate antiferromagnetism by a method that is intrinsically non-perturbative and includes effective collective bosonic fluctuations, namely the functional renormalization group for the “flowing action” (or “average action”) $\Gamma_k$ [@cw93; @berges_review02]. For this scale dependent effective action (or coarse grained free energy) the scale $k$ indicates an infrared cutoff such that only fluctuations with momenta larger than $k$ are effectively included. (Finally, one is interested in the limit $k\rightarrow0$, where $\Gamma_{k\rightarrow0}$ equals the effective action—the generating functional of 1PI-correlation functions—including all fluctuations.) We will work in a version where the dominant collective bosonic fluctuations are represented by bosonic fields [@bbw04; @bbw05]. Our model is equivalent to the purely fermionic Hubbard model from which it is derived by means of a Hubbard-Stratonovich transformation [@hubbardtransf; @stratonovich]. Earlier studies employing the present framework have focused on the temperature dependence of *commensurate* antiferromagnetic order [@bbw04], the Kosterlitz-Thouless transition in a more general class of Hubbard-type models [@kw07], and the generation of a coupling in the $d$-wave superconducting channel [@krahlmuellerwetterich]. The role of incommensurate antiferromagnetic fluctuations was not taken into account in this earlier work. Functional renormalization group treatments of the Hubbard model are more often given in a purely fermionic formulation, see [@zanchi1; @halbothmetzner; @halbothmetzner2; @honerkamp01; @salmhofer; @honerkampsalmhofer01]. Ref. is of particular interest since, in accordance with the results described here, it also reports on a region in the phase diagram where incommensurate spin density fluctuations dominate. The present paper also includes an independent computation of the size of the incommensurability that occurs.
Our ansatz for the flowing action includes contributions for the electrons, for the bosons in both the antiferromagnetic and d-wave superconducting channels, and for interactions between fermions and bosons:
$$\begin{aligned}
\label{eq:simplesttruncation}
\Gamma_k[\chi]=\Gamma_{F,k}[\chi]+\Gamma_{\mathbf a,k}[\chi]+\Gamma_{F\mathbf a,k}[\chi]+\Gamma_{d,k}[\chi]+\Gamma_{Fd,k}[\chi]
\,.\end{aligned}$$
The collective field $\chi=(\mathbf{a},d,d^*,\psi,\psi^*)$ describes fermion fields $\psi,\psi^*$, the “antiferromagnetic boson field” $\mathbf{a}$ and the complex field $d$ a finite expectation value of which signals $d$-wave superconductivity. The fermionic kinetic term $$\begin{aligned}
\label{eq:fermprop}
\Gamma_{F,k}=\sum_{Q}\psi^{\dagger}(Q)P_F(Q)\psi(Q)\end{aligned}$$ involves the inverse fermion propagator $$\begin{aligned}
\label{eq:PF}
P_{F}(Q)=Z_F(i\omega+\xi(\mathbf q))
\,,\end{aligned}$$ where $\xi(\mathbf q)=-\mu-2t(\cos q_x +\cos q_y)-4t' \cos q_x\cos q_y$ depends on the chemical potential $\mu$ and the nearest and next-to-nearest neighbor hopping parameters $t$ and $t'$ of the Hubbard model. We employ a compact notation $X=(\tau,\mathbf x)$, $Q=(\omega,\mathbf{q})$,$$\begin{aligned}
\label{eq:sumdefinition}
\sum\limits_X=\int\limits_0^\beta d\tau\sum\limits_{\mathbf{x}},\quad\sum\limits_Q=T\sum\limits_{n=-\infty}^\infty \int\limits_{-\pi}^\pi \frac{d^2q}{(2\pi)^2}\,,\nonumber\\
\delta(X-X')=\delta(\tau-\tau')\delta_{\mathbf{x},\mathbf{x'}}\,,\hspace{1.5cm}\nonumber\\
\delta(Q-Q')=\beta\delta_{n,n'}(2\pi)^2\delta^{(2)}(\mathbf{q}-\mathbf{q'})\,.\hspace{1cm}\end{aligned}$$ where all components of $X$ or $Q$ are measured in units of the lattice distance $\mathrm a$ or $\mathrm{a}^{-1}$. The discreteness of the lattice is reflected by the $2\pi$-periodicity of the momenta $\mathbf{q}$. A scale dependent fermionic wave function renormalization $Z_F$ is included in Eq. .
The purely antiferromagnetic bosonic term is described by a kinetic term and a local effective potential $$\begin{aligned}
\Gamma_{a,k}
=\frac{1}{2}\sum_{Q}\mathbf{a}^{T}(-Q)P_{a}(Q)\mathbf{a}(Q)
+\sum_XU_{a,k}[\mathbf{a}]
\,,\end{aligned}$$ where we employ a quartic effective potential $U_{\mathbf a}$ for $\mathbf a$: $$\begin{aligned}
\label{eq:Ueff}
U_{\mathbf a}[\mathbf{a}]&=&\bar{m}^2_{a}\alpha
+\frac{1}{2}\bar\lambda_a\alpha^2,
\,\end{aligned}$$ with $\alpha=\mathbf a^2/2$. The kinetic term $P_a$ involves the $Q$-dependent part of the inverse antiferromagnetic propagator and therefore contains the essential information about different kinds of magnetism. Our treatment of this term is discussed in detail below. Local antiferromagnetic order in domains of size $k^{-1}$ is signalled by a minimum of $\Gamma_{a,k}$ for $\mathbf{a}(Q)\neq0$. For $Q=0$ this describes commensurate antiferromagnetism, while a minimum for non-vanishing $\mathbf q$ in $Q=(0,\mathbf q)$ indicates incommensurate antiferromagnetism. A Yukawa-like interaction term couples the bosonic field to the fermions, $$\begin{aligned}
\label{eq:GFak}
&\Gamma_{F\mathbf a,k}=-\bar h_a\!\sum_{K,Q,Q'}\delta(K+\Pi-Q+Q')
\\
& \hspace{2.5cm}\times\;
\mathbf{a}(K)\cdot[\psi^{\dagger}(Q)\boldsymbol{\sigma}\psi(Q')]\nonumber
\,,\end{aligned}$$ where the momentum vector $\Pi$ is given by $\Pi=(0,\pi,\pi)$.
The bosonic field $d$ is associated to Cooper-pairs in the d-wave channel. It is described in more detail in [@kw07; @krahlmuellerwetterich] where the exact form of $\Gamma_{Fd}$ can be found. In this note we include the effect of d-wave fluctuations on the flow of the fermionic and “antiferromagnetic” part of $\Gamma_k$. We use $\Gamma_{d,k}=\sum_{Q} d^{*}(Q)P_d(Q)d(Q)+\sum_XU_{d,k}[d,d^*]$ with $U_{d}[d,d^*]=\bar{m}^2_{d}\delta+\bar\lambda_d\delta^2/2$ where $\delta=d^*d$. Here we focus exclusively on the emergence of (either commensurate or incommensurate) magnetic order which occurs in the vicinity of half-filling. The emergence of $d$-wave superconducting order at larger values of $|\mu|$ will be discussed in detail in a future publication. No superconductivity has been detected in the region of the phase diagram studied here.
The dependence of the flowing action on the scale $k$ is described by an exact flow equation [@cw93]. Our ansatz approximates the solutions to this functional differential equation. At the microscopic scale $k=\Lambda$ the flowing action must be equivalent to the microscopic action of the Hubbard model. Since we want to eliminate the (constant) four-fermion coupling $U$ at $k=\Lambda$ which, of course, has no contributions exhibiting $d$-wave symmetry, the repulsive interaction between the fermions must be contained in the antiferromagnetic Yukawa coupling $\bar h_a$. In the bosonized picture one has, instead of the original four-fermion coupling $U$ a boson-mediated interaction term $\bar h_a^2/\bar{m}_a^2$ which must be chosen proportional to $U$. Since an additional sum over spin directions has to be carried out it has to be chosen as $U/3$. Thereby we have simply transcribed the original model into an equivalent one using bosonic language. Since the original model does not contain any eight electron terms, no quartic bosonic coupling $\bar\lambda_a$ can exist at the UV scale $k=\Lambda$. In sum, a set of possible “initial conditions” for the flow of the coupling constants is given by $$\begin{aligned}
\label{eq:initialcond}
&\bar{m}_a^2|_{\Lambda}=U/3\,,\quad\bar{h}_a|_{\Lambda}=U/3\,,\quad\bar\lambda_a|_{\Lambda}=0\,,\quad P_a(Q)|_{\Lambda}=0,\nonumber\\
&\Gamma_{Fd}|_{\Lambda}=0\,,\quad \Gamma_{d}|_{\Lambda}=d^*d,\quad Z_{F}|_{\Lambda}=1
%\bar m_d^2|_{\Lambda}=1\,,\quad \bar h_d|_{\Lambda}=0\,,\quad\bar\lambda_d|_{\Lambda}=0\,,\quad P_d(Q)|_{\Lambda}=0
\,.\end{aligned}$$ These values specify the action $\Gamma_\Lambda$ (or, equivalently, the Hamiltonian) at the microscopic level.
At the microscopic scale $\Lambda$, the action for $\mathbf a$ is Gaussian such that $\mathbf a$ can be “integrated out” by solving its field equation as a functional of $\psi$. The $d$-boson decouples and becomes irrelevant. This demonstrates that $\Gamma_\Lambda$ indeed coincides with the purely fermionic Hubbard model with repulsive coupling $U>0$.
We still have to specify the truncation for the kinetic term of the $\mathbf a$-boson $P_a(Q)$. This is a central object of this paper, since incommensurate antiferromagnetic fluctuations will dominate if the minimum of $P_a(0,\mathbf q)$ occurs for nonzero $\mathbf q$. In order to gain some first information about the general shape of $P_a$ we compute the mean field contribution from the fermionic loop $$\begin{aligned}
\label{eq:rhoferm}
\Delta P_{a}(Q)=\sum_{P}
\frac{\bar{h}_{a}^2}
{P_F(Q+P+\Pi)P_F(P)} +(Q\rightarrow -Q)
\,.\end{aligned}$$ The general features of this mean field contribution are used in order to motivate the form of the bosonic propagator in our truncation. We observe close to half filling two qualitatively different situations. At half filling and for sufficiently high temperatures also close to half filling there is a pronounced minimum at $\mathbf{q}=0$, see Fig. \[fig:oneloopa\] (a). However, away from half filling the picture is different for sufficiently low temperatures, see Fig. \[fig:oneloopa\] (b). In the center at $\mathbf{q}=0$ there is a local maximum and there are four minima at positions $$\begin{aligned}
\label{eq:positionshatq}
\mathbf{q}_{1,2}=(\pm\hat q,0)\,,\quad \mathbf{q}_{3,4}=(0,\pm\hat q)
\,,\end{aligned}$$ where $\hat q$ is a function of $T$, $\mu$, and $t'$. This is a manifestation of the dominance of [*incommensurate*]{} antiferromagnetic fluctuations.
![[]{data-label="fig:oneloopa"}](fig1.eps){width="85mm"}
Once the minimal value of the inverse bosonic propagator $\left[P_a(0,\mathbf{q})+\bar m_a^2\right]$ at zero frequency becomes smaller than zero, the minimum of the free energy can no longer occur for $\langle\mathbf a(Q)\rangle=0$. One rather expects spontaneous symmetry breaking with a non-zero expectation value of $\langle\mathbf a\rangle$. As long as the minimum of $P_a(0,\mathbf{q})$ is located at $\mathbf{q}=0$, the order parameter $\langle|\mathbf a|\rangle\sim\delta(\mathbf q)$ indicates commensurate antiferromagnetism. However, for a minimum at $\mathbf{q}=\mathbf{q}_j\neq0$ the incommensurate antiferromagnetic order breaks further lattice symmetries. One of the pairs of minima is selected and the symmetry of rotations by $\pi/2$ around $\mathbf{q}=0$ in momentum space is spontaneously broken. The spins change sign between neighboring lattice sites only in one direction, the $x$-direction say, whereas in the orthogonal direction the periodicity corresponds to some momentum $\pi\pm \hat q$. The state with $\langle\mathbf a\rangle=0$ becomes unstable when $( P_{a})_{min}=-\bar m_a^2$. In case of a second order phase transition this occurs for the mean field critical temperature $T=T_{MFc}$. Figures \[fig:oneloopa\] (a) and (b) correspond to mean field critical temperatures. Note that the system selects one of the *pairs* $\mathbf{q}_{1,2}$ or $\mathbf{q}_{3,4}$ since $\mathbf a(X)$ is a real field. Therefore the system remains symmetric with respect to reflection about the axes.
We are interested in whether incommensurate antiferromagnetism persists if bosonic fluctuations are included. Taking into account bosonic fluctuations, the critical temperature vanishes in the infinite volume limit due to the Mermin-Wagner theorem. The destruction of local order by the long range fluctuations of the Goldstone bosons (antiferromagnetic spin waves) is only a logarithmic effect, however. For a probe of finite macroscopic size antiferromagnetic order can be observed and the critical temperature is nonzero [@bbw05]. Here the effective critical temperature $T_c$ is defined such that for $T<T_c$ the typical size of ordered domains exceeds the macroscopic size of the probe $l$. In other words, $\langle \mathbf a(k) \rangle$ differs from zero for $k_{ph}=l^{-1}$ if $T<T_c$, while for $T>T_c$ one has $\langle \mathbf a(k) \rangle=0$.
In this note we only study the pseudocritical temperature $T_{pc}$ which marks the onset of local ordering corresponding to a minimum of the flowing action $\Gamma_k$ for $\mathbf{a}(0,\mathbf q)$. Above this temperature, $\langle|\mathbf a|\rangle=0$ holds on all scales of the renormalization flow. For $T<T_{pc}$ local order sets in for $k=k_c>0$. In case of incommensurate antiferromagnetism we expect the formation of domain walls between regions where $\mathbf{\hat q}$ points in the $x$- or $y$-direction. This constrasts with commensurate antiferromagnetism where only a continuous symmetry is broken for $\mathbf a\neq0$. For $k<k_c$ the flow should then be continued in a regime with nonzero $\mathbf a$ in order to account properly for the Goldstone boson fluctuations. This has been investigated for the commensurate case in [@bbw05], but is not yet implemented for the incommensurate case in the present note. We note that $T_{pc}$ is the equivalent of the mean field critical temperature. For $T_c<T<T_{pc}$ the electron propagator does not exhibit a true gap, but it is suppressed for momenta corresponding to the inverse of length scales for which local order is present.
Inspired by the shape of $P_a$ in the mean field approximation we approximate the kinetic term for the antiferromagnetic boson by $$\begin{aligned}
\label{eq:apropparam}
P_{a,k}(Q)=Z_a\omega^2+A_aF(\mathbf q)\, ,\end{aligned}$$ where for the case of commensurate antiferromagnetism we choose for $F(\mathbf q)$ $$\begin{aligned}
\label{eq:apropparam1}
F_c(\mathbf q)=\frac{D^2[\mathbf{q}]^2}{D^2+[\mathbf{q}]^2}
\,.\end{aligned}$$ Here $[\mathbf{q}]^2$ is defined as $[\mathbf{q}]^2=q_x^2+q_y^2$ for $q_i\in [-\pi,\pi]$ and continued periodically otherwise. For small $\mathbf{q}^2$ the quadratic approximation $P_a=A_a\mathbf{q}^2$ describes a linear dispersion relation for the composite bosonic field, $\omega=\sqrt{A_a/Z_a}|\mathbf{q}|$, while for $\mathbf q$ near the boundary of the Brillouin zone the momentum dependence of $P_a$ ‘levels off’ as in Figs. \[fig:oneloopa\] (a), (b). For a suitable choice of $A_a$ and $D$ the shape of the mean field result for $P_a$ is well reproduced. Of course, due to the important contributions of bosonic fluctuations beyond mean field theory, the actual values of $A_a$ and $D$ will differ substantially from the mean field values.
Within the functional renormalization group approach, we describe the scale dependence of the bosonic kinetic term by flow equations for the parameters $A_a$ and $D$. For these purposes we define the gradient coefficient $A_a$ by $$\begin{aligned}
\label{eq:Aa}
A_a=\frac{1}{2}\frac{\partial^2}{\partial l^2}P_a(0,l,0)\big{|}_{l=\hat q}\end{aligned}$$ with $\hat q=0$ in the commensurate case. The shape coefficient $D$ is computed as $$\begin{aligned}
\label{eq:D}
D^2=\frac{1}{A_a}\big(P_a(0,\pi,\pi)-P_a(0,\hat q,0)\big).\end{aligned}$$ The flow equations for $A_a$ and $D$ can be extracted by inserting our truncation in the exact flow equations for the kinetic term .
![[]{data-label="fig:P_asym"}](fig2.eps){width="60mm"}
During the renormalization flow the gradient coefficient $A_a$ first increases, starting from $A_a=0$ at the scale $\Lambda$. At half filling and in the proximity of half filling for sufficiently high temperatures, $A_a$ either increases monotonically or at least remains larger than zero on all scales $k<\Lambda$, see Fig. \[fig:P\_asym\]. The minimum of $P_a$ occurs for $\mathbf q=0$ and commensurate antiferromagnetic fluctuations dominate.
However, for low enough temperatures and at sufficient distance from half-filling, $A_a$ becomes zero on a certain scale. If we continued to evaluate $A_a$ for $\mathbf q =0$ it would decrease to negative values for lower scales. This situation corresponds to the case of incommensurate antiferromagnetism. The ansatz for the function $F(\mathbf q)$ given in Eq. is no longer suitable. One has to allow for the existence of minima of $P_{a,k}(0,\mathbf{q})$ at nonzero $\mathbf{q}\neq0$. The ansatz for the inverse bosonic propagator employs now for $F(\mathbf q)$ $$\begin{aligned}
F_i(\mathbf q,\hat q)&=&\frac{D^2\tilde F(\mathbf q,\hat q)}{D^2+\tilde F(\mathbf q,\hat q)}
\,.\end{aligned}$$ The quadratic momentum dependence of the numerator in is replaced by an expression which is quartic in momentum and explicitly includes the incommensurability $\hat q$: $$\begin{aligned}
\label{eq:apropparaminkomm}
\tilde F(\mathbf q,\hat q)=\frac{1}{4\hat q^2}\big((\hat q^2-[\mathbf q]^2)^2+4[q_x]^2[q_y]^2\big)
\,.\end{aligned}$$ The first term in $\tilde F$ vanishes for $[\mathbf q]^2=\hat q^2$ and suppresses the propagator for $[\mathbf q]^2\neq\hat q^2$. The second term favours the minima as compared to a situation where rotation-symmetry in the $q_x-q_y$-plane is preserved. The prefactor is determined by Eq. . For $\hat q\rightarrow0$ one has $A_a\sim\hat q^2$ such that $P_a$ becomes quartic in $\mathbf q$. We compare in Fig. \[fig:menafieldapropmom\] the kinetic term $P_a(0,\mathbf{q})$ in mean field theory with the approximation from our ansatz which shows satisfactory agreement.
![[]{data-label="fig:menafieldapropmom"}](fig3.eps){width="85mm"}
![[]{data-label="fig:P_assb"}](fig4.eps){width="60mm"}
In Fig. \[fig:P\_assb\], a typical flow for $A_a$ and $\hat q$ in the incommensurate regime is displayed. For scales below the scale where $A_a$ becomes zero, $\hat q$ increases to a finite value and $P_a(0,\mathbf{q})$ has four degenerate minima at positions given by Eq. (\[eq:positionshatq\]). The solution $\hat q$ of Eq. (\[eq:calcqhat\]) at the end of the flow corresponds to the position of the minimum, e.g., at the positive $q_x$-axis .We next specify the flow in more detail.
The regulator function $R^a_k(Q)$ for the antiferromagnetic fluctuations should be adapted in order to allow for the dominance of incommensurate antiferromagnetism. We employ, similarly for the commensurate and incommensurate case, $$\begin{aligned}
\label{regulator}
R^a_k(Q)=A_{a}\cdot(k^2-F_{c,i}(\mathbf q,\hat q))\Theta(k^2-F_{c,i}(\mathbf q,\hat q))
\,,\end{aligned}$$ respectively. This generalizes the cutoff chosen in [@krahlmuellerwetterich].
The flow equation for the gradient coefficient is obtained by taking appropriate derivatives in one of the minima $$\begin{aligned}
\label{eq:flowAa}
\partial_kA_a&=&\sum_Q\bar{h}_a^2\tilde\partial_k\frac{\partial^2}{\partial l^2}\frac{1}{P^k_F(Q)P^k_F(K+Q+\Pi)}\Bigg{|}_{l=\hat q}\nonumber\\
&+&\sum_Q\bar{h}_a^2(\partial_k\hat q)\frac{\partial^3}{\partial l^3}\frac{1}{P^k_F(Q)P^k_F(K+Q+\Pi)}\Bigg{|}_{l=\hat q}
\,,\end{aligned}$$ where $K=(0,l,0)$. We have defined $P_{F,k}(Q)=P_F(Q)+R_k^F(Q)$, with fermion cutoff $R_k^F$ chosen as in [@krahlmuellerwetterich]. The first term in results from the change of the infrared cutoff in the fluctuations. The symbol $\tilde\partial_k$ means a formal derivative with respect to the cutoff function $R_k^F$. The second term in reflects the shift of the location of the minimum of $P_a$ at $(\hat q,0)$ and is absent if commensurate fluctuations dominate, $\hat q=0$.
A flow equation for the position of the minima $\hat q$ is derived from the condition $$\begin{aligned}
\frac{\partial}{\partial q_x}P_{a,k}(0,\mathbf q)\big{|}_{\mathbf q=(\hat q,0)}=0
\,.\end{aligned}$$ Taking the scale derivative of this equation one obtains the flow equation: $$\begin{aligned}
\label{eq:calcqhat}
(\partial_k\hat q)\frac{\partial^2}{\partial q_x^2}P_{a,k}(0,\mathbf q)\big{|}_{\mathbf q=(\hat q,0)}+\frac{d}{dk}\Big{|}_{\hat q}\frac{\partial}{\partial q_x}P_{a,k}(0,\mathbf q)\big{|}_{\mathbf q=(\hat q,0)}& &\\
=(\partial_k\hat q)2A_a+\frac{d}{dk}\Big{|}_{\hat q}\frac{\partial}{\partial q_x}P_{a,k}(0,\mathbf q)\big{|}_{\mathbf q=(\hat q,0)}&=&0\nonumber
\,.\end{aligned}$$
Flow equations for the other running couplings $Z_F,\bar m_a^2,\bar \lambda_a,\bar h_a,\bar m_d^2,\bar \lambda_d,\bar h_d,D$ are not given explicitly here, see [@krahlmuellerwetterich].
![[]{data-label="fig:phasediagronlyantif"}](fig5.eps){width="80mm"}
We now turn to the results obtained in our renormalization group scheme. An overview of the occurence of incommensurate antiferomagnetism is given in Fig. \[fig:phasediagronlyantif\], showing pseudocritical temperatures $T_{pc}$ for the different kinds of antiferromagnetic order in the presence of vanishing (upper panel) and nonvanishing (lower panel) next-to-nearest neighbor hopping $t'$. The solid line signals the onset of local commensurate, the long-dashed line the onset of local incommensurate antiferromagnetic order. Below the short-dashed line there is no local magnetic order but incommensurate fluctuations dominate. Below the point where the short-dashed line terminates at low temperatures, numerical solutions to the flow equations, as we have implemented them numerically, are no longer reliable. For both vanishing and non-vanishing $t'$, one observes commensurate antiferromagnetism for a certain range of chemical potential $\mu$, while for smaller and larger values of $\mu$ incommensurate fluctuations begin to dominate. For finite $t'$, however, the pseudocritical curve is no longer the same for positive and negative $\mu$ but, for negative $t'$, is shifted to more negative values of $\mu$.
The pseudocritical temperature is found to be substantially lower than according to the mean field computation. For $U=3t$, $t'=0$ and $\mu=0$, for example, the mean-field computation gives $T_{MFc}/t=0.205$, while we find $T_{pc}/t=0.0745$ when one takes into account bosonic fluctuations. By reducing the interaction, the shape of the pseudocritical curve remains the same but local order emerges only at lower temperatures.
![[]{data-label="fig:qhat"}](fig6.eps){width="65mm"}
With decreasing temperature the tendency towards incommensurate fluctuations is increased, which can be demonstrated by studying the dependence of $\hat q$ on $T$ at fixed chemical potential. It is shown for $\mu/t=-0.105$ and $\mu/t=-0.12$ in Fig. \[fig:qhat\]. For large enough temperatures one has $\hat q=0$, while below some $\mu$-dependent temperature incommensurate antiferromagnetism sets in. The temperature where this happens is indicated by the short-dashed line in Fig. \[fig:phasediagronlyantif\] (upper panel). For smaller $T$ the value of $\hat q$ increases, the final point of the $\mu/t=-0.105$-curve at low temperature corresponds to the long-dashed line in Fig. \[fig:phasediagronlyantif\]
As one can see from the curve representing $\mu/t=-0.12$ in Fig. \[fig:qhat\], at small temperatures the size of the incommensurability is approximately constant. Therefore we compare our result to the zero-temperature result obtained by [@schulz] saying that $\hat q=2\arcsin (|\mu|/2t)$ (which has also been used in the fermionic RG computation given in [@halbothmetzner2]). For $\mu/t=-0.12$ this formula gives $\hat q\approx0.120$ whereas we find $\hat q\approx0.132$. By taking into account fluctuations the incommensurability seems to be slightly enhanced. Agreement with the results displayed in [@mancini] obtained by means of the composite operator method is also satisfactory.
A dominance of incommensurate antiferromagnetic fluctuations can be observed in the momentum dependence of the magnetic susceptibility and the bosonic occupation number. The susceptibility is given by the bosonic propagator at zero frequency $P_a^{-1}(0,\mathbf q)$, while the occupation number is obtained by an additional sum over bosonic Matsubara frequencies, $n_a(\mathbf q)=T\sum_{\omega_B}(P_a(\omega_B,\mathbf q))^{-1}$. Fig. \[fig:suscocc\] shows that for parameters where the bosonic mass is small, here $\bar m_a^2/U\approx10^{-2}$, and thus close to the onset of local incommensurate order, both the magnetic susceptibility and the bosonic occupation number are peaked at $q_x=\pm\hat q$, signalling that incommensurate fluctuations strongly dominate. The situation is completely analogous for the $q_y$-dependence of the susceptibility at $q_x=0$, whereas both quantities do not have such a pronounced peak structure along the Brillouin zone diagonal.
![[]{data-label="fig:suscocc"}](fig7.eps){width="85mm"}
In those regions of the phase diagram in which (either commensurate or incommensurate) antiferromagnetic order exists on a certain legth scale $k$ our truncation becomes inapplicable in the regime below $k$. The simplest way of obtaining a glimpse at these regimes is by means of a mean field analysis, so before closing the discussion we briefly address this problem. A more extensive mean field treatment, if only with regards to the *commensurate* case but including a nonzero next-to-nearest neighbor hopping $t'$, is given in [@reiss]. Here one has to take into account that the periodicity of a system in the Néel state is changed resulting in a new “magnetic” Brillouin zone whose boundaries are given by the lines between the $(\pm\pi,0)$ and $(0,\pm\pi)$ points. Correspondingly, the mean field dispersion relation for a nonzero gap parameter $A=\bar h_a\langle |\mathbf{a}|\rangle$ has two branches $$\begin{aligned}
\label{eq:dispcomm}
E_\pm(\mathbf p)=\frac{1}{2}\left( \xi(\mathbf{p})+\xi(\mathbf{p}+\boldsymbol \pi)\pm\sqrt{(\xi(\mathbf{p})-\xi(\mathbf{p}+\boldsymbol \pi))^2+4 A^2} \right)
\,\end{aligned}$$ which, for finite $t'$, lead to an interestingly structured effective Fermi surface enclosing hole pockets around $(\pm\pi/2,\pm\pi/2)$ and electron pockets around $(\pm\pi,0)$ and $(0,\pm\pi)$, see the example drawn in Fig. \[fig:incommfermi\] (a), for further details see e.g. [@reiss].
![[]{data-label="fig:incommfermi"}](fig8.eps){width="85mm"}
In the presence of a nonzero expectation value $\langle\mathbf{a}(\mathbf{\hat q})\rangle$ with $\mathbf{\hat q}\neq0$, i.e. in the presence of incommensurate order, the inverse of the fermionic mean field propagator at zero frequency has contributions from Eqs. (with $Z_F=1$) and and is given by $$\begin{aligned}
&&P_F(\mathbf q, \mathbf{q}')=\xi(\mathbf q)\delta(\mathbf q-\mathbf q')\\&&\hspace{0.3cm}-\frac{\mathbf A\cdot\boldsymbol\sigma}{\sqrt{2}}\left( \delta(\mathbf q-\mathbf q'-\boldsymbol\pi+\mathbf{\hat q})+\delta(\mathbf q-\mathbf q'-\boldsymbol\pi-\mathbf{\hat q}) \right)\nonumber\end{aligned}$$ with $\mathbf{\hat q}=\mathbf{q}_{1,2}$ or $\mathbf{\hat q}=\mathbf{q}_{3,4}$ as defined in Eq. . The analogue of the Fermi surface corresponds to the zero eigenvalues of $P_F$. However, the corresponding eigenmodes are no longer momentum eigenstates. Nevertheless, if the gap parameter $A=|\mathbf A|$ is nonzero but small, many eigenvalues of $P_F(\mathbf q, \mathbf{q}')$ have most of their support each at a single momentum $\mathbf p$. This concerns all those momenta $\mathbf p$ for which the condition $$\label{eq:cond}
A\ll|\xi(\mathbf{p}+\boldsymbol \pi+\mathbf{\hat q})|,|\xi(\mathbf{p}+\boldsymbol \pi-\mathbf{\hat q})|$$ is fulfilled. With respect to these momenta the equation $$\begin{aligned}
\label{eq:disp}
\xi(\mathbf{p})-\frac{A^2}{2}\left(\frac{1}{\xi(\mathbf{p}+\boldsymbol \pi+\mathbf{\hat q})}+\frac{1}{\xi(\mathbf{p}+\boldsymbol \pi-\mathbf{\hat q})}\right)=0
\,\end{aligned}$$ defines an effective Fermi surface which is obtained by (approximately) diagonalizing $P_F(\mathbf q, \mathbf{q}')$ for small $A$. For large enough $A$ the effective Fermi surface vanishes completely because the number of solutions to Eq. that satisfy the condition rapidly goes down. In Fig. \[fig:incommfermi\] (b) the effective Fermi surface is shown for the incommensurate case with an order parameter $\langle\mathbf{a}(\mathbf{\hat q})\rangle$ where $\mathbf{\hat q}=\mathbf{q}_{1,2}$, i.e. the incommensurability is along the $x$-axis. The symmetry of rotations by $\pi/2$ is manifestly broken.
To summarize, we have shown that incommensurate antiferromagnetic order in the two-dimensional Hubbard model persists if bosonic fluctuations are taken into account. This phenomenon occurs at least in the form of local order for temperatures smaller than the pseudocritical temperature shown in Fig. \[fig:phasediagronlyantif\]. We speculate that for $T\rightarrow0$ the size of the incommensurate domains grows beyond the size of typical macroscopic probes, but this remains to be shown. If magnetic fluctuations play a role in the generation of d-wave superconducting order, the effect of incommensurability has to be taken into account.
[**Acknowledgments**]{}: HCK acknowledges financial support by the DFG research unit FOR 723 under the contract WE 1056/9-1. SF acknowledges support by the Studienstiftung des Deutschen Volkes.
[12]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{}
J. Hubbard, Proc. R. Soc. London, Ser. A **276**, 238 (1963).
J. Kanamori, Prog. Theor. Phys. **30**, 275 (1963).
M. C. Gutzwiller, Phys. Rev. Lett. **10**, 159 (1963).
P. W. Anderson, Science **235**, 1196 (1987).
D. J. Scalapino, Physics Reports **250**, 329 (1995).
Zaanen, J. and Gunnarsson, O. Phys. Rev. B **40**, 7391 (1989).
K. Machida, Physica C **158**, 192 (1989).
H. J. Schulz, Phys. Rev. Lett. **64**, 1445 (1990).
M. Kato, K. Machida, H. Nakanishi and M. Fujita, J. Phys. Soc. Jap. **59**, 1047 (1990).
P. Benard, L. Chen and A.-M. S. Tremblay, Phys. Rev. B **47**, 589 (1993).
A. V. Chubukov and K. A. Musaelian Phys. Rev. B **51**, 12605 (1995).
F. Mancini, D. Villani and H. Matsumoto, Phys. Rev. B **57**, 6145 (1998).
A. Moreo, D. J. Scalapino, R. L. Sugar, S. R. White and N. E. Bickers, Phys. Rev. B **41**, 2313 (1990).
N. Bulut, D. Hone, D. J. Scalapino and N. E. Bickers, Phys. Rev. Lett. **64**, 2723 (1990).
H. Kaga and M. Cyrot, Phys. Rev. B **58**, 12267 (1998).
S. Sarker, C. Jayaprakash, H.R. Krishnamurthy and W. Wenzel, Phys. Rev. B **43**, 8775 (1991).
C. Zhou and H. J. Schulz, Phys. Rev. B **52**, R11557 (1995).
C. Brügger, C. P. Hofmann, F. Kämpfer, M. Pepe and U.-J. Wiese, Phys. Rev. B **75**, 014421 (2007).
R. J. Birgeneau *et al.*, Phys. Rev. B **39**, 2868 (1989).
S.-W. Cheong *et al.*, Phys. Rev. Lett. **67**, 1791 (1991).
B. J. Sternlieb, J. M. Tranquada, G. Shirane, M. Sato and S. Shamoto, Phys. Rev. B **50**, 12915 (1994).
F. Yuan, S. Feng, Z. B. Su and L. Yu, Phys. Rev. B **64**, 224505 (2001).
J. Lorenzana and G. Seibold, Phys. Rev. Lett. **89**, 136401 (2002).
M. Fujita , H. Goka, K. Yamada, J. M. Tranquada and L. P. Regnault, Phys. Rev. B **70**, 104517 (2004).
G.Seibold and J. Lorenzana, Phys. Rev. Lett. **94**, 107006 (2005).
C. Wetterich, Phys. Lett. B **301**, 90 (1993).
J. Berges, N. Tetradis and C. Wetterich, Phys. Rep. **363**, 223 (2002).
T. Baier, E. Bick and C. Wetterich, Phys. Rev. B **70**, 125111 (2004).
T. Baier, E. Bick and C. Wetterich, Phys. Lett. B **605**, 144 (2005).
J. Hubbard, Phys. Rev. Lett. **3**, 77 (1959).
R. L. Stratonovich, Soviet. Phys. Doklady **2**, 416 (1958).
H. C. Krahl and C. Wetterich, Phys. Lett. A **367**, 263 (2007).
H. C. Krahl, J. A. Müller and C. Wetterich, Phys. Rev. B **79**, 094526 (2009).
D. Zanchi and H. J. Schulz, Z.Phys. **B103**, 339 (1997).
C. J. Halboth and W. Metzner, Phys. Rev. Lett. **85**, 5162 (2000).
C. J. Halboth and W. Metzner, Phys. Rev. B **61**, 7364 (2000).
M. Salmhofer and C. Honerkamp, Prog. Theor. Phys. **105**, 1 (2001).
C. Honerkamp, M. Salmhofer, N. Furukawa and T. M. Rice, Phys. Rev. B **63**, 035109 (2001).
C. Honerkamp and M. Salmhofer, Phys. Rev. Lett. **87**, 187004 (2001).
J. Reiss, D. Rohe and W. Metzner, Phys. Rev. B **75**, 075110 (2007).
|
---
abstract: 'We have calculated rates of $\beta^{-}$ decay to both continuum and bound states separately for some fully ionized (bare) atoms in the mass range A $\approx$ 60-240. One of the motivations of this work is that the previous theoretical calculations were very old and/or informatically incomplete. Probably no theoretical study on this subject has been done in the last three decades. For the calculation, we have derived a framework from the usual $\beta^{-}$ decay theory used by previous authors. Dependence of the calculated rates on the nuclear radius and neutral atom Q-value have been examined. We have used the latest experimental data for nuclear and atomic observables, such as $\beta^{-}$ decay Q-value, ionization energy, neutral atom $\beta^{-}$ decay branchings, neutral atom half-lives etc. Results of $\beta^{-}$ decay rates for decay to continuum and bound states and the enhancement factor due to the bound state decay for a number of nuclei have been tabulated and compared with the previously calculated values, if available. The effective rate or half-life calculated for bare atom might be helpful to set a limit for the maximum enhancement due to bound state decay. Finally, $\beta^{-}$ decay branching for bare atom has been calculated. The changes in branching in bare atom compared to that in the neutral atom and for the first time branching flip for a few cases have been obtained. Reason for this branching change has been understood in terms of Q-values of the transitions in the neutral and bare atoms. Verification of this branching change / flip phenomenon in bare atom decay might be of interest for future experiments.'
author:
- Arkabrata Gupta
- Chirashree Lahiri
- 'S. Sarkar'
title: ' Bound and continuum state $\beta^-$ decay of bare atoms: enhancement of decay rate and changes in $\beta^{-}$ decay branching'
---
[^1]
Introduction
============
It is well known that the usual theory of $\beta^{-}$ decay presumes that the decay of a neutron to proton is accompanied by the creation of an electron and an anti-neutrino in continuum states. However, in a stellar plasma where atoms get partially or fully ionized, this continuum decay is not the sole option. Nuclear $\beta ^{-}$ decay to the bound states of the ionized atom is another probable channel. Also bare atoms have been produced terrestrially and $\beta ^{-}$ decays have been studied in storage ring experiments. In 1947 Daudel [@daudel] first proposed the concept of bound state $\beta$ decay. This suggests that a nucleus has a possibility to undergo $\beta ^{-}$ decay by creating an electron in a previously unoccupied atomic orbital instead of the continuum decay. It is important to understand that the bound state decay process does not occur subsequently from the $\beta^{-}$ decay of an electron previously created in the continuum state, it is rather the direct creation of an electron in an atomic bound state accompanied by a mono-energetic anti-neutrino created in the free state carrying away the total decay energy. This process has been studied both theoretically as well as experimentally over the past seven decades.
In case of a neutral atom, available phase space for the creation of an electron in a vacant atomic orbital is very small and therefore the bound state decay is almost negligible compared to the contribution of the continuum decay. Contrarily, ionization of atoms may lead to drastic enhancement of bound state $\beta$ decay probability due to the availability of more unoccupied atomic levels. In some previous theoretical works from 60’s to 80’s, various groups have studied the continuum and bound state $\beta$ decay for neutron, tritium [@bahcall] and fully ionized (bare) heavy atoms [@takahashi; @takahashi1; @takahashi2]. However, in most cases, previous theoretical works were based on very old data and/or informatically incomplete. Simultaneously, the development of experimental techniques has served fruitfully to detect bound and continuum state $\beta$ decay channels of fully ionized atoms. In 1992, Jung first observed the bound state $\beta^{-}$ decay for the bare $^{163}$Dy atom [@jung] by storing the fully ionized parent atom in a heavy-ion storage ring. In the same decade, Bosch studied the bound state $\beta ^{-} $ decay for fully ionized $^{187}$Re [@bosch] which was helpful for the calibration of $^{187}$Re - $^{187}$Os galactic chronometer [@yokoi]. Further experiments with bare $^{207}$Tl [@ohtsubo] showed the simultaneous measurement of bound and continuum state $\beta^{-}$ decay. However, the authors have mentioned this decay as a single $\beta^{-}$ transition process to a particular daughter level with 100 % branching [@ohtsubo] whereas, the present data [@nndc] suggest three available levels among which the total $\beta^{-}$ decay is distributed.
In earlier studies, Takahashi and Yokoi [@takahashi; @takahashi2] had investigated $\beta$ transition (bound state $\beta^{-}$ decay and orbital electron capture) processes of some selected heavy nuclei suitable for s-process studies. However, in their work, they had not given separately the bound state decay rate of bare atoms. Further, in another work, Takahashi [@takahashi1] had studied the $\beta^{-}$ decay of some bare atoms for which bound state $\beta^{-}$ decays produce significant enhancement in decay rates and proposed measurement in storage ring experiment. However, they did not take into account the contribution of transitions to all possible energy levels of the daughter nucleus in total $\beta^{-}$ decay rate enhancement. As an example, according to the present $\beta^{-}$ decay data [@nndc], there are six possible $\beta^{-}$ transitions from the \[117.59 keV, $6^{+}$\] state of $^{110}$Ag to various states of $^{110}$Cd, but they had mentioned the contribution of only one transition.
With the availability of modern day experimental $\beta$ decay half-lives in terrestrial condition for the neutral atom, experimental Q-values for $\beta ^{-} $ decays and atomic physics inputs, it becomes inevitable to re-visit some of the earlier works. Moreover, in a previous work, Takahashi and Yokoi [@takahashi] addressed a few nuclei in their ‘case studies’, undergoing $\beta ^{-}$ transitions, as some of the essential turnabouts in $s$-process nucleosynthesis, where contributions from atoms with different states of ionization were considered. However, the explicit study of bound and continuum state $\beta ^{-}$ transitions of bare atoms for most of these nuclei remained unevaluated till date both experimentally as well as theoretically.
In the present work, our aim is to study the $\beta ^{-} $ decay of some elements, in the mass range (A $\approx$ 60 - 240) which might be of interest for future experimental evaluations using storage ring. In particular, calculations of $\beta^{-}$ decay rates to the continuum as well as bound state of these fully ionized atoms, where information for neutral atom experimental half-life and $\beta^{-}$ decay branchings are terrestrially available, have been performed. Most importantly the study of effective half-lives for bare atoms will be helpful to set a limit for the maximum enhancement in $\beta^{-}$ decay rate due to the effect of bound state decay channels. Moreover, we have also discussed the effect of different nuclear structure and decay inputs (Q value, radius etc.) over the bound to continuum decay rate ratio. In addition, some interesting phenomena of changes in $\beta^{-}$ decay branching for a number of bare atoms along with some notable change in branching (branching-flip) for a few of them, have been obtained. The branching-flip is obtained for the first time.
The paper is organized as follows: section \[2\] contains the methodology of our entire calculation for bound and continuum state $\beta^{-}$ decay rates for bare atom, as well as comparative half-life ($Log ft$) for neutral atom. In section \[3A\] we have discussed our results for the neutral atoms, whereas in section \[3B\] results for the bare atoms have been discussed. The phenomenon of change in $\beta^{-}$ decay branching for bare atom compared to that in neutral atom is also discussed explicitly in the section \[3B\]. Conclusion of our work has been described in section \[4\]. Finally, we present a table for the calculated $\beta^-$ decay rates in Appendix A followed by a discussion on the choice of spin-parity for unconfirmed states of neutral atom in Appendix B.
[Methodology]{} \[2\]
=====================
In this work, we have dealt with the allowed and first-forbidden $\beta ^{-} $ transitions for neutral and fully ionized atoms. The contributions of higher-order forbidden transitions are negligible in the determination of the final $\beta ^{-} $ decay rate and thus we have not tabulated the contributions for the same.
The transition rates (in $sec^{-1}$) for allowed (a), non-unique first-forbidden(nu) and unique first-forbidden(u) transitions are given by [@takahashi; @takahashi1; @takahashi2]
$$\begin{aligned}
\lambda = [(ln 2)/f_0t](f^{*}_{m}) ~~~~~~ \text {for m= a, nu} \\ \nonumber
=[(ln 2)/f_1t](f^{*}_{m})~~~~~~ \text{for m= u ~~ }.\end{aligned}$$
Here $t$ is the partial half-life of the specific parent-daughter energy level combination for which transition rate has to be calculated and $f^{*}_{m}$ is the lepton phase volume part described in detail, below in this section. For allowed and non-unique first-forbidden $\beta ^{-} $ decay, the expression for the decay rate function $f_0(Z,W_0)$ can be simplified to [@gove; @konopinski]
$$\begin{aligned}
f_0(Z,W_0) =\int^{W_0}_1\sqrt{(W^2-1)} W (W_0-W)^2\\ \nonumber
\times F_0(Z,W)L_0dW .\end{aligned}$$
The certain combinations of electron radial wave functions evaluated at nuclear radius R ( in the unit of $\hslash/m_ec$) were first introduced by Konopinski and Uhlenbeck [@konopinski] as $L_k$’s. The value for $k=0$ can be approximated as
$$L_0 = \dfrac{1+\sqrt{1-\alpha^2 Z^2}}{2}.$$
Here, $\alpha$ is the fine structure constant. In the work of Behrens and Jänecke [@behrens], the authors had taken a different form of $L_0$, which includes a slight dependence on the momentum. However, we find that the $L_0$ approximation, adopted in our calculation, is in good agreement with that from the Ref. [@behrens] within the considered energy window.
In Eq.(2), $W$ is the total energy of the $\beta^{-}$ particle for a $Z-1\rightarrow Z $ transition and $W_0 = Q_n/m_ec^2+1 $ is the maximum energy available for the $\beta^{-}$ particle. Here the mass difference between initial (parent) and final (daughter) states of neutral atoms are expressed as the decay $Q$ value ($Q_n$ in keV). The term $F_0(Z,W)$ is the Fermi function for allowed and non-unique transition, given by [@konopinski]
$$\begin{aligned}
F_0(Z,W)=\dfrac{4}{ \left|\Gamma \left( {1+2\sqrt{1-\alpha^2 Z^2 }}\right)\right|^2}\\ \nonumber
\left(2R\sqrt{W^2-1}\right)^{2\left(\sqrt{1-\alpha^2 Z^2} -1\right)}exp\left[\dfrac{\pi \alpha Z W}{\sqrt{W^2-1}}\right] \\ \nonumber
\times \left|{\Gamma{\left(\sqrt{1-\alpha^2 Z^2} + i \dfrac{\alpha Z W}{\sqrt{W^{2}-1}} \right)}}\right |^2.\end{aligned}$$
Similarly, for the unique first-forbidden transition the decay rate function $f_1(Z,W_0)$ has the form reduced from Refs. [@gove; @konopinski] is given by,
$$\begin{aligned}
f_1(Z,W_0) = \int^{W_0}_1\sqrt{(W^2-1)} W (W_0-W)^2 F_0(Z,W) \\ \nonumber
\times\left[(W_0-W)^2 L_0 + 9 L_1 \right]dW ,\end{aligned}$$
with $L_1$ given by,
$$L_1 = \dfrac{F_1(Z,W)}{F_0(Z,W)} \left(\dfrac{W^2-1}{9}\right) \dfrac{2+\sqrt{4-\alpha^2 Z^2}}{4}.$$
The term $F_1(Z,W)$ for unique $\beta ^{-} $ transition is given by [@konopinski],
$$\begin{aligned}
F_1(Z,W)=\dfrac{(4!)^2}{ \left|\Gamma \left( {1+2\sqrt{4-\alpha^2 Z^2 }} \right) \right|^2} \\ \nonumber
\left(2R\sqrt{W^2-1}\right)^{2\left(\sqrt{4-\alpha^2 Z^2} -2\right)}exp\left[\dfrac{\pi \alpha Z W}{\sqrt{W^2-1}}\right] \\ \nonumber
\times \left|{\Gamma{\left(\sqrt{4-\alpha^2 Z^2} + i \dfrac{\alpha Z W}{\sqrt{W^{2}-1}} \right)}}\right |^2.\end{aligned}$$
Eqs. (2) and (5) are general forms of $f_0(Z,W_0)$ and $f_1(Z,W_0)$. For more precise calculation of f-factor, one should in principle, include various corrections in the integrand of Eqs. (2) and (5). Corrections due to atomic physics effects, radiative correction and finite nuclear size effects might be important for such studies. For fully ionized atoms, corrections due to atomic physics effects, such as, imperfect overlap of initial and final atomic wave functions, exchange effects that comes from the anti-symmetrisation of the emitted electron with the atomic electrons [@bahcall2], screening of the nuclear charge due to the coulomb field of the atomic electronic cloud are not needed. For neutral atom, the decay to the atomic bound state should be negligible [@bahcall2]. Also, the screening and exchange corrections together cancel a large part of the overlap correction [@budick]. Further the non-orthogonality effect becomes rapidly smaller as Z increases [@takahashi1]. Some of the corrections have positive sign and some of them have negative sign. So unless all the corrections are taken together, the treatment for corrections to f- factor will not be consistent. Therefore we have neglected these contributions both for bare and neutral atoms. We have included the correction due to the extended charge distribution of the nucleus on the $\beta^{-}$ spectrum. This correction is $\Lambda_k(Z,W) \rightarrow \Lambda_k(1+\Delta\Lambda_k(Z,W)$), where the term $\Lambda_k$ can be written in terms of $L_k$ and $F_0(Z,W)$ as [@gove; @konopinski] $$\Lambda_k(Z,W)= F_0(Z,W)L_{k-1}\left[ \dfrac {(2k-1)!!}{(\sqrt{W^2-1})^{k-1}}\right]^2 ,$$
in such a way that it reduces to $\left[ F_0(Z,W)L_0\right]$ and $\left[ 9F_0(Z,W)L_1/(W^2-1)\right]$ for $k=1$ and $2$, respectively. The correction term is given by [@gove],
$$\begin{aligned}
\Delta\Lambda_k (Z,W) =(Z-50) \times \\ \nonumber
\left[ -25\times 10^{-4} - 4\times10^{-6} W \times (Z-50)\right] \\ \nonumber
\text { for } k = 1, Z > 50 , \\ \nonumber
= 0 ~~~~~~~~~~~~~~~~~~~~~ \text { for } k = 1 , Z \le 50, \\ \nonumber
= 0 ~~~~~~~~~~~~~~~~~~~~~~ \text {for } k > 1~~~~~~~~~~~.\end{aligned}$$
The screened energy of the emitted electron $(W')$ enters through $\Delta\Lambda_k(Z,W')$, where $W'=W-V_0(Z)$. We calculated $V_0(Z)$, following Gove and Martin [@gove], using expression from W. R. Garrett and C. P. Bhalla [@bhalla]. This correction to the integrand in Eqs. (2) and (5) has effect in the fourth decimal place of the f-factor and this is consistent with Ref. [@hayen] discussed for the allowed $\beta^{-}$ decay. So we have dropped $W'$ and used $W$ in the integrand.
It is to be noted that in the present work we have used experimental quantities, such as Q - value, half-life, branching, which have uncertainties even up to the first decimal place [@nndc; @nist]. So, in our treatment we have neglected the screening effect too for neutral atom. Therefore, by using Eqs. (8) and (9) in the integrand of Eq. (2) and Eq. (5) one can calculate the values for $f_0(Z,W_0)$ and $f_1(Z,W_0)$ incorporating only finite size correction.
In the work of A. Hayes [@hayes], the authors have taken a different form of the finite-size correction involving the charge density, which has a complicated radial dependency. However, we find that the results from the present calculation are in agreement with the available experimental data [@nndc].
Further, from the above expressions (Eqs.(4) and (7)), it is evident that the factors $F_0(Z,W)$ and $F_1(Z,W)$ depend on the radius, thereby making the terms $f_0$ and $f_1$ (Eqs.(2) and (5)), radius dependent. Thus, in our present study, we have used various radius values from different phenomenological models and experiments to study their effects on the final $ft$ values. In order to calculate $ft$ values for a nucleus, we have extracted the half-life $t$ for individual transition to daughter levels using the latest $\beta$ decay branching information available in the literature [@nndc].
The lepton phase volume $f^{*}_{m}$ [@takahashi2] for the continuum state $\beta^{-}$ decay can thus be expressed as,
$$\begin{aligned}
f^{*}_{m=a,nu}(Continuum) = \int^{W_c}_1\sqrt{(W^2-1)} \\ \nonumber
W (W_c-W)^2 F_0(Z,W) L_0 dW,\end{aligned}$$
and
$$\begin{aligned}
f^{*}_{m=u}(Continuum) = \int^{W_c}_1\sqrt{(W^2-1)} \\ \nonumber
W (W_c-W)^2 F_0(Z,W) \times \\ \nonumber
\left[(W_c-W)^2 L_0 + 9L_1\right] dW,\end{aligned}$$
Here $W_c = Q_c/m_ec^2 +1 $ is the maximum energy available to the emitted $\beta^{-}$ particle, and $Q_c$ is given by,
$$\begin{aligned}
Q_c = Q_n - \left[ B_n(Z+1) - B_n(Z)\right].\end{aligned}$$
The term $\left[ B_n(Z+1) - B_n(Z)\right]$ denotes the difference of binding energies for bound electrons of the daughter and the parent atom. The experimental values for all the atomic data (binding energies/ionization potential) are availed from Ref. [@nist].
Further, for the bound state $\beta^{-}$ decay of the bare atom $f^{*}_{m}$ takes the form [@takahashi2]
$$\begin{aligned}
f^{*}_{m=a,nu}(Bound) = \sum_x \sigma_x \left(\pi/2\right) \left[ f_x\text{ or }g_x \right]^2 b^2 \\ \nonumber
\left(\text {for } x=ns_{1/2},np_{1/2}\right),\end{aligned}$$
and
$$\begin{aligned}
f^{*}_{m=u}(Bound) = \sum_x \sigma_x \left(\pi/2\right) \left[ f_x\text{ or }g_x \right]^2 b^4 \\ \nonumber
\left(\text {for } x=ns_{1/2},np_{1/2}\right), \\ \nonumber
~~~~~~~~~~~~~~~~~~~~\\ \nonumber
= \sum_x \sigma_x \left(\pi/2\right) \left[ f_x \text{ or } g_x \right]^2 b^2 \left(9/R^2\right) \\ \nonumber
\left(\text{for } x=np_{3/2},nd_{3/2}\right).\end{aligned}$$
Here $\left[ f_x\text{ or }g_x \right]$ is the larger component of electron radial wave function evaluated at the nuclear radius $R$ of the daughter for the orbit $x$. The $\left[ f_x\text{ or }g_x \right]$ is obtained by solving Dirac radial wave equations using the Fortran subroutine RADIAL by Salvat [@cpc]. In our case, $\sigma_x$ is the vacancy of the orbit, chosen as unity and $b$ is equal to $Q_b/m_ec^2$ where, $$\begin{aligned}
Q_b = Q_n - \left[ B_n(Z+1) - B_n(Z)\right]-B_{shell}(Z+1).\end{aligned}$$
For example, in case of a bare atom, if the emitted $\beta^{-}$ particle gets absorbed in the atomic K shell, then the last term of Eq.(15) will be the ionization potential for the K electron denoted by $B_K(Z+1)$.
Results and Discussion
======================
In this work, we have calculated $\beta^-$ decay transition rates to bound and continuum states, for a number of fully ionized atoms in the mass range A $\approx$ 60-240. One of the motivations is that there are some evidences where earlier works were not equipped enough to address the entire $\beta^-$ decay scenario. This might be due to the unavailability of information about all the energy levels participating in transition processes.
As an example, Takahashi [@takahashi1] have considered transitions for allowed(a), first-forbidden non-unique(nu) and first-forbidden unique(u) decay of parent nuclei to a few energy levels of daughter nuclei. For instance, in the case of $^{228}$Ra nucleus, the authors have tabulated the decay from the ground state of the parent $[E$(keV), $J^\pi] = [0.0, 0^+]$ nucleus to $[6.3, 1^-]$ and $[33.1, 1^+]$ states of the daughter nucleus $^{228}$Ac. However, these two transitions cover only the 40% of the total $\beta^{-}$ decay branching of neutral $^{228}$Ra atom from the ground state. With the latest experimental data [@nndc], we find that there are two more available states of $^{228}$Ac where the rest amount of $\beta^{-}$ decay from the ground state of $^{228}$Ra occur. In this section, it will be shown that the contributions of all these four states are extremely important in the determination of effective enhancement of $\beta^{-}$ transition rates of bare $^{228}$Ra as well as to understand the phenomenon of branching-flip, discussed in section \[3B\].
For simplicity, this section is subdivided into two parts. The first subsection involves the calculation of $Log~ ft$ for the neutral atom, a necessary ingredient for the calculation of $\beta^-$ decay rate of the bare atom. In the next subsection, the $\beta^{-}$ decay transition rates of bare atoms have been discussed with a detailed explanation of TABLE [A.I]{}. The dependence of these decay rates on different parameters is also examined in the same subsection. Finally, we have shown and discussed the change in individual level branchings in fully ionized atoms.
[ $Log~ ft$ calculation for neutral atoms]{}\[3A\]
--------------------------------------------------
It is evident from Eqs.(1-9) that the calculation for $ft=f_0t (/f_1t)$ is one of the essential components in the determination of the transition rate $\lambda$, which in turn depends on radius R of the daughter nucleus. However, $Log~ ft$ data obtained from Ref. [@nndc] can not provide the information of the $R$ dependence of $Log~ ft$. As the present theoretical modelling for bare atom depends on radius (see section \[2\]), we find it more accurate to calculate $Log~ ft$ for neutral atom for different choices of radii. In Appendix \[lognu\], we present a table for bound and continuum state $\beta^-$ decay rates for bare atoms along with the values of $Log~ ft$ for corresponding neutral atoms at different radii and compare our calculations with existing theoretical as well as experimental results (see the supplemental material [@supl] for details). As explained in section \[2\], we have tabulated $Log~ ft$ values only for allowed (a), first-forbidden non-unique (nu) and first-forbidden unique (u) transitions.
Here, in TABLE [A.I]{}, $R_1$ is the phenomenological radius evaluated as $R_1=1.2 A^{1/3}$ fm, whereas $R_2$ is the nuclear charge radius in fm [@angeli] and $R_3$ is the half-density radius given by [@gove] $R_3=(1.123A^{1/3} - 0.941A^{-1/3})$ fm. We have calculated $Log~ ft$ values for $R_1$, $R_2$ and $R_3$ and compared them with the existing data [@nndc]. Besides, we have tabulated the available values from previous calculations of Takahashi [@takahashi1] in the same table.
One can see that the change in radius may cause a change in the $Log ~ft$ value mostly in the second decimal place. In the next subsection, we will show the effect of these variations on the transition rates for bare atoms.
Further, from TABLE [A.I]{} and the supplemental material [@supl], it can be noted that our calculation matches with the experimental $Log~ ft$ data [@nndc] in most cases up to the first decimal place. The agreement of our result with experimental data [@nndc] confirms the applicability of the methodology adopted in the present study.
[ Bound and Continuum decay rates of bare atoms]{}\[3B\]
--------------------------------------------------------
In the ninth and the eleventh column of TABLE [A.I]{} of Appendix \[lognu\], bound and continuum $\beta^-$ decay rates of bare atoms are presented, respectively. It is observed that the dependence on radius affect the bound ($\lambda_B$) and the continuum state ($\lambda_C$) decay rates in first or second decimal places, and the ratio $\lambda_B/\lambda_C$ remains almost unaffected up to the first decimal place for most of the examined cases.
Further, from TABLE [A.I]{} (also see the supplemental material [@supl]), we find that the values for $\lambda_B$ and $\lambda_C$ from our calculation agree with those of the existing theoretical results [@takahashi1] quite well. The possible reasons for the slight mismatch between our calculation and that from Takahashi [@takahashi1] are mainly due to (i) the effect of the nuclear radius, (ii) the adoption of present day Q values (for all $Q_n , Q_c$ and $Q_b$), (iii) availability of present day $\beta^-$ decay branching of neutral atoms and (iv) the choice of significant digits. Despite that, the overall success of our calculation in reproducing available $\lambda_B$ and $\lambda_C$ for bare atoms once again justify the extension of the present calculation to previously unevaluated cases.
[{width="15cm" height="11cm"}]{}
It can again be shown from TABLE [A.I]{} that in a transition from the parent nucleus $^AX_{Z-1}$ to different energy levels of the daughter nucleus $^AX_{Z}$, the ratio $\lambda_B/\lambda_C$ for all transitions are not same, rather it decreases with increasing $Q_n$ value. It can be understood from the expressions in Eqs.(10-15) where the factors $f^{*}_{Continuum}$ and $f^{*}_{Bound}$ depend on $Q_c$ and $Q_b$, respectively, which are again derived from the neutral atom Q value $Q_n$. Due to different $Q_n$ values for different transitions, $\lambda_B/\lambda_C$ can be identified as a function of $Q_n$. For the sake of understanding, in FIG. \[lblc\], we have plotted the ratio $\lambda_B/\lambda_C$ versus $ Q_n$ for the nuclei $^{115}$Cd, $^{123}$Sn, $^{136}$Cs and $^{152}$Eu. In each case, dependence on $Q_n$ is observed which can be fitted to the form
$$\begin{aligned}
\dfrac{\lambda_{B}}{\lambda_{C}}=a\times({Q_n})^b\end{aligned}$$
where a and b are the nucleus dependent constants given in TABLE \[ab\].
\[ab\]
The TABLE \[ab\] confirms that Eq.(16) is a characteristic feature of $\lambda_{B}/ \lambda_{C}$ ratio of the bare atom with particular Z and A values. If there is a mistake in the calculation of $f^{*}$ for $\lambda_{B}$ or $/$ and $\lambda_{C}$, then the ratio point will not fit to such a power law.
In the fourteenth column of TABLE [A.I]{}, the ratio of $\lambda_{Bare}(=\lambda_{B} + \lambda_{C})/\lambda_{Neutral}$ (called here rate enhancement factor) has been tabulated. It is evident from these values that there must be an enhancement in the decay rate for each transitions (i.e. $\lambda_{Bare}/\lambda_{Neutral} > 1$) because of the additional bound state decay channel.
In FIG. \[enh\], the ratio of $\lambda_{Bare}/\lambda_{Neutral}$ for $^{110}$Ag, $^{155}$Eu and $^{227}$Ac have been shown. From the figure, it can be noted that rate enhancements (a) are different for different transitions of a particular nucleus, (b) are dependent on $Q_n$ values : lower the $Q_n$, larger the enhancement. Moreover, this rate enhancement factor (c) also depends on Z and A of the atom; larger the value of Z and/or A, larger the enhancement.
Further, in TABLE [A.I]{}, we have tabulated effective $\beta^-$ decay half-lives for bare atoms and compared to those of neutral atoms. It should be noted that the neutral atom half-life given in the fifteenth column of the table is the total half-life corresponding to a, nu and u types of $\beta^{-}$ transitions only.
[![(Color online) Ratio of $\lambda_{Bare}/\lambda_{Neutral}$ Vs the neutral atom Q-value $Q_n$ (in keV) for various $\beta^{-}$ transitions (for the radius $R_{1}$). See text for details. \[enh\]](Figure_2b "fig:"){width="85mm" height="65mm"}]{}
[{width="19.5cm" height="7.5cm"}]{}
The dependence of the rate enhancement factor on $Q_n$ causes a change in $\beta^-$ branching for the bare atom. In bare atom, branchings similar to the neutral atom can only be achieved if the factor $\lambda_{Bare}/\lambda_{Neutral}$ remains constant with $Q_n$, which is obviously not the case (FIG. \[enh\]). In other words, this change can be understood to be an outcome of the non-uniformity of the $\lambda_B / \lambda_C $ ratio with $Q_n$. It is observed that the continuum decay rate for bare atom decreases with respect to that for the neutral atom (i.e. $\lambda_{C} < \lambda_{Neutral}$) due to the reduction of continuum Q value ($Q_c < Q_n$, Eq. (12)). Further from FIG. \[lblc\], it is clear that with the decrease in the $Q_n$ value, $\lambda_B$ dominates more over $\lambda_C$ and hence the effective decay rate of the bare atom $\lambda_{Bare}=\lambda_B + \lambda_C$ does not follow the same branching as that of the neutral atom.
Note: For the $\beta^{-}$ transition having very low $Q_n$ value, bound state decay may be the only path of $\beta^{-}$ decay. As an example, in the transition of $^{227}$Ac $[0.0,3/2^-]$ to $^{227}$Th $[37.9,3/2^-]$ with $Q_n = 6.9$ keV, $Q_c$ for continuum decay of the bare atom becomes $-13.1$ keV. As evident from Eqs.(10-12), due to the negative value of $Q_c$, the corresponding decay channel gets closed. On the other hand, as $(Q_b-Q_n) > 0$ for this transition, the total decay is governed by the bound state channel only.
As an example, in FIG. \[noflip\], we have compared branchings for neutral (left panel) and bare (right panel) $^{136}$Cs atom. It can be seen from FIG. \[noflip\] that the branchings for all $\beta^-$ transitions of the bare atom have been changed from that of the neutral atom. However, the ordering of each branch remains unaltered in both cases, i.e. the $[0.0, 5^+] \rightarrow [2207.1, 6^+]$ branch gets the maximum feeding followed by the $[0.0, 5^+] \rightarrow [1866.6, 4^+]$ and $[0.0, 5^+] \rightarrow [2140.2, 5^-]$ branches, whereas the minimum feed goes to $[0.0, 5^+] \rightarrow [2030.5, 7^-]$ channel for both the neutral and bare atoms.
Further, some notable observations and comments for some nuclei are given below.
$\bullet$ In case of neutral $^{207}$Tl atom in terrestrial condition, the $[0.0, 1/2^+]$ state of $^{207}$Tl decays to $[0.0, 1/2^-]$ state of $^{207}$Pb with 99.729% branching, whereas to $[569.6, 5/2^-]$ state of the daughter has the branching $>$0.00004% (in some places of Ref. [@nndc] this value is given as $<$0.00008%) and to $[897.8, 3/2^-]$ state has 0.271% branching [@nndc] (see supplemental material [@supl] for details). For bare atom, Ohtsubo [@ohtsubo] had observed bound state decay rate $\lambda_B= 4.29(29) \times 10^{-4}$ sec$^{-1}$ and continuum state decay rate $\lambda_C=2.29 (012) \times 10^{-3}$ sec$^{-1}$, by considering the transition to $[0.0, 1/2^-]$ state of $^{207}$Pb with 100% branching. In our calculation for bare atom, we have got bound state decay rate $\lambda_B = 4.15 \times 10^{-4}$ sec$^{-1}$ and continuum state decay rate $\lambda_C= 2.54 \times 10^{-3}$ sec$^{-1}$. The calculated branchings of bare $^{207}$Tl : $\sim$ 99.6 % to $[0.0, 1/2^-]$, $\sim$ 0.00005% - 0.0001% to $[569.6, 5/2^-]$ and $\sim$ 0.4 % to $[897.8, 3/2^-]$ states of the daughter $^{207}$Pb.
In our study, we found some special cases where effective branchings for the bare atom do not follow the same ordering as that of the neutral atom. This indicates a very interesting phenomenon of branching-flip, obtained for the first time in this work. Sometimes the additive contribution of $\lambda_B$ and $\lambda_C$ and the effect of these two competing channels can lead to this branching-flip. This can be understood from FIG. \[all\]. In FIG. \[all\], decay rates (sec$^{-1}$) for neutral ($\lambda_{Neutral}$) and bare ($\lambda_{Bare}$) atom along with all decay components ($\lambda_B$ and $\lambda_C$) of the bare atom versus $Q_n$ are shown for the ground state decay of $^{134}$Cs and $^{228}$Ra nuclei. One can see from FIG. \[all\] that the highest point corresponding to $\lambda_{Neutral}$ (i.e. maximum $\beta^{-}$ branching in neutral atom) and the highest point corresponding to $\lambda_{Bare}$ (i.e. maximum $\beta^{-}$ branching in bare atom) are coming from different transitions to the daughter nuclei (different $Q_n$ values), which clearly indicates the phenomenon of flip in the branching sequence.
$\bullet$ In the case of $^{134}$Cs, $\lambda_{Neutral}$ is maximum at $Q_{n} = 658.1$ keV, which is due to the maximum branching to the 1400.6 keV level (see supplemental material [@supl] for details) of $^{134}$Ba [@nndc]. In contrary, for the same nucleus, $\lambda_{Bare}$ is maximum at $Q_n = 88.8$ keV which therefore indicates the maximum branching to the 1969.9 keV level (see TABLE [A.I]{}) of the daughter $^{134}$Ba for bare atom.
$\bullet$ Similarly for $^{228}$Ra, the maximum branching for the bare atom ($(\lambda_{Bare})_{max}$ at $Q_n=12.7$ keV) shifts from that of the neutral atom ($(\lambda_n)_{max}$ at $Q_n=39.1$ keV). In FIG. \[flip\], we have shown the change and alteration of transition branchings for the $\beta^-$ decay of $^{228}$Ra. One can see the branching-flips of the participating levels of the $^{228}$Ac atom in FIGS. \[all\] and \[flip\]. In case of the neutral $^{228}$Ra atom, maximum branching is 40% for the $[6.7, 1^+]$ level of the daughter [@nndc]. After complete ionization, the major contribution of the total decay rate comes due to the bound state enhancement of $Q_n=$12.7 keV channel which has $\sim$ 84.07% decay to the $[33.1, 1^+]$ level (30% in neutral atom) of the daughter atom, whereas only $\sim$ 5.81% of the total decay branching is observed for the level $[6.7, 1^+]$.
There are a few more cases, where the branching-flips are observed. However, not necessarily, all the transition branches face the phenomenon of flip. It may also happen that only two or three branches change their sequence, whereas other branches remain in the same order as that of the neutral atom.
$\bullet$ In the $\beta^-$ decay of $^{152}$Eu $[45.5998, 0^{-}]$ (see Table 1 of the Ref. [@supl] for branching details), we find that in both cases (neutral and bare) the branching to $[0.0, 0^+]$ branch of the daughter dominate over the rest, whereas a branching-flip is observed between $[344.3,2^+]$ and $[1314.6, 1^-]$ states.
$\bullet$ Similarly for $^{227}$Ac, we find that there is a branching-flip between two transitions from $[0.0, 3/2^-]$ state of the parent to $[0.0, 1/2^+]$ and $[24.5, 3/2^+]$ states of the daughter atom. The ratio of branching for these two levels is 5.4:1 for neutral atom, which changes to 1:1.38 for bare atom.
It should be noted that the ultimate fate of individual branchings in the bare atom is decided by two factors: the initial branching (required to calculate $Log~t$ for each transition: a part of $Log~ft$ calculation) and Q value of the neutral atom. The competition between these two factors determines whether the branching-flip will occur or not.
**Effect of uncertainties:** Furthermore, in order to get the complete picture of $\beta^-$ decay for bare atom, effects due to uncertainties in $\beta^-$ decay half-life and Q value need to be considered. The effect of uncertainty is appreciable depending on the numerical value of the half-life and Q value. In case of atoms with the $\beta^-$ decay half-life of the order of seconds/minutes and having high Q value, no significant change is observed in the calculation of $Log~ ft$ due to the inclusion of experimental uncertainties. The contributions peek out for long lived nuclei with large uncertainty or for transitions of high Q value having large uncertainty. For example, in case of $^{93}$Zr atom, where the neutral atom half-life is equal to $1.61 \times 10^6(5)$ years, $Log~ ft$ for the transition $[0.0,5/2^+ \rightarrow 30.8,1/2^-]$ with the radius $R_1$ is given by ${10.234}^{+0.014}_{-0.013}$. Therefore, the final values for continuum and bare state $\beta^-$ transitions including the uncertainties can be written as $\lambda_C={6.87}_{-0.21}^{+0.22} \times 10^{-15}$ $sec^{-1}$ and $\lambda_B={6.13}_{-0.19}^{+0.20} \times 10^{-15}$ $sec^{-1}$, respectively.
[{width="17cm" height="6cm"}]{}
[{width="19cm" height="7cm"}]{}
[Conclusion]{}\[4\]
===================
To summarize, in this work we have calculated individual contributions of bound and continuum state $\beta^-$ decays to the effective $\beta^-$ decay rate of the bare atom in the A $\approx$ 60 to 240 mass range where earlier information were partial and/or old. Additionally, the dependence of transition rates over the nuclear radius and the Q value is illustrated clearly in the present study. We found a power law dependence of $\lambda_{B}/ \lambda_{C}$ of a bare atom on $Q_n$ for each value of Z and A. Along with the effective enhancement of transition rates, we have found that transition branchings for the bare atom differs from that of the neutral atom for all Z and A, which is an outcome of non-uniform enhancement amongst the participating branches. Most interestingly, we have found few nuclei, viz. $^{134}$Cs, $^{228}$Ra etc., where some flip in the branching pattern is found for their bare configuration. It will be interesting to see how these results help planning new experiments involving bare atoms. The calculations will be extended to partially ionized atoms which will provide decay rate as function of density and temperature of the stellar plasma and will be useful for calculation of nucleosynthesis processes.
ACKNOWLEDGEMENT {#acknowledgement .unnumbered}
===============
AG is grateful to DST-INSPIRE Fellowship (IF160297) for providing financial support. CL acknowledges the grant from DST-NPDF (No. PDF/2016/001348) Fellowship.
Table for $\beta^-$ decay {#lognu}
=========================
\[a1\] \[a2\]\
\[a3\] \[a4\]\
\[a5\] \[a6\]\
\[a7\] \[a8\]\
\[b2\]\
\[b3\] \[b4\]\
\[b5\] \[b6\]\
Here we present a table containing $Log~ f_{0}t(f_{1}t)$ values for neutral atoms, bound and continuum state $\beta^-$ decay rates for bare atoms along with the comparison with previous theoretical [@takahashi1] as well as existing data [@nndc], wherever available. Finally a comparative study on bare atom to neutral atom $\beta^{-}$ decay rates for different choices of radii has been presented.
[**EXPLANATION OF TABLE**]{}
:
$\bullet$ First column: $\beta^-$ decay transitions with Neutral atom branchings (FIGS.7 - 9)
$\bullet$ Second column: Participating parent - daughter energy levels in the transition.
$\bullet$ Third column: Transition types (a, nu and u).
$\bullet$ Forth column: Neutral atom Q value.
:
$\bullet$ Fifth column: $R_1$, $R_2$ and $R_3$ are given in consecutive rows.
:
$\bullet$ Sixth column: $Log~f_0(f_1)$ for different choices of radii (row-wise) calculated using Eq. (2) and (5).
$\bullet$ Seventh column: $Log~f_0t(f_1t)$ for radii $R_1$, $R_2$ and $R_3$ (row-wise), where $Log~ft = Log~f + Log~t$. Here $t$ is the partial half-life of individual transitions (second column) calculated using T$_{1/2}$ and branching of that particular level (first column).
$\bullet$ Eighth column: $Log~f_0t(f_1t)$ values from previous work [@takahashi1] (row 1) and existing data [@nndc](row 2).
:
$\bullet$ Ninth column: Bound state $\beta^-$ decay rates $\lambda_B$ for bare atoms for different choices of radii (row-wise) calculated using Eqs. (1) and (13-15).
$\bullet$ Tenth column: $\lambda_B$ from previous work [@takahashi1] (row 1).
$\bullet$ Eleventh column: Continuum state $\beta^-$ decay rates $\lambda_C$ for bare atoms for radii $R_1$, $R_2$ and $R_3$ (row-wise) calculated using Eqs. (1) and (10-12).
$\bullet$ Twelfth column: $\lambda_C$ from previous work [@takahashi1] (row 1).
$\bullet$ Thirteenth column: Ratio of bound and continuum state $\beta^-$ decay rates of bare atom for different choices of radii (row-wise).
:
$\bullet$ Fourteenth column: Ratio of Bare and Neutral atom decay rates for radii $R_1$, $R_2$ and $R_3$ (row-wise). Here $\lambda_{Bare} = \lambda_B + \lambda_C$. $\lambda_{Neutral}$ is obtained from parent level half-life (T$_{1/2}$) and $\beta^-$ branching (first column).
:
$\bullet$ Fifteenth column: Total $\beta^-$ decay (a, nu, u) half-life of the parent level for neutral atom, obtained from T$_{1/2}$ and $\beta^-$ branching (first column). $\bullet$ Sixteenth column: Total $\beta^-$ decay (a, nu, u) half-life of the parent level for bare atom for radii $R_1$, $R_2$ and $R_3$ (row-wise). It is obtained by the formula $0.693 \times 1/\sum_i(\lambda_{Bare})_i$, where [*i*]{} denotes all the possible a, nu and u type of $\beta^-$ transitions. Here, m: Minutes; h: Hours; d: Days; y : Years.
[\[TABLE -A.I\] TABLE A.I : $Log f_{0}t(f_{1}t)$ values , bound and continuum state $\beta^-$ decay rates (bare atom), comparison between neutral atom and bare atom $\beta^{-}$ decay rates for different choices of radii compared with the results of previous theoretical work and experimental data.]{}
TABLE A.I : (Cotnd.)
TABLE A.I : (Cotnd.)
TABLE A.I : (Cotnd.)
------------------------------------------------------------------------
$\ddagger$ Not calculated here, see Appendix \[spin\] for details.
TABLE A.I : (Cotnd.)
TABLE A.I : (Cotnd.)
TABLE A.I : (Cotnd.)
TABLE A.I : (Cotnd.)
TABLE A.I : (Cotnd.)
TABLE A.I : (Cotnd.)
------------------------------------------------------------------------
$\ddagger$ Not calculated here, see Appendix \[spin\] for details.
TABLE A.I : (Cotnd.)
------------------------------------------------------------------------
\* The mismatch between $\lambda_{B}$ value may arise from the typographical error in the tabulation of $\lambda_{B}$ in Ref. [@takahashi1].
TABLE A.I : (Cotnd.)
------------------------------------------------------------------------
\* Experimentally available values are given in the section \[3B\] of the main text.
Choice of Spin-Parity for unconfirmed states of neutral atom {#spin}
============================================================
Sometimes the comparison of the calculated $Log~ ft$ values with experimental data gives an idea about the spin-parity of participating energy levels where these quantities are still unconfirmed experimentally. We have identified a few such transitions in TABLE [A.I]{}. In the transition from $^{123}$Sn $[ 0.0, 11/2^-]$, there are a few states of the daughter $^{123}$Sb, where the spin values are not experimentally confirmed yet (identified as $(J)^\pi$ and/ or $(J^\pi)$ in the table). In the transition from $^{123}$Sn $[ 0.0, 11/2^-]$ to $E= 1181.3$ keV state of the daughter, if it chooses the decay channel with the spin-parity $J^\pi = (9/2)^+$ then the transition will be of the type (nu), whereas for the choice of spin $J^\pi = (7/2)^+$, the transition $[0.0, 11/2^- \rightarrow 1181.3, (7/2)^+]$ will be the (u) type. Now comparing with the available experimental $Log ~ft$ value, it seems from our calculation that the (nu) case is in good agreement whereas the (u) case deviates (difference $ \sim 0.4 $ ) from the same for all choices of the radius R.
Similarly, from TABLE [A.I]{}, our observations for other such transitions are given by (see the table for $Log ~ft$ comparison)\
$\bullet$ $^{123}$Sn $[ 0.0, 11/2^-] \rightarrow ^{123}$Sb $[1260.9, (9/2)^+]$: (nu),\
$\bullet$ $^{123}$Sn $[ 0.0, 11/2^-] \rightarrow ^{123}$Sb $[1337.4, 9/2^+]$: (nu),\
$\bullet$ $^{152}$Eu $[ 45.5998, 0^-] \rightarrow ^{152}$Gd $[1460.5, 1^+]$: (nu).
Note-1: This type of study is not conclusive in the transition from $[137.9, 5^-, 6^-]$ level of the $^{148}$Pm nucleus. Depending on the spin of the parent level $5^-/6^-$, all four transitions to the daughter level will either be of type (a) or (nu) and thus, the $Log~ ft$ value in each case will remain the same.
Note-2: In case of $^{152}$Eu $[45.5998, 0^-]$ to $^{152}$Gd $[1047.9, 0^+]$ transition our $Log ~ft$ differs from that of the experimental value [@nndc] by a difference of $\sim 0.16$ - $0.18$ (for different radii). However, we find a numerical mismatch in the tabulation for the experimental energy value of $[1047.9, 0^+]$ state of $^{152}$Gd [@nndc; @152].
Note-3: For the $\beta^{-}$ transitions from the first excited state of $^{95}$Nb, the parent level is mentioned as $[234.7, 1/2^{-}]$ in Ref. [@NDS95], whereas in Ref. [@nndc] this energy level is mentioned both as $[235.7, 1/2^{-}]$ and $[234.7, 1/2^{-}]$ at different places. However, our calculation of $Log ft$ matches with the reported $Log ft$ value only when we have taken the level energy as 234.7 keV.
[99]{} R. Daudel, M. Jean, and M. Lecoin, [*J. Phys. Radium*]{} [**8**]{}, 238 (1947). J. N. Bahcall, , 495 (1961). K. Takahashi and K. Yokoi, , 375 (1987). K. Takahashi, R. N. Boyd, G. J. Mathews, and K. Yokoi, , 1522 (1987). K. Takahashi and K. Yokoi, , 578 (1983). M. Jung , , 2164 (1992). F. Bosch , , 5190 (1996). K. Yokoi, K. Takahashi, and M. Arnould, [*Astronomy and Astrophysics* ]{}[**117**]{}, 65 (1983). T. Ohtsubo , , 052501 (2005). National Nuclear Data Center, (https://www.nndc.bnl.gov/). N. B. Gove and M. J. Martin, , 205 (1971). E. J. Konopinski and G. E. Uhlenbeck, , 308 (1941). H. Behrens and J. Jänecke, Numerical Tables for Beta-Decay and Electron Capture, Springer (1969). J. N. Bahcall, , 2683 (1963). B. Budick, , 1034 (1983). W. R. Garrett and C. P. Bhalla, [*Zeitschrift fur physik*]{} [**198**]{}, 453-460 (1967). L. Hayen , Rev. Mod. Phys. [**90**]{}, 015008 (2018). Atomic Spectra Database, (https://physics.nist.gov/PhysRefData/ASD/ionEnergy.html). A. C. Hayes, J. L. Friar, G. T. Garvey, Gerard Jungman and G. Jonkmans, , 202501 (2014). F. Salvat, J.M. Fernandez-Varea, and W. Williamson Jr, Comput. Phys. Commun. [**90**]{}, 151 (1995). See Supplemental Material at \[XXXX\] for the complete dataset. I. Angeli and K.P. Marinova, , 69 (2013). M. J. Martin, , 1497 (2013). S.K. Basu, G. Mukherjee, and A.A.Sonzogni, , 2555-2737 (2010).
[^1]: Corresponding author: ss@physics.iiests.ac.in
|
---
address: 'Department of Physics, University of Toronto, Toronto, Ontario, Canada M5S 1A7'
author:
- 'J. W. Moffat and I. Yu. Sokolov'
title: On Gravitational Collapse in the Nonsymmetric Gravitational Theory
---
The analytical structure of the difference between the static vacuum solution in the nonsymmetric gravitational theory (NGT) and the Schwarzschild solution of Einstein’s gravitational theory (EGT) is studied. It is proved that a smooth matching of the solutions does not exist in the range $0 < r \leq 2M$, for any non-zero values of the parameters $M$ and $s$ of the NGT solution. This means that one cannot consider the difference between the two solutions using perturbation theory in this range of $r$. Assuming that the exterior solution in gravitational collapse is a small, time dependent perturbation of the static solution for a non-zero, generic NGT source ($s\not=0$) and mass density, it is shown that the matching of the interior and exterior solutions will not lead to black hole event horizons.
In Einstein’s gravitational theory (EGT), the collapse of a star leads inevitably to the formation of a black hole event horizon and a singularity at the center of collapse [@HawkingPenrose]. The event horizon is an infinite red shift surface in any coordinate frame of reference, which separates the spacetime manifold into two causally disconnected pieces.
It has been conjectured that in the gravitational collapse of a star in the nonsymmetric gravitational theory (NGT), a black hole event horizon will not form and the appearance of a singularity at the center of collapse is not inevitable[@CornishMoffat1; @CornishMoffat2]. A detailed analysis of the gravitational collapse problem has been carried out[@Moffat1], using a new version of NGT which has a physically consistent linear approximation without ghost poles, tachyons and possesses good asymptotic behavior[@Moffat2; @Moffat3; @LegareMoffat]. An analysis of the spherically symmetric NGT system[@Moffat2; @Clayton], showed that a Birkhoff theorem does not exist in NGT, i.e., the spherically symmetric vacuum solution is time dependent.
Recently, Burko and Ori[@BurkoOri] have claimed that black holes can be anticipated in gravitational collapse in NGT. They used the linear approximation for $g_{[\mu\nu]}$, expanded about the background Schwarzschild solution of EGT.
In the following, we shall study the analytic properties of the static NGT solution and consider its consequences for the collapse of a star. This solution holds in the long-range approximation in which $\mu\approx 0$, where $\mu=1/r_0$ is a “mass" parameter and the long-range approximation holds for large $r_0$. Since we do not have a rigorous dynamical vacuum solution for NGT, we shall make the reasonable physical assumption that a quasi-static solution for the exterior of a star exists such that the limit of this solution to the static solution is smooth, i.e., the time dependent part of the solution is small and the dominant static piece determines the qualitative behavior of the solution[@Moffat1]. The NGT solution is a two-parameter static spherically symmetric solution, in which the parameter $M$ is the mass and $s$ is a dimensionless real parameter associated with the strength of the coupling of $g_{[\mu\nu]}$ to matter. We can model $s$ by the expression[@Moffat1]: $$\label{coupling}
s=\frac{g}{N^\beta},$$ where $g$ is a coupling constant, $N$ denotes the particle number of a star and $\beta$ is a dimensionless parameter. Thus, when $g$ is identically zero, the NGT vacuum solution reduces to the Schwarzschild solution of EGT.
In the case of a static spherically symmetric field, the canonical form of $g_{\mu\nu}$ in NGT is given by $$g_{\mu\nu}=\left(\matrix{-\alpha&0&0&w\cr
0&-\beta&f\hbox{sin}\theta&0\cr 0&-f\hbox{sin}\theta&
-\beta\hbox{sin}^2
\theta&0\cr-w&0&0&\gamma\cr}\right),$$ where $\alpha,\beta,\gamma, f$ and $w$ are functions of $r$ and $t$. In the new version of NGT, $w(r)$ does not satisfy the asymptotically flat boundary conditions in the limit $r\rightarrow\infty$[@Clayton]. Therefore, in the following, we shall set $w=0$ and only consider the unique two-parameter static spherically symmetric solution for $f$, first obtained by Wyman[@Wyman].
The NGT solution can be presented for $\beta=r^2$ as follows [@CornishMoffat1; @CornishMoffat2]:
$$\begin{aligned}
\label{gammaequation}
\gamma&=&e^\nu, \\
\alpha&=&\frac{M^2(\nu')^2 e^{-\nu}(1+s^2)}{(\cosh{(a\nu)}
-\cos{(b\nu)})^2},\\
\label{fequation}
f&=&\frac{2M^2e^{-\nu}[\sinh{(a\nu)}\sin{(b\nu)}
+s(1-\cosh{(a\nu)}\cos{(b \nu)})]}
{(\cosh{(a\nu)}-\cos{(b\nu)})^2},\\
a&=&\sqrt{{\sqrt{1+s^2}+1}\over 2},\\
b&=&\sqrt{{\sqrt{1+s^2}-1}\over 2}.\end{aligned}$$
Here, $\nu$ is given by the relation: $$\label{nuequation}
e^\nu (\cosh{(a\nu)}-\cos{(b \nu)})^2 {r^2\over {2M^2}} =
\cosh{(a\nu)}\cos{(b \nu)}-1 +s\sinh{(a\nu)}\sin{(b\nu)}.$$
For $2M/r \ll 1$ and $0 < sM^2/r^2 < 1$, the $\alpha, \gamma$ and $f$ take the approximate forms ($\mu^{-1} \gg 2M$): $$\gamma\approx\alpha^{-1}\approx 1-\frac{2M}{r},\, f\approx \frac{sM^2}{3}.$$ This result guarantees that all the experimental tests of EGT, based on the exterior point source Schwarzschild solution, are valid for large $r$ and a suitably chosen value of the parameter $s$.
In order to study the analytical structure of the static Wyman solution, we shall consider the expansion of the solution in a power series in $s$ about the Schwarzschild solution. Namely, we need to represent the nonsymmetric $g_{\mu\nu}$ solution in the following way: $$\label{difference}
g_{\mu \nu}=g_{\mu \nu}^S + \Delta_{\mu \nu},$$ where $g_{\mu \nu}^S$ is the Schwarzschild metric tensor and $\Delta_{\mu \nu}$ is the sought difference. The points where such an expansion does not exist will be specific points at which the NGT solution is a non-analytic function of the parameter $s$.
Let us consider first the difference $\Delta_{\mu \nu}$ inside the horizon ($r\leq
2M$). As was shown in[@CornishMoffat1; @CornishMoffat2], the diagonal components of $g_{\mu \nu}$ do not change their signs when crossing the point $r=2M$. Such a behavior does not depend on the parameter $s$. At the same time, the metric tensor $g^S_{\mu \nu}$ changes sign when crossing the horizon. It means that it is impossible to realize the sought expansion of the NGT solution near the Schwarzschild solution for $r\leq
2M$. However, such an expansion exists when we consider the analytical continuation of diagonal elements of $g_{\mu \nu}$ to negative values. This continuation exists on the field of complex numbers. In order to perform the continuation, it is enough to consider $g_{\mu\nu}$ as a complex-valued function, keeping the radius $r$, mass $M$ and the constant $s$ real.
We shall demonstrate the technique of analytical continuation for the example of the $g_{44}$-component of $g_{\mu \nu}$. It is seen from Eq.(\[gammaequation\]) that to have a negative $\gamma$, one should consider a pure imaginary $\nu$. For example, if $s=0$ (Schwarzschild solution), it leads to a real negative $\gamma$ for $r<2M$. Provided $s\not=0$ (NGT solution), the continuation for $\gamma$ becomes a complex valued function. To find this analytical continuation, one needs to use the known equality $$\ln \gamma = \ln |\gamma| +{\rm i}(2\pi k + \varphi ),$$ where $\varphi$ is the phase of $\gamma$ and $k$ is an integer number. Then, one sees that the continuation will be a multivalued function. Hereafter, for the sake of simplicity, let us put $k=0$.
Now one may find the continuation by solving Eq.(\[nuequation\]) with respect to complex $\nu$ (the connection between $\nu$ and $\gamma$ is given by Eq.(\[gammaequation\])). Regrettably, there is no explicit solution of this equation. Let us recall, however, that the region of interest is near $s=0$. So, we shall consider the analytical continuation of $\gamma$ for small $s$. Expanding $\gamma$ in a power series in $s$: $$\label{ser}
\gamma = c_0 + c_1 s + c_2 s^2 + \dots$$ and substituting (\[difference\]) into (\[nuequation\]), we can find the equations for the coefficients $c_0,\ c_1 \ \dots$. As a result, we get $$\label{delta_4}
\Delta_{44}=\gamma -1+\frac{2m}{r}=-(d_1+{\rm i}\ d_2)s^2+O[s^4],$$ where $$\begin{aligned}
d_1&=&\frac{\pi^2}{8}\;\biggl(\frac{M}{r}\biggr)\,\biggl[
1-3\biggl(\frac{r}{M}\biggr)
+\frac{3}{2}\biggl(\frac{r}{M}\biggr)^2+
\biggl(1+3\biggl(\frac{r}{M}\biggr)-6\biggl(\frac{r}{M}\biggr)^2\biggr)
\ln\biggl(\frac{2M}{r}-1\biggr)\biggr],\\
d_2&=&\frac{\pi}{8}\;\biggl(\frac{M}{r}\biggr)\,\biggl[3
-3\biggl(\frac{r}{M}\biggr)
+\biggl(-2+6\biggl(\frac{r}{M}\biggr)-3\biggl(
\frac{r}{M}\biggr)^2\biggr) \ln\biggl(\frac{2M}{r}-1\biggr)\biggr].\end{aligned}$$
It should be stressed that we have made no assumptions about the value of $r$ apart from $r \leq 2M$. Therefore, the expansion (\[delta\_4\]) must be valid for any $r \leq 2M$. One can then see that the continuation does not exist at the two points $r=0$ and $r=2M$, i.e., at the points of the origin and the horizon (logarithmic features). This behavior has been confirmed by an exact (numerical) calculation.
The same conclusion can be obtained for $\alpha$ and $f$. Using the aforementioned technique, we find for the difference between $\alpha^{-1}$ for NGT and the Schwarzschild solution: $$\begin{aligned}
\alpha^{-1} - (1 - \frac{2M}{r})=\frac{Ms^2}{16r}\biggl[
-44+28\frac{M}{r} +\pi^2 \biggl(6- 24\frac{M}{r}
+25\biggl(\frac{M}{r}\biggr)^2-7\biggl(\frac{M}{r}\biggr)^3\biggr)
\nonumber \\
+\biggl(22 -24\biggl(\frac{M}{r}\biggr)-22\biggl(\frac{M}{r}\biggr)^2
+14\biggl(\frac{M}{r}\biggr)^3\biggr)
\ln{\biggl(\frac{2M}{r} -1\biggr)}
\nonumber \\
+{\rm i}\pi\biggl\{34 -72\biggl(\frac{M}{r}\biggr)
+28\biggl(\frac{M}{r}\biggr)^2
+
(-12 +48\biggl(\frac{M}{r}\biggr)-50\biggl(\frac{M}{r}\biggr)^2\nonumber \\
+14\biggl(\frac{M}{r}\biggr)^3\biggr)\ln\biggl(\frac{2m}r -1\biggr)
\biggr\}\biggr]+O[s^4],\end{aligned}$$ and $$f\equiv{\Delta_{23}\over \sin{\theta}}={-\Delta_{32}\over \sin{\theta}}=
-\frac{1}{2}r^2 s\biggl[2-\biggl(\frac{r}{M}
-1\biggr)\ln\biggl(\frac{2m}{r}-1\biggr)
+{\rm i}\pi\biggl(\frac{r}{M}-1\biggr)\biggr] + O[{s}^5].$$ It should be noted that, in contrast to the other components of the analytical continuation of $g_{\mu\nu}$, the function $f$ is regular at $r=0$.
Now we shall consider the analytical structure of the difference beween the NGT and Schwarzschild solutions with $r\geq 2M$. There is no point in perfoming the continuation in the way that was done before, because the difference exists in real numbers. We can find it by substituting (\[ser\]) into Eqs.(\[gammaequation\])-(\[fequation\]). The result of the calculation is given by $$\begin{aligned}
\Delta_{44}&=&\frac{rs^2}{16 M [-4 +3 \left(\frac{r}{M}\right)^2]
\ln\biggl(1-\frac{2M}{r}\biggr)} + O[s^4],\\
\alpha^{-1}-1+{2M\over r}&=& \frac{Ms^2}{8r}
\biggl[22-14\frac{r}{M}\biggl(-11+12\biggl({r\over M}\biggr)\nonumber\\
&&+6\biggl({r\over M}\biggr)^2
-7\biggl({r\over M}\biggr)^3 \biggr)\ln{\biggl(1-{2M \over r}\biggr)}\biggr]
+O[s^4], \\
f&\equiv&{\Delta_{23}\over \sin{\theta}}\nonumber \\
&=&{-\Delta_{32}\over \sin{\theta}}=
\frac{1}{2} r^2 s\biggl[\biggl(1-\frac{r}{M}\biggr)\ln{\biggl(1-{2M \over r}
\biggr)}-2\biggr] +O[s^3].\end{aligned}$$
It should be noted that the asymptotics of these equations for $r/M \gg 2$ are in agreement with those presented in Refs.[@CornishMoffat1; @CornishMoffat2]. This result can also be corroborated by numerical calculations.
We reach the following conclusion: the difference between the NGT solution and the Schwarzschild solution is a regular function, complex-valued in the open range $0< r/M <2$ and real-valued in the range $r/M > 2$, it has a non-analytic logarithmic behaviour near $r/M=0,2$, i.e., at the origin and at the Schwarzschild horizon. In particular, this means that it is impossible to match smoothly the Schwarzschild and NGT solutions in the neighborhood of $r/M=0,2$ for any value of the parameter $s\not=0$. A small first order static $g_{[\mu\nu]}$ on a Schwarzschild background [*is not a global solution of the NGT static vacuum field equations*]{}. Moreover, if the time dependent part of $g_{[\mu\nu]}$ is small and is a smooth function on a Schwarzschild background, then this is not expected to be a global solution either for $s\not=0$.
We must now consider the matching of the interior and exterior solutions during the collapse of a star. The interior and exterior solutions match at the surface of the star, $r=r_0$, if we have[@Moffat1]
$$\begin{aligned}
\gamma(r_0,t)&=&\gamma_{\rm ext}(r_0,t),\\
\alpha(r_0,t)&=&\alpha_{\rm ext}(r_0,t),\\
\beta(r_0,t)&=&\beta_{\rm ext}(r_0,t),\\
f(r_0,t)&=&f_{\rm ext}(r_0,t),\end{aligned}$$
where $\gamma_{\rm ext}, \alpha_{\rm ext}, \beta_{\rm ext}$ and $f_{\rm ext}$ denote the non-vanishing components of $g_{\mu\nu}(r,t)$ for the exterior time dependent vacuum solution of the NGT field equations.
We shall expand the exterior time dependent solution as
$$\begin{aligned}
\gamma_{\rm ext}(r,t)&=&\gamma_{\rm ext}(r)+\delta\gamma_{\rm ext}(r,t),\\
\alpha_{\rm ext}(r,t)&=&\alpha(r)_{\rm ext}+\delta\alpha_{\rm ext}(r,t),\\
\beta_{\rm ext}(r,t)&=&\beta_{\rm ext}(r)+\delta\beta_{\rm ext}(r,t),\\
f_{\rm ext}(r,t)&=&f_{\rm ext}(r)+\delta f_{\rm ext}(r,t).\end{aligned}$$
Because we do not have an exact dynamical solution of the vacuum NGT field equations, let us assume that $\delta\gamma_{\rm ext}, \delta\alpha_{\rm ext},
\delta\beta_{\rm ext}$ and $\delta f_{\rm ext}$ are small quantities that can be neglected, without encountering any discontinuities when going to limit of the static part of $g_{\mu\nu}$. We shall refer to this as the quasi-static approximation of the vacuum field equations.
We have assumed that there exists a non-vanishing generic coupling of $g_{[\mu\nu]}$ to the matter composing the star and that there is no NGT neutral body in nature, i.e., the coupling (\[coupling\]) is always non-zero in the presence of a matter source. This guarantees the existence of a static exterior $g_{\mu\nu}$ which is given for a spherically symmetric star by the Wyman solution. Burko and Ori[@BurkoOri] assumed that $f(r,t)$ was small at the beginning of the collapse of a star and [*continued to be small of first order throughout the collapse*]{} (with the possible exception of $r=0$). Due to the assumption of the smallness of $f(r,t)$, they were implicitly assuming that the static Schwarzschild background solution dominated the collapse for $r \leq 2M$, i.e., they assumed that the exterior metric was described by a quasi-static approximation. We have proved that this cannot be a global solution of the NGT field equations; in particular, it fails to be a solution for $0 < r \leq 2M$.
From the static Wyman solution, we know that $f(r)$ has to be greater than 1 in Cartesian coordinates for $r\sim 2M$[@CornishMoffat1; @CornishMoffat2]. Therefore, according to our approximation scheme, the quasi-static solution for $f$ is also expected to give $f > 1$ in Cartesian coordinates. It follows that the claim by Burko and Ori that black holes can form in NGT for small enough $f$ fails to be true, for it is based on the validity of the linear approximation equation[@BurkoOri]: $$\label{flinear}
\frac{1}{2}\biggl(\frac{\ddot{f}}{\gamma}-\frac{f''}{\alpha}\biggr)
+\frac{f'}{\alpha
r}+\frac{1}{2}\frac{f'\alpha'}{\alpha^2}-2\frac{f\alpha'}{\alpha^2r}
=0,$$ which reduces in the limit $\dot{f}\rightarrow 0$ to the static equation for $f$[@Clayton; @Cornish]. The same arguments hold for the form of Eq.(\[flinear\]) in Kruskal-Szekeres coordinates[@BurkoOri].
An analogous situation holds in EGT, if we expand the metric tensor about Minkowski flat space. To first order we obtain the metric: $$ds^2=(1-\frac{2M}{r})dt^2-(1+\frac{2M}{r})dr^2-r^2(d\theta^2
+\sin^2\theta d\phi^2).$$ As the star collapses and $r\rightarrow 2M$, the linear expansion breaks down and we are required to solve the non-linear EGT field equations.
It is possible that there exists one exotic situation, namely, that there is no static part to $f$, i.e., $f$ is a purely source-free wave solution of the NGT field equations. Since we do not have a rigorous wave-type solution of the NGT field equations, we cannot at present know whether such a solution correctly describes the collapse problem near the Schwarzschild radius, $r\sim 2M$, when it is restricted to the linear approximation. A numerical solution of the NGT field equations may shed some light on this question. However, we expect that for a non-vanishing generic coupling of $f$ to matter, a static part of $f$ should exist for realistic collapse, and therefore our arguments about the non-formation of black holes would follow. We anticipate that black holes do not form, in NGT, for a general dynamical solution of the field equations.
This work was supported by the Natural Sciences and Engineering Research Council of Canada. We thank M. A. Clayton, N. J. Cornish, L. Demopoulos and M. Reisenberger for helpful discussions.
S. W. Hawking and R. Penrose, Proc. R. Soc. London A [**314**]{}, 529 (1970); S. W. Hawking and G. F. R. Ellis, Large Scale Structure of Space-Time, Cambridge University Press, 1973. N.J.Cornish and J.W.Moffat, Phys.Lett. [**B336**]{} 337 (1994). N.J.Cornish and J.W.Moffat, J.Math.Phys. [**35**]{}, 6628 (1994). J. W. Moffat, University of Toronto Report No. astro-ph/9510024. J. W. Moffat, Phys. Lett. [**B355**]{}, 447 (1995). J. W. Moffat, J. Math. Phys. [**36**]{}, 3722 (1995); [*errata*]{}, J. Math. Phys., to be published. J. W. Moffat, J. Math. Phys., to be published. J. Légaré and J. W. Moffat, Gen. Rel. Grav. [**27**]{}, 761 (1995). M. A. Clayton, J. Math. Phys., to be published. L. M. Burko and A. Ori, Phys. Rev. Lett. [**75**]{}, 2455 (1995). M. Wyman, Can. J. Math. [**2**]{}, 427 (1950). N. J. Cornish, University of Toronto Report No. gr-qc/9503034 (unpublished).
|
---
abstract: 'We present a compressive sensing approach for the long standing problem of Matsubara summation in many-body perturbation theory. By constructing low-dimensional, almost isometric subspaces of the Hilbert space we obtain optimum imaginary time and frequency grids that allow for extreme data compression of fermionic and bosonic functions in a broad temperature regime. The method is applied to the random phase and self-consistent $GW$ approximation of the grand potential and integration and transformation errors are investigated for Si and SrVO$_3$.'
author:
- Merzuk Kaltak
- Georg Kresse
bibliography:
- 'MyReferences.bib'
title: Minimax Isometry Method
---
Introduction: Matsubara Technique {#sec:MatsubaraTechnique}
=================================
The Matsubara technique is a way to formulate quantum field theory at finite-temperature. More precisely, it makes use of the Wick rotation[@PhysRev.96.1124], which transforms the real time axis of Minkowski spacetime to the imaginary time axis $t\to-i\tau$. Because real space remains unchanged by this transformation, spacetime becomes essentially euclidean, so that this approach is also known as euclidean quantum field theory (QFT). A summary on euclidean QFT is given in the appendix. Here we focus on the Matsubara method.
As Matsubara has shown, the imaginary time-integrals in finite-temperature perturbation theory are restricted to the interval $-\beta < \tau <
\beta$.[@Matsubara1955] This has the advantage that one can expand the imaginary time-dependence of the corresponding integrands into a Fourier series, such that imaginary-time integration becomes essentially an (infinite) series over discrete Fourier coefficients. The corresponding discrete frequencies are known as [*Matsubara frequencies*]{} and it is important for us to distinguish between [*fermionic*]{}, denoted by $\omega_n$ in the following, and [*bosonic*]{} Matsubara frequencies, denoted by $\nu_m$ in the remainder of this paper.
Fermionic Matsubara frequencies represent the non-zero Fourier modes of fermionic functions, while bosonic frequencies are the non-zero modes of bosonic functions. This is explained in more detail below by means of the free-electron Green’s function (Feynman propagator) and the irreducible polarizability, the building blocks of finite-temperature perturbation theory.
The free Feynman propagator in imaginary time $-i\tau$ represents a prototype of a fermionic function. In a one-electron basis the free propagator is diagonal $g_{\alpha\gamma}(-i\tau)=\delta_{\alpha\gamma}g_\alpha(-i\tau)$ and the entries read[@Negele1988; @Fetter2003] $$\label{eq:DefG0}
g_\alpha(-i\tau)=e^{-(\epsilon_\alpha-\mu)\tau}
\left[
(1-f_\alpha)\Theta(\tau)-f_\alpha\Theta(-\tau)
\right],$$ where $\epsilon_\alpha$, $\mu$ and $f_\alpha$ are the one electron energy, the chemical potential and the Fermi function (see Equ. \[eq:DefFermi\]), respectively. Here $\Theta$ is the Heaviside step function,[@NIST] and it is the reason why $g_\alpha(-i\tau)$ changes sign at $\tau=0$. Also, the presence of the step functions implies $$\label{eq:FermionicProperty}
g_\alpha(-i\beta+i\tau) = -g_\alpha(i\tau),\quad 0<\tau< \beta.$$ This anti-symmetric property has an important effect on the Fourier series in the interval $-\beta<\tau<\beta$ $$\begin{aligned}
\label{eq:FermionicSeries}
g_\alpha(-i\tau)&=\frac1\beta\sum\limits_{m=-\infty}^\infty \tilde
g_\alpha(i\omega_m)e^{-i\omega_m \tau}\\
\label{eq:FermionicCoefficients}
\tilde g_\alpha(i\omega_m)&=\int_{-\frac\beta2}^{\frac\beta2}
\mathrm{d}\tau
g_\alpha(-i\tau)e^{i\omega_m \tau},\end{aligned}$$ because it contains only fermionic frequencies $$\label{eq:FermionicFrequencies}
\omega_m=\frac{2m+1}{\beta}\pi,\quad m\in\mathbb{Z}.$$ This holds true for all fermionic functions on the imaginary time axis, including the self-energy (see appendix \[sec:ApproximationsToGrandPotential\]).
An example for a bosonic function is the independent particle polarizability, which is diagonal $\chi_{\alpha\gamma\alpha'\gamma'}=\delta_{\alpha\alpha'}\delta_{\gamma\gamma'}\chi_{\alpha\gamma}$ and has the entries $$\label{eq:IPPolarizability}
\chi_{\alpha\gamma}(-i\tau)=-
g_{\alpha}(-i\tau)g_{\gamma}(+i\tau).$$ However, in contrast to Equ. , bosonic functions do not change sign in imaginary time, but are symmetric $$\label{eq:BosonicProperty}
\chi_{\alpha\gamma}(-i\beta+i\tau) = \chi_{\alpha\gamma}(+i\tau),\quad 0<\tau<
\beta.$$ Consequently, the Fourier expansion $$\begin{aligned}
\label{eq:BosonicSeries}
\chi_{\alpha\gamma}(-i\tau)&=\frac1\beta\sum\limits_{n=-\infty}^\infty
\tilde \chi_{\alpha\gamma}(i\nu_n)e^{-i\nu_n \tau}\\
\label{eq:BosonicCoefficients}
\tilde \chi_{\alpha\gamma}(-i\nu_n)&=\int_{-\frac\beta2}^{\frac\beta2}
\mathrm{d}\tau
\chi_{\alpha\gamma}(-i\tau)e^{i\nu_n \tau}\end{aligned}$$ contains only bosonic frequencies $$\label{eq:BosonicFrequencies}
\nu_n=\frac{2n}{\beta}\pi,\quad n\in\mathbb{Z}.$$ It is often argued that bosonic (fermionic) functions are periodic (anti-periodic) in $\tau$. This is strictly speaking not correct. The free propagator, for instance, is defined [*a priori*]{} only in the fundamental imaginary time interval $|\tau|\le\beta$, because grows or decays exponentially for arguments outside (green lines in [Fig. \[fig:FermionicAndBosonicTimeDependence\]]{}). The same holds true for the irreducible polarizability. In fact, only the Fourier expansions and define periodic and anti-periodic functions in $\tau$ with (anti-) period $\beta$. This behavior is illustrated in [Fig. \[fig:FermionicAndBosonicTimeDependence\]]{} showing a typical fermionic and bosonic function. In practice, it is quite important to recall that exponentially growing terms in propagators are not present, because the $\tau$-integrations are performed over $0<\tau<\beta$ or, equivalently, over $-\beta/2 < \tau<\beta/2$. Due to consistency with our previous papers[@Kaltak2014; @Kaltak2015; @PhysRevB.94.165109] we work in the $[-\beta/2,\beta/2]$ interval. Furthermore, we mean by correlation function either a bosonic or fermionic function.
![(left) Fermionic function $g_1+g_2$ with $\epsilon_1=0.19,
\epsilon_2=9.12$ eV and $\beta=1$ eV$^{-1}$ in the fundamental interval (green line) and its corresponding Fourier series truncated after $m>10$ (blue line). (right) Corresponding bosonic function. Analytic fermionic and bosonic functions either increase or decrease exponentially for $|\tau|>\beta$ ([*e.g.*]{} Equ. ), while the corresponding Fourier expansion outside $[-\beta,\beta]$ is (anti-) periodic. []{data-label="fig:FermionicAndBosonicTimeDependence"}](FermionicAndBosonicTimeDependence.pdf){width="3.25in"}
The Matsubara summation has one major drawback. Matsubara series converge very slowly with the cutoff frequency, as shown in \[sec:FermionicFrequencyGrid\]. One thus requires compressed Matsubara representations of bosonic and fermionic correlation functions. For this purpose we studied isometries in metric spaces and developed the following formalism.
Minimax Isometry {#sec:MinimaxIsometry}
================
To keep the notation simple, we consider the case $\beta=1$ eV$^{-1}$ in the following. The general case $\beta\neq1$ follows from scaling relations that are discussed in \[sec:TimeGrid\] and \[sec:BosonicFrequencyGrid\].
Deriving an alternative to the Matsubara technique requires one additional abstraction level. For this purpose we write $x$ for the energies $\epsilon_\alpha-\mu$ and assume that $a\le x \le b$ ($b\to\infty$ is allowed in the following). Every energy $x$ is represented by a vector $\ket*{x}$ in a Hilbert space $\mathcal{H}$ and it is assumed that $\ket*{\tau}$ and $\ket*{n}$ are two complete basis sets in imaginary time and frequency, such that the identity operator ${1\!\!1}$ can be expressed as $$\begin{aligned}
\label{eq:Completeness1}
{1\!\!1}=&\int_0^{1/2}\mathrm{d}\tau \ket*{\tau}\bra*{\tau}\\
\label{eq:Completeness2}
{1\!\!1}=& \sum_{n\in\mathbb{Z}}\ket*{n}\bra*{n}\end{aligned}$$ for all $x\in[a,b]$ with $\mathbb{Z}$ representing the set of integers. From a functional analysis perspective, one says that the two spaces $\mathcal{U}=\mathrm{span}\lbrace \ket*\tau\rbrace_{\tau\in[0,1/2]}$ and $\mathcal{V}=\mathrm{span}\lbrace\ket*{n}\rbrace_{n\in\mathbb{Z}}$ are isometric w.r.t. to the scalar product induced norm $\|x\|_2=\sqrt{\mel*{x}x}$. This isometry is effectively a simple basis transformation that does not change the induced norm, since $$\label{eq:Conservation}
\begin{split}
\|x\|_2^2 = \mel*{x}{x} = &
\int_0^{1/2}\mathrm{d}\tau\mel*{x}\tau\mel*\tau{x}\\
=&
\sum_{n\in\mathbb{Z}}\mel*{x}n\mel*{n}{x}.
\end{split}$$ If $\ket*\tau$ is the time and $\ket*{n}=\ket*{\omega_n}$ the frequency basis, then $\mel*\tau{n}$ and $\mel*{n}\tau$ are the matrix elements of the forward and backward basis transformation, respectively as for instance given in Eqs. ,, and .[^1] Consequently, the two spaces $\mathcal{U}$ and $\mathcal{V}$ are equivalent and span the same Hilbert space $\mathcal{H}$. This equivalence holds true only if infinitely many basis vectors are considered; for finite dimensional subspaces the perfect isometry is violated.
One may illustrate the violation of the isometry with the discrete Fourier transform (DFT) having the bases $\ket*{\tau_k}=\ket*{\frac{1}{2N}(k-\frac12)}$ with $(k=1,\cdots,N)$ and $\ket*{n}=\ket{2\pi(n-1)}$ (truncated bosonic Matsubara grid). The corresponding completeness relations , become projectors onto finite dimensional subspaces $U\subset\mathcal{U},V\subset\mathcal{V}$ and have the form $$\begin{aligned}
\label{eq:DFTCompleteness1}
P=&\frac1{2N}\sum\limits_{k=1}^N\ket*{\tau_k}\bra*{\tau_k}\\
\label{eq:DFTCompleteness2}
\tilde{P}=& \sum_{n=-N}^N\ket*{n}\bra*{n}.\end{aligned}$$ Only in the limit $N\to\infty$ the projectors approach the identity operator ${1\!\!1}$. For finite $N$, the isometry is violated, but can be replaced by a so-called $\varepsilon$-isometry[@Fleming2002; @Ding1988] $$\label{eq:EpsilonIsometry}
\| P - \tilde P\|_\infty:=\max_{a\le x\le b}|\bra*{x} P - \tilde P \ket*{x}| \le
\varepsilon.$$ In the following, we will drop the subscript and write $\|\cdot\|$ for the maximum norm $\|\cdot\|_\infty$.
Of interest to us is the magnitude of $\varepsilon$ and especially how it decreases with increasing $N$. For instance, in the case of the DFT $\bra*{x}P\ket*{x}$ is the Riemann sum of the integral in of order $N$ and is known to be a poor method to evaluate integrals. As a consequence, $\varepsilon$ of the Matsubara grid is a weakly decaying function in $N$ and cannot be used for our purposes, as shown in section \[sec:MinimaxFourierTransformation\].
The following question naturally arises: how can one determine $\varepsilon$-isometric subspaces $U=\mathrm{span}\lbrace\ket*{\tau_k}\rbrace_{k=1}^N$, and $V=\mathrm{span}\lbrace\ket*{\omega_k}\rbrace_{k=1}^N$, such that the completeness relations and are approximated as good as possible for all vectors $\ket*{x}$ with $a\le x\le b$?
Using the notation in , the answer to this question are the solutions of following minimax problems: $$\begin{aligned}
\label{eq:MMU}
\min_{\sigma_k>0,\tau_k\in[0,1/2]} &
\left\|
{1\!\!1}- \sum\limits_{k=1}^N\sigma_k \ket*{\tau_k}\bra*{\tau_k}
\right\|
\\
\label{eq:MMV}
\min_{\lambda_k>0,\omega_k>0} &
\left\|
{1\!\!1}- \sum\limits_{k=1}^N\lambda_k \ket*{\omega_k}\bra*{\omega_k}
\right\|.\end{aligned}$$ Provided the solutions exist, they are known to yield errors $\varepsilon$ that decay exponentially with $N$.[@Braess1986] In the following we prove that and satisfy our requirements.
To prove the assertion above it suffices to show that the minimax errors are an upper bound for the isometry violation in . Therefore, assume $\lbrace\sigma_k^*,\tau_k^*\rbrace_{k=1}^N$ and $\lbrace\lambda^*_k,\omega_k^*\rbrace_{k=1}^N$ are the solutions of and and $P^*=\sum_{k=1}^N\sigma^*_k\ket*{\tau_k^*}\bra*{\tau_k^*}$ and $\tilde
P^*=\sum_{k=1}^N\lambda^*_k\ket*{\omega_k^*}\bra*{\omega_k^*}$ the corresponding projectors, respectively. Then a positive number $\varepsilon/2$ exists (for every given $N$) as an upper bound for and and one can write $$\label{eq:ProofEq1}
\begin{split}
\Big\|
{1\!\!1}- P^*
\Big\|
\le &\frac12 \varepsilon\\
\left\|
{1\!\!1}- \tilde P^*
\right\| =
\left\|
\tilde P^*-{1\!\!1}\right\|
\le &\frac12 \varepsilon.
\end{split}$$ Adding both inequalities in and using the triangle inequality $ \|f+g\|\le\|f\|+\|g\|$ (satisfied by every norm[@Boto2013]) one obtains $$\label{eq:ProofEq2}
\begin{split}
\underbrace{
\Big\|
{1\!\!1}-P^*
\Big{\|}
+
\left\|
\tilde P^*-{1\!\!1}\right\|
}_{
\left\|
{1\!\!1}-P^*+\tilde P^*-{1\!\!1}\right\|
\le
} \le \varepsilon.
\end{split}$$ Last inequality implies for the projectors $P^*,\tilde P^*$ and concludes our proof.
This is a quite remarkable result, because it means that the projectors $P$ and $P^*$ converge to the identity operator and, thus, define $\varepsilon$-isometric topological vector spaces $U^*,V^*$ that have the [*approximation property*]{}.[@Schaefer1999] A summary of $\varepsilon$-isometric bases is given in Tab. \[tab:IsometricBases\] and its selection is motivated below.
[0.49]{}[@ccccc]{} group & &$\mel*{\tau}x$ & $\mel*{\omega}x$ & $\| x\|_2^2$\
\
IA$_1$ & b & $\frac12\frac{\cosh\frac{x}{2}(1-2|\tau|)}{\cosh\frac{x}{2}}$ & $\frac{x\tanh\frac{x}{2}}{x^2+\omega^2}$ &\
IA$_2$ & f & $\frac{\mathrm{sgn}(\tau)}{2}\frac{\cosh\frac{x}{2}(1-2|\tau|)}{\cosh\frac{x}{2}}$ & $\frac{\omega}{x^2+\omega^2}$\
\
\
IB$_1$ & f &$\frac12\frac{\sinh\frac{x}{2}(1-2|\tau|)}{\cosh\frac{x}{2}}$ & $\frac{x}{x^2+\omega^2}$ &\
IB$_2$ & b &$\frac{\mathrm{sgn}(\tau)}{2}\frac{\sinh\frac{x}{2}(1-2|\tau|)}{\cosh\frac{x}{2}}$ &$\frac{\omega\tanh\frac{x}{2}}{x^2+\omega^2}$ &\
\
\
IC$_1$ & b & $\frac12 e^{-|x\tau|} $ & $\frac{|x|}{x^2+\omega^2}$ &\
IC$_2$ & f & $ \frac{\mathrm{sgn}(\tau)}{2}e^{-|x\tau|} $ & $\frac{\omega}{x^2+\omega^2}$ &\
\
\[tab:IsometricBases\]
Note that the discussion above does not give a prescription how to determine the transformation $U^*\to V^*$; a corresponding method is presented in section \[sec:MinimaxFourierTransformation\].
The proof above contains only an upper bound for the transformation error in . This upper bound $\varepsilon$ is inherited from the sum of the convergence rate of the minimax solutions in the $\tau$- and $\omega$-domain. This convergence rate has been studied by Braess and Hackbusch for the minimax problem IC in the $\tau$-domain listed in Tab. \[tab:IsometricBases\]. They obtained $\varepsilon(N)\approx6.7\log(2+N)e^{-\pi\sqrt{2N}}$ for $x\in [1,R_N]$, where $[1,R_N]$ belongs to the largest possible error for a given order $N$.[@Braess2005] Our numerical experiments discussed in section \[sec:Results\] indicate similar convergence rates for all other minimax problems in Tab. \[tab:IsometricBases\]. In contrast, the DFT or Matsubara grid has only a linear rate of convergence $\varepsilon(N)\propto N^{-1}$.
Motivation of basis functions {#sec:BasisFunctions}
-----------------------------
In this subsection, we motivate our choice of $\varepsilon$-isometric basis functions tabulated in Tab. \[tab:IsometricBases\]. We start with the IC bases, which has been used in previous publications by the authors to construct optimized minimax grids for low scaling random phase and $GW$ algorithms at zero temperature.[@Kaltak2014; @Kaltak2015; @PhysRevB.94.165109] Specifically, $\mel*{\tau}x$ of IC$_1$ describes the imaginary time dependence of the independent particle polarizability at zero temperature, while the corresponding $\omega$-basis functions describe its imaginary frequency dependence.[@Kaltak2014] Then isometric bases are the ideal choice for the zero-temperature formalism of quantum field theory, because the corresponding conserved $L^2$-norm (forth column of Tab. \[tab:IsometricBases\]) is the key quantity for the second order contributions to the correlation energy (see appendix or Ref. ). These contributions involve energy denominators of the form $1/(\epsilon_a+\epsilon_b-\epsilon_i-\epsilon_j)$ and are considered to be bound, [*i.e.*]{} virtual states with energies $\epsilon_a,\epsilon_b$ are separated by a band gap $\Delta$ from occupied states with energy $\epsilon_i,\epsilon_j$. The induced norm is important and essentially tells the optimization in the minimax problem, which contributions to the energy are most relevant. In our case contributions from small energy differences dominate over contributions from large energy differences.
The $\tau$-basis function of IC$_2$ has the same time dependence (apart from the opposite sign), while the imaginary frequency dependence of the cosine transformation differs considerably from the one obtained from the sine transform (compare $\mel*{\omega}x$ of IC$_1$ and IC$_2$). It comes with no surprise that the corresponding minimax grids for $\tau$ and $\omega$ differ too.[@PhysRevB.94.165109] However, the minimax isometry guarantees that one can map in time between IC$_1$ and IC$_2$ with high precision; a fact that has been exploited in a low scaling self-consistent $GW$ approach at zero temperature.[@PhysRevB.98.155143]
![(left) Fermionic basis functions $\mel*{\tau}{x}$ IA$_2$ (green line) and IB$_1$ (blue line) in imaginary time for $x=10$. (right) Corresponding bosonic basis functions IA$_1$ (green) and IB$_2$ (blue). []{data-label="fig:BasisFunctions"}](BasisFunctions.pdf){width="3.25in"}
Next, we consider the four basis functions of group IA and IB. They can be grouped into bosonic (IA$_1$ and IB$_2$) and fermionic (IA$_2$ and IB$_1$) pairs, where the $\omega$-bases for bosonic (fermionic) functions are defined for bosonic (fermionic) frequencies [*a priori*]{} (see next section for additional motivation of these functions). The symbol $\mel*{\omega}x$ denotes the Fourier transformations (cosine or sine) of these functions. When optimizing the frequency grids using the minimax algorithm, we allow $\omega$ in IA and IB to deviate from the corresponding Matsubara grid. Indeed, the corresponding Minimax solutions are non-uniformly distributed, but nevertheless closely match Matsubara frequencies at small $\omega$. It turns out, as shown in section \[sec:Results\], that this freedom allows us to describe the high frequency behavior of the correlation functions with high precision even in low dimensional subspaces $U^*, V^*$. The corresponding $\varepsilon$-isometric time basis functions have the fermionic anti-symmetry \[Equ. \] and bosonic symmetry \[Equ. \] for $-\beta/2\le \tau\le\beta/2$, respectively, and are illustrated in [Fig. \[fig:BasisFunctions\]]{}.
Similar to the zero-temperature case, the conserved $L^2$-norm describes the second order contribution to the correlation part of the grand-canonical potential (see appendix \[eq:DefMP2GrandPotentialTime\]). Thus, it can be employed for data compression when calculating the correlation part of the grand canonical potential in the random phase approximation (see appendix \[sec:ApproximationsToGrandPotential\]). The corresponding imaginary time and frequency grid are discussed in \[sec:TimeGrid\] and \[sec:BosonicFrequencyGrid\], respectively.
There is one further important point to note here. Some bosonic functions, such as the polarizability or screened potential can be entirely presented by IA$_1$ basis functions since $\chi(-i\tau)=\chi(-i\beta+i\tau)$ (green lines in [Fig. \[fig:BasisFunctions\]]{} right). This means that the second order part of the grand-canonical potential or the RPA correlation energy can rely on grids constructed for IA$_1$, only. This is not necessarily so for bosonic functions constructed e.g. by the product of the self-energy and Green’s functions, as is the case in the Galitskii-Migdal (GM) formula.
In fact, the minimax isometry method is not straightforwardly applicable to the GM formula of the grand canonical potential; here both, even and odd basis functions in the frequency domain (IB$_1$ and IA$_2$) contribute to the grand potential and have different $L^2$-norms (compare third column of IA and IB). We, therefore, propose an alternative approach in this case that is based on the minimization of the $L^1$-quadrature error instead (see section \[sec:FermionicFrequencyGrid\]).
Imaginary time grid {#sec:TimeGrid}
-------------------
To construct an imaginary time grid, we make use of the scaling properties $$\label{eq:ScalingTime}
\tau_j \to \beta \tau_j, \quad \sigma_j
\to \beta \sigma_j$$ that allow to recover the time quadrature for an arbitrary interval $[0,\beta/2]$ from the unscaled solution determined for $[0,1/2]$. Furthermore, we allow for infinitesimal small excitation energies $x=0$ and accordingly set $a=0$. The minimization interval is then characterized by one parameter $b$, as for the IC quadrature.[@Kaltak2014]
We now discuss how to choose the imaginary time grid. This remains a somewhat involved issue even with the yet gained insight. The main problem is that we want to use one and only one time grid, since this will allow us to represent the Green’s functions and the bosonic quantities on the same time grid permitting one to calculate for instance the polarizability as $g(-i \tau)g(+i
\tau)$ on that single grid. The even and odd basis functions of the IA and IB $\varepsilon$-isometry in Tab. \[tab:IsometricBases\] are obviously identical for fermions and bosons for positive $\tau$, $$\begin{aligned}
\label{eq:BasisTimeEven}
\phi(x,\tau)=&\frac12\frac{\cosh\frac{x}{2}(1-2\tau)}{\cosh\frac{x}{2}},\quad
\tau>0\\
\label{eq:BasisTimeOdd}
\psi(y,\tau)=&\frac12\frac{\sinh\frac{y}{2}(1-2\tau)}{\cosh\frac{y}{2}},\quad
\tau>0.\end{aligned}$$ Even without considering the related norm in Tab. \[tab:IsometricBases\], these functions seem to be a handy and intuitive choice, since they can be used to express the diagonal elements of the free propagator of $$\label{eq:G0InTermsOfEvenAndOdd}
g_\alpha(-i\tau) =
\psi\left(\beta\epsilon_\alpha,\frac{|\tau|}\beta\right)+\mathrm{sgn}(\tau)\phi\left(\beta\epsilon_\alpha,\frac{|\tau|}\beta\right)$$ as well as the matrix elements of the polarizability . Optimization of the time grid for even and odd functions, however, yields different time grids. As already motivated in the previous section, we have decided to use the optimal even time grid (IA$_1$) as a common grid for both fermionic and bosonic functions and repeat the arguments here. (i) The second order and RPA correlation energy depends only on bosonic functions, e.g. the polarizability. Hence the fermionic functions are only used at an intermediate stage. (ii) The polarizability possesses a special symmetry $\chi(-i \tau) = \chi(-i\beta + i \tau)$ matching the IA$_1$ $\varepsilon$-isometry in Tab. \[tab:IsometricBases\]. (iii) Finally, the imaginary time grid for the even functions $\phi$ yields a small minimax error also for the odd basis functions $\psi$ for the entire interval $x\in[0,b]$, with negligible deviations even for $x\to 0$. This follows from the fact that the corresponding $L^2-$ norms differ only for very small arguments (compare third column of IA and IB in Tab. \[tab:IsometricBases\]). Consequently, we solve the minimax problem only for $\phi_j(x)=\phi(x,\tau_j)=\mel*{\tau_j}{x}$ and use the same time grid points for the odd basis functions.
To obtain the time grid points $\tau_j$, it is convenient to rewrite the minimax problem into the following form $$\label{eq:MinimaxProblemTime}
\min_{\sigma_j>0,\tau_j\in(0,1/2)}
\max_{0\le x\le b}
\left|
\|x\|_2^2
-\sum\limits_{j=1}^{N}\sigma_j
\phi_j^2(x)
\right|$$ with $b=\beta\epsilon_{\max}$ and $\epsilon_{\max}$ the maximum one-electron energy considered. Then it becomes evident that is a non-linear fitting problem of separable type,[@Golub2003] which in general has only a solution, if every basis function $\phi_j$ is linearly independent and has less than $N-1$ zeros. The alternant theorem then implies[@Braess1986; @Haemmerlin1994] a set of points $\lbrace x^*_j\rbrace_{j=0}^{2N}$ (alternant) and a set of non-linear equations $$\label{eq:AlternantTheorem}
\|x_j^*\|_2^2-\sum\limits_{k=1}^N\sigma^*_k|\mel*{\tau_k^*}{x_j^*}|^2=E_N(-1)^j$$ with $$\label{eq:ErrorN}
E_N=\pm
\max_{a\le x\le b}
\left|
|\mel*\phi{x}|^2-\sum\limits_{k=1}^N\sigma_k\phi_k(x)\right|$$ being positive (negative) if the l.h.s. of is positive (negative) at $x=x_0^*$.
The alternant theorem provides the basis for the non-linear Remez algorithm that has been used successfully in other papers and yields the minimax solution $\lbrace\sigma_j^*,\tau_j^*\rbrace_{j=1}^N$.[@Braess2005; @Hackbusch2008; @Kaltak2014] The minimax solution yields abscissas in the unscaled interval $0\le\tau^*_j\le\frac12$. Furthermore, the corresponding weights $\sigma^*_j$ are positive and $\lim_{N\to\infty}\sum_{j=1}^N \sigma^*_j =1$ holds true. This is important for the application in many-body theory, since the conservation of particles is guaranteed with increasing $N$ including particles with energy $\epsilon_\alpha\approx\mu$.
Furthermore, the same minimax quadrature can be used to evaluate “off-diagonal terms”, since errors of the form $$\label{eq:2DErrorTime}
\eta(x,y)=\mel*{x}{y}
-\sum\limits_{k=1}^N\sigma^*_j \phi^*_k(x)\phi^*_k(y)$$ are exponentially suppressed with increasing quadrature order $N$. An example is given in [Fig. \[fig:2DTimeError\]]{} that demonstrates how the error is uniformly distributed in the entire region $(x,y)\in[a,b]\times[a,b]$ and bounded for $x=y=0$.
![Time error function \[Equ. \] of order $N=6$ for $b=100$ in two dimensions. For better visibility only the $[-20,20]\times[-20,20]$ region of the error surface is shown. []{data-label="fig:2DTimeError"}](2DTimeError.pdf){width="3.25in"}
Before we discuss the construction of the frequency grids, we note that a similar basis $e^{-x y}/\cosh\frac{x}2$ has been recently used to compress Green’s functions on the imaginary time axis in quantum Monte Carlo algorithms.[@PhysRevB.96.035147] We believe that there is a close connection to our method, since we found that the quadrature obtained from the solution of is also a good approximation to the solution for the corresponding problem for the odd basis function and both, $\psi$ and $\phi$ linear combined yield Shinaoka’s basis. However, Shinaoka [*et al.*]{} determine the grid as the solution of an integral equation and the connection to $\varepsilon$-isometric subspaces is not immediately evident. We, therefore, prefer to solve the non-linear optimization problem instead.
Bosonic Frequency Grid {#sec:BosonicFrequencyGrid}
----------------------
To construct the bosonic Matsubara grid we use the $\varepsilon$-isometric basis of the even time basis , specifically again the IA$_1$ basis $$\label{eq:BasisFreqEvenBos}
\tilde\phi_n(x)=\frac{x}{x^2+\nu_n^2}\tanh\frac{x}2.$$ The motivation behind this choice is three-fold. Firstly, it is obtained from the cosine transformation of the even time basis evaluated for bosonic Matsubara frequencies $\nu_n=2n\pi$. Thus it describes the imaginary frequency dependence of the polarizability that is of bosonic nature; the IA$_2$ basis is obtained from the sine transform of these functions and is evaluated at fermionic frequencies and hence irrelevant for the evaluation of bosonic integrals. Secondly, we can use the minimax isometry method to switch between the frequency and time representation of the polarizability with high precision. This follows from the theorem proved in section \[sec:MinimaxIsometry\] Equ. . Lastly, the infinite bosonic Matsubara series of the RPA grand potential can be evaluated with high precision without using any interpolation technique.
In practice, the unscaled bosonic frequency quadrature for $\beta=1$ is determined first and following scaling relations are used to obtain the result for arbitrary inverse temperatures $$\label{eq:ScalingFreq}
\nu_k \to \frac{\nu_k}\beta, \quad \lambda_k \to \frac{\lambda_k}\beta.$$
The corresponding minimax problem reads $$\label{eq:MinimaxProblemA1}
\min_{\lambda_k>0,\nu_k\in(0,\infty)}
\max_{0\le x\le b}
\left|
\|x\|_2^2- \sum\limits_{k=1}^{N}\lambda_k \tilde\phi^2_k(x)
\right|$$ where the $L^2$-norm is given in Tab. \[tab:IsometricBases\]. The solution $\lbrace \lambda_k^*,\nu_k^*\rbrace_{k=1}^N$ is called IA$_1$-quadrature in the following, in agreement with the notation used in Tab. \[tab:IsometricBases\].
Before we investigate the convergence of the IA$_1$-quadrature, we discuss the construction of the fermionic grid.
Fermionic Frequency Grid {#sec:FermionicFrequencyGrid}
------------------------
In this section, we discuss the construction of a compressed fermionic Matsubara quadrature. More precisely, we look for a quadrature that is converging exponentially with the number of grid points $N$ for the GM expression for the grand potential . In contrast to the polarization function and second order correlation energies, this requires an accurate handling of fermionic functions of the type IA$_2$ and IB$_1$ in the frequency domain, which is an intricate problem.
First, we consider the frequency dependence of the free propagator . The cosine and sine transformations of the odd and even time basis functions and for fermionic frequencies are determined as: $$\begin{aligned}
\label{eq:BasisFreqEvenFer}
{\rm IB}_1: \eqref{eq:BasisTimeOdd} \rightarrow u_m(x)=&\mel*{u_m}{x}=\frac{x}{x^2+\omega_m^2}\\
\label{eq:BasisFreqOddFer}
{\rm IA}_2: \eqref{eq:BasisTimeEven} \rightarrow v_m(x)=&\mel*{v_m}{x}=\frac{\omega_m}{x^2+\omega_m^2}.\end{aligned}$$ Then the non-interacting propagator on the fermionic Matsubara axis reads $$\label{eq:G0InFrequency}
\tilde g_\alpha(i\omega_m) =
u_m(\epsilon_\alpha-\mu)+ i v_m(\epsilon_\alpha-\mu).$$ Second, we observe that every fermionic function can be decomposed into terms that are even and odd in $\omega$, including the product of the propagator and self-energy as it appears in the GM grand potential . It is, obvious, that only the real part of the product $\tilde G\tilde \Sigma$ contributes to the total energy. Thus the most general matrix element, which gives a non-zero contribution to the GM grand potential has the form[^2] $$\label{eq:GMMatrixElement}
\begin{split}
\tilde G(i\omega_m)\tilde\Sigma(i\omega_m)=&\sum\limits_{-\Omega_{\max}\le x,y\le \Omega_{\max}}A(x)B(y)\\
\times&\left[u_m(x)u_m(y) -
v_m(x)v_m(y)\right],
\end{split}$$ where $x$ and $y$ are the poles of the Green’s function and the self-energy on the real-frequency axis and $A,B$ the spectral densities, respectively. Without loss of generality, we set $A=B=1$ and assume that the magnitudes of the poles are smaller than a positive number, [*i.e.*]{} $|x|,|y|\le \Omega_{\max}$.
Third, we note that the analogue of the IA$_1$-quadrature of bosonic functions for fermionic ones $$\label{eq:MinimaxProblemFerF2}
\min_{\sigma_k>0,\omega_k\in(0,\infty)}
\max_{0\le x\le b}
\left|
\|x\|^2_2 - \sum\limits_{k=1}^{N}\sigma_k u^2_k(x)
\right|$$ yields the IB$_1$-quadrature, see Tab.\[tab:IsometricBases\]. Unfortunately, the IB$_1$-quadrature only allows to evaluate the first term on the r.h.s. of accurately, but fails for the product of two odd functions $v$. Similarly, the IA$_2$-quadrature obtained from the minimax problem $$\label{eq:MinimaxProblemFerA2}
\min_{\sigma_k>0,\omega_k\in(0,\infty)}
\max_{0\le x\le b}
\left|
\|x\|^2_2 - \sum\limits_{k=1}^{N}\sigma_k v^2_k(x)
\right|$$ that approximates the same norm as the time and IA$_1$-quadrature, describes only the second term in . Consequently, neither the IB$_1$- nor the IA$_2$-quadrature can be used for our purposes.
A solution to this dilemma can be found by approximating the $L^1$-norm instead $$\label{eq:L1NormDef}
\|x\|_1=\sum\limits_{m\in\mathbb{Z}}|\mel*{u_m}x|=\frac12\tanh\frac{|x|}{2}.$$ A proof of this identity is found in the appendix \[app:ProofTanh\]. That is, we determine the solution of $$\label{eq:MinimaxProblemF}
\min_{\gamma_k,\omega_k>0}
\max_{0\le x\le b}
\left|
\|x\|_1-\sum\limits_{k=1}^{N}\gamma_k
u_k(x) \right|.$$ Because $\|x\|_2\le\|x\|_1$ holds true for any $0\le x\le b$,[@Fleming2002] the $L^1$ solution $\lbrace\gamma_k^*,\omega_k^*\rbrace_{k=1}^N$, called F-quadrature in the following, yields linearly independent basis functions $u^*_k$ that span a larger vector space than the basis obtained from . Our numerical experiments discussed below show that the F-quadrature evaluates the infinite sum over both terms in with high precision for increasing $N$.
A similar technique was employed by Ozaki[@Ozaki2007], who used the same basis functions $u$, but determined the abscissas from the partial fraction decomposition of the hyperbolic tangent in combination with a continued fraction representation of the hypergeometric function $_1F_0$ to derive a compressed form of .
Both, the F-quadrature as well as Ozaki’s hypergeometric quadrature (OHQ) use essentially a rational polynomial approximation to the hyperbolic tangent. In the following, we show why this approach provides also a good approximation of fermionic Matsubara series, such as the the density matrix for holes (upper sign) and electrons (lower sign) $$\label{eq:DensityMatrix}
\Gamma =\pm\lim_{\eta\to0\pm} G(-i\eta)=\pm\lim_{\eta\to0\pm}\frac1\beta\sum\limits_{n=-\infty}^{\infty}
\tilde G(i\omega_n)e^{-i\omega_n\eta}$$ where $G$ and $\tilde G$ is the interacting Green’s function in imaginary time and on the Matsubara axis, respectively. Specifically, we show that the last expression on the r.h.s. of can be approximated with the following quadrature formula $$\label{eq:DensityMatrixQuadrature}
\Gamma \approx \frac{\mathrm{sgn}(\eta)}{2}{1\!\!1}+ \sum\limits_{k=1}^N
\frac{\sigma_k}2\left[\tilde G(i\omega_k)+\tilde G(-i\omega_k)\right],$$ where ${1\!\!1}$ is the identity matrix in the considered basis and $\sigma_k,\omega_k$ are either the OHQ- or F-quadrature points.
![ Integration contours in (zigzag line) branch cut of $\tilde A$, (crosses) fermionic Matsubara frequencies $\omega_n$ correspond to poles $z=i\omega_n$ of auxiliary function $h_\eta(z)$ defined in and such that contour integral $\oint\mathrm{d}z \tilde A(z) h_\eta(z)$ for path $\mathcal{C}$ is zero. []{data-label="fig:Contour"}](Contour.pdf){width="3.25in"}
To understand Equ. and, in general, why an approximation to the hyperbolic tangent provides an excellent approach to compress fermionic Matsubara series, we consider a general correlation function $\tilde A$ that is analytic in the complex plane $z$ with a branch cut on the real axis and decays with $\mathcal{O}(|z|^{-1})$ or faster to zero for $|z|\to\infty$; for instance $\tilde G(z)$ or $\tilde G(z) \tilde \Sigma(z)$. Following Fetter and Walecka,[@Fetter2003] one introduces an auxiliary function $h_\eta(z)$ with an infinitesimal $\eta$ to force the complex contour integral over the infinite large outer circle $\mathcal{C}$ in [Fig. \[fig:Contour\]]{} to vanish, that is $$\label{eq:ComplexContourC}
\oint_{\mathcal{C}}\frac{\mathrm{d}z}{2\pi i}\tilde A(z)h_\eta(z) =0.$$ Regardless of the specific choice of $h_\eta(z)$ (discussed below), one can easily show using the residue theorem and the contours depicted in [Fig. \[fig:Contour\]]{} following identities: $$\label{eq:ComplexContourIdentities}
\begin{split}
\sum\limits_{n\in\mathbb{Z}}\operatorname*{Res}_{z=i\omega_n}
\left[\tilde A(z)h_\eta(z)\right]
=&\oint_{\mathcal{F}}\frac{\mathrm{d}z}{2\pi i}\tilde A(z)h_\eta(z)\\
=&-\oint_{\mathcal{B}}\frac{\mathrm{d}z}{2\pi i}\tilde A(z)h_\eta(z)\\
=&\int_{-\infty}^\infty \mathrm{d}\omega\frac1\pi\mathrm{Im}\left[\tilde
A(\omega)h_\eta(\omega)\right]
\end{split}$$ Apart from condition , the auxiliary function $h_\eta(z)$ has to be chosen such that the l.h.s. in gives the fermionic Matsubara series $\sum_{n\in\mathbb{Z}}\tilde A(i\omega_n)e^{-i\omega_n\eta}$, which imposes two conditions on $h_\eta$. Firstly, $h_\eta$ must have an infinite number of poles located at $z=i\omega_n$ (crosses in [Fig. \[fig:Contour\]]{}). Secondly, the corresponding residue has to be $\pm A(i\omega_n)e^{-i\omega_n\eta}$ for $\eta\to0\pm$.
If $\tilde A(z)$ is of order $\mathcal{O}(|z|^{-1-\delta}),\delta>0$ for $|z|\to\infty$ the contour integral is zero also for constant functions $h_\eta(z)$ in $\eta$ and one can choose $$\label{eq:AuxiliaryTanh}
\begin{split}
h_\eta(z)=&\frac12\tanh\frac{z}2\\
=&\frac12\sum\limits_{n\in\mathbb{Z}}\left[\frac1{z-i\omega_n}+\frac1{z+i\omega_n}\right],
\end{split}$$ where the last line follows from and shows the locations of the poles of $h_\eta(z)$. Thus for $\tilde A(z)=\tilde G(z)\tilde \Sigma(z)$ the fermionic Matsubara series is independent of the directional limit $\eta\to0\pm$. The replacement of the hyperbolic tangent by a rational polynomial with poles on the imaginary axis in shows that both terms in the GM energy can be well approximated using the F-quadrature.
In contrast, for correlation functions $\tilde A$ that decay only with $\mathcal{O}(|z|^{-1})$ at $|z|\to\infty$ the sign of the infinitesimal $\eta$ matters. This includes $\tilde G(z)$ as well as any mean field terms, [*e. g.*]{} Green’s function times the mean field Hamiltonian $\tilde G(z) H_0$. For instance, the limit $\eta\to0-$ in gives the electron density matrix, while $\eta\to0+$ gives the density matrix of holes. For functions of order $\mathcal{O}(|z|^{-1})$ at $|z|\to\infty$ one, therefore, has to add a term to the hyperbolic tangent. As can be shown easily, the form for $h_\eta(z)$ for which holds true is[^3] $$\label{eq:AuxiliaryForG}
h_\eta(z)=\left[\frac{\mathrm{sgn(\eta)}}{2}+\frac12\tanh\frac{z}2\right]e^{-z\eta}.$$ Inserting Equ. into the r.h.s. of yields $$\label{eq:ComplexContourDensity}
\begin{split}
\sum\limits_{n\in\mathbb{Z}}
\tilde A(i\omega_n)e^{-i\omega_n\eta}=&
\frac{\mathrm{sgn}(\eta)}2
\int_{-\infty}^\infty
\mathrm{d}\omega\frac1\pi\mathrm{Im}\left[\tilde A(z)e^{-z\eta}\right] \\
+&
\int_{-\infty}^\infty
\mathrm{d}\omega\frac1\pi\mathrm{Im}\left[\tilde
A(z)\frac12\tanh\frac{z}2e^{-z\eta}\right],
\end{split}$$ In the last term on the r.h.s. the evaluation of the limit $\eta\to 0$ can be performed before integration, because the integrand is of order $\mathcal{O}(|z|^{-2})$ for $|z|\to\infty$ \[see Equ. \]. The corresponding integral over the arch $\mathcal{C}$ vanishes, so that the last term in on the r.h.s. can be rewritten into the Matsubara series of $\tilde A$ that is independent of the sign of $\eta$. This is the [*convergent part*]{} of the Matsubara series and the term that can be evaluated using quadratures. Note that the first term on the r.h.s. of cannot be written into a Matsubara series, because the integrand diverges for $|z|\to\infty$ prohibiting the closure of the integration contour at infinity. However, for $\tilde A(z) = \tilde G(z)$ one has $$\label{eq:DivergentTermExplicit}
\lim_{\eta\to0\pm}\frac{\mathrm{sgn}(\eta)}{2}
\int_{-\infty}^\infty
\mathrm{d}\omega\frac1\pi\mathrm{Im}\left[\tilde G(z)e^{-z\eta}\right] =\pm
\frac{1}{2}{1\!\!1}.$$ This concludes our proof of Equ. . We call this term, therefore, the [*divergent part*]{} of the Matsubara series, although the term is finite in any practical calculation (number of electrons/holes is finite in practice). We use for the evaluation of the density matrix in self-consistent $GW$ calculations at finite temperature (see section \[sec:Results\]).
In summary, one can say that the approximation of the hyperbolic tangent by rational polynomials with poles only on the imaginary axis gives rise to fermionic Matsubara quadratures that describe the convergent part of the Matsubara representation. To obtain the directional limits $\eta\to0\pm$ of weakly decaying correlation functions, such as the propagator of electrons or holes, the integral over the spectral function has to be added or subtracted, respectively. The evaluation of the GM energy does not require this term, because $\tilde G(z)\tilde \Sigma(z)$ decays with $\mathcal{O}(|z|^{-2})$. Analogous bosonic quadratures can be obtained by approximating the $L^1$-norm of the hyperbolic cotangent, but this was not further investigated.
![Convergence of Matsubara grid (points), hypergeometric grid (diamonds) and F-grid (squares) for inverse temperatures $\beta=1,10,100$ eV$^{-1}$ (small, medium, large symbols). []{data-label="fig:MatsubaraConvergence"}](MatsubaraConvergence.pdf){width="3.25in"}
We have compared our F-quadrature with Ozaki’s hypergeometric quadrature (OHQ) by means of calculating the GM factor for a model that includes 50 randomly sampled poles in $-0.05\le x,y\le 0.05$ and 50 poles in $-50\le x,y\le 50$. The results for $\beta=1,10$ and $100$ eV$^{-1}$ are shown in [Fig. \[fig:MatsubaraConvergence\]]{} and are contrasted to the grid convergence for the ordinary fermionic Matsubara quadrature $\lbrace
\gamma_m=2,\omega_m=(2m+1)\pi/\beta\rbrace_{m=0}^\infty$. It can be seen that the F-quadrature outperforms the OHQ in all cases, especially for $\beta>1$ (low temperatures). This can be explained by the fact that the F-grid minimizes the quadrature error for all energies uniformly in the interval $|x|,|y|\le \Omega_{\max}$. The corresponding OHQ-quadrature error is non-uniformly distributed in the same interval and has the effect that at high $\beta$ values the convergence is almost linear with the number of grid points for small $N$. The same figure, also shows the linear convergence of the conventional Matsubara grid and demonstrates its pathology in practice.
Minimax Isometry Transformation {#sec:MinimaxFourierTransformation}
-------------------------------
We have seen how different basis functions for the time and frequency domain give rise to different grids. In this section we study the error made by transforming an object represented on the time grid $\lbrace\tau_1^*,\cdots,\tau^*_N\rbrace$ to the frequency axis. As a measure for the transformation error we use $$\label{eq:TransformationErrorPhi}
\tilde E(\omega)=
\min_{t_{\omega k\in\mathbb{R}}}
\left\|\mel*\omega{x}-\sum_{k=1}^Nt_{\omega k}\mel*{\tau_k^*}x
\right\|_2^2,$$ where $\mel*{\tau^*_k}x$ is either the even or odd time basis function , evaluated at the minimax time grid obtained from and $\mel*{\omega}x$ acts as a placeholder for one of the basis functions in the frequency domain listed in Tab. \[tab:IsometricBases\] with $\omega$ being a positive real number. The $L^2$-norm is evaluated by sampling the $x-$values with 100 points $X_j^*$ determined from the alternant $\lbrace x_j^*\rbrace_{j=0}^{2N}$ of the minimax time problem in and additional $(101-2N)/2N$ uniformly distributed points in each of the $2N$ sub-intervals $[x_j^*,x_{j+1}^*]$.
The solution of the ordinary least square problem for the frequency $\omega$ is then given by the corresponding normal equation[@NR2007] $$\label{eq:TransformationErrorSolution}
\begin{split}
&\sum_{i=1}^{100}\mel*\omega{X_i^*}\mel*{X_i^*}{\tau_k^*} = \\
&\sum_{j=1}^{N}t_{\omega
j}\sum_{i=0}^{100}\mel*{\tau_j^*}{X_i^*}\mel*{X_i^*}{\tau_k^*},\quad
k=1,\cdots,N.
\end{split}$$ The transformation error is rather insensitive to changes of the number of sampling points $X_j^*$; the $2N+1$ alternant points $x_j^*$ of the time grid also often suffice in practice.
Furthermore, Equ. allows to plot $\tilde E(\omega)$ as a function of the frequency. This gives independent insight, on which frequencies one is supposed to use in combination with a certain set of time basis functions, independent of the previous considerations (see [Fig. \[fig:TransformationErrors\]]{}).
Transformation to the IA$_1$ frequency basis functions (blue line), clearly shows that the error $\tilde E(\omega)$ is minimal at the previously determined IA$_1$ frequency points (blue triangles), and transformation to the IA$_2$ frequency basis functions (green line) shows that the error is smallest at the previously determined IA$_2$ frequency points (green diamonds). The reason for this behavior is due to the fact that the IA$_1$, IA$_2$ and the time quadrature for the even time basis $\phi(x,\tau)$ \[Equ. \] possess the same approximation property and span $N$-dimensional, almost isometric subspaces of the Hilbert space $\mathcal{H}$ as proven in section \[sec:MinimaxIsometry\]. The good agreement is a numerical confirmation that the previously determined frequency grids are optimal.
Thus, for polarizabilities and second order correlation energies, the optimal frequency points are clearly the IA$_1$-quadrature points (triangles). They approach the conventional bosonic frequencies $2n \pi$ \[Equ. \] at small $\omega$. The corresponding weights (not shown) approach $2$ (except for the first frequency $\nu^*_1=0,~\lambda^*_1=1$). Higher quadrature frequency points (as well as weights) deviate considerably from the conventional bosonic Matsubara points $\lbrace \nu_m = 2\pi m,
\lambda_m=2\rbrace_{m=0}^\infty$. This behavior is very similar to the bosonic grid presented by Hu [*et al.*]{} that is based on the continued fraction decomposition of the hyperbolic cotangent, the analogue of Ozaki’s method for bosons.[@Hu2010] However, the IA$_1$-quadrature has the advantage that the error is minimized uniformly for all transition energies $|x|\le \beta
\epsilon_{\max}$, while the continued fraction method yields non-uniformly distributed errors in general.
![ Transformation error $\tilde E(\omega)$ from even time basis functions to the IA$_1$ (blue) and IA$_2$ (green) frequency basis for $N=16$, $x_{\max}=1000$. The inset shows the low frequency regime. Points indicate minimax grid points in frequency domain for IA$_1$, IA$_2$, IB$_1$ and F (the abscissa corresponds to the optimal frequency, whereas the ordinate is given by the error $\tilde E(\omega)$ determined at the respective frequency point).[]{data-label="fig:TransformationErrors"}](TransformationErrors.pdf){width="3.25in"}
Transformation from the even time basis $\phi(x,\tau)$ \[Equ. \] to the fermionic frequency basis IA$_2$ yields further important insight. As already emphasized the minimax IA$_2$ frequency points match exactly those frequency points where the error for transformation into the IA$_2$ basis functions is minimal. On the other hand, the IB$_1$ and F minimax grid points are chosen to optimally present odd time basis functions $\psi(x,\tau)$ \[Equ. \] using the corresponding frequency basis \[Equ. \]. At small frequencies, these points are slightly shifted away from the optimal IA$_2$ frequency points, causing somewhat larger transformation errors for even functions to IA$_2$ basis functions. This is to be expected, since the points have been chosen to approximate a different scalar product than for IA$_2$ (and IA$_1$), see fourth column in Tab. . Specifically, the IB$_1$ minimax frequencies are by construction optimal to present odd time basis functions. Although, IB$_1$ and IA$_2$ minimax points are close at low frequencies, they progressively move away at higher frequencies, which prohibits the construction of a common frequency grid that can present the GM energy well (Sec. \[sec:FermionicFrequencyGrid\]). From this figure, it is somewhat unclear, why the F frequency grid works well, although it is noteworthy that the corresponding frequency points lie roughly at the positions where IA$_1$ and IA$_2$ errors intersect. This might implying an equally acceptable representation of odd and even functions.
### $\varepsilon$-isometric time grids of the F-quadrature {#sec:EpsilonIsometricTimeGrid}
To avoid ambiguity, we call the time grid defined in section \[sec:TimeGrid\] the IA time grid in the following, while the IB time grid is the same minimax solution , but with norm $\|x\|_2$ and basis function $\mel*{\tau}x$ of the IB isometry listed in Tab. \[tab:IsometricBases\].
Recapitulating the previous section, a natural question arises: Is there an optimum time grid for the F-quadrature? In analogy, to Equ. this grid may be defined by the minima of the inverse transformation error $$\label{eq:InverseTransformationError}
E(\tau)=
\min_{t_{\tau k\in\mathbb{R}}}
\left\|\mel*\tau{x}-\sum_{k=1}^Nt_{\tau k}\mel*{\omega_k^*}x
\right\|_2^2,$$ where $\omega_k^*$ are the abscissa of the F-quadrature. Table \[tab:IsometricBases\] shows that there are two possible choices for the time and frequency basis functions $\mel*{\tau}{x},\mel*{\omega_k^*}{x}$; one function describing the transformation error from odd to even basis functions (Equ. to Equ. ) and one from Equ. to Equ. . Both functions are plotted in [Fig. \[fig:TransformationErrorsF\]]{} (blue and green line), respectively.
![ Transformation error $E(\tau)$ from frequency F-grid to time domain for the IA$_2$ (blue line) and IB$_1$ (green line) $\varepsilon$-isometric basis functions in Tab. \[tab:IsometricBases\] for $N=6$, $x_{\max}=100$. Triangles and diamonds indicate the IA and IB time grids, respectively. []{data-label="fig:TransformationErrorsF"}](TransformationErrorsF.pdf){width="3.25in"}
The figure clearly shows that the minima of both error functions differ and implies two $\varepsilon$-isometric time grids for the frequency F-grid. This is analogous to the forward transformation errors discussed in the previous section, where the A$_1$- and A$_2$-frequency grids are $\varepsilon$-isometric to the IA time grid, respectively. For the F-grid, however, the IA and IB minimax solutions in time coincide with the minima of the error functions only for $\tau\approx0$, for larger values of $\tau$ the transformation error minima ($\varepsilon$-isometric grids) deviate from the IA and IB minimax grid points, respectively (compare minima of green and blue curve with triangles and diamonds in [Fig. \[fig:TransformationErrorsF\]]{}).
The small IB transformation error for $\tau=0$ follows from the fact that for small $\tau$ the time basis becomes $\mel*{\tau}x=\psi(x,\tau)\approx\tanh x/2$. Per construction (see ), the hyperbolic tangent function is approximated well by the basis using the F-grid. The IA transformation error (blue line), in contrast, is several orders of magnitude larger at $\tau=0$, since the time basis function is constant $\mel*{\tau=0}x=\phi(x,0)=1$ and the frequency basis functions $\mel*{\omega_k}x$ represent constants only poorly. The deviation of the IA and IB time grids from the $\varepsilon$-isometric time grids at higher $\tau$ values is not surprising, since the F-grid deviates from the $\varepsilon$-isometric frequency grids, that is the A$_2$ and B$_1$-grid discussed in section \[sec:BosonicFrequencyGrid\] and \[sec:FermionicFrequencyGrid\].
In summary, we recommend using the IA time grid presented in section \[sec:TimeGrid\] in combination with the A$_1$ frequency grid for bosonic functions (see \[sec:BosonicFrequencyGrid\]) and the F-grid for fermionic functions. First, the exact $\varepsilon$-isometric time points of the F-grid is only known numerically from inspection of the transformation error; an analogue to the minimax isometry method is not known to us. Second, Green’s function in imaginary time can be contracted without error, whilst at the same time transformation errors to the imaginary frequency axis are controlled. Before we demonstrate these advantages in section \[sec:Results\], we discuss the following details about our implementation in the Vienna ab initio software package (VASP).[@PhysRevB.59.1758]
Technical Details {#sec:TechnicalDetails}
=================
The implementation of the finite temperature RPA and $GW$ algorithms is the same as the zero-temperature ones,[@Kaltak2015; @PhysRevB.94.165109] with three exceptions.
- The zero-temperature frequency grid is replaced by the IA$_1$-grid discussed in \[sec:BosonicFrequencyGrid\] for the bosonic correlation functions $\tilde\chi,\tilde{W}$, while the F-quadrature from \[sec:FermionicFrequencyGrid\] replaces the grid for the fermionic functions $\tilde{G}$ and $\tilde\Sigma$.
- All correlation functions are evaluated on the same imaginary time grid presented in sec. \[sec:TimeGrid\].
- The occupied and unoccupied Green’s function $\underline{G},\overline{G}$ need to be set up carefully considering the partial occupancies $f_\mu$ in each system.
The last point requires some clarification. The Green’s function for positive $\overline G$ and negative times $\underline{G}$ can be combined to a full Green’s function using Heaviside theta functions $$\label{eq:DefGreensFunction}
G(-i\tau) = \Theta(\tau)\overline{G}(\tau)-\Theta(-\tau)\underline{G}(\tau).$$ At zero temperature $(\beta\to\infty)$ the occupied and unoccupied imaginary time Green’s function read[@Rojas1995; @Kaltak2015] $$\begin{aligned}
\label{eq:DefGoccT0}
\underline{G}(\tau)|_{\beta=\infty}
=&\sum\limits_{\alpha}\Theta(\epsilon_F-\epsilon_\alpha)e^{-(\epsilon_\alpha-\epsilon_F)\tau}
\\
\overline{G}(\tau)|_{\beta=\infty}
=&\sum\limits_{\alpha}\Theta(\epsilon_\alpha-\epsilon_F)e^{-(\epsilon_\alpha-\epsilon_F)\tau}
\label{eq:DefGunoT0}.\end{aligned}$$ Here the Fermi energy $\epsilon_F$ makes sure that $\overline{G}$ and $\underline{G}$ contains only unoccupied (occupied) one-electron states. This changes as temperature increases, which follows from $\lim_{\beta\to\infty}\mu=\epsilon_F$ and the limit $$\label{eq:HeavisideTheta}
\Theta(\pm\epsilon_F\mp\epsilon_\alpha)
= \lim\limits_{\beta\to\infty}\frac{1}{e^{\pm\beta(\epsilon_\alpha-\mu)}+1},$$ implying the form given in for the full Green’s function . Consequently, the Green’s function $\underline G$ needs to include also partially occupied states at finite temperature and vise versa, so that the positive and negative imaginary time Green’s functions $$\begin{aligned}
\label{eq:DefGneg}
\underline{G}(\tau)
=&\sum\limits_{\alpha}f_\alpha e^{-(\epsilon_\alpha-\mu)\tau},\quad\tau<0\\
\overline{G}(\tau)
=&\sum\limits_{\alpha}(1-f_\alpha) e^{-(\epsilon_\alpha-\mu)\tau},\quad\tau>0
\label{eq:DefGpos}\end{aligned}$$ are determined instead and include all considered one-electron states.
We emphasize that for $-\beta\le \tau\le \beta$ there are no exponentially growing terms, in neither of the two Green’s functions, because of the simple property of the Fermi function $$\label{eq:FermiFunctionReversal}
f_\alpha = (1-f_\alpha)e^{-(\epsilon_\alpha-\mu)\beta}.$$ In agreement with the Feynman-Stückelberg interpretation of QFT, every occupied state ($\epsilon_\alpha<\mu$) in the positive time Green’s function $\overline{G}$ is essentially a state propagating negatively in time $$\label{eq:TimeReversalUnoccupiedStates}
(1-f_\alpha)e^{-(\epsilon_\alpha-\mu)\tau} =
f_\alpha e^{-(\epsilon_\alpha-\mu)(\tau-\beta)}, \quad 0<\tau<\beta$$ and vise versa for $\epsilon_\alpha>\mu$ and negative times $$\label{eq:TimeReversalOccupiedStates}
f_\alpha e^{-(\epsilon_\alpha-\mu)\tau} =
(1-f_\alpha)e^{-(\epsilon_\alpha-\mu)(\tau+\beta)}, \quad -\beta<\tau<0.$$ Note, that all time points of the constructed time grid in section obey $0<\tau_j<\frac\beta2$, such that the restrictions for $\tau=\pm\tau_j$ in and are never violated, respectively.
Computational details {#sec:Details}
---------------------
The results presented in the following section have been obtained with VASP using a $\Gamma$-centered k-point grid of $4\times4\times4$ sampling points in the first Brillouin zone. To be consistent with the QFT formulation Fermi occupancy functions are forced by the code for all finite temperature many-body algorithms (selected with `LFINITE_TEMPERATURE=.TRUE.`), that is `ISMEAR=-1` and the temperature in eV is set via the k-point smearing parameter `SIGMA`. All calculations have been performed with experimental lattice constants of $a=5.431$ Å for Si[@Levinstein1999] and $a=3.842$ Å for SrVO$_3$[@Onoda1991], respectively. For both, Si as well as SrVO$_3$ the non-normconserving $GW$ potentials released with version 5.4.4, specifically `Si_sv_GW, Sr_sv_GW, V_sv_GW` and `O_s_GW` have been used and energy cutoffs of 475.1 eV and 434.4 eV for the basis set have been employed, respectively. This allows us to study the grid convergence in the presence of semi-core states and yields results that can be extrapolated to normconserving potentials with higher cutoffs. The independent electron basis required for RPA and $GW$ calculations has been determined with density functional theory in combination with the Perdew-Burke-Ernzerhof functional[@Perdew.105.9982] and the $q\to0$ convergence corrections have been neglected. Furthermore, because the polarizability converges faster with the number of plane waves considered compared to the wavefunction,[@PhysRevB.77.045136] smaller energy cutoffs of 316.6 eV and 289.6 eV for $\chi$ (set using `ENCUTGW`) for Si and SrVO$_3$ have been chosen, respectively.
Results {#sec:Results}
=======
Performance of IA$_1$-quadrature for RPA
------------------------------------------
We have used the IA$_1$-quadrature to generalize our cubic scaling RPA algorithm[@Kaltak2015] to finite temperatures in order to calculate the RPA grand potential for SrVO$_3$ and Si.
We emphasize that in the limit $\beta\to\infty$ all basis functions approach the IC basis functions used in the zero-temperature RPA algorithms.[@Kaltak2014; @Helmich2016; @Beuerle2018] That is at $T$=0 K, the bosonic and fermionic grid merge to the same frequency grid. However, the zero- and finite-temperature grids can be contrasted only for systems with a finite band gap, since the grid convergence of the former breaks down for metallic systems (the $L^2$-norm of IC in Tab. \[tab:IsometricBases\] diverges for $x\to0$). In contrast, the IA$_1$-quadrature is valid for all systems (including metals) at all finite temperatures (with exception of the $\beta\to\infty$ limit). Hence, the Kohn-Luttinger conundrum[@PhysRev.118.41] is circumvented, since the thermodynamic limit is performed at finite temperatures. Consequently, a comparison of the grid convergence to our zero temperature implementation of the RPA is useful only for systems with a finite band gap at $T$=0 K, like for instance Si. The corresponding comparisons are given in [Fig. \[fig:SiRPA\]]{}.
![ Grid convergence of RPA grand potential for Si at different temperatures (or k-point smearings). (empty symbols) corresponding $\beta=\infty$ implementation with same k-point smearing applied in the preceding Kohn-Sham groundstate calculation. Inverse temperatures are in eV$^{-1}$. []{data-label="fig:SiRPA"}](RPA-Si-NDep.pdf){width="3.25in"}
The exponential grid convergence of the IA$_1$-quadrature for finite temperatures is evident (solid lines). The required number of points for a given precision increases with decreasing temperature, because the minimization interval increases linearly with $\beta$ and therefore the quadrature error increases too. Not surprisingly, a similar grid convergence rate is observed for paramagnetic SrVO$_3$ as demonstrated in [Fig. \[fig:SrVO3RPA\]]{}. This system is known to be computationally challenging, because of the presence of several degenerate, partially populated states around the chemical potential, even in the limit $\beta\to\infty$.
![ Grid convergence of RPA grand potential for SrVO$_3$ at different inverse temperatures. Inverse temperatures are in eV$^{-1}$. []{data-label="fig:SrVO3RPA"}](RPA-SrVO3-NDep.pdf){width="3.25in"}
Comparing the IA$_1$-convergence rate with the zero temperature grid convergence for Si, a similar slope is observed for $\beta=100$ eV$^{-1}$, see empty triangles in [Fig. \[fig:SiRPA\]]{}. However, more IA$_1$-grid points for the same precision as in the $T=0$ case are required. The zero temperature quadrature, presented in another work of the authors,[@Kaltak2014] outperforms the finite temperature grid at $\beta=100$ eV$^{-1}$ corresponding to a sharp k-point smearing of $\beta^{-1}=0.01$ eV. The reason is that the zero temperature grid is “aware” of the band gap and designed to integrate that as well as the largest excitation energies exactly. The finite temperature grid is designed to work between 0 and the largest excitation energy (at a given $\beta$). As the temperature increases, partial occupancies are introduced. This has the effect that the exponential convergence rate of the $T=0$ grid deteriorates and causes the $T=0$ RPA algorithm even to converge towards a wrong limit that differs from the finite-temperature implementation (not shown). Only for $\beta=100$ eV$^{-1}$ we observed that both, the zero- and finite-temperature RPA implementations converge to the same result. This is not surprising, because as $\beta$ becomes smaller more states with energy around $\epsilon_F$ become fractionally populated; those states are described incorrectly by the zero-temperature algorithm. Thus, we recommend to use the finite-temperature RPA algorithm for systems with a small or zero band gap.
Performance of F-quadrature for $GW$
--------------------------------------
Next we study the grid convergence of the F-quadrature for Si and paramagnetic SrVO$_3$ by means of calculating the GM grand potential in the self-consistent $GW$ approximation. For demonstration purposes, we have fixed the chemical potential $\mu$ in the interacting Green’s function and self-energy to the value of the non-interacting Green’s function. Consequently, the interacting Green’s function for negative $\tau$ describes a system with a different number of electrons $N_e$ in the unit cell than the non-interacting counterpart.[^4]
![F-grid convergence for GM grand potential Si. Inverse temperatures are in eV$^{-1}$.[]{data-label="fig:GMSi"}](GM-Si-NDep.pdf){width="3.25in"}
![F-grid convergence for GM grand potential SrVO$_3$. Inverse temperatures are in eV$^{-1}$.[]{data-label="fig:GMSrVO3"}](GM-SrVO3-NDep.pdf){width="3.25in"}
The results for different values of $\beta$ of Si and SrVO$_3$ are given in [Fig. \[fig:GMSi\]]{} and [Fig. \[fig:GMSrVO3\]]{}, respectively. One recognizes that the grid convergence is very similar for both systems. Nevertheless, the convergence is worse compared to the RPA, because the F-quadrature error is larger compared to the IA$_1$-error.
However, our discussion in \[sec:MinimaxFourierTransformation\] shows that it is the best choice and the price one has to pay in order to use the same time grid for bosonic and fermionic functions. For practical applications, the error of roughly 1 $\mu$eV with 16 and more quadrature points is negligible. Other convergence parameters, such as the energy cutoff of the basis set, typically yield larger errors.[@PhysRevB.90.075125]
Last, we consider the electron number conservation of the F-quadrature, that is the difference of $|N_e-N'_e|$, where $N_e$ is the exact number of electrons in the unit cell and $N'_e$ has been calculated from the trace of Equ. . We have studied the non-interacting propagator $\tilde g_\alpha$ of SrVO$_3$. This corresponds to roughly $1056\times64$ poles of the Green’s function on the real-frequency axis in the regime $|x|\le 400\beta$. The error in the particle number with the number of F-quadrature points is shown in [Fig. \[fig:ChargeConservationSrVO3\]]{}.
![Particle number conservation error $|N_e-N_e'|$ of F-grid when calculating the electron density from the non-interacting Kohn-Sham propagator of SrVO$_3$ (see text). Inverse temperatures are in eV$^{-1}$.[]{data-label="fig:ChargeConservationSrVO3"}](N-SrVO3-NDep.pdf){width="3.25in"}
One can see that the convergence is exponential and increases and decreases with $\beta$ in the same way as the RPA and GM energies. Not surprisingly, the convergence is the same as compared to the case where the GM energy is used as measure (see [Fig. \[fig:GMSrVO3\]]{}). Also, the F-quadrature converges faster with the number of grid points compared to the OHQ-quadrature (not shown). For instance, the F-quadrature yields a precision of $10^{-10}$ states per unit cell for $\beta=10$ using $N=20$ quadrature points, while the same precision is reached with $N=118$ OHQ-quadrature points.
Conclusion {#sec:Conclusion}
==========
We presented an efficient method for the Matsubara summation of bosonic and fermionic correlation functions on the imaginary frequency axis. By constructing optimum subspaces of the considered Hilbert space of dimension $N$, we obtained imaginary time and frequency grids for all correlation functions appearing in finite-temperature perturbation theory. Furthermore, using the argument of $\varepsilon$-isometric spaces, we have shown that the transformation from imaginary time to imaginary frequency can be performed with high precision.
We implemented this technique in VASP to generalize our zero-temperature random phase approximation (RPA) and $GW$ algorithms to finite temperatures and obtained a similar exponential grid convergence for the RPA grand potential (see [Fig. \[fig:SiRPA\]]{}) as in the $T=0$ case.[@Kaltak2014] To reach $\mu$eV-accuracy, typically, less than 20 grid points are required. This holds true even for low temperatures, so that the RPA grand potential can be evaluated very efficiently for insulating as well as metallic systems with a computational complexity that grows only cubically with the number of electrons in the unit cell.
Furthermore, we showed how to choose the frequency grid for fermionic correlation functions and how to evaluate the Galitskii-Migdal grand potential at finite temperatures using the F-quadrature (see sections \[sec:FermionicFrequencyGrid\] and \[sec:MinimaxFourierTransformation\]). Here a compromise between $\varepsilon$-isometry and integration efficiency has to be made that deteriorates the grid convergence slightly compared to the RPA. For practical applications, however, the precision of the Matsubara summation is still sufficiently good. Other error sources, such as basis set errors will usually dominate.
In summary, we showed that optimized grids can be found for the accurate Matsubara summation of both, bosonic and fermionic functions, with roughly 20 grid points. The hypergeometric grids of Ozaki[@Ozaki2007] and Hu [*et. al.*]{}[@Hu2010] (see section \[sec:MinimaxIsometry\]) require roughly 100 and more points for the same precision at low temperatures and become competitive with our method only at high temperatures (where $\beta\ge1$).
Finite Temperature Formalism of Quantum Field Theory {#sec:Theory}
====================================================
This paper follows closely the formalism used in the book of Negele and Orland.[@Negele1988] We consider a system in thermal equilibrium, where the total number of particles is constant (electrons and holes). In contrast the number of electrons (or holes) depends on the chemical potential $\mu$, so that the system is described by the grand canonical ensemble with following partition function $$\begin{aligned}
\label{eq:DefPartitionFunction}
Z=\mathrm{Tr}
\left\lbrace
e^{-\beta( \hat H-\mu \hat N )} \right\rbrace\end{aligned}$$ and the corresponding grand potential $$\label{eq:DefGrandPotential}
\Omega = -\frac{1}{\beta}\ln Z.$$ Here the trace is performed w.r.t. an orthonormal basis of the full many-body Hamiltonian $\hat H$ $$\label{eq:DefHamiltonian}
\hat H=\hat H_0+\hat V$$ containing the Coulomb repulsion of all electrons $\hat V$, the particle number operator $\hat{N}$ and the non-interacting part $\hat H_0$ that can be expressed as sum of one-electron Hamiltonians $\hat h_i$. For the present paper, the exact form of $\hat h_i$ does not matter.[^5] We only assume that we have access to the one-electron eigensystem $\lbrace\ket*{\varphi_\alpha}, \epsilon_\alpha\rbrace$ of $\hat h_i$ and, thus, can determine the solutions of the non-interacting eigenvalue problem $$\label{eq:DefBasis}
\hat H_0\ket*{\Phi_A}=E_A\ket*{\Phi_A},$$ from a linear combination of Hartree products (Einstein summation assumed) $$\begin{aligned}
\label{eq:DefHartree}
\Phi_A(\mathbf{r}_1,\cdots,\mathbf{r}_M)=&c^{\alpha_1\cdots\alpha_M}\varphi_{\alpha_1}(\mathbf{r}_1)\cdots
\varphi_{\alpha_M}(\mathbf{r}_M)\end{aligned}$$ The non-interacting solutions $\ket*{\Phi_A}$, where $A$ numbers different (but orthogonal) eigenvalues of $\hat{H}_0$ provide a basis for our calculations and allow to write the number operator $\hat{N}$ and $\hat{H}_0$ in second quantization as $$\begin{aligned}
\label{eq:DefH0MinusMuN}
\hat H_0=& \sum\limits_\alpha \epsilon_\alpha \hat a_\alpha^\dagger \hat a_\alpha\\
\hat N =& \sum\limits_\alpha \hat a_\alpha^\dagger \hat a_\alpha.\end{aligned}$$ Furthermore, the non-interacting density operator $$\label{eq:DefDensityOp}
\hat\rho_0 = e^{-\beta(\hat{H}_0 - \mu \hat{N}-\Omega_0)}$$ and the non-interacting grand potential $$\label{eq:GrandPot0}
\begin{split}
\Omega_0 = &-\frac1\beta \ln Z_0\\
= &
-\frac1\beta \ln
\sum\limits_A \bra*{\Phi_A}
e^{-\beta(\hat{H}_0 - \mu\hat{N})}
\ket*{\Phi_A}\\
= & \frac1\beta\sum\limits_\alpha \ln f_\alpha,
\end{split}$$ provides access to expectation values of observables in the non-interacting system, such as the averaged number of electrons in state $\ket*{\varphi_\alpha}$ at a given temperature $T=\beta^{-1}$ (measured in units of eV) $$\label{eq:DefFermi}
f_\alpha=\mathrm{Tr}\left\lbrace \hat\rho_0
\hat{a}^\dagger_\alpha\hat{a}_\alpha\right\rbrace =
\frac{1}{e^{\beta(\epsilon_\alpha-\mu)}+1}.$$
Introducing thermal averages of an operator $\hat{\mathcal{O}}$ w.r.t. to the non-interacting system $$\label{eq:ThermalAv}
\langle \hat{\mathcal{O}} \rangle_\beta =
\sum\limits_A \bra*{\Phi_A}
\hat\rho_0
\hat{\mathcal{O}}
\ket*{\Phi_A},$$ the Gell-Mann and Low theorem[@PhysRev.84.350] can be used to relate all observables in the interacting system to the non-interacting one. More about this can be found in various books.[@Negele1988; @Abrikosov2012; @Fetter2003] For this paper, only the interacting grand potential is relevant $$\begin{aligned}
\label{eq:InteractingPotAll}
\Omega =& \Omega_0 + \Omega_v\\
\label{eq:InteractingPot}
\Omega_v = & -\frac1\beta \ln\left\langle \hat{T}_\tau
e^{-\int_{-\frac\beta2}^{\frac\beta2}\mathrm{d}\tau \hat{V}(-i\tau) }\right\rangle_\beta\end{aligned}$$ Here the interacting part $\Omega_v$ is expressed using the interaction picture (Heisenberg picture in imaginary time)[@Abrikosov2012] $$\label{eq:Heisenberg}
\hat{V}(-i\tau)=
e^{\beta(\hat{H}_0-\mu\hat{N})} \hat{V} e^{-\beta(\hat{H}_0-\mu\hat{N})},$$ the imaginary time ordering operator $$\label{eq:TimeOrder}
\hat{T}_\tau \hat{A}(\tau)\hat{B}(0)=
\left\lbrace
\begin{matrix}
\hat{A}(\tau)\hat{B}(0), & \tau>0\\
-\hat{B}(0) \hat{A}(\tau), & \tau<0
\end{matrix}
\right.$$ and the Coulomb operator $\hat{V}$ in second quantization with the matrix elements $$\label{eq:DefCoulombIntegrals}
V_{\alpha\alpha'\gamma\gamma'}=
\Big\langle\varphi_{\alpha},\varphi_\gamma\Big|
\frac{1}{|\mathbf{r}-\mathbf{r}'|}
\Big|\varphi_{\gamma'},\varphi_{\alpha'}\Big\rangle.$$ All approximations to the grand potential follow from an expansion of the exponential in and summation of specific terms. If one uses the prescription of Feynman diagrams to describe these terms, the natural logarithm in makes sure that only topologically connected diagrams contribute to the approximation (linked cluster theorem).
Approximations to the grand potential {#sec:ApproximationsToGrandPotential}
=====================================
Consider for instance the second order direct M[ø]{}ller-Plessett contribution to the grand potential $$\label{eq:DefMP2GrandPotentialTime}
\begin{split}
\Omega_c^{\rm MP2} =&
\frac12\int_{-\frac\beta2}^{\frac\beta2}\mathrm{d}\tau_1\int_{-\frac\beta2}^{\frac\beta2}\mathrm{d}\tau_2\\
\times& \mathrm{Tr}
\left\lbrace\mathbf{\chi}(-i\tau_1+i\tau_2)\mathbf{V}\mathbf{\chi}(-i\tau_2+i\tau_1)\mathbf{V}\right\rbrace,
\end{split}$$ where $\chi(-i\tau_1+i\tau_2)\mathbf{V}$ indicates the matrix multiplication with the Coulomb matrix elements and the trace is performed over the one-electron indices. Analyzing the integrand in , shows that no exponentially growing terms appear, since the imaginary time arguments always stay inside the fundamental interval $(-\beta,\beta)$. This also holds true for so-called metallic contributions, which are terms with $|\epsilon_\alpha-\epsilon_\gamma| \to 0$. Proper inspection of these terms shows that they are proportional to the inverse temperature of the system $\beta$ and only diverge as the temperature approaches 0 K.
Expression can be generalized to higher orders. For instance the contribution of order $k$ ($k$-th order ring diagram) contains $k$ convolutions in $\tau$ and is of the form $$\label{eq:MPNConvolutions}
\begin{split}
\Omega_{c}^{k}\propto&
\frac{1}{k}\int_{-\frac\beta2}^{\frac\beta2}\mathrm{d}\tau_1\cdots\int_{-\frac\beta2}^{\beta/2}\mathrm{d}\tau_k\\
&\times
\chi(-i\tau_1+\tau_2)\mathbf{V}\chi(-\tau_2+i\tau_3)\mathbf{V}\cdots\\
&\cdots\chi(-i\tau_k+i\tau_1)\mathbf{V}.
\end{split}$$ Here the Matsubara method shows its advantage, because it exploits the convolution theorem of harmonic analysis. A convolution in time, becomes a product in Fourier space and vice versa. Applying this theorem to results in a single series over Matsubara frequencies $$\label{eq:MPNConvolutionsMatsubara}
\begin{split}
\Omega_{c}^{k}= \frac{1}{\beta}\sum_{n\in\mathbb{Z}}\frac{1}{k}
\mathrm{Tr}\left\lbrace
\tilde\chi(i\nu_n)\mathbf{V}
\right\rbrace^k.
\end{split}$$ Similarly other approximated expressions of observables in euclidean QFT are tremendously simplified. In the following we discuss two well-known approximations to the correlation part of the grand potential. That is the random phase (RPA) and $GW$ approximation.
The RPA can be understood as an infinite sum of all possible ring diagrams (contribution of the form ) that becomes exact for the correlation energy of the interacting homogeneous electron gas at very high density as $T\to0$.[@PhysRev.106.364; @PhysRev.82.625; @PhysRev.85.338; @PhysRev.92.609] A closed form of the grand potential in the RPA can be found in Negele and Orland’s book[@Negele1988] and reads $$\label{eq:DefRPAGrandPotential}
\Omega_c^{\rm RPA} =
\frac1\beta\sum\limits_{n\in\mathbb{Z}}\mathrm{Tr}\left\lbrace
\ln\left[{1\!\!1}-\tilde{\mathbf{\chi}}(i\nu_n)\mathbf{V}\right]
-\tilde\chi(i\nu_m)\mathbf{V}
\right\rbrace$$ This expression stands for a typical Fourier series over bosonic frequencies and will provide a measure to test the IA$_1$-grid convergence in section \[sec:Results\].
The RPA is an approximative bosonization of the original problem. As an example of methods where bosonization is typically not applicable, we decided to evaluate the GM expression[@PhysRevLett.110.146403] for correlation part of the grand potential $$\label{eq:DefGMGrandPotential}
\Omega_c^{\rm GM} =
\frac1\beta\sum\limits_{m\in\mathbb{Z}} \mathrm{Tr}\left[
\tilde{\mathbf{G}}(i\omega_m)\tilde{\mathbf{\Sigma}}(i\omega_m)
\right].$$ Here $\tilde{\mathbf{G}}$ is the $GW$ dressed propagator and the solution of the Dyson equation $$\label{eq:DysonG}
\tilde{\mathbf{G}}(i\omega_m) =
\tilde{\mathbf{g}}(i\omega_m) +
\tilde{\mathbf{g}}(i\omega_m)
\tilde{\mathbf{\Sigma}}(i\omega_m)
\tilde{\mathbf{G}}(i\omega_m)$$ where $\mathbf{g}$ is the Hartree propagator (or Hartree Green’s function), $\tilde{\mathbf{\Sigma}}$ the $GW$ self-energy $$\label{eq:SelfEnergyGW}
\tilde{\mathbf{\Sigma}}(i\omega_m) = \int_{-\beta/2}^{\beta/2}
\mathrm{d}\tau
\mathbf{G}(-i\tau)\mathbf{W}(-i\tau)e^{i \omega_m \tau}$$ and $\tilde{\mathbf{W}}$ the RPA screened potential $$\label{eq:GWPotentialW}
\tilde{\mathbf{W}}(i\nu_n) =
\tilde{\mathbf{V}} +
\tilde{\mathbf{V}}
\tilde{\chi}(i\nu_n)
\tilde{\mathbf{W}}(i\nu_m).$$
Poisson summation and hyperbolic functions: A proof of Equ. {#app:ProofTanh}
============================================================
To proof identity , we use Poissons summation formula[@Higgins1985] $$\label{eq:Poisson}
\sum\limits_{n\in\mathbb{Z}}f(n) =
\sum\limits_{k\in\mathbb{Z}}\tilde f(k)$$ for a function $f(t)$ and its Fourier transform $\tilde f(k)$. Inserting $f(tz)
= e^{-2|tz|}$ into the l.h.s. of one obtains with the geometric series the hyperbolic cotangent $$\label{eq:IdentityCoth}
\sum\limits_{n\in\mathbb{Z}}e^{-2|n z|} =
\frac{1+e^{-2|z|}}{1-e^{-2|z|}}=\coth|z|.$$ Consequently, evaluating the Fourier integral gives $$\label{eq:FourierIntegral}
\tilde f(k)= \int_{-\infty}^\infty\mathrm{d}t e^{-2|tz|}
e^{i 2\pi k t}=\frac{|z|}{\pi^2 k^2+|z|^2},$$ which after inserting into the r.h.s. of yields the identity $$\label{eq:CothSeriesZ}
\coth|z|=\sum\limits_{k\in\mathbb{Z}}\frac{|z|}{\pi^2 k^2+|z|^2}.$$ On the one hand, replacing $|z|\to |z|/2$ and dividing by $2$, this identity becomes $$\label{eq:CothSeries}
\frac12\coth\frac{|z|}2=\sum\limits_{k\in\mathbb{Z}}\frac{|z|}{(\pi 2k)^2+|z|^2}.$$ On the other hand, the series on the r.h.s. can be split into a series over even and a series over odd integers $$\label{eq:SeriesCothSplit}
\coth|z|=\underbrace{\sum\limits_{k\in\mathbb{Z}}\frac{|z|}{(\pi2k)^2+|z|^2}}_{=\frac12\coth\frac{|z|}2}
+\sum\limits_{k\in\mathbb{Z}}\frac{|z|}{(\pi(2k+1))^2+|z|^2}$$ The first term on the r.h.s. follows from , while the second part is the l.h.s. of Equ. . After comparison with the well-known hyperbolic identity $$\label{eq:HyperbolicIdentity}
\coth z = \frac12\coth\frac{z}2 + \frac12\tanh\frac{z}2,$$ one identifies the second term in with $$\label{eq:TanhSeries}
\frac12\tanh\frac{|z|}2
=\sum\limits_{k\in\mathbb{Z}}\frac{|z|}{(\pi(2k+1))^2+|z|^2}$$ and Equ. is proven.
Note that the derivative of the l.h.s and r.h.s. of Eqs. and gives an alternative way to calculate the $L^2$-norms tabulated in Tab. \[tab:IsometricBases\].
[^1]: The isometry is known as Parseval theorem[@Boto2013] or, if $\ket*{n}$ is continuous, Plancherel theorem for Fourier transforms.[@Plancherel1910].
[^2]: The imaginary part is proportional to odd terms in the frequency $u_m(\epsilon_\alpha)v_m(\delta_\gamma)+v_m(\epsilon_\alpha)u_m(\delta_\gamma)$ and vanishes when the sum over all (positive and negative) fermionic Matsubara frequencies is carried out.
[^3]: Note, this expression is obtained from the free propagator by replacing $\epsilon_\alpha-\mu\to z$, $\tau\to\eta$ and using the identity $f(z)=\frac{1}{2}-\frac12\tanh\frac{z}2$.
[^4]: Fixing this requires the adjustment of the chemical potential and re-calculation of the interacting Green’s function $G$ until $N_e=\lim_{\tau\to0-}\mathrm{Tr}G(-i\tau)$ is satisfied.
[^5]: In textbooks the non-interacting term $\hat H_0$ is usually the Hartree-Fock Hamiltonian, which has the advantage that several so-called abnormal diagrammatic contributions to the total energy vanish[@Negele1988]. In practice, however, one typically uses the DFT Hamiltonian where those contributions are small but non-zero.[@Klimes2015]
|
---
abstract: |
\
Accurate quantum Monte Carlo calculations of ground and low-lying excited states of light p-shell nuclei are now possible for realistic nuclear Hamiltonians that fit nucleon-nucleon scattering data. At present, results for more than 30 different $(J^{\pi};T)$ states, plus isobaric analogs, in $A \leq 8$ nuclei have been obtained with an excellent reproduction of the experimental energy spectrum. These microscopic calculations show that nuclear structure, including both single-particle and clustering aspects, can be explained starting from elementary two- and three-nucleon interactions. Various density and momentum distributions, electromagnetic form factors, and spectroscopic factors have also been computed, as well as electroweak capture reactions of astrophysical interest.\
\
With permission from the Annual Review of Nuclear and Particle Science. Final version of this material is scheduled to appear in the Annual Review of Nuclear and Particle Science Vol. 51, to be published in December 2001 by Annual Reviews, http://AnnualReviews.org.
author:
- 'Steven C. Pieper and R. B. Wiringa'
date: '\'
title: Quantum Monte Carlo Calculations of Light Nuclei
---
psfig.sty
Nuclear many-body theory, structure, reactions, potentials
INTRODUCTION
============
A major goal of nuclear physics is to understand the stability, structure, and reactions of nuclei as a consequence of the interactions between individual nucleons. This is an extremely challenging many-body problem, exacerbated by the fact that we do not know in detail what the interactions are. Quantum chromodynamics, the fundamental theory for strong interactions, is so difficult to solve in the nonperturbative regime of low-energy nuclear physics that no one is currently able to quantitatively describe the interactions between two nucleons from first principles. However, we do have a tremendous amount of experimental data on nucleon-nucleon ($N\!N$) scattering and we can construct accurate representations of $N\!N$ interactions by means of two-body potentials. These potentials are complicated, depending on the relative positions, spins, isospins, and momenta of the nucleons, which makes the problem of finding accurate quantum-mechanical solutions of many-nucleon bound and scattering states a demanding task. Nevertheless, a variety of precise calculations have been carried out for the three- and four-nucleon systems which reveal an additional complication: two-nucleon forces alone are inadequate to quantitatively explain either the bound-state properties or a variety of $Nd$ and $Nt$ scattering data. This is not surprising, because meson-exchange theory and the composite nature of nucleons indicate that many-nucleon forces, at least three-nucleon forces, are important. However, these are even more difficult to construct from first principles. In like manner, meson-exchange introduces two-body charge and current operators, and calculations of electroweak transitions show that these are also necessary to achieve agreement with data.
Despite these difficulties, tremendous progress has been made in the past decade, both in the characterization of nuclear forces and currents, and in the development of accurate many-body techniques for evaluating them. A multienergy partial-wave analyses of elastic $N\!N$ scattering data below $T_{lab} = 350$ MeV was produced by the Nijmegen group in 1993 [@SKRS93], which demonstrated that over 4300 data points could be fit with a $\chi^2$ per datum $\approx$ 1. This was achieved by careful attention to the known long-range part of the $N\!N$ interaction and a rigorous winnowing of inconsistent data. Their work inspired the construction of a number of new $N\!N$ potential models which gave comparably good fits to the database.
Concurrently, several essentially exact methods have been developed for the study of few-nucleon ($A$ = 3,4) systems with such realistic $N\!N$ potentials. The first accurate, converged calculations of three-nucleon ($N\!N\!N$) bound states were obtained by configuration-space Faddeev methods in 1985 [@CPFG85], while the first accurate $^4$He bound states were obtained by the Green’s function Monte Carlo method in 1987 [@C87]. Other methods of comparable accuracy have been developed since then for both $A$ = 3,4 bound states and for $N\!N\!N$ scattering. A comprehensive review of many aspects of the few-nucleon problem can be found in the article by Carlson and Schiavilla [@CS98]. The result of all the progress in the last decade is that we have obtained an excellent understanding of few-nucleon systems, including both structure and reactions, although a few unanswered problems remain.
Until recently, nuclei larger than $A=4$ have been described within the framework of the nuclear shell model or mean-field pictures such as Skyrme models. The classic description of p-shell ($5 \leq A \leq 16$) nuclei was developed by Cohen and Kurath in the 1960s [@CK65]. In that work, a large number of experimental levels were fit in terms of a relatively small number of two-body matrix elements, but no direct connection was made to the underlying $N\!N$ force. Beginning with the work of Kuo and Brown [@KB66], approximate methods for relating the shell-model matrix elements to the underlying forces were developed and applied to even larger nuclei. However, in most calculations, only the valence nucleons, i.e., the nucleons in the last unfilled shell, are treated as active, while the core is inert. Recently the first “no-core” shell model calculations, in which all the nucleons are active, have been made in the p-shell with effective interactions that have been derived by a G-matrix procedure from a realistic underlying $N\!N$ force [@NVB00]. These calculations agree with the other exact methods for the $A=3,4$ nuclei, but fully-converged calculations for larger systems have not yet been obtained.
The present article focuses on recent developments in quantum Monte Carlo methods that are making the light p-shell nuclei accessible at a level of accuracy close to that obtained for the s-shell ($A \leq 4$) nuclei. The quantum Monte Carlo methods include variational Monte Carlo (VMC) and Green’s function Monte Carlo (GFMC) methods. The VMC is an approximate method [@LAPS81; @CPW83; @W91; @APW95] that is used as a starting point for the more accurate GFMC calculations. Because of its relative simplicity, the VMC has also been used to make first studies of various reactions, such as the nuclear response to electron [@WS98; @LWW99] or pion scattering [@LW01], and for electroweak capture reactions of astrophysical interest [@NWS01; @N01]. The GFMC is exact in the sense that binding energies can be calculated with an accuracy of better than 2%; it has been used to compute ground states and low-lying excitations of all the $A \leq 8$ nuclei [@C87; @PPCW95; @PPCPW97; @WPCP00]. These studies of light p-shell nuclei have shown that nuclear structure can be predicted from the elementary nuclear forces. As presently implemented, both the VMC and GFMC methods require significant additional computer resources for every nucleon added, so they probably will be limited to systems of $A \leq 12$ nuclei. A new method, whose accuracy lies between that of the VMC and GFMC methods, is auxiliary-field diffusion Monte Carlo (AFDMC); it has been implemented for large pure neutron systems [@SF99] and with further development should be capable of handling nuclei above $A=12$.
Most of the calculations discussed here have been made using Hamiltonians containing the Argonne $v_{18}$ $N\!N$ potential [@WSS95] alone or with one of the Urbana [@CPW83] or Illinois [@PPWC01] series of $N\!N\!N$ potentials. Argonne $v_{18}$ (AV18) is representative of the modern $N\!N$ potentials that give accurate fits to scattering data, while the Urbana and Illinois models are modern $N\!N\!N$ potentials based on meson-exchange interactions and fit to the binding energies of light nuclei. These models are described in some detail in Section 2 to show the complexity of modern nuclear forces, and the consequent challenge for many-body theory. The VMC method is presented in Section 3; it starts with the construction of a variational wave function of specified angular momentum, parity and isospin, $\Psi_V(J^{\pi};T)$, using products of two- and three-body correlation operators acting on a fully antisymmetrized set of one-body basis states coupled to specific quantum numbers. Metropolis Monte Carlo integration [@MR2T2] is used to evaluate $\langle \Psi_V | H | \Psi_V \rangle$, giving an upper bound to the energy of the state. The GFMC method, described in Section 4, is a stochastic method that systematically improves on $\Psi_V$ by projecting out excited state contamination using the Euclidean propagation $\Psi(\tau) = \exp [ - ( H - E_0) \tau ] \Psi_V$. In the limit $\tau \rightarrow \infty$, this leads to the exact $\langle | H | \rangle$.
Energy results for the ground states and many low-lying excited states of the $A\leq8$ nuclei are presented in Section 5. With the new Hamiltonians, the excitation spectra are reproduced very well, including the relative stability of different nuclei and the splittings between excited states. Energy differences between isobaric analog states and isospin-mixing matrix elements have also been computed to study the charge-independence-breaking of the strong and electromagnetic forces. A variety of density and momentum distributions, electromagnetic moments and form factors, transition densities, spectroscopic factors, and intrinsic deformation are presented in Section 6. One of the chief benefits of this approach is that absolute normalizations can be computed, and effective charges so commonly needed in traditional shell-model calculations of electroweak transitions are not required.
While the quantum Monte Carlo methods have been used primarily for stationary states, it is also possible to study scattering states, as discussed in Section 7. Some first calculations of astrophysical radiative capture reactions in the p-shell are presented in Section 8. Both of these applications are in their early stages, but promise to be major fields of endeavor for future work. In Section 9 we discuss neutron drops, systems of interacting neutrons confined in an artificial external well, that can serve as reference points for the development of Skyrme interactions for use in very large neutron-rich nuclei. Finally, in Section 10, we consider the outlook for future developments.
HAMILTONIAN AND CURRENTS
========================
The Hamiltonian used in this review use includes nonrelativistic kinetic energy, the AV18 $N\!N$ potential [@WSS95] and either the Urbana IX [@PPCW95] or one of the new Illinois series of $N\!N\!$ potential [@PPWC01]: $$H = \sum_{i} K_{i} + \sum_{i<j} v_{ij} + \sum_{i<j<k} V_{ijk} \ .
\label{eq:H}$$ The kinetic energy operator has charge-independent (CI) and charge-symmetry-breaking (CSB) components, the latter due to the difference between proton and neutron masses, $$K_{i} = K^{CI}_{i} + K^{CSB}_{i} \equiv -\frac{\hbar^2}{4}
(\frac{1}{m_{p}} + \frac{1}{m_{n}}) \nabla^{2}_{i}
-\frac{\hbar^2}{4} (\frac{1}{m_{p}} - \frac{1}{m_{n}})\tau_{zi}
\nabla^{2}_{i} ~,$$ where ${\mbox{\boldmath$\tau$}}_{i}$ is the isospin of nucleon $i$.
The AV18 potential is written as a sum of electromagnetic and one-pion-exchange terms and a shorter-range phenomenological part, $$v_{ij} = v^{\gamma}_{ij} + v^{\pi}_{ij} + v^{R}_{ij} \ .$$ The electromagnetic terms include one- and two-photon-exchange Coulomb interaction, vacuum polarization, Darwin-Foldy, and magnetic moment terms, all with appropriate proton and neutron form factors. The one-pion-exchange part of the potential includes the charge-dependent (CD) terms due to the difference in neutral and charged pion masses. It can be written as $$v^{\pi}_{ij} = f^{2} \left(\frac{m}{m_{s}}\right)^{2} {\textstyle{1\over3}}
mc^{2} \left[ X_{ij} {\mbox{\boldmath$\tau$}}_{i} \cdot {\mbox{\boldmath$\tau$}}_{j}
+ \tilde{X}_{ij} T_{ij} \right] \ ,
\label{eq:vpi}$$ where $T_{ij} = 3\tau_{zi}\tau_{zj}-{\mbox{\boldmath$\tau$}}_{i}\cdot{\mbox{\boldmath$\tau$}}_{j}$ is the isotensor operator and $$\begin{aligned}
&& X_{ij} = {\textstyle{1\over3}} \left( X^{0}_{ij} + 2 X^{\pm}_{ij} \right) , \\
&& \tilde{X}_{ij} = {\textstyle{1\over3}} \left( X^{0}_{ij} - X^{\pm}_{ij}
\right) , \\
&& X^{m}_{ij} = \left[ Y(mr_{ij}) {\mbox{\boldmath$\sigma$}}_{i} \cdot {\mbox{\boldmath$\sigma$}}_{j} +
T(mr_{ij}) S_{ij} \right] \ .\end{aligned}$$ Here $Y(x)$ and $T(x)$ are the normal Yukawa and tensor functions: $$\begin{aligned}
Y(x) &=& \frac{e^{-x}}{x} \ \xi_Y(r) \ ,
\label{eq:yij} \\
T(x) &=& \left( \frac{3}{x^2} + \frac{3}{x} + 1 \right) Y(x) \ \xi_T(r) \ .
\label{eq:tij}\end{aligned}$$ with short-range cutoff functions $\xi_Y(r)$ and $\xi_T(r)$, and the $X^{\pm,0}$ are calculated with $m = m_{\pi^{\pm}}$ and $m_{\pi^{0}}$. The coupling constant in Eq.(\[eq:vpi\]) is $f^2 = 0.075$ and the scaling mass $m_s = m_{\pi^{\pm}}$.
The remaining terms are of intermediate (two-pion-exchange) and short range, with some 40 adjustable parameters. The one-pion-exchange and the remaining phenomenological part of the potential can be written as a sum of eighteen operators, $$v^{\pi}_{ij} + v^{R}_{ij} = \sum_{p=1,18} v_{p}(r_{ij}) O^{p}_{ij} \ .$$ The first fourteen are CI operators, $$\begin{aligned}
O^{p=1,14}_{ij} = [1, {\mbox{\boldmath$\sigma$}}_{i}\cdot{\mbox{\boldmath$\sigma$}}_{j}, S_{ij},
{\bf L\cdot S},{\bf L}^{2},{\bf L}^{2}({\mbox{\boldmath$\sigma$}}_{i}\cdot{\mbox{\boldmath$\sigma$}}_{j}),
({\bf L\cdot S})^{2}]\otimes[1,{\mbox{\boldmath$\tau$}}_{i}\cdot{\mbox{\boldmath$\tau$}}_{j}] ~,
\label{eq:op14}\end{aligned}$$ while the last four, $$O^{p=15,18}_{ij} = [1, {\mbox{\boldmath$\sigma$}}_{i}\cdot{\mbox{\boldmath$\sigma$}}_{j},
S_{ij}]\otimes T_{ij} \ , {\rm and} \ (\tau_{zi}+\tau_{zj}) \ ,$$ are CD and CSB terms. The potential was fit directly to the Nijmegen $N\!N$ scattering data base [@SKRS93] containing 1787 $pp$ and 2514 $np$ data in the range 0–350 MeV, to the $nn$ scattering length, and to the deuteron binding energy, with a $\chi^2$ per datum of 1.09.
While the CD and CSB terms are small, there is a clear need for their presence. The Nijmegen group studied many $N\!N$ potentials from the 1980s and before, and found that potentials fit to $np$ data in $T=1$ states did not fit $pp$ data well even after allowing for standard Coulomb effects, and vice versa [@SS93-95]. All five modern $N\!N$ potentials, Reid93, Nijm I and II [@SKTS94], CD Bonn [@MSS96], and AV18, which fit $pp$ and $np$ scattering data with a $\chi^2/$point near 1, have CD components. The CSB term is required to fit the difference between the $pp$ and $nn$ scattering lengths, and is consistent with the mass difference between $^3$H and $^3$He, as shown below. The identification of the proper CI, CD, and CSB $N\!N$ force components is important in setting the proper baseline for the introduction of $N\!N\!N$ forces, which are required to fit the $^3$H and $^3$He binding energies as a primary constraint.
The Urbana series of three-nucleon potentials is written as a sum of two-pion-exchange P-wave and remaining shorter-range phenomenological terms, $$V_{ijk} = V^{2\pi,P}_{ijk} + V^{R}_{ijk} ~.$$ The structure of the two-pion P-wave exchange term with an intermediate $\Delta$ excitation (Fig. \[fig:vijk\]a) was originally written down by Fujita and Miyazawa [@FM57]; it can be expressed simply as $$V^{2\pi,P}_{ijk} = \sum_{cyc} \left( A^P_{2\pi}
\{X^{\pi}_{ij},X^{\pi}_{jk}\}
\{{\mbox{\boldmath$\tau$}}_{i}\cdot{\mbox{\boldmath$\tau$}}_{j},{\mbox{\boldmath$\tau$}}_{j}\cdot{\mbox{\boldmath$\tau$}}_{k}\}
+ C^P_{2\pi} [X^{\pi}_{ij},X^{\pi}_{jk}]
[{\mbox{\boldmath$\tau$}}_{i}\cdot{\mbox{\boldmath$\tau$}}_{j},{\mbox{\boldmath$\tau$}}_{j}\cdot{\mbox{\boldmath$\tau$}}_{k}] \right ) \ ,$$ where $X^{\pi}_{ij}$ is constructed with an average pion mass, $m_{\pi}={\textstyle{1\over3}}m_{\pi^0}+{\textstyle{2\over3}}m_{\pi^{\pm}}$, and $\sum_{cyc}$ is a sum over the three cyclic exchanges of nucleons $i,j,k$. For the Urbana models $C^P_{2\pi} = {\textstyle{1\over4}}A^P_{2\pi}$, as in the original Fujita-Miyazawa model [@FM57], while other potentials like the Tucson-Melbourne [@TM] and Brazil [@Brazil] models, have a ratio slightly larger than ${\textstyle{1\over4}}$. The shorter-range phenomenological term is given by $$V^{R}_{ijk} = \sum_{cyc} A_R T^2(m_{\pi}r_{ij}) T^2(m_{\pi}r_{jk}) \ .
\label{eq:vrijk}$$ For the Urbana IX (UIX) model, the parameters were determined by fitting the binding energy of $^3$H and the density of nuclear matter in conjunction with AV18.
As shown below, while the combined AV18/UIX Hamiltonian reproduces the binding energies of s-shell nuclei, it does not do so well for light p-shell nuclei. Recently a new class of $N\!N\!N$ potentials, called the Illinois models, has been developed to address this problem [@PPWC01]. These potentials contain the Urbana terms and two additional terms, resulting in a total of four coupling constants that can be adjusted to fit the data.
One term, $V^{2\pi,S}_{ijk}$, is due to $\pi N$ S-wave scattering as illustrated in Fig. \[fig:vijk\]b. It has been included in a number of $N\!N\!N$ potentials like the Tucson-Melbourne [@TM] and Brazil [@Brazil] models. The Illinois models use the form recommended in the latest Texas model [@Texas], where chiral symmetry is used to constrain the structure of the interaction. However, in practice, this term is much smaller than the $V^{2\pi,P}_{ijk}$ contribution and behaves similarly in light nuclei, so it is difficult to establish its strength independently just from calculations of energy levels.
A more important addition is a simplified form for three-pion rings containing one or two deltas (Fig. \[fig:vijk\]c,d). As discussed in Ref. [@PPWC01], these diagrams result in a large number of terms, the most important of which are used to construct the Illinois models: $$V^{3\pi,\Delta R}_{ijk} = \Athpi \left[ {\textstyle{50\over3}}S^I_{\tau}S^I_{\sigma}+
{\textstyle{26\over3}}A^I_{\tau}A^I_{\sigma} \right] \ .$$ Here the $S^I_x$ and $A^I_x$ are operators that are symmetric or antisymmetric under any exchange of the three nucleons, and the subscript $x=\sigma$ or $\tau$ indicates that the operators act on, respectively, spin and space or just isospin degrees of freedom.
The $S^I_\tau$ is a projector onto isospin-${\textstyle{3\over2}}$ triples: $$S^I_{\tau} = 2 + {\textstyle{2\over3}}\left({\mbox{\boldmath$\tau$}}_i \cdot {\mbox{\boldmath$\tau$}}_j
+{\mbox{\boldmath$\tau$}}_j \cdot {\mbox{\boldmath$\tau$}}_k + {\mbox{\boldmath$\tau$}}_k \cdot {\mbox{\boldmath$\tau$}}_i \right)
= 4 P_{T=3/2} \ .
\label{eq:sitau}$$ To the extent isospin is conserved, there are no such triples in the s-shell nuclei, and so this term does not affect them. It is also zero for $Nd$ scattering. However, the $S^I_{\tau}S^I_{\sigma}$ term is attractive in all the p-shell nuclei studied. The $A^I_\tau$ has the same structure as the isospin part of anticommutator part of $V^{2\pi,P}$, but the $A^I_{\tau}A^I_{\sigma}$ term is repulsive in all nuclei studied so far. In p-shell nuclei, the magnitude of the $A^I_{\tau}A^I_{\sigma}$ term is smaller than that of the $S^I_{\tau}S^I_{\sigma}$ term, so the net effect of the $V^{3\pi,\Delta R}_{ijk}$ is slight repulsion in s-shell nuclei and larger attraction in p-shell nuclei. The reader is referred to Ref. [@PPWC01], and its appendix, for the complete structure of $V^{3\pi,\Delta R}_{ijk}$.
Relativistic corrections to the Hamiltonian of Eq.(\[eq:H\]) have been studied in three- and four-body nuclei [@CPS93; @FPCS95; @FPA99]. We only briefly outline the results here. If a square-root kinetic energy, $$K^{rel}_i = \sqrt{ p^2_i + m^2_i } - m_i \ ,$$ where $p_i$ is the momentum of nucleon $i$, is introduced into the Hamiltonian, then the $N\!N$ potential must be readjusted to refit the $N\!N$ data. This can be done for AV18 with only small adjustments of the parameters, to produce a $v^{rel}$. VMC calculations show that the resulting $\langle K^{rel}+v^{rel} \rangle \sim \langle K+v \rangle$ for $^3$H and $^4$He; thus the square-root kinetic energy may be neglected in light nuclei.
A second relativistic correction, the so-called “boost correction,” arises from the fact that the two-nucleon interaction $v_{ij}$ depends both on the relative momentum ${\bf p} = ({\bf p}_i - {\bf p}_j)/2$ and the total momentum ${\bf P} = {\bf p}_i + {\bf p}_j $ of the interacting nucleons, $$v_{ij}({\bf p},{\bf P}) = \tvij({\bf p}) + \dvij({\bf p},{\bf P}) \ ,
\label{eq:tvdv}$$ where $\delta v_{ij}({\bf p},{\bf P}=0)=0$. Fitting the two-body potential to two-nucleon scattering data determines only $\tvij$. The $\dvij$ can be determined from $\tvij$; a suitable approximation [@FPA99; @FPF95; @F75] is $$\delta{v_{ij}}({\bf P})=-\frac{P^2}{8m^2}\tvij+\frac{1}{8m^2}\,\left[\ {\bf P}
\cdot{\bf r}\;{\bf P}\cdot\mbox{\boldmath$\nabla$}, \tvij\ \right] \ ,
\label{eq:friar}$$ where only the first six terms (the static terms) of $\tvij$ are retained and the gradient operators do not act on $\tvij$. The $\dvij$ is meant to be used only in first-order perturbation in a non-relativistic wave function. Its expectation values in light nuclei are nearly proportional to $\langle V^R_{ijk} \rangle$ [@PPWC01], and thus one can consider that part of $A_R$, Eq.(\[eq:vrijk\]), comes from $\dvij$. This proportionality does not, however, extend to nuclear matter or pure neutron systems. There one must use a reduced $A_R^*$ that does not contain the part ascribable to $\dvij$, and explicitly evaluate $\langle \dvij \rangle$; doing so results in a softer equation of state for dense matter.
VARIATIONAL MONTE CARLO
=======================
The variational method can be used to obtain approximate solutions to the many-body Schrödinger equation, $H\Psi = E\Psi$, for a wide range of nuclear systems, including few-body nuclei, light closed-shell nuclei, nuclear matter, and neutron stars [@W93]. A suitably parameterized wave function, $\Psi_V$, is used to calculate an upper bound to the exact ground-state energy, $$E_V = \frac{\langle \Psi_V | H | \Psi_V \rangle}
{\langle \Psi_V | \Psi_V \rangle} \geq E_0 \ .
\label{eq:expect}$$ The parameters in $\Psi_V$ are varied to minimize $E_V$, and the lowest value is taken as the approximate solution. Upper bounds to excited states are also obtainable, if they have different quantum numbers from the ground state, or from small-basis diagonalizations if they have the same quantum numbers. The corresponding $\Psi_V$ can be used to calculate other properties, such as particle density or electromagnetic moments, or to start a Green’s function Monte Carlo calculation. In this section we describe the [*ansatz*]{} for $\Psi_V$ for light nuclei and briefly review how the expectation value is evaluated and the parameters of $\Psi_V$ are fixed.
The best variational wave function has the form [@APW95] $$|\Psi_V\rangle = \left[1 + \sum_{i<j<k}(U_{ijk}+U^{TNI}_{ijk})
+ \sum_{i<j}U^{LS}_{ij} \right]
|\Psi_P\rangle \ ,
\label{eq:bestpsiv}$$ where the pair wave function, $\Psi_P$, is given by $$|\Psi_P\rangle = \left[ {\cal S}\prod_{i<j}(1+U_{ij}) \right]
|\Psi_J\rangle \ .
\label{eq:psip}$$ The $U_{ij}$, $U^{LS}_{ij}$, $U_{ijk}$, and $U^{TNI}_{ijk}$ are noncommuting two- and three-nucleon correlation operators, and ${\cal S}$ is a symmetrization operator. The form of the antisymmetric Jastrow wave function, $\Psi_J$, depends on the nuclear state under investigation. For the s-shell nuclei the simple form $$|\Psi_J\rangle = \left[ \prod_{i<j<k}f^c_{ijk} ({\bf r}_{ik},{\bf r}_{jk})
\prod_{i<j}f_c(r_{ij}) \right]
|\Phi_A(JMTT_{3})\rangle $$ is used. Here $f_c(r_{ij})$ and $f^c_{ijk}$ are central (spin-isospin independent) two- and three-body correlation functions and $\Phi_A$ is an antisymmetrized spin-isospin state, e.g., $$\begin{aligned}
&& |\Phi_{3}({\textstyle{1\over2}} {\textstyle{1\over2}} {\textstyle{1\over2}} {\textstyle{1\over2}}) \rangle
= {\cal A} |\uparrow p \downarrow p \uparrow n \rangle \ , \\
&& |\Phi_{4}(0 0 0 0) \rangle
= {\cal A} |\uparrow p \downarrow p \uparrow n \downarrow n \rangle \ ,\end{aligned}$$ with ${\cal A}$ the antisymmetrization operator.
The two-body correlation operators $U_{ij}$ and $U^{LS}_{ij}$ are sums of spin, isospin, tensor, and spin-orbit terms: $$\begin{aligned}
U_{ij} &=& \sum_{p=2,6} \left[ \prod_{k\not=i,j}f^p_{ijk}({\bf r}_{ik}
,{\bf r}_{jk}) \right] u_p(r_{ij}) O^p_{ij} \ , \\
U^{LS}_{ij} &=& \sum_{p=7,8} \left[ \prod_{k\not=i,j}f^p_{ijk}({\bf r}_{ik}
,{\bf r}_{jk}) \right] u_p(r_{ij}) O^p_{ij} \ ,\end{aligned}$$ where the $O^p_{ij}$ were introduced in Eq.(\[eq:op14\]). The spin-orbit correlations are only summed in Eq.(\[eq:bestpsiv\]) because of the extra computational expense of evaluating powers of ${\bf L\cdot S}$ that would occur if it was inserted in the symmetrized product of Eq.(\[eq:psip\]).
The central $f_c(r)$ and noncentral $u_p(r)$ pair correlation functions reflect the influence of the two-body potential at short distances, while satisfying asymptotic boundary conditions of cluster separability. Reasonable functions are generated by minimizing the two-body cluster energy of a somewhat modified $N\!N$ interaction; this results in a set of eight coupled, Schrödinger-like, differential equations corresponding to linear combinations of the first eight operators i of $v_{ij}$, with a number of embedded variational parameters [@W91]. The $f_c(r)$ is small at short distances, to reduce the contribution of the repulsive core of $v_{ij}$, and peaks at an intermediate distance corresponding to the maximum attraction of $v_{ij}$, as illustrated in Fig. \[fig:corr\] for several nuclei. For the s-shell nuclei, $f_c(r)$ falls off at larger distances to keep the system confined. For example, in $^4$He, $[f_c(r)]^3 \sim {\rm {exp}}(-\kappa r)/r$ at large r, as shown by the dashed line in Fig. \[fig:corr\], where $\kappa$ corresponds to an $\alpha \rightarrow t+p$ separation energy. The noncentral $u_p(r)$ are all relatively small; the most important is the long-range tensor-isospin part $u_{t\tau}(r)$, also shown in Fig. \[fig:corr\] for several nuclei, which is mainly induced by the one-pion-exchange part of $v_{ij}$.
The $f^c_{ijk}$, $f^p_{ijk}$, and $U_{ijk}$ are three-nucleon correlations also induced by $v_{ij}$ [@APW95]. The $U^{TNI}_{ijk}$ are three-body correlations induced by the three-nucleon interaction, which have the form suggested by perturbation theory: $$U^{TNI}_{ijk} = \sum_x \epsilon_x V_{ijk}(\tilde{ r}_{ij},
\tilde{r}_{jk}, \tilde{ r}_{ki}) \ ,
\label{eq:bestuijk}$$ with $\tilde{r}=yr$, $y$ a scaling parameter, and $\epsilon_x$ a (small negative) strength parameter. A somewhat simpler wave function to evaluate than $\Psi_V$ is given by: $$\begin{aligned}
|\Psi_{T}\rangle = \left[1 + \sum_{i<j<k}\tilde{U}^{TNI}_{ijk}\right]
|\Psi_P\rangle \ ,
\label{eq:psitgfmc}\end{aligned}$$ where $\tilde{U}^{TNI}_{ijk}$ is a slightly truncated TNI correlation. The $\Psi_T$ gives about $1-2$ MeV less binding than the full $\Psi_V$, but costs less than half as much to construct, making it a more efficient starting point for the GFMC calculation [@PPCPW97].
The Jastrow wave function for the light p-shell nuclei is more complicated, as a number of nucleons must be placed in the unfilled p-shell. The $LS$ coupling scheme is used to obtain the desired $JM$ value of a given state, as suggested by the shell-model studies of light p-shell nuclei [@CK65]. Different possible $LS$ combinations lead to multiple components in the Jastrow wave function. The possibility that the central correlations $f_{c}(r_{ij})$ could depend upon the shells (s or p) occupied by the particles and on the $LS$ coupling is also allowed for: $$\begin{aligned}
|\Psi_J\rangle &=& {\cal A} \left\{
\prod_{i<j<k}f^c_{ijk}
\prod_{i<j \leq 4}f_{ss}(r_{ij})
\prod_{k \leq 4 < l \leq A} f_{sp}(r_{kl}) \right. \\
&& \left. \sum_{LS[n]} \Big( \beta_{LS[n]} \prod_{4 < l < m \leq A}
f^{LS[n]}_{pp}(r_{lm})
|\Phi_{A}(LS[n]JMTT_{3})_{1234:56\ldots A}\rangle \Big) \right\} \ .
\nonumber
\label{eq:jastrow}\end{aligned}$$ The operator ${\cal A}$ indicates an antisymmetric sum over all possible partitions of the $A$ particles into 4 s-shell and $(A-4)$ p-shell ones. The central correlation $f_{ss}(r)$ is the $f_c(r)$ from the $^{4}$He wave function. The $f_{sp}(r)$, shown in Fig. \[fig:corr\] for Li nuclei, is similar to the $f_c(r)$ at short range, but with a long-range tail going to unity; this helps the wave function factorize to a cluster structure like $\alpha + d$ in $^6$Li or $\alpha + t$ in $^7$Li at large cluster separations. The $f^{LS[n]}_{pp}(r)$ is similar to the deuteron (triton) $f_c(r)$ in the case of $^6$Li ($^7$Li).
The $LS$ components of the single-particle wave function are given by: $$\begin{aligned}
&& |\Phi_{A}(LS[n]JMTT_{3})_{1234:56\ldots A}\rangle =
|\Phi_{\alpha}(0 0 0 0)_{1234}\rangle \prod_{4 < l\leq A}
\phi^{LS[n]}_{p}(R_{\alpha l}) \\
&& \left\{ [ \prod_{4 < l\leq A} Y_{1m_l}(\Omega_{\alpha l}) ]_{LM_L[n]}
\times [ \prod_{4 < l\leq A} \chi_{l}({\textstyle{1\over2}}m_s) ]_{SM_S}
\right\}_{JM}
\times [ \prod_{4 < l\leq A} \nu_{l}({\textstyle{1\over2}}t_3) ]_{TT_3}\rangle
\nonumber \ .
\label{eq:phi}\end{aligned}$$ The $\phi^{LS[n]}_{p}(R_{\alpha k})$ are p-wave solutions of a particle in an effective $\alpha + N$ potential and are functions of the distance between the center of mass of the $\alpha$ core and nucleon $k$; they may be different for different $LS[n]$ components. A Woods-Saxon potential well, $$V_{\alpha N}(r) = V^{LS}_p [1+exp(\frac{r-R_p}{a_p})]^{-1} \ ,
\label{eq:spwell}$$ where $V^{LS}_p$, $R_p$, and $a_p$ are variational parameters, and a Coulomb term if appropriate is used.
For $A \geq 7$ nuclei, with three or more p-shell particles there are multiple ways of coupling the orbital angular momentum, $L$, spin, $S$, and isospin, $T$, to obtain the total $(J^\pi;T)$ of the system. The spatial permutation symmetry, denoted by the Young pattern $[n]$, is used to enumerate the different possible terms, as discussed in Appendix 1C of Ref. [@BM69]. The different possible contributions to $A$ = 6–8 nuclei are given in Table \[tab:perms\], with the corresponding spin states of highest spatial symmetry for each nucleus.
[lllll]{} $A$ & $[n] $ & $L $ & $(T,S) $ & highest symmetry states\
$6$ & $[2] $ & $0,2 $ & $(1,0) (0,1) $ & $^6$He$(0^+,2^+)$, $^6$Li$(1^+,2^+,3^+)$\
& $[11] $ & $1 $ & $(1,1) (0,0) $ & $^6$He$(1^+)$\
$7$ & $[3] $ & $1,3 $ & $({\textstyle{1\over2}},{\textstyle{1\over2}}) $ & $^7$Li$({\textstyle{1\over2}}^-,{\textstyle{3\over2}}^-,{\textstyle{5\over2}}^-,{\textstyle{7\over2}}^-)$\
& $[21] $ & $1,2 $ & $({\textstyle{3\over2}},{\textstyle{1\over2}})
({\textstyle{1\over2}},{\textstyle{3\over2}})
({\textstyle{1\over2}},{\textstyle{1\over2}}) $ & $^7$He$({\textstyle{1\over2}}^-,{\textstyle{3\over2}}^-,{\textstyle{5\over2}}^-)$\
& $[111]$ & $0 $ & $({\textstyle{3\over2}},{\textstyle{3\over2}})
({\textstyle{1\over2}},{\textstyle{1\over2}}) $\
$8$ & $[4] $ & $0,2,4$ & $(0,0) $ & $^8$Be$(0^+,2^+,4^+)$\
& $[31] $ & $1,2,3$ & $(1,1) (1,0) (0,1) $ & $^8$Li$(0^+-4^+)$, $^8$Be$(1^+,3^+)$\
& $[22] $ & $0,2 $ & $(2,0) (1,1) (0,2) (0,0) $ & $^8$He$(0^+,2^+)$\
& $[211]$ & $1 $ & $(2,1) (1,2) (1,1) (1,0) (0,1) $ & $^8$He$(1^+)$\
After other parameters in the trial function have been optimized, a series of calculations are made in which the $\beta_{LS[n]}$ of Eq.(\[eq:jastrow\]) may be different in the left- and right-hand-side wave functions to obtain the diagonal and off-diagonal matrix elements of the Hamiltonian and the corresponding normalizations and overlaps. The resulting N$\times$N matrices are diagonalized to find the $\beta_{LS[n]}$ eigenvectors, using generalized eigenvalue routines because the correlated $\Psi_V$ are not orthogonal. This allows us to project out not only the ground states, but excited states of the same $(J^\pi;T)$ quantum numbers. For example, the ground states of $^6$Li, $^7$Li, and $^8$Li are obtained from 3$\times$3, 5$\times$5, and 7$\times$7 diagonalizations, respectively.
The energy expectation value of Eq.(\[eq:expect\]) is evaluated using Monte Carlo integration. A detailed technical description of the methods used can be found in Refs. [@W91; @CW91]. Monte Carlo sampling is done both in configuration space and in the order of operators in the symmetrized product of Eq.(\[eq:psip\]) by following a Metropolis random walk. The expectation value for an operator $O$ is computed with the expression $$\langle O \rangle = \frac
{ \sum_{p,q} \int d{\bf R}
\left[ \Psi_{p}^{\dagger}({\bf R}) O \Psi_{q}({\bf R}) /
W_{pq}({\bf R}) \right] W_{pq}({\bf R}) }
{ \sum_{p,q} \int d{\bf R}
\left[ \Psi_{p}^{\dagger}({\bf R}) \Psi_{q}({\bf R}) /
W_{pq}({\bf R}) \right] W_{pq}({\bf R}) } \ ,$$ where samples are drawn from a probability distribution, $W_{pq}({\bf R})$. The subscripts $p$ and $q$ specify the order of operators in the left- and right-hand-side wave functions, while the integration runs over the particle coordinates ${\bf R}=({\bf r}_1,{\bf r}_2,\ldots,{\bf r}_A)$. The probability distribution is constructed from the $\Psi_P$ of Eq.(\[eq:psip\]): $$W_{pq}({\bf R}) = | {\rm Re}( \langle \Psi_{P,p}^{\dagger}({\bf R})
\Psi_{P,q}({\bf R}) \rangle ) | \ .
\label{eq:vmc:weight}$$ This is much less expensive to compute than using the full wave function of Eq.(\[eq:bestpsiv\]) with its spin-orbit and operator-dependent three-body correlations, but it typically has a norm within 1–2% of the full wave function.
Expectation values have a statistical error which can be estimated by the standard deviation $\sigma$: $$\sigma = \left[ \frac{ \langle O^2 \rangle - \langle O \rangle ^2}
{ N-1 } \right] ^{1/2} \ ,$$ where $N$ is the number of statistically independent samples. Block averaging schemes can also be used to estimate the autocorrelation times and determine the statistical error.
The wave function $\Psi$ can be represented by an array of $2^A \times (^A_Z)$ complex numbers, $$\Psi({\bf R}) = \sum_{\alpha} \psi_{\alpha}({\bf R}) |\alpha\rangle \ ,
\label{eq:psivec}$$ where the $\psi_{\alpha}({\bf R})$ are the complex coefficients of each state $|\alpha\rangle$ with specific third components of spin and isospin. This gives vectors with 96, 1280, 4480, and 14336 complex numbers for $^4$He, $^6$Li, $^7$Li, and $^8$Li, respectively. The spin, isospin, and tensor operators $O^{p=2,6}_{ij}$ contained in the two-body correlation operator $U_{ij}$, and in the Hamiltonian are sparse matrices in this basis. For forces that are largely charge-independent, as is the case here, this charge-conserving basis can be replaced with an isospin-conserving basis that has $N(A,T) = 2^A \times I(A,T)$ components, where $$I(A,T) = \frac{2T+1}{{\textstyle{1\over2}}A+T+1}
\left(\begin{array}{c} A \\ {\textstyle{1\over2}}A+T \end{array}\right) \ .
\label{eq:numiso}$$ This reduces the number of vector elements to 32, 320, 1792, and 7168 for the cases given above — a significant savings. In practice, isospin operators are more expensive to evaluate in this basis, but the overall savings in computation is still large. Furthermore, if for even-$A$ nuclei the $M=0$ state is computed, only half the spin components need to be evaluated; the other ones can be found by time-reversal invariance.
Expectation values of the kinetic energy and spin-orbit potential require the computation of first derivatives and diagonal second derivatives of the wave function. These are obtained by evaluating the wave function at $6A$ slightly shifted positions of the coordinates ${\bf R}$ and taking finite differences, as discussed in Ref. [@W91]. Potential terms quadratic in [**L**]{} require mixed second derivatives, which can be obtained by additional wave function evaluations and finite differences.
The rapid growth in the size of $\Psi$ as $A$ increases is the chief limitation, both in computer time and memory requirements, on extending the current method to larger nuclei. The cost of an energy calculation for a given configuration also increases as the number of pairs, $P = {\textstyle{1\over2}}A(A-1)$, because of the number of pair operations required to construct $\Psi$, and roughly as the number of particles, $A$, due to the number of wave function evaluations required to compute the kinetic energy by finite differences. These factors are shown in Table \[tab:scaling\] for a number of nuclei, with a final overall product showing the cost relative to that for an $^8$Be calculation. Some initial $A=9,10$ calculations have been made, but are at the limit of current computer resources, so while $^{12}$C should be reached in a few years, it may be the practical limit for this approach. Calculations of nuclei like $^{16}$O or $^{40}$Ca, will require other methods such as auxiliary-field diffusion Monte Carlo, which samples the spin and isospin components of the wave function by introducing auxiliary fields, as the plain Monte Carlo samples the spatial part. Also shown in the table is the cost for several pure neutron problems, such as an eight-body drop (with external confining potential) or 14 or 38 neutrons in a box (with periodic boundary conditions) as neutron matter simulations. The $^8$n drop has been calculated [@PSCPPR96; @SRP97] by VMC and GFMC, and initial box simulations have been done for 14 neutrons by GFMC [@carlson] and up to 54 neutrons by AFDMC [@SF99].
[lrrll]{} & $A$ & $P$ & $2^A\times I(A,T)$ & $\prod / \prod(^8$Be)\
$^3$H & 3 & 3 & 8$\times$2 & 0.0004\
$^4$He & 4 & 6 & 16$\times$2 & 0.001\
$^5$He & 5 & 10 & 32$\times$5 & 0.020\
$^6$Li & 6 & 15 & 64$\times$5 & 0.036\
$^7$Li & 7 & 21 & 128$\times$14 & 0.66\
$^8$Be & 8 & 28 & 256$\times$14 & 1.\
$^9$Be & 9 & 36 & 512$\times$42 & 17.\
$^{10}$B & 10 & 45 & 1024$\times$42 & 24.\
$^{12}$C & 12 & 66 & 4096$\times$132 & 530.\
$^{16}$O & 16 & 120 & 65536$\times$1430 & $2 \times 10^5$\
$^{40}$Ca & 40 & 780 & 7$\times 10^{21}$ & $3 \times 10^{20}$\
$^8$n & 8 & 28 & 256$\times$1 & 0.071\
$^{14}$n & 14 & 91 & 16384$\times$1 & 26.\
$^{38}$n & 38 & 703 & 3$\times 10^{11}$ & $9 \times 10^9$\
A major problem arises in minimizing the variational energy for p-shell nuclei using the above wave functions: there is no variational minimum that gives reasonable rms radii. For example, the variational energy for $^6$Li is slightly more bound than for $^4$He, but is not more bound than for separated $^4$He and $^2$H nuclei, so the wave function is not stable against breakup into $\alpha + d$ subclusters. Consequently, the energy can be lowered toward the sum of $^4$He and $^2$H energies by making the wave function more and more diffuse. Such a diffuse wave function would not be useful for computing other nuclear properties, or as a starting point for the GFMC calculation, so the search for variational parameters is constrained by requiring the resulting point proton rms radius, $r_p$, to be close to the experimental values for $^6$Li and $^7$Li ground states. For other $A$ = 6–8 ground states, and all the excited states, the trial functions contain minimal changes to the $^6$Li and $^7$Li wave functions, with the added requirement that excited states should not have smaller radii than the ground states. The final step is always the diagonalization of the Hamiltonian in the $\beta_{LS[n]}$ mixing parameters.
GREEN’S FUNCTION MONTE CARLO
============================
The GFMC method [@C87; @C88] projects out the exact lowest-energy state, $\Psi_{0}$, for a given set of quantum numbers, using $\Psi_0 = \lim_{\tau \rightarrow \infty} \exp [ - ( H - E_0) \tau ] \Psi_T$, where $\Psi_{T}$ is an initial trial function. If the maximum $\tau$ actually used is large enough, the eigenvalue $E_{0}$ is calculated exactly while other expectation values are generally calculated neglecting terms of order $|\Psi_{0}-\Psi_{T}|^{2}$ and higher [@PPCPW97]. In contrast, the error in the variational energy, $E_{V}$, is of order $|\Psi_{0}-\Psi_{T}|^{2}$, and other expectation values calculated with $\Psi_{T}$ have errors of order $|\Psi_{0}-\Psi_{T}|$. In the following we present a brief overview of modern nuclear GFMC methods; much more detail may be found in Refs. [@PPCPW97; @WPCP00].
We start with the $\Psi_{T}$ of Eq.(\[eq:psitgfmc\]) and define the propagated wave function $\Psi(\tau)$ $$\begin{aligned}
\Psi(\tau) = e^{-({H}-E_{0})\tau} \Psi_{T}
= \left[e^{-({H}-E_{0})\triangle\tau}\right]^{n} \Psi_{T} \ ,\end{aligned}$$ where we have introduced a small time step, $\tau=n\triangle\tau$; obviously $\Psi(\tau=0) = \Psi_{T}$ and $\Psi(\tau \rightarrow \infty) = \Psi_{0}$. The $\Psi (\tau)$ is represented by a vector function of $\bf R$ using Eq.(\[eq:psivec\]), and the Green’s function, $G_{\alpha\beta}({\bf R},{\bf R}^{\prime})$ is a matrix function of $\bf R$ and ${\bf R}^{\prime}$ in spin-isospin space, defined as $$G_{\alpha\beta}({\bf R},{\bf R}^{\prime})= \langle {\bf
R},\alpha|e^{-({H}-E_{0})\triangle\tau}|{\bf R}^{\prime},\beta\rangle \ .
\label{eq:gfunction}$$ It is calculated with leading errors of order $(\triangle\tau)^{3}$. Omitting the spin-isospin indices $\alpha$, $\beta$ for brevity, $\Psi({\bf R}_{n},\tau)$ is given by $$\Psi({\bf R}_{n},\tau) = \int G({\bf R}_{n},{\bf R}_{n-1})\cdots G({\bf
R}_{1},{\bf R}_{0})\Psi_{T}({\bf R}_{0})d{\bf R}_{n-1}\cdots d{\bf R}_{1}d{\bf
R}_{0} \ ,
\label{eq:gfmcpsi}$$ with the integral being evaluated stochastically.
The short-time propagator should allow as large a time step $\triangle\tau$ as possible, because the total computational time for propagation is proportional to $1/\triangle\tau$. Earlier calculations [@PPCW95; @C87; @C88] used the propagator obtained from the Feynman formulae in which the kinetic and potential energy terms of $H$ are separately exponentiated. The main error in this approximation to $G_{\alpha,\beta}$ comes from terms in $e^{-{H}\triangle\tau}$ having multiple $v_{ij}$, like $v_{ij}Kv_{ij}(\triangle\tau)^{3}$, where $K$ is the kinetic energy, which can become large when particles $i$ and $j$ are very close due to the large repulsive core in $v_{ij}$. This requires a rather small $\triangle\tau \sim 0.1$ GeV$^{-1}$.
It has been found in studies of bulk helium atoms [@C95] that including the exact two-body propagator allows much larger time steps. This short-time propagator is $$\begin{aligned}
G_{\alpha\beta}({\bf R},{\bf R}^{\prime}) &=& G_{0}({\bf R},{\bf
R}^{\prime})\langle\alpha|\left[{\cal S}\prod_{i<j}\frac{g_{ij}({\bf
r}_{ij},{\bf r}_{ij}^{\prime})}{g_{0,ij}({\bf r}_{ij},{\bf r}_{ij}^{\prime})}
\right] |\beta\rangle \ ,\end{aligned}$$ where $$\begin{aligned}
G_{0}({\bf R},{\bf R}^{\prime}) &=& \langle {\bf R}|e^{-{K}\triangle\tau}|{\bf
R}^{\prime}\rangle = \left[ \sqrt{\frac{m}
{2\pi\hbar^{2} \triangle\tau}}\, \right]^{3A}\exp\left[\frac{-({\bf R}-{\bf
R}^{\prime})^2}{2\hbar^{2}\triangle\tau/m}\right] \ ,
\label{eq:propagator2}\end{aligned}$$ $g_{ij}$ is the exact two-body propagator, $$\begin{aligned}
g_{ij}({\bf r}_{ij},{\bf r}_{ij}^{\prime}) = \langle{\bf
r}_{ij}|e^{-H_{ij}\triangle\tau}|{\bf r}_{ij}^{\prime}\rangle \ ,
\label{eq:gij}\end{aligned}$$ and $g_{0,ij}$ is the free two-body propagator. All terms containing any number of the same $v_{ij}$ and $K$ are treated exactly in this propagator, as we have included the imaginary-time equivalent of the full two-body scattering amplitude. It still has errors of order $(\triangle\tau)^{3}$, however they are from commutators of terms like $v_{ij}Kv_{ik}(\triangle\tau)^{3}$ which become large only when both pairs $ij$ and $ik$ are close. Because this is a rare occurrence, a five times larger time step, $\triangle\tau \sim 0.5$ GeV$^{-1}$, can be used [@PPCPW97].
Including the three-body forces and the $E_{0}$ in Eq.(\[eq:gfunction\]), the complete propagator is given by $$\begin{aligned}
G_{\alpha\beta}({\bf R},{\bf R}^{\prime})& = &e^{E_{o}\triangle\tau}G_{0}({\bf
R},{\bf R}^{\prime})\exp[{- \sum (V^{R}_{ijk}({\bf R})+ V^{R}_{ijk}({\bf
R^{\prime}}))\frac{\triangle\tau}{2}}] \nonumber \\
& & \langle\alpha|I_{3}({\bf R})|\gamma\rangle\langle\gamma|\left[{\cal
S}\prod_{i<j}\frac{g_{ij}({\bf r}_{ij},{\bf r}_{ij}^{\prime})}
{g_{0,ij}({\bf r}_{ij},{\bf r}_{ij}^{\prime})} \right]
|\delta\rangle\langle\delta|I_{3}({\bf
R}^{\prime})|\beta\rangle~,
\label{eq:fullprop}\end{aligned}$$ where $$\begin{aligned}
I_{3}({\bf R}) = \left[1 - \frac{\triangle\tau}{2}\sum V^{\pi}_{ijk}({\bf
R})\right]~.\end{aligned}$$ and $V^{\pi}_{ijk}=V^{2\pi}_{ijk}+V^{3\pi}_{ijk}$ represents, in general, all non-central $V_{ijk}$ terms. With the exponential of $V^{\pi}_{ijk}$ expanded to first order in $\triangle\tau$, there are additional error terms of the form $V^{\pi}_{ijk}V^{\pi}_{i'j'k'}(\triangle\tau)^{2}$. However, they have negligible effect because $V^{\pi}_{ijk}$ has a magnitude of only a few MeV.
Quantities of interest are evaluated in terms of a “mixed” expectation value between $\Psi_T$ and $\Psi(\tau)$: $$\begin{aligned}
\langle O \rangle_{\rm Mixed} & = & \frac{\langle \Psi_{T} | O |
\Psi(\tau)\rangle}{\langle \Psi_{T} | \Psi(\tau)\rangle} \nonumber \\
& = & \frac{ \int d {\bf P}_n
\Psi_{T}^{\dagger}({\bf R}_{n}) O G({\bf R}_{n},{\bf
R}_{n-1})\cdots G({\bf R}_{1},{\bf R}_{0})\Psi_{T}({\bf R}_{0})}
{\int d{\bf P}_{n} \Psi_{T}^{\dagger}
({\bf R}_{n})G({\bf R}_{n},{\bf R}_{n-1}) \cdots G({\bf
R}_{1},{\bf R}_{0})\Psi_{T}({\bf R}_{0})}~,
\label{eq:expectation}\end{aligned}$$ where ${\bf P}_{n} = {\bf R}_{0},{\bf R}_{1},\cdots,{\bf R}_{n}$ denotes the ‘path’, and $d{\bf P}_{n} = d{\bf R}_{0} d{\bf R}_{1}\cdots d{\bf R}_{n}$ with the integral over the paths being carried out stochastically. The desired expectation values would, of course, have $\Psi(\tau)$ on both sides; by writing $\Psi(\tau) = \Psi_{T} + \delta\Psi(\tau)$ and neglecting terms of order $[\delta\Psi(\tau)]^2$, we obtain the approximate expression $$\begin{aligned}
\langle O (\tau)\rangle =
\frac{\langle\Psi(\tau)| O |\Psi(\tau)\rangle}
{\langle\Psi(\tau)|\Psi(\tau)\rangle}
\approx \langle O (\tau)\rangle_{\rm Mixed}
+ [\langle O (\tau)\rangle_{\rm Mixed} - \langle O \rangle_T] ~,
\label{eq:pc_gfmc}\end{aligned}$$ where $\langle O \rangle_T$ is the variational expectation value. More accurate evaluations of $\langle O (\tau)\rangle$ are possible [@K67], essentially by measuring the observable at the mid-point of the path. However, such estimates require a propagation twice as long as the mixed estimate and require separate propagations for every $\langle O \rangle$ to be evaluated. The nuclear calculations published to date use the approximation of Eq.(\[eq:pc\_gfmc\]).
A special case is the expectation value of the Hamiltonian. The $\langle{H}(\tau)\rangle_{\rm Mixed}$ can be re-expressed as [@CK79] $$\begin{aligned}
\langle{H}(\tau)\rangle_{\rm Mixed} = \frac{\langle \Psi_{T} |
e^{-({H}-E_{0})\tau /2}{H} e^{-({H}-E_{0})\tau /2} |
\Psi_{T}\rangle}{\langle \Psi_{T} |e^{-({H}-E_{0})\tau /2}
e^{-({H}-E_{0})\tau /2}| \Psi_{T}\rangle} \geq E_{0}~,\end{aligned}$$ since the propagator $\exp [ - (H - E_0) \tau ] $ commutes with the Hamiltonian. Thus $\langle{H}(\tau)\rangle_{\rm Mixed}$ approaches $E_{0}$ in the limit $\tau\rightarrow\infty$, and the expectation value obeys the variational principle for all $\tau$.
The AV18 interaction contains terms that are quadratic in orbital angular momentum, $L$. These terms are, in essence, state- and position-dependent modifications of the mass of the nucleons. If they are included in the calculation of $g_{ij}$, then the ratio $g_{ij}/g_{0,ij}$ in Eq.(\[eq:propagator2\]) becomes unbounded for large $|{\bf r}_{ij}|$ or $|{\bf r}_{ij}^{\prime}|$ and the Monte Carlo statistical error will also be unbounded. Hence the GFMC propagator is constructed with a simpler isoscalar interaction, $H^{\prime}$, with a $v^{\prime}_{ij}$ that has only eight operator terms, $[1, {\mbox{\boldmath$\sigma$}}_{i}\cdot{\mbox{\boldmath$\sigma$}}_{j}, S_{ij}, {\bf L\cdot S}]\otimes
[1, {\mbox{\boldmath$\tau$}}_{i}\cdot{\mbox{\boldmath$\tau$}}_{j}]$, chosen such that it equals the CI part of the full interaction in all S- and P-waves and in the deuteron. The $v^{\prime}_{ij}$ is a little more attractive than $v_{ij}$, so a $V^{\prime}_{ijk}$ adjusted so that $\langle H^{\prime} \rangle \approx \langle H \rangle$ is also used. This ensures the GFMC algorithm will not propagate to excessively large densities due to overbinding. Consequently, the upper bound property applies to $\langle{H}^{\prime}(\tau)\rangle$, and $\langle H-H^{\prime} \rangle$ must be evaluated perturbatively.
Another complication that arises in the GFMC algorithm is the “fermion sign problem”. This arises from the stochastic evaluation of the matrix elements in Eq.(\[eq:expectation\]). The $G({\bf R}_{i},{\bf R}_{i-1})$ is a local operator and can mix in the boson solution. This has a (much) lower energy than the fermion solution and thus is exponentially amplified in subsequent propagations. In the final integration with the antisymmetric $\Psi_T$, the desired fermionic part is projected out, but in the presence of large statistical errors that grow exponentially with $\tau$. Because the number of pairs that can be exchanged grows with $A$, the sign problem also grows exponentially with increasing $A$. For $A{\geq}8$, the errors grow so fast that convergence in $\tau$ cannot be achieved.
For simple scalar wave functions, the fermion sign problem can be controlled by not allowing the propagation to move across a node of the wave function. Such “fixed-node” GFMC provides an approximate solution which is the best possible variational wave function with the same nodal structure as $\Psi_T$. However, a more complicated solution is necessary for the spin- and isospin-dependent wave functions of nuclei. In the last few years a suitable “constrained path” approximation has been developed and extensively tested, first for condensed matter systems [@ZCG95] and more recently for nuclei [@WPCP00]. The basic idea of the constrained-path method is to discard those configurations that, in future generations, will contribute only noise to expectation values. If the exact ground state $| \Psi_0 \rangle$ were known, any configuration for which $$\Psi({\bf P}_n)^\dagger \Psi_0({\bf R}_n) = 0 \ ,
\label{eq:gfmc:const_config}$$ where a sum over spin-isospin states is implied, could be discarded. Here ${\bf P}_n$ designates the complete path, $({\bf R}_0, ..., {\bf R}_n)$ that has led to ${\bf R}_n$. The sum of these discarded configurations can be written as a state $| \Psi_d \rangle$, which obviously has zero overlap with the ground state. The $\Psi_d$ contains only excited states and should decay away as $\tau \rightarrow \infty$, thus discarding it is justified. Of course the exact $\Psi_0$ is not known, and so configurations are discarded with a probability such that the average overlap with the trial wave function, $$\langle \Psi_d | \Psi_T \rangle = 0 \ .$$ Many tests of this procedure have been made [@WPCP00] and it usually gives results that are consistent with unconstrained propagation, within statistical errors. However a few cases in which the constrained propagation converges to the wrong energy (either above or below the correct energy) have been found. Therefore a small number, $n_u=10$ to 20, unconstrained steps are made before evaluating expectation values. These few unconstrained steps, out of typically 400 total steps, appear to be enough to damp out errors introduced by the constraint, but do not greatly increase the statistical error.
Figure \[fig:8be-e\_of\_tau\] shows the progress with increasing $\tau$ of typical constrained GFMC calculations, in this case for various states of $^8$Be. The values shown at $\tau=0$ are the VMC values using $\Psi_T$; VMC values with the best known $\Psi_V$ are about 1.5 MeV below these. The GFMC very rapidly makes a large improvement on these energies; by $\tau=0.01$ MeV$^{-1}$, the $\Psi_T$ energies have been reduced by 6.5 to 8 MeV. This rapid improvement corresponds to the removal of small admixtures of states with excitation energies $\sim~1$ GeV from $\Psi_T$. Averages over typically the last nine $\tau$ values are used as the GFMC energy. The standard deviation, computed using block averaging, of all of the individual energies for these $\tau$ values is used to compute the corresponding statistical error. The solid lines show these averages; the corresponding dashed lines show the statistical errors. The g.s., $1^+$, and $3^+$ states of $^8$Be have widths less than 250 keV and their $E(\tau)$ appear to be converged and constant over the averaging regime. The $2^+$ state has a width of 1.5 MeV and the $4^+$ state’s width is 3.5 MeV; the $E(\tau)$ for the $2^+$ state might be converged, but that of the $4^+$ state is clearly steadily decreasing. Reliable estimates of the energies of broad resonances requires the use of scattering-state boundary conditions (see Sec. \[sec:scat\]) which have not yet been implemented for more than five nucleons.
Tests with different $\Psi_T$ as starting points for the GFMC calculation are described in Ref. [@PPCPW97]. The most crucial aspect in choosing $\Psi_T$ is that it have the correct mix of spatial symmetries, i.e., the $\beta_{LS[n]}$ admixtures. This is because the limited propagation time is not sufficient for the GFMC algorithm to filter out low-lying excitations with the same quantum numbers. This issue is unimportant in $^4$He, where the first 0$^+$ excited state is near 20 MeV, but in $^6$Li, the first 1$^+$ excited state is at only 5.65 MeV; other $p$-shell nuclei have similar low-lying excitations. Otherwise, the GFMC algorithm is able to rapidly correct for some very poor $\Psi_T$ whose variational energies are actually positive. However, a good $\Psi_T$ helps to keep the error bars small at larger $\tau$. The most efficient balance between speed of construction of $\Psi_T$ and smallest number of samples needed to achieve a given error bar seems to be given by Eq.(\[eq:psitgfmc\]). A number of other tests of the GFMC algorithm and its ability to determine nuclear radii are described in Refs. [@PPCPW97; @WPCP00].
The calculations described here are computationally intense, and would not have been possible without the advent of parallel supercomputers. The initial studies [@PPCW95] of $^6$Li required $\sim 2,000$ node hours on an IBM SP1 to propagate 10,000 configurations to $\tau$ = 0.06 MeV$^{-1}$. Today, with improvements in the algorithm and after much effort to optimize the computer codes, the same calculation requires about 15 node hours on a third generation IBM SP. However, a present calculation for $^8$Li, requiring 10,000 configurations to get a reasonable error bar, takes 1,000 node hours, so forefront computer resources remain essential for this program.
ENERGY RESULTS
==============
There are a number of accurate many-body methods for evaluating the binding energy of the s-shell nuclei, $^3$H, $^3$He, and $^4$He, using realistic nuclear forces. These include Faddeev in configuration space [@FPSS93] (FadR), Faddeev and Faddeev-Yakubovsky in momentum space [@NKG00] (FadQ), and hyperspherical harmonics [@HH] (HH), correlated hyperspherical harmonics [@VKR95] (CHH) and pair-correlated hyperspherical harmonics [@KVR93] (PHH). Some results of these methods for the AV18 and AV18/UIX Hamiltonians are shown in Table \[tab:s-shell\]. We observe that the VMC upper bounds are generally 1.5–2.0% above the GFMC energies, which in turn are 0.25–1.0% above the Faddeev results. The disagreement between GFMC and the other methods may be attributable to the fact that GFMC is propagated with an $H'$ and a small piece of the Hamiltonian is computed in perturbation. At this fine level of comparison, one needs to worry about how features such as $T={\textstyle{3\over2}}$ admixtures and $n-p$ mass differences in the trinucleon ground state are treated in each calculation.
Table \[tab:s-shell\] also shows the necessity of including (at least) a three-nucleon potential in the Hamiltonian in order to reproduce the A=3,4 experimental binding energies. This is true for all the other modern $N\!N$ potentials, as shown in Refs. [@FPSS93; @NKG00]. The local potentials AV18, Reid93, and Nijm II, give very similar results, while the slightly nonlocal Nijm I gives 1–3% more binding, and the more nonlocal CD-Bonn gives 5–8% more binding, or about halfway between the local potential values and experiment. At present, no two- plus three-nucleon potential combination gives an exact fit to both trinucleon and $^4$He energies, but several combinations, like AV18/UIX, come quite close. It also appears that it may be easier to fit $^3$He and $^4$He simultaneously, rather than $^3$H and $^4$He, perhaps because of the better data constraints on the $pp$ interaction compared to the $nn$ interaction [@NKG00]. In principle, there should also be four-body forces, but their contribution must be quite small, and it is impossible to unambiguously identify a need for such terms until a more thorough survey of possible three-nucleon potentials is made and the discrepancies between the various many-body methods are resolved.
[llrrrrrrr]{} Hamiltonian & nucleus & VMC & GFMC & FadQ & FadR & HH & Expt.\
AV18 & $^3$H & –7.50(1) & –7.61(1) & –7.623 &–7.62 &–7.618 & 8.482\
& $^4$He &–23.72(3) &–24.07(4) &–24.28 & &–24.18 & 28.296\
AV18/UIX & $^3$H & –8.32(1) & –8.46(1) & –8.478 & &–8.475 & 8.482\
& $^4$He &–27.78(3) &–28.33(2) &–28.50 & &–28.1 & 28.296\
Figure \[fig:vmc-gfmc-exp\] compares VMC (using the full $\Psi_V$ of Eq.(\[eq:bestpsiv\])) and GFMC calculations for the AV18/UIX Hamiltonian, and also shows experimental energies. All GFMC energies in this review are from Ref. [@PPWC01]; VMC energies are from Refs. [@WPCP00; @PPCPW97]. All particle-stable or narrow-width ($\Gamma<150$ keV) states for $4\leq{A}\leq8$ are shown, except isobaric analogs. We see that the VMC calculations get progressively worse in the p-shell compared to the s-shell, being on the order of 10–15% above the final GFMC results, for nuclei with $N \sim Z$. In absolute terms, $\Psi_V$ misses roughly an extra 1.5 MeV of binding for each p-shell nucleon that is added. The VMC results also fail to reproduce important qualitative features of the GFMC calculations; for example $^6$Li is stable against breakup into $\alpha$+d for this Hamiltonian but the VMC calculation shows it unbound by $\sim$2 MeV (the black dashed lines show the indicated thresholds computed using the sub-cluster energies appropriate to each calculation or experiment). Clearly, there is some significant feature of light p-shell nuclei that is not yet included in the trial wave functions.
Table \[tab:energy\] shows the GFMC energy values for several Hamiltonians. The table and Fig. \[fig:vmc-gfmc-exp\] show that the AV18/UIX Hamiltonian, which gives excellent energies, compared to experiment, for the s-shell nuclei, significantly underbinds the p-shell nuclei. The Li isotopes are stable against breakup into subclusters with AV18/UIX, but progressively more underbound as $A$ increases. There is also an isospin problem, in that the He isotopes are even further off from the experimental values, and in fact do not show stability against breakup into subclusters.
[lcccl]{} & AV18 & AV18/UIX & AV18/IL2 & Expt.\
$^3$H(${\textstyle{1\over2}}^+$) & $-$7.61(1) & $-$8.46(1) & $-$8.43(1) & $-$8.48\
$^4$He(0$^+$) & $-$24.07(4) & $-$28.33(2) & $-$28.37(3) & $-$28.30\
$^6$He(0$^+$) & $-$23.9(1) & $-$28.1(1) & $-$29.4(1) & $-$29.27\
$^6$Li(1$^+$) & $-$26.9(1) & $-$31.1(1) & $-$32.3(1) & $-$31.99\
$^7$He(${\textstyle{3\over2}}^-$) & $-$21.2(2) & $-$25.8(2) & $-$29.2(3) & $-$28.82\
$^7$Li(${\textstyle{3\over2}}^-$) & $-$31.6(1) & $-$37.8(1) & $-$39.6(2) & $-$39.24\
$^8$He(0$^+$) & $-$21.6(2) & $-$27.2(2) & $-$31.3(3) & $-$31.41\
$^8$Li(2$^+$) & $-$31.8(3) & $-$38.0(2) & $-$42.2(2) & $-$41.28\
$^8$Be(0$^+$) & $-$45.6(3) & $-$54.4(2) & $-$56.6(4) & $-$56.50\
$^7$n(${\textstyle{1\over2}}^-$) & $-$33.47(5) & $-$33.2(1) & $-$35.8(2) &\
$^8$n(0$^+$) & $-$39.21(8) & $-$37.8(1) & $-$41.1(3) &\
The new Illinois three-nucleon potentials were constructed to solve the p-shell binding problems. Figure \[fig:av18-il2-exp\] and Table \[tab:energy\] compare GFMC calculations for AV18 with no $V_{ijk}$ and AV18/IL2 to the same experimental energies shown in Fig. \[fig:vmc-gfmc-exp\]. The AV18/IL2 Hamiltonian does a good job of reproducing these energies; the rms deviation from experiment for these levels is only 360 keV, while it is 2.3 MeV for AV18/UIX. The AV18 values with no $V_{ijk}$ show the large contribution that three-nucleon potentials make to these binding energies; for $A=8$ the IL2 increases the binding energy by more than 10 MeV.
Table \[tab:sosplit\] shows some splittings for states that in a simple shell-model picture have the same $L$ and $S$ but different total $J$. In all cases the lowest state for each $J$ is used. Some of the states, particularly the $^5$He states, have large experimental widths; calculations such as those described in Sec. \[sec:scat\] would be more reliable. The splittings are computed for the AV18, AV18/UIX, and AV18/IL2 Hamiltonians and are compared to experimental values. The AV18 with no $V_{ijk}$ significantly underpredicts all the splittings except the ${\textstyle{1\over2}}^- - {\textstyle{3\over2}}^-$ doublet in $^7$Li; however in this case the relatively large statistical error for the small splitting makes any conclusion difficult (the computed splittings are the difference of two independent GFMC calculations whose statistical errors must be added in quadrature). The AV18/UIX Hamiltonian substantially improves the $^5$He splitting, and a VMC study showed that an earlier Urbana $V_{ijk}$ significantly increased the computed splitting in $^{15}$N, resulting in much better agreement with experiment [@PP93]. However the other splittings in the table are still underpredicted with AV18/UIX. The AV18/IL2 results in good predictions of all the splittings except for the 1$^+-2^+$ doublet in $^8$Li which is overpredicted.
[lccccccc]{} & &$L$& $S$ & AV18 & AV18/UIX & AV18/IL2 & Expt.\
$^5$He & ${\textstyle{1\over2}}^- - {\textstyle{3\over2}}^-$ & 1 & ${\textstyle{1\over2}}$ & 0.6(1) & 1.1(2) & 1.3(2) & 1.20\
$^6$Li & 2$^+ - 3^+$ & 2 & 1 & 0.8(1) & 0.9(1) & 2.2(2) & 2.12\
$^7$Li & ${\textstyle{1\over2}}^- - {\textstyle{3\over2}}^-$ & 1 & ${\textstyle{1\over2}}$ & 0.5(2) & 0.3(2) & 0.6(3) & 0.47\
$^7$Li & ${\textstyle{5\over2}}^- - {\textstyle{7\over2}}^-$ & 3 & ${\textstyle{1\over2}}$ & 0.7(2) & 0.8(2) & 2.2(3) & 2.05\
$^8$Li & 1$^+ - 2^+$ & 1 & 1 & 0.2(4) & 0.6(2) & 1.7(4) & 0.98\
$^7$n & ${\textstyle{3\over2}}^- - {\textstyle{1\over2}}^-$ & 1 & ${\textstyle{1\over2}}$ & 1.65(7) & 1.5(1) & 2.8(3) &\
A detailed breakdown of the GFMC ground-state energies for AV18/IL2 into kinetic, two- and three-nucleon interactions is given in Table \[tab:gfmc\]. Because of the extrapolation of the mixed expectation values, Eq.(\[eq:pc\_gfmc\]), these components are not as accurate as the total energy, and do not add up to the full amount; indeed the sum of the individual kinetic and potential energies differs from $\langle H \rangle$ by the same amount that the GFMC has improved the $\Psi_T$ energy. Nevertheless, they give a good idea about the relative size of the different terms involved. There is a big cancellation between the kinetic and two-body potential terms. Consequently, while $V_{ijk}$ is less than 8% of $v_{ij}$ in magnitude, its expectation value is up to 50% of the net binding (however the net effect of the IL2 $V_{ijk}$, defined as the difference of the binding energies computed without and with $V_{ijk}$, is at most 30%, as can be seen in Table \[tab:energy\]). This difference between expectation value and net effect is due to the large change that $V_{ijk}$ induces in $\langle K+v_{ij} \rangle$.
[lccccccl]{} & $K$ & $v_{ij}$ & $V_{ijk}$ & $v^{\gamma}_{ij}$ & $v^{\pi}_{ij}$ & $V^{2\pi}_{ijk}$ & $V^{3\pi}_{ijk}$\
$^3$H & 51. & $-$59. & $-$1.5 & 0.04 & $-$45. & $-$3.0 & 0.18(1)\
$^4$He & 115.(1) & $-$136.(1) & $-$8.4(1) & 0.86 & $-$105. & $-$16.3(1) & 0.63(1)\
$^6$He & 147.(2) & $-$171.(2) & $-$11.5(3) & 0.87 & $-$127.(1) & $-$20.3(4) & $-$0.91(6)\
$^6$Li & 160.(2) & $-$187.(2) & $-$11.1(3) & 1.73 & $-$150.(1) & $-$19.8(4) & $-$0.44(5)\
$^7$He & 175.(3) & $-$199.(3) & $-$16.3(4) & 0.89 & $-$145.(2) & $-$25.(1) & $-$3.1(1)\
$^7$Li & 199.(3) & $-$232.(3) & $-$14.5(4) & 1.80 & $-$178.(2) & $-$25.6(6) & $-$1.1(1)\
$^8$He & 190.(3) & $-$218.(3) & $-$16.3(5) & 0.89 & $-$153.(1) & $-$25.6(6) & $-$4.0(1)\
$^8$Li & 242.(2) & $-$278.(2) & $-$20.6(4) & 1.93 & $-$211.(1) & $-$34.2(5) & $-$3.8(1)\
$^8$Be & 256.(4) & $-$303.(3) & $-$21.(1) & 3.32 & $-$234.(2) & $-$38.5(9) & $-$0.9(2)\
$^7$n & 105.(1) & $-$59.(1) & $-$3.6(3) & 0.07 & $-$10. & $-$0.1(1) & $-$5.4(3)\
$^8$n & 122.(1) & $-$73.(1) & $-$3.0(3) & 0.09 & $-$12. & 0.3(1) & $-$5.9(4)\
Among the subcomponents of $v_{ij}$, the one-pion exchange dominates, being 70–80% of the total $v_{ij}$ for $A\geq3$ nuclei. Similarly, the two-pion exchange is the dominant component of $V_{ijk}$. The three-pion exchange term is small and repulsive in s-shell nuclei, but attractive in the p-shell nuclei, which have $T={\textstyle{3\over2}}$ triples. It is this term which results in the large improvement of the Illinois models over the Urbana models for p-shell nuclei; however its contributions are always less than 15% of the two-pion $V_{ijk}$. Finally, we note that the electromagnetic $v^\gamma_{ij}$ is dominated by the Coulomb interaction between protons, but about 17% (8%) of its total contribution comes from the magnetic moment and other terms in He (Li) isotopes.
[lcrccrr]{} $a_n(A,T)$ & $K^{CSB}$& $v_{C1}(pp)$ & $v^{\gamma,R}$ & $v^{CSB}+v^{CD}$ & Total & Expt.\
$a_1(3,{\textstyle{1\over2}})$ & 14 & 649(1) & 29 & 64(0) & 757(1) & 764\
$a_1(6,1)$ & 16 & 1091(5) & 18 & 47(1) & 1172(6) & 1173\
$a_2(6,1)$ & & 166(1) & 19 & 107(13) & 293(13) & 223\
$a_1(7,{\textstyle{1\over2}})$ & 22 & 1447(6) & 40 & 79(2) & 1588(7) & 1644\
$a_1(7,{\textstyle{3\over2}})$ & 18 & 1337(6) & 12 & 52(1) & 1420(8) & 1373\
$a_2(7,{\textstyle{3\over2}})$ & & 137(1) & 7 & 36(6) & 180(7) & 175\
$a_1(8,1)$ & 23 & 1686(5) & 24 & 76(1) & 1810(6) & 1770\
$a_2(8,1)$ & & 141(1) & 4 & $-$3(8) & 143(8) & 128\
$a_1(8,2)$ & 18 & 1528(7) & 17 & 59(1) & 1622(8) & 1659\
$a_2(8,2)$ & & 136(1) & 6 & 38(5) & 180(5) & 153\
Energy differences among isobaric analog states are probes of the charge-indepen-dence-breaking parts of the Hamiltonian. The understanding of these “Nolen-Schiffer energies” has been a theoretical problem for 30 years [@NS69]. The energies for a given isomultiplet of states can be expanded as $$E_{A,T}(T_z) = \sum_{n\leq 2T} a_n(A,T) Q_n(T,T_z) \ ,
\label{eq:isoexpand}$$ where $Q_0=1$, $Q_1=T_z$, and $Q_2={\textstyle{1\over2}}(3T_z^2-T^2)$ are isoscalar, isovector, and isotensor terms [@P60]. The isovector and isotensor coefficients $a_n(A,T)$, and various contributions to them, are given in Table \[tab:analog\]. These were calculated as expectation values in the $T_z=-T$ GFMC wave functions for AV18/IL2. The individual terms are: 1) the effect of the neutron-proton mass difference on the kinetic energy, $K^{CSB}$; 2) the proton-proton Coulomb potential, including the AV18 form factor, $v_{C1}(pp)$; 3) all other electromagnetic terms such as vacuum polarization and magnetic-moment terms, $v^{\gamma,R}$; and 4) the strong-interaction contributions, $v^{CSB}$ which contributes to the isotensor coefficients and $v^{CD}$ which contributes to the isovector coefficients.
The $^3$H–$^3$He mass difference is 757 keV with the AV18/IL2 Hamiltonian, in excellent agreement with the experimental value of 764 keV. The bulk of the difference is the $pp$ Coulomb energy, but 108 keV comes from the other terms. Because of its very long range, the $v_{C1}(pp)$ dominates the $a_1(A,T)$ of heavier nuclei even more and small errors in the rms radius of the nucleus can result in changes in the $v_{C1}(pp)$ contribution that can totally mask the effects of the other, much smaller, terms.
[lcccccc]{} $J^{\pi}$ & $K^{CSB}$& $v_{C1}(pp)$ & $v^{\gamma,R}$ & $v^{CSB}$ & $E_{01}$ & Expt.\
2$^+$ & 2 & 62 & 19 & 26 & 109(4) & 149\
1$^+$ & 1 & 39 & 0 & 15 & 55(2) & 120\
3$^+$ & 1 & 33 & 15 & 13 & 62(2) & 63\
\[table:mixing\]
Two (2$^+$;0+1) states occur very close together in the spectrum of $^8$Be at 16.6 and 16.9 MeV excitation; these isospin-mixed states come from blending the (2$^+$;1) isobaric analog of the $^8$Li ground state with the second (2$^+$;0) excited state. There are also fairly close (1$^+$;0,1) and (3$^+$;0,1) pairs at slightly higher energies in the $^8$Be spectrum. The isospin-mixing matrix elements that connect these pairs of states, $$E_{01}(J) = \langle \Psi(J^+;0) | H | \Psi(J^+;1) \rangle \ ,$$ have been computed using VMC wave functions for the AV18/UIX Hamiltonian. Results for the $E_{01}(J)$ are given in Table \[table:mixing\]. The experimental values are determined from the observed decay widths and energies [@B66]. The dominant contribution, from the Coulomb potential, typically accounts for less than half of the matrix element. We see that the remaining part of the electromagnetic interaction and the strong CSB interaction can provide a significant boost, although the experimental mixing is still underpredicted by $\sim$20%. It appears that these mixing elements are a more sensitive test of the small CSB components of the Hamiltonian than are the isobaric analog energy differences.
DENSITY AND MOMENTUM DISTRIBUTIONS
==================================
The one- and two-nucleon distributions of light $p$-shell nuclei are interesting in a variety of experimental settings. For example, the Borromean $^6$He and $^8$He nuclei are popular candidates for study as ‘halo’ nuclei whose last neutrons are weakly bound. In addition, the polarization densities of $^6$Li and $^7$Li are important because of possible applications in polarized targets.
[cllll]{} & &\
& GFMC & Expt. & GFMC & Expt.\
$^3$H & 1.59(0) & 1.60 & &\
$^3$He & 1.76(0) & 1.77 & &\
$^4$He & 1.45(0) & 1.47 & &\
$^6$He & 1.91(1) & & &\
$^6$Li & 2.39(1) & 2.43 & –0.32(6) & –0.083\
$^7$Li & 2.25(1) & 2.27 & –3.6(1) & –4.06\
$^7$Be & 2.44(1) & & –6.1(1) &\
$^8$He & 1.88(1) & & &\
$^8$Li & 2.09(1) & & 3.2(1) & 3.11(5)\
$^8$B & 2.45(1) & & 6.4(1) & 6.8(2)\
Point proton rms radii and quadrupole moments are shown in Table \[tab:radii\], as computed by GFMC for the AV18/IL2 model, along with experimental values. The experimental charge radii have been converted to point proton radii by removing the proton and neutron $\langle r^2 \rangle$ of 0.743 and –0.116 fm$^2$, respectively. The radii are in good agreement with experiment, thanks at least in part to the fact that the AV18/IL2 model reproduces the binding energies of these nuclei very well. The quadrupole moments have been calculated in impulse approximation; two-body charge contributions are expected to provide only a few percent correction. The agreement with experiment is again fairly good, with the exception of the $^6$Li quadrupole moment, which involves a delicate cancellation between the contributions from the deuteron quadrupole moment and the D-wave part of the $\alpha$+$d$ relative wave function. In general, quadrupole moments are difficult to calculate accurately with quantum Monte Carlo methods because they are dominated by the long-range parts of the wave functions, which contribute very little to the total energy that VMC and GFMC are both designed to optimize.
[ccrrrr]{} & T & &\
& & GFMC & Expt. & GFMC & Expt.\
$^3$He$-^3$H & ${\textstyle{1\over2}}$ & 0.403(0) & 0.426 & $-$4.330(1) & $-$5.107\
$^6$Li & 0 & 0.817(1) & 0.822\
$^7$Be$-^7$Li & ${\textstyle{1\over2}}$ & 0.894(1) & 0.929 & $-$3.93(1) & $-$4.654\
$^8$B$-^8$Li & 1 & 1.276(1) & 1.345 & 0.369(9) & $-$0.309\
Calculated and experimental isoscalar and isovector magnetic moments are shown in Table \[tab:mag-mom\], as defined with the convention used in Eq.(\[eq:isoexpand\]); thus the isovector values for the $T={\textstyle{1\over2}}$ cases are $-$2 times those often quoted in the literature. (For the $A=8$, $T=1$ nuclei the experimental isoscalar and isovector moments are obtained from the sum and difference of the values for B and Li, since the magnetic moment of the $T=1$ $J^{\pi}=2^+$ state in Be is not measured.) The moments were calculated as expectation values in the GFMC wave functions with $T_z=-T$ in impulse approximation. These impulse isoscalar moments are quite close to experiment and it is expected that two-body current contributions are small [@SPR89]. For isovector moments, however, there can be significant corrections from meson-exchange contributions. We note that the corrections needed for the $A$=7,8 isovector moments are the same sign and magnitude as for $A$=3, so an exchange-current model that fixes the s-shell [@MRS98] is likely to work for the light p-shell nuclei also.
In Fig. \[fig:he-rho1\], we present proton and matter (proton$+$neutron) densities for the stable helium isotopes, as calculated with GFMC for AV18/IL2. Nucleon densities are calculated as simple $\delta$-function expectation values, with possible spin and/or isospin projectors: for example, the proton density is given by $$\rho_{p}(r) \ = \ \frac{1}{4 \pi r^2}
\langle \Psi | \sum_i \
\ \frac { 1 + \tau_{zi}}{2} \ \delta ( r - | {\bf r}_i - {\bf R}_{cm} | )
\ | \Psi \rangle \ .$$ The blue symbols and curves show results for $^6$He, red ones for $^8$He; open symbols give the proton distributions which are also interpreted as the alpha “core” density, full symbols are the total matter densities. It can be seen that, as more neutrons are added, the tails of the matter distributions broaden considerably because of the relatively weak binding of the p-shell neutrons. In addition, the central neutron and proton densities decrease rather dramatically. This effect does not necessarily require any changes to the $\alpha$ core, but can be understood at least partially from the fact that the $\alpha$ no longer sits at the center of mass of the entire system. The motion relative to the center of mass spreads out the mass distribution relative to that of $^4$He.
The figure also shows two attempts to extract the densities of $^{6,8}$He from scattering of beams of these short-lived nuclei from proton targets [@Alkhazov]. The analysis of Alkhazov, et al. (dashed curves) used Glauber theory and assumed proton (core) and matter density distributions which were varied to fit the measured cross sections. The later work of Al-Khalili and Tostevin [@Al-Khalili] (dot-dashed curves) improved on this by doing the Glauber theory using model wave functions which contained correlations between the valence neutrons. The GFMC calculations definitely prefer the latter analysis.
The proton-proton distribution function is defined by $$\rho_{pp} (r) = \frac{1}{4 \pi r^2}
\langle \Psi | \sum_{i<j} \ \frac{1 + \tau_{zi}}{2}
\ \frac { 1 + \tau_{zj}}{2} \ \delta ( r - |{\bf r}_i - {\bf r}_j|)|
\Psi \rangle \ .$$ These distributions are directly related to the Coulomb sum measured in inclusive longitudinal electron scattering; such measurements in $^3$He have been used to put constraints on the $\rho_{pp} (r_{ij})$, and realistic calculations agree with the experimental results[@SWC93]. The behavior of $\rho_{pp} (r)$ at short distances is largely determined by the repulsive core of the $N\!N$ potential and is nearly independent of the nucleus, but at larger distances it is determined by the size of the nucleus.
It is interesting to compare $\rho_{pp}$ for $^4$He to that of $^6$He and $^8$He. These distribution functions are shown in Fig. \[fig:he-pp\], again calculated with GFMC for the AV18/IL2 model. These nuclei each have just one $pp$ pair which presumably is in the “alpha core” of $^{6,8}$He. Unlike the one-body densities, these distributions are not sensitive to center of mass effects, and thus if the alpha core of $^{6,8}$He is not distorted by the surrounding neutrons, all three $\rho_{pp}$ distributions in the figure should be the same. We see that the $pp$ distribution spreads out slightly with neutron number in the helium isotopes, with an increase of the pair rms radius of approximately 4% in going from $^4$He to $^6$He, and 8% to $^8$He. While this could be interpreted as a swelling of the alpha core, it might also be due to the charge-exchange ($\tau_i \cdot \tau_j$) correlations which can transfer charge from the core to the valence nucleons. Since these correlations are rather long-ranged, they can have a significant effect on the $pp$ distribution. VMC calculations of $^4$He with wave functions modified to give $\rho_{pp}$ distributions close to those of $^{6,8}$He suggest that the alpha cores of $^{6,8}$He are excited by $\sim80$ and $\sim350$ keV, respectively.
In general, VMC calculations give one-body densities very similar to the GFMC results, although the two-body densities may differ by up to 10% in the peak, with the GFMC having a somewhat sharper structure. Because the VMC wave functions are simpler and less expensive to construct they have been used in a number of applications where the single-particle structure is dominant. All these calculations were carried out with wave functions for the AV18/UIX Hamiltonian, but we expect that GFMC calculations with the improved AV18/IL2 model will not qualitatively alter the results. (Remember that the VMC wave functions are constrained to reproduce available experimental rms radii.)
VMC calculations of elastic and inelastic electromagnetic form factors for $^6$Li are shown in Fig. \[fig:li6ff\]. These have been made in impulse approximation (IA) and with meson-exchange contributions (MEC) to the charge and density operators [@WS98]. The comparison with data for the elastic longitudinal form factor, $F_L(q^2)$, is excellent, as is the E2 transition to the 3$^+$ first excited state. The MEC corrections are small, but stand out noticeably in the first minimum where they significantly improve the fit to data. The elastic transverse form factor, $F_T(q^2)$, is good up to the first zero, but is a little too large in the region of the second maximum. The M1 transition to the 0$^+$ isobaric analog state is also reproduced reasonably well. This kind of quantitative agreement with data has not been achieved in the past with either shell model or $\alpha$+$d$ cluster wave functions. In fact, shell model calculations normally require the introduction of effective charges, typically adding a charge of $\sim 0.5e$ to both the proton and neutron, to obtain reasonable transition rates [@PWK67]. No effective charges are used in the VMC calculations.
While electron scattering from nuclei is primarily sensitive to the proton distributions in nuclei, $\pi^-$ scattering is most sensitive to the neutron distributions. Pion inelastic scattering at medium energy ($80 \leq E_{lab} \leq 300$ MeV) is dominated by strong absorption due to excitation of the $\Delta$ resonance, and is well described in distorted-wave impulse approximation (DWIA). An analysis of meson factory data for p-shell nuclei based on the Cohen-Kurath shell model [@CK65] showed that reasonable agreement with data could be achieved if the quadrupole excitation operator was enhanced by a factor $\sim$ 1.75–2.5 [@LK80]. Figure \[fig:pipip\] shows a calculation of the differential cross sections for $^7$Li($\pi,\pi^\prime$) scattering to the first three excited states using VMC transition densities as input, with no enhancement factors [@LW01]. Solid red lines show the full results, which are in good agreement with the data, while the dashed blue lines give the contribution coming from protons only. Because of the strong isospin dependence of the $\pi N$ scattering $t$-matrix, this reaction can be a stringent test of the neutron densities predicted by the quantum Monte Carlo calculations.
The VMC wave functions for the AV18/UIX model have also been used to calculate single-nucleon momentum distributions in many nuclei, and a variety of cluster-cluster overlap wave functions, such as $\langle d p | t \rangle$, $\langle d d | \alpha \rangle$, and $\langle
\alpha d | ^6{\rm Li} \rangle$ [@SPW86; @FPPWSA96]. Recently the overlaps $\langle^6{\rm He}(J^{\pi})+p(\ell_j)|^7{\rm Li}\rangle$ for all possible p-shell states in $^6$He were studied [@LWW99]. The spectroscopic factors obtained from these overlaps are 0.41 to the ground state of $^6$He and 0.19 to the 2$^+$ first excited state. These factors are significantly smaller than the predictions of the Cohen-Kurath shell model [@CK67], which gives values of 0.59 and 0.40, respectively. The CK shell model requires that the possible $^6$He$+p$ states sum to unity within the p-shell, whereas in the VMC calculation, the correlations in the wave function push significant strength to higher momenta that cannot be represented as a $^6$He state plus p-wave proton. The VMC overlaps were used as input to a Coulomb DWIA analysis of recent $^7$Li$(e,e^{\prime}p)^6$He data taken at NIKHEF [@LWW99] and found to give an excellent fit to the data, as shown in Fig. \[fig:he6p\].
The VMC wave functions are based on one-body parts that have a shell-model structure, namely four nucleons in an $\alpha$ core coupled to $(A-4)$ one-body $\ell=1$ wave functions. However the low-lying states of $^8$Be exhibit a rotational spectrum and are believed to be well approximated as two $\alpha$’s rotating around their common center of mass. It is possible to recover this picture from the VMC wave functions by a modified Monte Carlo density calculation [@WPCP00].
The standard Monte Carlo method for computing one-body densities, $\rho({\bf r})$, is to make a random walk that samples $|\Psi({\bf r}_1,{\bf r}_2,\cdots,{\bf r}_A)|^2$ and to bin ${\bf r}_1,{\bf r}_2,\cdots,{\bf r}_A$ for each configuration in the walk. The density is then proportional to the number of samples in each bin. In the case of a $J$ = 0 nucleus, this “laboratory” density will necessarily be spherically symmetric.
The intrinsic density in body-fixed coordinates can be approximated by computing the moment of inertia matrix, ${\cal M}$, of the $A$ positions for each configuration. The eigenvalues and eigenvectors of ${\cal M}$, are found and a rotation to those principal axes is made. The resulting ${\bf r}_1^\prime,{\bf r}_2^\prime,\cdots,{\bf r}_A^\prime$ is then binned. The eigenvector with the largest eigenvalue is chosen as the ${\bf z}^\prime$ axis. This procedure will not produce a spherically symmetric distribution, even if there is no underlying deformed structure, because almost every random configuration will have principal axes of different lengths and the rotation will always orient the longest principal axis in the ${\bf z}^\prime$ direction. However no artificial structure is introduced [@WPCP00].
When the above procedure is applied to the three lowest $^8$Be states, a dramatic intrinsic structure is revealed, as shown in Fig. \[fig:be8\] for the ground state. The figure shows contours of constant density plotted in cylindrical coordinates. The left side of the figure shows the standard, lab frame, density calculation. For the $J$ = 0 ground state, this is spherically symmetric as shown. The right side of the figure shows the intrinsic density. It is clear that the intrinsic density has two peaks, with the neck between them having only one-third the peak density; we regard these as two $\alpha$’s. This assignment is strengthened by making the same construction for the $J=2^+$ and $4^+$ states; although the laboratory densities for these states (in $M=J$ states) are quite different, the intrinsic densities are, within statistical errors, the same as the $J=0$ intrinsic density [@WPCP00].
If the $0^+$, $2^+$, and $4^+$ states are generated by rotations of a common deformed structure, then their electromagnetic moments and transition strengths should all be related to the intrinsic moments which can be computed by integrating over the projected body-fixed densities. This is explored in Ref. [@WPCP00] and a generally consistent picture is shown.
LOW-ENERGY NUCLEON-NUCLEUS SCATTERING {#sec:scat}
=====================================
The calculations presented so far have treated resonant nuclear states as if they were particle stable; that is the VMC trial wave functions decay exponentially at large distances and the GFMC propagation does not impose a scattering-state boundary condition. As was shown in Fig. \[fig:8be-e\_of\_tau\], this GFMC propagation converges for unstable states that are narrow, but for wide states, such as $^8$Be(4$^+$), the $E(\tau)$ keeps decreasing for increasing $\tau$. Such states should be computed using scattering-wave boundary conditions.
The $^5$He(${\textstyle{1\over2}}^-$, ${\textstyle{3\over2}}^-$) states have been studied in VMC [@csk87] and GFMC [@cs94] using scattering-wave boundary conditions. The VMC calculations use a trial function that is a straight-forward generalization of the $\Psi_V$ given in Sec. 3. In it the $\Psi_J$ contains one p-wave single-particle function that goes to zero at a specified (large) radius, $R_n$. The variational energy is minimized subject to this boundary condition. The phase shift at the resulting scattering energy, $E_s=E(^5\mbox{He}(R_n))-E(^4\mbox{He})$ is then obtained from $$\tan(\delta_l) = \frac{j_l(k R_n)}{n_l(k R_n)},$$ where $k = (2 \mu E_s)^{1/2} / \hbar$, $\mu$ is the $\alpha$+$n$ reduced mass, and $j_l$ and $n_l$ are the spherical Bessel functions (in this case $l=1$). It is possible to generalize this formulation by specifying a logarithmic-derivative at $R_n$, rather than having the wave function go to zero there. This method requires the computation of the energy difference, $E_s$. If this is done with two different VMC calculations, one for $^5$He with the specified boundary condition, and one for $^4$He, then the statistical errors of these two calculations must be added in quadrature. Instead one can directly evaluate the energy difference in a VMC random walk that is controlled by the $^5$He wave function. In practice this gives $E_s$ with a smaller statistical error than either of the separate statistical errors [@csk87]. The GFMC calculations start with the scattering-state trial function and use a modified propagator that preserves the boundary condition [@cs94]. Both methods result in a phase shift and energy corresponding to a given boundary condition. By repeating the calculations for different boundary conditions, the phase shift as a function of energy can be mapped out.
The VMC calculations of Ref. [@csk87] used older two- and three-nucleon potentials than have been used for the other calculations reported in this review. They reproduced the qualitative features of the experimental $\alpha$+$n$ ${\textstyle{1\over2}}^-$ and ${\textstyle{3\over2}}^-$ phase shifts, but, in particular, gave a spin-orbit splitting about 1 MeV too small. The GFMC calculations [@cs94] used a different, but still somewhat old Hamiltonian and obtained values that are closer to experiment. These calculations need to be repeated with the modern Hamiltonians described in this review, and for other systems such as $\alpha$+$d$, $^6$He+$n$, and $\alpha$+$\alpha$.
ASTROPHYSICAL ELECTROWEAK REACTIONS
===================================
Most of the key nuclear reactions in primordial nucleosynthesis [@NB00] and solar neutrino production [@A+98] involve only the $A \leq 8$ nuclei. Many of these are electroweak capture reactions that are difficult or impossible to measure in the laboratory. With the high-precision few-body methods, such as PHH and Faddeev, or new effective field theory treatments, the reactions to s-shell final states have been calculated with unprecedented accuracy in the last few years. These include the $^1$H$(p,e^+\nu_e)^2$H [@SS+98] and $^3$He$(p,e^+\nu_e)^4$He [@MSVKR00] weak capture reactions, and the $^1$H$(n,\gamma)^2$H [@R00], $^2$H$(n,\gamma)^3$H and $^2$H$(p,\gamma)^3$He [@VSK96] radiative capture reactions.
Recently, the VMC method has been applied to some of the radiative captures to p-shell final states, including $^2$H($\alpha,\gamma)^6$Li, $^3$H($\alpha,\gamma)^7$Li, and $^3$He($\alpha,\gamma)^7$Be [@NWS01; @N01]. As discussed in the previous section, many-nucleon scattering states can be constructed within the quantum Monte Carlo framework, but it has not yet been done for $A \geq 6$ nuclei. Thus for these first studies, the appropriate electromagnetic matrix elements (primarily E1 and E2) are evaluated between a VMC six- or seven-nucleon final state, and a correlated cluster-cluster scattering state which is not variationally improved.
Since much of the capture takes place at long range, an important ingredient in these calculations is an asymptotically-correct description of the six- or seven-body final state. This feature has to be implemented in the initial variational wave function because, as remarked before, the VMC and GFMC methods find their best wave functions by optimizing the energy, which is not sensitive to the long-range behavior. The asymptotic behavior can be imposed by requiring the one-body wave function in Eq.(\[eq:phi\]) to satisfy the condition: $$\label{eqn:asymptotic}
[\phi^{LS[n]}_{p}(r\rightarrow\infty)]^n \propto W_{km}(2\gamma r)/r \ ,$$ where $W_{km}(2\gamma r)$ is the Whittaker function for bound-state wave functions in a Coulomb potential and $n$ is the number of p-shell nucleons. Here $\gamma^2 = 2\mu_{4n} B_{4n}/\hbar^2$, with $\mu_{4n}$ and $B_{4n}$ the appropriate two-cluster effective mass and binding energy. For $^6$Li, $B_{42} = 1.47$ MeV; for $^7$Li or $^7$Be, capture can be to the ${\textstyle{3\over2}}^-$ ground or ${\textstyle{1\over2}}^-$ excited state with corresponding values for $B_{43}$.
The initial-state wave functions are taken as elastic-scattering states of the form $$\label{eqn:scatstate}
|\psi_{\alpha\tau}; LSJM \rangle =
{\cal A}\left\{\phi_{\alpha\tau}^{JL}(r_{\alpha\tau})Y_{LM_L}({\bf\hat{r}}_{\alpha\tau})
\prod_{ij}G_{ij}|\psi_\alpha\psi_\tau^{m_S}\rangle\right\}_{LSJM},$$ where curly braces indicate angular momentum coupling, $\cal{A}$ antisymmetrizes between clusters, $\psi_\alpha$ is the $^4$He ground state, and $\psi_\tau^{m_S}$ is the deuteron or trinucleon ground state in spin orientation $m_S$. The $G_{ij}$ are identity operators if the nucleons $i$ and $j$ are in the same cluster, else, they are a set of pair correlation operators, including both central and spin-isospin dependent terms, which introduce distortions in each cluster, under the influence of individual nucleons from the other cluster. They are similar to the correlations discussed in Sec. 3, except that they revert to the identity operator at pair separations beyond about 2 fm. The correlations $\phi_{\alpha\tau}^{JL}$ are generated from optical potentials that describe the experimental phase shifts in cluster-cluster scattering; see Refs. [@NWS01; @N01] for details. Because the $G_{ij}$ go to unity, the $\psi_{\alpha\tau}$ has the same phase shifts as those generated by the optical potential.
Evaluation of electromagnetic matrix elements between the initial scattering and final bound states can be split into two parts by noting that all of the energy dependence is contained in the relative wave function $\phi_{\alpha \tau}^{JL}$ and the transition operators. Thus, using a technique first applied in a VMC calculation of $d$+$d$ radiative capture [@APS91], the matrix element for a given scattering partial wave can be written as $$\begin{aligned}
\label{eqn:integrand}
T^{LSJ_iJ_f}_\ell(q) &=& \int_0^\infty dx\, x^2 \, \phi_{\alpha\tau}^{J_iL}(x)
\\
&\times& \langle \psi_{A}^{J_fm_f}|
T_{\ell \lambda}(q) {\cal A}\left\{\delta(x-r_{\alpha\tau} )
Y_L^{M_L}({\bf\hat{r}}_{\alpha\tau}))
\prod_{ij}G_{ij}|\psi_\alpha\psi_\tau^{m_S}\rangle\right\}_{LSJ_iM_i} \ ,
\nonumber\end{aligned}$$ where $T_{\ell \lambda}$ denotes any of the standard $E_{\ell}$ or $M_{\ell}$ operators. The integration over all coordinates except $x$ can be calculated just once for each partial wave by Monte Carlo sampling, and the result can then be used to compute the full integral for as many energies as desired by recomputing $\phi_{\alpha\tau}^{J_iL}(x)$ only.
The result for the $^2$H($\alpha,\gamma)^6$Li capture $S$-factor is shown in Fig. \[fig:sfactor\], where it is compared to a collection of direct and indirect data. The $S$-factor falls rapidly at low-energy because a pseudo-orthogonality between the initial and final states suppresses normal S-wave capture. Whenever there is such a suppression, it is important to consider higher-order terms that might contribute, including relativistic corrections and two-body charge and current operators [@CS98]. This is possible with the VMC calculation because of the fully correlated $A$-body wave functions that are used. In this case, a relativistic center-of-energy correction leads to an E1 contribution that dominates at low energy, while the resonance region and above is primarily E2. Results for $^3$H($\alpha,\gamma)^7$Li, and $^3$He($\alpha,\gamma)^7$Be can be found in Ref. [@N01]
This kind of VMC calculation is only a first investigation into the p-shell astrophysical reactions, and should be improved in the future by using the more exact GFMC wave functions for both bound and scattering states, and the Hamiltonians with improved $N\!N\!N$ potentials.
NEUTRON DROPS
=============
Neutron-rich nuclei are interesting both because of their importance in various astrophysical contexts, such as the r-process or neutron-star crusts, and the current interest in experiments with radioactive beams. These nuclei are often studied within the framework of mean-field models using Skyrme or other potential models. The parameters of these models are fit to experimentally known binding energies, that is for situations with $N{\sim}Z$. In particular the isospin dependence of the spin-orbit component of such potentials is considered to be not strongly constrained. Neutron drops offer the possibility of theoretical guidance for the isotopic dependence of such parameters.
Neutron drops are systems of interacting neutrons confined in an artificial external well. Eight neutrons form a closed shell and single-particle spin-orbit splittings can be studied in drops of seven or nine neutrons. Pairing energies can be studied if six-neutron drops are also computed. Calculations of systems of seven and eight neutrons interacting with AV18/UIX were used as a basis for comparing Skyrme models of neutron-rich systems with microscopic calculations based on realistic interactions [@PSCPPR96]. The external one-body well used is a Woods-Saxon: $$V_1 (r) = \sum_i \frac{V_0}{1 + \exp [ - (r_i - r_0)/a_0 ]} \ ;$$ the parameters are $V_0 = -20$ MeV, $r_0 = 3.0$ fm, and $a_0 = 0.65$ fm. Neither the external well nor the total internal potential ($v_{ij}+V_{ijk}$) are individually attractive enough to produce bound states of seven or eight neutrons; however the combination does produce binding.
Tables \[tab:energy\]-\[tab:gfmc\] show results for the neutron drops with this external well. The $T={\textstyle{3\over2}}$ nature of the $S^I_{\tau}S^I_{\sigma}$ term of $V^{3\pi}$ results in large contributions in the neutron drops. As a result the seven-neutron drops computed with AV18/IL2 have double the spin-orbit splitting predicted by AV18/UIX. Thus the spin-orbit splitting in neutron-rich systems depends strongly on the Hamiltonian used.
CONCLUSIONS AND OUTLOOK
=======================
Quantum Monte Carlo methods can now be used to obtain accurate (within 2%) energies for light p-shell nuclei up to $A$ = 8. Initial calculations of $A$=9,10 nuclei have been made by the authors, and studies of $A$=11,12 nuclei should be feasible in the next few years by variational and Green’s function Monte Carlo methods. This progress in the nuclear many-body problem is due both to the rapid growth of computational power and the continuing evolution of algorithms. The new auxiliary-field diffusion Monte Carlo method, which samples spins and isospins by auxiliary fields, and space by standard diffusion Monte Carlo, may be the key to doing even larger nuclear systems [@SF99].
Studies of the p-shell nuclei allow us to test nuclear forces in new ways not accessible in s-shell nuclei, particularly the odd partial waves of $N\!N$ scattering and the $T={\textstyle{3\over2}}$ triples for $N\!N\!N$ forces. They also give us many more cases in which to examine charge-dependent and charge-symmetry-breaking interactions. With these calculations of nuclear spectra, we see for the first time that nuclear structure, including both single-particle and clustering aspects, really can be explained directly in terms of bare nuclear forces that fit $N\!N$ data. A crucial ingredient for quantitative agreement is the addition of realistic $N\!N\!N$ forces, including at least two terms beyond the standard long-range two-pion-exchange potential [@PPWC01].
Many aspects of nuclear structure and reactions are described quantitatively in these studies. The energy differences between isobaric analog states are explained well once the complete electromagnetic and strong charge-independence-breaking forces deduced from $N\!N$ scattering are included. Charge radii and quadrupole moments agree with experiment, and we expect magnetic moments will also once the important two-body exchange currents are included, as they have been in s-shell nuclei [@MRS98]. Elastic and transition form factors measured in electron scattering, and transition densities that are tested in pion scattering, are accurately predicted without the introduction of effective charges. Spectroscopic factors are naturally quenched by the correlations in $A$-body wave functions and we see that those correlations build up intrinsic cluster structure, as in the case of $^8$Be.
Initial calculations of nucleon-nucleus scattering states and electroweak capture reactions are encouraging. These two related problems will be a major activity for the quantum Monte Carlo studies over the next several years, with studies of scattering states in nuclei like $^7$He and $^9$He, and of the cluster-cluster scattering states that enter into the astrophysical capture reactions. Additional reactions like $^7$Be$(p,\gamma)^8$B and even $^8$Be$(\alpha,\gamma)^{12}$C should become feasible. Still other projects will include the study of weak decays and, going beyond the p-shell, looking at the unnatural-parity intruder states which start to dip into the low-energy spectrum with the $A=9$ nuclei, and can be particle stable by $A=10$.
Refining the nuclear Hamiltonian will remain a major aspect of future work. Extensive new $N\!N$ scattering data taken since the Nijmegen partial-wave analysis has increased the database to more than 6000 points, and a new CD-Bonn 2000 potential has been constructed that again achieves a $\chi^2/$datum $\approx 1$ [@M01]. Also, intensive searches are being made to look for $N\!N\!N$ force signatures in $N+d$ scattering experiments [@K+01; @W+01], which may well indicate a need for additional terms beyond those contained in the Illinois models. Our ability to test new force models, as they become available, in light nuclei by accurate quantum Monte Carlo methods will continue to be an important tool for nuclear physics.
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors thank J. Carlson, A. Kievsky, V. R. Pandharipande, R. Schiavilla, J. P. Schiffer and K. E. Schmidt for useful communications and discussions. This work was supported by the U.S. Department of Energy, Nuclear Physics Division, under contract No. W-31-109-ENG-38.
[99]{}
Stoks VGJ, Klomp RAM, Rentmeester MCM, de Swart JJ. [*Phys. Rev. C*]{} 48:792 (1993)
Chen CR, Payne GL, Friar JL, Gibson BF. [*Phys. Rev. C*]{} 31:2266 (1985)
Carlson J. [*Phys. Rev. C*]{} 36:2026 (1987)
Carlson J, Schiavilla R. [*Rev. Mod. Phys.*]{} 70:743 (1998)
Cohen S, Kurath D. [*Nucl. Phys.*]{} 73:1 (1965)
Kuo TTS, Brown GE. [*Nucl. Phys.*]{} 85:40 (1966)
Navratil P, Vary JP, Barrett BR. [*Phys. Rev. Lett.*]{} 84:5728 (2000)
Lomnitz-Adler J, Pandharipande VR, Smith RA. [*Nucl. Phys.*]{} A361:399 (1981)
Carlson J, Pandharipande VR, Wiringa RB. [*Nucl. Phys.*]{} A401:86 (1983)
Wiringa RB. [*Phys. Rev. C*]{} 43:1585 (1991)
Arriaga A, Pandharipande VR, Wiringa RB. [*Phys. Rev. C*]{} 52:2362 (1995)
Wiringa RB, Schiavilla R. [*Phys. Rev. Lett.*]{} 81:4317 (1998)
Lapikás L, Wesseling J, Wiringa RB. [*Phys. Rev. Lett.*]{} 82:4404 (1999)
Lee T-SH, Wiringa RB. [*Phys. Rev. C*]{} 63:014006 (2001)
Nollett KM, Wiringa RB, Schiavilla R. [*Phys. Rev. C*]{} 63:024003 (2001)
Nollett KM. nucl-th/0102022, submitted to [*Phys. Rev. C*]{}
Pudliner BS, Pandharipande VR, Carlson J, Wiringa RB. [*Phys. Rev. Lett.*]{} 74:4396 (1995)
Pudliner BS, et al. [*Phys. Rev. C*]{} 56:1720 (1997)
Wiringa RB, Pieper SC, Carlson J, Pandharipande VR. [*Phys. Rev. C*]{} 62:014001 (2000)
Schmidt KE, Fantoni S. [*Phys. Lett. B*]{} 446:99 (1999)
Wiringa RB, Stoks VGJ, Schiavilla R. [*Phys. Rev. C*]{} 51:38 (1995)
Pieper SC, Pandharipande VR, Wiringa RB, Carlson J, nucl-th/0102004, submitted to [*Phys. Rev. C*]{}
Metropolis N, et al. [*J. Chem. Phys.*]{} 21:1087 (1953)
Stoks VGJ, de Swart JJ. [*Phys. Rev. C*]{} 47:761 (1993); [*Phys. Rev. C*]{} 52:1698 (1995)
Stoks VGJ, Klomp RAM, Terheggen CPF, de Swart JJ. [*Phys. Rev. C*]{} 49:2950 (1994)
Machleidt R, Sammarruca F, Song Y. [*Phys. Rev. C*]{} 53:R1483 (1996)
Fujita J, Miyazawa H. [*Prog. Theor. Phys.*]{} 17:360 (1957)
Coon SA, et al. [*Nucl. Phys.*]{} A317:242 (1979)
Coelho HT, Das TK, Robilotta MR. [*Phys. Rev. C*]{} 28:1812 (1983)
Friar JL, Hüber D, van Kolck U. [*Phys. Rev. C*]{} 59:53 (1999)
Carlson J, Pandharipande VR, Schiavilla R. [*Phys. Rev. C*]{} 47:484 (1993)
Forest JL, Pandharipande VR, Carlson J, Schiavilla R. [*Phys. Rev. C*]{} 52:576 (1995)
Forest JL, Pandharipande VR, Arriaga A. [*Phys. Rev. C*]{} 60:014002 (1999)
Forest JL, Pandharipande VR, Friar JL. [*Phys. Rev. C*]{} 52:568 (1995)
Friar JL. [*Phys. Rev. C*]{} 12:695 (1975)
Wiringa RB. [*Rev. Mod. Phys.*]{} 65:231 (1993)
Bohr A, Mottelson BR. [*Nuclear Structure Volume I*]{}. New York: W. A. Benjamin (1969)
Carlson JA, Wiringa RB. [*Computational Nuclear Physics 1*]{}, ed. K Langanke, JA Maruhn, SE Koonin. Berlin: Springer (1991)
Pudliner BS, et al. [*Phys. Rev. Lett.*]{} 76:2416 (1996)
Smerzi A, Ravenhall DG, Pandharipande VR. [*Phys. Rev. C*]{} 56:2549 (1997)
Carlson J. private communication
Carlson J. [*Phys. Rev. C*]{} 38:1879 (1988)
Ceperley DM. [*Rev. Mod. Phys.*]{} 67:279 (1995)
Kalos MH. [*J. Comp. Phys.*]{} 2:257 (1967)
Ceperley DM, Kalos MH. [*Monte Carlo Methods in Statistical Physics*]{}, ed. K. Binder. Heidelberg: Springer (1979)
Zhang S, Carlson J, Gubernatis JE. [*Phys. Rev. Lett.*]{} 74:3652 (1995)
Friar JL, Payne GL, Stoks VGJ, de Swart JJ. [*Phys. Lett. B*]{} 311:4 (1993)
Nogga A, Kamada H, Glöckle W. [*Phys. Rev. Lett.*]{} 85:944 (2000)
Kievsky A, Marcucci LE, Rosati S, Viviani M. [*Few-Body Syst.*]{} 22:1 (1997)
Viviani M, Kievsky A, Rosati S. [*Few-Body Syst.*]{} 18:25 (1995)
Kievsky A, Viviani M, Rosati S. [*Nucl. Phys.*]{} A551:241 (1993)
Kievsky A. private communication
Tilley DR, Weller HR, Hasan HH. [*Nucl. Phys.*]{} A474:1 (1987)
Tilley DR, Weller HR, Hale GM. [*Nucl. Phys.*]{} A541:1 (1992)
Ajzenberg-Selove F. [*Nucl. Phys.*]{} A490:1 (1988)
Pieper SC, Pandharipande VR. [*Phys. Rev. Lett.*]{} 70:2541 (1993)
Tilley DR, et al. TUNL preprint “Energy Levels of Light Nuclei A = 5,6,7” (2001)
Nolen JA, Schiffer, JP. [*Ann. Rev. Nucl. Sci.*]{} 19:471 (1969)
Peshkin M. [*Phys. Rev.*]{} 121:636 (1960)
Barker FC. [*Nucl. Phys.*]{} 83:418 (1966)
De Vries H, De Jager CW, De Vries C. [*At. Data Nucl. Data Tables*]{} 36:495 (1987)
Shiner D, Dixson R, Vedantham V. [*Phys. Rev. Lett.*]{} 74:3553 (1995)
Raghavan P. [*At. Data Nucl. Data Tables*]{} 42:189 (1989);\
Stone N. www.nndc.bnl.gov/nndc/stone\_moments/moments.html (1997)
Schiavilla R, Pandharipande VR, Riska DO. [*Phys. Rev. C*]{} 40:2294 (1989)
Marcucci LE, Riska DO, Schiavilla R. [*Phys. Rev. C*]{} 58:3069 (1998)
Alkhazov GD, et al.. [*Phys. Rev. Lett.*]{} 78:2313 (1997)
Al-Khalili JS, Tostevin JA. [*Phys. Rev. C*]{} 57:1846 (1998)
Schiavilla R, Wiringa RB, Carlson J. [*Phys. Rev. Lett.*]{} 70:3856 (1993)
Li GC, Sick I, Whitney RR, Yearian MR. [*Nucl. Phys.*]{} A162:583 (1971)
Lapikás L. [*Proceedings of the Conference on Modern Trends in Elastic Electron Scattering*]{}, ed. C De Vries. Amsterdam:NIKHEF-K (1978)
Bergstrom JC, Kowalski SB, Neuhausen R. [*Phys. Rev. C*]{} 25:1156 (1982)
Rand RE, Frosch R, Yearian MR. [*Phys. Rev.*]{} 144:859 (1966)
Bergstrom JC, Deutschmann U, Neuhausen R. [*Nucl. Phys.*]{} A327:439 (1979)
Bergstrom JC, Tomusiak EL. [*Nucl. Phys.*]{} A262:196 (1976)
Bergstrom JC, Auer IP, Hicks RS. [*Nucl. Phys.*]{} A251:401 (1975)
Poletti AR, Warburton EK, Kurath D. [*Phys. Rev.*]{} 155:1096 (1967)
Bolger J, et al. [*Meson-Nuclear Physics 1979; AIP Conf. Proc. No. 54*]{}, ed. EV Hungerford III. New York:AIP (1979)
Lee T-SH, Kurath D. [*Phys. Rev. C*]{} 21:293 (1980); [*Phys. Rev. C*]{} 21:1670 (1980)
Schiavilla R, Pandharipande VR, Wiringa RB. [*Nucl. Phys.*]{} A449:219 (1986)
Forest JL, et al. [*Phys. Rev. C*]{} 54:646 (1996)
Cohen S, Kurath D. [*Nucl. Phys.*]{} A101:1 (1967)
Carlson J, Schmidt KE, Kalos MH. [*Phys. Rev. C*]{} 36:27 (1987)
Carlson J, Schiavilla R. [*Few-Body Syst. Suppl.*]{} 7:349, ed. BLG Bakker, R van Dantzig. Wien: Springer (1994)
Nollett KM, Burles S. [*Phys. Rev. D*]{} 62:123505 (2000)
Adelberger E, et al. [*Rev. Mod. Phys.*]{} 70:1265 (1998)
Schiavilla R, et al. [*Phys. Rev. C*]{} 58:1263 (1998)
Marcucci LE, et al. [*Phys. Rev. C*]{} 63:015801 (2000)
Rupak G. [*Nucl. Phys.*]{} A678:405 (2000)
Viviani M, Schiavilla R, Kievsky A. [*Phys. Rev. C*]{} 54:534 (1996)
Arriaga A, Pandharipande VR, Schiavilla R. [*Phys. Rev. C*]{} 43:983 (1991)
Robertson R, et al. [*Phys. Rev. Lett.*]{} 47:1867 (1981)
Mohr P, et al. [*Phys. Rev. C*]{} 50:1543 (1994)
Kiener J, et al. [*Phys. Rev. C*]{} 44:2195 (1991)
Machleidt R. [*Phys. Rev. C*]{} 63:024001 (2001)
Kievsky A, et al. [*Phys. Rev. C*]{} 63:024005 (2001)
Wita[ł]{}a H, et al. [*Phys. Rev. C*]{} 63:024007 (2001)
|
---
abstract: 'Newton’s problem of minimal resistance is one of the first problems of optimal control: it was proposed, and its solution given, by Isaac Newton in his masterful *Principia Mathematica*, in 1686. The problem consists of determining, in dimension three, the shape of an axis-symmetric body, with assigned radius and height, which offers minimum resistance when it is moving in a resistant medium. The problem has a very rich history and is well documented in the literature. Of course, at first glance, one suspects that the two dimensional case should be well known. Nevertheless, we have looked into numerous references and ask at least as many experts on the problem, and we have not been able to identify a single source. Solution was always plausible to everyone who thought about the problem, and writing it down was always thought not to be worthwhile. Here we show that this is not the case: the two-dimensional problem is more rich than the classical one, being, in some sense, more interesting. Novelties include: (i) while in the classical three-dimensional problem only the restricted case makes sense (without restriction on the monotonicity of admissible functions the problem doesn’t admit a local minimum), we prove that in dimension two the unrestricted problem is also well-posed when the ratio height versus radius of base is greater than a given quantity; (ii) while in three dimensions the (restricted) problem has a unique solution, we show that in the restricted two-dimensional problem the minimizer is not always unique – when the height of the body is less or equal than its base radius, there exists infinitely many minimizing functions.'
author:
- |
Cristiana J. Silva\
`cjoaosilva@mat.ua.pt`
- |
Delfim F. M. Torres\
`delfim@mat.ua.pt`
date: |
Department of Mathematics\
University of Aveiro\
3810-193 Aveiro, Portugal
title: |
Two-dimensional Newton’s Problem\
of Minimal Resistance[^1]
---
Newton’s problem of minimal resistance, dimension two, calculus of variations, optimal control.
49K05 (76G25, 76M30).
Introduction
============
Newton’s aerodynamical problem, in dimension three, is a classic problem (see [@CD:Amaral:1913; @CD:Veubeke:1966; @CD:Kneser:1913]). It consists in joining two given points of the plane by a curve’s arc that, while turning around a given axis, generate the body of revolution offering the least resistance when moving in a fluid in the direction of the axis. Newton has considered several hypotheses: that the body moves with constant velocity, and without rotation, on a very rare and homogeneous medium of particles which are all equal; that the axis-symmetric body is inscribed in a cylinder of height $H$ and radius $r$; that the particles of the medium are infinitesimally small and immovable (there exists no temperature motion of particles); that collisions of the particles with the body are absolutely elastic. Newton has indicated in the *Mathematical principles of natural philosophy* the correct solution to his problem. He has not explained, however: how such solution can be obtained; how the problem is formulated in the language of mathematics. This has been the work of many mathematicians since Newton’s time (see [@MR56:4953; @CD:Tikhomirov:1990; @CD:arXiv:math.OC/0404237]). Extensions of Newton’s problem is a topic of current intensive research, with many questions remaining open challenging problems. Recent results, obtained by relaxing Newton’s hypotheses, include: non-symmetric bodies [@CD:BK:1993]; one-collision non-convex bodies [@CD:CL:2001]; collisions with friction [@CD:friction:2002]; multiple collisions allowed [@CD:P1:2003]; temperature noise of particles [@CD:arXiv:math.OC/0404194; @math.OC/0407406]. Here we are interested in the classical problem, under the classical hypotheses considered by Newton. Our main objective is to study the apparently more simpler Newton’s problem of minimal resistance for a two-dimensional body moving with constant velocity in a homogeneous rarefied medium of particles. The first work on a two-dimensional Newton-type problem seems to be [@CD:arXiv:math.OC/0404194], where the authors study the problem in a chaotically moving media of particles (in the classical problem particles are immovable). The results in [@CD:arXiv:math.OC/0404194] were later generalized to dimension three [@math.OC/0407406]. This paper is motivated by the results in [@math.OC/0407406]: when one considers temperature motion of particles, the three-dimensional problem admits only two types of solutions; while the two-dimensional case is more rich, showing solutions of five distinct types. Here we prove that in the classical framework, with an immovable media of particles, also the two-dimensional case is more rich: in certain cases of input of data (height $H$ and radius $r$ of the body) the problem is well-posed (admit a local minima) without imposing the restriction $\dot{y}(x) \geq 0$ on the admissible curves $y(\cdot)$. This is different from the three-dimensional classical problem or the problem in higher-dimensions, where the restriction $\dot{y}(x) \geq 0$ is always necessary for the problem to make sense: without it there exists no strong and no weak local minimum for Newton’s problem of minimal resistance (see [@CD:Veubeke:1966; @CD:CJSilvaMScThesis]). We show that for $H >
\frac{\sqrt 3}{3}r$ the function $\hat{y}(x)= \frac{H}{r}x$ is a local minimum for the unrestricted Newton’s problem of minimal resistance in dimension two. In the restricted case, while in dimension three (or higher-orders) the problem has always a unique solution, we prove that infinitely many different minimizers appear in dimension two for $r \ge H$. These simple facts seem to be new in the literature, and never noticed before.
Restricted and unrestricted problems
====================================
In the classical three dimensional Newton’s problem of minimal aerodynamical resistance, the resistance force is given by $R\left[ \dot{y}(\cdot)\right] = \int_0^r \frac{x}{1 + \dot{y}(x)^2} \, dx$. Minimization of this functional is a typical problem of the calculus of variations. Most part of the old literature wrongly assume the classical Newton’s problem to be “one of the first applications of the calculus of variations”. The truth, as Legendre first noticed in 1788 (see [@CD:belloniKawohl:1997]), is that some restrictions on the derivatives of admissible trajectories must be imposed: $\dot{y}(x) \geq 0$, $x \in [0, r]$. The restriction is crucial, because without it there exists no solution, and the problem suffers from Perron’s paradox [@CD:MR41:4337 §10]: since the *a priori* assumption that a solution exists is not fulfilled, does not make any sense to try to find it by applying necessary optimality conditions. It turns out that, with the necessary restriction, the problem is better considered as an optimal control one (see [@Tikhomirov2 p. 67] and [@CD:arXiv:math.OC/0404237]). Correct formulation of Newton’s problem of minimal resistance in dimension three is ( [@CD:Veubeke:1966; @CD:Tikhomirov:1990]): $$\begin{gathered}
\mathcal{R}\left[u(\cdot)\right] = \int_0^r \frac{x}{1 +
u(x)^2} dx \longrightarrow \min \, , \\
\dot{y}(x) = u(x) \, , \quad u(x) \geq 0 \, ,\\
y(0) = 0 \, , \quad y(r) = H \, , \quad H > 0 \, ,
\end{gathered}$$ where we minimize the resistance $\mathcal{R}$ in the class of continuous functions $y : [0,r] \rightarrow \mathbb{R}$ with piecewise continuous derivative. Here we consider Newton’s problem of minimal resistance in dimension two (see [@CD:arXiv:math.OC/0404237]): $$\label{eq:R-2-dim}
\begin{gathered}
R\left[ u(\cdot)\right] = \int_0^r \frac{1}{1 + u(x)^2} dx
\longrightarrow \min \, , \\
\dot{y}(x) = u(x) \, , \quad u(x) \in \Omega \, ,\\
y(0) = 0 \, , \quad y(r) = H \, , \quad H > 0 \, .
\end{gathered}$$ We consider two cases: (i) unrestricted problem, where no restriction on the admissible trajectories $y(\cdot)$ other than the boundary conditions $y(0) = 0$, $y(r) = H$ is considered ($\Omega = \mathbb{R}$); (ii) restricted problem, where the admissible functions must satisfy the restriction $\dot{y}(x) \geq
0$, $x \in [0, r]$ ($\Omega = \mathbb{R}_0^+$). While for the classical three-dimensional problem only the restricted problem admits a minimizer, we prove in §\[sec:UP\] that the two-dimensional problem is more rich: the unrestricted case also admits a local minimizer when the given height $H$ of the body is big enough. In §\[sec:RP\] we study the restricted problem. Also in the restricted case the two-dimensional problem is more interesting: if $r \ge H$, then infinitely many different minimizers are possible, while in the classical three-dimensional problem the minimizer is always unique.
General results for both problems
=================================
The central result of optimal control theory is the Pontryagin Maximum Principle [@CD:MR29:3316b], which gives a generalization of the classical necessary optimality conditions of the calculus of variations. The following results are valid for both restricted and unrestricted problems: respectively $\Omega = \mathbb{R}_0^+$ and $\Omega = \mathbb{R}$ in .
\[PontMP-d-dim\] If $(y(\cdot),u(\cdot))$ is a minimizer of problem , then there exists a non-zero pair $(\psi_0,\psi(\cdot))$, where $\psi_0 \leq 0$ is a constant and $\psi(\cdot) \in PC^1\left([0, r];\,\mathbb{R}\right)$, such that the following conditions are satisfied for almost all $x$ in $[0,r]$:
- the Hamiltonian system
$$\begin{cases}
\dot{y}(x)&=\frac{ \partial{{\cal H}} }{\partial {\psi}}(u(x),
\psi_0, \psi(x)) \quad \text{(control equation $\dot{y} = u$)} \, , \\
\dot{\psi}(x) &= -\frac{ \partial{{\cal H}} }{\partial {y}}(u(x),
\psi_0, \psi(x)) \quad \text{(adjoint system $\dot{\psi} = 0$)} \, ;
\end{cases}$$
- the maximality condition $$\label{condMax} {\cal H}(u(x),\psi_0,\psi(x))
={ \mathop {\max}\limits_{u \in \Omega}} {\cal H}(u, \psi_0, \psi(x)) \, ;$$
where the Hamiltonian ${\cal H}$ is defined by $$\label{eq:def:ham}
{\cal H}(u,\psi_0,\psi) = \psi_0 \frac{1}{1+u^2}+\psi u \, .$$
The adjoint system asserts that $\psi(x) \equiv c$, with $c$ a constant. From the maximality condition it follows that $\psi_0 \ne 0$ (there are no abnormal extremals for problem ).
\[pont-ext-d-dim\] All the Pontryagin extremals $\left(y(\cdot), u(\cdot), \psi_0, \psi(\cdot) \right)$ of problem are normal extremals ($\psi_0 \ne 0$), with $\psi(\cdot)$ a negative constant: $\psi(x) \equiv
-\lambda$, $\lambda >0$, $x \in [0,r]$.
The Hamiltonian ${\cal H}$ for problem , $\mathcal{H}\left(u,\psi_0,\psi\right) = \psi_0 \frac{1}{1+u^2} + \psi u$, does not depend on $y$. Therefore, by the adjoint system we conclude that $$\dot{\psi}(x)= - \frac{\partial \mathcal{H}}{\partial y}
\left(u(x),\psi_0,\psi(x) \right)=0\, ,$$ that is, $\psi(x)
\equiv c$, $c$ a constant, for all $x \in [0,r]$. If $c=0$, then $\psi_0 < 0$ (because one can not have both $\psi_0$ and $\psi$ zero) and the maximality condition simplifies to $$\label{eq:maxCondAppNP2}
\frac{\psi_0}{1 + u^2(x)}
= { \mathop {\max }\limits_{u \in \Omega}}\left\{\frac{\psi_0}{1 + u^2} \right\} \, .$$ From we conclude that the maximum is not achieved ($u \rightarrow \infty$). Therefore $c \neq 0$. Similarly, for $c>0$ the maximum $$\frac{\psi_0}{1+u^2(x)} + cu(x)
= {\mathop {\max}\limits_{u \in \Omega}}\left\{\frac{\psi_0}{1+u^2}+ cu\right\}$$ does not exist, and we conclude that $c<0$. It remains to prove that $\psi_0 \neq 0$. Let us assume $\psi_0 = 0$. Then the maximality condition reads $$\label{eq:td2c}
c u(x) = { \mathop {\max }\limits_{u \in \Omega }} \{c u \} \, ,
\quad c < 0 \, .$$ For $\Omega = \mathbb{R}$ the maximum does not exist, and we conclude $\psi_0 \neq 0$. For $\Omega = \mathbb{R}_0^+$ imply $u(x) \equiv 0$ and $y(x) \equiv w$, $w$ a constant ($\dot{y}(x) = u(x)$). This is not possible, given the boundary conditions $y(0)=0$ and $y(r)=H$ with $H>0$. Therefore $\psi_0 \neq 0$: there exists no abnormal Pontryagin extremals.
\[remark-d-dim\] If $\left(y(\cdot), u(\cdot), \psi_0,\psi(\cdot) \right)$ is an extremal, then $\left(y(\cdot), u(\cdot), \gamma \psi_0, \gamma \psi(\cdot)
\right)$ is also a Pontryagin extremal, for all $\gamma > 0$. Therefore one can fix, without loss of generality, $\psi_0 = -1$.
From Proposition \[pont-ext-d-dim\] and Remark \[remark-d-dim\] it follows that the Hamiltonian takes the form $$\label{eq:HamCNMC}
{\mathcal H}\left(u\right)
= -\frac{1}{1+u^2} - \lambda u \, , \quad \lambda > 0 \, .$$
It is not easy to prove the existence of a solution for problem with classical arguments. We will use a different approach. We will show, following [@CD:arXiv:math.OC/0404237], that for problem the Pontryagin extremals are absolute minimizers. This means that to solve problem it is enough to identify its Pontryagin extremals.
\[TeormExtPont-d-dim\] Pontryagin extremals for problem are absolute minimizers.
Let $\hat{u}(\cdot)$ be a Pontryagin extremal control for problem . We want to prove that $$\int_0^r \frac{1}{1+u^2(x)} dx \ge
\int_0^r \frac{1}{1+\hat{u}^2(x)} dx$$ for any admissible control $u(\cdot)$. Given , we conclude from the maximality condition that $$\label{cdmaxPN} - \frac{1}{1+\hat{u}^2(x)} - \lambda \hat{u}(x)
\geq - \frac{1}{1+u^2(x)} - \lambda u(x)$$ for all $u(\cdot) \in PC \left([0,r], \Omega \right)$. Having in mind that all the admissible processes $\left(y(\cdot),
u(\cdot)\right)$ of satisfy $$\int_0^r u(x) dx = \int_0^r \dot{y}(x) dx = y(r) - y(0) = H \, ,$$ we only need to integrate to conclude that $\hat{u}(\cdot)$ is an absolute control minimizer: $$\begin{split}
& \int_0^r \left( - \frac{1}{1+\hat{u}^2(x)} - \lambda
\hat{u}(x) \right) dx \geq \int_0^r \left(- \frac{1}{1+u^2(x)}
- \lambda u(x) \right) dx\\
& \Leftrightarrow \int_0^r \frac{1}{1+\hat{u}^2(x)} dx
+ \lambda \int_0^r \hat{u}(x) dx \leq \int_0^r \frac{1}{1+u^2(x)} dx
+ \lambda \int_0^r u(x) dx \\
&\Leftrightarrow \int_0^r \frac{1}{1+\hat{u}^2(x)} dx + \lambda H
\leq \int_0^r \frac{1}{1+u^2(x)} dx + \lambda H \\
&\Leftrightarrow \int_0^r \frac{1}{1+\hat{u}^2(x)} dx
\leq \int_0^r \frac{1}{1+u(x)^2} dx \, .
\end{split}$$
Roughly speaking, Theorem \[TeormExtPont-d-dim\] reduces the infinite dimension optimization problem to the study of a one-dimension maximization problem: $$\label{eq:opt:dim1}
\max_{u \in \Omega} {\mathcal H}\left(u\right)
= \max_{u \in \Omega} \left\{ -\frac{1}{1+u^2} - \lambda u \right\}
\, , \quad \lambda > 0 \, .$$
Unrestricted problem {#sec:UP}
====================
The following standard result of calculus (see [@fenske]) will be used in the sequel.
\[th:cns:calc:ho\] Let $n \ge 2$ and $\Omega \subseteq \mathbb{R}$ be an open set. If $f : \Omega \rightarrow \mathbb{R}$ is $n-1$ times differentiable on $\Omega$ and $n$ times differentiable at some point $a \in \Omega$ where $f^{(k)}(a) = 0$ for $k = 0,\ldots,n-1$ and $f^{(n)}(a) \ne 0$, then:
- either $n$ is even, and $f(\cdot)$ has an extremum at $a$, that is a maximum in case $f^{(n)}(a) < 0$ and a minimum in case $f^{(n)}(a) > 0$;
- or $n$ is odd, and $f(\cdot)$ does not attain a local extremum at $a$.
We are considering now the unrestricted two-dimensional Newton’s problem of minimal resistance, that is, $\Omega = \mathbb{R}$ in . A necessary (sufficient) condition for $u$ to be a local maximizer for problem is given by ${\mathcal H}'\left(u\right) = 0$ and ${\mathcal H}''\left(u\right) \le 0$ (${\mathcal H}''\left(u\right) < 0$), where $$\begin{gathered}
{\mathcal H}'\left(u\right) = {\frac {2 u}{ \left( 1+{u}^{2} \right) ^{2}}}-\lambda \, ,\\
{\mathcal H}''\left(u\right) = -2\,{\frac {3\,{u}^{2}-1}{ \left( 1+{u}^{2} \right) ^{3}}} \, .\end{gathered}$$ From the first order condition (maximality condition ) it follows that $$\label{LCN-d-2-naoparam}
\frac{u(x)}{\left(1+u^2(x) \right)^2}=\frac{\lambda}{2}
\Leftrightarrow
\frac{\dot{y}(x)}{\left(1+ \dot{y}^2(x) \right)^2}= \frac{\lambda}{2} \, .$$ Using the boundary conditions $y(0)=0$ and $y(r)=H$, we conclude that $y(x)=\frac{H}{r}x$ ($u = \frac{H}{r}$) is a local candidate for the solution of the unrestricted problem ($\lambda = {\frac{2{r}^{3} H}{\left( {r}^{2}+{H}^{2} \right)^{2}}}$). However, by Theorem \[th:cns:calc:ho\], we conclude that such $u$ is a maximizer only when $H > \frac{\sqrt 3}{3}r$. For $H < \frac{\sqrt 3}{3}r$ the value $u = \frac{H}{r}$ corresponds to a local minimizer of ${\mathcal H}\left(u\right)$ since ${\mathcal H}''>0$; for $H = \frac{\sqrt 3}{3}r$ function ${\mathcal H}\left(u\right)$ has neither local maximum nor minimum since ${\mathcal H}''\left(\frac{\sqrt 3}{3}r\right) = 0$ and ${\mathcal H}'''\left(\frac{\sqrt 3}{3}r\right) = -{\frac {27\sqrt{3}}{16}} \ne 0$.
If $H > \frac{\sqrt 3}{3}r$, then function $y(x)= \frac{H}{r}x$ is a (local) minimum for the unrestricted problem . For $H \le \frac{\sqrt 3}{3}r$ the problem has no solution.
The unrestricted problem does not admit global minimum. Take indeed, for large values of the parameter $a$, the control function $$\tilde{u}(x) =
\begin{cases}
a & \text{ if } \quad 0 \le x \le \frac{r}{2} + \frac{H}{2a} \\
-a & \text{ if } \quad \frac{r}{2} + \frac{H}{2a} < x \le r \, .
\end{cases}$$ This gives $R[\tilde{u}(\cdot)] = \frac{r}{1 + a^2}$ which vanishes as $a \rightarrow + \infty$, showing that no global solution can exist.
By the symmetry with respect to the $yy$ axis, a local solution to the unrestricted two-dimensional Newton’s problem of minimal resistance with $H > \frac{\sqrt 3}{3}r$ is a triangle, with value for resistance $R$ equal to $\frac{r^3}{r^2 + H^2}$.
Restricted problem {#sec:RP}
==================
We now study problem with $\Omega = \mathbb{R}_0^+$. In this case the optimal control can take values on the boundary of the admissible set of control values $\Omega$ ($u = 0$). If the optimal control $u(\cdot)$ is always taking values in the interior of $\Omega$, $u(x) > 0$ $\forall$ $x \in [0,r]$, then the optimal solution must satisfy and it corresponds to the one found in §\[sec:UP\]: $$\label{eq:t}
u(x) = \frac{H}{r} \, , \quad \forall x \in [0,r] \, ,$$ with resistance $$\label{eq:rt}
R = \frac{r^3}{r^2 + H^2} \, .$$ We show next that this is solution of the restricted problem only for $H \ge r$: for $H \le r$ the minimum value for the resistance is $R = r - \frac{H}{2}$.
It is clear, from the boundary conditions $y(0) = 0$, $y(r) = H$, $r > 0$, $H > 0$, that $u(x) = 0$, $\forall$ $x \in [0,r]$, is not a possibility: there must exist at least one non-empty subinterval of $[0,r]$ for which $u(x) > 0$ (otherwise $y(x)$ would be constant, and it would be not possible to satisfy simultaneously $y(0)=0$ and $y(r)=H$). The simplest situations are given by $$\label{eq:i}
u(x) =
\begin{cases}
0 & \text{ if } \quad 0 \le x \le \xi \, , \\
\frac{H}{r - \xi} & \text{ if } \quad \xi \le x \le r \, ,
\end{cases}$$ or $$\label{eq:f}
u(x) =
\begin{cases}
\frac{H}{\xi} & \text{ if } \quad 0 \le x \le \xi \, , \\
0 & \text{ if } \quad \xi \le x \le r \, .
\end{cases}$$ We get from taking $\xi = 0$; from with $\xi = r$. For the resistance is given by $R(\xi) = \xi + \frac{(r-\xi)^3}{(r-\xi)^2 + H^2}$, that has a minimum value for $\xi = r - H \ge 0$: $R(r-H) = r - \frac{H}{2}$, $$\label{eq:io}
u(x) =
\begin{cases}
0 & \text{ if } \quad 0 \le x \le r-H \, , \\
1 & \text{ if } \quad r-H \le x \le r \, .
\end{cases}$$ For $r = H$ coincides with ; for $r > H$ $$\left(r - \frac{H}{2}\right) - \left(\frac{r^3}{r^2 + H^2}\right)
= - \frac{H(r-H)^2}{2(r^2+H^2)} < 0 \, ,$$ and is better than . Similarly, for the resistance is given by $$\label{eq:rf}
R(\xi) = \frac{\xi^3}{\xi^2 + H^2} + r - \xi \, ,$$ that has minimum value for $\xi = H > 0$: $$\label{eq:fo}
u(x) =
\begin{cases}
1 & \text{ if } \quad 0 \le x \le H \, , \\
0 & \text{ if } \quad H \le x \le r \, ,
\end{cases}$$ $R(H) = r - \frac{H}{2}$, which coincides with the value for the resistance obtained with . If one compares directly with one get the conclusion that is better than precisely when $r < H$: $$\label{eq:difrtrf}
\frac{r^3}{r^2 + H^2} - \left(\frac{\xi^3}{\xi^2 + H^2}+r-\xi\right)
= \frac{\xi H^2 \left(r^2 - r\xi - H^2\right)}{\left[(r-\xi)^2+H^2\right](r^2+H^2)} \, ,$$ and since $-H^2 \le r^2 - r\xi - H^2 \le r^2 - H^2$, is negative if $r < H$, that is, for $r < H$ is better than . For $r = H$ coincide with , for $r > H$ is better than and as good as .
We now show that for $r > H$ it is possible to obtain the resistance value $r - \frac{H}{2}$ from infinitely many other ways, but no better (no less value) than this quantity. Generic situation is given by $$\label{eq:ug}
u_n(x) =
\begin{cases}
0 & \text{ if } \quad \xi_{2i} \le x \le \xi_{2i + 1} \, ,
\quad i = 0,\ldots,n \, ,\\
\frac{\mu_{i+1}-\mu_i}{\xi_{2i+2}-\xi_{2i+1}} &
\text{ if } \quad \xi_{2i+1} \le x \le \xi_{2i+2} \, ,
\quad i = 0,\ldots,n-1 \, ,
\end{cases}$$ where $n \in \mathbb{N}$, $0 = \xi_0 \le \xi_1 \le \cdots \le \xi_{2n+1} = r$, $0 = \mu_0 \le \mu_1 \le \cdots \le \mu_{n} = H$. We remark that for the simplest case $n = 1$ simplifies to $$u_1(x) =
\begin{cases}
0 & \text{ if } \quad 0 \le x \le \xi_1 \, , \\
\frac{H}{\xi_2 - \xi_1} & \text{ if } \quad \xi_1 \le x \le \xi_2 \, , \\
0 & \text{ if } \quad \xi_2 \le x \le r \, ,
\end{cases}$$ which covers all the previously considered situations: for $\xi_1 = 0$, $\xi_2 = r$ we obtain ; for $\xi_2 = r$ ; and for $\xi_1 = 0$ one obtains . All Pontryagin control extremals of the restricted problem are of the form , and by Theorem \[TeormExtPont-d-dim\] also the minimizing controls. The resistance force $R_n$ associated with is given by $$\begin{gathered}
\label{eq:Rnug}
R_n\left(\xi_0,\ldots,\xi_{2n+1},\mu_0,\ldots,\mu_n\right) \\
= \sum_{i=0}^{n} \left(\xi_{2i+1}-\xi_{2i}\right)
+ \sum_{i=0}^{n-1} \frac{\left(\xi_{2i+2}-\xi_{2i+1}\right)^3}{\left(\xi_{2i+2}
-\xi_{2i+1}\right)^2 + \left(\mu_{i+1}-\mu_i\right)^2} \, .\end{gathered}$$ It is a simple exercise of calculus to see that function has three critical points: two of them not admissible, the third one a minimizer. The first critical point is defined by $\mu_i = 0$, $i = 0,\ldots,n$, which is not admissible given the fact that $\mu_n = H > 0$. The second critical point is given by $\mu_i - \mu_{i-1} = \xi_{2i-1} - \xi_{2i}$, $i = 1,\ldots,n$, which is not admissible since $\mu_i - \mu_{i-1} \ge 0$, $\xi_{2i-1} - \xi_{2i} \le 0$, and $\mu_i = \mu_{i-1}$, $i = 1,\ldots,n$, is not a possibility given $\mu_n = H > \mu_0 = 0$. The third critical point is $$\label{eq:minug}
\mu_i - \mu_{i-1} = \xi_{2i} - \xi_{2i-1}\, , \quad i = 1,\ldots,n \, ,$$ which is a minimizer for $H \le r$. Thus, all the minimizing controls for the restricted two-dimensional problem with $H \le r$ are of the following form: $$\label{eq:minContug}
u_n(x) =
\begin{cases}
0 & \text{ if } \quad \xi_{2i} \le x \le \xi_{2i + 1} \, ,
\quad i = 0,\ldots,n \, ,\\
1 & \text{ if } \quad \xi_{2i+1} \le x \le \xi_{2i+2} \, ,
\quad i = 0,\ldots,n-1 \, ,
\end{cases}$$ $n = 1,2,\ldots$, $0 = \xi_0 \le \xi_1 \le \cdots \le \xi_{2n+1} = r$. For $u_n(x)$ given by the resistance reduces to $R_n = r - \frac{H}{2}$, $\forall$ $n \in \mathbb{N}$.
The restricted two-dimensional Newton’s problem of minimal resistance admit always a solution:
- the unique solution associated to control , when $H > r$;
- infinitely many solutions associated to the controls , when $H \le r$.
In the case $H > r$ the minimum value for the resistance is $\frac{r^3}{r^2+H^2}$, otherwise $r - \frac{H}{2}$.
Conclusion
==========
Newton’s classical problem of minimal resistance offer two interesting situations to be studied: the problem in dimension two; and the problem in dimension $d$, $d$ a real number greater or equal than three. While second situation is well studied in the literature, and well understood, the first one has been ignored. In the classical three-dimensional Newton’s problem of minimal resistance, only the problem with restriction $u(x) = \dot{y}(x)
\geq 0$ makes sense (without the restriction the problem has no local minimum). In the two-dimensional case, we have proved that the unrestricted case is also a well defined problem when $H >
\frac{\sqrt{3}}{3} r$, the minimum value for the resistance being $\frac{r^3}{r^2+H^2}$. The local minimizer is a triangle. The two-dimensional problem with restriction $u(x) = \dot{y}(x) \geq
0$ has always a solution: a unique solution (a triangle) when $H >
r$, with value for resistance equal to the unrestricted case; infinitely many alternative solutions for $r \ge H$, the minimal aerodynamical resistance being $r - \frac{H}{2}$.
Acknowledgments {#acknowledgments .unnumbered}
===============
This study was proposed to the authors by Alexander Plakhov. The authors are grateful to him for stimulating discussions. CS acknowledges the support of the Department of Mathematics of the University of Aveiro for participation in the 4th Junior European Meeting; DT the support from project FCT/FEDER POCTI/MAT/41683/2001.
[99]{}
I. M. Azevedo do Amaral. Note sur la solution finie d’un problème de Newton, Ann. Ac. Pol. Porto, Vol. 8, pp. 207–209, 1913.
M. Belloni, B. Kawohl. A paper of Legendre revisited, Forum Mathematicum, Vol. 9, pp. 655–668, 1997.
A. E. Bryson, Yu Chi Ho. Applied optimal control, Hemisphere Publishing Corp. Washington, D. C. Optimization, estimation, and control, Revised printing, 1975.
G. Buttazzo, B. Kawohl. On Newton’s problem of minimal resistance, Math. Intelligencer, Vol. 15, No. 4, pp. 7–12, 1993.
M. Comte, T. Lachand-Robert. Newton’s problem of the body of minimal resistance under a single-impact assumption, Calc. Var. Partial Differential Equations, Vol. 12, No. 2, pp. 173–211, 2001.
C. C. Fenske. Extrema in case of several variables, The Mathematical Intelligencer, Vol. 25, No. 1, pp. 49–51, 2003.
B. Fraeijs de Veubeke. Le problème de Newton du solide de révolution présentant une traînée minimum, Acad. Roy. Belg. Bull. Cl. Sci., Vol. 52, No. 5, pp. 171–182, 1966.
D. Horstmann, B. Kawohl, P. Villaggio. Newton’s aerodynamic problem in the presence of friction, NoDEA Nonlinear Differential Equations Appl., Vol. 9, No. 3, pp. 295–307, 2002.
A. Kneser, E. Zermelo, H. Hahn, and M. Lecat. Problème de Newton et questions analogues – Surfaces propulsives, Encyclopédie des sciences mathématiques pures et appliquées, Édition Française, Tome II, Vol. 6, Fasc. 1, Calcul des variations, Paris: Gauthier Villars, Leipzig: B. G. Teubner, pp. 243–250, 1913.
A. Yu. Plakhov. Newton’s problem of the body of minimal aerodynamic resistance, Doklady of the Russian Academy of Sciences, Vol. 390, No. 3, pp. 1–4, 2003.
A. Yu. Plakhov, D. F. M. Torres. Two-dimensional problems of minimal resistance in a medium of positive temperature, Proceedings of the 6th Portuguese Conference on Automatic Control - Controlo 2004, pp. 488–493, 2004.
A. Yu. Plakhov, D. F. M. Torres. *Newton’s aerodynamic problem in media of chaotically moving particles*, Sbornik: Mathematics, Volume 196 (2005), Number 6, pp. 885–933.
L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, E. F. Mishchenko. The mathematical theory of optimal processes, Interscience Publishers John Wiley & Sons, Inc. New York-London, 1962.
C. J. Silva. Abordagens do Cálculo das Variações e Controlo Óptimo ao Problema de Newton de Resistência Mínima, M.Sc. thesis (supervisor: Delfim F. M. Torres), Univ. of Aveiro, Portugal, June 2005.
V. M. Tikhomirov. Stories about maxima and minima, American Mathematical Society, Providence, RI, 1990.
V. M. Tikhomirov. Extremal problems - past and present. In *The Teaching of Mathematics*, Vol. 2, pp. 59-69, 2002.
D. F. M. Torres, A. Yu. Plakhov. Optimal control of Newton-type problems of minimal resistance, Rend. Semin. Mat. Univ. Politec. Torino Vol. 64 (2006), no. 1, pp. 79–95.
L. C. Young. Lectures on the calculus of variations and optimal control theory, W. B. Saunders Co., Philadelphia, 1969.
[^1]: Presented at the 4th Junior European Meeting on “Control and Optimization”, Institute of Mathematics and Physics, Białystok Technical University, Białystok, Poland, 11-14 September 2005. Research report CM06/I-01. Accepted (06-July-2006) to Control & Cybernetics.
|
harvmac
=manfnt \#1
-1.5pt
c
\#1\#2\#3\#4\#5\#6[ ]{}
\#1\#2\#3[ [(\#3)]{} ]{}
=cmss10 =cmss10 at 7pt \#1 \#1
=cmss10 =cmss10 at 7pt
\#1\#2\#1\#2
-\#1
------------------------------------------------------------------------
height\#1 depth\#2 -\#2
\#1\#2[-\#1[0=\#10 by \#2 width0]{}-\#2]{}
\#1\#2[ [\#1]{} \_]{}
\#1\#2[ [\#1]{} \_\^\~]{}
Juan Maldacena,$^{1,2}$ Gregory Moore,$^{1,3}$ and Andrew Strominger$^2$
$^1$ School of Natural Sciences
Institute for Advanced Study
Princeton, NJ 08540
$^2$ Department of Physics
Harvard University
Cambridge, MA 02138
$^3$ Department of Physics
Yale University
New Haven, CT 06511
.2in We derive a $U$-duality invariant formula for the degeneracies of BPS multiplets in a D1-D5 system for toroidal compactification of the type II string. The elliptic genus for this system vanishes, but it is found that BPS states can nevertheless be counted using a certain topological partition function involving two insertions of the fermion number operator. This is possible due to four extra toroidal U(1) symmetries arising from a Wigner contraction of a large $\CN=4$ algebra $\CA_{\kappa,\kappa'}$ for $\kappa'
\rightarrow \infty$. We also compare the answer with a counting formula derived from supergravity on $AdS_3\times S^3 \times T^4$ and find agreement within the expected range of validity.
Supersymmetric indices have proven to be invaluable in the program of accounting for black-hole entropy using D-branes . In particular, in those cases where the computation of BPS black holes can be related to counting functions in a conformal field theory, the elliptic genus has been of particular use. Nevertheless, there are examples, notably toroidal compactification of type II string, where the relevant elliptic genus vanishes, thus giving little indication about the D-brane BPS state degeneracies. Perhaps surprisingly, the degeneracies are therefore more subtle for compactification on $T^4$ than for $K3$. These degeneracies were first seriously investigated in . In this paper we study these degeneracies further in the case of the three charge system of consisting of $Q_1$ $D1$-branes, $Q_5$, $D5$-branes and momentum $N$. Using a function closely related to the elliptic genus we derive $E_{6,6}(\IZ)$ $U$-dual expressions for the case of primitive charges, i.e., charges such that $gcd(Q_1, Q_5, N)=1$. The formula is given in equations $(6.2),(6.3)$ below and is easily derived from our central result, the counting formula for 1/8 BPS states given in equation (5.9) below, valid for $gcd(Q_1, Q_5)=1$.
Our approach to the problem is to define an “index” in the same spirit as the “new supersymmetric index” of . These authors investigated the traces in supersymmetric quantum mechanics defined by: where $H$ is the Hamiltonian and $F$ is a fermion number operator. For $\CN=2$ supersymmetric theories one can take $F$ to be the generator of a $U(1)$ invariance and the “index” with $\ell=1$ is invariant under perturbations of $D$-terms (but not $F$-terms). Moreover, in general $\CE_{\ell}$ has no special invariances for $\ell \geq 2$. In this paper we consider the case $\ell=2$ in the context of certain conformal field theories. In the problem of interest we have some extra symmetry, namely the four U(1) translation symmetries of the torus. The full symmetry is a Wigner contraction of the large $\CN=4$ supersymmetry algebra $\CA_{\kappa,\kappa'}$ . We show that the presence of this large $\CN = 4$ algebra leads to invariance of $\CE_{\ell=2}$ under a class of perturbations discussed below. From the point of view of the five dimensional theory these indices are particular cases of supertrace formulas , which are invariant under deformations of the theory.
We will consider black strings in 6D compactification of IIB theory on $T^4$ and the black holes in 5D compactifications on $T^5$ obtained by wrapping these strings. In this section we summarize some standard facts about $U$-duality. See for background.
The low energy theory of $IIB$ on $T^5$ is given by the 32-supercharge supergravity supermultiplet. This has $27$ gauge fields and $42$ scalars. The scalar moduli space is $E_{6,6}(\IR)/USp(8)$. We will work in a regime of moduli space where $T^5 = S^1 \times T^4$ is metrically a product with a large radius for the $S^1$. Moreover, we assume there are no Wilson lines (of 6D gauge fields) along the $S^1$. This submanifold of moduli space is described by the moduli of 6D compactification where the last factor is the radius of the large $S^1$. A subgroup of the $U$-duality group preserving this submanifold is $O(5,5;\IZ)$ ([*not*]{} to be confused with the Narain duality group in 5D).
In 5D there are particles charged under the 27 gauge fields. Their charges form the $\IZ^{27}$ representation of $E_{6,6}(\IZ)$. Since the $U$-duality symmetry is broken to $O(5,5;\IZ)$ along the 5D particle charges accordingly decompose as the representation: of $O(5,5;\IZ)$. These representations have the following interpretations. The lattice $II^{5,5}$ is the electric/magnetic charge lattice of 6D strings. The representation $\IZ^{16}$ corresponds to the 6D charges of particles. Finally, the singlet $\IZ$ is the momentum along the large circle. We will denote a 5d charge vector in this decomposition as $\gamma= (S;P;N)$.
We are interested in charged black holes arising from wrappings of 6D strings on the large $S^1$, and in their BPS excitations. In the following sections we will count these BPS excitations using a mapping to instanton moduli space sigma models. We will then verify that this counting is invariant under a certain subgroup of the $U$-duality group $E_{6,6}(\IZ)$. To explain this subgroup we need to understand the physics of the three summands in .
The first summand is the charge lattice of 6D strings (general considerations show it is a lattice, i.e., has a symmetric nondegenerate bilinear form ). We can write $II^{5,5} \cong H_{\rm even}(T^4) \oplus II^{1,1}$. Corresponding to the decomposition in terms of D-branes and (fundamental strings, wrapped NS5 branes), respectively. We can further decompose $ H_{\rm even}(T^4) = (H_0 \oplus H_4) \oplus H_2
\cong II^{1,1} \oplus II^{3,3}$ corresponding to a natural basis of $D1$ strings parallel to the large $S^1$, wrapped $D5$ branes, and wrapped $D3$-branes, respectively.
The particle charges $P$ in 6D form the spinor representation $\IZ^{16}$ of $O(5,5;\IZ)$. Writing the decomposition under the $O(4,4;\IZ)$ Narain subgroup this decomposes as $\IZ^{16} = II^{4,4} \oplus H_{\rm odd}(T^4;\IZ)$, corresponding to momentum, fundamental string winding, and wrapping of D1, D3 branes. In this paper we often take $P=0$.
Now let us consider $U$-duality. Let us first assume the string charge $S\in II^{5,5}$ is a primitive vector. It is then a standard result of lattice theory (see, e.g. , Theorem 1.1.2 or Theorem 1.14.4) that all primitive vectors $S\in II^{5,5}$ of a given length are equivalent under $O(5,5;\IZ)$. Since uses some heavy machinery it is worth giving the following elementary example of this phenomenon. We may identify the lattice $II^{2,2}$ with the set of integral $2\times 2$ matrices. The signature $(2,2)$ quadratic form is simply the determinant. The $O(2,2;\IZ)$ automorphism group acts by left- and right-multiplication by $SL(2,\IZ)$: Now, using the standard fact that if $gcd(a,b)=1$ then there exist $p,q$ with $a p + bq =1 $, it is easy to show that $M$ can be bidiagonalized over $SL(2,\IZ) \times SL(2,\IZ)$ to Smith normal form: Thus, if $M$ is primitive then the only invariant is the determinant, i.e., the norm-square. In a similar way, if $S\in II^{5,5}$ is primitive we can, without loss of generality, put it in the form $S=(Q_1,Q_5)\oplus \vec 0 \oplus(0,0)$ with $gcd(Q_1,Q_5)=1$ (These are the cases for which there is a sigma model description). In other words, we can map any general string charge into a D1-D5 system. We then simply write $S=(Q_1,Q_5)$ and henceforth consider the charge vectors
Charge vectors of the form are special because states with these charges can be described using an instanton sigma model as in the original discussion of . It follows that invariance of physical quantities under $U$-duality transformations which preserve the form can lead to nontrivial predictions for the instanton sigma model. For simplicity we will henceforth consider only those charges $\gamma$ which can be mapped to the standard 3-charge system $\gamma = (Q_1, Q_5; \vec 0 ; N)$ of .
The $U$-duality transformations preserving the 3-charge system $\gamma = (Q_1, Q_5; \vec 0 ; N)$ form a subgroup This group is generated by 3 transformations: The transformation $\CR$ is simply a rotation by $\pi$ and is certainly an invariance of the sigma-model. Also, $\CT$ is an order two element of the Narain duality group $O(4,4;\IZ)$ corresponding to $T$-duality in all four directions. This is supposed to be a symmetry of the conformal field theory on the instanton moduli space. However $\CT'$ is [*not*]{} an invariance of the instanton sigma model. This is an “STS” type transformation in 5D which is not in $O(5,5;\IZ)$. Thus, the nontrivial predictions of $E_{6,6}(\IZ)$ $U$-duality for the instanton sigma model are reduced to checking invariance under . This is what we will check below for degeneracies of BPS states, when $Q_1,Q_5$ are relatively prime.
Now we consider the standard $D1D5$ system as an effective string in the $05$ direction. For the present purposes we will approximate the CFT for the low energy excitations of the theory on the string by a supersymmetric sigma model : Here $\sigma(X)$ denotes a supersymmetric sigma model with target space $X$, and $k=Q_1 Q_5$. The factor $\sigma\bigl[ \IR^4 \times T^4 \bigr] $ is the free sigma model from the diagonal $U(1)$ factor in the $U(Q_1) \times U(Q_5)$ gauge symmetry. The other degrees of freedom come from the hypermultiplets of interacting $D1D5$ degrees of freedom. Their target space is approximated by ${\rm Hilb}^k(T^4)$, the Hilbert scheme of $k$ points on $T^4$. This is a smooth resolution of the singular orbifold , and is endowed with a smooth hyperkähler metric.
It is important to realize that the innocent-looking has several subtleties. First of all, there should be an orbifold by certain translation symmetries. Because of a restriction to a charge zero sector, described below, this can be ignored. Furthermore, we will be working at a point in moduli space where the D1 branes cannot leave the fivebranes. At some special points in moduli space, for example when all B-fields are zero, the D1 branes can leave the system and the CFT becomes singular.
The symmetries of the CFT can be deduced from standard Dbrane technology. We assume the D1 string is in the 05 direction and the D5 wraps the $T^4$ and is in the 056789 direction. The spinors, which initially transform in the $16_+$ of the ten dimensional Lorentz group now transform under Note that the last factor is not really a full symmetry of the CFT since we are on $T^4$, but it is useful to classify spinors. The ten-dimensional supersymmetries are in the $16_+$ but only those invariant under $SU(2)^-_{6789}$ survive, i.e. only the ones with positive chirality in the 051234 directions. Thus the unbroken supersymmetry is in the representation In 1+1 dimensions $\pm \half$ chiralities correspond to left and right movers, so we see that we get (4,4) supersymmetry. We also see that spacetime rotations in the directions 1234 act as R-symmetries of this conformal field theory. Since we have 8 supersymmetries we can denote the two possible multiplets as vectors and hypers. From the center-of-mass (COM) CFT $\sigma(\IR^4 \times T^4)$ we get a vector and a hyper. The vector describes motion in $\IR^4$ and the hyper describes motion in $T^4$.
The left-moving part of the vector multiplet transforms as: and similarly for the right-moving part exchanging the $SU(2)_{1234}$ factors from .
The left-moving part of the hypermultiplet describing motion on $T^4$ transforms as The D1D5 strings give hypermultiplets $(h,\psi)$ transforming as The full CFT has a global $SU(2)^+_{1234} \times SU(2)^-_{1234}$ symmetry corresponding to spacetime rotations. This is the massive little group of particles in 5D and will be used below to enumerate BPS representations. The quantum numbers of the fields under this symmetry follow from . Note that for the $\IR^4$ factor the bosons transform under the global symmetry. Note also that all hypermultiplets transform in the same way under $ SO(1,1) \times SU(2)^+_{1234} \times
SU(2)^-_{1234}$ and in a different way from the vector multiplets . This difference is what distinguishes a vectormultiplet from a hypermultiplet in 1+1 dimensions.
For the $T^4$ and factors the $SU(2)^+_{1234}\times SU(2)^-_{1234}$ are zeromodes of left and right-moving $SU(2)$ current algebras of level $k$ which are part of the left- and right-moving $\CN=4$ superconformal algebra. In fact, in the example of toroidal compactification there is a larger superconformal algebra. This arises because there is a $U(1)^4$ current algebra which commutes with the $SU(2)_k$, and can be understood as follows. The unsymmetrized product of $k$ copies of $T^4$ has four currents which generate simultaneous translation along the four axes of all $k$ copies of $T^4$. These four currents are permutation invariant and therefore descend to four $U(1)$ currents in the orbifold theory on . The resolved conformal field theory on is determined by twenty parameters (= $4 h_{1,1}$) which determine the complex structure, Kahler class and $B$-fields . Sixteen of these are essentially associated to each $T^4$ and the last four are involved in blowing up the orbifold points. The values of these $20$ parameters are invariant under the $U(1)^4$ action. Therefore, the $U(1)^4$ current algebra survives the resolution of to .
Put more geometrically, the resolution $p: \hilbk \rightarrow \symk $ only depends on local data (such as the direction along which points approach each other at the orbifold loci) so the obvious translation symmetry of $\symk$ lifts to an action of $U(1)^4$ on $\hilbk$. $U(1)^4$ can be regarded as the $\kappa^\prime \to \infty$ limit of $SU(2)_{\kappa^\prime} \times U(1)$. Since the large $\CN=4$ current algebra is $SU(2)_{\kappa}
\times SU(2)_{\kappa^\prime} \times U(1)$, we conclude that conformal field theory has a degenerate large $\CN=4$ algebra, $\CA_{\kappa ,\infty}$. (In the following we will sometimes abuse language and refer to $\CA_{\kappa,\infty}$ as a large $\CN=4$ algebra.)
In the study of 5D black holes in $S^1 \times K3$ compactifications a key role was played by the elliptic genus for $N=2 $ conformal field theories defined by where $J_0^3$ and $\tilde J_0^3$ are the half-integral left and right $U(1)$ charges. Here and henceforth we normalize $L_0$ so that the Ramond ground states have $L_0=0$. The elliptic genus $\CE$ is a useful object because it is invariant under all smooth deformations of the theory. The trace is taken in the RR sector of the conformal field theory. Of course, it can also be defined for ${\cal N}=4$ theories by embedding the $U(1)$ charges in $SU(2)$. But in theories having large ${\cal N}=4$ symmetry it is not useful because it always vanishes. We will now show that the modified partition function is an analogous topological invariant for theories with the large $\CN=4$ symmetry.(Note that $\CE_1=0$, and indeed, $\Tr (J_0^3)^n = 0 $ in any $SU(2)$ representation, for $n$ odd.) This amounts to showing that the massive representations of this degenerate large $\CN=4$ algebra do not contribute to $\CE_2$. Consider the subalgebra generated by the Ramond-sector zero mode generators $G_0^{\pm \pm}$, $Q_0^{\pm\pm}$, $J_0^3$ and $L_0$. Since $L_0$ commutes with the rest of the generators we can just think of it as a c-number. The relevant commutation relations are The rest of the commutators, including those of $G$’s with $Q$’s, vanish if we consider states neutral with respect to $U(1)^4$, $i.e.$ with no momentum or winding on $T^4$. The general case will be discussed momentarily.
For a massive representation, by definition $L_0 >0$. This implies that the commutation relations of the $G$’s and $Q$’s are those of fermionic creation and annihilation operators. We have four creation operators $b_i^\dagger$ which we choose to have $J_0^3 =
1/2$. The annihilation operators then have $J_0^3=-1/2$. Let $|0,j \rangle $ denote the state that is annihilated by all the annihilation operators and obeys $J_0^3|0,j \rangle=j|0,j \rangle
$ for some $j$. Acting with the creation operators we get four states with $J_0^3 = j+1/2$, six states with $J_0^3= j+1$, four with $J_0^3=j +3/2$ and one with $J_0^3 = j+2$. The fermion numbers of these states alternate. It is easy to check that the traces over this zero mode representation $\Tr_j(-1)^F = \Tr_j
(-1)^{2J_0^3}$ as well as $\Tr_j(-1)^{2J_0^3} J_0^3$ vanish. One also finds by direct computation We conclude the massive representations do not contribute to $\CE_2$.
If we now relax the assumption that the $U(1)^4$ charges vanish, then the anti commutation relations of the $G$’s and $Q$’s (denoted collectively as $b_i, b_i^\dagger$, $i=1,\cdots, 4$) are of the form where $M_{ij}$ is an Hermitian matrix which depends on $L_0$ and the four $U(1)$ charges. We can diagonalize $M$ by a unitary transformation. If the eigenvalues of $M$ are all non-zero, then the $b$’s are usual creation and annihilation operators and the trace of $(-1)^{2 J_3} J_3^2 $ vanishes. This is the case when $L_0 > \sum_{i=1}^4 u_i^2 $ where $u_i$ are the eigenvalues of the four U(1) charges (appropriately normalized). If $M$ has zero eigenvalues, this is no longer the case. This happens when $L_0 =
\sum_{i=1}^4 u_i^2 $. It would be very interesting to understand these BPS states carrying additional charges. In this paper, however, we concentrate on the case where all these charges are zero.
For non-degenerate large $\CN=4$ algebras $\CA_{\kappa,\kappa'}$ with finite $SU(2)$ levels $\kappa $ and $\kappa^\prime$, the commutators of $G_0$ and $Q_0$ have $SU(2)_\kappa\times SU(2)_{\kappa^\prime}$ current algebra zero modes on the right hand side. This complicates the preceding argument. However in this case one may conclude from direct examination of formulae in that the massive characters do not contribute to $\CE_2$. This implies that the index will be useful to analyze the conformal field theory related to $AdS_3 \times S^3 \times S^3 \times
S^1 $ . In fact it would be very interesting to compute the supergravity result since it could teach us something about the dual conformal field theory.
As we shall see shortly, the massless characters with $L_0=0$ do contribute to $\CE_2$. This contribution is independent of the continuous parameters describing the resolution of to . Hence we can compute $\CE_2$ for all cases from the limiting case of .
In this section we explain the spacetime interpretation of . The $D1D5$ system on $S^1 \times T^4$ and its excitations describe particles in 5 dimensions. These all transform in representations of the 5d Poincaré supersymmetry algebra with 32 real supercharges. The different representations can be characterized by the $Spin(4)_{1234}$ characters of the representation of the little superalgebra. The long representations built with 32 active (i.e. broken) supercharges have character Here the subscript indicates the number of preserved supercharges, $(j_L, j_R)$ are arbitrary half-integral spins, and
The BPS states we will encounter in the $D1D5$ system come in three kinds of short representations:
A. $M = Z_1$. $M \not= \vert Z_i\vert, i>1 $, where $Z_i$ are the skew eigenvalues of the central charge matrix. The characters are
B. If instead $M=-Z_1$ we get
C. Finally, a shorter representation has character U-duals of massive Dabholkar-Harvey states turn out to be in representations of type C. There are also other BPS states in other representations for example $1/2$ BPS states, etc.
We now discuss how these characters show up in CFT partition functions. In general for the CFT $\sigma(X)$ we denote For the conformal field theory this trace is a product of two factors: One for the COM degrees of freedom and one for the CFT $\sigma({\rm Hilb}^k(T^4))$. The first factor can be computed straightforwardly in terms of oscillators using the quantum numbers . We will discuss the second factor in section five.
The trace for the CFT can be decomposed in terms of the characters of the massive little superalgebra: where $(\Delta,\bar \Delta)$ run over the massive spectrum of the CFT .
In the $D$’s measure the degeneracies of various types of representations of the spacetime $D=5, \CN=4$ superalgebra. In particular, $D_{8/32}(Q_1,Q_5; j_L, j_R)$ is the number of BPS multiplets of charge $(Q_1, Q_5, N=0)$ in the representation . $D_{4/32}^+(Q_1,Q_5,N; j_L, j_R)$ is the number of BPS multiplets of charge $(Q_1, Q_5, N>0)$ in the representation . These are macroscopically black holes with positive horizon area, etc.
Note that part of the structure of as a function of $y, \tilde y$ follows from the representation theory of the algebra $\CA_{\kappa,\infty}$. From the COM sigma model we have an overall factor of $(y^{1/2} - y^{-1/2})^4
(\tilde y^{1/2} - \tilde y^{-1/2})^4$. Then, in the sigma model we have $\Tr(-1)^F F^{\ell} = 0 $ for $\ell=0,1$ and therefore there is an extra factor of $(y^{1/2} - y^{-1/2})^2
(\tilde y^{1/2} - \tilde y^{-1/2})^2$ coming from this piece. For massive reps $\Delta >0$ of $\CA_{\kappa,\infty}$ we showed in section three that in fact $\Tr(-1)^F F^{\ell} = 0 $ for $\ell=0,1,2,3$ and therefore reps with $\Delta >0, \bar \Delta =0$ give a factor of $(y^{1/2} - y^{-1/2})^4
(\tilde y^{1/2} - \tilde y^{-1/2})^2$, etc.
In order to give a counting formula for BPS multiplets we should take a derivative of by ${ 1 \over 6!} \bigl( {d \over d\tilde y} \bigr)^6$ at $\tilde y=1$. From the CFT of the sigma model $\sigma(\IR^4\times T^4) \times \sigma(\symk)$ we need 4 derivatives to act on the COM part of the sigma model and 2 derivatives to act on the $S^k(T^4)$ part. There is a surprising cancellation of the COM contributions from $\IR^4$ and $T^4$ after setting $\tilde y = 1$ and the $D1D5$ CFT gives simply: Comparing with we finally obtain the desired counting formula for representations:
In this section we evaluate $\CE_2$ more explicitly. In a general formula was derived relating the partition function for a conformal field theory with target $X$ to that of a conformal field theory whose target is the orbifold ${\rm Sym}^k(X)$. The partition function for a single copy of $X$ defines the degeneracies $c(\Delta,{\bar \Delta}, \ell,
{\tilde \ell})$ via: Here the trace is in the RR sector. The spectrum of $U(1)$ charges $\ell,\tilde \ell$ is integer or half-integer, according to the parity of the complex dimension of $X$ and $\Delta,\bar \Delta$ runs over the spectrum of $L_0, \bar L_0$. The values of $\Delta,\bar\Delta$ are in general arbitrary nonnegative real numbers, although the difference $\Delta - \bar \Delta$ is integral.
In terms of $c$, the partition function over ${\rm Sym}^k(X)$ may be derived using a small modification of the discussion in , and is: where the prime on the product indicates that $\Delta,{\bar \Delta}$ are restricted so that $ {\Delta-{ \bar \Delta} \over n}$ is an integer.
We now specialize to a target space such that $Z(X) \vert_{\tilde y=1} = Z(X)' \vert_{\tilde y=1} =0$ (as, for example, in the case $X=T^4$ due to fermion zero modes.). Thus we have Moreover, we assume that the conformal field theory for $X$ has a realization of the superconformal algebra $\CA_{\kappa,\kappa'}$ or its $\kappa' \rightarrow \infty$ contraction. In this case we may use the results of the previous section to obtain the identity (Recall that we are taking the $U(1)$ charges to be zero.)
Let us now compute ${\cal Z}''$. It follows from that $\cal Z$ in is equal to one for $\tilde y =1$. Differentiating with respect to ${\tilde y}$ gives If we set ${\tilde y } = 1$ we get zero by . Next we compute the second derivative of with respect to ${\tilde y}$ and set $\tilde y =1$. If the second derivative acts on the ${\cal Z}$ factor in the result vanishes when we set ${\tilde y} =1 $. So the second derivative must act on the sum in . After setting ${\tilde y } =1$ one finds that the sum over ${\bar \Delta}$ drops out, since only the ${\bar \Delta}=0 $ term contributes by . This implies that $\Delta$ is integral and divisible by $n$: $\Delta = n m$ with $m=0,1,2 \cdots$. So we get Here we have defined
Expanding yields where $s,n\geq 1, m\geq 0, \ell\in \IZ$. Collecting powers of $p,q,y$ we finally obtain our counting formula: We stress that this formula is only applicable for $gcd(Q_1, Q_5) = 1$ (i.e., for a primitive vector in the string charge lattice) because otherwise the possibility of bound states at threshold obscures the relationship between the sigma model and the $D1D5$ system.
Let us now specialize to the particular example of $X=T^4$. The partition function becomes Here $\Gamma^{4,4}$ is a lattice of zeromodes. We will be interested in states with zero $U(1)$ charges and as we discussed above this implies that only states with $\tilde L_0 = 0 $ will contribute to the index we will be computing. This implies that $p_R =0$ for each copy of the symmetric product of $T^4$’s. For generic values of the $T^4$ moduli this implies that also $p_L = 0$. If we go to the particular values where we have additional values of $p_L$ allowed we see that they should appear in pairs so that their contribution to the index cancels. We will therefore drop the lattice sum in .
Taking explicit derivatives and using the product formula: with $y = e^{2 \pi i z}$ gives: The left hand side is a weak Jacobi form of weight $-2$ and index $1$. Therefore, the coefficients $\hat c(n,\ell)$ are actually functions of only one variable $\hat c(n,\ell) = \hat c(4 n - \ell^2)$ . (This can also be seen by bosonizing the U(1) current.) Using the sum formula and $\chi_{j=n}(y^{1/2}) = \chi_{j=n/2}(y) + \chi_{j=(n-1)/2}(y)$ (valid for $n>0$ and integral) and expanding one easily derives explicit formulae for the expansion coefficients: Note that the positive powers of $q$ have an extra factor of $(y^{1/2}- y^{-1/2})^2$, and that $(y^{1/2}- y^{-1/2})^2= \chi_{1/2}- 2\chi_0$ allowing a decomposition into $SU(2)$ characters.
Finally, let us close with two remarks.
[1.]{} First, the expression $\CZ''$ in can be interpreted more geometrically. Recall that there is an action of $T^4$ on $\hilbk$ lifting the action by translation on $\symk$. The quotient space $\hiltil:= \hilbk/T^4$ is a simply connected irreducible hyperkähler manifold. Working in the charge zero sector the partition function factorizes: for $k\geq 1$. Therefore, so $\CZ''$ is essentially just the generating function for elliptic genera of the hyperkähler spaces $\hiltil$.
[2.]{} Second, similar multiplet counting formulae apply to compactifications on $S^1 \times K3$. In this case, $K3$ breaks half of the supersymmetries, the BPS multiplets are smaller, the sigma model is now $\sigma(\IR^4)
\times \sigma({\rm Sym}^k(K3))$ and the analog of is obtained by taking $\half {d^2 \over d \tilde y^2}\vert_{\tilde y =
1}$ to get: We see that in this case the center of mass sigma model contributes to the index for spacetime BPS states.
$U$-duality has interesting implications in connection with the long-string picture of . The six dimensional $O(5,5;\IZ)$ $U$-duality group does not transform $N$, but, as mentioned above can be used to put the string charge $S$ in a canonical form, which we take to be $S= (Q_1 Q_5, 1)$. By a permutation like we can then map to $S=(Q_1 Q_5, N)$. Then, if $N$ and $Q_1 Q_5$ are relatively prime we can again use $U$-duality to map to a charge vector of the form $\gamma = (1,1;\vec 0; Q_1 Q_5 N)$. This state is just a single D1 and a single D5 with momentum $N'= Q_1 Q_5 N$, and its degeneracy is the same as that of a single long string. This implies that if we think in terms of strings in the fivebrane , only the long string contributes and all other contributions cancel. It can be seen from that indeed in this case only the term with $s=1$ contributes to .
This description in terms of a long string applies when we take $N$ to be coprime with $Q_1Q_5$. However, given $k=Q_1Q_5$ we should consider all possible values of $N$ and the structure of the Hilbert space is of the form: where $\sum r k_r =k$, and $\CH_{r}(X) $ is the single string Hilbert space for a string of length $r$. The sectors which contribute to $\CZ''$ are of the form $ \oplus_{r \vert k} {\rm Sym}^{k/r} (\CH_{r}(X) ) \oplus
\cdots$ and correspond to collections of strings of a single length $k/r$.
In this section we extend the counting formula from the case $gcd(Q_1,Q_5)=1$ to all primitive vectors equivalent to the three charge system.
$U$-duality under the transformation $\CT$ is obvious. To check U-duality under $\CT'$ we should remember that our formula is valid only if $gcd(Q_1,Q_5)=1$, therefore we can compute the right hand side of only if $gcd(N,Q_5) =1$ as well. In that case it is easy to see that the sum over $s$ is such that the two results agree. If one drops the restriction $gcd(Q_1,Q_5)=1$ then is [*not*]{} $U$-duality invariant. As a simple example, let $p_1, p_2$ be two distinct primes. Then $\gamma = (p_1, p_1;0;p_2)$ on the RHS of gives $\sum_\ell y^\ell \hat c(p_1^2 p_2, \ell)$ while $\gamma = (p_2,p_1;0;p_1) $ gives $\sum_\ell y^\ell \hat c(p_1^2 p_2, \ell)
+ \sum_\ell p_1 y^{\ell p_1} \hat c(p_1 p_2, \ell)$.
To cure this problem we begin by noting that, in close analogy to the remark of , the expression on the RHS of is just a transform by a Hecke operator $V_{Q_1 Q_5}$ applied to a Jacobi form . Since $V_{Q_1 Q_5} = V_{Q_1} V_{Q_5}$ for $Q_1, Q_5$ relatively prime one might wonder if the general formula is given by $V_{Q_1} V_{Q_5}$. Indeed, this guess has some very attractive features. Following , pp. 44-45 one can write (for any $Q_1, Q_5$, not necessarily relatively prime): where $N(s)$ is the number of integral divisors $\delta$ of It follows that $V_{Q_1}V_{Q_5} = V_{Q_5} V_{Q_1}$ and, more importantly, that is [*completely symmetric*]{} in $Q_1, Q_5, N$. Thus, this is a natural $U$-duality invariant ansatz for the general case. Indeed, it is the unique $U$-duality invariant extension to primitive 3-charge systems.
In this section we compute $\CE_2$ by summing over multiparticle supergravity excitations of $AdS_3\times S^3\times T^4$ and using the AdS/CFT correspondence . A similar comparison with the elliptic genus for the $K3$ case was made by de Boer .
The supergravity computation is in the NS sector while the CFT partition function is normally calculated in the R sector. Under spectral flow between the sectors a state with weight $h_R$ and (half-integer) U(1) charge $j_R$ is mapped into a state with weights where $k = c/6$ ($c=6Q_1Q_5$ is the central charge of the CFT). We use a convention such that $h_{NS} =-k/4 $ for the NS vacuum. The partition function in the NS sector can be obtained from the partition function in the R sector by the following replacements
In principle one expects agreement with supergravity only for small conformal weights, not much bigger than the NS vacuum $h_{NS} = - k/4 $. When conformal weights are of order $k$ the stringy exclusion principle is relevant and supergravity breaks down. We shall in fact find agreement for all negative values of $h_{NS}$, $-k/4 \leq h_{NS} <0$.
For the CFT we start from in the RR sector and we find the NS-NS partition function
Now we concentrate on the terms in this expansion with negative powers of $q$, relevant for the comparison to supergravity. The only possibility is $n=1,~m=0,~l=-1$, since $\hat c(r) =0$ for $r< -1$. Using $\hat c(-1) = 1, ~
\hat c(0) =-2 $ this gives where the dots involve non-negative powers of $q$.
Now we consider the supergravity calculation. We need to define an appropriate notion of a “supergravity elliptic genus” $Z_{\rm sugra}(p,q,y)$. We will follow the proposal of de Boer . The single particle supergravity Hilbert space can be derived by group theory and Kaluza-Klein reduction. It decomposes as a representation of $SU(2\vert 1,1) \times SU(2 \vert 1,1)$: Short $SU(2\vert 1,1)$ reps are labelled by the maximal spin, i.e., a nonnegative half-integer $j$. The highest weight has $h= j$. Label it by $(j)$. It turns out that single particle states are always products of short representations. There is no analog of the $ long \otimes short $ of CFT. (These latter come from multiparticle supergravity states.) It turns out that the degeneracies in can be read off from the identity of where $h(X)= \sum_{r, \tilde r}(-1)^{r+\tilde r}
h^{r,\tilde r} y^r \tilde y^{\tilde r}$ is the Hodge polynomial. The generating function counts $(c,c)$ primaries. Each $(c,c)$ primary in turn corresponds to a short $SU(2\vert 1,1) \times SU(2 \vert 1,1)$ representation. De Boer proposes to associate a new quantum number to the supergravity states, the degree, in order to take into account the exclusion principle. The degree is the power of $p$ multiplying the various factors in . Thus, representations are now labelled by $(r,\tilde r;d)$ where $d$ is the degree. Notice that this assignment of degree breaks the $SO(4,5)$ continuous U-duality symmetry of supergravity on $AdS_3\times S^3 \times T^4$.
With this innovation the single-particle Hilbert space is: where $h^{r,\tilde r}(X) $ are the Hodge numbers of $X=K3, T4$. For the torus the Hodge polynomial factorizes as $(1-y)^2(1-\tilde y)^2$ so we can introduce the useful device for the torus Hilbert space: Here $d(0)=d(2) = 1, d(1)=-2$ (the sign is for a fermionic representation). Notice that we are including the identity.
We now define the “supergravity elliptic genus” as the free field theory partition function for the Fock space built up from $\CH_{\rm single\ particle} $: (here it is more convenient to use $\ell = 2j$ which is integral). Since we will eventually set $\tilde y=1$ and expect only holomorphic quantities from left chiral primaries we will temporarily suppress $\bar q$. This is not totally innocent, and we will return to the $\bar q$-dependence at the end of this section. Suppressing $\bar q, \tilde y$, we can rewrite as a product of factors $(1-p^n q^h y^\ell)^{ - c_s(n,h,\ell) } $ where $c_s(n,h,\ell)$ is the number of single particle states with $L_0 = h$, $U(1)$ charge $=\ell$ and ’degree’ $n$. (As usual $c <0$ for fermions). Here we are measuring $L_0$ relative to the NS vacuum, as is conventional in $AdS$ discussions. So in order to compare with the above formulae we need to replace $p \to p q^{-1/4}$. The effects of the exclusion principle are approximated by truncating the supergravity spectrum to states with total degree $k=Q_1Q_5$.
The full exclusion-principle-modified supergravity partition function is thus Of course, as written $\CZ_{\rm sugra}=1$ for $T^4$ at $\tilde y=1$. We therefore need to put back $\tilde y$ and take derivatives to get a nontrivial quantity. The $\tilde y$ that appears in the R partition function differs from the one appearing in the NS partition function by a factor of $ {\bar q}^{1/2}$ arising in the spectral flow. So after differentiating twice we set $\tilde y = {\bar q}^{1/2}$. This selects the chiral primaries. Manipulations similar to those in section five then lead to where $\hat c_s(n,h,\ell) = \sum_{\tilde \ell}
{\tilde \ell}^2
c_s(n,h,\ell,\tilde \ell)$ counts the number of right chiral primaries with the given properties and the sum over $\tilde \ell$ runs over all the chiral primaries of degree $n$.
Next we need a good way to enumerate chiral primaries in this theory. Using above we can perform the sum over ${\tilde \ell }^2$ over chiral primaries of given degree. It is easy to see that $\sum_r d(\tilde
r) = 0$, $\sum d(\tilde r) ( n + \tilde r) =0$ and $\sum_r
d(\tilde r) (n+\tilde r)^2 = 2$. This final sum is independent of $n$ and just gives an overall factor, as in the CFT result. We now need to compute $c(n,h,\ell)$ just for the left-moving piece. Ignoring for a moment the sum over $s$ we see that we have where $r=0,1,2$ as above and $t=0,1,2$ takes into account the descendants of the form $G_{-1/2}$, etc. The sum over $k$ takes into account the descendants of the form $L_{-1}^k$, $k =0,1,2$. The sum over $\ell$ is in steps of 2. We have replaced $p \to p
q^{-1/4}$ to take into account the ground state energy so that we can compare to . The sum over $s$ is taken into account by replacing $(p,q,y) \to ( p^s, q^s,y^s) $ multiplying by $s$ and then summing over $s$. We are interested in terms with negative powers of $q$. This requires The only possibility is $k=r=t=n=0$, and this reproduces . Hence the supergravity and CFT calculations of $\z$ agree exactly for all negative powers of $q$. Notice that basically only the ground state is contributing to . So the agreement boils down to the statement that all the gravity contributions cancel at low enough energies.
It is not hard to see that the agreement does not persist for nonnegative powers of $q$ (indeed, there is a discrepancy at order $q^0$). This is not surprising because supergravity becomes strongly coupled before this point. Indeed a black hole which is a left-chiral primary appears at this level. This black hole is an extremal rotating black hole with angular momentum on $S^3$.
Finally, let us return to the issue of the $\bar q$ dependence of $\CZ_{\rm sugra}''$. In fact if $\bar q$ is reinstated, one finds at these excited levels dependence on positive powers of $\bar q$. This might seem to be a contradiction because we argued in section 3 that the large $\CN=4$ algebra forbids $\bar q$-dependence of $\z$. What happens is that the implementation of the exclusion principle as a cutoff on supergravity states breaks the large $\CN=4$, which for example maps single particle states below the cutoff to multi-particle states above the cutoff. Hence this implementation, while very successful at low energies, is too naive to describe the Hilbert space at high energies. Indeed, the $\bar q$ dependence at order $p^N$ first shows up at order $\bar q^{N/4}$. Thus, in the large $N$ limit the action of the large $\CN=4$ algebra is restored, in accord with the AdS/CFT correspondence.
As we have stressed, is false when $Q_1, Q_5$ have common factors, i.e., when the Mukai vector of the instanton moduli space is not primitive. This is not terribly surprising since it is known that the moduli space is singular under such circumstances, and there are even resolutions of the space not equivalent to the Hilbert scheme of points . Physically, nonprimitive vectors are associated with the possibility of boundstates at threshold so we expect subtleties in counting BPS states. Very similar subtleties were found already in the work of Vafa and Witten in . In view of this one should be cautious about the existing formulae for BPS states in $S^1 \times K3$ compactification for nonprimitive Mukai vectors. Unfortunately, $U$-duality is not a useful tool for probing this question.
In we extended to all primitive 3-charge systems, but this leaves open the question of what the degeneracies really are when $\gamma$ is not primitive. Because of bound-states at threshold this question requires careful definition. One way to approach this question is to use the trick in , compactifying on another circle and turning on a charge to remove the boundstates at threshold.
The results of this paper raise some interesting open problems. It is natural to expect that the full set of BPS states for toroidally compactified type II string is counted by some interesting automorphic functions transforming nontrivially under $E_{d,d}(\IZ)$. One might hope that such forms might appear in quantum corrections along the lines of the BPS counting formulae appearing in quantum corrections. (See , p.10 for a list of references.) At present such automorphic forms remain part of the Great Unkown.
Finally it would be interesting to compute this index for supergravity on $AdS_3 \times S^3 \times S^3
\times S^1$ in order to see what we can learn about the conformal field theory.
**Acknowledgements**
We would like to thank R. Dijkgraaf, B. Gross, N. Seiberg, C. Vafa and E. Witten for very helpful discussions and correspondence. GM is supported by DOE grant DE-FG02-92ER40704 and the Monell Foundation. JM and AS are supported by DOE grant DE-FG02-92ER40559. JM is also supported by NSF PHY-9513835, the Sloan Foundation and the David and Lucille Packard Foundation. JM is also a Raymond and Beverly Sackler Fellow.
|
---
abstract: 'The entropy production rate for an open quantum system with a classically chaotic limit has been previously argued to be independent of $\hbar$ and $D$, the parameter denoting coupling to the environment, and to be equal to the sum of generalized Lyapunov exponents, with these results applying in the near-classical regime. We present results for a specific system going well beyond earlier work, considering how these dynamics are altered for the Duffing problem by changing $\hbar,D$ and show that the entropy dynamics have a transition from classical to quantum behavior that scales, at least for a finite time, as a function of $\hbar^2/D$.'
author:
- 'Arnaldo Gammal$^{(a)}$ and Arjendu K. Pattanayak$^{(b)}$'
title: Quantum entropy dynamics for chaotic systems beyond the classical limit
---
Consider a quantum system with a nonlinear classical limit: Non-classical effects depend on the size of Planck’s constant $\hbar$ compared to the characteristic action. Further, the system-environment interaction as measured through some parameter $D$, is crucial[@zurek-rmp]. The dynamics of the classical limit of the problem are important[@ClassicalLimit] particularly through the classical Lyapunov exponents $\lambda$. It has recently been found that the quantum entanglement rate for chaotic systems shows a valuable speed-up[@entanglement-chaos] but this is to be balanced against the observed enhanced decoherence effects for chaotic systems in the classical limit[@zp; @akp; @MP]. However, there is increased stability against fidelity decay deep in the quantum parameter regime[@Prosen], leading to the proposal to ‘chaoticize’ quantum computation[@Prosen-qc]. This complex multi-parameter quantum-classical transition is fundamental, poorly understood, and also valuable in understanding the behavior of quantum devices.
A recent analysis[@akp-03], summarized below, suggested that headway could be made in characterizing the full range of behavior by considering composite parameters and scaling. That is, the quantum-classical difference as measured by some quantity $QC_d(\hbar,D,\lambda)$ should be the simpler function $QC_d'(\zeta)$ of a single composite parameter $\zeta =\hbar^\alpha D^\beta\lambda^\gamma$. Evidence has begun to accumulate[@scaling-new] supporting this perspective. These come mostly from studying the effect of changing $D,\hbar$ on time-independent (usually from $t\to\infty$) measures $QC_d$. The change with $\lambda$ is harder to study since the classical phase-space changes along with $\lambda$. A different but related issue is the non-equilibrium statistical mechanics of a nonlinear quantum system as measured through the system’s entropy dynamics. A powerful result of broad interest is that the entropy production rate for an open quantum system with a classically chaotic limit is independent of $\hbar$ and $D$ and is equal to the sum of generalized Lyapunov exponents[@zp; @akp; @MP]. However, this has been verified only in the classical limit, and despite the considerable interest in this, there are few useful results away from this limit.
In this Letter, we start with the argument that quantum-classical distance can be measured sensibly with a quantum system’s linear entropy. We then study the entropy dynamics for the chaotic Duffing oscillator as a function of $\hbar, D$ to obtain several novel results that considerably extend results on entropy decay as well as generalize the scaling results. Specifically, the Lyapunov exponent dependence is shown to be valid only for a small parameter range and for times. We look, more usefully, at the time-dependent entropy itself which unexpectedly shows scaling with a single parameter $\zeta_0 = \hbar^2/D$, thus generalizing previous results from time-independent measures[@akp-03; @scaling-new]. That is, behavior from widely varied $\hbar,D$ collapse onto curves that depend only on $\zeta_0$, which we explain on the basis of an expansion in $\zeta_0$, as well as direct comparison of dynamics. This enables the characterization of entropy dynamics over a much wider range of parameters and times than previously attempted. We show dynamical regimes which we term (I) classical, (II) semi-classical and (III) quantum, with a smooth transtion between these regimes with increasing $\zeta_0$.
We begin with the Master equation for the evolution of a quantum Wigner quasi-probability $\rho_W$ under Hamiltonian flow with potential $V(q)$ while coupled to an external environment [@zp]: $$\begin{aligned}
{\partial \rho_W\over\partial t}
&=& L_c + L_{q} + T \\
&=&\{H,\rho_W\} \nonumber \\
&+&\sum_{n \geq 1}\frac{\hbar^{2n}(-1)^n}{2^{2n} (2n +1)!}
\frac{\partial^{2n+1} V(q)}{\partial q^{2n+1}}\;
\frac{\partial^{2n+1} \rho_W}{\partial p^{2n+1}}\nonumber \\
&+& 2\gamma\partial_p(p\rho_W) + D \partial_p^2\rho_W
\label{wigner}\end{aligned}$$ The first term, the Poisson bracket $L_c$, generates the classical evolution for $\rho_W$. The terms in $\hbar$ are the quantal ‘correction’ terms (denoted $L_{q}$). The environmental coupling ($T$) is modelled by the diffusive $D$ term and the dissipative $\gamma$ term. We assume, as typical, short time-scales or high temperatures such that the $\gamma$ term is negligible. A $QC_d$ can then be considered by propagating the same initial condition with $L_q+L_c+T$, compared to using only $L_c$, or more appropriately using $L_c +T$ from above.
If $QC_d$ is the difference between the expectation values of an observable, it becomes strongly dependent on the observable. For example, even when the centroids of a quantum and classical distribution are behaving identically, differences exist in higher-order moments. Further, measures such as the time when the $QC_d$ hits a pre-defined value introduce subjectivity. Moreover, while powerful in the abstract, it is inherently unphysical to propagate something both classically and quantally. Some of these problems can be avoided by monitoring the quantum entropy, which does not measure distances but directly addresses relevant issues of information. The linear or Renyi entropy of second order $S_2$ is also the natural logarithm of the purity $P$ as $S_2 = \ln(P)= \ln[ Tr\{\hat\rho^2\}]$. Note that $P={2\pi\hbar}Tr\{\rho_W^2\}$ where the ${Tr}$ now represents integration over all phase-space variables. This has been extensively studied and for a system with a classically chaotic limit, it has been argued[@zp; @MP] that in the weak-noise, small-$\hbar$ classical limit, $-dS_2/dt$ equals the sum of the positive classical Lyapunov exponents. More careful considerations generalize this to a weighted sum over Lyapunov exponents[@akp]. For the classical limit itself, this should arguably be further generalized to time-dependent versions[@Vulpiani05]. That is, although the previous results apply in some limits or special cases, even the classical behavior is not fully understood. Less is known about the quantum system, particularly the impact of changing scale or noise through $\hbar, D$, which is what we address below. We work with the Hamiltonian $H = \frac{p^2}{2m} -Bx^2 + \frac{C}{2}x^4+ Ax\cos(\omega t)$. This is the Duffing oscillator, which as a 1-dimensional driven problem with a quartic nonlinearity is one of the simplest flows with a rich phase-space structure and hence is a paradigmatic problems in Hamiltonian chaos. The quantum version has also been frequently studied, including for decoherence issues[@MP; @Shiz].
![Entropy production rate $\dot{S_2}$ (where $S_2=\ln(P)$ for states with purity $P$) for the quantum Duffing oscillator with $m=1,B=10,C=1,A=1,\omega=5.35$ and $\hbar,D$ as indicated, showing a wide variation in behavior. The initial conditions are Gaussians in the chaotic region with $\langle x\rangle_{t=0}=1.0$, the spread $\sigma^2_x =0.05,
\langle p\rangle_{t=0}=0.0$ and the spread $\sigma^2_p$ is set by the constraint of defining a minimum-uncertainty state and hence by the particular value of $\hbar$. []{data-label="figone"}](fig1i.eps){width="9.5cm" height="7.5cm"}
We briefly review the behavior of the classical density $\rho_c$ in the limit of only the $L_c +T$ evolution. As a result of chaos due to $L_c$ alone, $\rho_c$ increases fine-scale structure exponentially rapidly, with a rate given by a generalized Lyapunov exponent. When the structure gets to sufficiently fine scales, the noise $T$ becomes important, and it acts to decrease, or coarse-grain, fine-scale structure. These competing effects can be profitably studied using the measure $$\chi^2 \equiv -\frac{{\rm Tr}[\rho_c \nabla^2\rho_c]}{{\rm Tr}
[(\rho_c)^2]} = \frac{{\rm Tr}[|\nabla \rho_c|^2]}{{\rm Tr}
[(\rho_c)^2]}
\label{chi-def}$$ whence $\chi^2$ is approximately the mean-square radius of the Fourier transform of $\rho$, measuring the structure in the distribution [@97_1]. Most importantly, Eq. (\[wigner\]) yields the identity $dS_2/dt = -2D\chi^2$[@footnote-nabla; @MP] with this valid classically or quantum mechanically, that is, with both $S_2,\chi^2$ computed for $\rho_c$ or $\rho_W$[@akp] respectively. For a classically uniformly chaotic system, the dynamics of $\chi^2$ can be written approximately [@physicaD] as a competition between chaos and diffusion as $$\label{chi-dyn}
\frac{d \chi^2}{dt} \approx 2\Lambda\chi^2 - 4D\chi^4.$$ This implies that $\chi^2$ settles after a transient to the metastable (that is, constant for finite-time) value $$\chi^{2*} = \Lambda/2D
\label{meta}$$ where $\Lambda$ is a $\rho$ dependent generalized Lyapunov exponent [@schlogl; @physicaD]. This [*classical*]{} argument leads to the argument[@zp; @akp; @MP] that quantum entropy-production rates are equal to generalized Lyapunov exponents. This applies to a greater range of parameters than might be anticipated because decoherence suppresses quantum effects. While this behavior has been shown in several instances, it does not capture the complete picture, particularly the effect of changing $\hbar,D$. We show this in Fig. (\[figone\]) plotting $dS_2/dt$ for the Duffing problem with $m=1,B=10, C=1, A=1,\omega = 5.35$, as previously used[@MP]. The behavior, over a wide parameter and time range is quite complicated. If a subset (all of those with $\hbar=0.1$) are plotted for a short time ($t<15$) as in[@MP], they show the classical Lyapunov exponent entropy-production behavior[@zp; @akp; @MP]. This is valid only for some small range of parameters and short times. There has been a suggestion of a superposition of classical and quantal exponential decay[@petit] for the purity. This would lead to a crossover transition within a fairly narrow range from one constant value to another in Fig. (1), which we do not see. Other ways of considering the data (as in Fig. (2) below) also do not support this. In general the search for these small regimes of linear decay for entropy is not as helpful as understanding the broader parameter dependence.
To do this, consider as in Fig. (2), $Tr\{\rho_W^2(t)\}/Tr\{\rho_W^2(0)\}$. Since the $y$ axis is logarithmic, we are effectively looking at $\ln(Tr\{\rho_W^2(t)\})- \ln(Tr\{\rho_W^2(0)\}) = S_2(t) - S_2(0) = S(t)$ for our pure-state Gaussians. This shows useful organization invisible in Fig. (1) due to the small-scale variation in a narrow range. Most interestingly, the entropy dynamics for the wide variety of parameters considered is captured entirely for the times shown by the composite parameter $\hbar^2/D\equiv\zeta_0$, even though a wide range of behavior, not obviously characterized as exponential decay, is seen as $\zeta_0$ is varied. Larger $\zeta_0$ corresponds to high $\hbar$ or low noise $D$ or both, and remains closer to a pure quantum state for longer times, which makes physical sense. Note also that there is some $\zeta_0$ dependence for the time-scale of scaling, with a long-term separation of curves.
![Evolution of the normalized purity for the same states as Fig. (1) in the quantum Duffing oscillator. Scaling is observed relative to the parameter $\zeta_0\equiv\hbar^2/D$. (I)’Classical’: $\zeta_0=2$ a)$\hbar=0.1, D=5\times10^{-4}$ b)$\hbar=0.2, D=2\times10^{-2}$, c)$\hbar=0.5, D=0.125$, (II)’Semi-classical’: $\zeta_0=40$ d)$\hbar=0.004^{1/2}, D=10^{-4}$, e)$\hbar=0.1, D=2.5\times 10^{-4}$, f)$\hbar=0.2, D=10^{-3}$, (III)’Quantum’: $\zeta_0=100$ g)$\hbar=0.1, D=10^{-4}$, h)$\hbar=0.5, D=2.5\times 10^{-3}$, i)$\hbar=1, D=10^{-2}$. []{data-label="figtwo"}](fig2.eps){width="9.5cm" height="7.5cm"}
We understand this $\zeta_0$ dependence by considering quantum corrections to the classical dynamics, which depend (see Eq. (\[wigner\])) on the derivatives of the Wigner function. Given that the second derivatives $\partial^2\rho_W \propto \chi^2$, these corrections scale as $$L_{q} \approx \hbar^{2n} \frac{\partial^{2n+1} V(q)}{\partial q^{2n+1}}\;
\frac{\partial^{2n+1} \rho_W}{\partial p^{2n+1}} \approx
\hbar^{2n}\chi^{2n+1}V^{(2n+1)}(x) ,
\label{quant-correction}$$ where $V^{(r)}$ denotes the $r$th derivative of $V$. When the phase-space distribution hits a metastable state such that $\chi^2$ settles to the fixed value $\Lambda/2D$, the difference between the quantum and classical evolution may be estimated to depend on $$\zeta\equiv\hbar^{2n} \Lambda^{n+1/2}D^{-(n+1/2)} V^{(2n+1)}(x)
\label{zeta-estimate}$$ where, since $\chi$ is a ’length’ in Fourier space, we have that $x\approx \chi^{-1}=\sqrt{2D/\Lambda}$. This is essentially the same result as that derived in Ref. [@Shiz] from a completely different perspective and is also the root of the suggestion in Ref. [@akp-03] to search for scaling. Therefore, the first order quantum corrections in a semi-classical regime should scale, in complete generality, with the single parameter $\zeta$. The particular form of $\zeta$ is decided by the details of the Hamiltonian and the difference between the quantal and classical propagators. For the Duffing problem, the only quantum term of Eq. (\[quant-correction\]) comes from the $3$rd derivative of the quartic term whence Eq. (\[zeta-estimate\]) gives that the quantum term goes as $\zeta = \hbar^2\chi^2$; for any other form of the potential, we expect different corrections and hence different scaling as below.
We now use this in an expansion technique for entropy dynamics that may be applied in general. In the Duffing problem, even though $\dot S_2$ is not a simple function of $\zeta$, a scaling relationship still obtains in the two parameters $\hbar,D$ as follows. To zeroth order, the classical and quantal phase-space distributions are the same, $\rho_{W0} \approx \rho_c$, and $\chi_{q0}^2 = \chi_c^2$, where the entropy production rate $\dot S_{2q0} = -2 D \chi_{q0}^2$ and the numerical subscripts on $\chi_q, \rho_W, \dot S_{2q}$ indicates the order of the approximation. We now use the results from Eq. (\[quant-correction\],\[zeta-estimate\]) that the quantum-classical distance for this system behaves as $\hbar^2\chi^2$. To first order we insert the zeroth order solution in this to write $$\label{correction}
\rho_{W1} \approx \rho_{W0} + a\hbar^2\chi^2\rho_{W0} = \rho_{c} +
a\hbar^2\chi^2\rho_{c}$$ where $a$ is constant for the meta-stable state, but time-dependent in general. We substitute this in Eq. (\[chi-def\]) to get that $\chi_{q1}^2 \approx \chi_c^2 + a\hbar^2\chi_c^4.$ Corrections from the denominator of Eq. (\[chi-def\]) are of higher order, and also tend to cancel the higher order corrections from the numerator. We insert this first order quantally corrected form for the dynamical term into Eq. (\[chi-dyn\]) to get that to first order in $\hbar^2$, $\chi^2$ obeys $$\label{q1-chi-dyn}
\frac{d \chi^2}{dt} \approx 2\Lambda(\chi^2 + a\hbar^2\chi^4) - 4D\chi^4$$ and in parallel to Eq. (\[meta\]) we get that $$\label{q1-chi}
\chi^{2*} = \frac{\Lambda}{2D(1 - \frac{a\hbar^2}{4D})}$$ leading finally to $\dot S_{2q1} = -2D\chi^2_{q1} = -\Lambda(1 + \frac{a\hbar^2}{4D})$ that is, the quantum correction scales as $\hbar^2/D$. This expansion around the metastable state can occur only when the growth of structure is balanced by noise, only when $\chi^2$ is large enough that the diffusion term becomes relevant. Since $a$ is in general time-dependent, at each value of $\zeta_0$ we expect a different entropy dynamics, as in fact we see. In sum, this expansion for the entropy dynamics around the metastable state yields a $\hbar^2/D$ dependence for entropy, although the time-dependence itself is not easy to extract.
This expansion must fail for arbitrarily large $\zeta_0$, in the quantum regime. Here an alternate approach applies: the Poisson bracket term is neglected and the dynamics are given approximately by the competition between the $L_q$ and $T$ terms alone. To compare them, consider $L_q$: For the Duffing system, there is a third-derivative of $\rho_W$ multiplied by $x$ (resulting in this acting like a $2$nd derivative overall), compared to the $2$nd derivative from the diffusion term[@footnote-III]. This means that the terms have essentially the same scale, with quantum dynamics continuing to add structure and the noise smoothing it out. The entropy-production then depends only on the ratio of the parameters multiplying these terms which is again $\hbar^2/D\equiv \zeta_0$. This last parameter regime is consistent with recent results[@petit; @Prosen] et al. Finally, consider some details of the time-dependence: The rate of purity decay decreases with $\zeta_0$. Physically, the time-asymptotic dynamics are dominated by essentially classical diffusive behavior, with a common final state (the natural invariant measure) for all $\rho_W$. Since $Tr\{\rho_W^2(0)\}\propto \hbar^{-1}$ (see above), the time-asymptotic value of $Tr\{\rho_W^2(t)\}/Tr\{\rho_W^2(0)\} \propto \hbar^{-1}$. With the different rates of purity decay, the system approaches the time-asymptotic state later as $\zeta_0$ increases. Further, within each $\zeta_0$, the different values of $\hbar$ separate out from the scaling curve as the final diffusive regime kicks in, as seen in Fig. (2).
The values of $\zeta_0$ where these regimes change is in general determined by the parameters of the potential, i.e. by the quantity labeled as $a$ in Eq. (\[correction\]) above. Given the continuous behavior as a function of $\zeta_0$ the actual transition is subjective. In Fig. (2) we label what corresponds to rapidly decohering and hence essentially classical behavior as (I), the relatively slowly decohering and hence deep quantum behavior as (III) and in-between ‘semi-classical’ behavior as (II) in the three sets of curves with $(\zeta_0 = 2, 100, 40)$ respectively. That is, for this potential, empirically $\zeta_0 = \zeta_c \approx 10$ sets the approximate upper limit of the rapidly decohering regime (I), and by extension the quantum regime (III) kicks in at $\zeta_0 =\zeta_q \approx \zeta^2_c \approx 100$. We note the same scaling also holds (results not shown) for other diagnostics as well as for very different parameters for the Duffing oscillator, $A=10,\omega=6.07$, a regime of significantly increased chaos[@MP].
In conclusion, our results strengthen the argument that it is valuable to study the behavior of nonlinear open quantum systems through the scaling behavior of appropriate diagnostics, as recently suggested[@akp-03]. In particular, this is used to study the non-equilibrium statistical mechanics of an open quantum system with a classically chaotic counterpart over a wide parameter range in $\hbar,D$. We show that the entropy dynamics of this system can be dramatically different from the broadly-accepted Lyapunov exponent dependence which is only valid in the classical limit (and is itself arguably suspect[@Vulpiani05]). We show a $\hbar^2/D$ scaling in the time-dependent entropy dynamics, although the particular form of the scaling is expected to depend on the form of the nonlinearity in general.
[*Acknowledgements*]{}: A.G. is partially supported by FAPESP (Brazil) and CNPq (Brazil). A.K.P. acknowledges a CCSA Award from Research Corporation, the SIT, Wallin, and Class of 1949 Fellowships from Carleton, and hospitality from CiC (Cuernavaca) during this work.
[99]{}
W.H. Zurek, , 715 (2003).
See [*e.g.*]{} A.R.R. de Carvalho and A. Buchleitner, , 204101 (2004).
P.A. Miller and S. Sarkar, , 1542 (1999); A. Lakshminarayan, , 036207 (2001); X.G. Wang et al. , 016217 (2004).
W.H. Zurek and J.P. Paz, , 2508 (1994); Physica [**83 D**]{}, 300 (1995).
A.K. Pattanayak, Phys. Rev. Lett. [**83**]{}, 4526 (1999).
D. Monteoliva and J.P. Paz, , 3373 (2000); , 056238 (2001).
T. Prosen and M. Znidaric, New J.Phys. [**5**]{} , 109 (2003); , 044101 (2005).
T. Prosen and M. Znidaric, J.Phys.A [**34**]{}, L681-687 (2001).
A.K. Pattanayak et al., , 014103 (2003).
N. Wiebe and L. Ballentine, , 022109 (2005); F. Toscano et al., , 010101 (R) (2005); A.R.R. Carvalho et al., Phys. Rev. E 70, 026211 (2004).
C. Petitjean and Ph. Jacquod, , 194103 (2006); Ph. Jacquod, , 150403 (2004).
S. Habib et al., , 4361 (1998).
M. Falcioni et al., , 016118 (2005).
A. K. Pattanayak and P. Brumer, , 5174 (1997); Also see Yuan Gu, Phys.Lett.A [**149**]{}, 95 (1990).
A.K. Pattanayak, Physica D 148, 1 (2001).
Our argument assumes environmental coupling symmetrically to all phase-space variables, with $T$ of the form $\nabla^2$ rather than $\partial_p^2$. This simplifies calculations, but note that for a chaotic system there is no difference between the behavior of these two forms.
C. Beck and F. Schlögl, [*Thermodynamics of chaotic systems*]{}, (Cambridge University Press, N.Y., 1993).
Again, this should be different for different potentials and result in different scaling functions in this regime.
|
---
abstract: 'We present the possibility of tuning the spin-wave band structure, particularly the bandgaps in a nanoscale magnonic antidot waveguide by varying the shape of the antidots. The effects of changing the shape of the antidots on the spin-wave dispersion relation in a waveguide have been carefully monitored. We interpret the observed variations by analysing the equilibrium magnetic configuration and the magnonic power and phase distribution profiles during spin-wave dynamics. The inhomogeneity in the exchange fields at the antidot boundaries within the waveguide is found to play a crucial role in controlling the band structure at the discussed length scales. The observations recorded here will be important for future developments of magnetic antidot based magnonic crystals and waveguides.'
author:
- 'D. Kumar$^1$, P. Sabareesan$^1$, W. Wang$^2$, H. Fangohr$^2$,'
- 'A. Barman$^1$'
title: 'Effect of Hole Shape on Spin-Wave Band Structure in One-Dimensional Magnonic Antidot Waveguide'
---
Copyright (2013) American Institute of Physics. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics. The following article appeared in J. Appl. Phys. 114, 023910 (2013) and may be found at <http://dx.doi.org/10.1063/1.4813228>
Introduction {#Sec:I}
============
Magnonics [@Kruglyak2010; @Serga2010; @Lenk2011; @Demokritov2013] is an emerging sub-field of solid state physics, which studies the propagation of spin-waves (SWs) in micro- and nanoscale magnetic structures. Magnonic devices [@Khitun2008] aim to use the information carried by SWs to perform their respective designated tasks. Waveguides [@Venkat2013], SW interferometers [@Choi2006; @Podbielski2006; @Schneider2008], phase shifters [@Au2012] and magnonic crystals (MCs) [@Lenk2011; @Nikitov2001; @Neusser2009; @Chumak2010] are some of the important components of magnonic devices. Knowledge of spin-wave dispersion within such structures is necessary for their design and operation. An MC can be realized by a combination of periodic modulation of structural and material parameters of a known magnetic material and a control over the external bias magnetic field.[@Sykes1976; @Bayer2006; @Kruglyak2005] These periodic modulations of magnetic potentials within an MC interact with the spin-waves eventually yielding a characteristic dispersion relation comprising of stop and pass bands. Most MCs that form the topic of current research in magnonics are either one-dimensional (1D) [@Chumak2008; @Wang2009] or two-dimensional (2D) [@Mamica2012; @Vysotskii2010; @Tacchi2010; @Tacchi2011; @Saha2013; @Wang2013] as they are easier to fabricate on a wafer when compared to three-dimensional (3D) MCs. Nevertheless, few theoretical reports on the study of dispersion of SWs in 3D MCs have been made.[@Dyson1956; @Krawczyk2008]
Structured magnonic waveguides [@Lee2009; @Kim2009; @Ciubotaru2012] have recently attracted considerable attention due to their selective transmission of microwave bands in the micro- and nano-scales and their potential applications in on-chip microwave signal processing and communication. Magnetic antidots have emerged as an important system of MCs; and a thorough investigation of high frequency magnetization dynamics in them have been reported in the literature. [@Pechan2005; @Neusser2010; @Neusser2011; @Tacchi2012; @Hu2011; @Mandal2012] Magnonic antidot waveguide (MAW) is an attractive option for manipulation of transmitted spin waves towards the above application, but it has only recently been started to be explored. [@Ma2011; @Kumar2012; @Klos2012] So far a study of the dependence of spin-wave dispersion on the shapes of the antidots has not been reported. More importantly, how changes in the exchange field distribution around the antidot boundary can alter the characteristic dispersion of exchange or dipole-exchange SWs in a MAW, has never been observed before.
This article aims to help fill that gap in research by numerically simulating the magnonic dispersion in 1D MAW lattices with different geometric shapes of the antidots. We also study the spatial magnetization distribution for different frequencies and wavevectors of the observed dispersion modes. We further plot exchange and demagnetization fields to examine how they change with differing antidot shapes. We have used antidots, which are $n$ sided regular convex polygons inscribed within a circumcircle of radius,$$\begin{aligned}
r_n=\sqrt{\frac{2fA}{n}\text{cosec}(\frac{2\pi}{n})};\label{rn}\end{aligned}$$ such that, the filling fraction $f$, the ratio of area of the hole to the area $A$ of the unit cell, remains a constant. Micromagnetic simulations were performed for $n=3$ (triangular), $4$ (square), $5$ (pentagonal) and $6$ (hexagonal antidots) in Object-Oriented Micromagnetic Framework (OOMMF).[@Donahue2002] The case of $n=\infty$ (circular antidots) was simulated using Nmag.[@Fischbacher2007] The paper is organized as follows. The geometrical structure of the waveguide and method used for calculating dispersion are described in greater detail in Sec. \[Sec:II\]. Section \[Sec:III\] presents the results and analysis linking the ground state field distribution with changes in the observed SW dispersion modes. Section \[Sec:IV\] contains the concluding remarks.
MAW and The Numerical Method {#Sec:II}
============================
MAW Structural and Material Parameters
--------------------------------------
![(Top panel) A part of the 1D MAW structure showing square antidots (white holes in grey magnetic region) disposed along the central axis of the waveguide of width, $w=24$ nm and lattice constant, $a=24$ nm. The square antidots are inscribed within a circle of radius, $r_4$. (Bottom panel) Other examined antidot shapes inscribed within their respective imaginary circumcircles. For $n\in\{3, 4, 5, 6, \infty\}$, $r_n$ is given by Eqn. \[rn\], where filling fraction $f=0.25$ and unit cell area $A=wa$.[]{data-label="Fig:structure"}](Fig1.pdf){width="8.5"}
Figure \[Fig:structure\] depicts the MAW structures under investigations. The MAWs had both width, $w$ and lattice constant, $a$ set to $24$ nm and a length, $l$ and thickness, $s$ of $2.4$ $\mu$m and $3$ nm in all cases. For $f=0.25$, $A=wa$ and $n\in\{3, 4, 5, 6, \infty\}$, Eq. \[rn\] dictates $r_n$ as $21.06, 16.97, 15.56, 14.89$ and $13.54$ nm, respectively. The material parameters similar to that of permalloy (Py: Ni$_{80}$Fe$_{20}$) were used during simulations (exchange constant, $A=13{\times}10^{-12}$ J/m, saturation magnetization, $M_s=0.8{\times}10^6$ A/m, gyromagnetic ratio, $\bar{\gamma}=2.21{\times}10^5$ m/As and no magnetocrystalline anisotropy).
Micromagnetic Simulations
-------------------------
Micromagnetic simulations [@Fidler2000] are done with the help of the finite difference method (FDM) based OOMMF (for $n=3, 4, 5$ and $6$) or the finite element method (FEM) based Nmag (for $n=\infty$). For the cell size used here, Nmag reproduces the circular shape much better than that obtained in OOMMF. The use of two different simulation packages also ensures that the established results are independent of the spatial discretisation. Both these open source platforms solve the Landau-Lifshitz-Gilbert (LLG) equation [@Landau1935; @Gilbert2004]:
$$\begin{aligned}
\frac{d {\bf M}}{dt}&=&-\bar{\gamma}{\bf M}\times {\bf H}_{\text{eff}}-\frac{\alpha \bar{\gamma}}{M_{\text{s}}}{{\bf M}}\times\left({\bf M}\times {\bf H}_{\text{eff}}\right)\text{.}\label{eq:ll}\end{aligned}$$
In order to obtain the SW dispersion relations, a 2D discrete Fourier transform (DFT) was performed on the obtained results.[@Kumar2012] Before simulating the SW dynamics, a magnetic steady state was achieved by subjecting the MAWs to an external bias of $1.01$ T (along the length of the waveguide) under a Gilbert damping constant, $\alpha=0.95$. This high external field saturates the magnetization of MAWs. To observe sharper dispersion peaks $\alpha$ was artificially reduced to $10^{-4}$ during simulation of the dynamics. For simulations done in OOMMF, cuboidal cells of dimensions $dx=dy=d=1$ nm and $dz=s=3$ nm were used to span the MAWs. The resultant gridding of antidot edges which are not aligned with $X$ or $Y$ axes may cause the entire hole geometry to move towards one of the edges of the MAW. How this intrinsic mirror symmetry breaking affects the SW dispersion relations is the subject of a separate study.[@Klos2013] Nmag, being FEM based, uses adaptive meshing and hence, its outputs do not suffer from this issue. However, spatial interpolation needs to be done in order to obtain magnetization values at every $1$ nm interval before performing the DFT. Data was collected every $dt=1$ ps for both OOMMF and Nmag for a total duration of $4$ ns. This gives us a sampling frequency, $f_s=1000$ GHz. The excitation signal, $H_z$ is normal to the plane of the MAWs and is given by:
$$\begin{aligned}
H_z=H_0\left({\frac{\sin(2{\pi}f_{c}(t-t_0)}{2{\pi}f_{c}(t-t_0)}}\right) \times \left(\frac{\sin(2{\pi}k_{c}(x-x_0)}{2{\pi}k_{c}(x-x_0)}\right) \label{eq:signal}
\times\left(\sum_{i=1}^{w/dy}\sin(i{\pi}y/w)\right). \nonumber\end{aligned}$$
Here ${\mu}_0H_0=6$ mT, $f_c=490$ GHz, $t_0=1/(f_s-2f_c)=50$ ps, $k_c={\pi}/a$ and $x_0=l/2=1$ $\mu$m. This form of excitation signal will excite both symmetric and antisymmetric modes of the dispersion relations in a width confined MAW. The aliasing associated with DFT is mitigated by the fact that the signal given by Eq. \[eq:signal\] carries no power beyond $f_c$ in the frequency domain. Similarly, power in the wavevector domain is limited to the first Brillouin zone (BZ) from $-k_c$ to $k_c$.
We also calculated the SW power and phase distribution profiles (PPDPs) for a given ($k$, $f$) pair of any dispersion relation. It was done by masking the obtained relation with a suitable mask in wavevector domain followed by doing an inverse Fourier transform in the same domain to yield data in physical space. For example, in order to obtain these results for ($k$, $f$) = ($K$, $F$) a mask, $D_{\text{m}}$ was created to span the entire $k$ vs. $f$ space such that:
$$\begin{aligned}
D_{\text{m}}(k, f)=
\begin{cases}
1 & \text{ if } k=2c\pi/a{\pm}K\text{: }c\text{ is an integer}\\
0 & \text{ elsewhere.}
\end{cases}
\label{eq:mask}\end{aligned}$$
After multiplying $D_{\text{m}}$ with the obtained dispersion relations we then take an inverse Fourier transform in $k$-space to arrive at the desired PPDPs. This mask is designed to include power only from $k=K$ and nullify the power present in the rest of the wavevector domain. Simply performing the inverse transform in $k$-space without using such a mask will allow power from the entire wavevector range to distort the results.
Results and Observations {#Sec:III}
========================
![(Colour Online) SW dispersion results of MAW structures marked with their respective antidot shapes as insets. Indexed band gaps are highlighted with horizontal bars.[]{data-label="Fig:dispersion"}](Fig2.pdf){width="8.5"}
The calculated dispersion relations are tabulated in Fig. \[Fig:dispersion\]. Frequency ranges from $0$ to $120$ GHz and wavevector $k$ ranges from $0$ to the first BZ boundary ($\pi/a$) are displayed. As the bias field is kept constant at $1.01$ T, a forbidden region is observed in all the cases up to the ferromagnetic resonance mode of about $39$ GHz. SW of any $k$ is not allowed in this region. Bandgap I is also present in all the cases. For triangular, square, pentagonal, hexagonal and circular antidots, its respective values are $4.3$ GHz ($43$ GHz to $47.3$ GHz), $5.6$ GHz ($44.1$ GHz to $49.7$ GHz), $4.4$ GHz ($44.5$ GHz to $48.9$ GHz), $4.4$ GHz ($44.8$ GHz to $49.2$ GHz) and $3.5$ GHz ($44.9$ GHz to $48.4$ GHz). In the case where the square antidots were tilted by $45^{\circ}$ (diamond shaped antidots), bandgaps I & II were observed; and their respective values were $3.6$ GHz ($44.2$ GHz to $47.8$ GHz) and $3.5$ GHz ($57.8$ GHz to $61.3$ GHz). An additional bandgap (III) of $6.6$ GHz ($94$ GHz to $100.6$ GHz) was observed in the case of triangular antidots. Bandgaps II & III are direct but bandgap I is indirect suggesting a difference in their origin which can be studied by looking at the spatial PPDPs for the modes between which they exist.
![(Colour Online) Power (first and third column) and phase (second and fourth column) distribution profiles corresponding to marked $(k,f)$ locations ( to ) in Fig. \[Fig:dispersion\] for MAWs with triangular, diamond, square and hexagonal shaped antidots. Power is presented on an arbitrary logarithmic colour map while the phase profile representations use a cyclic colour map.[]{data-label="Fig:powerPhase"}](Fig3.pdf){width="16"}
Figure \[Fig:powerPhase\] shows the spatial SW PPDPs for the marked ($k$, $f$) values in the Fig. \[Fig:dispersion\]. Only a part of the entire MAW structures have been shown for convenience. Mode appears to describe the uniform mode showing insignificant power or phase variation in the medium. The power distribution profile (PoDP) of mode , being at the BZ boundary, features narrow vertical nodal lines at $x=x_0{\pm}(c+1/2)a$; where $c$ is an integer. The regions joining these nodal lines are $\pi$ radians out of phase with each other. This suggests that the positions of the phase boundaries in the phase distribution profiles (PhDP) depend on the location of the signal $x_0$ used in Eq. \[eq:signal\]. Power distribution profiles for mode contains a horizontal nodal line right down the centre of the MAWs in all cases. The upper and lower parts of the waveguide are again $\pi$ radians out of phase with each other. This hints at the fact that modes and correspond to zero and first order modes along the width due to the lateral confinement of the waveguide.[@Guslienko2002] Modes and are calculated at $k=\pi/2a$ as they become nearly degenerate at the BZ boundary for square and hexagonal antidots. This degeneracy can lead one of the modes to effect the results of the other. Vertical nodal lines for both these modes are now located at $x=x_0{\pm}(2c+1)a$. Yet again, the positions of the phase boundaries appear to be controlled by the location of the signal at $x_0$. The periodicity of these nodal lines $2a$ is understandable given the location of modes (half way from BZ boundary). Slight curvature is observed in all the nodal lines for triangular antidots. We attribute this to the lack of mirror symmetry within the hole geometry along a vertical axis. Similar curvature of nodal lines was detected for the MAW with pentagonal antidots (not shown) which also lacked such a symmetry. Belonging to the same dispersive branch of the spectrum, modes and share a horizontal nodal line which stems from the aforementioned lateral confinement. The observed effects of such confinement and the shape of dispersion curve to which modes through belong reminds us of the first two (nearly) parabolic dispersion curves observed in the case of a uniform waveguide.[@Venkat2013] In contrast, mode belongs to dispersive branch in the spectrum, which curves downwards. This branch is formed by the anti-crossing of lowest energy modes originating in the two neighbours of a BZ; and as such mode unlike modes and does not show any horizontal nodal lines. Since the first two lowest energy branches share the same upward curvature, only indirect bandgap originating in the same BZ is possible. The third lowest energy branch of a BZ which originates in its two neighbouring BZs (aided by zone folding) has downward curvature. Thus, only a direct bandgap can be supported between this and the second lowest energy curve at the BZ boundary.
A quick visual comparison of different dispersion relations displayed in Fig. \[Fig:dispersion\] reveals a qualitative convergence of dispersion modes starting as early as $n=4$ (square antidots). No new band gaps open or close. Reference talks about such similarities between results from square and circular antidot based MAWs and how this convergence, or insensitivity towards the shape of the hole is desirable for the functioning of MAWs. However, note that when the square antidots are tilted by $45^{\circ}$ (diamond shaped antidots) (see Fig. \[Fig:dispersion\], left column middle row), one of the band gaps from $n=3$ case is partially restored. The computations of the exchange and the dipole field profiles (EFPs and DFPs) are done to help understand the cause for this observation. These profiles are shown in Fig. \[Fig:field\]. It may be noted how the EFP around the square antidots matches to that around the hexagonal antidots. They have similar field orientations and cover similar regions in space. Maximum value of the this field is of the order of $20\%$ of $M_s$. However, their demagnetizing field profiles do not match well. On the other hand, the demagnetizing field profile around the tilted square antidots matches better with the same around the hexagonal antidots (similar field orientations and elongated coverage in space and comparable maxima of the order of $50\%$ of $M_s$). Hence, the demagnetizing field or its corresponding potential distribution, may not be the cause of the observed changes in the band structure. Dipole dominated SWs, which occur in much larger structural dimensions are more likely to be affected by the demagnetizing field distribution. To further test the postulate, that the dispersion in considered MAWs is largely dependent upon the exchange field distribution, the case of diamond shaped antidots was considered. It was anticipated that these antidots will produce elongated regions of inhomogeneous exchange fields (similar to what is observed along the slanting edges of the triangular antidots) as opposed to chiefly circular ones (which is seen in the case of square antidots). Surly enough, the exchange field profiles of triangular, diamond shaped and square antidots were remarkably different from each other (as one of the edges of triangular antidot is vertical). This establishes a correlation of observed SW dispersion on their exchange instead of their demagnetizing field distribution.
![(Colour Online) Exchange (left column) and demagnetization (right column) field profiles at $t=0$ for $n=3, 4$ & $6$ (marked by insets).[]{data-label="Fig:field"}](Fig4.pdf){width="8.5"}
Exchange energy density, $E_{\text{exch}}\left(\bf{r}_i\right)$, which contributes to the total energy $\bf{M}\cdot\bf{H}_{\text{eff}}$, is isotropic in a homogeneous magnetic medium with uniform exchange coefficient $A$. This field is calculated in OOMMF [@Donahue2002] as given below:
$$\begin{aligned}
E_{\text{exch}}\left({\bf r}_i\right)=A{\bf m}\left({\bf r}_i\right)\cdot\sum_{{\bf r}_j}\frac{{\bf m}\left({\bf r}_i\right)-{\bf m}\left({\bf r}_j\right)}{\left|{\bf r}_i-{\bf r}_j\right|^2}\text{.}
\label{eq:exchange}\end{aligned}$$
Where ${\bf r}_j$ enumerates the region in the immediate neighbourhood of ${\bf r}_i$. In the absence of SW dynamics ${\bf m}({\bf r}_i)-{\bf m}({\bf r}_j){\simeq}0$ except where ${\bf r}_j$ lies close to antidot boundary. Therefore, by changing its geometrical boundary, the exchange field distribution around an antidot can be changed. This can conceivably scatter exchange dominated SWs differently and alter their resultant dispersion relation.
It also needs to be considered if the simulations represent the physical reality. Particularly, how can FDM or FEM based ordinary differential equation solvers like OOMMF or Nmag, which necessarily discretize the continuous sample, calculate the isotropic exchange energy and the demagnetization energy [@Donahue2007] with good accuracy? Reference concludes that the discrete representations should yield accurate results for ${\pi}d/a={\pi}/24\text{ }{\ll}\text{ }1$. This was further confirmed by the fact that using $d=0.5$ nm for the MAW with tilted square antidots did not alter the exchange field distribution significantly.
Conclusions {#Sec:IV}
===========
We have discussed the dispersion of spin-waves in nanoscale one-dimensional magnonic antidot waveguides. In particular we have observed how an antidot’s geometry can affect the said dispersion. By dint of power and phase distribution profiles of different spin-wave modes, we have explored the origin of direct and indirect bandgaps that were encountered in the obtained dispersion relations. This understanding can be used, for example, to more readily design for the direct bandgaps and avoid the indirect ones. We have also studied the degree and nature of the inhomogeneity in the exchange field distribution around the edges of an antidot. Apart from offering a way to control the band structure of the exchange dominated spin-waves, we have also demonstrated their dependence on the exchange field profile around the antidots. We demonstrated that useful direct bandgaps can be opened at the same filling fraction without removing additional material during fabrication. Demagnetizing field profile, whose intensity here reached over $0.5 M_{\text{s}}$, is expected to affect the dispersion relations on (thousand times) greater length scales. Without considering the changes in the exchange field distribution, the same has been established by Ref. in two-dimensional magnonic crystals where the hole is filled up by another magnetic material. However, forbiddingly vast computational resources will be required to obtain those results with good frequency and wavevector domain resolutions without compromising the accuracy of the dynamics.
We acknowledge the financial support from the Department of Science and Technology, Government of India (grant no. INT/EC/CMS 24/233552), Department of Information Technology, Government of India (grant no. 1(7)/2010/M&C), the European Community’s Seventh Framework Programme FP7/2007-2013 (Grant Agreement no. 233552 for DYNAMAG and Grant Agreement no. 228673 for MAGNONICS) and the EPSRC Doctoral Training Centre Grant EP/G03690X/1. D. K. would like to acknowledge financial support from CSIR - Senior Research Fellowship (File ID: 09/575/(0090)/2011 EMR-I) and useful discussions with M. Donahue, M. Krawczyk and J. W. K[ł]{}os.
[10]{}
V. V. Kruglyak, S. O. Demokritov, and D. Grundler, J. Phys. D: Appl. Phys. **43**, 264001 (2010).
A. A. Serga, A. V. Chumak, and B. Hillebrands, J. Phys. D: Appl. Phys. **43**, 264002 (2010).
B. Lenk, H. Ulrichs, F. Garbs, and M. Münzenberg, Phys. Rep. **507**, 107 (2011).
S. O. Demokritov and A. N. Slavin, . (Springer, 2013).
A. Khitun, M. Bao, and K. L. Wang, IEEE Trans. Mag. **44**, 2141 (2008).
G. Venkat, D. Kumar, M. Franchin, O. Dmytriiev, M. Mruczkiewicz, H. Fangohr, A. Barman, M. Krawczyk, and A. Prabhakar, IEEE Trans. Mag. **49**, 524 (2013).
S. Choi, K.-S. Lee, and S.-K. Kim, Appl. Phys. Lett. **89**, 062501 (2006).
J. Podbielski, F. Giesen, and D. Grundler, Phys. Rev. Lett. **96**, 167207 (2006).
T. Schneider, A. A. Serga, B. Leven, B. Hillebrands, R. L. Stamps, and M. P. Kostylev, Appl. Phys. Lett. **92**, 022505 (2008).
Y. Au, M. Dvornik, O. Dmytriiev, and V. V. Kruglyak, Appl. Phys. Lett. **100**, 172408 (2012).
S. A. Nikitov, Ph. Tailhades, and C. S. Tsai, J. Magn. Magn. Mater. **236**, 320 (2001).
S. Neusser and D. Grundler, Adv. Mater. **21**, 2927 (2009).
A. V. Chumak, V. S. Tiberkevich, A. D. Karenowska, A. A. Serga, J. F. Gregg, A. N. Slavin, and B. Hillebrands, Nat. Commun. **1**, 141 (2010).
C. G. Sykes, J. D. Adam, and J. H. Collins, Appl. Phys. Lett. **29**, 388 (1976).
C. Bayer, M. P. Kostylev, and B. Hillebrands, Appl. Phys. Lett. **88**, 112504 (2006).
V. V. Kruglyak, R. J. Hicken, A. N. Kuchko, and V. Yu. Gorobets, J. Appl. Phys. **98**, 014304 (2005).
A. V. Chumak, A. A. Serga, B. Hillebrands, and M. P. Kostylev, Appl. Phys. Lett. **93**, 022508 (2008).
Z. K. Wang, V. L. Zhang, H. S. Lim, S. C. Ng, M. H. Kuok, S. Jain, and A. O. Adeyeye, Appl. Phys. Lett. **94**, 083112 (2009).
S. Mamica, M. Krawczyk, and J. W. K[ł]{}os, Adv. Cond. Matt. Phys. **2012**, 161387 (2012).
S. L. Vysotskii, S. A. Nikitov, E. S. Pavlov, and Y. A. Filimonov, J. Commun. Technol. Electron. **55**, 800 (2010).
S. Tacchi, M. Madami, G. Gubbiotti, G. Carlotti, H. Tanigawa, T. Ono, and M. P. Kostylev, Phys. Rev. B **82**, 024401 (2010).
S. Tacchi, F. Montoncello, M. Madami, G. Gubbiotti, G. Carlotti, L. Giovannini, R. Zivieri, F. Nizzoli, S. Jain, A. O. Adeyeye, and N. Singh, Phys. Rev. Lett. **107**, 127204 (2011).
S. Saha, R. Mandal, S. Barman, D. Kumar, B. Rana, Y. Fukuma, S. Sugimoto, Y. Otani, and A. Barman, Adv. Funct. Mater. **23**, 2378 (2013).
Q. Wang, Z. Zhong, L. Jin, X. Tang, F. Bai, and H. Zhang, J. Appl. Phys. **113**, 153905 (2013).
F. J. Dyson, Phys. Rev. **102**, 1230 (1956).
M. Krawczyk and H. Puszkarski, Phys. Rev. B **77**, 054437 (2008).
K.-S. Lee, D.-S. Han, and S.-K. Kim, Phys. Rev. Lett. **102**, 127202 (2009).
S.-K. Kim, K.-S. Lee, and D.-S. Han, Appl. Phys. Lett. **95**, 082507 (2009).
F. Ciubotaru, A. V. Chumak, N. Y. Grigoryeva, A. A. Serga, and B. Hillebrands, J. Phys. D: Appl. Phys. **45**, 255002 (2012).
M. J. Pechan, C. Yu, R. L. Compton, J. P. Park, and P. A. Crowell, J. Appl. Phys. **97**, 10J903 (2005).
S. Neusser, G. Duerr, H. G. Bauer, S. Tacchi, M. Madami, G. Woltersdorf, G. Gubbiotti, C. H. Back, and D. Grundler, Phys. Rev. Lett. **105**, 067208 (2010).
S. Neusser, H. G. Bauer, G. Duerr, R. Huber, S. Mamica, G. Woltersdorf, M. Krawczyk, C. H. Back, and D. Grundler, Phys. Rev. B **84**, 184411 (2011).
S. Tacchi, B. Botters, M. Madami, J. W. Kłos, M. L. Sokolovskyy, M. Krawczyk, G. Gubbiotti, G. Carlotti, A. O. Adeyeye, S. Neusser, and D. Grundler, Phys. Rev. B **86**, 014417 (2012).
C.-L. Hu, R. Magaraggia, H.-Y. Yuan, C. S. Chang, M. Kostylev, D. Tripathy, A. O. Adeyeye, and R. L. Stamps, Appl. Phys. Lett. **98**, 262508 (2011).
R. Mandal, S. Saha, D. Kumar, S. Barman, S. Pal, K. Das, A. K. Raychaudhuri, Y. Fukuma, Y. Otani, and A. Barman, ACS Nano, **6**, 3397 (2012).
F. S. Ma, H. S. Lim, Z. K. Wang, S. N. Piramanayagam, S. C. Ng, and M. H. Kuok, Appl. Phys. Lett. **98**, 153107 (2011).
D. Kumar, O. Dmytriiev, S. Ponraj, and A. Barman, J. Phys. D: Appl. Phys. **45**, 015001 (2012).
J. W. K[ł]{}os, D. Kumar, J. Romero-Vivas, H. Fangohr, M. Franchin, M. Krawczyk, and A. Barman, Phys. Rev. B **86**, 184433 (2012).
M. Donahue and D. G. Porter, National Institute of Standards and Technology, Gaithersburg, MD, Tech. Rep. (1999). (Online). Available: http://math.nist.gov/oommf/doc/.
T. Fischbacher, M. Franchin, G. Bordignon, and H. Fangohr, IEEE Trans. Mag. **43**, 2896 (2007).
J. Fidler and T. Schrefl, J. Phys. D: Appl. Phys. **33**, R135 (2000).
L. D. Landau and E. M. Lifshitz, Phys. Z. Sowjetunion **8**, 153 (1935).
T.L. Gilbert, IEEE Trans. Mag. **40**, 3443 (2004).
J. W. K[ł]{}os, D. Kumar, M. Krawczyk, and A. Barman, Cond. Matt: arXiv:1304.1799 (2013).
K. Y. Guslienko, S. O. Demokritov, B. Hillebrands, and A. N. Slavin, Phys. Rev. B **66**, 132402 (2002).
M. J. Donahue and R. D. McMichael, IEEE Trans. Mag. **43**, 2878 (2007).
M. J. Donahue and R. D. McMichael, Physica B: Cond. Matt. **233**, 272 (1997).
J. W. K[ł]{}os, M. L. Sokolovskyy, S. Mamica and M. Krawczyk, J. Appl. Phys. **111**, 123910 (2012).
|
---
abstract: 'We show that if the dwarf-nova disc instability model includes the effects of heating by stream impact and tidal torque dissipation in the outer disc, the calculated properties of dwarf-nova outbursts change considerably, and several notorious deficiencies of this model are repaired. In particular: (1) outside-in outbursts occur for mass transfer rates lower than in the standard model as required by observations; (2) the presence of long (wide) and short (narrow) outbursts with similar peak luminosities is a natural property of the model. Mass-transfer fluctuations by factors $\sim$ 2 can explain the occurrence of both long and short outbursts above the cataclysmic variable period gap, whereas below 2 hr only short normal outbursts are expected (in addition to superoutbursts which are not dealt with in this article). With additional heating by the stream and tidal torques, such fluctuations can also explain the occurrence of both outside-in and inside-out outbursts in and similar systems. The occurrence of outside-in outbursts in short orbital-period, low mass-transfer-rate systems requires the disc to be much smaller than the tidal-truncation radius. In this case the recurrence time of both inside-out and outside-in outbursts have a similar dependence on the mass-transfer rate $\dot{M}_{2}$. accretion, accretion discs – instabilities – (Stars:) novae, cataclysmic variables – (stars:) binaries : close'
author:
- 'Valentin Buat-Ménard, Jean-Marie Hameury'
- 'Jean-Pierre Lasota'
date: 'Received / Accepted'
title: The nature of dwarf nova outbursts
---
Introduction
============
Dwarf novae (DN) are erupting cataclysmic variable stars. The 4 - 6 mag outbursts last from few days to more than a month and the recurrence times can be as short as a few days and as long as 30 years (see e.g. [@w95]). In cataclysmic variables, a Roche-lobe filling low-mass secondary star is losing mass which is accreted by a white dwarf primary. In DNs the matter transfered from the secondary forms an accretion disc around the white dwarf; it is this disc which is the site of the outbursts. It is believed that an accretion disc instability due to partial hydrogen ionization is triggering the outbursts. The disc instability model (DIM; see Cannizzo 1993, Lasota 2000 for reviews) is supposed to describe the whole DN outburst cycle. This model assumes the $\alpha$-prescription ([@shak73]) for the viscous heating and the angular momentum transport in the disc. In such a framework, the outbursts are due to propagating heating and cooling fronts, while during quiescence the disc is refilling the mass lost during the outburst. In the standard version of the model, the mass transfer rate is assumed to be constant during the whole outburst cycle. In order to reproduce outburst amplitudes and durations, the $\alpha$-parameter has to be larger in outburst than in quiescence (Smak 1984).
The DIM is widely considered to be the model of DN outbursts because it identifies the physical mechanism giving rise to outbursts, and with the above mentioned assumptions it produces outburst cycles which roughly look like the real thing. A closer look, however, shows that the model fails to reproduce some of the fundamental properties of dwarf-nova outbursts. The main deficiencies of the DIM were recently summarized by Smak (2000). First, models predict a significant increase of the disc luminosity during quiescence (see e.g. Fig. \[fig3\]), an effect which is not observed. Second, in many systems the distribution of burst widths is bimodal (van Paradijs 1983). The DIM can reproduce the width bimodality but, contrary to observations, its narrow outbursts have lower amplitudes than the wide ones. Third, observations clearly show that during outbursts the secondary is irradiated and the mass transfer rate increases. The standard DIM does not take this into account. To these problems pointed out by Smak one can add several other deficiencies. The model predicts outbursts beginning in the outer disc regions (‘outside-in’) only for high mass transfer while observations show the presence of such outbursts at apparently low mass transfer rates. In most cases the model predicts quiescent accretion rate which are at least two orders of magnitude lower than the rates deduced from quiescent luminosities. Finally, it is clear that superoutbursts observed in SU UMa stars and other systems (Smak 2000) cannot be explained by the standard DIM.
All this leads to the obvious conclusion that the standard version of the DIM has to be enriched by inclusion of physical processes which might be important but which, for various reasons, were not taken into account in the original model. Osaki (1989) proposed that superoutbursts are due to a tidal-thermal instability. Smak (2000) suggested that problems of this model could be solved by a hybrid which would combine it with the radiation-induced enhanced mass transfer model (Osaki 1985; Hameury, Lasota & Huré 1997). He included mass transfer enhancements during outburst into the standard model and suggested that this could explain properties of the narrow and wide outbursts which alternate in many systems (Smak 1999a). Hameury, Lasota & Warner (2000), extending previous work by Hameury et al. (1997) and Hameury, Lasota & Dubus (1999), showed that inclusion of effects such as mass transfer variations, formation of holes in the inner disc and irradiation of both the disc and the secondary allows to reproduce many properties of the observed outbursts.
Some questions, however, were left without satisfactory answers. First, the problem of ‘outside-in’ outbursts occurring for too high mass-transfer rates was still unsolved. Second, Smak (1999a) obtained narrow and wide outbursts by including irradiation induced mass transfer fluctuations but wide outbursts in his model result from fluctuations whose amplitude bring the disc to a stable state. In such a case the width of wide outbursts would not be determined by the disc structure alone. There is nothing fundamentally wrong with such a model and long standstills of Z Cam systems are most probably due to such fluctuations, as proposed by Meyer & Meyer-Hofmeister (1983) and shown by Buat-Ménard et al. (2000). However, one should check that the model is complete, that no other mechanism can explain outburst width distribution.
In the present paper we study two mechanisms which dissipate energy in the outer parts of an accretion disc. They are: (1) the impact on the outer boundary of the accretion disc of the gas stream which leaves the secondary through the $L_1$ critical point of the Roche lobe; this impact zone is observed in the light-curves as a bright (‘hot’) spot; (2) the viscous dissipation induced by the tidal torques which are responsible for the disc outer truncation. These mechanisms which may heat up the disc’s outer regions have been known for a long time ([@pacz77; @pap-pring77; @meyer83]), but have been neglected in most of DIM calculations. The exception is Ichikawa & Osaki (1992) who included both effects in their model. They used, however, a viscosity prescription specially designed to suppress inside-out outbursts and did not perform a systematic study of the effects as a function of the mass-transfer rate, size of the disc etc.
We add stream-impact and tidal torque heating effects to the Hameury et al. (1998) version of the DIM, presented in Sect. 2. We use a 1D approximation because the dynamical time-scale is shorter than the thermal time-scale. We study the effect of the stream impact (Sect. 3) and the tidal torques (Sect. 4) on outburst properties in short and long period DN systems. Both effects change the energy equation of the standard DIM. In both cases we find similar results: when compared to the standard case, ‘outside-in’ outbursts occur for lower mass transfer rate. The recurrence time is decreased by both the stream impact and dissipation of the tidal torques. We conclude, therefore, that heating of the outer disc by the stream impact and the rate of work by tidal torques provides a satisfactory explanation of the presence of ‘outside-in’ outbursts at realistic mass-transfer rates.
The most important result of our investigation is that when these effects are included one can obtain two types of outbursts: short (‘narrow’) and long (‘wide’) (Sect. 5). We find a critical mass transfer rate for which these two type of outburst ‘alternate’. This mass transfer rate is very close to that deduced from evolutionary calculations for systems with orbital periods longer than 3 hours (‘above the period gap’). For lower mass-transfer rates only narrow outbursts are present; for higher ones, outbursts are only of the wide type. Fluctuations of the mass-transfer rate allow to expand the range of orbital periods for which the bimodal outburst width distribution is occurring. We conclude that the observed bimodal outburst width distribution is caused by the outer-disc heating.
In Sect. 6 we discuss conditions for which outside-in outbursts can occur. We show that one can naturally account for the occurrence of both types of outbursts in if the mass transfer rate fluctuates by factors $\lta 2$ around a mean value which is close to what is generally assumed as the average mass transfer rate in this system. Outside-in outbursts in systems below the period gap, which have low mass transfer rates, can be explained only if the disc is significantly smaller than the tidal truncation radius. We show that this is in agreement with observations and consistent with expectations of the tidal-thermal instability model, at least during the first half of the supercycle of SU UMa systems. We also point out the ambiguity in observational determination of outburst ‘outside-in’ or ‘inside-out’ type, also respectively called A and B types. These types are generally assigned to outbursts according to their shape and the UV delay. We question the general validity of such criteria and argue that the eclipse profile observations during outbursts is the only reliable diagnostics of the outburst type.
The model
=========
We use the recent version of the disc instability model described in Hameury et al. (1998). We assume here that $\alpha=\alpha_{\rm
cold}=0.04$ in quiescence and $\alpha = \alpha_{\rm hot} = 0.2$ in outburst ([@smak99b]). Difficulties in reproducing the observed UV delays (see Sect. 6 for more details) and accounting for observed quiescent X-ray fluxes well above the values predicted by the DIM, led to the suggestion that ‘holes’ are present in the center of quiescent accretion discs ([@meyer94; @lasota95]). Smak (1998) pointed out, however, that the failure to reproduce observed UV delays by numerical codes was largely due to the assumption of a fixed outer disc radius, which prevented the occurrence of outside-in outbursts. In addition, the discs used in such calculations were too small to give the required time-scales. When one uses reasonable outer boundary conditions, long UV delays can be observed. However, if one wishes to reproduce both the delays [*and*]{} the light-curve shapes, a truncated disc might be still a necessary ingredient of the model (Hameury et al. 1999). Such a hole could be due to magnetic disruption in the case of a magnetic white dwarf or to evaporation in the general case. Because we are not interested here in the exact values of the rise time and the UV delay, we use a fixed size hole only for numerical convenience. The inner disc radius is fixed at $r_{\rm in}=10^9$ cm, while $5 \times 10^8$ cm is approximately the radius of a 1.2 M$_\odot$ white dwarf and $6.9 \times
10^8$ cm correspond to the other mass we use: 0.8 M$_\odot$. The outer boundary condition depends on the circularization radius $r_{\rm circ}$ at which a particle that leaves the Lagrangian point $L_1$ with angular momentum $j = \Omega_{\rm orb} b^2$ (where $\Omega_{\rm orb}$ is the orbital angular velocity and $b$ the distance between the primary and the $L_1$ point), would stay in circular orbit if there were no accretion disc. From $j = \Omega_{\rm K} r_{\rm circ}^2$, where $\Omega_{\rm K}$ is the Keplerian angular frequency, one gets: $r_{\rm
circ} \simeq \Omega_{\rm orb}^2 b^4 / G M$. When one takes the gravitational potential of the secondary into account, these relations are slightly modified, and we linearly interpolate the table from Lubow & Shu (1975) to calculate $r_{\rm circ}$. The mean outer radius $<
r_{\rm out} >$ is taken to be the average of the three radii $r_1$, $r_2$ and $r_{\rm max}$ calculated in table 1 of Paczyński (1977).
We consider in this paper the effects of the impact of the gas stream coming from the secondary (Sect. 3) and of the dissipation induced by tidal torques (Sect. 4) in both short and long period dwarf novae (thus for both small and large accretion discs) using two characteristic sets of parameters taken from Ritter & Kolb (1998): those of and (Table \[table1\]). For the short period systems, although we take the parameters of SU UMa, we include neither the tidal instabilities (suggested by Osaki 1989) to be responsible for superoutbursts in SU UMa), nor the combination of irradiation, mass transfer variation and evaporation used by Hameury, Lasota & Warner (2000) to obtain long lasting outbursts. Therefore, we do not expect to reproduce superoutbursts. We also neglect irradiation of secondary star which in principle may have significant impact on some of our results. We will consider this effect in a future work.
Long period Short Period
---------------------------------- ------------- --------------
$M_1 / $M$_\odot$ 1.2 0.8
$M_2 / $M$_\odot$ 0.7 0.15
$P_{\rm orb}$ (hr) 6.6 1.83
$r_{\rm circ}$ ($10^{10}$ cm) 1.14 0.91
$< r_{\rm out} >$ ($10^{10}$ cm) 5.4 2.3
: \[table1\] Parameters adopted for long and short period systems.
Stream impact
=============
Description
-----------
The gas stream from the secondary hits the accretion disc creating the observed bright spot. As the dynamical time is smaller than the thermal time ($t_{\rm dyn} / t_{\rm th} = \alpha < 1$), the temperature is roughly constant around each annulus of the accretion disc. Thus, the effect of stream impact could be approximatively axisymmetric and we can use a 1D approximation. This assumption is obviously very rough. Spruit & Rutten (1998) have shown that the stream impact region in is not at all axisymmetric (see also Marsh & Horne 1990 for ). Moreover, if we refer to the generally assumed values of $\alpha$, $t_{\rm dyn} / t_{\rm th}$ is not that small in outburst; in addition, as the hot spot temperature $T_{\rm hs} \gg T_{\rm disc}$ in the outer layers, the actual thermal time is much smaller than the unperturbed equilibrium value. Finally, we do not know exactly how the stream impact interacts with the disc and where precisely its energy deposited in the disc; e.g. in the simulations of Armitage & Livio (1996, 1998) a significant amount of the stream material can ricochet off the disc edge and overflow toward smaller radii.
It is therefore difficult to model properly the hot-spot contribution to the energy balance of the outer layers of an accretion disc. Here we consider that the stream impact heats an annulus $\Delta r_{\rm hs}$ of the disc with an ‘efficiency’ $\eta_{\rm i}$. The coefficient $\eta_{\rm
i}$ is supposed to represent the rather complex way the outer disc is heated by the stream. The energy per unit mass released in the shock between the stream and the disc is $({\bf V_2 - V_1})^2$, where $\bf V_2$ is the keplerian velocity at the outer disc edge, $|{\bf V_2}| = G M_1/(2r_{\rm out})$, and $\bf V_1$ is the stream velocity at the position of impact. As the angle between $\bf V_1$ and $\bf
V_2$ is large, and $|{\bf V_1}|$ is comparable to $|{\bf V_2}|$, $|{\bf V_2 -
V_1}|$ is of order of $G M_1/(2r_{\rm out})$.
The heating rate $Q_{\rm i}$ is therefore taken as: $$Q_{\rm i}(r) = \eta_{\rm i} \frac{G M_1 \dot{M_2}}{2 r_{\rm
out}} \frac{1}{2 \pi r_{\rm out} \Delta r_{\rm hs}}
\exp \left(-\frac{r_{\rm out}-r}{\Delta r_{\rm hs}}\right)
\label{eq:qi}$$ where $M_1$ is the primary’s mass, $\dot{M_2}$ the mass transfer rate from the secondary and $r_{\rm out}$ the accretion disc outer radius. This assumes that the difference between the stream and the Keplerian kinetic energy is released in a layer of width $\Delta r_{\rm hs}$ with an exponential attenuation. Note that $\eta_{\rm i}$ can be larger than unity, as $|{\bf V_2
- V_1}|$ can be larger than the Keplerian velocity. In the following, we take $\eta_{\rm i}$ = 1.
The energy equation then becomes: $$\begin{aligned}
\frac{\partial T_{\rm c}}{\partial t} = \frac{2(Q^+ + 1/2 Q_{\rm i} - Q^- +
J)} {C_P \Sigma} - \frac{\mathcal{R} T_{\rm c}}{\mu C_P} \frac{1}{r}
\frac{\partial r v_r}{\partial r} \nonumber \\
- v_r \frac{\partial T_{\rm c}}{\partial r}\end{aligned}$$ Here, $T_{\rm c}$ is the central temperature, $Q^+ = (9/8) \nu \Sigma
\Omega_{\rm K}^2$ and $Q^- = \sigma T_{\rm eff}^4$ the heating and cooling rate respectively, $\Sigma$ the surface density, $\nu$ the kinematic viscosity coefficient, $J$ is a term that accounts for the radial energy flux carried either by viscous processes or by radiation (see Hameury et al. 1998), and $v_r$ is the radial velocity. $C_P$ is the specific heat at constant pressure, $\mu$ the mean molecular weight, and $\mathcal{R}$ the perfect gas constant.
The cooling rate $Q^-(\Sigma , T_{\rm c}, r)$ is obtained from interpolation in a grid obtained by solving the vertical structure of the disc ([@jmh98]). These structures are calculated assuming a steady state, the departure from thermal equilibrium being accounted for by an effective viscosity parameter $\alpha_{\rm eff}$ different from the actual viscosity parameter $\alpha$. We assume here that the effect of the stream impact on the vertical structure can also be accounted for in this way. This would be perfectly correct if the local heating term were proportional to pressure, as is viscosity; if on the other hand it were localized at the disc surface, the problem would resemble that of disc illumination, and also in such a case, the use of an effective $\alpha_{\rm eff}$ leads to qualitatively correct results: Stehle & King (1999), who used this approximation, and Dubus et al. (1999), who calculated the exact vertical structure of irradiated discs, obtained similar modifications of the S curves by irradiation of the disc.
Results
-------
Fig. \[fig1\] displays the $\Sigma - T_{\rm eff}$ curves, calculated at two different radii with and without the heating by the stream impact. These are the well known ’S-curves’ in which three branches can be distinguished: the stable, low- and high-state branches where $T_{\rm eff}$ increases with $\Sigma$, and the unstable middle branch where $T_{\rm eff}$ decreases with $\Sigma$ increasing. The latter branch is delimited by two critical points $(\Sigma_{\rm
min},T_{\rm min})$ and $(\Sigma_{\rm max},T_{\rm max})$. To compute these S-curves, a steady state was assumed so $T_{\rm eff}$ is proportional to $\dot{M}^{1/4}$. As mentioned above, we take $\eta_{\rm i}$ = 1.
When stream impact is taken into account one observes at large radii significant deviations from the standard case. It must, however, be stressed that $Q_{\rm i}$ is a non-local quantity proportional to $\dot{M}_2$ and thus depending on the outer boundary condition, whereas the usual S-curves are locally defined. Hence, the interpretations of these curves is not straightforward. For a fixed $\Sigma$, there is a factor $\Delta \log T_{\rm eff} \sim 0.5$ between curves 2 and 2$^{'}$. For a fixed $T_{\rm eff}$, $\Sigma$ is reduced in the outer disc region when heating by the stream impact is included. It should therefore be easier, for a fixed mass transfer rate, to trigger outbursts starting at the outer edge of the disc when heating by the stream impact is included. One also expects the disc to be less massive. Moreover, the $\Sigma_{\rm min} - \Sigma_{\rm max}$ interval decreases (see Fig. \[fig2\]) so that the recurrence time should decrease too.
Fig. \[fig3\] shows light curves calculated without and with heating for a long period system (“SS Cyg"). One can see that the recurrence time is reduced in the model with the stream impact and that the quiescent luminosity is increased, as expected, since the stream impact heats up the outer parts of the disc which dominate the quiescent luminosity. During quiescence, however, the luminosity still increases in contrary to observations. Figures \[fig4\] and \[fig5\] show how the outburst type and the recurrence times depend on the mass-transfer rate. Including the additional heating by the stream impact implies that outside-in outbursts are triggered for lower mass transfer rates than in the standard DIM. Also the upper critical rate $\dot{M}_{\rm c}$, above which the disc is in a steady hot state, is lower as predicted by Meyer & Meyer-Hofmeister (1983).
For larger $\Delta r_{\rm hs}$, the quiescent luminosity is smaller, because the additional amount of energy per unit surface decreases in the outer parts of the disc (note also that the integral of $Q_{\rm i}$ over the disc surface depends weakly on $\Delta r_{\rm hs}$ because of the simplified form of Eq. \[eq:qi\] where terms of order of $\Delta
r_{\rm hs} / r_{\rm out}$ have been neglected). For smaller $\eta_{\rm
i}$, the recurrence time and the quiescent magnitude are higher. As short period systems have small discs, one might have expected that the stream impact would have stronger effects in these systems, but the results are really similar as can be seen in Figs. \[fig4\] and \[fig5\]. However, in these calculations we assumed that the average disc outer radius is equal to the tidal radius. This is an overestimate of the disc’s size, especially for the SU UMa stars. Realistically small discs will be discussed in Sect. \[oi\].
The main effects of the stream-impact heating seen on these figures are: (1) The critical accretion rate $\dot{M}_{\rm c}\left(\Sigma_{\rm
min}\right)$ above which the disc is stable, is decreased by a factor of more than 3 ; (2) the transition between inside-out and outside-in outburst (’A-B transition’) occurs for a transfer rate $\dot{M}_{\rm
A-B}$ that is decreased by a factor $\leq 2$ ; (3) the recurrence time is almost unchanged. As expected (Osaki 1996), the quiescence time is approximately constant when the outbursts are of the inside-out type, and decreases (very roughly as $\dot{M}_2^{-2}$) for outside-in outbursts, although for the smaller disc some deviations from this relation can be seen (see Sect. 6). (Note that we plot here the time in quiescence and not the recurrence time.)
For systems with parameters close to that of SS Cyg, the inclusion of the stream impact heating lowers the critical rate for outside-in outburst from $\dot{M}_{\rm A-B} \simeq 2.5 \times 10^{17}$ g s$^{-1}$ to $\dot{M}_{\rm A-B} \simeq 1.7 \times 10^{17}$ g s$^{-1}$ . The quiescence time is also shorter but only by a few days as long as $\dot{M}_2$ is not too close to $\dot{M}_{\rm c}$ (see Fig. \[fig4\]).
For short period systems, the critical mass transfer is $\dot{M}_{\rm c}
\simeq 5 \times 10^{16}$ g s$^{-1}$, however, the transition between inside-out and outside-in outbursts still occurs for mass-transfer rates ($ \dot{M}_{\rm A-B} \simeq 2 \times 10^{16}$ g s$^{-1}$) much higher that what is estimated for these systems (less than a few $ \times
10^{15}$ g s$^{-1}$; see e.g. Warner 1995). We will discuss this problem in Sect. \[oi\].
Tidal dissipation
=================
Description
-----------
Accretion discs in close binary systems are truncated by the tidal effect of the secondary star at a radius close to the Roche radius, the so-called tidal truncation radius: angular momentum transported outwards by viscous torques is returned to the binary orbital motion by the tidal torques acting on the disc’s outer edge. In the standard DIM model, tidal torques can be included in the angular momentum equation in the following way (Smak 1984; Hameury et al. 1998): $$\frac{\partial \Sigma}{\partial t} = - \frac{1}{r} \frac{\partial}{\partial r} (-
\frac{3}{2} r^2 \Sigma \nu \Omega_{\rm K}) - \frac{1}{2 \pi r} T_{\rm tid}$$ where the tidal torque per unit of area can be written as $$T_{\rm tid} = c r \nu \Sigma \Omega_{\rm orb} \left(\frac{r}{a}\right)^n
\label{eq:tid}$$ where $a$ is the orbital separation of the binary and $c$ a numerical constant that allows to adjust the tidal truncation radius $< r_{\rm
out} >$ that the disc would have had if it were steady (this gives an average value of the outer radius). The $r^n$ term used in this formula determines the disc fraction that is significantly affected by the tidal-torque dissipation (it corresponds to radii where the $\Sigma_{\rm
min}$ and $\Sigma_{\rm max}$ curves are decreasing with radius (Fig. \[fig2\])). We use $n=5$ which is quite an uncertain value, and we discuss in Sect. 6 the effects of changing it.
As pointed out by Smak (1984) and discussed in detail by Ichikawa & Osaki (1994), tidal torques (as viscous torques) induce a viscous dissipation ([@pap-pring77]): $$Q_{\rm tid} = (\Omega_{\rm K} - \Omega_{\rm orb}) T_{\rm tid}$$ This tidal dissipation is usually neglected as it is small compared to $Q^+$ except near the outer radius. During outbursts, the disc radius varies significantly, so does the tidal-torque dissipation which can be large enough to modify the outburst behaviour. We therefore include this term in the energy equation: $$\begin{aligned}
\frac{\partial T_{\rm c}}{\partial t} = \frac{2(Q^+ + 1/2 Q_{\rm tid} + 1/2
Q_{\rm i} - Q^- + J)}{C_{\rm P} \Sigma} \nonumber & \\
- \frac{\mathcal{R} T_{\rm c}}{\mu C_{\rm P}}
\frac{1}{r} \frac{\partial r v_{\rm r}}{\partial r}
- v_{\rm r} {\partial T_{\rm c} \over \partial r} &\end{aligned}$$
Standard DIM plus tidal dissipation
-----------------------------------
We first discuss the tidal effect alone, without including the heating by the stream impact. We find that type A outbursts are obtained for lower mass transfer rates than in the standard case; $\dot{M}_{\rm A-B}$ is only slightly larger than in the case where the stream impact is included. For long period systems, we found $\dot{M}_{\rm A-B} \simeq 1.8 \times 10^{17}$g s$^{-1}$. The critical mass transfer rate $\dot{M}_{\rm c}$ is also slightly larger than in the model with stream impact.
The major difference with previous calculations is the presence of a critical mass transfer rate $\dot{M}_{\rm SL}$, for which one obtains a sequence of alternating short (narrow) and long (wide) outbursts with [*roughly the same amplitudes*]{} (see Fig. \[fig6\]). At low mass transfer rates only short outbursts are present whereas for mass transfer rates higher than $\dot{M}_{\rm SL}$ all outbursts are long. In a small range of mass transfer rates outbursts of both durations are present (see Fig. \[fig7\]). We return to this point in the next section.
$\dot{M}_{\rm SL}$ is comparable to $\dot{M}_{\rm A-B}$; for the parameters used in Fig. \[fig6\], $\dot{M}_{\rm SL} < \dot{M}_{\rm
A-B}$, and short outbursts are all of the inside-out type. They are similar to those produced in the standard model, except that the quiescent flux is slightly larger as a result of viscous dissipation in the outer disc (as in the case of stream impact). As for the stream-impact heating case, the outburst peak luminosity is very slightly lower than in the standard model for the same mass transfer rate. The recurrence time scales are not affected by tidal dissipation when only short outbursts are produced. Long outbursts are slightly brighter than the short ones as observed ([@oppen98]), and for this set of parameters they can be either of the inside-out or outside-in type because the critical mass-transfer rate above which they occur $\dot{M}_{\rm SL}$ is less than $\dot{M}_{\rm A-B}$.
Long outbursts appear because near the outburst maximum the outer radius increases, so does tidal dissipation. Tidal dissipation lowers the value of $\Sigma_{\rm min}$ so that it takes longer to reach this critical density at which a cooling-front starts to propagate and shuts off the outburst. Outbursts therefore last longer than in the standard case and have the characteristic “flat top" shape. The amplitude of the outer-disc variations increases with the mass-transfer rate. At low mass-transfer rate these variations are too small to produce an observable effect on the outburst properties. The critical rate $\dot{M}_{\rm SL}$ corresponds to the case when the disc expansion begins to affect the outburst duration. Short outburst follow long ones because the tidal dissipation lowers both the critical and the actual post-outburst surface-density so that during the next outburst the time to empty the disc is shorter. Sequences of alternating short and long outbursts are present for rather narrow interval of mass-transfer rates. At higher mass-transfer rates outbursts are of the outside-in type and therefore long (see Hameury et al. 1998).
Compared to the standard case, the recurrence time is increased when long outbursts are present, but the duration of quiescence is unchanged: the duration of the outburst is the only modified quantity.
When stream impact heating and the tidal dissipative process are both included in the model, their effects cumulate: $\dot{M}_{\rm SL}$, $\dot{M}_{\rm A-B}$ and $\dot{M}_{\rm c}$ are smaller by about 15 % than what one gets when including only one effect (see Fig. \[fig4\] for $\dot{M}_{\rm A-B}$ and $\dot{M}_{\rm c}$).
The nature of long and short outbursts
======================================
We have therefore a natural explanation of the coexistence of short and long outbursts in long period ($>$ 3 hr) systems: for which the mass transfer is large enough for the additional heating to produce this effect; in short period system, $\dot{M_2}$ corresponds to short outbursts only. Heating by stream impact and tidal-torque dissipation account also for the occurrence of outside-in outbursts in , provided that the mass transfer rate happens to be large enough (approximatively two to three times larger than average).
In many cases, a bimodal distribution of the outbursts is observed ([@vanP83]). In the case of , Cannizzo & Mattei (1992) identified short outbursts with a duration between 5 and 10 days, and long ones with a duration between 13 and 17 days. Cannizzo (1993) obtained in his model a sequence of alternating short and long outbursts, but this was due to the use of a fixed outer radius which is an incorrect (‘brick wall’) boundary condition ([@jmh98; @smak98]). In any case his sequence looks different from the observed one and all his outbursts are inside-out (type B).
When tidal dissipation is included, one obtains, for a SS Cyg-like system, an alternating sequence of narrow and wide outbursts for a mass transfer rate $\dot{M}_2 \sim \dot{M}_{\rm SL} \sim 1.5 \times 10^{17}$g s$^{-1}$ (the value used in Fig. \[fig6\]). As the short/long sequence of outbursts is not uncommon, a small mass transfer rate variation around $\dot{M}_{\rm SL}$ is the most likely cause for the occurrence of both narrow and wide outbursts. Moreover, if one believes that both A and B outburst types occur in (or another long period system), $\dot{M}_2$ must also vary around $\dot{M}_{\rm A-B}$.
The value of $\dot{M}_{\rm SL}$ depends on the binary parameters and for a given orbital period one obtains a small range of mass transfer rates for which both short and long outbursts occur. Fig. \[fig7\] shows this range as a function of the orbital period for a standard dwarf nova with a primary mass $M_1 = 0.6 $M$_\odot$ and a main-sequence secondary mass $M_2 = 0.11 \times P_{\rm hr} \;$M$_\odot$. Note that now the primary’s mass is different from the values used previously. $M_1 = 0.6
$M$_\odot$ corresponds to the average white-dwarf mass in CVs. As mentioned earlier, $< r_{\rm out} >$ is taken from a table in Paczyński (1977). The lower boundary of this range is $\dot{M}_{\rm
SL}$ which varies by a factor of about 4 from a 2 hour period to a 7 hour period system. For short period systems, the mass transfer rate required to obtain both narrow and wide outburst is larger than $2
\times 10^{16}$g s$^{-1}$. This is much larger than the expected secular values of a few times $10^{15}$g s$^{-1}$ and can explain why narrow and wide outburst alternation is not observed in short period dwarf novae. Superoutbursts in SU UMa’s, which are different from long outbursts (Warner 1995; Smak 2000), occur for mass transfer rates lower than $\dot{M}_{\rm SL}$. On the other hand there exist a subclass of the SU UMa stars, the ER UMa systems, which have very short supercycles, and are in outburst most of the time. They are presumed to have mass transfer rates as large as $10^{16}$g s$^{-1}$, of order of or above the value of $\dot{M}_{\rm SL}$ corresponding to their orbital periods, and should therefore show the alternation of long and short outbursts described here. This effect could add to the enhancement of the mass-transfer rate resulting from the illumination of the secondary that also cause sequences of long and short outbursts in short period systems (Hameury et al. 2000).
Fig. \[fig7\] shows also $\dot{M}_{\rm A-B}$; in this case we find that $\dot{M}_{\rm SL}$ is larger than $\dot{M}_{\rm A-B}$, which means that all long outbursts are now of the outside-in type. However, the resulting light curves in which an alternance of short and long outbursts is found do not differ much from the one shown in Fig. \[fig6\], except that the rise times are shorter.
We propose that stochastic variations of the mass-transfer rate for the secondary cause the sequence of alternating short and long, as well as type A and type B outbursts. For the parameters of SS Cyg, we need mass-transfer rate variations in the range $\sim 1 - 2 \times 10^{17}$ g s$^{-1}$ to obtain the four types in the same light curve; this range depends on the system parameters, and in particular on the orbital period (see Fig. \[fig7\]), and on the mass of the primary. Because outburst duration depends mainly on the disc’s size (Smak 1999b) for short outbursts whose duration is independent of the mass transfer rate we obviously reproduce the correlation (van Paradijs 1983) between the outburst width and the orbital period. It is much more difficult to reproduce the observed correlation for long outbursts, since their duration is quite sensitive to $\dot{M}_2$, whose variation with the orbital period is not well known. One must, however, be cautious when interpreting this correlation, as (i) the statistics is small, and (ii) this correlation may result in part from the very definition of “long" outbursts: they must last longer than short outbursts, whose duration increases with the orbital period simply because of the increase of the disc’s size (Smak 1999b). In the model by Smak (1999a), long outbursts have to be interrupted by $\dot{M}_2$ returning to a low value since otherwise the disc would remain steady. This requires the action of an external physical mechanism with an unspecified free parameter. If for long outbursts the correlation is real, in Smak’s model the mechanism terminating long outbursts would have to depend on the orbital period.
Since $\dot{M}_{\rm SL}$ is much larger than the mass-transfer rates expected in systems below the period gap, we do not expect long outbursts for periods shorter than 2 hours. This agrees with observations. Superoutbursts observed in SU UMa systems must have a different explanation; they are either due to a tidal instability, as proposed by Osaki (1996), and/or to the irradiation of the secondary, which is quite efficient in these short period systems (Hameury et al. 2000).
We still cannot, however, account for the presence of outside-in outbursts in short period systems, whereas normal outbursts in SU UMa systems are generally believed to be of this type (Osaki 1996). There are two possible ways out of this difficulty: first, as discussed in the Sect. 6, SU UMa normal outbursts occur in truncated discs and could be of the inside-out type; second, the disc outer radius could be much smaller than what the standard DIM predict (Smak 2000); indeed, observations of eclipsing dwarf novae by Harrop-Allin and Warner (1996) tend to substantiate this idea. The second option is also favoured by the ‘direct’ observations of outside-in outbursts in two eclipsing SU UMa systems (see Sect. \[obs\] for references).
Inside-out versus outside-in outbursts {#oi}
======================================
Outbursts of SS Cyg are believed to be of both A and B types ([@smak84; @can86; @mauche96; @smak98]). Thus, the mass transfer rate from the secondary probably varies around the A-B transition. In models, outside-in outbursts are usually obtained only for mass transfer rates large compared to those expected from binary evolution models ([@bk1999]). Moreover, disc model computations with such values of $\dot{M}_2$ predict recurrence times that are much shorter than observed. Ichikawa & Osaki (1992) managed to get rid of this problem by using an $r$-dependent $\alpha$-prescription specially designed to suppress type B outbursts. Their viscous diffusion time is then almost independent of the radius and inside-out outbursts occur only at very small mass transfer rates.
As mentioned earlier, including the stream-impact heating lowers $\dot{M}_{\rm A-B}$ without changing the $\alpha$-prescription. We obtain $\dot{M}_{\rm A-B} = 1.7 \times 10^{17}$ g s$^{-1}$, 1.4 times smaller than in the standard case; this is reasonably close to estimates of SS Cyg average mass transfer rate $\dot{M}_2 = 6 \times 10^{16}$g s$^{-1}$ (e.g. [@pat84]). An increase of the mass transfer rate by a factor 2 – 3 is therefore sufficient to provoke outside-in outbursts. On the other hand, for short period systems, the transition between inside-out and outside-in outburst still occurs for much too large mass transfer rates ($2 \times 10^{16}$ g s$^{-1}$), whereas estimates of mass transfer in these systems do not exceed a few $ \times 10^{15}$ g s$^{-1}$ (Warner 1995; [@bk1999]).
Predictions of the DIM
----------------------
Outside-in outbursts occur if the time scale for matter accumulation at the outer disc edge $t_{\rm accum}$ is shorter than the viscous drift time scale $t_{\rm drift}$. These times are (Osaki 1996): $$t_{\rm accum} = {4 \pi^2 r^2 \nu \Sigma_{\rm max}^2(r) \over \dot{M}^2} \\
\label{tacc}$$ $$t_{\rm drift} = r^2 \delta \nu^{-1}$$ where $\nu = 2/3 \alpha (\mathcal{R }/\mu) T/\Omega$ is the kinematic viscosity, and $\delta \sim 0.05$ is a numerical correction factor. Using the analytical fits of $\Sigma_{\rm max}$ given by Hameury at al. (1998), the condition for outside-in outbursts is: $$\dot{M} > 4.35 \times 10^{15} \; \alpha_{\rm c}^{0.17} \left( {T \over 4000 \; \rm K}
\right) M_1^{-0.88} r_{10}^{2.64} \; \rm g \; s^{-1}
\label{eq:oi}$$ where $r_{10}$ is the radius measured in 10$^{10}$ cm and where we have assumed $\mathcal{R}/\mu = 5 \times 10^7$. This condition can be compared with our numerical results. In the case without the additional effects, our critical rate is $2.5 \times 10^{17}$ g s$^{-1}$ for long period systems, whereas Eq. (\[eq:oi\]) gives $1.8 \times 10^{17}$ g s$^{-1}$; for short period systems, these two numbers are $2.5 \times 10^{16}$ and $2.8 \times
10^{16}$ g s$^{-1}$ respectively. It must be noted that the quality of the agreement should not be a surprise, since the quantity $\delta$ was introduced by Osaki (1996) precisely for obtaining good fits.
When heating of the outer parts of the disc is included Eq. (\[eq:oi\]) is no longer valid. Osaki (1996) assumed that in quiescence the surface density is parallel to the critical density profiles, that $\Sigma\sim 2 \Sigma_{\rm min}$, and he used the standard critical densities. Things are different when heating of the outer disc is included. Near the disc outer rim both $\Sigma_{\rm min}$ and $\Sigma_{\rm max}$ are considerably reduced as is the distance between them. These effects reduce the value of $\dot{M}_{\rm A-B}$.
This reduction is not sufficient to obtain desired values of the mass-transfer rate, i.e. to $\sim 3 \times 10^{15}$ g s$^{-1}$, which is typical of the mass transfer rate in SU UMa systems. The only hope is in the strong dependence of $\dot{M}_{\rm A-B}$ on the outer disc radius. The mean radius we have assumed for short period systems ($2.3 \times
10^{10}$ cm) is probably too large for most SU UMa’s since it exceeds in many cases the 3:1 resonance radius, which is of order of 0.46 $a$, i.e. $2 \times 10^{10}$ cm for a total mass of 0.7 M$_\odot$ and a period of 1.6 hr characteristic of SU UMa stars. The 3:1 resonance radius should be reached only during superoutbursts and should be much smaller (0.3 $a$, i.e. $1.3 \times 10^{10}$ cm) during the first few outbursts of a supercycle . For such small radii $\dot{M}_{\rm A-B}$ would then be reduced by a factor $\sim$ 4 – 5, i.e. to a value slightly larger than the estimated mass transfer rate in SU UMa systems.
In Fig. \[fig9\] we show the quiescence time and the radius at which the outburst starts versus mass transfer rate, for a short period system with a mean outer disc radius $\sim 1.3 \times 10^{10}$ cm. This model corresponds to a relaxed disc, i.e. a disc whose mass is constant over an outburst cycle. Such small discs are expected to exist only during the first half of a supercycle and later should grow in size. We have assumed a large coefficient $c$ in Eq. \[eq:tid\], in order to obtain a small disc, but we have not included heating by tidal torques that would have resulted in much too large a dissipation: the dissipation of tidal torques must be negligible in discs much smaller than the truncation radius. As can be seen, outside-in outbursts occur for mass transfer rates as low as $5 - 6 \times 10^{15}$ g s$^{-1}$, reasonably close to the expected value in these systems.
It is worth noting that the recurrence time depends on the mass transfer rate, even in the case of inside-out outbursts. A comparison of Figs. \[fig4\], \[fig5\] and \[fig9\] shows that this effect is related to the size of the disc. This is due to the fact that a local dimensional analysis becomes invalid when the radius of the outer edge of the disc is not much larger than the inner radius (for example, in the case of steady discs, the inner boundary condition introduces a factor $[1 -(r_{\rm in}/r)^{1/2}]$). Osaki (1996) asserts that the correlation $t_{\rm s} \propto t_{\rm n}^{0.5}$ found between the supercycle recurrence time $t_{\rm s}$ and the recurrence time of normal outbursts $t_{\rm n}$ in SU UMa systems results from Eq.(\[tacc\]) and the relation $t_{\rm s} \propto \dot{M}^{-1}$ which results from the assumption that super-outburst occur when sufficient mass is accumulated (Osaki 1996; Hameury et al. 2000). Clearly, this cannot be the case if one uses the standard disc model in which $\alpha$ is not forced to adopt a functional dependence that would give the desired effect but is just kept constant. In any case, the mass transfer rate cannot be assumed to be the only variable on which recurrence times depend. For example Menou (2000) showed that both the normal outburst and super-outburst recurrence times strongly depend on the secondary to primary mass ratio.
Small discs are also expected if one reduces the index $n$ in Eq. (\[eq:tid\]), since the smaller $n$, the larger the amplitude of disc radius variations during outbursts; for a given maximum disc radius (some fraction of the Roche radius), the disc radius during quiescence is therefore expected to be smaller for smaller $n$.
Observations {#obs}
------------
Campbell (1934) identified four distinct types of outburst in light curves which have a large variety of characteristic rise times. Since then, several techniques have been used to investigate the physical outburst properties: multi-wavelength photometry, spectrometry, eclipse mapping and model fitting. The first disc instability models have predicted the outburst types A (outside in) and B (inside-out) ([@smak84]), and showed that the first category rises earlier at optical wavelengths and have a more asymmetric shape. As a result, asymmetric outbursts are usually supposed to be of type A. However asymmetric outbursts can also be of the inside-out type. For instance, in Fig. \[fig6\], long outbursts are not symmetric but are of type B.
The UV delay is the observed lag between the rise in optical and the rise in UV. Its duration is believed to be a signature of the outburst starting region. In the naive picture since the hottest parts of the disc associated with the shortest wavelengths are close to the white dwarf, there should be almost no UV delay if the outburst starts in the inner part of the accretion disc, whereas one should expect an important UV delay for outside-in outbursts. However, as discussed by Smak (1998), things are more complicated than what the naive picture would suggest, and long delays can also exist for inside-out outbursts if the outer disc radius expansion during outburst and the correct size of the disc are taken into account. For a given binary system the UV delay will always be longer for an outside-in outburst than for an inside-out one, but the length of the delay is not sufficient by itself to determine the outburst type.
Smak (1998) proposed that the ratio of the UV delay and the outburst duration is a good indicator of the outburst type. However, the existence of both long and short outbursts at a given orbital period makes this test very uncertain. Indeed Fig. \[fig6\] shows a probing counter-example where both short and long outbursts are of type B and for which Smak’s ratio would be quite different.
In addition Hameury et al. (1999), using SS Cyg parameters, reproduced the observed ([@mauche96]) asymmetric outburst and 1-day EUV delays for an [*inside-out*]{} outburst starting to propagate at the inner edge of a truncated disc. This obviously raises the question of the real relationship between the outburst type and the UV delay.
It has been known for some time that eclipse profiles allow to determine the propagation direction of an outburst (Smak 1971 for U Gem; Vogt 1984 and Rutten, van Paradijs &Tinbergen 1992 for OY Car). For example in the case of an accretion disc whose luminosity is much larger than both the white dwarf and the hot spot luminosity, the eclipse profile is almost symmetric. This is the case of , a SU UMa system with a period of 106 minutes ([@ioanou99]). The authors found that during the rise of the burst, the eclipse is shallow and its width is large. This means that most of the flux originates from the outer parts of the disc. They observed that the width of the eclipse decreases after the outburst peak, which is in perfect agreement with the predicted decrease of the disc radius during decline, therefore proving the outside-in nature of the outburst. Webb et al. (1999) and Baptista, Catalán & Costa (2000) used the same technique to bring evidences of the inside-out nature of an outburst of and respectively.
At present, the only way to be sure of the outburst type is to use, whenever possible, eclipse profiles. As eclipses are not present in all systems it would be worthwhile to test Smak’s assumption on the UV delay in eclipsing dwarf novae.
Conclusions
===========
We have shown that the inclusion of additional heating effects such as the stream impact and tidal dissipation might solve several problems of the standard DIM.
The most important result is that when the tidal torque dissipation in the outer disc regions is included, one obtains for a certain range of mass-transfer rates an alternation of short and long outbursts which is observed in long orbital period system such as ([@can92]) or ([@oppen98]). The long outbursts are slightly brighter than short outbursts as observed. Moderate fluctuations of the mass-transfer rate would extend the range for which such alternating outbursts are present. For systems with mass transfer rates close to the stability limit, such fluctuations will produce standstills as shown in Buat-Ménard et al. (2000). This solution of the long/short outburst problem (Smak 2000) is different from the one suggested by Smak (1999a). In our model mass-transfer fluctuations play only an auxiliary role and long outbursts do not result from fluctuations bringing the disc to a steady state.
We showed that additional heating effects make the outside-in outburst possible at low mass transfer rates without changing the $\alpha$-prescription. Our results on mass transfer rates and outburst types are in good agreement with most estimates in the case of long period systems. In the case of short period systems of the SU UMa type, for outside-in outburst to occur the disc would have to be smaller than usually assumed. For small discs the recurrence times of both the inside-out and outside-in outbursts is $\dot{M}_2$-dependent. It must be stressed out that there are still many uncertainties on the observational determination of outburst types. For instance, the UV delay is not necessarily a good criterion for determining the outburst type; the eclipse profile during outburst is a much better criterion, but unfortunately restricted to a small subset of dwarf novae.
There are still several points that need to be clarified. First, the stream impact effect geometry is not well known and is considered here in a 1D approximation (as for the tidal dissipation). We do not know how far and how efficiently the energy propagates in the disc. The expression of the tidal torque can also be subject to discussion. Moreover, we did not include irradiation here, and it would be interesting to study the combined effects of all phenomena. Finally, as raised above, the existing discrimination criteria on the outburst types A and B have to be tested and improved.
We thank Guillaume Dubus for helpful comments. This work was supported in part by a grant from [*Programme National de Physique Stellaire*]{} of the CNRS.
Armitage P.J., Livio M., 1996, ApJ 470, 1024
Armitage P.J., Livio M., 1998, ApJ 493, 898
Baptista R., Catalán M.S., Costa L., 2000, MNRAS, 316, 529
Baraffe I., Kolb U., 1999, MNRAS 309, 1034
Buat-Ménard V., Hameury J.-M., Lasota J.-P., 2000, A&A submitted
Campbell L., 1934, AHCO 90, 93
Cannizzo J.K. 1993, ApJ 419, 318
Cannizzo J.K., Mattei J.A. 1992, ApJ 401, 642
Cannizzo J.K., Wheeler J.C., Polidan R.S., 1986, ApJ 301, 634
Dubus G., Lasota J.-P., Hameury J.-M., Charles P., 1999, MNRAS 303, 139
Hameury J.-M., Lasota J.-P., Huré J.-M., 1997, MNRAS 287, 937
Hameury J.-M., Menou K., Dubus G., Lasota J.-P., Huré J.-M., 1998, MNRAS 298, 1048
Hameury J.-M., Lasota J.-P., Dubus G., 1999, MNRAS 303, 39
Hameury J.-M., Lasota J.-P., Warner B., 2000, A&A 353, 244
Harrop-Allin M.K., Warner B., 1996, MNRAS 279, 228
Ichikawa S., Osaki Y., 1992, PASJ 44, 15
Ichikawa S., Osaki Y., 1994, PASJ 46, 621
Ioannou Z., Naylor T., Welsh W.F., Catalán M.S., Worraker W.J., James N.D., 1999, MNRAS 310, 398
Lasota J.-P., 2000, New Astronomy Reviews, submitted
Lasota J.-P., Hameury J.-M., Huré J.-M., 1995, A&A 302, L29
Livio M., Pringle J., 1992, MNRAS 259, 23
Lubow S.H., Shu F.H., 1975, ApJ 198, 383
Marsh T.R., Horne K., 1990, ApJ 349, 593
Mauche C.W., 1996, in Astrophysics in the Extreme Ultraviolet, eds. S. Bowyer and R.F. Malina, Dordrecht, Kluwer, p. 317.
Menou K., 2000, Science, in press
Meyer F., Meyer-Hofmeister E. 1983, A&A 121, 29
Meyer F., Meyer-Hofmeister E. 1994, A&A 228, 175
Oppenheimer B.D., Kenyon S.J., Mattei J.A., 1998, AJ 115, 1175
Osaki Y., 1985, A&A 144, 369
Osaki Y., 1989, PASJ 41, 1005
Osaki Y., 1996, PASP 108, 39
Paczyński B., 1977, ApJ 216, 822
Papaloizou J., Pringle J.E., 1977, MNRAS, 181, 441
Patterson J., 1984, ApJS 54, 443
Ritter H., Kolb U., 1998, A&AS 129, 83
Rutten R.G.M., van Paradijs J., Tinbergen J., 1992, A&A, 260, 213
Shakura N.I., Sunyaev R.A. 1973, A&A 24, 337
Smak J., 1971, Acta Astr. 21, 15
Smak J., 1984, Acta Astron. 34, 161
Smak J., 1998, Acta Astron. 48, 677
Smak J., 1999a, Acta Astron. 49, 383
Smak J., 1999b, Acta Astron. 49, 391
Smak J., 2000, New Astronomy Reviews 44, 171
Spruit H.C., Rutten R.G.M. 1998, MNRAS, 299 768
Stehle R., King A.R., 1999, MNRAS 304, 698
van Paradijs J., 1983, A&A 125, L16
Vogt N., 1983, A&A, 128, 29
Warner B., 1994, Ap&SS 226, 187
Warner B., 1995, Cataclysmic Variable Stars. Cambridge University Press, Cambridge
Webb N.A., Naylor T., Ioannou Z., Worraker W.J., Stull J., Allan A., Fried R., James N.D., Strange D., 1999, MNRAS 310, 407
|
---
author:
- 'Jocelyn Simmonds Shoham Ben-David Marsha Chechik'
bibliography:
- 'combined.bib'
title: |
Optimizing Computation of Recovery Plans\
for BPEL Applications
---
|
---
abstract: 'We prove the following indistinguishability theorem for $k$-tuples of trees in the uniform spanning forest of $\Z^d$: Suppose that $\sA$ is a property of a $k$-tuple of components that is stable under finite modifications of the forest. Then either every $k$-tuple of distinct trees has property $\sA$ almost surely, or no $k$-tuple of distinct trees has property $\sA$ almost surely. This generalizes the indistinguishability theorem of the author and Nachmias (2016), which applied to individual trees. Our results apply more generally to any graph that has the Liouville property and for which every component of the USF is one-ended.'
author:
- '[**Tom Hutchcroft**]{}'
title: '**Indistinguishability of collections of trees in the uniform spanning forest**'
---
Introduction
============
The **uniform spanning forests** are infinite-volume analogues of uniform spanning trees, and can be defined for any connected, locally finite graph $G$ as weak limits of the uniform spanning trees on certain finite graphs derived from $G$. These limits can be taken with respect to two extremal boundary conditions, leading to the **free uniform spanning forest** (FUSF) and **wired uniform spanning forest** (WUSF). For many graphs, such as the hypercubic lattice $\Z^d$, the FUSF and WUSF coincide and we speak simply of the USF. Being an open condition, connectivity is not necessarily preserved by taking weak limits, and it is possible for the USF to be disconnected. Indeed, Pemantle [@Pem91] proved that the USF of $\Z^d$ is a.s. connected if and only if $d\leq 4$. More generally, Benjamini, Lyons, Peres, and Schramm [@BLPS; @lyons2003markov] proved that the WUSF of an infinite graph $G$ is a.s. connected if and only if two independent random walks on $G$ intersect infinitely often a.s.
This disconnectivity leads us to consider the following natural question: If the USF is disconnected, how different can the different components of the forest be? For instance, is it possible that some are recurrent while others are transient? Similarly, could it be possible that there exists a single “thick" component that occupies a positive density of space, while all other components are “thin" and have zero spatial density? Benjamini, Lyons, Peres, and Schramm [@BLPS] conjectured the following answer to questions of this nature: If $G=(V,E)$ is *transitive* and *unimodular* (e.g., if $G$ is a Cayley graph of a finitely generated group) and $\F$ is either the WUSF or FUSF of $G$, then the components of $\F$ are *indistinguishable* from each other. This means that for every measurable set $\sA \subseteq \{0,1\}^E$ of subgraphs of $G$ that is invariant under the automorphisms of $G$, either every component of $\F$ is in $\sA$ a.s. or none of the components of $\F$ are in $\sA$ a.s. This conjecture followed earlier work of Lyons and Schramm [@LS99], who proved an analogous theorem in the context of Bernoulli percolation. The conjecture regarding the USF was verified (in slightly greater generality) by the author and Nachmias [@HutNach2016a], while partial progress was also made in the independent work of Timár [@timar2015indistinguishability]. In the setting of Bernoulli percolation, various extensions and generalizations of the Lyons-Schramm theorem have subsequently been obtained by Aldous and Lyons [@AL07], Martineau [@martineau2012ergodicity], and Tang [@tang2018heavy]. Besides their intrinsic probabilistic interest, such indistinguishability theorems have also found applications in ergodic theory, see e.g. [@MR3621428; @TDprep; @gaboriau2009measurable].
In this paper, we are interested in a form of indistinguishability that holds not only for individual components of the forest, but rather for arbitrary finite collections of components. Our results are motivated by our work with Yuval Peres on the adjacency structure of the trees in the USF of $\Z^d$ [@hutchcroft2017component], in which we use the results of this paper as a zero-one law to boost positive-probability statements to almost-sure statements. A remarkable feature of the results we obtain is that, unlike in [@LS99; @HutNach2016a], we do not require any kind of homogeneity assumptions on $G$ (such as transitivity or unimodularity), nor do we require any kind of automorphism-invariance type assumptions on the properties we consider. Rather, the primary assumption we make on $G$ is that it is *Liouville*, i.e., does not admit non-constant bounded harmonic functions. We also restrict attention to *tail* properties of tuples of components, which are stable under finite modifications of the forest. Heuristically, one can think of our proof as lifting the tail triviality of the random walk (which is equivalent to the Liouville property) to indistinguishability of trees in the USF, which is itself a strong form of tail triviality.
Let us now give the definitions required to state our main theorems. Let $G$ be a graph, and let $k\geq 1$. We equip the set $\Omega_{k}(G):=\{0,1\}^E\times V^k$ with its product topology and associated Borel $\sigma$-algebra. We think of this set as the set of subgraphs of $G$ rooted at an ordered $k$-tuple of vertices. A measurable set $\sA \subseteq \Omega_k(G)$ is said to be a $k$-**component property** if $$(\omega,(u_i)_{i=1}^k)\in \sA \Longrightarrow (\omega,(v_i)_{i=1}^k)\in \sA \,\,\,
\begin{array}{l}
\text{for all } (v_i)_{i=1}^k \in V^k \text{ such that $u_i$ is} \\\text{connected to $v_i$ in $\omega$ for each $i=1,\ldots,k$}.
\end{array}$$ In other words, $\sA$ is a $k$-component property if it is invariant under replacing the root vertices with other root vertices from within the same components. We call $\sA$ a **multicomponent property** if it is a $k$-component property for some $k$. Given a $k$-component property $\sA$, we say that a $k$-tuple of components $K_1,\ldots,K_k$ of a configuration $\omega \in \{0,1\}^E$ **has property** $\sA$ if $(\omega,(v_i)_{i=1}^k) \in \sA$ whenever $u_1,\ldots,u_k$ are vertices of $G$ such that $u_i \in K_i$ for every $1 \leq i \leq k$.
Given a vertex $v$ of $G$ and a configuration $\omega \in \{0,1\}^E$, let $K_\omega(v)$ denote the connected component of $\omega$ containing $v$. We say that a $k$-component property $\sA$ is a **tail** $k$-component property if $$(\omega,(v_i)_{i=1}^k)\in \sA \Longrightarrow (\omega',(v_i)_{i=1}^k)\in \sA \,\,\,
\begin{array}{l}
\forall \omega' \in \{0,1\}^E \text{ such that } \omega {\hspace{.1em}\triangle\hspace{.1em}}\omega' \text{ is finite and }\\
K_\omega(v_i){\hspace{.1em}\triangle\hspace{.1em}}K_{\omega'}(v_i) \text{ is finite for every $ i =1,\ldots,k$,}
\end{array}$$ where ${\hspace{.1em}\triangle\hspace{.1em}}$ denotes the symmetric difference. In other words, tail multicomponent properties are stable under finite modifications to $\omega$ that result in finite modifications to each of the components of interest $K_\omega(v_1),\ldots,K_\omega(v_k)$. For example, if $G$ is locally finite then the set $\sA_{k}$ of $(\omega,(v_i)_{i=1}^k)$ such that $K_\omega(v_i)$ contains infinitely many vertices adjacent to $K_\omega(v_j)$ for every $1 \leq i < j \leq k$ is a tail $k$-component property for each $k \geq 2$.
We now state our result in the case $G=\Z^d$. The general result is given below.
\[thm:indist\] Let $d\geq 5$ and let $\F$ be the uniform spanning forest of $\Z^d$. Then for each $k\geq 1$ and each tail $k$-component property $\sA \subseteq \Omega_k(\Z^d)$, either every $k$-tuple of distinct connected components of $\F$ has property $\sA$ almost surely or no $k$-tuple of distinct connected components of $\F$ has property $\sA$ almost surely.
The general form of our result will require the underlying graph to be *Liouville*, i.e., not admitting any non-constant bounded harmonic functions. Note that if $G$ is Liouville then its free and wired uniform spanning forests coincide [@BLPS Theorem 7.3], so that we may speak simply of the USF of $G$. Our proof will also require that every component of the USF of $G$ is *one-ended* almost surely. Here, an infinite graph is said to be **one-ended** if deleting any finite set of vertices from the graph results in at most one infinite connected component. In particular, a tree is one-ended if it does not contain a simple bi-infinite path. It is known that every component of the wired uniform spanning forest is one-ended almost surely in several large classes of graphs [@Pem91; @BLPS; @AL07; @LMS08; @H15; @HutNach2016b; @hutchcroft2015interlacements], including all transitive graphs not rough-isometric to $\Z$ [@LMS08]. In particular, \[thm:indist\] applies to all of the transitive graphs of polynomial growth that are studied in [@hutchcroft2017component]. (The condition that $G$ is one-ended in the following theorem is in fact redundant, being implied by the other hypotheses.)
\[thm:indistgeneral\] Let $G=(V,E)$ be a infinite, one-ended, connected, locally finite graph, and suppose that $G$ is Liouville. Let $\F$ be the uniform spanning forest of $G$, and suppose further that every component of $\F$ is one-ended almost surely. Then for each $k\geq 1$ and each tail $k$-component property $\sA \subseteq \Omega_k(G)$, either every $k$-tuple of distinct connected components of $\F$ has property $\sA$ almost surely or no $k$-tuple of distinct connected components of $\F$ has property $\sA$ almost surely.
The special case of \[thm:indist\] concerning a single component (i.e., $k=1$) is essentially equivalent to [@BeKePeSc04 Theorem 4.5], which was not phrased in terms of indistinguishability. \[thm:indist\] is also closely related to [@HutNach2016a Theorem 1.20], which implies in particular that components of the wired uniform spanning forest of any transitive graph are indistinguishable from each other by automorphism-invariant tail properties. See \[sec:closing\] for a discussion of how \[thm:indistgeneral\] can fail in the absence of the assumption that $G$ is Liouville, even if we require that $G$ is a Cayley graph and $\sA$ is automorphism invariant.
We will assume that the reader is familiar with the definition of the uniform spanning forest and with Wilson’s algorithm, referring them to e.g. [@LP:book] for background otherwise.
Proof {#sec:indist}
=====
Indistinguishability is closely related to tail-triviality. Let $\Omega$ be a measurable space, let $I$ be a countable set, and let $\Omega^I = \{(\omega_i)_{i\in I} : \omega_i \in \Omega \text{ for every $i \in I$}\}$ be equipped with the product $\sigma$-algebra $\cF$. For each subset $J$ of $I$, we define $\cF_J$ to be the sub-$\sigma$-algebra of $\cF$ of events depending only on $(\omega_i)_{i\in J}$. The **tail $\sigma$-algebra** of $\Omega^I$ is defined to be the intersection $$\mathcal{T}=\bigcap \left\{\cF_J : I \setminus J \text{ is finite} \right\}.$$ An $\Omega^I$-valued random variable $A=(A_i)_{i\in I}$ is said to be **tail-trivial** if it has probability either zero or one of belonging to any set in the tail $\sigma$-algebra $\cT$. The following was proven by Benjamini, Lyons, Peres, and Schramm [@BLPS Theorem 8.3], generalizing a result of Pemantle [@Pem91].
\[thm:USFtailtriviality\] Let $G=(V,E)$ be an infinite, connected, locally finite graph, and let $\F \in \{0,1\}^E$ be either the free or wired uniform spanning forest of $G$. Then $\F$ is tail-trivial.
In particular, if the USF of $G$ has only one component a.s. then \[thm:indistgeneral\] is implied by \[thm:USFtailtriviality\]. Thus, it suffices to prove \[thm:indistgeneral\] in the case that the USF of $G$ has more than one component with positive probability, in which case $G$ must be transient.
Recall that the **lazy random walk** on a graph is the random walk that stays where it is with probability $1/2$ at each step, but otherwise chooses a uniform edge emanating from its current location just as the usual random walk does. Note that we can use lazy random walks instead of simple random walks when sampling the WUSF of a graph using Wilson’s algorithm, since doing so does not affect the law of the resulting forest. This will be useful to us thanks to the following well-known theorem due to Blackwell [@Blackwell55] and Derriennic [@Derriennic80]; see also [@LP:book Corollary 14.13 and Theorem 14.18]. (Laziness is used in this theorem to avoid parity issues.)
Let $G=(V,E)$ be an infinite, connected, locally finite graph, and let $X \in V^\N$ be a lazy random walk on $G$ started at some vertex $v$. Then $G$ is Liouville if and only if $X$ is tail-trivial.
It will also be useful for us to use the following well-known equivalence between tail-triviality and asymptotic independence: See e.g. [@LP:book Proposition 10.17] or [@GeorgiiBook Proposition 7.9].
\[lem:tailtriviality\] Let $\Omega$ be a measurable space, let $I$ be a countable set, let $A=(A_i)_{i \in I}$ be an $\Omega^I$-valued random variable with law $\P$, and let $(K_n)_{n\geq0}$ be an increasing sequence of finite subsets of $I$ such that $\bigcup_{n\geq0} K_n = I$. Then $A$ is tail-trivial if and only if for every event $\sA \subseteq \Omega^I$ for which $\P(A \in \sA)>0$ we have that $$\label{eq:tailtrivlem0}
\lim_{n\to\infty} \sup \left\{ \left |\P\left(A \in \sA \text{ and } A \in \sB \right) - \P\left( A \in \sA\right)\P\left( A \in \sB\right)\right| : \sB \in \cF_{I\setminus K_n} \right\} = 0
$$ and hence that $$\label{eq:tailtrivlem}
\lim_{n\to\infty} \sup \left\{ \left |\P\left(A \in \sB \mid A \in \sA \right) - \P\left( A \in \sB\right)\right| : \sB \in \cF_{I \setminus K_n} \right\} = 0.
$$ In other words, $A$ is tail-trivial if and only if the total variation distance between the distribution of $(A_i)_{i \in I \setminus K_n}$ and the conditional distribution of $(A_i)_{i \in I \setminus K_n}$ given $\{A \in \sA\}$ converges to zero as $n\to\infty$ for every event $\sA \subseteq \Omega^I$.
Let us note also that if $\Omega_1$, $\Omega_2$ are measurable spaces, $I$ is a countable set, $A$ is an $\Omega_1$ valued random variable and $B^1,\ldots,B^k$ are independent, tail-trivial, $\Omega_2^I$-valued random variables, then $(X_i)_{i\in I} =((A_i,(B_i^j)_{j=1}^k))_{i \in I}$ is a tail trivial $(\Omega_1 \times \Omega_2^k)^I$-valued random variable.
Our first step towards \[thm:indist\] is the following lemma. Given $\mathbf{u}=(u_1,\ldots,u_k)$, we write $\sW(\mathbf{u})$ for the event that the vertices $u_1,\ldots,u_k$ are all in distinct components of $\F$.
\[lem:indistlem\] Let $G=(V,E)$ be an infinite, transient, Liouville graph, let $\F$ be the uniform spanning forest of $G$, and suppose that every component of $\F$ is one-ended almost surely. Let $\mathbf{u}=(u_1,\ldots,u_k)$ be a $k$-tuple of vertices of $G$, and let $\mathbf{X}=(X^1,\ldots,X^k)$ be a $k$-tuple of independent lazy random walks on $G$, independent of $\F$, such that $\mathbf{X}_0=\mathbf{u}$. If $\P(\sW(\mathbf{u}))>0$, then for every tail $k$-component property $\sA$ we have that $$\vspace{0.2cm}
\P\left((\F,\mathbf{u}) \in \sA \mid \sW(\mathbf{u}) \right) = \lim_{m\to\infty} \P\left((\F,\mathbf{X}_m) \in \sA\right).
\label{eq:indist3}$$ In particular, the right hand limit exists.
Before beginning the proof of this lemma, let us note the following. Suppose that $G$ is an infinite, transient, Liouville graph whose USF is disconnected with positive probability. Then whenever $X$ and $Y$ are independent lazy random walks on $G$, we have by [@BLPS Theorem 9.2] that $X$ and $Y$ intersect only finitely often with positive probability. Since the event that both walks are transient and intersect only finitely often is a tail event, we deduce that $X$ and $Y$ intersect only finitely often almost surely. It follows that if $X^1, \ldots, X^k$ are independent lazy random walks on $G$, then $$\label{eq:liouvilleintersections}
\lim_{m\to\infty} \P\left( (X^i_{n+m})_{n \geq 0} \text{ and } (X^j_{n+m})_{n \geq 0} \text{ intersect for some $1 \leq i < j \leq k$}\right)=0.$$ In particular, it follows from [@BLPS Theorem 9.4] that the USF of $G$ has infinitely many connected components almost surely.
The proof of \[lem:indistlem\] will also apply the following simple measure-theoretic lemma.
\[lem:measuretheory\] Let $(X_i)_{i\geq 1}$ and $X$ be random variables defined on a shared probability space $(\Omega,\P)$ and taking values in a locally compact Hausdorff space $\mathbb{X}$. Let $(B_i)_{i\geq 1} \subseteq \Omega$ and $B_i \subseteq \Omega$ be measurable with $\P(B)>0$. Suppose further that the following hold:
1. $X_i$ and $X$ have the same distribution for every $i\geq 1$.
2. $X_i$ converges to $X$ in probability as $i\to\infty$.
3. $\P(B_i {\hspace{.1em}\triangle\hspace{.1em}}B) \to 0$ as $i\to \infty$.
Then $\P( X \in A \mid B) = \lim_{i\to\infty}\P( X_i \in A \mid B_i )$ for every Borel set $A \subseteq \mathbb{X}$.
By [@MR924157 Theorem 3.14], for every ${\varepsilon}>0$ there exists a continuous function $f_{\varepsilon}:\mathbb{X}\to\R$ such that $
\E\left[|f_{\varepsilon}(X)-\mathbbm{1}(X\in A)|\right] \leq {\varepsilon}$. We have by the triangle inequality that $$\begin{aligned}
|\P(X\in A,B) - \P(X_i\in A, B_i)|&\leq
\E\left[ |\mathbbm{1}(X\in A) \mathbbm{1}(B)- \mathbbm{1}(X_i\in A) \mathbbm{1}(B_i)|\right]\\&\leq \E\left[ |\mathbbm{1}(X\in A) - f_{\varepsilon}(X)| \mathbbm{1}(B)\right] + \E\left[ f_{\varepsilon}(X)|\mathbbm{1}(B) -\mathbbm{1}(B_i)|\right]\\
&\hspace{1cm}+\E\left[ |f_{\varepsilon}(X)-f_{\varepsilon}(X_i)| \mathbbm{1}(B_i)\right]
+\E\left[ |f_{\varepsilon}(X_i)-\mathbbm{1}(X_i\in A)| \mathbbm{1}(B_i)\right]\\
&\leq 2
\E\left[ |\mathbbm{1}(X\in A) - f_{\varepsilon}(X)|\right] + \E\left[ f_{\varepsilon}(X)|\mathbbm{1}(B) -\mathbbm{1}(B_i)|\right]\\
&\hspace{1cm}+\E\left[ |f_{\varepsilon}(X)-f_{\varepsilon}(X_i)| \right],\end{aligned}$$ where we used the fact that $X$ and $X_i$ have the same distribution in the third inequality. Applying the dominated convergence theorem we deduce that $$\limsup_{i\to\infty}
|\P(X\in A,B) - \P(X_i\in A, B_i)| \leq 2 \E\left[ |\mathbbm{1}(X\in A) - f_{\varepsilon}(X)|\right] \leq 2{\varepsilon}.$$ The claim follows since ${\varepsilon}>0$ was arbitrary.
Let $(v_i)_{i\geq0}$ be an enumeration of the vertices of $G$ and let $( Y^{i,m} )_{i\geq 0, m\geq 0}$ be a collection of independent lazy random walks, independent of $\mathbf{X}$ and of $\F$, such that $Y^{i,m}$ is started at $v_i$ for every $i\geq 0$ and $m \geq 0$. Let $\fG$ be a uniform spanning forest of $G$ sampled using Wilson’s algorithm, beginning with the walks $X^1,\ldots, X^k$, and then using the walks $Y^{0,0},Y^{1,0},\ldots$. Similarly, for each $m\geq 1$, let $\fG_m$ be a uniform spanning forest of $G$ sampled using Wilson’s algorithm, beginning with the walks $(X^1_{n+m})_{n \geq 0},\ldots, (X^k_{n+m})_{n\geq 0}$ and then using the walks $Y^{0,m},Y^{1,m},\ldots$. We clearly have that $\left( \fG_m,\, \mathbf{X}_m\right)$ and $\left(\F,\, \mathbf{X}_m \right)$ have the same distribution for every $m\geq0$. Unlike $\F$, the forests $\fG_m$ are *not* independent of the random walks $\mathbf{X}$.
For each $m \geq 0$, we will define a sequence of forests $\F_{m,R}$ which interpolate between $\F$ and $\fG_m$. The **future** of a vertex $v$ in $\F$, denoted $\operatorname{fut}_\F(v)$, is defined to be the set of vertices on the unique infinite simple path starting at $v$ in $\F$, including $v$ itself. The **past** of a vertex $v$ in $\F$ is defined to be the set of all vertices $u$ such that $v$ is in the future of $u$. For each integer $R\geq 0$, let $\F_R$ be the subgraph of $\F$ induced by the set $$\bigcup\left\{ \operatorname{fut}_\F(v) : v \in V \setminus B(u_1,R)\right\},$$ where $B(u_1,R)$ denotes the graph-distance ball of radius $R$ around $u_1$ in $G$. For each $R \geq r \geq 0$, let $\sC_{r,R}$ be the event that $\F_R$ does not intersect the set $\bigcup_{i=1}^k B(u_i,r)$. A vertex $v$ of $G$ is contained in the forest $\F_R$ if and only if its past intersects $V \setminus B(u_1,R)$. Since every component of $\F$ is one-ended almost surely, the past of each vertex of $G$ is finite almost surely. We deduce that $\bigcap_{R\geq0} \F_R=\emptyset$ almost surely and hence that $\lim_{R\to\infty}\P\left(\sC_{r,R}\right)=1$ for every $r\geq 0$.
Now, for each $m\geq 0$ and $R \geq 0$, we define a forest $\F_{m,R} = \bigcup_{i\geq0} \F_{m,R}^i$, where the forests $\F_{m,R}^i$ are defined recursively as follows. Let $\F_{m,R}^0=\F_R$. For each $1 \leq i \leq k$, given $\fF^{i-1}_{m,R}$, stop the random walk $(X^i_{n+m})_{n\geq 0}$ when it first visits the set of vertices already included in $\fF^{i-1}_{m,R}$. Take the loop-erasure of this stopped path, and and let $\F_{m,R}^{i}$ be the union of $\F_{m,R}^i$ with this loop-erased path. Similarly, if $i > k$, stop the random walk $(Y^{i-k-1,m}_{n})_{n\geq 0}$ when it first visits the set of vertices already included in $\fF^{i-1}_{m,R}$, take the loop-erasure of this stopped path, and and let $\F_{m,R}^{i}$ be the union of $\F_{m,R}^{i-1}$ with this loop-erased path. We refer to this procedure as **completing the run** of Wilson’s algorithm. It follows from [@HutNach2016a Lemma 4.1] that each of the forests $\F_{m,R}$ is distributed as the uniform spanning forest of $G$. (Indeed, we can think of the forests $\F_{m,R}$ as being sampled using Wilson’s algorithm, except that we are choosing which vertices to start our random walks from using a well-ordering of the vertex set that is not an enumeration.)
It is not hard to see that $\F_{m,R}$ converges to $\fG_m$ almost surely as $R \to \infty$ (with respect to the product topology on $\{0,1\}^E$). Indeed, it follows from the transience of $G$ that the loop-erasure of $(X^1_{n+m})_{n\geq 0}$ stopped when it first hits $\F_R$ converges almost surely to the loop-erasure of the unstopped walk as $R\to\infty$, and applying a similar argument inductively we deduce that $\F^i_{m,R}$ converges almost surely as $R\to\infty$ to the forest generated by the first $i$ steps of the application of Wilson’s algorithm used to generate $\fG_m$; the claim that $\F_{m,R}$ converges to $\fG_m$ almost surely follows by taking $i$ to infinity.
For each $m\geq 0$ and $R\geq0$, let $\sW_m$ be the event that the vertices $\{X^i_m : 1\leq i \leq k\}$ are all in different components of $\fG_{m}$, and let $\sB_{m,R}$ to be the event that the vertices $\{ X^i_m : 1 \leq i \leq k\}$ are all in different components of the forest $\F_{m,R}\setminus \F_R$. Thus, the event $\sB_{m,R}$ occurs if and only if for each $1 \leq i \leq k$ the walk $(X^i_{n+m})_{n\geq0}$ first hits the set of vertices included in $\F^{i-1}_{m,R}$ at a vertex of $\F_R$. It is easily seen that $\lim_{R\to\infty}\P(\sW_m {\hspace{.1em}\triangle\hspace{.1em}}\sB_{m,R})=0$ for each $m\geq 0$.
Now, for each $m\geq 0,$ $R\geq 0$ and $1 \leq \ell \leq k$, let $\tau_\ell(m,R)$ be the first time after time $m$ that the walk $X^\ell$ hits the set of vertices included in $\F_R$. Write $\mathbf{\tau}(m,R)=(\tau_\ell(m,R))_{\ell=1}^k$ and $\mathbf{X}_{\mathbf{\tau}(m,R)}=(X^\ell_{\tau_\ell(m,R)})_{\ell=1}^k$. On the event $\sB_{m,R}$, the vertex $X^i_m$ is connected in $\F_{m,R}$ to the vertex $X^i_{\tau_i(m,R)}$. Since $\sA$ is a tail property, we deduce that $$\sB_{m,R} \cap \Bigl\{
(\F_{m,R},\mathbf{X}_m) \in \sA
\Bigr\} =
\sB_{m,R} \cap \Bigl\{
\bigl(\F_{R},\mathbf{X}_{\mathbf{\tau}(m,R)}\bigr) \in \sA
\Bigr\}$$ up to a null set. Moreover, observe that $\tau_\ell(m,R)=\tau_\ell(0,R)$ for every $1 \leq \ell \leq k$ and every $m \leq r$ on the event $\sC_{r,R}$, so that if $r\geq m$ then $$\begin{gathered}
\sC_{r,R}\cap \sB_{0,R}\cap\sB_{m,R} \cap \Bigl\{
(\F_{m,R},\mathbf{X}_m) \in \sA
\Bigr\}\\ =
\sC_{r,R}\cap \sB_{0,R}\cap \sB_{m,R} \cap \Bigl\{
\bigl(\F_{R},\mathbf{X}_{\mathbf{\tau}(m,R)}\bigr) \in \sA
\Bigr\}\\
= \sC_{r,R}\cap \sB_{0,R}\cap\sB_{m,R} \cap \Bigl\{
(\F_{m,R},\mathbf{u}) \in \sA
\Bigr\}\end{gathered}$$ up to null sets, and taking probabilities we have that $$\P\left(
(\F_{m,R},\mathbf{u}) \in \sA
\mid \sC_{r,R}\cap \sB_{0,R}\cap\sB_{m,R} \right)
=
\P\left(
(\F_{m,R},\mathbf{X}_m) \in \sA
\mid \sC_{r,R}\cap \sB_{0,R}\cap\sB_{m,R} \right).$$ Using \[lem:measuretheory\] to take the limit as $R \to \infty$ on both sides, we obtain that $$\P((\fG_0,\mathbf{u}) \in \sA \mid \sW_0,\, \sW_m)
= \P((\fG_m,\mathbf{X}_m) \in \sA \mid \sW_0,\, \sW_m).$$ for every $m \geq 0$. The event $\sW_m$ is contained in the event that none of the random walks $(X^i_{n+m})_{n\geq0}$ intersect each other. If $k=1$, then $\sW_m$ trivially has probability one for every $m \geq 0$. Otherwise, $k>1$ and our assumption that $\P(\sW)>0$ implies that $\F$ is disconnected with positive probability, so that the event $\sW_m$ has probability converging to $1$ as $m\to\infty$ by . In either case, we deduce that $$\P((\fG_0,\mathbf{u}) \in \sA \mid \sW_0) = \lim_{m\to\infty}
\P((\fG_m,\mathbf{X}_m) \in \sA \mid \sW_0).
\label{eq:indist1}$$ In particular, the right-hand limit exists.
Since $G$ is Liouville, the sequence of random variables $((\mathbf{X}_{n+m})_{n\geq 0})_{m\geq 0}$ is tail-trivial. Moreover, for each $m\geq 0$ the forest $\fG_m$ is conditionally independent given the random variables $(\mathbf{X}_{n+m})_{n\geq 0}$ of the walks $(\mathbf{X}_n)_{n=0}^m$ and the forests $(\fG_i)_{i=1}^{m-1}$, and it is easily deduced that the sequence of random variables $\left(\fG_m,\, (\mathbf{X}_{n+m})_{n\geq0} \right)_{m \geq0}$ is also tail-trivial. Applying \[lem:tailtriviality\] we deduce that $$\lim_{m\to\infty}\left|
\P((\fG_m,\mathbf{X}_m) \in \sA \mid \sW_0)-\P((\fG_m,\mathbf{X}_m) \in \sA)\right| =0.
\label{eq:indist2}$$ The claim follows by combining and and using that $(\F,\mathbf{X}_m)$ and $(\fG_m,\mathbf{X}_m)$ are equidistributed.
\[lem:constantprobs\] Let $G=(V,E)$ be an infinite, transient, Liouville graph, let $\F$ be the uniform spanning forest of $G$, and suppose that every component of $\F$ is one-ended almost surely. Then for each tail $k$-component property $\sA$ there exists a constant $P(\sA)$ such that $$\vspace{0.2cm}
\P\left((\F,\mathbf{u}) \in \sA \mid \sW(\mathbf{u}) \right) = P(\sA).
\label{eq:indist1000}$$ for every $\mathbf{u}\in V^k$ with $\P(\sW(\mathbf{u}))>0$.
Let $\sA$ be a tail $k$-component property. Let $\mathbf{u}=(u_1,\ldots,u_k)$ and $\mathbf{u}'=(u_1',\ldots,u'_k)$ be two $k$-tuples of vertices of $G$ such that $\P(\sW(\mathbf{u})),\P(\sW(\mathbf{u}'))>0$. Let $\mathbf{X}=(X^1,\ldots,X^k)$ be a $k$-tuple of independent lazy random walks, independent of $\F$, with $\mathbf{X}_0=\mathbf{u}$. Let $M=\max d(u_i,u_i')$, and let $\sM$ be the event that $\mathbf{X}_M=\mathbf{u}'$, which is easily seen to have positive probability. It follows by \[lem:indistlem\], the Liouville property, \[lem:tailtriviality\], and the Markov property of the lazy random walk that $$\begin{aligned}
\P\left( (\F,\mathbf{u})\in \sA \mid \sW(\mathbf{u})\right)
&=\lim_{m\to\infty} \P\left((\F,\mathbf{X}_m)\in \sA \right)=\lim_{m\to\infty} \P\left((\F,\mathbf{X}_m)\in \sA \mid \sM \right)\\
&= \P\left( (\F,\mathbf{u}')\in \sA \mid \sW(\mathbf{u}')\right).\end{aligned}$$ as claimed.
It remains to prove that $P(\sA)\in \{0,1\}$ for every tail $k$-component property $\sA$.
Our next goal is to establish a conditional version of \[lem:indistlem\]. Let $r\geq 1$. By a slight abuse of notation, write $\F \cap B(u_1,r)$ to denote the set of edges of $\F$ that have both endpoints in the graph-distance ball $B(u_1,r)$ of radius $r$ around $u_1$, and $B(u_1,r) \setminus \F$ to denote the set of edges that have both endpoints in the ball $B(u_1,r)$ and are not contained in $\F$. For each $r\geq 1$, let $\cG_r$ be the $\sigma$-algebra generated by $\F \cap B(u_1,r)$. Similarly, for each $R\geq 1$, let $\cG^R$ be the $\sigma$-algebra generated by the restriction of $\F$ to the *complement* of the ball $B(u_1,R)$.
\[lem:indistlem2\] Let $G=(V,E)$ be an infinite, one-ended, transient, Liouville graph, let $\F$ be the uniform spanning forest of $G$, and suppose that every component of $\F$ is one-ended almost surely. Let $\mathbf{u}=(u_1,\ldots,u_k)$ be a $k$-tuple of vertices of $G$, and let $\mathbf{X}=(X^1,\ldots,X^k)$ be a $k$-tuple of independent lazy random walks on $G$, independent of $\F$, such that $\mathbf{X}_0=\mathbf{u}$. If $\P(\sW(\mathbf{u}))>0$, then for every tail $k$-component property $\sA$ we have that $$\vspace{0.2cm}
\P\left((\F,\mathbf{u}) \in \sA \mid \cG_r,\, \sW(\mathbf{u}) \right) = \lim_{m\to\infty} \P\left((\F,\mathbf{X}_m) \in \sA \mid \cG_r \right) \quad \text{ a.s.}
\label{eq:indist3}$$ In particular, the right hand limit exists almost surely.
Before beginning the proof, let us recall that the Liouville property can equivalently be defined in terms of the triviality of *invariant* events, rather than tail events. The **invariant $\sigma$-algebra** of $V^\N$ is defined to be the set of all measurable sets $\sI \subseteq V^\N$ such that $(v_i)_{i\geq 1}\in \sI$ if and only if $(v_{i+1})_{i\geq 1}\in \sI$. A graph is Liouville if and only if the lazy random walk has probability either zero or one of belonging to any set in the invariant $\sigma$-algebra [@LP:book Corollary 14.13].
Write $\sW=\sW(\mathbf{u})$. Let $A \subset E$ be such that $\F \cap B(u_1,r)=A$ with positive probability, and let $B$ be the set of edges that have both endpoints in $B(u_1,r)$ but are not in $A$. Let $\widehat G = (\widehat V, \widehat E)$ be the graph obtained from $G$ by contracting every edge in $A$ and deleting every edge in $B$. Note that, since $G$ is one-ended and every component of $\F$ is almost surely infinite, every two vertices of $G$ are almost surely connected by a path in $G$ that does not use any edges of $B(u_1,r)\setminus \F$, and it follows that $\widehat G$ is connected. Let $\widehat{\F}$ a wired uniform spanning forest of $\widehat G$ independent of $\F$. By the spatial Markov property of the uniform spanning forest (see e.g. [@HutNach2016b Section 2.2.1]), the conditional distribution of $\F$ given $\F \cap B(u_1,r)=A$ coincides with the distribution of $\widehat \F \cup A$ (after appropriate identification of edges).
Let $\pi: V \to \widehat V$ be the function sending each vertex of $V$ to its image following the contraction, and for each $v \in \widehat V$ let $\pi^{-1}(v)$ be an arbitrarily chosen vertex of $G$ such that $\pi(\pi^{-1}(v))=v$. Let $\pi^{-1}(\mathbf{v}) =(\pi^{-1}(v_1),\ldots,\pi^{-1}(v_k))$ for each $\mathbf{v}\in V^k$ and define $$\widehat{\hspace{0cm}\sA} = \left\{(\omega,\mathbf{v}) \in \{0,1\}^{\widehat E} \times \widehat V^k : \left(\omega \cup A , \pi^{-1}(\mathbf{v})\right) \in \sA \right\},$$ which does not depend on the arbitrary choices used to define $\pi^{-1}$ since $\sA$ is a multicomponent property. It is easily verified that $\widehat{\hspace{0cm}\sA}$ is a tail $k$-component property, and that $$\P\left((\F,\mathbf{u}) \in \sA \mid \F \cap B(u_1,r)=A,\, \sW\right) = \P\left((\widehat {\F} , \pi(\mathbf{u})) \in \widehat{\hspace{0cm}\sA} \mid \widehat{\hspace{0cm}\sW}\right),$$ where $\widehat{\hspace{0cm}\sW}$ is the event that $\pi(u_1),\ldots,\pi(u_k)$ are all in different components of $\widehat \F$.
Let $\widehat{\mathbf{X}}=(\widehat{X}^\ell)_{\ell=1}^k$ be independent lazy random walks on $\widehat G$ that are conditionally independent of $\F$ and $\widehat \F$ given $\cG_r$ and satisfy $\widehat{\mathbf{X}}_0=\pi(\mathbf{u})$. Let $T_\ell$ and $\widehat T_\ell$ be the last times that the walks $ X^\ell$ and $\widehat X^\ell$ visit $B(u_1,r)$ and $\pi(B(u_1,r))$ respectively, and write $\mathbf{T}=(T_1,\ldots,T_k)$ and $\widehat{\mathbf{T}}=(\widehat T_1,\ldots,\widehat T_k)$. Define $\mathbf{X}_{\mathbf{T}+1}:=(X^\ell_{T_\ell+1})_{\ell = 1}^k$ and $\widehat{\mathbf{X}}_{\widehat{\mathbf{T}}+1}:=(\widehat{X}^\ell_{\widehat{T}_\ell+1})_{\ell = 1}^k$. Observe that, since $\widehat G$ is connected, the supports of the random variables $\mathbf{X}_{\mathbf{T}+1}$ and $\pi^{-1}(\widehat{\mathbf{X}}_{\widehat{\mathbf{T}}+1})$ are both equal to the set of vertices $v\in V\setminus B$ for which the random walk started at $v$ has a positive probability not to hit $B(u_1,r)$. Similarly, the support of the random variable $(\mathbf{T},\mathbf{X}_{\mathbf{T}+1})$ is contained in the support of $(\widehat{\mathbf{T}},\pi^{-1}(\widehat{\mathbf{X}}_{ \widehat{\mathbf{T}}+1}))$. Furthermore, for every $\mathbf{t}\in \N^k$ and $\mathbf{v}=(v_1,\ldots,v_k)\in V^k$ such that $\mathbf{T}=\mathbf{t}$ and $\mathbf{X}_{\mathbf{T}+1} = \mathbf{v}$ with positive probability, we have the equality of conditional distributions $$\begin{gathered}
\label{eq:conditionallaws}
\bigl(\text{Law of } \bigl(\pi^{-1}(\widehat{\mathbf{X}}_{\widehat{\mathbf{T}}+n})\bigr)_{n\geq 1} \text{ given $\widehat{\mathbf{T}}=\mathbf{t}$ and $\pi^{-1} (\widehat{\mathbf{X}}_{\widehat{\mathbf{T}}+1} )= \mathbf{v}$}\bigr)\\
= \left(\text{Law of } (\mathbf{X}_{\mathbf{T}+n})_{n\geq 1} \text{ given $\mathbf{T}=\mathbf{t}$ and $\mathbf{X}_{\mathbf{T}+1} = \mathbf{v}$}\right).
\end{gathered}$$ Similarly, for every $\mathbf{v}=(v_1,\ldots,v_k)\in V^k$ such that $\mathbf{T}=\mathbf{t}$ and $\mathbf{X}_{\mathbf{T}+1} = \mathbf{v}$ with positive probability, we have the equality of conditional distributions $$\begin{gathered}
\label{eq:conditionallaws2}
\bigl(\text{Law of } \bigl(\pi^{-1}(\widehat{\mathbf{X}}_{\widehat{\mathbf{T}}+n})\bigr)_{n\geq 1} \text{ given $\pi^{-1}(\widehat{\mathbf{X}}_{\widehat{\mathbf{T}}+1}) = \mathbf{v}$}\bigr)
= \left(\text{Law of } (\mathbf{X}_{\mathbf{T}+n})_{n\geq 1} \text{ given $\mathbf{X}_{\mathbf{T}+1} = \mathbf{v}$}\right).
\end{gathered}$$ Indeed, both sides of both and are equal to the law of a $k$-tuple of independent lazy random walks on $G$, started at the vertices $(v_\ell)_{\ell=1}^k$ and conditioned not to return to $B(u_1,r)$. We deduce from that $\widehat G$ is Liouville: If $\sI \subseteq \widehat V^{\N}$ is an invariant event, then $\sI' = \{ (v_n)_{ n \geq 0} \in V^{\N} : (\pi(v_n))_{n \geq 0} \in \sI\}$ is also an invariant event, and hence that $$\P((X^1_n)_{n\geq 1} \in \sI')=\P((X^1_{T_1+n})_{n\geq 1} \in \sI') \in \{0,1\}$$ since $G$ is Liouville, from which it follows that $\P((X^1_{T_1+n})_{n\geq 1} \in \sI' \mid X^1_{T_1}) \in \{0,1\}$ almost surely. The equality of the supports of $X^1_{T_1+1}$ and of $\widehat X ^1_{\widehat T_1 +1 }$ and of the conditional laws then implies that $\P(\widehat X^1 \in \sI)$ is also equal to either zero or one, and since $\sI$ was arbitrary it follows that $\widehat G$ is Liouville as claimed. Thus, applying \[lem:indistlem\] to both $G$ and $\widehat G$ yields that $$ \P\left((\F,\mathbf{u}) \in \sA \mid \sW \right) = \lim_{m\to\infty} \P\left((\F,\mathbf{X}_m) \in \sA\right)
\label{eq:indistlem2eq4}
$$ and $$\begin{aligned}
\P\Bigl(\bigl(\widehat\F,\pi(\mathbf{u})\bigr) \in \widehat{\hspace{0cm}\sA} \mid \widehat{\hspace{0cm}\sW}\Bigr)
=
\lim_{m\to\infty}
\P\Bigl(\bigl(\widehat\F, \widehat{\mathbf{X}}_m\bigr) \in \widehat{\hspace{0cm}\sA} \Bigr).
\label{eq:indistlem2eq5}\end{aligned}$$
Let $\mathbf{t}$ and $\mathbf{v}$ be such that $\mathbf{T}=\mathbf{t}$ and $
\mathbf{X}_{\mathbf{T}+1} = \mathbf{v}$ with positive probability. Conditioning on $\F$ and applying \[lem:tailtriviality\] we deduce that $$\lim_{m\to\infty}\bigl|\P\left(( \F,\mathbf{X}_m) \in \sA \mid \F,\,
\mathbf{T} = \mathbf{t},\,
\mathbf{X}_{\mathbf{T}+1} = \mathbf{v}\right)
-
\P\left(( \F,\mathbf{X}_m) \in \sA \mid \F \right)\bigr| =0$$ almost surely, and hence that $$\begin{gathered}
\label{eq:tv1}
\lim_{m\to\infty}\bigl|\P\left(( \F,\mathbf{X}_m) \in \sA \mid \F \cap B(u_1,r) =A,\,
\mathbf{T} = \mathbf{t},\,
\mathbf{X}_{\mathbf{T}+1} = \mathbf{v}\right)\\
-
\P\left(( \F,\mathbf{X}_m) \in \sA \mid \F \cap B(u_1,r) =A \right)\bigr| =0.\end{gathered}$$ Applying a similar analysis to $\widehat G$ yields that $$\lim_{m\to\infty}\bigl|\P\bigl(( \widehat \F, \widehat{\mathbf{X}}_m) \in \widehat{\hspace{0cm}\sA} \mid
\widehat{\mathbf{T}} = \mathbf{t},\,
\pi^{-1}(\widehat{\mathbf{X}}_{\widehat{\mathbf{T}}+1}) = \mathbf{v}\bigr)- \P\bigl(( \widehat \F, \widehat{\mathbf{X}}_m) \in \widehat{\hspace{0cm}\sA} \bigr)\bigr| =0.
\label{eq:tv2}$$ On the other hand, the spatial Markov property of the USF and the equality of conditional laws implies that $$\begin{gathered}
\P\left(( \F,\mathbf{X}_m) \in \sA \mid \F \cap B(u_1,r) =A,\,
\mathbf{T} = \mathbf{t},\,
\mathbf{X}_{\mathbf{T}+1} = \mathbf{v}\right)\\
=
\P\bigl(( \widehat{\F}, \widehat{\mathbf{X}}_m) \in \widehat{\hspace{0cm}\sA} \mid
\widehat{\mathbf{T}} = \mathbf{t},\,
\pi^{-1}(\widehat{\mathbf{X}}_{\widehat{\mathbf{T}}+1}) = \mathbf{v}\bigr)
\label{eq:tv3}\end{gathered}$$ for every $m\geq 1+\max_{1\leq i \leq k} t_i$. Together, , , and imply that $$\lim_{m\to\infty}\Bigl| \P\left(( \F,\mathbf{X}_m) \in \sA \mid \F \cap B(u_1,r) =A \right) - \P\bigl((\widehat \F,\widehat{\mathbf{X}}_m) \in \widehat{\hspace{0cm}\sA} \bigr)\Bigr| = 0
\label{eq:indistlem2eq2}$$ almost surely, which yields the claim when combined with and .
We are now ready to complete the proof of \[thm:indist\].
By \[lem:constantprobs\], it remains to prove only that $P(\sA)\in \{0,1\}$ for every tail $k$-component property $\sA$. We continue to use the notation of \[lem:indistlem,lem:indistlem2\]. Since $\bigcup_{r\geq0} \cG_r$ generates the product $\sigma$-algebra of $\{0,1\}^E$, it suffices to prove that $$\label{eq:indistclaim1}
\P\left((\F,\mathbf{u}) \in \sA \mid \cG_r,\, \sW\right) = \P\left((\F,\mathbf{u}) \in \sA \mid \sW\right) =P(\sA)
\quad \text{ a.s.}$$ for every tail $k$-component property $\sA$, every $\mathbf{u}=(u_1,\ldots,u_k)\in V^k$ with $\P(\sW(\mathbf{u}))>0$, and every $r \geq 1$. By \[lem:indistlem,lem:indistlem2\], to prove it suffices to prove that $$\label{eq:indistclaim3}
\lim_{m\to\infty} \P\left((\F,\mathbf{X}_m) \in \sA \right) = \lim_{m\to\infty}\P\left((\F,\mathbf{X}_m) \in \sA \mid \cG_r\right) \quad \text{ a.s.}$$
Let $\sD_{m,R}$ be the event that the future of $X^\ell_m$ in $\F$ is contained in the complement of $B(u_1,R)$ for every $1\leq \ell \leq k$, and let $\sE_{m,R} = \{ (\F,\mathbf{X}_m) \in \sA\} \cap \sD_{m,R}$. In particular, $X^\ell_m \in V \setminus B(u_1,R)$ for every $1\leq \ell\leq k$ on the event $\sE_{m,R}$. Moreover, since $\F$ is one-ended almost surely and $G$ is transient, we have that $\P(\sD_{m,R}) \to 1$ as $m\to\infty$ and hence that $$\lim_{m\to\infty} \left| \P(\sE_{m,R}) - \P\left(\left(\F,\mathbf{X}_m\right) \in \sA\right)\right|=0$$ for every $R\geq 1$. Thus, there exists a sequence $m(R)$ growing sufficiently quickly that $$\label{eq:EmRR}
\lim_{R\to\infty} \Bigl| \P(\sE_{m(R),R}) - \P\bigl((\F,\mathbf{X}_{m(R)}) \in \sA\bigr)\Bigr|=0.$$
Let $A$ be a set of edges such that $\P(\F \cap B(u_1,r) = A ) >0$. Since $\sA$ is a tail property and every component of $\F$ is one-ended almost surely, the event $\sE_{m,R}$ is measurable (up to a null set) with respect to the $\sigma$-algebra generated by $\cG^R$ and $\mathbf{X}_m$. Thus, by tail-triviality of the uniform spanning forest, conditioning on $\mathbf{X}$ and applying \[lem:tailtriviality\] yields that $$\lim_{R\to\infty} \left|\P\left(\left\{\F \cap B(u_1,r)=A\right\} \cap \sE_{m(R),R} \mid \mathbf{X}\right) - \P\left(\F \cap B(u_1,r)=A\right)\P\left(\sE_{m(R),R} \mid \mathbf{X} \right)\right| = 0$$ almost surely. Taking expectations over $\mathbf{X}$, we deduce that $$\lim_{R\to\infty} \left|\P\left(\left\{\F \cap B(u_1,r)=A\right\} \cap \sE_{m(R),R}\right)- \P(\F \cap B(u_1,r)=A)\P\left(\sE_{m(R),R}\right)\right| = 0.$$ Dividing through by $\P(\F \cap B(u_1,r) = A)$ then yields that $$\begin{aligned}
\lim_{R\to\infty}
\left|\P\left( \sE_{m(R),R} \mid \F \cap B(u_1,r) =A \right) - \P\left( \sE_{m(R),R}\right)\right| =0\end{aligned}$$ and applying we deduce that $$\lim_{R\to\infty}
\left|\P\left( \left(\F,\mathbf{X}_{m(R)}\right) \in \sA \mid \F \cap B(u_1,r) =A \right) - \P\left( \left(\F,\mathbf{X}_{m(R)}\right) \in \sA\right)\right| =0.$$ Since $A$ was arbitrary, the claimed equality follows.
The Liouville property is necessary {#sec:closing}
===================================
Suppose $G$ is a non-Liouville graph such that every component of the USF of $G$ is one-ended almost surely. Let $\sA$ be a non-trivial invariant event for the random walk on $G$ and let $h(v)={\mathbf{P}}_v(\sA)$ be the associated bounded harmonic function. It follows from the martingale convergence theorem that $h(X_n) \to \mathbbm{1}(A)$ almost surely as $n\to\infty$ whenever $X$ is a random walk on $G$. Thus, by Wilson’s algorithm, almost surely for every tree of $\F$, the value of $h$ converges as we move progressively higher up the tree. Thus, we can assign a value of either zero or one to each tree of $\F$, and the value of the tree is a tail component property. Moreover, it is easy to see that there must be trees with both values zero and one a.s. This shows that, without the Liouville condition, \[thm:indist\] always fails even in the case $k=1$.
If $G$ is a transitive graph, it is natural to consider tail multicomponent properties that are invariant under the automorphisms of $G$. In this case, [@HutNach2016a Theorem 1.20] implies the indistinguishability of individual components of the WUSF by automorphism invariant tail properties, corresponding to the case $k=1$ of \[thm:indist\]. The question of whether a similar result holds for larger $k$ seems to depend on the symmetries of $G$. For example, if $G$ is the $7$-regular triangulation, then we can consider the circle packing of $G$ into the unit disc (see e.g. [@Rohde11] and references therein), which is unique up to Möbius transformations and reflections. Every tree in the wired uniform spanning forest of $G$ has a unique limit point in the unit circle under this embedding [@BS96a], and we can define a tail $4$-component property by asking whether, given some $4$-tuple of trees, the cross-ratio of their limit points has absolute value greater than one. Some $4$-tuples of trees in the WUSF will satisfy this property while others will not, so that it is possible to distinguish $4$-tuples of distinct trees in the WUSF via tail $4$-component properties. On the other hand, the unit circle is the Poisson boundary of $G$ [@ABGN14] and hence, intuitively, all the tail information about a collection of trees should be contained in their collection of limit points. Since the group of Möbius transformations of the unit disc acts $3$-transitively on the unit circle, it is plausible that it should be impossible to distinguish $3$-tuples of distinct trees in the WUSF of this graph via multicomponent properties.
Acknowledgments {#acknowledgments .unnumbered}
---------------
This work took place while the author was an intern at Microsoft Research, Redmond.
|
---
abstract: 'Boron Carbide exhibits a broad composition range, implying a degree of intrinsic substitutional disorder. While the observed phase has rhombohedral symmetry (space group $R\bar{3}m$), the enthalpy minimizing structure has lower, monoclinic, symmetry (space group $Cm$). The crystallographic primitive cell consists of a 12-atom icosahedron placed at the vertex of a rhombohedral lattice, together with a 3-atom chain along the 3-fold axis. In the limit of high carbon content, approaching 20% carbon, the icosahedra are usually of type B$_{11}$C$^p$, where the $p$ indicates the carbon resides on a polar site, while the chains are of type C-B-C. We establish an atomic interaction model for this composition limit, fit to density functional theory total energies, that allows us to investigate the substitutional disorder using Monte Carlo simulations augmented by multiple histogram analysis. We find that the low temperature monoclinic $Cm$ structure disorders through a pair of phase transitions, first via a 3-state Potts-like transition to space group $R3m$, then via an Ising-like transition to the experimentally observed $R\bar{3}m$ symmetry. The $R3m$ and $Cm$ phases are electrically polarized, while the high temperature $R\bar{3}m$ phase is nonpolar.'
address:
- 'Department of Physics, Carnegie Mellon University, 5000 Forbes Ave, Pittsburgh, PA, 15232, United States of America, 412-268-7645'
- 'Department of Mechanical Engineering and Materials Science, Duke University, Durham, NC, 27708, United State of America'
author:
- Sanxi Yao
- 'W. P. Huhn'
- 'M. Widom'
title: 'Phase Transitions of Boron Carbide: Pair Interaction Model of High Carbon Limit'
---
Boron Carbide ,density functional theory ,multi-histogram method ,3-state Potts like transition ,Ising like transition
Introduction
============
The phase diagram of boron carbide is not precisely known, with both qualitative and quantitative discrepancies among the different research groups [@Samsonov58; @Ekbom81; @Beauvy83; @Schwetz91; @Okamoto92; @Domnich11; @Rogl14]. The most widely accepted diagram of Schwetz [@Schwetz91; @Okamoto92] displays a single boron carbide phase at temperature above 1000C, coexisting with elemental boron and graphite. The carbon concentration covers the range 9$\%$-19.2$\%$ carbon, falling notably short of the 20% carbon fraction at which the electron count is believed to be optimal [@Longuet-Higgins55; @Lipscomb81; @Balakrishnarajan07]. More interestingly, the nearly temperature independent behavior of the phase boundaries are thermodynamically improbable, and the broad composition range suggests substitutional disorder at 0K, in apparent violation of the 3rd law. Since so much remains unknown, and experiment can only be assured of reaching equilibrium at high temperature, theoretical calculation offers hope for resolving the behavior at lower temperatures, in addition to interpreting the disorder at high temperatures.
As determined crystallographically [@GWill76; @GWill79; @Kwei96], boron carbide has a 15-atom primitive cell, consisting of an icosahedron and 3-atom chain, in a rhombohedral lattice with symmetry $R\bar{3}m$. At 20$\%$ carbon a proposed B$_4$C structure featured pure boron icosahedra B$_{12}$ with a C-C-C chain [@Clark43]. Although this structure exhibits $R\bar{3}m$ symmetry, later experimental work [@Kwei96; @Schmechel00] suggested that the icosahedron should be B$_{11}$C instead of B$_{12}$ and the chain should be C-B-C instead of C-C-C. For other compositions, the icosahedra can be B$_{12}$, B$_{11}$C, or even B$_{10}$C$_2$ (the bi-polar defect [@Mauri01]), and the chain can be C-B-C, C-B-B, B-B$_2$-B [@Yakel; @Shirai14] or B-V-B (V means vacancy). Fig. \[fig:structure\] illustrates the rhombohedral cell, the C-B-C chain and the icosahedron. The 12 icosahedral sites are categorized into 2 classes: equatorial and polar. We further classify the polar sites into north and south. Icosahedra are connected along edges of the rhombohedral lattice, which pass through the polar sites.
0.5cm ![Primitive cell of boron carbide showing C-B-C chain at center along the 3-fold axis. The icosahedron (not to scale) occupies the cell vertex. Equatorial sites of the icosahedron are shown in red, labeled “e”, the north polar sites are shown in green and labeled $p_0$, $p_1$ and $p_2$, while the south polar sites are shown in cyan and labeled as $p_0'$, $p_1'$ and $p_2'$.[]{data-label="fig:structure"}](structure.eps "fig:"){width="0.8\linewidth"}
Density functional theory studies of a large number of possible arrangements of boron and carbon atoms [@Mauri01; @Widom12; @Bylander90] identified four stable phases: pure $\beta$-Boron, rhombohedral B$_{13}$C$_2$, monoclinic B$_4$C, and graphite. The stable rhombohedral phase consists of B$_{12}$ icosahedra with C-B-C chains, giving full rhombohedral symmetry $R\bar{3}m$, while the stable monoclinic phase has B$_{11}C^p$ icosahedra and C-B-C chains. Carbon occupies the same polar site (e.g. $p_0$) in every icosahedron, resulting in symmetry $Cm$. Introducing disorder in the occupation of polar sites (i.e. randomly choosing one polar site to occupy with carbon) can restore $R\bar{3}m$ symmetry. In general, we define site occupations $m_i$ ($i=0, 1, 2, 0', 1', 2'$) corresponding to the mean occupation of the sites $p_i$. The experimentally observed phase has all $m_i=1/6$. We call this orientational disorder. The the stable monoclinic phase has one large order parameter (e.g. $m_0\sim 1$) and the remaining $m_i\sim 0$, which we call orientational order. Hence it was proposed [@Huhn12] that a temperature-driven order-disorder phase transition is responsible for the high symmetry seen in experiments that are likely in equilibrium only at high temperature.
According to Landau’s theory of phase transitions [@Wooten08], the space groups of structures linked by continuous or at most weakly first order phase transitions should obey group-subgroup relationships. Additionally, the subgroup should be maximal, again provided the transition is continuous or at most weakly first order. Typically the high temperature phase possesses the higher symmetry, as this permits a higher entropy. In regard to boron carbide, the high temperature phase has space group $R\bar{3}m$ (group \#166) and the low temperature phase has group $Cm$ (group \#8). However, $Cm$ is not a maximal subgroup of $R\bar{3}m$, suggesting the possible existence of an intermediate phase. Two sequences of transitions obey the maximality requirement: $R\bar{3}m\rightarrow R3m\rightarrow Cm$ and $R\bar{3}m\rightarrow C2/m\rightarrow Cm$ [@Hahn84]. The two corresponding intermediate symmetries $R3m$ and $C2/m$ have space group numbers \#160 and \#12, respectively.
In terms of the distribution of carbon atoms on icosahedra, $R3m$ breaks the inversion symmetry, and hence corresponds to occupying one pole (e.g. the north pole) more heavily than the other, so that $m_i\sim 1/3$ ($i=0, 1, 2$) with the remaining $m_i\sim 0$. In contrast, $C2/m$ breaks the 3-fold rotational symmetry but preserves inversion. Thus the carbon atoms preferentially occupy a pair of diametrically opposite polar sites, e.g. $m_0=m_0'\sim 1/2$ while the remaining $m_i\sim 0$.
Since the carbon atom draws charge from surrounding borons, the phases that break inversion symmetry possess an electric dipole moment. Hence we name the $R3m$ state “polar”. The possibility of a polar phase was independently suggested recently [@Ektarawong14]. We name the $Cm$ state “tilted polar” because the broken 3-fold symmetry creates a component of polarization in the $xy$ plane. Although it lacks a net dipole moment, we name the $C2/m$ state “bipolar” because it is reminiscent of the bipolar defect [@Mauri01]. Finally, we name the the high symmetry phase $R\bar{3}m$ “nonpolar”.
To identify phase transitions, and to determine which symmetry-breaking sequence occurs, we perform Monte Carlo simulations. Strictly speaking, phase transitions occur only in the thermodynamic limit of large system size, which is beyond the reach of density functional theory calculations. Hence we construct a classical interatomic interaction model, with parameters fit to density functional theory energies. For simplicity we consider only the high carbon limit where every icosahedron contains a single polar carbon (i.e. we essentially project the composition range onto the $x_C=0.2$ line). We analyze our simulation results with the aid of the multiple histogram technique [@Swendsen88; @Swendsen89]. In the end we indeed discover a sequence of two phase transitions. One arises from the breaking of 3-fold symmetry linking $Cm$ to $R3m$ that is first order, similar to the 3-state Potts model [@Potts52] in three dimensions. The other corresponds to the breaking of inversion symmetry linking $R3m$ to $R\bar{3}m$ that is in the Ising universality class.
Methods
=======
Pair interaction model
----------------------
Given that every primitive cell contains a B$_{11}$C$^p$ icosahedron and a C-B-C chain, the configuration can be uniquely specified by assigning a 6-state variable $\sigma$ to each cell, corresponding to which of the six polar sites holds the carbon atom. The relaxed total energy of a specific configuration can be expressed through a cluster expansion in terms of pairwise, triplet and higher-order interactions [@Sanchez84; @Walle02] of these variables. As shown below, truncating at the level of pair interactions provides sufficient accuracy for present purposes. Further, we observe that symmetry-inequivalent pairs are in nearly one-to-one correspondence with the inter-carbon separation $R_{ij}=|\bR_i-\bR_j|$ where the $\bR_i$ are the initial positions prior to relaxation and belong to a discrete set of fixed possible values $\{R_k\}$, arranged in order of increasing length. Note that we need not concern ourselves with interactions of polar carbons with chain carbons, as the number of such pairwise interactions is conserved across configurations. Thus our total energy can be expressed as $$\label{eq:bondmodel}
E(N_1,\dots,N_m)=E_0+\sum_{i=1}^m a_k N_k$$ where $N_k$ is the number of intercarbon separations of length $R_k$, and $m$ is the number of such separations we choose to treat in our model.
We use the density functional theory-based Vienna ab initio simulation package (VASP) [@Kresse93; @Kresse94; @Kresse961; @Kresse962] to calculate the total energies within the projector augmented wave (PAW) [@Blochl94; @Kresse99] method utilizing the PBE generalized gradient approximation [@Perdew96; @Perdew97] as the exchange-correlation functional. We construct a variety of 2x2x2 and 3x3x3 supercells, which we relax with increasing $k$-point meshes until convergence is reached at the level of 0.1 meV/atom, holding the plane-wave energy cutoff fixed at 400 eV.
Using our DFT energies, we fit the $m+1$ parameters $E_0$ and $\{a_i\}$ in our bond interaction model Eq. (\[eq:bondmodel\]), increasing $m$ until we are satisfied with the quality of the fit, at $m=10$. The shortest bond included has length $R_1=1.732$ Å, corresponding to carbons at polar sites joined by an intericosahedral bond. For example, $p_0$ and $p_0'$ carbons on icosahedra joined by a bond in the $p_0$ direction. This bond has the largest strength, with $a_1=1.126$ eV. Bond strength rapidly diminishes with separation $R$. Our longest bonds included are $R_9=5.174$ Å (the lattice constant separating neighboring rhombohedral vertices) and $R_{10}=5.465$ Å (the second neighbor rhombohedral vertex separation).
Our fitting procedure minimizes the mean-square deviation of model energy from calculated DFT energy, supplemented by a small contribution from the $L_1$ norm of the set of coefficients $\{a_k\}$. Including the $L_1$ norm regularizes the expansion in a manner similar to compressive sensing [@Hart13; @Wakin08] and improves transferability to larger cell sizes. For our fitted data set of 188 independent 2x2x2 supercells (see Fig. \[fig:fitting\]) we obtain an RMS error of 0.474meV/atom. Checking our fit on a different set of 47 $2\times2\times2$ supercells with energy below 5.7 meV/atom we obtain RMS error of 0.233meV/atom, while checking on 57 $3\times3\times3$ supercells yielded RMS error of 0.405meV/atom. Note that 5.7 meV/atom corresponds to $15\times 5.7=86$ meV/cell corresponding to $kT$ per degree of freedom at T=1000K.
1.0cm ![Fit of bond interaction model to DFT-calculated total energies in $2\times2\times2$ supercells. Inset shows cross validation check of $2\times2\times2$ supercells transferability to $3\times3\times3$ supercells.[]{data-label="fig:fitting"}](fitting.eps "fig:"){width="0.8\linewidth"}
Symmetry and order parameters
-----------------------------
Within Landau theory, each possible symmetry breaking is quantified by an order parameter whose transformation properties match irreducible representations of the parent symmetry group. Space group $R\bar{3}m$ contains point group $D_{3d}$, which is the symmetry group of the triangular antiprism formed by the six polar sites of the icosahedron. Important elements include 3-fold rotation about the $z$-axis, reflection in a vertical plane containing the $z$-axis, and inversion through the center.
The longitudinal polarization $$P_z=m_0+m_1+m_2-m_0'-m_1'-m_2'$$ transforms as the one dimensional irreducible representation $A_{2u}$, which breaks inversion symmetry, while preserving rotation and reflection, and hence is suitable for characterizing the transition $R\bar{3}m\rightarrow R3m$. The pair of functions $$P_{xz}=(m_0-m_0')+\frac{1}{2}(m_1'+m_2'-m_1-m_2), ~~~
P_{yz}={\sqrt{3}\over 2}(m_1+m_2'-m_1'-m_2)$$ transform as the two dimensional irreducible representation $E_g$, which additionally breaks both rotational symmetry, and hence characterizes the further transition $R3m\rightarrow Cm$. Since we will not care which specific orientation is selected at low temperature, we take the norm of the two dimensional representation, and define $P_{xy}=\sqrt{P_{xz}^2+P_{yz}^2}$. Although we shall not need it, we note that the functions $$P_x=(m_0+m_0')-\frac{1}{2}(m_1+m_1'+m_2+m_2'), ~~~
P_y={\sqrt{3}\over 2}(m_1+m_1'-m_2-m_2'),$$ which transform as the irrep $E_u$, characterize the transformation $R\bar{3}m\rightarrow C2/m$.
As examples of the use of these order parameters, consider a fully disordered nonpolar state of symmetry $R\bar{3}m$ in which all $m_i=1/6$. All the above order parameters vanish in this state. Now let $m_i=1/3$ while $m_i'=0$, and note that $P_z=1$, while $P_{xz}=P_{yz}=P_x=P_y=0$, so that $P_z$ indeed characterizes the polar state $R3m$. Completing the symmetry breaking so that $m_0=1$ while all others vanish, we have both $P_z=1$ and $P_{xz}=1$, so the state is both polar and tilted, with symmetry $Cm$. Finally, take $m_0=m_0'=1/2$ and all other $m_i$ and $m_i'=0$ and note that $P_x\ne 0$, while $P_z=P_{xz}=P_{yz}=0$, as expected for the bipolar state of symmetry $C2/m$.
Monte Carlo simulation and multi-histogram method
-------------------------------------------------
We perform conventional Metropolis Monte Carlo simulations in $L\times L\times L$ supercells of the rhombohedral primitive cell, with $L$ ranging from 3 to 12. Our basic move is a “rotation” in which we randomly select an icosahedron, then randomly displace the carbon from its current polar site to a randomly chosen alternate polar site. The move is then accepted or rejected according to the Boltzmann factor for the energy change $\Delta E$. Following an equilibration period, we begin recording the total energy $E$ and the occupations $m_i$ ($i=0, 1, 2, 0', 1', 2'$) of the polar sites for each subsequent configuration. After a run at one temperature is completed, we take the final configuration as the initial configuration for another run at a nearby temperature.
At a given simulation temperature $T_s$, a histogram of configuration energies $H_{T_s}(E)$ (see Fig. \[fig:histograms8\]) can be converted into a density of states $W(E)=H_{T_s}(E) \exp{(E/kT_s)}$, which is accurate over the energy range that has been well sampled at temperature $T$. This density of states can be used to calculate the partition function $$Z(T)=\sum_E W(E) e^{-E/kT}$$ which is accurate over a range of temperatures close to $T_s$ [@Swendsen88]. The logarithm of $Z(T)$ yields the free energy, and derivatives of the free energy yield other quantities such as internal energy, entropy and specific heat (see Fig. \[fig:cv\]). Alternatively, we may take moments of the energy distribution, $$\label{eq:avE}
\avg{E^q} = \sum_E W(E) E^q e^{-E/k_BT}.$$ The first moment ($q=1$) yields the thermodynamic internal energy, while the fluctuations $$\label{eq:Cv}
c_v(T)=\frac{\avg{E^2}-\avg{E}^2}{k_BT^2}$$ give the specific heat.
Moreover, by combining histograms taken at temperatures chosen so that the tails of the histograms overlap, the density of states can be self-consistently reconstructed [@Swendsen89] so that the free energy becomes accurate over all intervening temperatures. Inspecting the histograms shown in Fig. \[fig:histograms8\], rapid evolution is apparent with between temperatures 710 and 730K, which can be an indication of a phase transition. As supercell size increases the histograms narrow, requiring additional simulation temperatures to maintain the degree of overlap seen here.
0.5cm ![Multiple histograms for the 8x8x8 supercell. The ground state $Cm$ configuration is taken as the zero of energy.[]{data-label="fig:histograms8"}](totalhist.eps "fig:"){width="0.8\linewidth"}
This notion can be extended to multidimensional histograms in which the density of states is further broken down according to values of order parameters of interest. For instance, average powers of the longitudinal polarization can be evaluated as $$\label{eq:avPz}
\avg{|P_z|^q}(T) = \sum_{E, P_z} W(E, P_z) |P_z|^q e^{-E/kT},$$ where $W(E, P_z)$ is the joint distribution of energy and longitudinal polarization, and we take the absolute value of $P_z$ because in a well equilibrated simulation both positive and negative values of $P_z$ occur with equal frequency. The first power gives the mean polarization, while from the first and second powers together we obtain the longitudinal susceptibility $$\label{eq:chiz}
\chi_z(T)=N\frac{\avg{|P_z|^2} - \avg{|P_z|}^2}{ k_BT},$$ where $N$ is the number of atoms. The susceptibility $\chi_{xy}(T)$ is obtained in similar fashion. The units of $\chi_z$ and $\chi_{xy}$ are $eV^{-1}/atom$.
Results and Discussion
======================
Order parameters
----------------
Plotting the order parameters vs. temperature provides a quick way to determine the sequence of phases and transitions. As Fig. \[fig:4pics\] shows, $\avg{|P_z|}$ passes through two regimes of anomalous behavior. As the supercell size grows, the average longitudinal polarization $\avg{|P_z|}$ vanishes for $T\gtrsim 790$K but approaches to finite values for $T\lesssim 790$K. At $T=790$K the slope of $\avg{|P_z|}(T)$ diverges. An even stronger divergence of slope occurs at $T\approx 717$K. Meanwhile, $\avg{P_{xy}}$ decreases with increasing supercell size for $T\gtrsim 717$K but approaches finite values for $T\lesssim 717$K. The diverging slope at $T\approx 717$K is consistent with an emerging discontinuity in $\avg{P_{xy}}(T)$.
On the basis of the order parameters, we judge there are three phases separated by two phase transitions. The high temperature phase has symmetry $R\bar{3}m$ both $P_z$ and $P_{xy}$ vanish. Below 790K a longitudinal polarization grows continuously, and we enter a phase of symmetry $R3m$, having lost inversion symmetry. Around 717K the polarization suddenly tilts off the $z$-axis and we enter the tilted polar phase of symmetry $Cm$.
Specific heat and susceptibility
--------------------------------
Having explored the order parameters, which can be considered as first derivatives of the free energy with respect to applied fields, we now consider the specific heat and susceptibilities. The specific heat corresponds to a second derivative of free energy with respect to temperature, while the susceptibilities are second derivatives with respect to conjugate fields. All are evaluated from Monte Carlo data via the fluctuations formulas such as Eqs. (\[eq:Cv\]) and (\[eq:chiz\]).
Specific heat for a series of increasing supercell sizes is shown in Fig. \[fig:cv\]. In addition to a strong peak around T=717K, a weak peak around T=790K can be seen growing for larger supercell sizes in the inset. The growing peaks converge to temperatures that roughly correspond to the order parameter anomalies seen above. Fig. \[fig:4pics\] shows the longitudinal and perpendicular (i.e. in-plane) susceptibilities, $\chi_z$ and $\chi_{xy}$ respectively. Evidentally the high temperature specific heat peak coincides with the peak in $\chi_z$, and hence relates to the fluctuations associated with onset of longitudinal polarization. Similarly, the low temperature specific heat peak coincides with the peak in $\chi_{xy}$, and hence relates to fluctuations associated with the tilt of polarization off the 3-fold axis.
0.5cm ![Specific heat for 3x3x3 to 12x12x12 supercells.[]{data-label="fig:cv"}](TCv1.eps "fig:"){width="80.00000%"}
### Ising-like transition
As the high temperature transition from $R\bar{3}m$ to $R3m$ coincides with a breaking of inversion symmetry, we expect the transition to be in the universality class of the three-dimensional Ising model. Some associated critical exponents are $\alpha=0.110$ (specific heat), $\gamma=1.2372$ (susceptibility) and $\nu=0.6301$ (correlation length) [@Pelissetto02]. Applying finite size scaling theory [@Landau05], we note that the specific heat peak height should diverge with increasing supercell size as $L^{\alpha/\nu}$, where $\alpha/\nu=0.175$. The small value of this exponent explains the weak divergence seen around 790K in Fig. \[fig:cv\]. Similarly the susceptibility $\chi_z$ should diverge as $L^{\gamma/\nu}$, with $\gamma/\nu=1.963$.
Validation of the size- and temperature-dependence of $\chi_z$ requires a finite-size scaling collapse of axes, plotting the scaled susceptibility $\chi_z/L^{\gamma/\nu}$ as a function of an expanded temperature scale $\epsilon L^{1/\nu}$ where $\epsilon=(T-T_c)/T_c$. When plotted in this manner as seen in Fig. \[fig:Isingscale\] the finite size susceptibilities converge to a common scaling function $\chi_0$, supporting the proposed Ising universality class of this continuous phase transition as well as yielding an improved estimate for $T_c=793.7$K.
0.5cm ![Validation of universality classes. (left) Ising scaling function for $\chi_z$; (right) Lee-Kosterlitz histograms of $P_\perp$.[]{data-label="fig:Isingscale"}](Isingscaling.eps "fig:"){width="0.4\linewidth"} ![Validation of universality classes. (left) Ising scaling function for $\chi_z$; (right) Lee-Kosterlitz histograms of $P_\perp$.[]{data-label="fig:Isingscale"}](rwthistpxy.eps "fig:"){width="40.00000%"}
### 3-state Potts-like transition
Once a direction for longitudinal polarization has been chosen at the high temperature Ising-like transition (e.g. north), the remaining orientational ordering requires selecting a particular in-plane direction (e.g. $i=0, 1$ or 2), resulting in a breaking of 3-fold rotational symmetry. Thus we expect the low temperature transition to be in the universality class of the 3-state Potts model. In three dimensions this transition is expected to be weakly first order [@Wu82].
Because the order parameter jumps discontinuously at a first order transition, the fluctuations per atom of energy and polarization should grow proportionally to the number of atoms, i.e. as $L^3$. When peak heights of $c_v$ and $\chi_{xy}$ are plotted on a log-log plot vs. $L$, we expect a straight line in the asymptotic limit of large $L$ whose slope should be 3. Unfortunately, our largest supercell size $L=12$ has not yet reached this limit, with the slopes around 2 and 2.8 seen for $c_v$ and $\chi_{xy}$ respectively, clearly tending to increase with $L$.
The Lee-Kosterlitz criterion [@Landau05; @Lee90] is an alternative method to confirm a first order transition. Because the two coexisting phases exhibit finite differences in properties such as energy and polarization, probability distributions of such properties should then be bimodal, with each peak sharpening as system size grows. Fig. \[fig:Isingscale\] illustrates this distribution for $P_{xy}$. This distribution is obtained by marginalizing the joint energy and polarization histogram $H_{T_s}(E,P_{xy})$ over energy, then reweighting with the factor $\exp(E/k_BT_e-E/k_BT_s)$, where the temperature $T_e$ is chosen so as to make the heights of the two peaks equal. Clearly the distributions of polarization illustrate coexistence of a state with $P_{xy}=0$ and a state with $P_{xy}\sim 0.5$. Thus we conclude the transition is first order, as expected for symmetry-breaking of the 3-state Potts type in three dimensions.
Electric dipole moments
-----------------------
Because of the charge imbalance created by the polar carbons, the polar and tilted polar states must exhibit electric dipole moments, while the nonpolar state does not. We constructed specific representative structures of each of the three phases to calculate these dipole moments. Taking a single hexagonal unit cell containing three primitive cells, we constructed the tilted polar $Cm$ state by placing carbon at each of the three $p_0'$ sites. We then constructed a polar $R3m$ state by placing the polar carbon at $p_0'$ in one cell, $p_1'$ in another and $p_2'$ in the third. Finally we took a $2\times1\times1$ supercell of the hexagonal unit cell, and inverted the polarization of every second carbon (e.g. we replace the $p_0'$ in the first hexagonal cell by $p_0$, and did the same for $p_1'$ and $p_2'$ in the second and third hexagonal cell), resulting in a nonpolar state that locally resembles $R\bar{3}m$. Electric dipole moments as calculated by VASP are given in Table \[tab:dipole\].
Phase symmetry group $p_x$ $p_y$ $p_z$
-------------- ---------------- ---------- --------- -------
tilted polar $Cm$ -0.63437 0.36626 1.13
polar $R3m$ 0.00 0.00 1.11
nonpolar $R\bar{3}m$ 0.00 0.00 0.00
: Electric dipole moments of the 3 phases (units are $e$Å, where $e$ is the magnitude of the charge on an electron. The polar phase has p0’,p1’,p2’ carbons, resulting in dipole moment along +z direction. For $Cm$ phase, the projection onto xy-plane of dipole moment is along the projection of the vector $\bP0'$ (1.894,-1.093,-2.666) from the center of icosahedra to p0’.[]{data-label="tab:dipole"}
Conclusion
==========
We construct an artificial model inspired by boron carbide by placing an orientational degree of freedom at the vertices of a rhombohedral lattice, mimicking the distribution of carbon sites among polar vertices of B$_{11}$C$^p$ icosahedra. Because this model is restricted to 20% carbon it cannot capture the broad composition range of true boron carbide, but it can reveal orientational order and disorder similar to what might be seen in experiment. A pairwise interaction model counting bonds of specific type between polar carbons fits well to density functional theory total energies, and is transferable between supercells of differing sizes.
Monte Carlo simulations utilizing this model reveal three distinct phases separated by a pair of phase transitions. The high temperature phase has symmetry group $R\bar{3}m$, similar to what is observed experimentally in boron carbide. As temperature falls to 790K, inversion symmetry is lost via a continuous phase transition, resulting in a polarized state of symmetry $R3m$, which is a maximal subgroup of $R\bar{3}m$. The possible existence of such a state was independently suggested [@Ektarawong14], although at a much higher temperature. Finally, as temperature falls below 717K, 3-fold rotational symmetry is broken via a first order transition, resulting in a tilted polar phase of monoclinic symmetry $Cm$, which is a maximal subgroup of $R3m$. When fully orientationally ordered, this state matches the previously known ground state of B$_4$C [@Mauri01; @Widom12; @Bylander90].
The universality classes of each transition follow the expectations based on the type of symmetry breaking. The continuous 790K transition, which breaks inversion symmetry, is shown to fall in the Ising universality class because of the finite size scaling collapse of longitudinal susceptibility $\chi_z$ as shown in Fig. \[fig:Isingscale\]. The 717K transition, which breaks 3-fold rotation symmetry, is shown by the Lee-Kosterlitz criterion to be weakly first order, consistent with expectations for the 3-state Potts universality class.
Acknowledgements
================
We thank Robert H. Swendsen, David P. Landau and James P. Sethna for helpful discussions.
References
==========
[10]{} url \#1[`#1`]{}urlprefixhref \#1\#2[\#2]{} \#1[\#1]{}
Samsonov G.V., The present state of the investigation of the boron-carbon diagram, Zh. Fiz. Khim., Vol. 32, 1958, p 2424-2429 (in Russian).
Ekbom, Lars B., Amundin, Carl Olof, Microstructural evaluation of sintered boron carbides with different compositions, Science of Ceramics, 11, p 237-243.
Beauvy M., Stoichiometric limits of carbon-rich boron carbide phases, J. Less-Common Met., Vol. 90, 1983, p 169-175.
K. A. Schwetz, P. Karduck, Investigations of the boron-carbon system with the aid of electron probe microanalysis, J. Less Common Met. 175 (1991) 1–100.
H. Okamoto, - (boron-carbon), J. Phase Equil. 13 (1992) 436.
Domnich, V., Reynaud, S., Haber, R.A., Chhowalla, M. Boron carbide: Structure, properties, and stability under stress, Journal of the American Ceramic Society (2011), 94 (11), p 3605-3628.
Peter F. Rogl, Jan Vřešt’' al, Takaho Tanaka and Satoshi Takenouchi, The B-rich side of the B-C phase diagram, Calphad 44 (2014) 3-9.
H. C. Longuet-Higgins and M. de V. Roberts, The electronic structure of an icosahedron of boron atoms, Proc. R. Soc. Lond. A 1955 230, 110-119.
W. N. Lipscomb, Borides and boranes, J. Less-Common Metals 82 (1981) 1-20.
Musiri M. Balakrishnarajan,z Pattath D. Pancharatna and Roald Hoffmann, Structure and bonding in boron carbide: The invincibility of imperfections, New J. Chem. 31 (2007) 473-85.
G. Will, K. H. Kossobutzki, An x-ray structure analysis of boron carbide, b13c2, J. Less-Common Met. 44 (1976) 87.
G. Will, A. Kirfel, A. Gupta, E. Amberger, Electron density and bonding in b13c2, J. Less-Common Met. 67 (1979) 19-29.
G. H. Kwei, B. Morosin, Structures of the boron-rich boron carbides from neutron powder diffraction: Implications for the nature of the intericosahedral chains, J. Phys. Chem. 100 (1996) 8031-9.
H. K. Clark, J. L. Hoard, The crystal structure of boron carbide, J. Am. Chem. Soc. 65 (1943) 2115-9.
R. Schmechel, H. Werheit, Structural defects of some icosahedral boron-rich solids and their correlation with the electronic properties, J. Solid State Chem. 154 (2000) 61-7.
F. Mauri, N. Vast, C. J. Pickard, Atomic structure of icosahedral b4c boron carbide from a first principles analysis of nmr spectra, Phys. Rev. Lett. 87 (2001) 085506.
Yakel, H. L., The crystal structure of a boron-rich boron carbide, Acta Crystallogr. B 31, 1797 (1975).
Koun Shirai, Kyohei Sakuma and Naoki Uemura, Theoretical study of the structure of boron carbide B$_13$C$_2$, Phys. Rev. B 90 (2014).
M. Widom and W. Huhn, Prediction of Orientational Phase Transition in Boron Carbide, Solid State Sciences 14, 1648 (2012), ISSN 1293-2558.
D. M. Bylander, L. Kleinman, S. Lee, Self-consistent calculations of the energy bands and bonding properties of b12c3, Phys. Rev. B 42 (1990) 13941403.
W.P. Huhn and M. Widom, A free energy model of boron carbide, J. Stat. Phys. 150 (2012) 432-41.
El-Batanouny M, Wooten F. Symmetry and condensed matter physics: a computational approach \[M\]. Cambridge University Press, 2008.
Hahn, T, and Paufler, P. International tables for crystallography, Vol. A. space-group symmetry D. REIDEL Publ. 1984.
A. Ektarawong, S. I. Simak, L. Hultman, J. Birch, and B. Alling, First-principles study of configurational disorder in B4C using a superatom-special quasirandom structure method, Phys. Rev. B 90 (2014) 024204.
A. M. Ferrenberg and R. H. Swendsen, Phys. Rev. Lett. 61, 2635 (1988).
A. M. Ferrenberg and R. H. Swendsen, Phys. Rev. Lett. 63, 1195 (1989).
R. B. Potts, Some generalized order-disorder transformations, Mathematical Proceedings of the Cambridge Philosophical Society. Cambridge University Press, 1952, 48(01): 106-109.
J. M. Sanchez, F. Ducastelle and D. Gratias, Generalized cluster description of multicomponent systems, Physica 128A (1984) 334-50.
A. van de Walle, M. Asta and G. Ceder, The alloy theoretic automated toolkit: A user guide, Calphad 26 (2002) 539-53.
G. Kresse and J. Hafner. Ab initio molecular dynamics for liquid metals. Phys. Rev. B, 47:558, 1993.
G. Kresse and J. Hafner. Ab initio molecular-dynamics simulation of the liquid-metal-amorphous-semiconductor transition in germanium. Phys. Rev. B, 49:14251, 1994.
G. Kresse and J. Furtmuller. Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set. Comput. Mat. Sci., 6:15, 1996.
G. Kresse and J. Furthmuller. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys. Rev. B, 54:11169, 1996.
P. E. Blochl. Projector augmented-wave method. Phys. Rev. B, 50:17953, 1994.
G. Kresse and D. Joubert. From ultrasoft pseudopotentials to the projector augmented-wave method. Phys. Rev. B, 59:1758, 1999.
J. P. Perdew, K. Burke, and M. Ernzerhof. Generalized gradient approximation made simple. Phys. Rev. Lett., 77:3865, 1996.
J.P. Perdew, J.A. Chevary, S.H. Vosko, K.A. Jackson, M.R. Pederson, D.J. Singh, and C. Fiolhais. Erratum: Atoms, molecules, solids, and surfaces: Applications of the generalized gradient approximation for exchange and correlation. Phys. Rev. B, 48:4978, 1993.
Nelson L J, Hart G L W, Zhou F, et al. Compressive sensing as a paradigm for building physics models\[J\]. Physical Review B, 2013, 87(3): 035125.
Candès E J, Wakin M B. An introduction to compressive sampling\[J\]. Signal Processing Magazine, IEEE, 2008, 25(2): 21-30.
A. Pelissetto and E. Vicari, Critical phenomena and renormalization-group thoery, Physics Reports 368, 549 (2002).
D. Landau and K. Binder, A Guide to Monte Carlo Simulations in Statistical Physics, Cambridge University Press, New York, NY, USA, 2005, ISBN 0521842387.
F. Y. Wu, The Potts model, Rev. Mod. Phys. 54, 235 (1982).
J. Lee and J. M. Kosterlitz, New numerical method to study phase transitions, Phys. Rev. Lett. 65, 137 (1990).
{width="1.1\linewidth"}
|
---
author:
- |
and Hideo Nakajima$^\dagger$\
$^*$ School of Sci.& Engr., Teikyo Univ., Utsunomiya, 320-8551 Japan\
E-mail:\
$^\dagger$ Dept. of Infor. Sci., Utsunomiya Univ., Utsunomiya, 320-8585 Japan\
E-mail:
title: |
The running coupling in lattice Landau gauge\
with unquenched Wilson fermion and KS fermion
---
Introduction
============
The mechanism of dynamical chiral symmetry breaking and confinement is one of the most fundamental problem of hadron physics. The propagator of dynamical quarks in the infrared region provides information on dynamical chiral symmetry breaking and confinement. In the previous paper[@FN04], we measured gluon propagators and ghost propagators of unquenched gauge configurations obtained with quark actions of Wilson fermions (JLQCD/CP-PACS) and those of Kogut-Susskind(KS) fermions (MILC) [@MILC1; @MILC2] in Landau gauge and observed that the configurations of the KS fermion are closer to the chiral limit than those of Wilson fermions.
In the analysis of running coupling obtained from the gluon propagator and the ghost propagator, with use of the operator product expansion of the Green function, we observed possible contribution of the quark condensates and $A^2$ condensates in the configurations of the KS fermion[@FN04]. The quark propagator of quenched KS fermion was already measured in [@ABB], and possible contribution of these condensates are reported. Unquenched KS fermion propagator of $20^3\times 64$ lattice (MILC$_c$) was measured in [@bhlpwz], but to distinguish the gluon condensates and the quark condensates, it is desirable to measure the quark propagator of larger lattice (MILC$_f$) and to compare with data of MILC$_c$. We measure quark propagator of gauge configuration of 1) MILC$_c$ $20^3\times 64$, $\beta=6.76$ and 6.83 and 2) MILC$_f$ $28^3\times 96$, $\beta=7.09$ and 7.11, using the Staple+Naik action[@OT].
The running coupling in $\widetilde{MOM}$ scheme
================================================
The running coupling in $\widetilde{MOM}$ scheme is given with use of the vertex renormalization factor $\tilde Z_1$ as $$\label{alphadef}
\alpha_s(q)=\alpha_R(\mu^2)Z_R(q^2,\mu^2){G_R}(q^2,\mu^2)^2\nonumber\\
=\frac{\alpha_0(\Lambda_{UV})}{{\tilde Z_1(\beta,\mu)}^2} Z(q^2,\beta){G}(q^2,\beta)^2$$ where $Z$ and $G$ are the gluon and the ghost dressing function, respectively, and $\tilde Z_1$ is the vertex renormalization factor. \[alp709\_711\]
Our data suggests that the gluon propagator is infrared finite as in Dyson-Schwinger equation[@Blo1]. The running coupling in the infrared is suppressed as shown in Figure 2,but the main origin is the suppression of the ghost propagator in the infrared.
We parametrize the difference of the lattice data and the pQCD 4-loop result[@ChRe; @chet] in the 1GeV$<q<6$GeV region in the form, with a minor correction term $d$ as $$\Delta \alpha_s(q)=\alpha_{s,latt}(q)-\alpha_{s,pert}(q)=\frac{c_1}{q^2}+\frac{c_2}{q^4}+d,$$ where the $A^2$ condensates gives $c_1$ and the gluon condensates and/or quark condensates gives $c_2$. Although statistics is not large, running coupling of CP-PACS suggests $c_1\sim 2$GeV. The MILC data suggests $c_1\sim 4$GeV and $c_2\sim -2$GeV. There is an analytical calculation that suggests correlation between the $A_2$ condensates and the the horizon function parameter[@dssv].
The quark wave function renormalization
=======================================
We renormalize the quark field as $\psi_{bare}=\sqrt{Z_2}\psi_R$, and define the colorless vector current vertex[@Orsay1] $$\Gamma_\mu(q,p)=S^{-1}(q)G_\mu(q,p)S^{-1}(q+p)$$ where $$G_\mu(p,q)=\int d^4 x d^4 ye^{ip\cdot y+iq\cdot x}\langle q(y)\bar q(x)\gamma_\mu q(x)\bar q(0)\rangle$$ and $S(q)$ is the quark propagator.
The vertex of the vector current with $p=0$ is written as $$\Gamma_\mu(q)=\delta_{a,b}\{g_1(q^2)\gamma_\mu+ig_2(q^2)p_\mu+g_3(q^2)q_\mu{\ooalign{\hfil/\hfil\crcr$q$}}+ig_4(q^2)[\gamma_\mu,{\ooalign{\hfil/\hfil\crcr$q$}}]\}$$ The Ward identity implies $$Z_V^{\widetilde{MOM}}\Gamma_\mu (q)=-i\frac{\partial}{\partial q^\mu} S^{-1}(q)$$ where $Z_V^{\widetilde{MOM}}g_1(q^2)=Z_\psi(q^2)$ and $Z_V^{\widetilde{MOM}}=1$ when there is no lattice artefact.
The running coupling $g$ of the ghost, anti-ghost, gluon coupling $$g(q)=\tilde Z_1^{-1} Z_3^{1/2}(\mu^2,q^2) {\tilde Z}_3(\mu^2,q^2)$$ and that of quark, gluon coupling $$g(q)=Z_1^{-1} Z_3^{1/2}(\mu^2,q^2) Z_2(\mu^2,q^2)$$ are identical due to the Slavnov-Taylor identity. At the renormalization point $q=\mu$, we fix $Z_2(\mu^2,\mu^2)=1$ and $\tilde Z_3(\mu^2,\mu^2)=1$, and so $\tilde Z_1^{-1} Z_3^{1/2}{\tilde Z}_3=Z_1^{-1} Z_3^{1/2} Z_2$ implies $\tilde Z_1=Z_1$. On the other hand $Z_\psi(q^2)=Z_V^{\widetilde{MOM}}g_1(q^2)$ where for 16 tastes $$g_1(q^2)=\frac{1}{48N_c}Tr[\Gamma_\mu(q,p=0)(\gamma_\mu-q_\mu\frac{{\ooalign{\hfil/\hfil\crcr$q$}}}{q^2})]$$ When $Z_V^{\widetilde{MOM}}=1$, $g_1(\mu^2)$ is identical to $\tilde Z_1$ which is defined by the renormalization of the running coupling on the lattice defined at $\mu\sim 6$GeV and summarized in Table \[z1fac\].
$\beta_1/K_{sea 1}$ $\beta_2/K_{sea 2}$ average configurations
---------- --------------------- --------------------- --------- -----------------------------
CP-PACS 1.07(8) 1.21(10) 1.14 $K_{sea 1,2}=0.1357,0.1382$
MILC$_c$ 1.49(11) 1.43(10) 1.46 $\beta_{1,2}=6.83, 6.76$
MILC$_f$ 1.37(9) 1.41(12) 1.40 $\beta_{1,2}=7.11, 7.09$
: The $1/\tilde Z_1^2$ factor of the unquenched SU(3).[]{data-label="z1fac"}
\[z2\_MILC\]
The MILC configurations are produced by the Asqtad action with the bare masses 13.6MeV,
27.2MeV and 68.0MeV in the MILC$_f$, and 11.5MeV, 65.7MeV and 82.2MeV in the MILC$_c$. The gluon wave function renormalization $Z_3$ of the Asqtad action is renormalized by $1/u_0^2$, where $u_0$ is the fourth root of the plaquette value. The tadpole renormalization on quark wave function is $u_0$ and we normalize $S(q)$ by multiplying the average of $u_0\tilde Z_1$ i.e. 1/1.36 for MILC$_f$ and 1/1.38 for MILC$_c$. The quark propagators $Z_2(q)$ after this renormalization are infrared suppressed as shown in Figures 3 and 4. The apparent difference in the formulae of [@bhlpwz] and our work are only in the expression and in fact they are equivalent and the results agree with [@bhlpwz].
Using the pQCD result of the inverse quark propagator[@ChRe; @chet] and $\langle A^2\rangle$ and $\bar c_2$ (the contribution from the mixed condensate $\langle \bar q{\ooalign{\hfil/\hfil\crcr$A$}}q\rangle$[@LO]) as fitting parameters, we calculate $$Z_\psi(q)=\frac{1}{Z_2(q)}=Z_\psi^{pert}(q^2)+\frac{\left(\frac{\alpha(\mu)}{\alpha(q)}\right)^{{-\gamma_0+\gamma_{A^2}}\over{\beta_0}}}{q^2}\frac{\langle A^2(\mu)\rangle}{4(N_c^2-1)}{Z_\psi}^{pert}(\mu^2)+\frac{\bar c_2}{q^4}$$ where $\alpha(q)$ are data calculated in the $\widetilde{MOM}$ scheme using the same MILC$_f$ gauge configurations[@FN04]. We estimate that $\bar c_2$ is small. In Figure 5, we show a fit of the MILC$_f$ $m_0=13.6$MeV data, by taking the renormalization point at $\mu\sim 3.8$GeV with use of $\langle A^2(\mu)\rangle\sim 1.6(3)$GeV$^2$ and $\bar c_2=0$. These parameters are consistent with [@Orsay1].
In $q>1.5GeV$ region, dynamical mass of a quark in pQCD is expressed as[@ABB] $$\label{massf}
M(q)=-\frac{4\pi^2 d_M\langle \bar q q\rangle_\mu [\log (q^2/\Lambda_{QCD}^2)]^{d_M-1}}{3q^2 [\log (\mu^2/\Lambda_{QCD}^2)]^{d_M}}+\frac{m(\mu^2)[\log (\mu^2/\Lambda_{QCD}^2)]^{d_M}}{[\log (q^2/\Lambda_{QCD}^2)]^{d_M}},$$ where $d_M=\frac{12}{33-2N_f}$. The second term is the contribution of the massive quark. In the analysis of the lattice data, we observe that the quark condensates $-\langle \bar q q\rangle_\mu$ and $\Lambda_{QCD}$ roughly satisfy $-\langle \bar q q\rangle_\mu=(0.70\Lambda_{QCD})^3$[@Blo1], with $\Lambda_{QCD}=0.69$GeV.
For the global fit of $M(q)$, we try the phenomenological monopole type[@SkW] $$M(q)=\frac{c\Lambda^3}{q^2+\Lambda^2}+m_0$$ where $m_0$ is the bare quark mass.
Figures 6 and 7 show the mass function $M(0)$ of MILC$_f$ and MILC$_c$, respectively.
\[massfunc\]
We observe that the product $c\Lambda$ becomes larger as the bare quark mass becomes heavy and it depends on $\beta$ in the case of MILC$_f$ but not in the case of MILC$_c$. In the case of MILC$_f$ $m_0=13.6$MeV, the lowest three momentum points of $M(q)$ are systematically smaller than the other points. Ignoring these points we find $c\Lambda$ of $\beta=7.09$ is smaller than that of 7.11 and that of MILC$_c$ as shown in Figure \[chiralmass\]. In the chiral limit $m_0\to 0$, we obtain $M(0)=0.35\sim 0.37$GeV, which is larger than [@bhlpwz] and that of the Wilson fermion[@SkW] by about 20%.
Discussion and conclusion
=========================
We measured running coupling of unquenched Wilson fermion and KS fermion and the quark wave function renormalization factor and mass function of the KS fermion. The renormalization factor $Z_\psi$ obtained as 1/$Z_2$ is infrared finite. The Kugo-Ojima confinement criterion favours infrared vanishing of $Z_\psi$ and $g_1(q^2)$ approximated by the running coupling suggests this behavior, but it could be a lattice artefact. With infrared finite $Z_2$, infrared vanishing of $Z_{1\psi}$ is necessary for the confinement criterion to be satisfied.
We thank the MILC collaboration and JLQCD/
CP-PACS collaboration for supplying their gauge configurations in the data bases. We acknowledge discussion with Tony Williams, Christian Fischer and Patrick Bowman. H.N. is supported by the JSPS grant in aid of scientific research in priority area No.13135210. This work is supported by the KEK supercomputing project 05-128.
[999]{} S. Furui and H. Nakajima, [*Infrared features of the Kogut-Susskind fermion and the Wilson fermion in Lattice Landau Gauge QCD*]{}, [hep-lat/0503029]{}. C. Bernard et al., [*Quenched hadron spectroscopy with improved staggered quark action*]{}, [**58**]{}[(1998)]{}[014503]{}. K. Orginos, D. Toussaint and R.L. Sugar, [*Variants of fattening and flavor symmetry restoration*]{} [**60**]{}[(1999)]{}[054503]{}. E.R. Arriola, P.O. Bowman and W. Broniowski, [*Landau-gauge condensates from quark propagator on the lattice*]{},[hep-ph/0408309]{}. P.O. Bowman et al., [*Unquenched quark propagator in Landau gauge*]{},[**71**]{}[(2005)]{}[054507]{}, [hep-lat/0402032]{}. K. Orginos and D. Toussaint, [*Tests of Improved Kogut-Susskind Fermion Actions*]{},(Proc. Suppl.)[**73**]{}[(1999)]{}[909]{}. Ph. Boucaud, F. de Soto, A. Le Yaouanc, J. Micheli, H. Moutarde, O Pène and J. Rodríguez-Quintero, [*Artefacts and $\langle A^2\rangle$ power corrections: revisiting the MOM $Z_\psi(p^2)$ and $Z_V$*]{},[hep-lat/0504017]{}. M.J. Lavelle and M. Oleszczuk,[*Gauge-dependent condensates and the quark propagator*]{}, [**275**]{}[(1992)]{}[133]{}. J.C.R. Bloch, [*Multiplicative renormalizability and quark propagator*]{}, Few Body Syst [**33**]{}[(2003)]{}[111]{}. K.G. Chetyrkin and A. Rétay,[*Three-loop Three-Linear Vertices and Four-Loop $\widetilde{MOM}$ $\beta$ functions in massless QCD*]{}, [hep-ph/0007088]{}. K.G. Chetyrkin, [*Four-loop renormalization of QCD*]{}, [hep-ph/0405193]{} v3(2005). D. Dudal, R.F. Soreiro, S.P. Sorella and H. Verschelde, [*The Gribov parameter and the dimension two gluon condensates in Eucledian Yang-Mills theories in the Landau gauge*]{}, [hep-th/0502183]{}. J.I. Skullerud and A.G. Williams, [*Quark propagator in the Landau gauge*]{},[**63**]{}[(2001)]{}[054508]{}.
|
---
abstract: 'We develop a new method in order to classify the Bianchi I spacetimes which admit conformal Killing vectors (CKV). The method is based on two propositions which relate the CKVs of 1+(n-1) decomposable Riemannian spaces with the CKVs of the (n-1) subspace and show that if 1+(n-1) space is conformally flat then the (n-1) spacetime is maximally symmetric. The method is used to study the conformal algebra of the Kasner spacetime and other less known Bianchi type I matter solutions of General Relativity.'
author:
- Michael Tsamparlis
- Andronikos Paliathanasis
- 'Leonidas Karpathopoulos.'
date: 'Received: date / Accepted: date'
title: Exact solutions of Bianchi I spacetimes which admit Conformal Killing vectors
---
Introduction
============
The Bianchi models are spatially homogeneous spacetimes which admit a group of motions $G_{3}$ [@kramer-stephani; @wald] acting on spacelike hypersurfaces. These spacetimes include the non-isotropic generalizations of the Friedman-Robertson-Walker (FRW) space-time and have been used in the discussion of anisotropies in a primordial universe and its evolution towards the observed isotropy of the present epoch [jacobs,narlikar-hoyle-misner]{}
The simplest type of these spacetimes are the Bianchi I models for which $G_{3}$ is the abelian group of translations of the three dimensional Euclidian space $E^{3}$. In synchronous coordinates the metric of Bianchi I spacetimes is: $$ds^{2}=-dt^{2}+A^{2}\left( t\right) dx^{2}+B^{2}\left( t\right)
dy^{2}+C^{2}\left( t\right) dz^{2} \label{sx1.1}$$where $A(t),B(t),C(t)$ are functions of the time coordinate only and the corresponding KVs are $\left\{ \partial _{x},\partial _{y},\partial
_{y}\right\} $. When two of the metric functions are equal, e.g. $%
A^{2}\left( t\right) =B^{2}\left( t\right) $, a Bianchi I spacetime ([sx1.1]{}) reduces to the important class of Locally Rotational Symmetric (LRS) spacetimes [@kramer-stephani].
A Conformal Killing Vector (CKV) $X^{a}$ is defined by the requirement $%
\mathcal{L}_{X}g_{ab}=2\psi g_{ab}$ and reduces to a Killing vector (KV) ($%
\psi =0$), to a Homothetic Killing Vector (HV) ($\psi _{;a}=0$), and to a Special Conformal Killing Vector (SCKV) ($\psi _{;ab}=0$). The effects of these vectors can be seen at all levels of General Relativity, that is, geometry, kinematics and dynamics. At the geometry level the knowledge of a CKV makes possible the choice of coordinates so that the metric is simplified, in the sense that one of the metric components is singled out [@Petrov; @Tsamp1]. At the level of kinematics the CKVs impose restrictions on the kinematic variables (rotation, expansion and shear) and produce well known results (see for example [maartens-mason-tsamparlis,mason-maartens,maartens-maharaj-tupper,coley-tupper1]{}). Finally at the level of dynamics the CKVs can (and have) been used in various directions, for example to obtain new solutions of the field equations with (hopefully) better physical properties (see for example [coley-tupper1,coley-tupper2,coley-tupper3,coley-tupper4,herrera-leon]{}). It becomes evident that it is important that we know the conformal algebra of a given spacetime.
In [@Apost-Tsamp1] all LRS spacetimes which admit CKVs have been determined. In the following we determine all Bianchi I spacetimes which are not reducible to LRS spacetimes and admit CKVs.
The general Bianchi I spacetime (\[sx1.1\]) does not admit CKVs. However, as we will show, there are two families of Bianchi I spacetimes which admit CKVs. One family consists of the conformally flat Bianchi I spacetimes, which admit 15 CKVs and are conformally related[^1] to Rebouças and Tiommo (RT) and Rebouças and Teixeira (ART) [@RT; @ART]) spacetimes. The second family contains the not conformally flat Bianchi I spacetimes, which admit only one proper CKV. In the determination of the CKVs we use the Bilyanov - Defrise - Carter theorem which relates the conformal algebra of conformally related metrics (for details see [@Defrise-Carter] [@Hall-Steele]).
In the literature one finds very few cases of Bianchi I spacetimes which admit proper CKVs. For example even the CKV found by Maartens and Mellin [@maartens-mellin] is really a CKV in an LRS spacetime and not in a Bianchi I spacetime [@Apost-Tsamp1]. The difficulty lies in the fact that the direct solution of the conformal equations in Bianchi I spacetimes is a major task. Thus an alternative simpler method is needed to solve this problem and this is what it is developed in the following sections. It is to be noted that using the Petrov classification and the Bilyanov - Defrise - Carter theorem McIntosh and Steele [@McSteele] have determined all vacuum Bianchi I spacetimes which admit a homothety.
One extra advantage of the proposed method is that one can use it to prove/test if a given Bianchi I spacetime admits a CKV or not. For example as it will be shown the two well known anisotropic Bianchi I solutions that is, the Kasner solution [@kramer-stephani] and the anisotropic dust solution[@ellis-hawking], which have formed the basis of many studies of anisotropic universes, do not admit a proper CKV; in particular the Kasner spacetime admits a HV.
The structure of the paper is as follows. In section \[preliminaries\] we present two propositions required for the computation of the CKVs in Bianchi I spacetimes. In sections \[classAC\] and \[classBC\] we apply the results of section \[preliminaries\] and we determine all Bianchi I spacetimes which admit CKVs. In section \[Exact\] we consider the application of these results in various Bianchi I metrics found in the literature. Finally in section \[Dis\] we discuss our results.
Preliminaries
=============
As it has been remarked in the last section the computation of CKVs of Bianchi I spacetimes by direct solution of the conformal equations is a difficult task. Thus we have developed an indirect method which is based on the two Propositions discussed below. The first is proposition \[prop21\] which has been given in [@Tsamp-Nikol-Apost] for spacetimes ($n=4$) and below is generalized[^2] to $n$- dimensional Riemannian spaces as follows.
\[prop21\] A decomposable $1+\left( n-1\right)$ ($n\ge 3$ ) Riemannian space $g_{ab}$ with line element (Greek indices take the values 1,...,n and Latin indices the values 0,...,n) $$ds^{2}=\varepsilon dt^{2}+h_{\mu \nu }\left( x^{\sigma }\right) dx^{\mu
}dx^{\nu } \label{sx2.1}$$admits a proper CKV $X^{a}$ if and only if the $\left( n-1\right) $ space $%
h_{\mu \nu }\left( x^{\sigma }\right) $ admits a gradient proper CKV $\xi
^{\mu }$. In particular the two vector fields are related as follows$$X^{a}=-\frac{\varepsilon }{p}\dot{\lambda}\left( t\right) \psi \left(
x^{\sigma }\right) \partial _{t}+\frac{1}{p}\lambda \left( t\right) \xi
^{\mu }\left( x^{\sigma }\right) +H^{\mu }\left( x^{\sigma }\right)
\label{sx2.0a}$$where:
- $p$ is a non vanishing constant,
- $\psi \left( x^{\sigma }\right) $ is the conformal factor of the CKV $\xi
^{\mu }$ and satisfies the condition$$\psi _{;\mu \nu }=p\psi h_{\mu \nu } \label{sx2.0b}$$that is, $\psi _{;\mu}$ is a gradient CKV of the $n-1$ space
- $\lambda \left( t\right) $ satisfies the linear second order equation $$\ddot{\lambda}\left( t\right) +\varepsilon p\lambda \left( t\right) =0~
\label{sx.02c}$$where $\dot{\lambda}=\frac{d\lambda }{dt}$.
- $H^{\mu }$ is a KV or a HV of the $n-1$ metric $h_{\mu \nu }\left(
x^{\sigma }\right) .$
From proposition \[prop21\] follows that a proper CKV of the $n-1$ metric $%
h_{\mu \nu }$ generates two proper CKVs for the $n$ metric $g_{ab}$.
The crucial result of proposition \[prop21\] is that the gradient CKVs of the $(n-1)$ space are of the specific form (\[sx2.0b\]). Furthermore if the $(n-1)$ space has constant *non vanishing* Ricciscalar $R$, then the constant $p$ is given by the expression $$p=\frac{R}{(n-1)(n-2)}\qquad (n\geq 3). \label{Sx.1.0}$$
The second Proposition \[prop22\] concerns the $1+\left( n-1\right) $ decomposable spacetimes which admit CKVs[^3].
\[prop22\] The metric (\[sx2.1\]) is conformally flat if and only if the $(n-1)$ metric $h_{\mu \nu }\left( x^{\sigma }\right)
$ is the metric of a space of constant curvature ($n\geq 3$).
In addition to these propositions we recall the following result (see [Tsamp-Nikol-Apost]{})
*The metric of a space of constant non-vanishing curvature *of dimension* $n$ admits* $n+1$* gradient CKVs*.
From proposition \[prop22\] it follows that as far as the admittance of CKVs is concerned, the connected $1+(n-1)$ decomposable spaces are classified in two major classes.
i)
: Class A: The $1+(n-1)$ space is conformally flat. Then the $(n-1)$ space is not conformally flat and the $1+(n-1)~$space admits $\frac{%
(n+1)(n+2)}{2}$ CKVs
ii)
: Class B: The $1+(n-1)$ space is not conformally flat. Then the $%
(n-1)$ space is not a space of constant curvature.
For a space conformally related to a $1+(n-1)$ decomposable space this classification of CKVs remains the same since all conformally related spaces admit the same conformal algebra.
We conclude that the parameter in the classification of the connected $1+(n-1)$ spacetimes which admit CKVs is the constancy or not of the curvature scalar of the $(n-1)$ space. Using this observation and propositions \[prop21\] and \[prop22\] we are able to determine all Bianchi I spacetimes which admit CKVs.
CKVs of Bianchi I spacetimes {#CKVsB1}
----------------------------
In the generic line element (\[sx1.1\]) of Bianchi I spacetime, we consider the coordinate transformation $dt=C\left( \tau \right) d\tau $ and get$$ds^{2}=C^{2}\left( \tau \right) \left( dz^{2}+ds_{(3)}^{2}\right)
\label{sx4.1}$$where $ds_{\left( 3\right) }^{2}$ is the three dimensional metric $$ds_{\left( 3\right) }^{2}=-d\tau ^{2}+B_{1}^{2}(\tau )dy^{2}+A_{1}^{2}(\tau
)dx^{2} \label{sx4.2}$$with $A_{1}^{2}\left( \tau \right) =\frac{A^{2}\left( \tau \right) }{%
C^{2}\left( \tau \right) }$, $B_{1}^{2}\left( \tau \right) =\frac{%
B^{2}\left( \tau \right) }{C^{2}\left( \tau \right) }$. Applying a second transformation $d\tau =B_{1}^{2}\left( \bar{\tau}\right) d\bar{\tau}$ and $%
\Gamma ^{2}\left( \bar{\tau}\right) =\frac{A_{1}^{2}\left( \bar{\tau}\right)
}{B_{1}^{2}\left( \bar{\tau}\right) }$ the three dimensional metric ([sx4.2]{}) becomes$$ds_{\left( 3\right) }^{2}=B_{1}^{2}(\bar{\tau})ds_{1+2}^{2} \label{sx4.3}$$where$$ds_{1+2}^{2}=dy^{2}+ds_{\left( 2\right) }^{2} \label{sx4.4}$$and $$ds_{\left( 2\right) }^{2}=-d\bar{\tau}^{2}+\Gamma ^{2}(\bar{\tau})dx^{2}.
\label{sx4.5}$$The two dimensional metric (\[sx4.5\]) is conformally flat[^4]. Indeed if we introduce the new variable $d\bar{\tau}=\Gamma \left( \hat{\tau}\right) d\hat{\tau}$ the metric $ds_{\left( 2\right) }^{2}$ becomes$$ds_{\left( 2\right) }^{2}=\Gamma ^{2}(\hat{\tau})(-d\hat{\tau}^{2}+dx^{2}).
\label{sx4.6}$$
The $2d$ metric $\eta _{AB}=diag\left( -1,1\right) $ admits the three KVs
$$\mathbf{P}_{\hat{\tau}}=\partial _{\hat{\tau}}\qquad \mathbf{P}_{x}=\partial
_{x}\qquad \mathbf{r}=x\partial _{\hat{\tau}}+\hat{\tau}\partial _{x}
\label{sx4.10}$$
and the gradient HV $$\mathbf{H}=\hat{\tau}\partial _{\hat{\tau}}+x\partial _{x}~,~\psi _{H}=1.
\label{sx4.11}$$ The curvature scalar $R_{(2)}$ of the 2d-metric (\[sx4.5\]) is calculated to be: $$R_{\left( 2\right) }=2\frac{\Gamma _{,\bar{\tau}\bar{\tau}}}{\Gamma }.
\label{sx4.14}$$
According to proposition \[prop22\] the condition that the $(1+2)$d - metric (\[sx4.4\]) - and consequently the 3d-metric $ds_{3}^{2}$ - is conformally flat, is that the 2d-metric (\[sx4.6\]) is a the metric of a space of constant curvature. We set $R_{(2) }=const.=2c$ and find that that this is the case when $\Gamma _{,\bar{\tau}\bar{\tau}}=c\Gamma $.
On the other hand when $ds_{3}^{2}$ is of constant curvature then by means of the inverse of proposition \[prop22\] the metric $ds_{1+3}^{2}$ is conformally flat hence the metric $ds^{2}$ is also conformally flat.
We conclude that the classification of Bianchi I spacetimes which admit CKVs is done in two classes:
*Class A* : Contains all Bianchi I spacetimes which are conformally flat. According to proposition \[prop22\] in this case the 3d-metric $%
ds_{\left( 3\right) }^{2}$ is of constant curvature and the form of the metric functions $A_{1}(\tau )$, $B_{1}(\tau )$ is fixed.
*Class B* : Contains all Bianchi I spacetimes which are not conformally flat therefore the decomposable metric is not conformally flat. According to the inverse of proposition \[prop22\] in this class the 3-d metric $%
ds_{\left( 3\right) }^{2}$ is not the metric of a space of constant curvature.
In Class B there are two cases to be considered.Case B1: The 3d-metric $ds_{\left( 3\right) }^{2}$ is not conformally flat in which case the scalar curvature of the 2d-metric $R_{(2)}\neq const.$Case B2: The 3d-metric $ds_{\left( 3\right) }^{2}$ is conformally flat hence according to proposition \[prop22\] the 2d-metric $ds_{\left( 2\right)
}^{2}$ is of constant curvature i.e. $R_{(2)}=const.$
In the following we consider each Class and derive the corresponding Bianchi I spacetimes together with the CKV(s). We ignore the cases $%
A_{1}=B_{1}\Leftrightarrow \Gamma =const.$ which lead to LRS spacetimes whose CKVs have already been found in [@Apost-Tsamp1].
Class A: The conformally flat Bianchi I spacetimes {#classAC}
==================================================
Demanding that the Weyl tensor of the metric (\[sx4.1\]) vanishes we find the following conditions on the metric functions $A_{1},B_{1}$:
$$\begin{aligned}
A_{1}\ddot{B}_{1}+B_{1}\ddot{A}_{1}-2\dot{A}_{1}\dot{B}_{1} &=&0
\label{sx4.14a1} \\
A_{1}\ddot{B}_{1}-2B_{1}\ddot{A}_{1}+\dot{A}_{1}\dot{B}_{1} &=&0
\label{sx4.14a2} \\
\ddot{A}_{1}B_{1}-2A_{1}\ddot{B}_{1}+\dot{A}_{1}\dot{B}_{1} &=&0
\label{sx4.14a3}\end{aligned}$$
where a dot over a symbol denotes differentiation with respect to coordinate $\tau $. We note that only two of these three equations are independent.
Using (\[sx4.14a1\])-(\[sx4.14a3\]) we can prove that the 3-metric ([sx4.2]{}) is the metric of a 3-space of constant curvature $%
R_{(3)}=6\varepsilon a^{2}$ where $\varepsilon =\pm 1$ and $a\ne 0$ is a constant. There are only two such spacetimes the RT spacetime [@RT] and the ART spacetime [@ART] mentioned above.
The RT and the ART spacetimes in isochronous coordinates have the line element$$ds_{RT}^{2}=-dt^{2}+\sin ^{2}(t/a)dx^{2}+\cos ^{2}(t/a)dy^{2}+dz^{2}
\label{sx3.3}$$$$ds_{ART}^{2}=-dt^{2}+\sinh ^{2}(t/a)dx^{2}+\cosh ^{2}(t/a)dy^{2}+dz^{2}
\label{sx3.4}$$respectively. These spacetimes are 1+3 decomposable spaces whose three dimensional space is a space of constant curvature. They admit a 15 dimensional conformal algebra with a seven dimensional Killing subalgebra, which has been given in [@Apost-Tsamp1]. For the completeness of the paper in appendix \[appendixB\] we give the conformal algebra of the RT and the ART spacetimes in a convenient form.
Class B: The non-conformally flat Bianchi I spacetimes {#classBC}
======================================================
In this class there are two subcases to be considered depending on $%
R_{\left( 2\right) }=const$ and $R_{\left( 2\right) }\neq const$ where $%
R_{\left( 2\right) }$ is the Ricciscalar of the two dimensional space ([sx4.6]{}).
Case B.I: $R_{\left( 2\right) }\neq const.$
-------------------------------------------
In this case we are interested only for the KVs and the HV of $ds_{\left(
2\right) }^{2}$ since if there exist a proper CKV which satisfy condition (\[sx2.0b\]) of Proposition \[prop21\], then the two dimensiona space is of constant curvature. From the CKVs of $d\hat{s}_{\left( 2\right) }^{2}=(-d%
\hat{\tau}^{2}+dx^{2})$ only the ones which do not contain terms $f(\hat{\tau%
})g(x)\partial _{\hat{\tau}}$ with $f(\hat{\tau})\neq \hat{\tau}$ can satisfy this property. It is well known that the two dimensional space $d%
\hat{s}_{\left( 2\right) }^{2}$ admits infinity CKVs. However, the vector fields which do not contain the terms $f(\hat{\tau})g(x)\partial _{\hat{\tau}%
}$ with $f(\hat{\tau})\neq \hat{\tau}$ are the two vector fields $\mathbf{P}%
_{\hat{\tau}}$ and the $\mathbf{H}$.
The conformal factor of $\mathbf{P}_{\hat{\tau}}$ of the metric $ds_{\left(
2\right) }^{2}$ is: $$\psi (\mathbf{P}_{\hat{\tau}})=\Gamma _{,\bar{\tau}}.$$
If we demand $\psi (\mathbf{P}_{\hat{\tau}})=0$ (the case of KVs) then we get $A_{1}^{2}=B_{1}^{2}$, i.e.$~$the LRS case which we ignore. If we demand $\psi (\mathbf{P}_{\hat{\tau}})=const$ then we find $\Gamma _{,\bar{\tau}%
\bar{\tau}}=0$ which implies by (\[sx4.14\]) that $R_{\left( 2\right) }=0$ i.e constant which contradicts our assumption. Therefore $\mathbf{P}_{\hat{%
\tau}}$ produces nothing relevant.
The HV $\mathbf{H~}$ has conformal factor $$\psi (\mathbf{H})=\Gamma _{,\bar{\tau}}\int \frac{d\bar{\tau}}{\Gamma }+1.
\label{sx4.15}$$The requirement that $\mathbf{H}$ is a KV of the 2-metric $ds_{\left(
2\right) }^{2}$ gives $\hat{\tau}\Gamma _{,\hat{\tau}}+\Gamma =0,$ hence $%
\Gamma \left( \hat{\tau}\right) =\frac{\Gamma _{0}}{\hat{\tau}}$ which implies $R_{\left( 2\right) }=const.$ and it is excluded. The requirement that $\mathbf{H}$ is a HV with conformal factor $\alpha _{2}(\neq 0)$ gives: $$\Gamma =c_{1}\hat{\tau}^{\alpha _{2}-1} \label{sx4.16}$$where $c_{1}=const$. This HV is acceptable provided that $\alpha _{2}\neq 1$ in order to avoid the LRS case. By proposition \[prop21\] this gives the following HV for the 1+2 metric (\[sx4.4\]): $$\mathbf{H}_{1}=\alpha _{2}y\partial _{y}+\hat{\tau}\partial _{\hat{\tau}%
}+x\partial _{x} \label{sx4.17}$$with conformal factor $$\psi (\mathbf{H}_{1})=\alpha _{2}. \label{sx4.18}$$
This vector is a non-gradient CKV for the metric (\[sx4.3\]) with conformal factor: $$\bar{\psi}(\mathbf{H}_{1})=\hat{\tau}(\ln A_{1})_{,\hat{\tau}}+\alpha _{2}.
\label{sx4.19}$$We are interested in KVs and HVs (we show in the Appendix that the gradient CKVs of the form $\lambda (\mathbf{\xi })_{|\alpha \beta }=p\lambda (\mathbf{%
\xi })g_{\alpha \beta }$ imply that the 3-metric (\[sx4.3\]) is of constant curvature) thus we examine possible reductions of this CKV to a KV or a HV.
If $\mathbf{H}_{1}$ is a KV then $\bar{\psi}(\mathbf{H}_{1})=0$ and this gives $A_{1}=c_{2}\hat{\tau}^{-\alpha _{2}}$. From (\[sx4.6\]) and ([sx4.16]{}) we obtain $B_{1}=\frac{c_{1}c_{2}}{\hat{\tau}}$ which implies $%
\hat{\tau}=c_{3}e^{\tau /c}$ where $c=c_{1}c_{2}$. Thus we have the following KV : $$X_{B_{1}}=\alpha _{2}y\partial _{y}+c\partial _{\tau }+x\partial _{x}
\label{sx4.20}$$for the three dimensional metric: $$ds_{\left( 3\right) }^{2}=-d\tau ^{2}+c_{2}^{2}c_{3}^{-2\alpha
_{2}}e^{-2\alpha _{2}\tau /c}dy^{2}+\left( \frac{c}{c_{3}}\right)
^{2}e^{-2\tau /c}dx^{2}. \label{sx4.21}$$Due to proposition \[prop21\] this is also a KV for the metric $%
ds_{1+3}^{2}=dz^{2}+ds_{(3) }^{2}$ hence a proper CKV for the metric ([sx4.1]{}) with conformal factor (note that $\partial _{\tau }=A_{1}\partial
_{t}$) $$\psi (X_{B_{1}})=c(C\left( t\right) )_{,t}. \label{sx4.22}$$The metric $ds^{2}$ is given in (\[sx4.1\]) and describes a family of Bianchi I metrics parameterized by the function $C(t)$.
When $\mathbf{H}_{1}$ is a HV from equation (\[sx4.19\]) we obtain ($%
\alpha _{3}=const.$): $$\hat{\tau}(\ln A_{1})_{,\hat{\tau}}+\alpha _{2}=\alpha _{3}\Leftrightarrow
A_{1}=c_{2}\hat{\tau}^{\alpha _{3}-\alpha _{2}} \label{sx4.23}$$and $$B_{1}=c_{1}c_{2}\hat{\tau}^{\alpha _{3}-1} \label{sx4.24}$$therefore we have that $$\hat{\tau}=\left( \frac{\alpha _3}{c_1c_2}\right) ^{1/\alpha _3}\tau
^{1/\alpha _3}. \label{sx4.25}$$ Eventually we have the CKV: $$X_{B_{1}}=\alpha _{2}y\partial _{y}+\alpha _{3}\tau \partial _{\tau
}+x\partial _{x}+\alpha _{3}z\partial _{z} \label{sx4.26}$$for the Bianchi I metric: $$ds^{2}=C^{2}(\tau )\left[ dz^{2}-d\tau ^{2}+c_{2}^{2}\left( \frac{\alpha _{3}%
}{c_{1}c_{2}}\right) ^{2\frac{(\alpha _{3}-\alpha _{2})}{\alpha _{3}}}\tau
^{2\frac{(\alpha _{3}-\alpha _{2})}{\alpha _{3}}}dy^{2}+c_{1}^{2}c_{2}^{2}%
\left( \frac{\alpha _{3}}{c_{1}c_{2}}\right) ^{2\frac{(\alpha _{3}-1)}{%
\alpha _{3}}}\tau ^{2\frac{(\alpha _{3}-1)}{\alpha _{3}}}dx^{2}\right]
\label{sx4.27}$$ with conformal factor: $$\psi (X_{B_{2}})=\alpha _{3}\left[ 1+\tau (\ln \left\vert C\right\vert
)_{,\tau }\right] . \label{sx4.28}$$
Case B.II: $R_{\left( 2\right) }=const$
---------------------------------------
We consider the subcases: $R_{\left( 2\right) }=0$, and $R_{\left( 2\right)
}\neq 0$.
When $R_{\left( 2\right) }=0$ from (\[sx4.14\]) we have that $$\Gamma =b_{0}\bar{\tau}\Leftrightarrow B_{1}=b_{0}\bar{\tau}A_{1}.
\label{sx4.29}$$Equation (\[sx4.29\]) implies that the 3-metric (\[sx4.4\]) has the form (we ignore the unimportant integration constant $b_{0}$): $$ds_{1+2}^{2}=dy^{2}-d\bar{\tau}^{2}+\bar{\tau}^{2}dx^{2}. \label{sx4.30}$$The CKVs of the flat 3-metric $ds^{2}=-d\tilde{t}^{2}+d\tilde{x}^{2}+d\tilde{%
y}^{2}$ are known [@choquet-bruhat]. Using the transformation $\tilde{t}=%
\bar{\tau}\cosh x,\tilde{x}=\bar{\tau}\sinh x,\tilde{y}=y$ we obtain the 3-metric (\[sx4.30\]) from which we obtain the following conformal algebra (we ignore the KVs $\partial _{x},\partial _{y};$ $i=1,2,3,4;$ $\alpha
=1,2,3 $.):
- Four KVs$$\mathbf{X}_{1}=\cosh x\partial _{\bar{\tau}}-\frac{\sinh x}{\bar{\tau}}%
\partial _{x}~~$$$$\mathbf{X}_{2}=\sinh x\partial _{\bar{\tau}}-\frac{\cosh x}{\bar{\tau}}%
\partial _{x}$$$$\mathbf{X}_{3}=y\sinh x\partial _{\bar{\tau}}-y\frac{\cosh x}{\bar{\tau}}%
\partial _{x}+\bar{\tau}\sinh x\partial _{y}$$$$\mathbf{X}_{4}=y\cosh x\partial _{\bar{\tau}}-y\frac{\sinh x}{\bar{\tau}}%
\partial _{x}+\bar{\tau}\cosh x\partial _{y}$$
- one gradient HV$$\mathbf{X}_{7}=\bar{\tau}\partial _{\bar{\tau}}+y\partial _{y}~,~\psi (%
\mathbf{X}_{7})=1$$- three special CKVs$$\mathbf{X}_{8}=(y^{2}+\bar{\tau}^{2})\cosh x\partial _{\bar{\tau}}+\frac{%
\bar{\tau}^{2}-y^{2}}{\bar{\tau}}\sinh x\partial _{x}+2y\bar{\tau}\cosh
x\partial _{y}$$$$\mathbf{X}_{9}=(y^{2}+\bar{\tau}^{2})\sinh x\partial _{\bar{\tau}}+\frac{%
\bar{\tau}^{2}-y^{2}}{\bar{\tau}}\cosh x\partial _{x}+2y\bar{\tau}\sinh
x\partial _{y}$$$$\mathbf{X}_{10}=2\bar{\tau}y\partial _{\bar{\tau}}+(y^{2}+\bar{\tau}%
^{2})\partial _{y}$$
with corresponding conformal factors: $$\psi \left( \mathbf{X}_{8}\right) =2\bar{\tau}\cosh x~,~\psi \left( \mathbf{X%
}_{9}\right) =2\bar{\tau}\sinh x~,~\psi \left( \mathbf{X}_{10}\right) =2y
\label{sx4.34}$$
These vectors are also CKVs for the metric (\[sx4.3\]) but with conformal factors: $$\psi ^{\prime }(\mathbf{X}_{A})=\mathbf{X}_{A}(\ln A_{1})+\psi (\mathbf{X}%
_{A}) \label{sx4.35}$$where $A=1,2,...,10$. The possible vectors $\mathbf{X}_{A}$ which give $\psi
^{\prime }(\mathbf{X}_{A})=const.$ are the KVs and the HV which do not contain terms of $f(\bar{\tau})g(x)\partial _{\bar{\tau}}$. The only such vector is the HV $\mathbf{X}_{7}$.
The case that $\mathbf{X}_{7}$ is a KV for the metric (\[sx4.3\]) gives $%
B_{1}=const.$ and we ignore it. We set $\psi ^{\prime }(\mathbf{X}%
_{A})=\alpha _{4}$ and we obtain, after standard calculations, that the vector $\mathbf{X}_{7}=\alpha _{4}\tau \partial _{\tau }+y\partial _{y}$ is a HV for the 3-metric: $$ds^{2}=-d\tau ^{2}+b_{1}^{2}\left( \frac{\alpha _{4}}{b_{1}}\right) ^{2\frac{%
\alpha _{4}-1}{\alpha _{4}}}\tau ^{2\frac{\alpha _{4}-1}{\alpha _{4}}%
}dy^{2}+\alpha _{4}^{2}\tau ^{2}dx^{2} \label{sx4.36}$$with conformal factor $\alpha _{4}$. This vector is extended to a HV for the 1+3 metric $ds_{1+3}^{2}=dz^{2}+ds_{(3)}^{2}$ which is of the form: $$X_{B_{3}}=\alpha _{4}\tau \partial _{\tau }+y\partial _{y}+\alpha
_{4}z\partial _{z}. \label{sx4.37}$$The Bianchi I metric (\[sx4.36\]) and the CKV (\[sx4.37\]) are obtained from the metric (\[sx4.27\]) and the CKV (\[sx4.26\]) if we set $a_{1}=0$ and interchange the coordinates $x,y.$ Therefore it is not a new case. A detailed study of the subcase $R_{\left( 2\right) }\neq 0$ shows that there are no more new Bianchi I metrics which admit CKVs. The calculations are rather standard and similar to the ones above and are omitted.
We conclude that there are two families of metrics in B.II class parameterized by the function $C(\tau )$. Each family admits one proper CKV and have as follows:
Metrics B$_{1}$ with $(\alpha _{1}\neq 0,1~,~c\neq 0)~$
$$ds^{2}=C^{2}(\tau )\left[ -d\tau ^{2}+e^{-\frac{2}{c}\tau }dx^{2}+e^{-\frac{%
2\alpha _{1}}{c}\tau }dy^{2}+dz^{2}\right] \label{sx3.21}$$
and corresponding CKV
$$X_{B_{1}}=c\partial _{\tau }+x\partial _{x}+\alpha _{1}y\partial _{y}
\label{sx3.22}$$
$$\psi (X_{B_{1}})=c\left( \ln \left\vert C\right\vert \right) _{,\tau }
\label{sx3.23}$$
Metrics B$_{2}$ with $(\alpha _{2}\neq 0,1)$ and $(\alpha _{1}\neq \alpha
_{2})$
$$ds^{2}=C^{2}(\tau )\left[ -d\tau ^{2}+\tau ^{2\frac{\alpha _{2}-1}{\alpha
_{2}}}dx^{2}+\tau ^{2\frac{\alpha _{2}-\alpha _{1}}{\alpha _{2}}%
}dy^{2}+dz^{2}\right] \label{sx3.24}$$
and corresponding CKV $$X_{B_{2}}=\alpha _{2}\tau \partial _{\tau }+\alpha _{1}y\partial
_{y}+x\partial _{x}+\alpha _{2}z\partial _{z} \label{sx3.25}$$ with conformal factor $$\psi (X_{B_{2}})=\alpha _{2}\left[ 1+\tau (\ln \left\vert C\right\vert
)_{,\tau }\right]. \label{sx3.26}$$
We observe that the CKV $X_{B_{1}}$ of the metric B$_{1}$ becomes a HV when $%
\left( \ln \left\vert C\right\vert \right) _{,\tau }=\psi _{0}$, i.e. $%
C\left( \tau \right) =e^{\psi _{0}\tau }$. In that case the metric ([sx3.21]{}) becomes ($e^{\psi _{0}\tau }=t$) $$ds^{2}=-dt^{2}+t^{-\frac{2}{c\psi _{0}}}dx^{2}+t^{-\frac{2a_{1}}{c\psi _{0}}%
}dy^{2}+t^{2}dz \label{sx3.27}$$where we substitute $e^{\psi _{0}\tau }=t.$ Furthermore the metric $B_{2}$ admits a HV when $C\left( \tau \right) =\tau ^{\psi _{0}-1}$. In that case the line element (\[sx3.24\]) becomes$$ds^{2}=-dt^{2}+t^{2\frac{\alpha _{2}-1}{\psi _{0}\alpha _{2}}}dx^{2}+t^{2%
\frac{\alpha _{2}-\alpha _{1}}{\psi _{0}\alpha _{2}}}dy^{2}+t^{2\frac{\left(
\psi _{0}-1\right) }{\psi _{0}}}dz^{2} \label{sx3.28}$$
Therefore from the spacetimes (\[sx3.27\]) and (\[sx3.28\]) we have that the Bianchi I spacetimes (\[sx1.1\]) which admit a proper HV are the spacetimes with power law coefficients. As it has been noted in the introduction all vacuum Bianchi I spacetimes which admit a Homothetic vector have been determined in [@McSteele].
In the following section we study the CKVs of some well known exact solutions of Einstein field equations in a Bianchi I spacetime.
Exact Bianchi I solutions and conformal symmetries {#Exact}
==================================================
One can apply the results of the last section to determine if a given Bianchi I metric admits or not CKVs and at the same time determine the exact form of the CKVs and their conformal factors. The method of work is simple and consists of the following steps.
From the given Bianchi metric one computes the traceless projection tensor $%
\Delta _{ab}^{cd}=g_{ab}g^{cd}-\frac{1}{4}\delta _{a}^{c}\delta _{b}^{d}$ and demands that $\Delta _{ab}^{cd}X_{c;d}=0$ where $X_{c}$ is any of the CKVs defined in (\[sx3.6\]), (\[sx3.8\]), (\[sx3.13\]), (\[sx3.15\]) (conformally flat case) and (\[sx3.22\]), (\[sx3.25\]) (non-conformally flat case). If this condition cannot be satisfied for any values of the parameters of the metric then the metric does not admit a CKV otherwise it does. It is possible that the conformal factors are constants in which case the CKVs reduce to HVs.
Before one proceeds with the above it is convenient to compute the Weyl tensor and examine if the space is conformally flat or not. If it is not there is no need to consider the vectors (\[sx3.6\]), (\[sx3.8\]), ([sx3.13]{}), (\[sx3.15\]) whereas if it is there is no need to consider the vectors (\[sx3.22\]), (\[sx3.25\]).
In the following section we apply the above method to various anisotropic Bianchi I metrics which we have traced in the literature. We present the derivation of the results for the Kasner type metrics in some detail whereas the for rest of the metrics we give only the results of the calculations.
Kasner type metrics
-------------------
The Kasner type metrics are defined by the line element: $$ds^2=-dt^2+t^{2p}dx^2+t^{2q}dy^2+t^{2r}dz^2 \label{sx3.30}$$ where $p,q,r\,$ are different constants (otherwise the metric reduces to an LRS metric (two of the constants equal) or to a FRW metric (all constants equal). The well known Kasner spacetime - which has been used extensively in the literature in the discussion of anisotropies of the Universe - is a vacuum solution of Einstein’s field equations with the parameters $p,q,r$ restricted by the relations: $$\begin{aligned}
p+q+r &=&1 \label{sx3.31} \\
p^2+q^2+r^2 &=&1. \notag\end{aligned}$$
Kasner spacetime is vacuum so if conformally flat it is flat therefore we have a non-conformally flat case. Condition $\Delta
_{ab}^{cd}X_{c;d}=0$ for the vector fields (\[sx3.22\]),(\[sx3.25\]) yields in turn:
:
We find $r=1,$ $c=1,\alpha _{1}=\frac{q-1}{p-1},$ $\tau d\tau =$ $%
\frac{1}{p-1}tdt$ ($p\neq 1$ otherwise we have an LRS spacetime) from which follows that the Kasner type metric: $$ds^2=-dt^2+t^{2p}dx^2+t^{2q}dy^2+t^2dz^2 \label{sx3.32}$$ admits the HV [@kramer-stephani]: $$X_{B_{1}}=\frac{1}{1-p}t\partial _{t}+\frac{q-1}{p-1}y\partial
_{y}+x\partial _{x};\;\psi (X_{B_{1}})=\frac{1}{1-p} \label{sx3.33}$$
:
We find $r\neq 1,\alpha _{1}=\frac{q-1}{p-1},$ $\alpha _{2}=\frac{r-1}{p-1}$ ($p\neq 1)$ from which we conclude that the Kasner type metric (\[sx3.30\]) with $r\neq 1,p\neq 1$ admits the HV: $$X_{B_{2}}=\frac{r-1}{p-1}\tau \partial _{\tau }+x\partial _{x}++\frac{q-1}{%
p-1}y\partial _{y}+\frac{q-1}{p-1}z\partial _{z};\;\psi (X_{B_{2}})=\frac{1}{%
1-p}\text{ } \label{sx3.34}$$We emphasize that due to conditions (\[sx3.31\]) the Kasner spacetime admits only the HV $X_{B_{2}}$. These results agree with those of [McSteele]{}.
Bianchi I shear free spacetimes
-------------------------------
This class contains many well known solutions of the field equations. The general form of the spacetime metric is $$ds^{2}=-dt^{2}+S^{2}(t)f^{2p}(t)dx^{2}+S^{2}(t)f^{2q}(t)dy^{2}+S^{2}(t)f^{2r}(t)dz^{2}
\label{sx3.35}$$where the functions $S(t),f(t)$ are general functions. The various known solutions of this form are perfect fluid solutions with vanishing and non-vanishing cosmological constant $\Lambda $. These solutions are:
a\. Dust solution
$\Lambda =0$ [@ellis-hawking].
$$S^3(t)=\frac 92Mt(t+\Sigma );f(t)=\frac{t^{2/3}}{S(t)};p=2\sin \alpha
,q=2\sin (\alpha +\frac{2\pi }3),r=2\sin (\alpha +\frac{4\pi }3)
\label{sx3.36}$$
The constant $\alpha $ is the angle where the anisotropy is maximal ($-\frac
\pi 2<\alpha <\frac \pi 2$) and $\Sigma ,M$ are constants with $\Sigma >0$.
$\Lambda \neq 0$ [@kramer-stephani]
$$S^{3}(t)=\left\{
\begin{array}{c}
a\sinh \omega t+\frac{M}{2\Lambda }(\cosh \omega t-1)\qquad \text{for}\qquad
\Lambda >0 \\
a\sin \omega t+\frac{M}{2\Lambda }(\cos \omega t-1)\qquad \text{for}\qquad
\Lambda <0%
\end{array}%
\right\} \label{sx3.37}$$
$$f(t)=\left\{
\begin{array}{c}
\frac{\cosh \omega t-1}{S^{3}(t)}\qquad \text{for}\qquad \Lambda >0 \\
\frac{1-\cos \omega t}{S^{3}(t)}\qquad \text{for}\qquad \Lambda <0%
\end{array}%
\right\} . \label{sx3.38}$$
b\. Perfect fluid solutions with an equation of state $p=(\gamma -1)\mu ~$[@jacobs],[@kramer-stephani]
$$S^{3}(t)=\left\{
\begin{array}{c}
c\sinh \omega t\qquad \text{for}\qquad \Lambda >0 \\
\sqrt{3(3+M)}t\qquad \text{for}\qquad \Lambda =0 \\
c\sin \omega t\qquad \text{for}\qquad \Lambda <0%
\end{array}%
\right\} \label{sx3.39}$$
$$f(t)=\left\{
\begin{array}{c}
\left( \tanh \frac{\omega t}{t}\right) ^{b}\qquad \text{for}\qquad \Lambda >0
\\
t^{b}\qquad \text{for}\qquad \Lambda =0 \\
\left( \tan \frac{\omega t}{t}\right) ^{b}\qquad \text{for}\qquad \Lambda <0%
\end{array}%
\right\} \label{sx3.40}$$
where $b=\left( \frac{3}{3+M}\right) ^{1/2}$ and $c=\left( \frac{3+M}{%
\Lambda }\right) ^{1/2}$.
For $\mu =0$ we take the vacuum solutions for $\Lambda =,>,<0$. In [kramer-stephani]{} one can find the form of the solutions for various values of $\gamma .$
Einstein-Maxwell solutions
--------------------------
We have found two solutions describing cosmological models with an electromagnetic field satisfying the Rainich conditions. These are:
Data solution [@Datta]:
$$ds^2=A^{-1}(-dt^2+A^2dx^2+ABdy^2+ACdz^2) \label{sx3.41}$$
where:
$$\begin{aligned}
A &=&c_{1}t^{\mu }+c_{2}t^{-\mu } \\
AB &=&t^{\lambda }\quad \text{and}\quad AC=t^{2-\lambda }\end{aligned}$$
and $c_{1},c_{2},\mu ,\lambda $ are constants with $c_{1}c_{2}\neq 0$.
Rosen solution [@Rosen]:
$$ds^2=-\frac{b_1^2(\tan \frac 12t)^{2(b_2+b_3)}}{\sin ^4t}dt^2+\sin ^2tdx^2+%
\frac{(\tan \frac 12t)^{2b_2}}{\sin ^2t}dy^2+\frac{(\tan \frac 12t)^{2b_3}}{%
\sin ^2t}dz^2 \label{sx3.42}$$
where $b_1,b_2,b_3$ are constants and $b_2b_3=1$.
Using the criterion $\Delta _{ab}^{cd}X_{c;d}=0$ for each of the above spacetimes, we find, after standard but lengthy computations, the results of Table \[Table1\].
[ccc]{} **Spacetime** & **CKVs** & **Conformal factor**\
Datta Solution & $\nexists $ & $\nexists $\
Rosen Solution & $\nexists $ & $\nexists $\
Kasner-type & (\[sx3.33\]) / (\[sx3.34\]) & constant\
Shear free spacetimes &
------------------------------
$\nexists $for $\Lambda >0$
(\[sx3.33\]) / (\[sx3.34\])
$\nexists $for $\Lambda <0$
------------------------------
: Exact solutions of Bianchi I spacetimes which admit CKVs
&
-----------------------------
$\nexists $for $\Lambda >0$
constant
$\nexists $for $\Lambda <0$
-----------------------------
: Exact solutions of Bianchi I spacetimes which admit CKVs
\
Dust solution & $\nexists $ & $\nexists $\
\[Table1\]
Discussion {#Dis}
==========
In this work we studied the CKVs of proper (that is the LRS case is excluded) Bianchi I spacetimes. We have shown that there are only four families of Bianchi type I spacetimes which admit CKVs. Two of these families concern conformally flat spacetimes and two non-conformally flat spacetimes. The non-conformally flat families, to the best of our knowledge, are new.
One important aspect of these metrics is the symmetry inheritance of the CKVs by the 4-velocity $u^{a}=\delta _{0}^{a}$ of the comoving observers. This property is important because it assures that Lie dragging along the CKVs, fluid flow lines transform onto fluid flow lines thus giving rise to dynamical conservation laws [maartens-mason-tsamparlis,mason-maartens,coley-tupper1,coley-tupper2,coley-tupper3,coley-tupper4]{}.
The application of the general results of this work to the widely known Bianchi I metrics (\[sx3.32\]) and (\[sx3.35\]) has shown that these spacetimes do not belong to the solutions we have found. More specifically the Kasner type spacetimes (\[sx3.32\]) and (\[sx3.36\]) admit at most a HV while the Bianchi type I dust solution (\[sx3.35\]) does not admit even a HV.
The families of Bianchi I metrics we have found contain many anisotropic matter solutions which was not possible to be found before due to the complexity of the conformal equations for Bianchi I spacetimes. It is hoped that these new solutions will have at least equally interesting properties as the classical Bianchi I metrics and will make possible the production of new results mainly at the kinematical level where CKVs play a significant role.
A final remark concerns the Lie and the Noether point symmetries of differential equations. Indeed it has been shown that for a general class of second order partial differential equations the Lie point symmetries are related to the conformal algebra of the underlying geometry[@IJGMMP]. This class of equations contains among others the heat equation and the Klein Gordon equation. Therefore one is possible to use the CKVs we have determined and construct conservation laws or to solve explicitly this type of differential equations in the corresponding Bianchi I spacetimes.
We would like to thank the anonymous referee for helpful comments which have improved the manuscript. AP acknowledge financial support of INFN
Proof of Proposition \[prop22\] {#appendixA}
===============================
In this appendix we give the direct and the inverse proof of Proposition [prop22]{}.
**Direct Proof:** First recall the decomposition of the curvature tensor [@kramer-stephani]:$$R_{abcd}=C_{abcd}+\frac{2}{n-2}\left( g_{c[a}R_{b]d}+g_{d[b}R_{a]c}\right) -%
\frac{R}{(n-1)(n-2)}g_{abcd} \label{sx2.2}$$where $g_{abcd}=g_{ac}g_{bd}-g_{ad}g_{bc}$ and the dimension of space is $%
n\geqslant 4.$ Furthermore in a $1+(n-1)$ decomposable space holds that [kramer-stephani]{}:$$\overset{n}{R}_{abcd}=\delta _{a}^{\alpha }\delta _{b}^{\beta }\delta
_{c}^{\gamma }\delta _{d}^{\sigma }\overset{n-1}{R}_{\alpha \beta \gamma
\sigma };\qquad \overset{n}{R}_{ab}=\delta _{a}^{\alpha }\delta _{b}^{\beta }%
\overset{n-1}{R}_{\alpha \beta };\qquad \overset{n}{R}=\overset{n-1}{R}.
\label{sx2.4}$$We consider cases.
Case 1: $n\geqslant 5$
Assume the metric $g_{ab}$ to be conformally flat; then $C_{abcd}=0$. Replacing $\overset{n}{R}_{abcd},$ $\overset{n}{R}_{ab},$ $\overset{n}{R}$ in (\[sx2.2\]) and taking into account that $C_{abcd}=0$ we find $$\overset{n-1}{R}_{\alpha \beta \gamma \delta }=\frac{2}{n-2}\left( g_{\gamma
\lbrack \alpha }\overset{n-1}{R_{\beta ]\delta }}+g_{\delta \lbrack \beta }%
\overset{n-1}{R_{\alpha ]\gamma }}\right) -\frac{\overset{n-1}{R}}{(n-1)(n-2)%
}g_{\alpha \beta \gamma \delta }. \label{sx2.4.1}$$where $g_{\alpha \beta \gamma \sigma }$ is defined similarly to $g_{abcd}$.
From (\[sx2.2\]) we conclude that $\overset{n-1}{C}_{\alpha
\beta \gamma \delta }$ (because $n-1\geqslant 4)$ therefore the $n-1$ space is conformally flat. Contracting with $%
g^{\alpha \gamma }$ we get:$$\overset{n-1}{R}_{\alpha \beta }=\frac{\overset{n-1}{R}}{n-1}g_{\alpha \beta
} \label{sx2.5}$$and the $(n-1)$ space is also an Einstein space. We conclude that the $n-1$ space is a space of constant curvature [@Eisenhart].
In order to compute the constant $p$ we insert (\[sx2.5\]) back to ([sx2.2]{}) and find:$$\overset{n-1}{R}_{\alpha \beta \gamma \sigma }=\frac{\overset{n-1}{R}}{%
(n-1)(n-2)}g_{\alpha \beta \gamma \sigma } \label{sx2.6}$$
The $n$ space being conformally flat admits CKVs. According to proposition \[prop21\] these vectors are found from the gradient CKVs of the $(n-1)$ space of the form (\[sx2.0b\]). Ricci identity for the CKV $\psi _{,\mu }$ gives:$$\psi _{|\mu \nu \sigma }-\psi _{|\mu \sigma \nu }=\overset{n-1}{R}_{\sigma
\nu \mu \delta }\psi ^{,\delta }. \label{sx2.8}$$Using (\[sx2.6\]) and (\[sx2.0b\]) in equation (\[sx2.8\]) we obtain:$$\left[ \frac{\overset{n-1}{R}}{(n-1)(n-2)}+p\right] g_{\alpha \beta \gamma
\delta }\psi ^{,\delta }=0. \label{sx2.9}$$from which follows:$$\overset{n-1}{R}=-p(n-1)(n-2) \label{sx2.10}$$and $$p=-\frac{\overset{n-1}{R}}{(n-1)(n-2)}.$$Case 2: $n=4$
In this case relation (\[sx2.2\]) still applies and (\[sx2.4.1\]) becomes:$$\overset{3}{R}_{\alpha \beta \gamma \delta }=\left( g_{\gamma \lbrack \alpha
}\overset{3}{R_{\beta ]\delta }}+g_{\delta \lbrack \beta }\overset{3}{R}%
_{\alpha ]\gamma }\right) -\frac{\overset{3}{R}}{6}g_{\alpha \beta \gamma
\delta } \label{sx2.10.1}$$where now the Greek indices take the values 1,2,3. Contracting with $%
g^{\alpha \gamma }$ we find$$\overset{3}{R}_{\beta \delta }=\frac{\overset{3}{R}}{3}g_{\beta \delta }
\label{sx2.10.2}$$which implies that the 3d space is an Einstein space. Although the 3d - space is an Einstein space of curvature $\overset{3}{R}=const.$ we cannot conclude that it is a space of constant curvature before we prove that it is conformally flat. The condition for this is that the Cotton - York tensor $$C_{\beta }^{\alpha }=2\varepsilon ^{\alpha \gamma \delta }\left( \overset{3}{%
R}_{\beta \gamma }-\frac{1}{4}g_{\beta \gamma }\overset{3}{R}\right)
_{;\delta }$$vanishes [@Eisenhart]. Replacing $\overset{3}{R}_{\beta \delta }$ from (\[sx2.10.2\]) we find$$C_{\beta }^{\alpha }=\frac{1}{6}\varepsilon ^{\alpha \gamma \delta }g_{\beta
\gamma }\overset{3}{R}_{;\delta }. \label{sx2.10.3}$$
We replace $\overset{3}{R}_{\beta \delta }$ from (\[sx2.10.2\]) in ([sx2.10.1]{}) and find$$\overset{3}{R}_{\alpha \beta \gamma \delta }=\left( g_{\gamma \lbrack \alpha
}\overset{3}{R}_{\beta ]\delta }+g_{\delta \lbrack \beta }\overset{3}{R}%
_{\alpha ]\gamma }\right) -\frac{\overset{3}{R}}{6}g_{\alpha \beta \gamma
\delta }=\frac{\overset{3}{R}}{6}g_{\alpha \beta \gamma \delta }.$$Ricci identity for the gradient CKV $\psi _{,\mu }$ gives:$$\psi _{|\mu \nu \sigma }-\psi _{|\mu \sigma \nu }=\overset{3}{R}_{\sigma \nu
\mu \delta }\psi ^{,\delta }=-\frac{\overset{3}{R}}{6}g_{\mu \nu \sigma
\delta }\psi ^{,\delta }$$Using (\[sx2.6\]) and (\[sx2.0b\]) in equation (\[sx2.8\]) we obtain:$$\left[ \frac{\overset{3}{R}}{6}+p\right] g_{\alpha \beta \gamma \sigma }\psi
^{,\delta }=0$$from which follows $\overset{3}{R}_{;\delta }=0$ hence $C_{\beta }^{\alpha
}=0,$ which completes the proof.
Case 3: $n=3$
In this case the space $3-1=2$ is conformally flat and admits gradient CKVs hence the curvature scalar is a constant and the space is a space of constant curvature.
**Inverse Proof: **Suppose the $(n-1)$ space of the $1+(n-1)
$ space (\[sx2.1\]) is a space of constant curvature. Then it is conformally flat and by (\[sx2.2\]),(\[sx2.4\]) and (\[sx2.5\]) the $%
1+(n-1)$ space is conformally flat. This completes the proof of proposition \[prop22\]
The conformal algebra of RT and ART spacetimes {#appendixB}
==============================================
The eight proper CKVs of the RT spacetime are$$X_{(k)\mu }=a^{2}A_{k,\mu }~~,~X_{(k)z}=-a^{2}A_{k,z} \label{sx3.6}$$$$X_{(k+4)\mu }=a^{2}B_{k,\mu }~,~X_{(k+4)z}=-a^{2}B_{k,z} \label{sx3.8}$$where $\mu =t,x,y$ and the corresponding conformal factors are $$\psi _{X_{k}}=A_{k}~,~\psi _{X_{k+4}}=B_{k}$$where the fields $A_{k},B_{k}$ are given by the expressions$$A_{k}=\cos (\frac{\tau }{a})\left\{ \cosh (\frac{y}{a})\left[ \sin (\frac{z}{%
a}),\cos (\frac{z}{a})\right] ,\sinh (\frac{y}{a})\left[ \sin (\frac{z}{a}%
),\cos (\frac{z}{a})\right] \right\}$$$$B_{k}=\sin (\frac{\tau }{a})\left\{ \cosh (\frac{x}{a})\left[ \sin (\frac{z}{%
a}),\cos (\frac{z}{a})\right] ,\sinh (\frac{x}{a})\left[ \sin (\frac{z}{a}%
),\cos (\frac{z}{a})\right] \right\} .$$
The eight proper CKVs of the ART spacetime are $$Y_{(k)\mu }=-a^{2}\bar{A}_{k,\mu }~~,~Y_{(k)z}=a^{2}\bar{A}_{k,z}
\label{sx3.13}$$$$Y_{(k+4)\mu }=-a^{2}\bar{B}_{k,\mu }~,~Y_{(k+4)z}=a^{2}\bar{B}_{k,z}
\label{sx3.15}$$where $\mu =t,x,y$ and the corresponding conformal factors are $$\psi _{Y_{k}}=A_{k}~,~\psi _{Y_{k+4}}=B_{k}$$where the fields $\bar{A}_{k},\bar{B}_{k}$ are $$\bar{A}_{k}=\cosh (\frac{\tau }{a})\left\{ \cos (\frac{y}{a})\left[ \sinh (%
\frac{z}{a}),\cosh (\frac{z}{a})\right] ,\sin (\frac{y}{a})\left[ \sinh (%
\frac{z}{a}),\cosh (\frac{z}{a})\right] \right\} \label{sx3.17}$$$$\bar{B}_{k}=\sinh (\frac{\tau }{a})\left\{ \cos (\frac{x}{a})\left[ \sinh (%
\frac{z}{a}),\cosh (\frac{z}{a})\right] ,\sin (\frac{x}{a})\left[ \sinh (%
\frac{z}{a}),\cosh (\frac{z}{a})\right] \right\} .$$For easy reference in tables \[RTCKVS\] and \[ARTCKVS\] we give the explicit form of the CKVs for the RT spacetime and the ART spacetime respectively.
Furthermore, the RT spacetime (\[sx3.3\]) admits a seven dimensional Killing algebra, the three vector fields are the KVs $\left\{ \partial
_{x},\partial _{y},\partial _{z}\right\} $ and the four extra KVs are$$\mathbf{\xi }_{4,RT}=\sinh (\frac{y}{a})\cosh (\frac{x}{a})\partial _{\tau
}-\cot (\frac{\tau }{a})\sinh (\frac{y}{a})\cosh (\frac{x}{a})\partial
_{x}+\tan (\frac{\tau }{a})\sinh (\frac{y}{a})\cosh (\frac{x}{a})\partial
_{y}$$$$\xi _{5,RT}=\sinh (\frac{y}{a})\sinh (\frac{x}{a})\partial _{\tau }-\cot (%
\frac{\tau }{a})\sinh (\frac{y}{a})\sinh (\frac{x}{a})\partial _{x}+\tan (%
\frac{\tau }{a})\sinh (\frac{y}{a})\sinh (\frac{x}{a})\partial _{y}$$$$\xi _{6,RT}=\cosh (\frac{y}{a})\cosh (\frac{x}{a})\partial _{\tau }-\cot (%
\frac{\tau }{a})\cosh (\frac{y}{a})\cosh (\frac{x}{a})\partial _{x}+\tan (%
\frac{\tau }{a})\cosh (\frac{y}{a})\cosh (\frac{x}{a})\partial _{y}$$$$\xi _{7,RT}=\cosh (\frac{y}{a})\sinh (\frac{x}{a})\partial _{\tau }-\cot (%
\frac{\tau }{a})\cosh (\frac{y}{a})\sinh (\frac{x}{a})\partial _{x}+\tan (%
\frac{\tau }{a})\cosh (\frac{y}{a})\sinh (\frac{x}{a})\partial _{y}$$
Similarly for the ART spacetime (\[sx3.4\]), the four extra KVs are$$\mathbf{\xi }_{4,ART}=\sin (\frac{y}{a})\cosh (\frac{x}{a})\partial _{\tau
}-\coth (\frac{\tau }{a})\sin (\frac{y}{a})\sinh (\frac{x}{a})\partial
_{x}+\tanh (\frac{\tau }{a})\cos (\frac{y}{a})\cosh (\frac{x}{a})\partial
_{y}$$$$\mathbf{\xi }_{5,ART}=\sin (\frac{y}{a})\sinh (\frac{x}{a})\partial _{\tau
}-\coth (\frac{\tau }{a})\sin (\frac{y}{a})\cosh (\frac{x}{a})\partial
_{x}+\tanh (\frac{\tau }{a})\cos (\frac{y}{a})\sinh (\frac{x}{a})\partial
_{y}$$$$\mathbf{\xi }_{6,ART}=\cos (\frac{y}{a})\cosh (\frac{x}{a})\partial _{\tau
}-\coth (\frac{\tau }{a})\cos (\frac{y}{a})\sinh (\frac{x}{a})\partial
_{x}-\tanh (\frac{\tau }{a})\sin (\frac{y}{a})\cosh (\frac{x}{a})\partial
_{y}$$$$\mathbf{\xi }_{7,ART}=\cos (\frac{y}{a})\sinh (\frac{x}{a})\partial _{\tau
}-\coth (\frac{\tau }{a})\cos (\frac{y}{a})\cosh (\frac{x}{a})\partial
_{x}-\tanh (\frac{\tau }{a})\sin (\frac{y}{a})\sinh (\frac{x}{a})\partial
_{y}$$
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
$\mathbf{X}$ $\mathbf{X}_{\tau }$ $\mathbf{X}_{x}$ $\mathbf{X}_{y}$ $% **Conformal factor** $\mathbf{\psi }$
\mathbf{X}_{z}$
-------------- ----------------------------------------------------------------- ------------------------------------------------------------------------------- --------------------------------------------------------------------------- ----------------------------------------------------------------- --------------------------------------------------------------------------
$X_{1}$ $a\sin \left( \frac{\tau }{a}\right) \cosh \left( \frac{y}{a}% $0$ $\frac{a\sinh \left( \frac{y% $-a\cos \left( \frac{\tau }{a}\right) \cosh \left( \frac{y}{a}% $\cos \left( \frac{\tau }{a}%
\right) \sin \left( \frac{z}{a}\right) $ }{a}\right) \sin \left( \frac{z}{a}\right) }{\cos \left( \frac{\tau }{a}% \right) \cos \left( \frac{z}{a}\right) $ \right) \cosh \left( \frac{y}{a}\right) \sin \left( \frac{z}{a}\right) $
\right) }$
$X_{2}$ $a\sin \left( \frac{\tau }{a}\right) \cosh \left( \frac{y}{a}% $0$ $\frac{a\sinh \left( \frac{y% $a\cos \left( \frac{\tau }{a}\right) \cosh \left( \frac{y}{a}% $\cos \left( \frac{\tau }{a}%
\right) \cos \left( \frac{z}{a}\right) $ }{a}\right) \cos \left( \frac{z}{a}\right) }{\cos \left( \frac{\tau }{a}% \right) \sin \left( \frac{z}{a}\right) $ \right) \cosh \left( \frac{y}{a}\right) \cos \left( \frac{z}{a}\right) $
\right) }$
$X_{3}$ $a\sin \left( \frac{\tau }{a}\right) \sinh \left( \frac{y}{a}% $0$ $\frac{a\cosh \left( \frac{y% $-a\cos \left( \frac{\tau }{a}\right) \sinh \left( \frac{y}{a}% $\cos \left( \frac{\tau }{a}%
\right) \sin \left( \frac{z}{a}\right) $ }{a}\right) \sin \left( \frac{z}{a}\right) }{\cos \left( \frac{\tau }{a}% \right) \cos \left( \frac{z}{a}\right) $ \right) \sinh \left( \frac{y}{a}\right) \sin \left( \frac{z}{a}\right) $
\right) }$
$X_{4}$ $a\sin \left( \frac{\tau }{a}\right) \sinh \left( \frac{y}{a}% $0$ $\frac{a\cosh \left( \frac{y% $a\cos \left( \frac{\tau }{a}\right) \sinh \left( \frac{y}{a}% $\cos \left( \frac{\tau }{a}%
\right) \cos \left( \frac{z}{a}\right) $ }{a}\right) \cos \left( \frac{z}{a}\right) }{\cos \left( \frac{\tau }{a}% \right) \sin \left( \frac{z}{a}\right) $ \right) \sinh \left( \frac{y}{a}\right) \cos \left( \frac{z}{a}\right) $
\right) }$
$X_{5}$ $-a\cos \left( \frac{\tau }{a}\right) \cosh \left( \frac{x}{a}% $\frac{a\sinh \left( \frac{x}{a}% $0$ $-a\sin \left( \frac{\tau }{a}\right) \cosh \left( \frac{x}{a}% $\sin \left( \frac{\tau }{a}%
\right) \sin \left( \frac{z}{a}\right) $ \right) \sin \left( \frac{z}{a}\right) }{\sin \left( \frac{\tau }{a}\right) } \right) \cos \left( \frac{z}{a}\right) $ \right) \cosh \left( \frac{x}{a}\right) \sin \left( \frac{z}{a}\right) $
$
$X_{6}$ $-a\cos \left( \frac{\tau }{a}\right) \cosh \left( \frac{x}{a}% $\frac{a\sinh \left( \frac{x}{a}% $0$ $a\sin \left( \frac{\tau }{a}\right) \cosh \left( \frac{x}{a}% $\sin \left( \frac{\tau }{a}%
\right) \cos \left( \frac{z}{a}\right) $ \right) \cos \left( \frac{z}{a}\right) }{\sin \left( \frac{\tau }{a}\right) } \right) \sin \left( \frac{z}{a}\right) $ \right) \cosh \left( \frac{x}{a}\right) \cos \left( \frac{z}{a}\right) $
$
$X_{7}$ $-a\cos \left( \frac{\tau }{a}\right) \sinh \left( \frac{x}{a}% $\frac{a\cosh \left( \frac{x}{a}% $0$ $-a\sin \left( \frac{\tau }{a}\right) \sinh \left( \frac{x}{a}% $\sin \left( \frac{\tau }{a}%
\right) \sin \left( \frac{z}{a}\right) $ \right) \sin \left( \frac{z}{a}\right) }{\sin \left( \frac{\tau }{a}\right) } \right) \cos \left( \frac{z}{a}\right) $ \right) \sinh \left( \frac{x}{a}\right) \sin \left( \frac{z}{a}\right) $
$
$X_{8}$ $-a\cos \left( \frac{\tau }{a}\right) \sinh \left( \frac{x}{a}% $\frac{a\cosh \left( \frac{x}{a}% $0$ $a\sin \left( \frac{\tau }{a}\right) \sinh \left( \frac{x}{a}% $\sin \left( \frac{\tau }{a}%
\right) \cos \left( \frac{z}{a}\right) $ \right) \cos \left( \frac{z}{a}\right) }{\sin \left( \frac{\tau }{a}\right) } \right) \sin \left( \frac{z}{a}\right) $ \right) \sinh \left( \frac{x}{a}\right) \cos \left( \frac{z}{a}\right) $
$
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
: Proper CKVs of the RT spacetime (\[sx3.3\])
\[RTCKVS\]
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
$\mathbf{X}$ $\mathbf{X}_{\tau}$ $\mathbf{X}_{x}$ $\mathbf{X}_{y}$ $% **Conformal factor** $\mathbf{\psi }$
\mathbf{X}_{z}$
-------------- ---------------------------------------------------------------- ------------------------------------------------------------------------- --------------------------------------------------------------------------- ------------------------------------------------------------ ---------------------------------------------------------------------------
$X_{1}$ $a\sinh \left( \frac{\tau }{a}\right) \cos \left( \frac{y}{a}% $0$ $\frac{a\sin \left( \frac{y% $a\cosh \left( \frac{\tau }{a}\right) \cos \left( \frac{% $\cosh \left( \frac{\tau }{a%
\right) \sinh \left( \frac{z}{a}\right) $ }{a}\right) \sinh \left( \frac{z}{a}\right) }{\cosh \left( \frac{\tau}{% y}{a}\right) \cosh \left( \frac{z}{a}\right) $ }\right) \cos \left( \frac{y}{a}\right) \sinh \left( \frac{z}{a}\right) $
\alpha }\right) }$
$X_{2}$ $a\sinh \left( \frac{\tau }{a}\right) \cos \left( \frac{y}{a}% $0$ $\frac{a\sin \left( \frac{y% $a\cosh \left( \frac{\tau }{a}\right) \cos \left( \frac{% $\cosh \left( \frac{\tau }{a%
\right) \cosh \left( \frac{z}{a}\right) $ }{a}\right) \cosh \left( \frac{z}{a}\right) }{\cosh \left( \frac{\tau}{% y}{a}\right) \sinh \left( \frac{z}{a}\right) $ }\right) \cos \left( \frac{y}{a}\right) \cosh \left( \frac{z}{a}\right) $
\alpha }\right) }$
$X_{3}$ $a\sinh \left( \frac{\tau }{a}\right) \sin \left( \frac{y}{a}% $0$ $-\frac{a\cos \left( \frac{% $a\cosh \left( \frac{\tau }{a}\right) \sin \left( \frac{% $\cosh \left( \frac{\tau }{a%
\right) \sinh \left( \frac{z}{a}\right) $ y}{a}\right) \sinh \left( \frac{z}{a}\right) }{\cosh \left( \frac{\tau}{% y}{a}\right) \cosh \left( \frac{z}{a}\right) $ }\right) \sin \left( \frac{y}{a}\right) \sinh \left( \frac{z}{a}\right) $
\alpha }\right) }$
$X_{4}$ $a\sinh \left( \frac{\tau }{a}\right) \sin \left( \frac{y}{a}% $0$ $-\frac{a\cos \left( \frac{% $a\cosh \left( \frac{\tau }{a}\right) \sin \left( \frac{% $\cosh \left( \frac{\tau }{a%
\right) \cosh \left( \frac{z}{a}\right) $ y}{a}\right) \cosh \left( \frac{z}{a}\right) }{\cosh \left( \frac{\tau}{% y}{a}\right) \sinh \left( \frac{z}{a}\right) $ }\right) \sin \left( \frac{y}{a}\right) \cosh \left( \frac{z}{a}\right) $
\alpha }\right) }$
$X_{5}$ $a\cosh \left( \frac{\tau }{a}\right) \cos \left( \frac{x}{a}% $\frac{a\sin \left( \frac{x}{a}% $0$ $a\sinh \left( \frac{\tau }{a}\right) \cos \left( \frac{x% $\sinh \left( \frac{%
\right) \sinh \left( \frac{z}{a}\right) $ \right) \sinh \left( \frac{z}{a}\right) }{\sinh \left( \frac{\tau }{a}% }{a}\right) \cosh \left( \frac{z}{\alpha }\right) $ \tau }{a}\right) \cos \left( \frac{x}{a}\right) \sinh \left( \frac{z}{a}%
\right) }$ \right) $
$X_{6}$ $a\cosh \left( \frac{\tau }{a}\right) \cos \left( \frac{x}{a}% $\frac{a\sin \left( \frac{x}{a}% $0$ $a\sinh \left( \frac{\tau }{a}\right) \cos \left( \frac{x% $\sinh \left( \frac{%
\right) \cosh \left( \frac{z}{a}\right) $ \right) \cosh \left( \frac{z}{a}\right) }{\sinh \left( \frac{\tau }{a}% }{a}\right) \sinh \left( \frac{z}{\alpha }\right) $ \tau }{a}\right) \cos \left( \frac{x}{a}\right) \cosh \left( \frac{z}{a}%
\right) }$ \right) $
$X_{7}$ $a\cosh \left( \frac{\tau }{a}\right) \sin \left( \frac{x}{a}% $-\frac{a\cos \left( \frac{x}{a}% $0$ $a\sinh \left( \frac{\tau }{a}\right) \sin \left( \frac{x% $\sinh \left( \frac{%
\right) \sinh \left( \frac{z}{a}\right) $ \right) \sinh \left( \frac{z}{a}\right) }{\sinh \left( \frac{\tau }{a}% }{a}\right) \cosh \left( \frac{z}{\alpha }\right) $ \tau }{a}\right) \sin \left( \frac{x}{a}\right) \sinh \left( \frac{z}{a}%
\right) }$ \right) $
$X_{8}$ $a\cosh \left( \frac{\tau }{a}\right) \sin \left( \frac{x}{a}% $-\frac{a\cos \left( \frac{x}{a}% $0$ $a\sinh \left( \frac{\tau }{a}\right) \sin \left( \frac{x% $\sinh \left( \frac{%
\right) \cosh \left( \frac{z}{a}\right) $ \right) \cosh \left( \frac{z}{a}\right) }{\sinh \left( \frac{\tau }{a}% }{a}\right) \sinh \left( \frac{z}{\alpha }\right) $ \tau }{a}\right) \sin \left( \frac{x}{a}\right) \cosh \left( \frac{z}{a}%
\right) }$ \right) $
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
: The proper CKVs of the ART spacetime (\[sx3.4\])
\[ARTCKVS\]
[99]{} D. Kramer, H. Stephani, M. MacCallum and E. Herlt, *Exact Solutions of Einstein’s Field Equations *(Cambridge, Cambridge University Press, 1980)
R.M. Wald, *General Relativity* (Chicago University Press, 1984)
K.C. Jacobs, Astrophys J **151 (**1968) 431; Astrophys J **153** (1968)** **661
F. Hoyle and J.V. Narlikar, Proc R Soc **A273** (1963) 1; C.W. Misner, Ap. J. **151** (1968) 431
A.Z. Petrov, *Einstein Spaces* Pergamon, (Oxford University Press, 1969)
M. Tsamparlis, Class.Quantum Grav **15** (1998) 2901
R. Maartens, D.P. Mason and M. Tsamparlis, J. Math. Phys. **27 (**1986) 2987
D.P. Mason and R.J Maartens, Math. Phys. **28** (1986)** **2511
R. Maartens, S.D. Maharaj and B.O.J. Tupper Class.Quantum Grav **12(**1995) 2577; R. Maartens, S.D. Maharaj and B.O. J. Tupper Class.Quantum Grav **13 (**1996) 317
A.A. Coley and B.O.J. Tupper, Class.Quantum Grav **7 (**1990) 1961
A.A. Coley and B.O.J. Tupper, Class.Quantum Grav **7 (**1990) 2195
A.A. Coley and B.O.J. Tupper, Math. Phys. **33** (1992)1754
A.A. Coley and B.O.J. Tupper, Class.Quantum Grav **11 (**1994) 2553
L. Herrera, J. Ponce de Leon J.Math.Phys **26 (**1985) 2332; L. Herrera, J. Ponce de Leon J.Math.Phys **26 (**1985) 2018
P.S. Apostolopoulos and M. Tsamparlis, Class. Quantum Grav. **18** (2001) 3775-3790
M.J.Rebouças and J. Tiommo, Phys. Rev. D **28** (1983) 1251
M.J. Rebouças and A.F.F. Teixeira, J.Math.Phys **33(**1992) 2855
L. Defrise-Carter, Commun.Math.Phys. **40** (1975) 273
G.S. Hall and J.D. Steele, J.Math.Phys 1991 **32 (**1991) 1847
R. Maartens and C. M. Mellin Class.Quantum Grav **13 (**1996) 1571.
C.B.G. McIntosh and J.D. Steele, Class. Quantum Grav. 8 (1991) 1173
S.W. Hawking and G.F.R. Ellis *The large scale structure of space-time* (Cambridge University Press, Cambridge 1973)
M. Tsamparlis, D. Nikolopoulos and P. S. Apostolopoulos, Class. Quantum Grav. **15** (1998) 2909
Y. Choquet-Bruhat, C. DeWitt-Morette and M. Dillard-Bleick *Analysis, Manifolds and Physics* (Amsterdam: North Holland 1977)
G.F.R. Ellis in Cargese Lectures in Physics, Vol 6, edited by E.Schatznan (Gordon and Breach, New York, 1971)
B.K. Datta, Nuovo Cimento **36** (1965)** **109
G.J. Rosen, Math Phys **3** (1962) 313
M. Tsamparlis and A. Paliathanasis, Gen. Relativ. Gravit. 42 (2010) 2957
A. Paliathanasis and M. Tsamparlis, Int. J. Geom. Methods Mod. Phys. 11 (2014) 1450037
M. Tsamparlis, A. Paliathanasis and A. Qadir, Int. J. Geom. Methods Mod. Phys. 15 (2015) 1550003
L.P. Eisenhart *Riemannian Geometry* (Princeton University Press, Princeton 1964)
[^1]: These spacetimes are 1+3 spacetimes in which the 3d hypersurface is a maximally symmetric space with positive and negative curvature scalar respectively
[^2]: The generalization of the proof of [@Tsamp-Nikol-Apost] to $n$ dimensions is similar to the one for $n=4$ and we omit it.
[^3]: The proof of Proposition \[prop22\] is given in appendix \[appendixA\]
[^4]: All $2d$ metrics are conformally flat.
|
---
abstract: 'A new numerical method is presented for solving the rotating shallow water equations on a rotating sphere using quasi-uniform polygonal meshes. The method uses special families of finite element function spaces to mimic key mathematical properties of the continuous equations and thereby capture several desirable physical properties related to balance and conservation. The method relies on two novel features. The first is the use of [*compound finite elements*]{} to provide suitable finite element spaces on general polygonal meshes. The second is the use of [*dual finite element spaces*]{} on the dual of the original mesh, along with suitably defined discrete Hodge star operators to map between the primal and dual meshes, enabling the use of a finite volume scheme on the dual mesh to compute potential vorticity fluxes. The resulting method has the same mimetic properties as a finite volume method presented previously, but is more accurate on a number of standard test cases.'
address:
- 'College of Engineering, Mathematics and Physical Sciences, University of Exeter, Exeter, UK'
- 'Department of Mathematics, Imperial College, London, UK'
author:
- John Thuburn
- Colin J Cotter
title: 'A primal-dual mimetic finite element scheme for the rotating shallow water equations on polygonal spherical meshes'
---
compound finite element ,dual finite element ,mimetic ,shallow water
Introduction {#sec_introduction}
============
In order to exploit the new generation of massively parallel supercomputers that are becoming available, weather and climate models will require good parallel scalability. This requirement has driven the development of numerical methods that do not depend on the orthogonal coordinate system and quadrilateral structure of the longitude-latitude grid, whose polar resolution clustering is predicted to lead to a scalability bottleneck. A significant challenge is to obtain good scalability without sacrificing accuracy; in particular conservation, balance, and wave propagation are important for accurate modelling of the atmosphere [@staniforth2012].
Building on earlier work [@ringler2010; @thuburn2012], @thuburn2014 presented a finite volume scheme for the shallow water equations on polygonal meshes. They start from the continuous shallow water equations in the so-called vector invariant form: $$\begin{aligned}
\phi_t + \nabla \cdot \mathbf{f} & = & 0, \label{ctsmass} \\
\mathbf{u}_t + \mathbf{q}^{\perp} + \nabla (\phi_{\mathrm{T}} + k) & = & 0,
\label{ctsvel}\end{aligned}$$ where $\phi$, the geopotential, is equal to the fluid depth times the gravitational acceleration, $\phi_{\mathrm{T}} = \phi +
\phi_{\mathrm{orog}}$ is the total geopotential at the fluid’s upper surface including the contribution from orography, $\mathbf{u}$ is the velocity, $\mathbf{f} = \mathbf{u} \phi$ is the mass flux, and $k = |\mathbf{u}|^2/2$. The $\perp$ symbol is defined by $\mathbf{u}^{\perp} = \mathbf{k} \times \mathbf{u}$ where $\mathbf{k}$ is the unit vertical vector. Finally, $\pi = \zeta/\phi$ is the potential vorticity (PV), where $\zeta = f + \xi$ is the absolute vorticity, with $f$ the Coriolis parameter and $\xi = \mathbf{k} \cdot \nabla \times \mathbf{u}$ the relative vorticity, and $\mathbf{q} = \mathbf{f} \pi$ is the PV flux. By the use of a C-grid placement of prognostic variables, and by ensuring that the numerical method mimics key mathematical properties of the continuous governing equations (hence the term ‘mimetic’), the scheme was designed to have good conservation and balance properties. These good properties were verified in numerical tests on hexagonal and cubed sphere spherical meshes. However, their scheme has a number of drawbacks. Most seriously, the Coriolis operator, whose discrete form is essential to obtaining good geostrophic balance, is numerically inconsistent and fails to converge in the $L_\infty$ norm [@weller2014; @thuburn2014]. Also, although the gradient and divergence operators are consistent, their combination to form the discrete Laplacian operator also fails to converge in the $L_\infty$ norm in some cases. These inaccuracies are clearly visible in idealized convergence tests, and give rise to marked ‘grid imprinting’ for initially symmetrical flows. Although they are less conspicuous in more complex flows, they are clearly undesirable.
@cotter2012 [see also @mcrae2014; @cotter2014] showed that the same mimetic properties can be obtained using a certain class of mixed finite element method. The mimetic properties follow from the choice of an appropriate hierarchy of function spaces for the prognostic and diagnostic variables (e.g. section \[sec\_compound\] below), which also provides a finite element analogue of the C-grid placement of variables, or a higher-order generalization. (The use of such a hierarchy goes by various names in the literature, including ‘mimetic finite elements’, ‘compatible finite elements’, and ‘finite element exterior calculus’; see @cotter2014 for a discussion of the shallow water equation case in the language of exterior calculus.) Importantly, the resulting schemes are numerically consistent.
While the mimetic finite element approach appears very attractive, it is not yet clear which particular choice of mesh and function spaces is most suitable. Standard finite element methods use triangular or quadrilateral elements. For the lowest-order mimetic finite element scheme on triangles, the dispersion relation for the linearized shallow water equations suffers from extra branches of inertio-gravity waves, which are badly behaved numerical artefacts [@leroux2007], analogous to the problem that occurs on the triangular C-grid [@danilov2010]. Higher-order finite element methods also typically exhibit anomalous features in their wave dispersion relations, such as extra branches, frequency gaps, or zero group velocity modes. Some progress has been made in reducing these problems, at least on quadrilateral meshes, through the inclusion of dissipation or modification of the mass matrix [e.g. @melvin2013; @ullrich2013], though the remedies are somewhat heuristic except in the most idealized cases. Finally, coupling to subgrid models of physical processes such as cumulus convection or cloud microphysics may be less straightforward with higher-order elements (P. Lauritzen, pers. comm.). These factors suggest that it may still be worthwhile investigating lowest-order schemes on quadrilateral and hexagonal meshes.
The above arguments raise two related questions. Can the mimetic finite element method inspire a development to fix the inconsistency of the mimetic finite volume method? Alternatively, can the mimetic finite element method at lowest order be adapted to work on polygonal meshes such as hexagons? Below we answer the second question by showing that the mimetic finite element method can indeed be adapted. In fact, from a certain viewpoint the mimetic finite volume and mimetic finite element schemes have very similar mathematical structure. The notation below is chosen to emphasize this similarity[^1]. Moreover, the similarity is sufficiently strong that much of the code of the mimetic finite volume model of @thuburn2014 could be re-used in the model presented below. This, in turn, facilitates the cleanest possible comparison of the two approaches.
The adaptation of the mimetic finite element method employs two novel features. The first is the definition of a suitable hierarchy of finite element function spaces on polygonal meshes. This is achieved by defining compound elements built out of triangular subelements, and is described in section \[sec\_compound\]. The second ingredient is the introduction of a dual family of function spaces that are defined on the dual of the original mesh. This permits the definition of a spatially averaged mass field that lives in the same function space as the vorticity and potential vorticity fields; this, in turn, enables the use of an accurate finite volume scheme on the dual mesh for advection of potential vorticity, and keeps the formulation of the finite element model as close as possible to that of the finite volume model.
Meshes and dual meshes {#sec_meshes}
======================
The scheme described here is suitable for arbitrary two-dimensional polygonal meshes on flat domains or, as used here, curved surfaces approximated by planar facets. Two particular meshes are used to obtain the results in section \[sec\_results\], namely the same variants of the hexagonal-icosahedral mesh and the cubed sphere mesh used by @thuburn2014, in order to facilitate comparison with their results. Coarse-resolutions versions are shown in Fig. \[fig\_grids\].
![Left: a hexagonal–icosahedral mesh with 162 cells and 642 degrees of freedom. Right: a cubed-sphere mesh with 216 cells and 648 degrees of freedom. Continuous lines are primal mesh edges, dotted lines are dual mesh edges.[]{data-label="fig_grids"}](f01a.eps "fig:"){width="60mm"} ![Left: a hexagonal–icosahedral mesh with 162 cells and 642 degrees of freedom. Right: a cubed-sphere mesh with 216 cells and 648 degrees of freedom. Continuous lines are primal mesh edges, dotted lines are dual mesh edges.[]{data-label="fig_grids"}](f01b.eps "fig:"){width="60mm"}
Any polygonal mesh has a corresponding dual mesh. (We will refer to the original mesh as the ‘primal’ mesh where necessary to distinguish it from the dual.) Each primal cell contains one dual vertex; each dual cell contains one primal vertex; each primal edge corresponds to one dual edge and these usually cross each other. Figure \[fig\_grids\] shows both primal and dual edges for the two meshes.
Function spaces and compound finite elements {#sec_compound}
============================================
![ Schematic showing the function spaces used in the scheme and the relationships between them. Primal function spaces are on the bottom row and dual function spaces are on the top row.[]{data-label="fig_spaces"}](schematic2.eps){width="150mm"}
The mimetic properties of the scheme arise from the relationships between the finite element function spaces. Three function spaces are used on the primal mesh ($\mathbf{V}_0$, $\mathbf{V}_1$, and $\mathbf{V}_2$), and three on the dual mesh ($\mathbf{V}^2$, $\mathbf{V}^1$, and $\mathbf{V}^0$). Figure \[fig\_spaces\] indicates that $\nabla^\perp$ (i.e. $\mathbf{k} \times \nabla$) maps from $\mathbf{V}_0$ to $\mathbf{V}_1$ and $\nabla \cdot$ maps from $\mathbf{V}_1$ to $\mathbf{V}_2$[^2]. More precisely, the primal function spaces satisfy the following properties.
**Property List 1**
- $\mathbf{u} \in \mathbf{V}_1 \ \ \Rightarrow \ \
\nabla \cdot \mathbf{u} \in \mathbf{V}_2$.
- $\phi \in \mathbf{V}_2$ with $\int \phi \, dA = 0 \ \ \Rightarrow \ \ \exists \,
\mathbf{u} \in \mathbf{V}_1 \mathrm{\ st\ } \nabla \cdot \mathbf{u} = \phi$.
- $\psi \in \mathbf{V}_0 \ \ \Rightarrow \ \ \nabla^\perp \psi \in \mathbf{V}_1$.
- $\forall \psi \in \mathbf{V}_0 , \ \nabla \cdot \nabla^\perp \psi = 0$, and $\forall \mathbf{u} \in \mathbf{V}_1 \ \mathrm{st} \ \nabla \cdot \mathbf{u} = 0, \
\exists \, \psi \in \mathbf{V}_0 \ \mathrm{st} \ \mathbf{u} = \nabla^\perp \psi$. That is, $\nabla^\perp$ maps onto the kernel of $\nabla \cdot$ .
The second condition assumes spherical geometry so that there are no lateral boundaries. The same assumption will be made throughout this paper; in particular, no boundary terms will arise when integrating by parts[^3].
In a similar way, Fig. \[fig\_spaces\] indicates that $\nabla$ maps from $\mathbf{V}^0$ to $\mathbf{V}^1$ and $\mathbf{k} \cdot \nabla \times$ maps from $\mathbf{V}^1$ to $\mathbf{V}^2$. More precisely, the dual function spaces satisfy the following properties.
**Property List 2**
- $\hat{\mathbf{u}} \in \mathbf{V}^1 \ \ \Rightarrow \ \
\mathbf{k} \cdot \nabla \times \hat{\mathbf{u}} \in \mathbf{V}^2$.
- $\hat{\xi} \in \mathbf{V}^2$ with $\int \hat{\xi} \, dA = 0 \ \ \Rightarrow \ \ \exists \,
\hat{\mathbf{u}} \in \mathbf{V}^1 \mathrm{\ st\ }
\mathbf{k} \cdot \nabla \times \hat{\mathbf{u}} = \hat{\xi}$.
- $\hat{\chi} \in \mathbf{V}^0 \ \ \Rightarrow \ \ \nabla \hat \chi \in \mathbf{V}^1$.
- $\forall \hat{\chi} \in \mathbf{V}^0, \ \mathbf{k} \cdot \nabla \times \nabla \hat{\chi} = 0$, and $\forall \hat{\mathbf{u}} \in \mathbf{V}^1 \ \mathrm{st} \ \mathbf{k} \cdot \nabla \times
\hat{\mathbf{u}} = 0, \ \exists \, \hat{\chi} \in \mathbf{V}^0 \ \mathrm{st} \
\hat{\mathbf{u}} = \nabla \hat{\chi}$. That is, $\nabla$ maps onto the kernel of $\mathbf{k} \cdot \nabla \times$ .
As noted earlier, standard finite element schemes in two dimensions typically use triangular or quadrilateral elements. Several families of mixed finite elements that satisfy Property List 1 on such meshes are known. However, in order to apply our scheme on more general polygonal meshes we will need to define families of mixed finite elements satisfying Property List 1 on those meshes. One way to do this is to use *compound elements*. Any polygonal element can be subdivided into a number of triangular subelements. A basis function on the polygonal element can then be defined as a suitable linear combination of basis functions on the subelements. The allowed linear combinations are determined by the requirement to satisfy Property List 1 or 2; see below.
The desire to use a dual mesh increases the need for finite element spaces on polygons, and hence for compound elements. Only in special cases (such as the cubed sphere, Fig. \[fig\_grids\]) can both the primal and dual meshes be built of triangles and quadrilaterals; other cases require higher degree polygons for either the primal or dual mesh (or both).
For a triangular primal mesh, @buffa2007 describe a scheme for the construction of a dual hierarchy of function spaces. The dual mesh elements are compound elements, similar, though not identical, to those used here. However, their scheme is limited to the case of a triangular primal mesh and a barycentric refinement for the construction of the dual.
In a complementary study, @christiansen2008 describes how finite element basis functions satifying Property List 1 may be constructed on arbitrary polygonal elements, without the need to divide into subelements, through a process of *harmonic extension*. For example, let $\gamma_j$ be a basis function for $\mathbf{V}_0$ associated with primal vertex $j$. Define $\gamma_j$ to equal $1$ at vertex $j$ and zero at all other vertices. Next extend $\gamma_j$ harmonically along primal mesh edges; that is, its second derivative should vanish so that its gradient is constant along each edge. Then extend $\gamma_j$ harmonically into the interior of each element; that is, solve $$\label{harmonicV0}
\nabla^2 \gamma_j = 0$$ subject to the Dirichlet boundary conditions given by the known values of $\gamma_j$ on element edges. In a similar way, let $\mathbf{v}_e$ be a basis function for $\mathbf{V}_1$ associated with edge $e$. Define the normal component of $\mathbf{v}_e$ to be a nonzero constant along edge $e$ (some arbitrary sign convention must be chosen to define the positive direction) and zero at all other edges. Then extend $\mathbf{v}_e$ harmonically into the interior of each element; that is, solve $$\label{harmonicV1a}
\nabla \left( \nabla \cdot \mathbf{v}_e \right) = 0$$ and $$\label{harmonicV1b}
\nabla^\perp \left( \mathbf{k} \cdot \nabla \times \mathbf{v}_e \right) = 0$$ subject to the known values of the normal component at element edges, for example by writing $\mathbf{v}_e = \nabla \phi + \nabla^\perp \psi$, implying $\nabla^2 \phi = c_1$ and $\nabla^2 \psi = c_2$ for constants $c_1$ and $c_2$. The boundary conditions determine the value of $c_1$, but not $c_2$. However, condition [[(\[harmonicV0\])]{}]{} along with the fourth property in List 1 implies that we must choose $c_2 = 0$, so that [[(\[harmonicV1b\])]{}]{} reduces to $$\label{harmonicV1c}
\mathbf{k} \cdot \nabla \times \mathbf{v}_e = 0 .$$ For the last function space $\mathbf{V}_2$ the basis function associated with cell $i$ is defined to be a nonzero constant in cell $i$ and zero in all other cells. It may then be verified that the properties in List 1 do indeed hold for the spaces spanned by these basis functions.
Although the harmonic extension approach provides a general method for constructing the lowest order mimetic finite element spaces on polygonal meshes, its drawback is that, except for the simplest element shapes, the basis functions cannot be found analytically. Even if they are found numerically, the inner products required for the finite element method cannot be computed exactly, either analytically or by numerical quadrature.
Here we take inspiration from both @buffa2007 and @christiansen2008 to construct spaces of compound finite elements for arbitrary polygonal primal and dual meshes, by a process that might be called *discrete harmonic extension*. For the function spaces on the primal mesh, in effect, we solve a finite element discretization of [[(\[harmonicV0\])]{}]{}, [[(\[harmonicV1a\])]{}]{}, and [[(\[harmonicV1c\])]{}]{} on the mesh of triangular subelements in order to construct the compound basis elements for the original polygonal mesh. For this discretization we use the lowest order mimetic finite element spaces on the triangular subelements, in which $\mathbf{V}_0$ comprises continuous piecewise linear elements, $\mathbf{V}_1$ comprises the lowest order Raviart-Thomas elements, and $\mathbf{V}_2$ comprises piecewise constant elements; $\text{P1-RT0-P0}^\text{DG}$ in standard shorthand. Although only discrete versions of [[(\[harmonicV0\])]{}]{}, [[(\[harmonicV1a\])]{}]{}, and [[(\[harmonicV1c\])]{}]{} are solved, it may be verified that the properties in List 1 hold exactly. The basis functions on the triangular subelements are known analytically, and the compound elements are linear combinations of these; therefore, integrals of products of basis functions, for example to compute entries of a mass matrix, can all be computed exactly.
The resulting compound elements provide a generalization to polygonal meshes of the $\text{P1-RT0-P0}^\text{DG}$ hierarchy of spaces, so we will refer to them as compound $\text{P1-RT0-P0}^\text{DG}$ elements. Like the non-compound spaces described by @christiansen2008, the expansion coefficients for $\mathbf{V}_0$ correspond to mesh vertices, for $\mathbf{V}_1$ to edges, and for $\mathbf{V}_2$ to cells. Thus, this hierarchy provides a finite element analogue of the polygonal C-grid if we choose to represent velocity in $\mathbf{V}_1$ and the mass variable in $\mathbf{V}_2$.
The construction of basis elements for the dual spaces proceeds in a very similar way, except that the basis function for $\mathbf{V^1}$ is given by $\mathbf{k} \times$ the solution of [[(\[harmonicV1a\])]{}]{} and [[(\[harmonicV1c\])]{}]{}. This gives rise to a compound $\text{P1-N0-P0}^\text{DG}$ hierarchy of spaces, where N0 refers to the lowest order two-dimensional Nédélec elements.
An important detail concerns the number of subelements needed. It may appear natural to subdivide an $n$-gon cell into $n$ triangular subelements. However, it will be necessary to calculate integrals on the overlap between primal and dual elements (section \[massmatrices\]). In order to be able to do this when the domain is a curved surface approximated by plane triangular subelement facets, both the primal and dual compound element meshes must be built from triangular subelements of the the same supermesh. To achieve this we divide $n$-gon cells (whether primal or dual) into $2n$ subelements (Fig. \[fig\_supermesh\]).
![ Example of part of a hexagonal primal mesh (solid lines) with its triangular dual mesh (dashed lines) and the supermesh of triangular subelements (all lines) used to construct the compound elements.[]{data-label="fig_supermesh"}](supermesh.eps){width="100mm"}
It is convenient to normalize the basis functions as follows: $$\begin{aligned}
\alpha_i \in \mathbf{V}_2 : & \ \ \ \ \ &
\int_{\mathrm{cell}\ i'} \alpha_i \, dA = \delta_{i \, i'} ; \\
\mathbf{v}_e \in \mathbf{V}_1 : & \ \ \ \ \ &
\int_{\mathrm{edge}\ e'} \mathbf{v}_e \cdot \mathbf{n} \, dl = \delta_{e \, e'} ; \\
\gamma_j \in \mathbf{V}_0 : & \ \ \ \ \ &
\left. \gamma_j \right|_{\mathrm{vertex}\ j'} = \delta_{j \, j'} ; \\
\beta_j \in \mathbf{V}^2 : & \ \ \ \ \ &
\int_{\mathrm{dual cell}\ j'} \beta_j \, dA = \delta_{j \, j'} ; \\
\mathbf{w}_e \in \mathbf{V}^1 : & \ \ \ \ \ &
\int_{\mathrm{dual edge}\ e'} \mathbf{w}_e \cdot \mathbf{m} \, dl = \delta_{e \, e'} ; \\
\chi_i \in \mathbf{V}^0 : & \ \ \ \ \ &
\left. \chi_i \right|_{\mathrm{dual\ vertex}\ i'} = \delta_{i \, i'} .\end{aligned}$$ Here $\mathbf{n}$ is the unit normal vector to primal edge $e$ and $\mathbf{m}$ is the unit tangent vector to dual edge $e$, with $\mathbf{m}$ and $\mathbf{n}$ pointing in the same sense (i.e. $\mathbf{n} \cdot \mathbf{m} > 0$, though they need not be parallel if the dual edges are not orthogonal to the primal edges), as in @thuburn2012. The normalization is chosen so that degrees of freedom for fields in $\mathbf{V}_2$ and $\mathbf{V}^2$ correspond to area integrals of scalars over primal cells and dual cells, respectively, degrees of freedom for a field in $\mathbf{V}_1$ correspond to normal fluxes integrated along primal edges, degrees of freedom in $\mathbf{V}^1$ correspond to circulations integrated along dual edges, and degrees of freedom for fields in $\mathbf{V}_0$ and $\mathbf{V}^0$ correspond to nodal values of scalars at primal vertices and dual vertices respectively. Again, this corresponds closely to the framework of @thuburn2012.
@melvin2014 have analyzed the wave dispersion properties for finite element discretizations of the linear shallow water equations using these compound elements. That paper gives explicit expressions for the $\mathbf{V}_1$ and $\mathbf{V}_2$ compound element basis functions for the cases of a square mesh and a regular hexagonal mesh on a plane. For more general meshes it is straightforward and convenient to construct the compound element basis functions numerically. Figure \[fig\_elements\] shows typical basis elements for the three primal mesh function spaces for quadrilateral and hexagonal cells.
![ Typical compound basis elements of the function spaces on a square mesh (left) and a hexagonal mesh (right). structures. Top row $\alpha_i \in \mathbf{V}_2$; middle row $\mathbf{v}_e \in \mathbf{V}_1$; bottom row $\gamma_j \in \mathbf{V}_0$. In the middle row, at subelement edges the normal components of the basis vectors are continuous. []{data-label="fig_elements"}](V_quad.eps "fig:"){width="50mm"} ![ Typical compound basis elements of the function spaces on a square mesh (left) and a hexagonal mesh (right). structures. Top row $\alpha_i \in \mathbf{V}_2$; middle row $\mathbf{v}_e \in \mathbf{V}_1$; bottom row $\gamma_j \in \mathbf{V}_0$. In the middle row, at subelement edges the normal components of the basis vectors are continuous. []{data-label="fig_elements"}](V_hex.eps "fig:"){width="50mm"}
![ Typical compound basis elements of the function spaces on a square mesh (left) and a hexagonal mesh (right). structures. Top row $\alpha_i \in \mathbf{V}_2$; middle row $\mathbf{v}_e \in \mathbf{V}_1$; bottom row $\gamma_j \in \mathbf{V}_0$. In the middle row, at subelement edges the normal components of the basis vectors are continuous. []{data-label="fig_elements"}](compound_v1_square.eps "fig:"){width="60mm"} ![ Typical compound basis elements of the function spaces on a square mesh (left) and a hexagonal mesh (right). structures. Top row $\alpha_i \in \mathbf{V}_2$; middle row $\mathbf{v}_e \in \mathbf{V}_1$; bottom row $\gamma_j \in \mathbf{V}_0$. In the middle row, at subelement edges the normal components of the basis vectors are continuous. []{data-label="fig_elements"}](compound_v1_hex.eps "fig:"){width="70mm"}
![ Typical compound basis elements of the function spaces on a square mesh (left) and a hexagonal mesh (right). structures. Top row $\alpha_i \in \mathbf{V}_2$; middle row $\mathbf{v}_e \in \mathbf{V}_1$; bottom row $\gamma_j \in \mathbf{V}_0$. In the middle row, at subelement edges the normal components of the basis vectors are continuous. []{data-label="fig_elements"}](E_quad.eps "fig:"){width="60mm"} ![ Typical compound basis elements of the function spaces on a square mesh (left) and a hexagonal mesh (right). structures. Top row $\alpha_i \in \mathbf{V}_2$; middle row $\mathbf{v}_e \in \mathbf{V}_1$; bottom row $\gamma_j \in \mathbf{V}_0$. In the middle row, at subelement edges the normal components of the basis vectors are continuous. []{data-label="fig_elements"}](E_hex.eps "fig:"){width="60mm"}
The fields used in the computation are represented as expansions in terms of these basis elements. For example, $$\begin{aligned}
\phi = \sum_i \phi_i \alpha_i & & \ \ \ \ \ \in \mathbf{V}_2, \label{eqn_exp_v2}\\
\mathbf{u} = \sum_e u_e \mathbf{v}_e & & \ \ \ \ \ \in \mathbf{V}_1 \label{eqn_exp_v1}\end{aligned}$$ for the prognostic geopotential and velocity fields, and $$\xi = \sum_j \xi_j \gamma_j \ \ \ \ \ \in \mathbf{V}_0 \label{eqn_exp_v0}$$ for the relative vorticity field. Here the sums are global sums over all basis elements in the relevant spaces. In some cases it will be useful to introduce dual space representations of fields; these will be indicated by a hat symbol where necessary to distinguish them from the corresponding primal space representations. For example, $$\begin{aligned}
\hat{\phi} = \sum_i \hat{\phi}_i \chi_i & & \ \ \ \ \ \in \mathbf{V}^0, \label{eqn_exp_V0}\\
\hat{\mathbf{u}} = \sum_e \hat{u}_e \mathbf{w}_e & & \ \ \ \ \ \in \mathbf{V}^1, \label{eqn_exp_V1}\\
\hat{\xi} = \sum_j \hat{\xi}_j \beta_j & & \ \ \ \ \ \in \mathbf{V}^2. \label{eqn_exp_V2}\end{aligned}$$ The fields $\phi$ and $\hat{\phi}$ have the same number of degrees of freedom, and it is possible construct a well-conditioned and reversible map between them by demanding that they agree when integrated against any test function in the primal space $\mathbf{V}_2$. Similarly, the fields $\mathbf{u}$ and $\hat{\mathbf{u}}$ have the same number of degrees of freedom, and it is possible construct a well-conditioned and reversible map between them by demanding that they agree when integrated against any test function in the primal space $\mathbf{V}_1$. It will also be useful to introduce spatially averaged versions of some fields. For example, $$\begin{aligned}
\widetilde{\phi} = \sum_j \widetilde{\phi}_j \gamma_j & & \ \ \ \ \ \in \mathbf{V}_0, \label{eqn_exp_v0_bar}\\
\overline{\phi} = \sum_j \overline{\phi}_j \beta_j & & \ \ \ \ \ \in \mathbf{V}^2. \label{eqn_exp_V2_tilde}\end{aligned}$$ Here, $\widetilde{\phi}$ and $\overline{\phi}$ have the same number of degrees of freedom, and it is possible construct a well-conditioned and reversible map between them by demanding that they agree when integrated against any test function in the primal space $\mathbf{V}_0$. $\widetilde{\phi}$ or $\overline{\phi}$ can be obtained from $\phi$ by demanding that they agree when integrated against any test function in $\mathbf{V}_0$; in effect this provides an averaging operation from $\mathbf{V}_2$ to $\mathbf{V}_0$ or $\mathbf{V}^2$. (However, we should not expect to be able to obtain $\phi$ from $\widetilde{\phi}$ or $\overline{\phi}$, as this would require an un-averaging operation, which will be ill-conditioned if it exists at all.)
It will be convenient to be able to refer to the vector of degrees of freedom for any field. To do this, we will use the same letter (with hat, tilde or bar if needed) but in upper case. Thus, for example, $\Phi$ will be the vector of values $(\phi_1, \phi_2, \dots )^T$, $\hat{U}$ will be the vector of values $( \hat{u}_1 , \hat{u}_2 , \ldots )^T$, etc.
Finite element scheme {#sec_FE_scheme}
=====================
Finite element schemes solve the governing equations by approximating the solution in the chosen function spaces, written as expansions in terms of basis functions (e.g. [[(\[eqn\_exp\_v2\])]{}]{}, [[(\[eqn\_exp\_v1\])]{}]{}), and demanding that the equations be satisfied in weak form, that is, when multiplied by any test function in the appropriate space and integrated over the domain. In this approach [[(\[ctsmass\])]{}]{} becomes $$\int \alpha_i \left( \phi_t + \nabla \cdot \mathbf{f} \right) \, dA
= 0 \ \ \ \ \forall \alpha_i \in \mathbf{V}_2 ,$$ or, regarding the integral as an inner product for which we introduce angle backet notation, $$\label{weakmass}
\langle \alpha_i , \phi_t \rangle
+
\langle \alpha_i , \nabla \cdot \mathbf{f} \rangle
= 0 \ \ \ \ \forall \alpha_i \in \mathbf{V}_2 .$$ Similarly, [[(\[ctsvel\])]{}]{} becomes $$\int \mathbf{v}_e \cdot
\left\{ \mathbf{u}_t + \mathbf{q}^{\perp} + \nabla (\phi_{\mathrm{T}} + k) \right\} \, dA
= 0 \ \ \ \ \forall \mathbf{v}_e \in \mathbf{V}_1 ,$$ or $$\label{weakvel}
\langle \mathbf{v}_e , \mathbf{u}_t \rangle
+
\langle \mathbf{v}_e , \mathbf{q}^{\perp} \rangle
+
\langle \mathbf{v}_e , \nabla (\phi_{\mathrm{T}} + k) \rangle = 0 .$$ (The construction of the nonlinear terms $\mathbf{f}$, $\mathbf{q}$ and $k$ is discussed in section \[nonlinearswe\] below.) The method generally leads to a system of algebraic equations for the unknown coefficients in the expansion of the solution.
The following subsections show how the mimetic finite element method can be re-expressed in terms of certain matrix operators acting on the coefficient vectors $\Phi$, $U$, etc. The notation is chosen to highlight the similarity to the finite volume scheme of @thuburn2014.
Matrix representation of derivatives – strong derivatives {#deriv1}
---------------------------------------------------------
The velocity basis elements are constructed and normalized so as to have constant divergence over the cell upwind of the edge where the degree of freedom resides, with area integral equal to $1$, and constant divergence over the cell downwind of this edge, with area integral equal to $-1$, with zero velocity and hence zero divergence in all other cells. Thus $$\nabla \cdot \mathbf{v}_e
=
\sum_i n_{e \, i} \alpha_i \ \ \ \ \ \in \mathbf{V}_2,$$ where $n_{e \, i}$ is equal to $1$ when the normal at edge $e$ points out of cell $i$, equal to $-1$ when the normal at edge $e$ points into cell $i$, and is zero otherwise. We will write $D_2$ for the matrix whose transpose has components $n_{e \, i}$. $D_2$ is called an incidence matrix because it describes some aspects of the grid topology. Hence, the divergence $\delta$ of an arbitrary velocity field $\mathbf{u}$ is $$\sum_i \delta_i \alpha_i = \delta = \nabla \cdot \mathbf{u}
= \sum_e u_e \nabla \cdot \mathbf{v}_e
= \sum_{e \, i} u_e n_{e \, i} \alpha_i . \label{eqn_div}$$ Equating coefficients of $\alpha_i$ gives $$\delta_i = \sum_e n_{e \, i} u_e,$$ or, in matrix-vector notation $$\Delta = D_2 U .$$ Note we could have demanded that [[(\[eqn\_div\])]{}]{} should hold when integrated against any test function in $\mathbf{V}_2$, to obtain the same result. However, this would obscure the fact that [[(\[eqn\_div\])]{}]{} actually holds at every point in the domain (except on cell edges where all terms are discontinuous), not just when integrated against a test function. In this sense, $\nabla \cdot \, : \mathbf{V}_1 \rightarrow \mathbf{V}_2$ is a *strong* derivative operator.
Similarly, the basis elements in $\mathbf{V}_0$ are constructed so that $$\nabla^\perp \gamma_j
=
\sum_e - t_{e \, j} \mathbf{v}_e ,$$ where $t_{e \, j}$ is defined to equal $1$ if edge $e$ is incident on vertex $j$ and the unit tangent vector $\mathbf{t}$ at edge $e$ points towards vertex $j$, $-1$ if it points away from vertex $j$, and zero otherwise. The unit normal and unit tangent at any edge are related by $\mathbf{t} = \mathbf{k} \times \mathbf{n}$. Hence, a stream function $\psi$ is related to the corresponding rotational velocity field $\mathbf{u}$ by $$\sum_e u_e \mathbf{v}_e
=
\mathbf{u}
=
\nabla^\perp \psi
=
\sum_j \psi_j \nabla^\perp \gamma_j
=
\sum_{j \, e} -\psi_j t_{e \, j} \mathbf{v}_e . \label{eqn_gradperp}$$ Equating coefficients of $\mathbf{v}_e$ and defining $D_1$ to be the matrix whose entries are $t_{e \, j}$ gives the matrix-vector form $$U = - D_1 \Psi .$$ Equation [[(\[eqn\_gradperp\])]{}]{} holds pointwise (again with the exception of discontinuities), so $\nabla^\perp : \mathbf{V}_0 \rightarrow \mathbf{V}_1$ is a strong derivative operator.
The matrices $D_1$ and $D_2$ are exactly the same as in the finite volume framework of @thuburn2012. In particular, they have the property that $$D_2 D_1 \equiv 0,$$ giving a discrete analogue of the continuous property $\nabla \cdot \nabla^\perp \equiv 0$.
Analogous relations hold on the dual spaces. $$\nabla \chi_i
=
- \sum_e n_{e \, i} \mathbf{w}_e$$ implies that the discrete analogue of $$\hat{\mathbf{u}} = \nabla \hat{p}$$ is $$\hat{U} = \overline{D}_1 \hat{P} ,$$ where $\overline{D}_1 = -D_2^T$. Similarly $$\mathbf{k} \cdot \nabla \times \mathbf{w}_e
=
\sum_j t_{e \, j} \beta_j$$ implies that the discrete analogue of $$\hat{\xi} = \mathbf{k} \cdot \nabla \times \hat{\mathbf{u}}$$ is $$\hat{\Xi} = \overline{D}_2 \hat{U},$$ where $\overline{D}_2 = D_1^T$. Again, these are strong derivative operators.
The matrices $\overline{D}_1$ and $\overline{D}_2$ have the property $$\overline{D}_2 \overline{D}_1 \equiv 0,$$ giving a discrete analogue in the dual space of the continuous relation $\mathbf{k} \cdot \nabla \times \nabla \equiv 0$.
Mass matrices and other operators {#massmatrices}
---------------------------------
Define the following mass matrices for the primal function spaces: $$\begin{aligned}
\label{massL}
L_{i \, i'} = \langle \alpha_i , \alpha_{i'} \rangle
=
\int \alpha_i \alpha_{i'} \, dA ,
& \ \ \ \ \ & (\mathbf{V}_2 \rightarrow \mathbf{V}_2), \\
\label{massM}
M_{e \, e'} = \langle \mathbf{v}_e , \mathbf{v}_{e'} \rangle
=
\int \mathbf{v}_e \cdot \mathbf{v}_{e'} \, dA ,
& \ \ \ \ \ & (\mathbf{V}_1 \rightarrow \mathbf{V}_1), \\
\label{massN}
N_{j \, j'} = \langle \gamma_j , \gamma_{j'} \rangle
=
\int \gamma_j \gamma_{j'} \, dA ,
& \ \ \ \ \ & (\mathbf{V}_0 \rightarrow \mathbf{V}_0).\end{aligned}$$ The expressions in parentheses indicate that $L$ maps $\mathbf{V}_2$ to itself, etc. (Analogous mass matrices may be defined for the dual spaces; however, they will not be needed here.)
The following matrices are also needed. $$\begin{aligned}
\label{operR}
R_{j \, i} = \langle \gamma_j , \alpha_{i} \rangle ,
& \ \ \ \ \ & (\mathbf{V}_2 \rightarrow \mathbf{V}_0), \\
\label{operW}
W_{e \, e'} = - \langle \mathbf{v}_e , \mathbf{v}_{e'}^\perp \rangle
= - W_{e' \, e} ,
& \ \ \ \ \ & (\mathbf{V}_1 \rightarrow \mathbf{V}_1), \\
H_{e \, e'} = \langle \mathbf{v}_e , \mathbf{w}_{e'} \rangle ,
& \ \ \ \ \ & (\mathbf{V}^1 \rightarrow \mathbf{V}_1), \\
J_{j \, j'} = \langle \gamma_j , \beta_{j'} \rangle ,
& \ \ \ \ \ & (\mathbf{V}^2 \rightarrow \mathbf{V}_0).\end{aligned}$$ For completeness we may also define $$I_{i \, i'} = \langle \alpha_i , \chi_{i'} \rangle ,
\ \ \ \ \ (\mathbf{V}^0 \rightarrow \mathbf{V}_2),$$ though we will not need to employ this matrix in the shallow water scheme.
One further operator will be needed to construct the kinetic energy per unit mass. It is $$T_{i \, e \, e'} = \int_{\mathrm{cell}\ i} \mathbf{v}_e \cdot \mathbf{v}_{e'} \, dA
= A_i \langle \alpha_i , \mathbf{v}_e \cdot \mathbf{v}_{e'} \rangle
\ \ \ \ \ (\mathbf{V}_1 \otimes \mathbf{V}_1 \rightarrow \mathbf{V}_2).$$ where $A_i = (L_{i \, i})^{-1}$ is the area of primal cell $i$.
All of these matrices can be precomputed, so that no quadrature needs to be done at run time. Moreover, they are all sparse, so they can be efficiently stored as lists of stencils and coefficients.
Let $U^\perp$ be the coefficients of the expansion of the projection of $\mathbf{u}^\perp$ into $\mathbf{V}_1$: $$\label{uperp}
\langle \mathbf{v}_e , \sum_{e'} U^\perp_{e'} \mathbf{v}_{e'} \rangle
=
\langle \mathbf{v}_e , \sum_{e'} U_{e'} \mathbf{v}^\perp_{e'} \rangle
=
\langle \mathbf{v}_e , \mathbf{u}^\perp \rangle
~~~~~~~~\forall \mathbf{v}_e \in \mathbf{V}_1 .$$ Using [[(\[massM\])]{}]{} and [[(\[operW\])]{}]{} gives the discrete version of the $\perp$ operator: $$\label{perp}
M U^\perp = - W U .$$
Demanding agreement between [[(\[eqn\_exp\_v2\])]{}]{} and [[(\[eqn\_exp\_V0\])]{}]{} when integrated against any test function in $\mathbf{V}_2$ leads to $$L \Phi = I \hat{\Phi} .
\label{eqn_hodge02}$$ Similarly, demanding agreement between [[(\[eqn\_exp\_v1\])]{}]{} and [[(\[eqn\_exp\_V1\])]{}]{} when integrated against any test function in $\mathbf{V}_1$ gives $$M U = H \hat{U} ,
\label{eqn_hodge11}$$ while demanding agreement between [[(\[eqn\_exp\_v0\])]{}]{} and [[(\[eqn\_exp\_V2\])]{}]{} when integrated against any test function in $\mathbf{V}_0$ gives $$N \Xi = J \hat{\Xi} .
\label{eqn_hodge20}$$ The relations [[(\[eqn\_hodge02\])]{}]{}, [[(\[eqn\_hodge11\])]{}]{}, [[(\[eqn\_hodge20\])]{}]{} provide invertible maps between the primal and dual function spaces. Thus, they are examples of discrete Hodge star operators [e.g. @hiptmair2001]. They may be contrasted with the analogous relations employed by @thuburn2012 and @thuburn2014 for the finite volume case, which do not involve mass matrices.
Demanding agreement between [[(\[eqn\_exp\_v2\])]{}]{}, [[(\[eqn\_exp\_v0\_bar\])]{}]{}, and [[(\[eqn\_exp\_V2\_tilde\])]{}]{} when integrated against any test function in $\mathbf{V}_0$ leads to $$N \widetilde{\Phi} = J \overline{\Phi} = R \Phi .
\label{eqn_avephi}$$ This is the matrix representation of the averaging operator discussed in section \[sec\_compound\].
Matrix representation of derivatives – weak derivatives {#deriv2}
-------------------------------------------------------
A field in $\mathbf{V}_2$ is discontinuous, so its gradient in $\mathbf{V}_1$ can only be defined in a *weak* sense, by integrating against all test functions in $\mathbf{V}_1$. For example, $$\mathbf{g} = \nabla \phi$$ must be approximated as $$\langle \mathbf{v}_e , \mathbf{g} \rangle
=
\langle \mathbf{v}_e , \nabla \phi \rangle
\ \ \forall \mathbf{v}_e \in \mathbf{V}_1,$$ where $\phi \in \mathbf{V}_2$, $\mathbf{g} \in \mathbf{V}_1$. Expanding both $\phi$ and $\mathbf{g}$ in terms of basis elements and integrating by parts then leads to the matrix form $$M G = \overline{D}_1 L \Phi.$$
Similarly, the curl of a vector field in $\mathbf{V}_1$ must be defined by integration against all test functions in $\mathbf{V}_0$. For example, the discrete analogue of $$\xi = \mathbf{k} \cdot \nabla \times \mathbf{u},$$ after expanding in basis functions and integrating by parts, is $$N \Xi = \overline{D}_2 M U. \label{xi}$$
Combining these two results, the discrete analogue of $$z = \mathbf{k} \cdot \nabla \times \nabla \phi$$ is $$N Z = \overline{D}_2 \overline{D}_1 L \Phi ,$$ which is identically zero.
These derivative operators can be combined to obtain the Laplacian of a scalar. For a scalar $\phi \in \mathbf{V}_2$, the discrete Laplacian is $D_2 M^{-1} \overline{D}_1 L \Phi$. For a scalar $\psi \in \mathbf{V}_0$, the discrete Laplacian is $ - N^{-1} \overline{D}_2 M D_1 \Psi $. The operators introduced above lead to a discrete version of the Helmholtz decomposition, in which an arbitrary vector field is decomposed into its divergent and rotational parts: $$U = M^{-1} \overline{D}_1 L \Phi - D_1 \Psi .$$ Figure \[opermap\] summarizes how the operators introduced here map between the different function spaces.
Some operator identities {#identities}
------------------------
The operators defined above satisfy some key relations that underpin the mimetic properties of the scheme. We have already seen that $${D}_2 {D}_1 \equiv 0$$ and $$\overline{D}_2 \overline{D}_1 \equiv 0,$$ leading to discrete analogues of $\nabla \cdot \nabla^\perp \equiv 0$ and $\mathbf{k} \cdot \nabla \times \nabla \equiv 0$.
Next, note that the basis elements $\gamma_j$ give a partition of unity, that is $$\sum_j \gamma_j = 1$$ at every point in the domain. Consequently $$\label{sumR}
\sum_j R_{j \, i} = \langle 1 , \alpha_i \rangle = 1$$ and $$\sum_j J_{j \, j'} = \langle 1 , \beta_{j'} \rangle = 1.$$
Now let $$\psi = \sum_j \psi_j \gamma_j \ \ \in \mathbf{V}_0 .$$ By considering the projection of $\nabla \psi$ into $\mathbf{V}_1$ $$\langle \mathbf{v}_e , (\nabla^\perp \psi)^\perp \rangle =
- \langle \mathbf{v}_e , \nabla \psi \rangle$$ and integrating by parts and using the matrices defined in sections \[deriv1\] and \[massmatrices\], we obtain $$- \overline{D}_2 W = R D_2. \label{TRiSK}$$ This identity is key to obtaining the steady geostrophic mode property (section \[geostrophic\] below). A rough interpretation is that averaging velocities to construct the Coriolis terms ($W$) then taking their divergence ($\overline{D}_2$) gives the same result as computing the velocity divergence ($D_2$) followed by averaging to $\mathbf{V}_0$ ($R$). One consequence is $\overline{D}_2 W D_1 = - R D_2 D_1 \equiv 0$.
An identical formula to [[(\[TRiSK\])]{}]{} relating $R$ and $W$ was obtained by @thuburn2012 for the finite volume case. The result was originally derived for the construction of the Coriolis terms on orthogonal grids by @thuburn2009, and @thuburn2012 showed that it could be embedded in a more general framework applicable to nonorthogonal grids. Moreover, @thuburn2009 showed that, for any given $R$ with the appropriate stencil (which we have here) and satisfying [[(\[sumR\])]{}]{}, there is a unique antisymmetric $W$ satisfying [[(\[TRiSK\])]{}]{}, and gave an explicit construction for $W$ in terms of $R$. Thus, although the context and interpretation are slightly different here, we can, nevertheless, use the Thuburn et al. construction in implementing the mixed finite-element version of the $W$ operator!
Now consider the two representations of any vector field $\mathbf{u} \in \mathbf{V}_1$, $\hat{\mathbf{u}} \in \mathbf{V}^1$ related by $$\langle \mathbf{v} , \mathbf{u} \rangle
=
\langle \mathbf{v} , \hat{\mathbf{u}} \rangle \ \ \forall \mathbf{v} \in \mathbf{V}_1,$$ so that $$\label{uuhat}
M U = H \hat{U}.$$ Since $\nabla^\perp \gamma_j \in \mathbf{V}_1$, $$\langle \nabla^\perp \gamma_j , \mathbf{u} \rangle
=
\langle \nabla^\perp \gamma_j , \hat{\mathbf{u}} \rangle
\ \ \forall \gamma_j \in \mathbf{V}_0,$$ and integrating by parts gives $$- \langle \gamma_j , \mathbf{k} \cdot \nabla \times \mathbf{u} \rangle
=
- \langle \gamma_j , \mathbf{k} \cdot \nabla \times \hat{\mathbf{u}} \rangle
\ \ \forall \gamma_j \in \mathbf{V}_0.$$ Hence $$- \overline{D}_2 M U = - J \overline{D}_2 \hat{U} .$$ Finally, substituting from [[(\[uuhat\])]{}]{} and noting that $\hat{U}$ is arbitrary gives $$\label{JDeqDH}
\overline{D}_2 H = J \overline{D}_2 .$$ The interpretation of this identity is that, for a velocity field in $\mathbf{V}^1$, taking the curl followed by mapping to the primal space is equivalent to mapping the velocity field to the primal space then taking its curl. One consequence is that $\overline{D}_2 H \overline{D}_1 = J \overline{D}_2 \overline{D}_1 \equiv 0$.
Finally, let $\chi \in \mathbf{V}_2$ and $\hat{\chi} \in \mathbf{V}^0$ be two discrete representations of a scalar field related by $$\langle \alpha , \chi \rangle = \langle \alpha , \hat{\chi} \rangle
\ \ \forall \alpha \in \mathbf{V}_2 ,$$ so that $$\label{xxhat}
L X = I \hat{X} .$$ Since $\nabla \cdot \mathbf{v}_e \in \mathbf{V}_2$ for any $\mathbf{v}_e \in \mathbf{V}_1$, we have $$\begin{aligned}
\langle \nabla \cdot \mathbf{v}_e , \chi \rangle
& = &
\langle \nabla \cdot \mathbf{v}_e , \hat{\chi} \rangle , \nonumber \\
\langle \mathbf{v}_e , \nabla \chi \rangle
& = &
\langle \mathbf{v}_e , \nabla \hat{\chi} \rangle , \nonumber \\
\overline{D}_1 L X & = & H \bar{D}_1 \hat{X},\end{aligned}$$ or, using [[(\[xxhat\])]{}]{} and noting that $\hat{X}$ is arbitrary, $$\label{DIeqHD}
\overline{D}_1 I = H \overline{D}_1 .$$
Using these identities and the Hodge star operators, it can be seen that taking a weak derivative in the primal space is equivalent to applying a Hodge star to map to the dual space, taking a strong derivative in the dual space, and applying another Hodge star to map back to the primal space: $$\begin{aligned}
M^{-1} \overline{D}_1 L & = & (M^{-1} H) \, \overline{D}_1 \, (I^{-1} L) ; \\
N^{-1} \overline{D}_2 M & = & (N^{-1} J) \, \overline{D}_2 \, (H^{-1} M) \end{aligned}$$ [@cotter2014]. Thus, certain paths in Fig. \[opermap\] commute. Weak derivative operators in the dual space can be defined by demanding a similar equivalence with primal space strong derivatives; however, the resulting formulas are less elegant and, in any case, will not be needed.
Linear shallow water equations {#linearswe}
------------------------------
We first examine the spatial discretization of the linear shallow water equations to illustrate how some key conservation and balance properties arise. The rotating shallow water equations [[(\[ctsmass\])]{}]{}, [[(\[ctsvel\])]{}]{} when linearized about a resting basic state with constant geopotential $\phi_0$ and with constant Coriolis parameter $f$ become $$\begin{aligned}
\phi_t + \nabla \cdot ( \phi_0 \mathbf{u} ) & = & 0 , \\
\mathbf{u}_t
+ f \mathbf{u}^\perp + \nabla \phi & = & 0.\end{aligned}$$ By writing these in weak form (analogous to [[(\[weakmass\])]{}]{} and [[(\[weakvel\])]{}]{}), expanding $\phi$ and $\mathbf{u}$ in terms of basis functions, and using the notation and operators defined above, we obtain $$\begin{aligned}
\dot{\Phi} + \phi_0 D_2 U & = & 0 , \\
M \dot{U} - f W U + \overline{D}_1 L \Phi & = & 0 .\end{aligned}$$
### Mass conservation {#linearmass}
Mass conservation is trivially satisfied (for both the linear and nonlinear equations) because the discrete divergence is a strong operator, so the domain integral of the discrete divergence of any vector field vanishes.
### Energy conservation {#energy}
For the linearized equations the total energy is given by $$\begin{aligned}
E & = &
\frac{1}{2} \int \phi^2 + \phi_0 \mathbf{u} \cdot \mathbf{u} \, dA \nonumber \\
& = &
\frac{1}{2} \Phi^T L \Phi + \frac{1}{2} \phi_0 U^T M U .\end{aligned}$$ Hence, the rate of change of total energy is $$\begin{aligned}
\frac{d E}{d t} & = &
\Phi^T L \dot{\Phi} + \phi_0 U^T M \dot{U} \nonumber \\
& = &
-\phi_0 \Phi^T L D_2 U + \phi_0 U^T (f W U - \overline{D}_1 L \Phi) \nonumber \\
& = & 0 ,\end{aligned}$$ where we have used the fact that $L$ and $M$ are symmetric, $W$ is antisymmetric, and $D_2^T = - \overline{D}_1$.
### Steady geostrophic modes {#geostrophic}
The linear shallow water equations support steady non-divergent flows in geostrophic balance. A numerical method must respect this property in order to be able to represent geostrophic balance. However, it is non-trivial to achieve this property because several ingredients must fall into place.
- The geopotential $\phi$ must be steady. The steadiness of $\phi$ follows immediately from the assumption that $\nabla \cdot \mathbf{u} = 0$.
- The relative vorticity $\xi$ must be steady; neither the pressure gradient nor the Coriolis term should generate vorticity. First note that, from Property List 1, $U = - D_1 \Psi$ for some $\Psi$. Taking the curl of the momentum equation then gives $$N \dot{\Xi} = \overline{D}_2 M \dot{U}
= \overline{D}_2 (- fW D_1 \Psi - \overline{D}_1 L \Phi).
\label{linvort}$$ The pressure gradient term does not contribute because $\overline{D}_2 \overline{D}_1 \equiv 0$, and the Coriolis term does not contribute because $\overline{D}_2 W D_1 \equiv 0$.
- There must exist a geopotential $\phi$ that balances the Coriolis term so that the divergence is steady. Taking the divergence of the momentum equation gives the divergence tendency $$\dot{\Delta} = D_2 M^{-1} (- f W D_1 \Psi - \overline{D}_1 \Phi) .$$ If we define $\Phi = f L^{-1} R^T \Psi$ and use the transpose of [[(\[TRiSK\])]{}]{} we find that $\dot{\Delta}$ does indeed vanish; thus the required $\phi$ does exist.
Consequently, the scheme does support steady geostrophic modes for the linearized equations. (Note, it is not necessarily true that any given $\phi$ field can be balanced by some non-divergent velocity field. On some meshes, particularly those with triangular primal cells, there might not be enough velocity degrees of freedom to balance all possible $\phi$ fields.)
### Linear PV equation {#linPV}
A generalization of the steady geostrophic mode property is that the scheme should have a suitable PV equation. In this section we consider the linear case; the nonlinear case is dealt with in section \[nonlinearPV\].
The mass field $\phi$ and the vorticity field $\xi$ live in different spaces. To construct a suitable discrete PV we need an averaged mass field $\widetilde{\phi}$ that lives in the same space as $\xi$. The linearized PV should be independent of time. For this to hold, $\xi$ and $\widetilde{\phi}$ must see the same divergence field.
For a general (possibly divergent) velocity field $\mathbf{u}$, the vorticity equation [[(\[linvort\])]{}]{} becomes $$N \dot{\Xi} = f \overline{D}_2 W U .$$ Define $\widetilde{\phi}$ using [[(\[eqn\_avephi\])]{}]{}. Then the evolution of $\widetilde{\phi}$ is given by $$\begin{aligned}
N \dot{\widetilde{\Phi}} & = & R \dot{\Phi} \nonumber \\
& = & - \phi_0 R D_2 U \nonumber \\
& = & \phi_0 \overline{D}_2 W U ,\end{aligned}$$ (using [[(\[TRiSK\])]{}]{}). Thus, $\Xi$ and $\widetilde{\Phi}$ see the same divergence $- \overline{D}_2 W U$; consequently the linearized PV $\Xi / \phi_0 - f \widetilde{\Phi} / \phi_0^2$ is independent of time.
Nonlinear shallow water equations {#nonlinearswe}
---------------------------------
The nonlinear rotating shallow water equations are [[(\[ctsmass\])]{}]{} and [[(\[ctsvel\])]{}]{}. Writing these in weak form [[(\[weakmass\])]{}]{} and [[(\[weakvel\])]{}]{}, and letting $F$, $Q$, and $K$ be the vectors of coefficients for the discrete representations of the mass flux $\mathbf{f} = \mathbf{u} \phi$, the PV flux $\mathbf{q} = \mathbf{f} \pi$, and the kinetic energy per unit mass $k = \mathbf{u} \cdot \mathbf{u} / 2$, the nonlinear discretization becomes $$\begin{aligned}
\label{phidot}
\dot{\Phi} + D_2 F & = & 0 , \\
\label{mudot}
M \dot{U} + M Q^\perp + \overline{D}_1 L ( \Phi_T + K ) & = & 0 .\end{aligned}$$ The remaining issue is how to construct suitable values of the three nonlinear terms $K$, $F$, and $Q^\perp$.
### Constructing $K$
The discretization of $k$ follows the standard finite element construction, which is to project $\nabla k$ into $\mathbf{V}_1$. It may easily be verified that this is equivalent to projecting $k$ into $\mathbf{V}_2$ before taking the weak gradient. Using the $T$ operator defined in section \[massmatrices\], the expansion coefficients $K$ of the projected $k$ are given by $$k_i = \frac{1}{2} \sum_{e \, e'} T_{i \, e \, e'} u_e u_{e'} .$$
### Constructing $F$
Because the $\phi$ field is approximated as piecewise constant, its degrees of freedom can be interpreted as primal cell integrals. Similarly, the degrees of freedom of the $\mathbf{u}$ field are the integrals of the normal velocity fluxes across primal cell edges, and the $D_2$ operator looks exactly like a finite volume divergence operator. Thus, it is straightforward to use a finite volume advection scheme for advection of $\phi$. The mass flux is constructed using a forward in time advection scheme, identical to that used by @thuburn2014, using the fluxes $U$ and the mass field $\Phi$ as input. We write this symbolically as $$\label{adv1}
F = \mathrm{adv}_1 (U,\Phi) .$$ The subscript $1$ indicates that this version of the advection scheme operates on the primal mesh and works with densities or concentrations.
### Constructing $Q^\perp$ {#nonlinearPV}
So far we have not needed to use the dual mesh representation of any field. However, in order to use the same finite volume advection scheme as @thuburn2014 to compute PV fluxes, we need a piecewise constant representation of the PV field on dual cells, and a representation of the mass flux field in terms of components normal to dual cell edges. These are naturally given by the dual function spaces: $$\pi = (f + \hat{\xi}) / \bar{\phi}~~~~~~~~\in \mathbf{V^2} ,$$ and $$\widehat{\mathbf{f}^\perp}~~~~~~~~\in \mathbf{V}^1 .$$ Applying [[(\[perp\])]{}]{} followed by [[(\[eqn\_hodge11\])]{}]{} to the mass flux gives $$\label{hatfperp}
H \widehat{F^\perp} = M F^\perp = -W F .$$
Now consider the evolution of the dual mass field $\bar{\phi}$. $$J \dot{\bar{\Phi}} =
N \dot{\widetilde{\Phi}} =
\overline{D}_2 W F =
- \overline{D}_2 H \widehat{F^\perp} =
- J \overline{D}_2 \widehat{F^\perp} ,$$ i.e. $$\label{phibar}
\dot{\bar{\Phi}} + \overline{D}_2 \widehat{F^\perp} = 0 .$$ Since $\overline{D}_2$ acts exactly like a finite volume divergence operator on the dual mesh, $\bar{\phi}$ behaves exactly as if it were evolving according to a finite volume advection scheme.
Next, in order for PV to evolve in a way consistent with the mass field $\bar{\phi}$, we construct PV fluxes in $\mathbf{V}^1$ using the dual mesh finite volume advection scheme: $$\label{adv2}
\widehat{Q^\perp} = \mathrm{adv}_2 (\widehat{F^\perp} , \Pi)$$ The subscript 2 indicates that this version of the advection scheme operates on the dual mesh and works with quantities analogous to mixing ratios (such as PV $\pi$). Finally, these dual mesh PV fluxes are mapped to the primal mesh for use in the momentum equation: $$H \widehat{Q^\perp} = M Q^\perp .$$
It may be verified that the resulting vorticity equation for $\dot{\hat{\Xi}}$ is indeed analogous to [[(\[phibar\])]{}]{}, involving the potential vorticity flux $\widehat{Q^\perp}$. Using [[(\[eqn\_hodge20\])]{}]{}, [[(\[xi\])]{}]{}, [[(\[mudot\])]{}]{} and [[(\[JDeqDH\])]{}]{}, we have $$\begin{array}{ccccccc}
J \dot{\hat{\Xi}} & = & N \dot{\Xi}
& = & \overline{D}_2 M \dot{U}
& = & - \overline{D}_2 M Q^\perp \nonumber \\
& = & - \overline{D}_2 H \widehat{Q^\perp}
& = & - J \overline{D}_2 \widehat{Q^\perp} . & &
\end{array}$$ Hence, $$\label{dualPV}
\dot{\hat{\Xi}} + \overline{D}_2 \widehat{Q^\perp} = 0,$$ which is of the desired form. The similarity of [[(\[dualPV\])]{}]{} and [[(\[phibar\])]{}]{} means that it is possible to construct PV fluxes from the dual mass fluxes $\widehat{F^\perp}$ such that the evolution of the PV is consistent with the evolution of $\bar{\Phi}$.
Time integation scheme
----------------------
The same time integration scheme as in @thuburn2014 is used. $$\begin{aligned}
\label{timephi}
\Phi^{n+1} - \Phi^n + D_2 \widetilde{F} & = & 0, \\
\label{timeu}
M U^{n+1} - M U^n + H \widetilde{\widehat{Q^\perp}}
+ \overline{\overline{D}_1 L (\Phi + K)}^t & = & 0.\end{aligned}$$ Here, $\overline{()}^t$ indicates the usual (possibly off-centred) Crank-Nicolson approximation to the integral over one time interval: $$\overline{\psi}^t = (\alpha \psi^{n+1} + \beta \psi^n) \Delta t$$ (for any field $\psi$) where $\alpha + \beta = 1$. All results presented below use $\alpha = \beta = 0.5$.
$\widetilde{F}$ is an approximation to the time integral of the mass flux across primal cell edges computed using the advection scheme. The velocity field used for the advection is $\overline{U}^t$. We write this symbolically as $$\widetilde{F} = \mathrm{Adv}_1 (\overline{U}^t,\Phi^n) .$$ (The notation $\mathrm{Adv}_1$, as distinct from $\mathrm{adv}_1$ in [[(\[adv1\])]{}]{}, indicates that here we are working with time integrals $\overline{U}^t$ and $\widetilde{F}$.)
Finally, $\widetilde{\widehat{Q^\perp}}$ is an approximation to the time integral of the PV flux across dual edges computed using the advection scheme. Dual grid time integrated mass fluxes are calculated from the primal grid time integrated mass fluxes as $$\label{eq123}
H \widetilde{\widehat{F^\perp}} = -W \widetilde{F} .$$ These are then used in the dual grid advection scheme to compute the time integrated PV fluxes: $$\widetilde{\widehat{Q^\perp}} = \mathrm{Adv}_2 (\widetilde{\widehat{F^\perp}},\Pi^n) .$$
Incremental iterative solver
----------------------------
The system [[(\[timephi\])]{}]{}, [[(\[timeu\])]{}]{} is nonlinear in the unknowns $\Phi^{n+1}$, $U^{n+1}$. It can be solved efficiently using an incremental method; this may be viewed as a Newton method with an approximate Jacobian. After $l$ iterations [[(\[timephi\])]{}]{} and [[(\[timeu\])]{}]{} will not be satisfied exactly but will have some residuals $R_\Phi$, $R_U$ defined by: $$\begin{aligned}
\label{resphi}
R_\Phi & = & \Phi^{(l)} - \Phi^n + D_2 \widetilde{F} , \\
\label{resu}
R_U & = & M U^{(l)} - M U^n + H \widetilde{\widehat{Q^\perp}}
+ \overline{\overline{D}_1 L (\Phi + K)}^t .\end{aligned}$$ Here $\Phi^{(l)}$ and $U^{(l)}$ are the approximations after $l$ iterations to $\Phi^{n+1}$ and $U^{n+1}$ and it is understood that these have been used in evaluating $\widetilde{F}$, $\widetilde{\widehat{Q^\perp}}$, and $\overline{\overline{D}_1 L (\Phi + K)}^t$. We then seek updated values $$\label{increment}
\Phi^{(l+1)} = \Phi^{(l)} + \Phi', \ \ \ \ \ U^{(l+1)} = U^{(l)} + U',$$ that will reduce the residuals, where the increments $\Phi'$, $U'$ satisfy $$\Phi' + \alpha \Delta t D_2 \phi^* U' = -R_\Phi,$$ $$\label{uinc}
U' + \alpha \Delta t \mathcal{M}^{-1} \overline{D}_1 L \Phi' = - \mathcal{M}^{-1} R_U.$$ Here, $\phi^*$ is a reference value of $\phi$; in the current implementation it is given by $\phi^n$ interpolated to cell edges. To avoid the appearance of the non-sparse matrix $M^{-1}$ in the Helmholtz problem below, a sparse approximation $\mathcal{M}^{-1}$ has been introduced. The construction of $\mathcal{M}^{-1}$ is briefly discussed in the Appendix.
Eliminating $U'$ leaves a Helmholtz problem for $\Phi'$: $$\alpha^2 \Delta t^2 D_2 \phi^* \mathcal{M}^{-1} \overline{D}_1 L \Phi'
- \Phi' =
R_\Phi - \alpha \Delta t D_2 \phi^* \mathcal{M}^{-1} R_U .$$ In the current implementation, the Helmholtz problem is solved using a single sweep of a full multigrid algorithm. This gives sufficient accuracy to avoid harming the convergence rate of the Newton iteration. Once $\Phi'$ is found, $U'$ is obtained by backsubstitution in [[(\[uinc\])]{}]{}. Finally, [[(\[increment\])]{}]{} is used to obtain improved estimates for the unknowns.
Testing to date has given satisfactory results with 4 Newton iterations. The algorithm requires the inversion of several of the linear operators represented as matrices above. The appendix describes how this is done.
Results {#sec_results}
=======
The same tests were applied to the finite element shallow water model as were applied to the finite volume model of @thuburn2014. Only a subset of results are shown here to emphasize the differences between the two models. Other aspects are the following.
- [**Stability.**]{} All experimentation to date suggests the two models have the same stability limit: with no temporal off-centring ($\alpha = \beta = 0.5$) the models are stable for large gravity wave Courant numbers and advective Courant numbers less than 1.
- [**Advection.**]{} The same advection scheme is used in the two models to compute mass, PV, and tracer fluxes on primal and dual meshes. In particular, the models share the consistency between mass and PV, between mass and tracers, and between primal mass and dual mass discussed by @thuburn2014.
- [**Balance.**]{} The balance test discussed in section 6.8 of @thuburn2014 was repeated for the finite element shallow water model. The results on both the hexagonal and cubed sphere meshes were very similar to those for the finite volume model and the ENDGame semi-implicit semi-Lagrangian model [@zerroukat2009], implying that any spurious numerical generation of imbalance is extremely weak.
- [**Computational Rossby modes.**]{} The experiment to test the ability of the scheme on hexagonal meshes to handle grid-scale vorticity features was not repeated here. However, given the general arguments in @thuburn2014 [see also @weller2012], and the similarities between the numerics of the finite volume and finite element models, the results are expected to be very similar for the finite element model.
For the remaining tests discussed below, the same mesh resolutions and time steps were used as in @thuburn2014.
Convergence of the Laplacian
----------------------------
The discrete Laplacian defined in section \[deriv2\] was applied to the $\mathbf{V}_2$ representation of the field $\cos \varphi \sin \lambda$ on the unit sphere, where $\varphi$ is latitude and $\lambda$ is longitude, and the $L_\infty$ and $L_2$ errors computed on different resolution meshes. The results are shown in table \[tab:laplacian\].
-------- ---------------- ----------- -------- ---------------- -----------
Hex Cube
Ncells $L_\infty$ err $L_2$ err Ncells $L_\infty$ err $L_2$ err
42 0.14 0.074 54 0.12 0.064
162 0.033 0.019 216 0.030 0.016
642 0.0090 0.0049 864 0.0077 0.0043
2562 0.0026 0.0012 3456 0.0038 0.0012
10242 0.00082 0.00031 13824 0.0022 0.00037
40962 0.00036 0.000081 55296 0.0012 0.00012
163842 0.00018 0.000022 221184 0.00062 0.000039
-------- ---------------- ----------- -------- ---------------- -----------
: Convergence of the scalar Laplacian on hexagonal and cubed sphere grids.[]{data-label="tab:laplacian"}
On both the hexagonal and cubed sphere meshes the $L_\infty$ errors converge at first order. On the hexagonal mesh the $L_2$ errors converge at close to second order, while on the cubed sphere mesh the convergence rate is between first and second order. For the cubed sphere mesh the convergence of the discrete scalar Laplacian is significantly better than for the finite volume scheme of @thuburn2014 (their table 4).
Convergence of the Coriolis operator
------------------------------------
The convergence of the Coriolis operator was investigated as follows. A stream function equal to $\cos \varphi \sin \lambda$ was sampled at dual vertices ($\hat{\Psi}$), enabling exact dual edge normal fluxes $\widehat{U^\perp} = - \overline{D}_1 \hat{\Psi}$ to be computed. The same stream function was also sampled at primal vertices ($\Psi$), enabling exact primal edge normal fluxes $U = - D_1 \Psi$ to be calculated; approximate dual edge normal fluxes are then given by the Coriolis operator: $H \widehat{U^\perp}_\mathrm{approx} = M U^\perp_\mathrm{approx} = -W U = W D_1 \Psi$. The difference between the two estimates $W D_1 \Psi + H \overline{D}_1 \hat{\Psi}$ gives a measure of the error in the Coriolis operator.
-------- ---------------- ----------- -------- ---------------- -----------
Hex Cube
Ncells $L_\infty$ err $L_2$ err Ncells $L_\infty$ err $L_2$ err
42 0.018 0.0092 54 0.0079 0.0049
162 0.0049 0.0026 216 0.0055 0.0021
642 0.0018 0.00066 864 0.0039 0.00092
2562 0.00078 0.00017 3456 0.0022 0.00037
10242 0.00036 0.000042 13824 0.0012 0.00014
40962 0.00017 0.000011 55296 0.00060 0.000050
-------- ---------------- ----------- -------- ---------------- -----------
: Convergence of the Coriolis operator on hexagonal and cubed sphere grids.[]{data-label="tab:coriolis"}
Values of the error at different resolutions on the two meshes are shown in table \[tab:coriolis\]. On both meshes the $L_\infty$ errors converge at first order. The $L_2$ errors converge at second order on the hexagonal mesh and between first and second order on the cubed sphere mesh. This consistency of the Coriolis operator, in contrast to the finite volume scheme of @thuburn2014, was one of the primary motivations for investigating the finite element approach.
Solid body rotation {#sbr}
-------------------
Test case 2 of @williamson1992 tests the ability of models to represent large-scale steady balanced flow. The exact solution is known, allowing errors in $\phi$ and $u$ to be computed. The errors on the two meshes after 5 days are given in table \[tab:sbr\], along with the time steps used at different resolutions.
Ncells $\Delta t \ \ \mathrm{(s)}$ $L_2(\phi)$ $L_\infty(\phi)$ $L_2(u)$ $L_\infty(u)$
-------- ----------------------------- ------------- ------------------ ---------- ---------------
Hex
642 7200 19.62 43.40 0.290 0.774
2562 3600 8.59 14.52 0.0940 0.217
10242 1800 2.27 4.01 0.0244 0.0551
40962 900 0.584 1.13 0.00609 0.0144
Cube
864 7200 35.04 87.48 0.212 0.569
3456 3600 10.16 18.06 0.0754 0.235
13824 1800 2.57 4.65 0.0194 0.0692
55296 900 0.639 1.17 0.00484 0.0257
: Geopotential errors ($\mathrm{m}^2\mathrm{s}^{-2}$) and velocity errors ($\mathrm{m}\mathrm{s}^{-1}$) for the solid body rotation test case.[]{data-label="tab:sbr"}
On the hexagonal mesh the convergence rate is close to second order or better. On the cubed sphere mesh it is between first and second order for $L_\infty(u)$ and close to second order for the other error measures. The errors are considerably smaller than for the finite volume scheme of @thuburn2014 (their table 6).
Figure \[fig:sbr\] shows the pattern of geopotential errors after 5 days at the second highest resolution in the table. The errors clearly reflect the mesh structure, showing a zonal wavenumber 5 pattern on the hexagonal mesh and a zonal wavenumber 4 pattern on the cubed sphere mesh. However, in contrast to the finite volume model, which shows errors concentrated along certain features of the mesh, the error pattern here is large scale and almost smooth.
![Geopotential error ($\mathrm{m}^2\mathrm{s}^{-2}$) after 5 days for the solid body rotation test case. Left: hexagonal mesh, 10242 cells. Right: cubed sphere mesh, 13824 cells. In each case 11 evenly space contours (i.e. 10 intervals) are used between the minimum and maximum values. (The coarse resolution meshes shown as background are for orientation only.) []{data-label="fig:sbr"}](SBR_phierr_fem_hex_10242.eps "fig:"){width="65mm"} ![Geopotential error ($\mathrm{m}^2\mathrm{s}^{-2}$) after 5 days for the solid body rotation test case. Left: hexagonal mesh, 10242 cells. Right: cubed sphere mesh, 13824 cells. In each case 11 evenly space contours (i.e. 10 intervals) are used between the minimum and maximum values. (The coarse resolution meshes shown as background are for orientation only.) []{data-label="fig:sbr"}](SBR_phierr_fem_cube_13824.eps "fig:"){width="65mm"}
Flow over an isolated mountain {#tc5}
------------------------------
Test case 5 of @williamson1992 involves an initial solid body rotation flow impinging on a conical mid-latitude mountain, leading to the generation of gravity and Rossby waves and, eventually, a complex nonlinear flow. There is no analytical solution for this test case, so a high-resolution reference solution was generated using the semi-implict, semi-Lagrangian ENDGame shallow water model [@zerroukat2009]. The finite element model runs stably with the time steps given in table \[tab:sbr\], but, as discussed by @thuburn2014 for the finite volume model and for ENDGame itself, the errors are then dominated by the semi-implicit treatment of the large amplitude gravity waves present in the solution. At any given resolution, the errors look almost identical for all combinations of model and mesh tested. The test was therefore repeated with the time steps reduced by a factor 4. The resulting height errors at day 15 are shown in table \[tab:tc5\]. The errors on the two meshes are generally very similar, and in most cases are a little smaller than those produced by the finite volume model [@thuburn2014 table 7].
Ncells $\Delta t \ \ \mathrm{(s)}$ $L_1(h)$ $L_2(h)$ $L_\infty(h)$
-------- ----------------------------- ---------- ---------- --------------- --
Hex
642 1800 36.37 50.91 191.47
2562 900 11.62 15.83 66.84
10242 450 3.12 4.11 15.06
40962 225 1.27 1.82 9.28
Cube
864 1800 44.11 64.93 291.35
3456 900 17.57 25.14 100.66
13824 450 3.75 5.25 21.42
55296 225 1.08 1.46 6.47
: Height errors ($\mathrm{m}$) for test case 5.[]{data-label="tab:tc5"}
Maps of height error at day 15 are shown in Fig. \[fig:tc5\]. The errors produced by the finite element model are of comparable size to those from ENDGame, though the error patterns are different in the three cases. Comparison with figure 6 of @thuburn2014 confirms that the errors in the finite element model are somewhat smaller than those in the finite volume model.
![Height errors ($\mathrm{m}$) at day 15 for the isolated mountain test case. Top: hexagonal mesh, 10242 cells. Middle: cubed sphere mesh, 13824 cells. Bottom: ENDGame on a regular longitude-latitude mesh, $160 \times 80$ cells. The contour interval is $6\,\mathrm{m}$, and zero and negative contours are bold. The bold circle indicates the position of the mountain. []{data-label="fig:tc5"}](TC5_herr_fem_hex_10242.eps){width="100mm"}
![Height errors ($\mathrm{m}$) at day 15 for the isolated mountain test case. Top: hexagonal mesh, 10242 cells. Middle: cubed sphere mesh, 13824 cells. Bottom: ENDGame on a regular longitude-latitude mesh, $160 \times 80$ cells. The contour interval is $6\,\mathrm{m}$, and zero and negative contours are bold. The bold circle indicates the position of the mountain. []{data-label="fig:tc5"}](TC5_herr_fem_cube_13824.eps){width="100mm"}
![Height errors ($\mathrm{m}$) at day 15 for the isolated mountain test case. Top: hexagonal mesh, 10242 cells. Middle: cubed sphere mesh, 13824 cells. Bottom: ENDGame on a regular longitude-latitude mesh, $160 \times 80$ cells. The contour interval is $6\,\mathrm{m}$, and zero and negative contours are bold. The bold circle indicates the position of the mountain. []{data-label="fig:tc5"}](TC5_herr_eg_160x080.eps){width="100mm"}
This test case was also run to 50 days at the highest resolutions in table \[tab:tc5\] and several diagnostics relevant to the mimetic properties of the scheme were calculated. The results are very similar to those shown in figure 8 of @thuburn2014. They confirm that mass is conserved to within roundoff error, and that changes in the total available energy (available potential energy plus kinetic energy) are much smaller than the conversions between available potential energy and kinetic energy. The dissipation of available energy and potential enstrophy is associated almost entirely with the inherent scale-selective dissipation in the advection scheme; it is very small, of order 1 part per thousand, during the first 20 days, but increases subsequently as PV contours begin to wrap up and nonlinear cascades become significant, implying that the inherent dissipation adapts automatically to the flow complexity in a reasonable way. A dual-mass-like tracer remains consistent with the diagnosed dual mass field $\bar{\phi}$ to within $2$ parts in $10^4$, and a PV-like tracer remains consistent with the diagnosed PV field, to within $3$ parts in $10^3$. The small errors result from imperfect convergence of various iterative aspects of the solver, and can be reduced by taking more iterations.
Barotropically unstable jet {#galewsky}
---------------------------
The test case proposed by @galewsky2004 follows the evolution of a perturbed barotropically unstable jet. The case tests the ability of models to handle the complex small scale vorticity features produced by the rapidly growing instability. The results are very sensitive to spurious triggering of the instability by error patterns related to the mesh structure.
Figure \[fig:galewsky\] shows the relative vorticity field at day 6 for the hexagonal mesh with 10242 cells and 163842 cells, the cubed sphere mesh with 13824 cells and 221184 cells, and, for comparison, from ENDGame on a $640 \times 320$ longitude-latitude mesh. In all cases the vorticity field is free of noise and spurious ripples. However, at coarse resolution the finite element model solutions show distinct ‘grid imprinting’, with a zonal wavenumber 5 pattern on the hexagonal mesh and a zonal wavenumber 4 pattern on the cubed sphere mesh. At finer resolution the solutions are more similar to the ENDGame solution, but still show significant development in the longitude range $\pi/2$ to $\pi$ where the jet in the ENDGame solution remains quiescent. The solutions on the hexagonal mesh, especially at the finer resolution, are remarkably similar to those from the finite volume model [@thuburn2014 figure 9]. On the other hand, the solutions on the cubed sphere mesh show some noticable differences from the finite volume model. At the finer resolution, outside the region strongly affected by the spurious devleopment, the structure of vorticity features is slightly more accurate in the finite element model.
![Relative vorticity field at day 6 for the barotropic instability test case. Row 1: hexagonal mesh, 10242 cells, $\Delta t = 900\,\mathrm{s}$. Row 2: hexagonal mesh, 163842 cells, $\Delta t = 225\,\mathrm{s}$. Row 3: cubed sphere mesh, 13824 cells, $\Delta t = 900\,\mathrm{s}$. Row 4: cubed sphere mesh, 221184 cells, $\Delta t = 225\,\mathrm{s}$. Row 5: ENDGame, $640 \times 320$ cells, $\Delta t = 225\,\mathrm{s}$. The plotted region is $0^\mathrm{o}$ to $360^\mathrm{o}$ longitude, $10^\mathrm{o}$ to $80^\mathrm{o}$ latitude. The contour interval is $2 \times 10^{-5}\,\mathrm{s}^{-1}$. []{data-label="fig:galewsky"}](galewsky_fem_hex_10242.eps){width="150mm"}
![Relative vorticity field at day 6 for the barotropic instability test case. Row 1: hexagonal mesh, 10242 cells, $\Delta t = 900\,\mathrm{s}$. Row 2: hexagonal mesh, 163842 cells, $\Delta t = 225\,\mathrm{s}$. Row 3: cubed sphere mesh, 13824 cells, $\Delta t = 900\,\mathrm{s}$. Row 4: cubed sphere mesh, 221184 cells, $\Delta t = 225\,\mathrm{s}$. Row 5: ENDGame, $640 \times 320$ cells, $\Delta t = 225\,\mathrm{s}$. The plotted region is $0^\mathrm{o}$ to $360^\mathrm{o}$ longitude, $10^\mathrm{o}$ to $80^\mathrm{o}$ latitude. The contour interval is $2 \times 10^{-5}\,\mathrm{s}^{-1}$. []{data-label="fig:galewsky"}](galewsky_fem_hex_163842.eps){width="150mm"}
![Relative vorticity field at day 6 for the barotropic instability test case. Row 1: hexagonal mesh, 10242 cells, $\Delta t = 900\,\mathrm{s}$. Row 2: hexagonal mesh, 163842 cells, $\Delta t = 225\,\mathrm{s}$. Row 3: cubed sphere mesh, 13824 cells, $\Delta t = 900\,\mathrm{s}$. Row 4: cubed sphere mesh, 221184 cells, $\Delta t = 225\,\mathrm{s}$. Row 5: ENDGame, $640 \times 320$ cells, $\Delta t = 225\,\mathrm{s}$. The plotted region is $0^\mathrm{o}$ to $360^\mathrm{o}$ longitude, $10^\mathrm{o}$ to $80^\mathrm{o}$ latitude. The contour interval is $2 \times 10^{-5}\,\mathrm{s}^{-1}$. []{data-label="fig:galewsky"}](galewsky_fem_cube_13824.eps){width="150mm"}
![Relative vorticity field at day 6 for the barotropic instability test case. Row 1: hexagonal mesh, 10242 cells, $\Delta t = 900\,\mathrm{s}$. Row 2: hexagonal mesh, 163842 cells, $\Delta t = 225\,\mathrm{s}$. Row 3: cubed sphere mesh, 13824 cells, $\Delta t = 900\,\mathrm{s}$. Row 4: cubed sphere mesh, 221184 cells, $\Delta t = 225\,\mathrm{s}$. Row 5: ENDGame, $640 \times 320$ cells, $\Delta t = 225\,\mathrm{s}$. The plotted region is $0^\mathrm{o}$ to $360^\mathrm{o}$ longitude, $10^\mathrm{o}$ to $80^\mathrm{o}$ latitude. The contour interval is $2 \times 10^{-5}\,\mathrm{s}^{-1}$. []{data-label="fig:galewsky"}](galewsky_fem_cube_221184.eps){width="150mm"}
![Relative vorticity field at day 6 for the barotropic instability test case. Row 1: hexagonal mesh, 10242 cells, $\Delta t = 900\,\mathrm{s}$. Row 2: hexagonal mesh, 163842 cells, $\Delta t = 225\,\mathrm{s}$. Row 3: cubed sphere mesh, 13824 cells, $\Delta t = 900\,\mathrm{s}$. Row 4: cubed sphere mesh, 221184 cells, $\Delta t = 225\,\mathrm{s}$. Row 5: ENDGame, $640 \times 320$ cells, $\Delta t = 225\,\mathrm{s}$. The plotted region is $0^\mathrm{o}$ to $360^\mathrm{o}$ longitude, $10^\mathrm{o}$ to $80^\mathrm{o}$ latitude. The contour interval is $2 \times 10^{-5}\,\mathrm{s}^{-1}$. []{data-label="fig:galewsky"}](galewsky_eg_640x320.eps){width="150mm"}
Conclusions and discussion {#conclusions}
==========================
A method of constructing low-order mimetic finite element spaces on arbitrary two-dimensional polygonal meshes, using compound elements, has been presented, along with corresponding discrete Hodge star operators for mapping between primal and dual function spaces. The method has been used as the basis of a numerical model to solve the shallow water equations on a rotating sphere. The model has the same mimetic properties, which underpin the ability to capture important physical properties, as the finite volume model of @thuburn2014, but with improved accuracy.
The finite volume model of @thuburn2014 relies on certain properties of the mesh for accuracy, namely the @heikes1995b optimization on the hexagonal mesh and the placement of primal vertices relative to dual vertices on the cubed sphere mesh. Although identical meshes have been used here to ensure the cleanest possible comparison, the mimetic finite element scheme does not depend on such mesh properties for accuracy; thus it provides greater flexibility in the choice of mesh.
An important practical consideration is the computational cost of the method. As a rough guide, the cost of the finite element model on a single processor varied between 3.3 and 4.6 times the cost of ENDGame for the cubed sphere mesh and between 4.2 and 7.3 times the cost of ENDGame for the hexagonal mesh, at the resolutions tested[^4]. (For comparison, the cost of the finite volume model varied between 2.7 and 3.7 times the cost of ENDGame for the cubed sphere mesh and between 3.3 and 4.9 times the cost of ENDGame for the hexagonal mesh.) The greater cost on the hexagonal mesh compared to the cubed sphere results from a combination of a greater stencil size for some operators and, in the current implementation, a less cache-friendly mesh numbering (the latter could straightfowardly be optimized). Given the potential to optimize the implementation and the expected gains in parallel scalability from the quasi-uniform mesh, these figures suggest that, despite the need for indirect addressing and the need to invert several linear operators, the finite element method need not be prohibitively expensive compared to methods currently used for operational forecasting, typified by ENDGame.
Computing integrals over compound elements is more complex and costly than for the usual triangular or quadrilateral elements. In the current implementation, all the operators $L$, $M$, $H$, $J$, $R$, $W$, and $T$ are precomputed, thus avoiding the need for any run-time quadrature in the finite element parts of the calculations[^5]. (Also, once these operators are computed, there is no need to retain the details of how the compound elements were built from subelements.) This precomputation is possible because all but one of these operators are linear; the only nonlinear term (other than advection) is a simple quadratic nonlinearity in the kinetic energy. In a system with more complex nonlinearities, such as the pressure gradient term in a compressible three-dimensional fluid, precomputation might not be possible and some run time quadrature would be unavoidable. Even so, in a high performance computing environment it is not clear whether precomputation or run-time quadrature would be most efficient, given the relative cost of memory access and computation (David Ham, pers. comm.).
The mathematical similarity of the finite element and finite volume formulations has been emphasized, the principal difference being the appearance of mass matrices in the finite element formulation. The similarity is made even clearer if we use [[(\[eqn\_hodge02\])]{}]{}, [[(\[eqn\_hodge11\])]{}]{} and [[(\[DIeqHD\])]{}]{} to rewrite [[(\[mudot\])]{}]{} in the equivalent dual space form $$\label{mudotdual}
\dot{\hat{U}} + \widehat{Q^\perp} + \overline{D}_1 ( \hat{\Phi} + \hat{K} ) = 0 .$$ The velocity degrees of freedom then correspond to dual edge circulations, and $\hat{U}$ can be identified with the $V$ of @thuburn2014. Equations [[(\[phidot\])]{}]{} and [[(\[mudotdual\])]{}]{} explicitly involve only the topological derivative operators $D_2$ and $\overline{D}_1$; the metric enters through the Hodge star operators needed to map between primal and dual function spaces. This approach of isolating the metric from the purely topological operators in order to construct numerical methods with mimetic properties on complex geometries or meshes has been advocated by several authors [e.g. @bossavit1998; @hiptmair2001; @palha2014 and references therein].
It is also worth emphasizing that the roles of primal and dual function spaces are not symmetrical here. Although any given field may be represented in both the primal and dual spaces, with a reversible Hodge star map between them, only primal space test functions are ever used, and so only primal space mass matrices appear, and dual space weak derivatives are never needed. (An interesting alternative would be to use only dual space test functions; then the prognostic equations remain [[(\[phidot\])]{}]{} and [[(\[mudotdual\])]{}]{}, but [[(\[eqn\_hodge02\])]{}]{}-[[(\[eqn\_hodge20\])]{}]{} are replaced by $$I^T \Phi = \hat{L} \hat{\Phi} ,
\label{eqn_alt_hodge02}$$ $$H^T U = \hat{M} \hat{U} ,
\label{eqn_alt_hodge11}$$ $$J^T \Xi = \hat{N} \hat{\Xi} ,
\label{eqn_alt_hodge20}$$ where $\hat{L}$, $\hat{M}$ and $\hat{N}$ are the mass matrices for the spaces $\mathbf{V}^0$, $\mathbf{V}^1$ and $\mathbf{V}^2$, respectively.)
Only the lowest order polygonal finite element spaces are used here: compound $\text{P1-RT0-P0}^\text{DG}$. An interesting question is whether the approach can be extended to higher order. The harmonic extension idea of @christiansen2008 has been extended to higher order by @christiansen2010. It appears plausible that higher order compound elements could be built from constrained linear combinations of, for example, the $\text{P2}^+\text{-BDFM1-P1}^\text{DG}$ elements recommended by @cotter2012, but the details have yet to be worked out. A more subtle question is whether suitable higher order dual spaces can be constructed.
Another, more straightforward, extension of the compound element approach is to three dimensions. The compound elements used here can be extruded into polygonal prisms; we have made some initial progress in working out the details of using such a scheme for the compressible Euler equations. (In atmosphere and ocean models it is desirable, for several reasons, to use a columnar mesh.) Fully three-dimensional compound elements can also be constructed using the discrete harmonic extension approach. These might be useful, for example, to implement a finite element version of the cut cell method for handling bottom topography [e.g. @lock2012 and references therein] while retaining a columnar mesh.
Besides their ability to use arbitrary polygonal meshes, another potentially useful property of the compound elements used here is that the function spaces are built directly in physical space, without the need for Piola transforms. Thus, for example, globally constant functions are always contained in $\mathbf{V}_2$. In this way, the compound elements avoid the reduced convergence rate, and even loss of consistency, discussed by @arnold2014, and so provide an alternative to the [*rehabilitation*]{} technique of @bochev2008.
[**Acknowledgements**]{}
We are grateful to Thomas Dubos for drawing our attention to the work of @buffa2007 and @christiansen2008, to Martin Schreiber and David Ham for valuable discussions on code optimization, and to Nigel Wood for helpful comments on a draft of this paper. This work was funded by the Natural Environment Research Council under the “Gung Ho” project (grants NE/I021136/1, NE/I02013X/1, NE/K006762/1 and NE/K006789/1).
Operator inverses and sparse approximate $M$ inverse {#lumped}
====================================================
Inverses of the $H$ and $J$ operators are needed at the beginning of every time step, and inverses of $H$ and $M$ are needed at every Newton iteration. These are computed by (under- or over-relaxed) Jacobi iteration based on a diagonal approximation of the relevant operator. E.g., to solve $A x = R$, define $$\label{jacobi_fg}
x^{(1)} = (A^{*})^{-1} R$$ where $A^{*}$ is a diagonal approximation to $A$, then iterate: $$x^{(l+1)} = x^{(l)} + \mu (A^{*})^{-1} (R - A x^{(l)}).$$
A diagonal approximation $J^{*}$ to the operator $J$ is defined by demanding that, for every dual cell $j$, $J^{*}$ and $J$ should give the same result in dual cell $j$ when acting on the $\mathbf{V}^2$ representation of a constant scalar field. A diagonal approximation $M^{*}$ to the velocity mass matrix $M$ is defined by demanding that, for every edge $e$, $M^{*}$ and $M$ should give the same result at edge $e$ when acting on the $\mathbf{V}_1$ representation of a solid body rotation velocity field whose maximum velocity is normal to primal edge $e$. A diagonal approximation $H^{*}$ to the operator $H$ is defined by demanding that, for every edge $e$, $H^{*}$ and $H$ should give the same result at edge $e$ when acting on the $\mathbf{V}^1$ representation of a solid body rotation velocity field whose maximum velocity is tangential to dual edge $e$.
Optimal values of the relaxation parameter $\mu$ were found to depend on the operator and mesh structure. The values used are given in table \[tab:relax\].
Grid $J^{-1}$ $M^{-1}$ $H^{-1}$
------ ---------- ---------- ----------
Hex 1.4 1.4 1.4
Cube 1.4 0.9 1.4
: Relaxation parameters used for Jacobi iteration for operator inverses.[]{data-label="tab:relax"}
For the inverses that occur once per time step, 10 Jacobi iterations are used. For those that occur at every Newton iteration, 2 Jacobi iterations are used taking the solution obtained at the previous Newton iteration as the first guess (or [[(\[jacobi\_fg\])]{}]{} on the first Newton iteration).
A sparse approximate inverse $\mathcal{M}^{-1}$ of the $\mathbf{V}_1$ mass matrix is needed for the Helmholtz problem. On the hexagonal mesh it is sufficient to use a diagonal approximation $$\mathcal{M}^{-1} = (M^{*})^{-1}.$$ However, on the cubed sphere mesh, whose dual and primal edges are not mutually orthogonal, such a diagonal approximation is less accurate and limits the convergence of the Newton iterations. Therefore we use instead an approximation based on a single Jacobi iteration towards the inverse of $M$: $$\mathcal{M}^{-1} = (M^{*})^{-1} \left\{ (1 + \mu)\mathrm{Id} - \mu M (M^{*})^{-1} \right\},$$ where $\mathrm{Id}$ is the identity matrix. This approximate inverse is not diagonal but has the same stencil as $M$ itself.
[**References**]{}
[00]{}
Arnold, D. N., Boffi, D. and Bonizzoni, F.: Finite element differential forms on curvilinear cubic meshes and their approximation properties, Numer. Math., DOI:10.1007/s00211-014-0631-3, 2014.
Rehabilitation of the lowest-order Raviart-Thomas element on quadrilateral grids, SIAM J. Numer. Anal., 47, 487–507, 2008.
Bossavit, A.: Computational Electromagnetism: Variational Formulation, Complementarity, Edge Elements. Academic Press Electromagnetism Series, Number 2 San Diego: Academic Press, 1998.
Buffa, A., and Christiansen, S. H.: A dual finite element complex on the barycentric refinement, Math. of Comput., 76, 1743–1769, 2007.
Christiansen, S. H.: A construction of spaces of compatible differential forms on cellular complexes, Math. Models and Methods in Appl. Sci., 18, 739–757, 2008.
Christiansen, S. H.: Minimal mixed finite elements on polyhedra, C. R. Acad. Sci. Paris, 348, 217–221, 2010.
Cotter, C. J. and Shipton, J.: Mixed finite elements for numerical weather prediction, J. Comput. Phys., 231, 7076–7091, 2012.
Cotter, C. J. and Thuburn, J.: A finite element exterior calculus framework for the rotating shallow-water equations, J. Comp. Phys., 257, 1506–1526, 2014. Danilov, S.: On the utility of triangular C-grid type discretization for numerical modeling of large-scale ocean flows, Ocean Dyn., 60, 1361–1369, 2010.
Galewsky, J., Scott, R. K., and Polvani, L. M.: An initial value problem for testing numerical models of the global shallow water equations, Tellus A, 56, 429–440, 2004.
Heikes, R. and Randall, D.: Numerical integration of the shallow-water equations on a twisted icosahedral grid, Part II: A detailed description of the grid and analysis of numerical accuracy, Mon. Weather Rev., 123, 1881–1997, 1995b.
Hiptmair, R.: Discrete Hodge operators, Numer. Math., 90, 265–289, 2001.
Le Roux, D. Y., Rostand, V., and Pouliot, B.: Analysis of numerically induced oscillations in 2D finite-element shallow-water models. Part I: Inertia-gravity waves, SIAM J. Sci. Comp., 29, 331–360, 2007.
Lock, S.-J., Bitzer, H.-W., Coals, A., Gadian, A., and Mobbs, S.: Demonstration of a cut-cell representation of 3D orography for studies of atmospheric flows over very steep hills, Mon. Weather Rev., 140, 411–424.
McRae, A. T. T. and Cotter, C.: Energy- and enstrophy-conserving schemes for the shallow-water equations, based on mimetic finite elements, Submitted to Q. J. R. Meteorol. Soc. (available at http://dx.doi.org/10.1002/qj.2291 ).
Melvin, T., Staniforth, A., and Cotter, C.: A two-dimensional mixed finite-element pair on rectangles, Quart. J. Roy. Meteorol. Soc., 140, 930–942, 2013.
Melvin, T. and Thuburn, J.: Wave dispersion properties of compound finite elements, submitted to J. Comput. Phys.
Physics-compatible discretization techniques on single and dual grids, with application to the Poisson equation of volume forms. J. Comput. Phys., 257, 1394–1422, 2014.
Ringler, T. D., Thuburn, J., Klemp, J. B., and Skamarock, W. C.: A unified approach to energy conservation and potential vorticity dynamics for arbitrarily structured C-grids, J. Comput. Phys., 229, 3065–3090, 2010.
Staniforth, A. and Thuburn, J.: Horizontal grids for global weather and climate prediction models: a review, Q. J. Roy. Meteor. Soc., 138, 1–26, 2012.
Thuburn, J. and Cotter, C. J.: A framework for mimetic discretization of the rotating shallow water equations on arbitrary polygonal grids, SIAM J. Sci. Comput., 34, 203–225, 2012.
Thuburn J., Cotter, C. J., and Dubos, T.: A mimetic, semi-implicit, forward-in-time, finite volume shallow water model: comparison of hexagonal–icosahedral and cubed sphere grids, Geosci. Model Dev., 7, 909–929, 2014.
Thuburn, J., Ringler, T. D., Skamarock, W. C., and Klemp, J. B.: Numerical representation of geostrophic modes on arbitrarily structured C-grids, J. Comput. Phys., 228, 8321–8335, 2009.
Ullrich, P.A.: Understanding the treatment of waves in atmospheric models. Part 1: The shortest resolved waves of the 1D linearized shallow-water equations, Quart. J. Roy. Meteorol. Soc., 140, 1426–1440, 2014.
Weller, H.: Controlling the computational modes of the arbitrarily structured C-grid, Mon. Weather Rev., 140, 3220–3234, 2012.
Weller, H.: Non-orthogonal version of the arbitrary polygonal C-grid and a new diamond grid, Geosci. Model Dev., 7, 779–797, 2014.
Williamson, D. L., Drake, J. B., Hack, J. J., Jakob, R., and Swartztrauber, P. N.: A standard test set for numerical approximations to the shallow water equations in spherical geometry, J. Comput. Phys., 102, 211–224, 1992.
Zerroukat, M., Wood, N., Staniforth, A., White, A. A., and Thuburn, J.: An inherently mass-conserving semi-implicit semi-Lagrangian discretization of the shallow water equations on the sphere, Quart. J. Roy. Meteorol. Soc., 135, 1104–1116, 2009.
[^1]: Readers wishing to compare the two formulations should note that a different sign convention is used for the expansion coefficients of $\mathbf{k}\times$ any vector, such as $U^\perp$ in [[(\[uperp\])]{}]{}.
[^2]: Note $\nabla^\perp$ and $\mathbf{k} \cdot \nabla \times$ (like $\nabla$ and $\nabla \cdot$) can both be defined as intrinsic operations on a curved surface, without reference to $\mathbf{k}$ or a third dimension.
[^3]: The most general form of the Helmholtz-Hodge decomposition of a vector field $\mathbf{u}$ in 2D is $$\mathbf{u} = \nabla \phi + \nabla^\perp \psi + \mathbf{h},$$ where $\phi$ is a potential, $\psi$ is a stream function, and $\mathbf{h}$ is a harmonic vector field, i.e. one satisfying $$\nabla \nabla \cdot \mathbf{h} +
\nabla^\perp \mathbf{k} \cdot \nabla \times \mathbf{h} =
\mathbf{0}.$$ The fourth condition in Property List 1 implies that all nondivergent fields $\mathbf{u}$ can be written as $\nabla^\perp \psi$, which rules out the possibility of harmonic vector fields. This is appropriate for spherical geometry, since there exist no non-zero harmonic vector fields on the sphere. However, for a doubly period plane, for example, for which a constant vector field is harmonic, we would have to extend the fourth condition to allow for harmonic vector fields. This issue does not affect any of the discussion below except for the discrete Helmholtz decomposition (section \[deriv2\]), which would only need to be extended in the obvious way to allow for harmonic vector fields.
[^4]: Martin Schreiber (pers. comm.) reports that the cost of the finite element model can be significantly reduced, by roughly a factor 2, by reordering the dimensions of a couple of key arrays to improve cache usage.
[^5]: Some run-time quadrature is done in the advection scheme to compute swept area integrals.
|
---
abstract: 'A $\k$-configuration is a set of points $\X$ in $\mathbb{P}^2$ that satisfies a number of geometric conditions. Associated to a $\k$-configuration is a sequence $(d_1,\ldots,d_s)$ of positive integers, called its type, which encodes many of its homological invariants. We distinguish $\k$-configurations by counting the number of lines that contain $d_s$ points of $\X$. In particular, we show that for all integers $m \gg 0$, the number of such lines is precisely the value of $\Delta \H_{m\X}(m d_s -1)$. Here, $\Delta \H_{m\X}(-)$ is the first difference of the Hilbert function of the fat points of multiplicity $m$ supported on $\X$.'
address:
- |
Department of Mathematics and Statistics\
McMaster University, Hamilton, ON, L8S 4L8
- 'Department of Mathematics, Sungshin Women’s University, Seoul, Korea, 136-742'
- |
Department of Mathematics and Statistics\
McMaster University, Hamilton, ON, L8S 4L8
author:
- Federico Galetto
- 'Yong-Su Shin'
- Adam Van Tuyl
title: 'Distinguishing $\k$-configurations'
---
[^1]
Introduction {#sec:intro}
============
In the late 1980’s, Roberts and Roitman [@RR] introduced special configurations of points in $\mathbb{P}^2$ which they named $\k$-configurations. We recall this definition:
\[kdefn\] A $\k$-configuration of points in $\mathbb{P}^2$ is a finite set $\mathbb{X}$ of points in $\mathbb{P}^2$ which satisfies the following conditions: there exist integers $1\leqslant d_1 < \cdots < d_s$, subsets $\mathbb{X}_1, \ldots, \mathbb{X}_s$ of $\mathbb{X}$, and distinct lines $\mathbb{L}_1, \ldots, \mathbb{L}_s \subseteq \mathbb{P}^2$ such that:
1. $\mathbb{X} = \bigcup_{i=1}^s \mathbb{X}_i$;
2. $|\mathbb{X}_i| = d_i$ and $\mathbb{X}_i \subseteq \mathbb{L}_i$ for each $i=1,\ldots,s$, and;
3. $\mathbb{L}_i$ ($1< i \leqslant s$) does not contain any points of $\mathbb{X}_j$ for all $1{\leqslant}j<i$.
In this case, the $\k$-configuration is said to be of type $(d_1,\ldots,d_s)$.
This definition was first extended to $\mathbb{P}^3$ by Harima [@H], and later to all $\mathbb{P}^n$ by Geramita, Harima, and Shin (see [@GHS:1; @GHS:5]). As shown by Roberts and Roitman [@RR Theorem 1.2], all $\k$-configurations of type $(d_1,\ldots,d_s)$ have the same Hilbert function (which can be computed from the type). This result was later generalized by Geramita, Harima, and Shin [@GHS:2 Corollary 3.7] to show that all the graded Betti numbers of the associated graded ideal $I_\X$ only depend upon the type.
Interestingly, $\k$-configurations of the same type can have very different geometric properties. Figure \[fig:kconfig-123\] shows various examples of $\k$-configurations of type $(1,2,3)$. Note that the different shapes correspond to different sets of $\X_i$, i.e., the star is the point of $\X_1$, the squares are the two points of $\X_2$, and the circles are the three points $\X_3$.
at (-2,6) [$\L_1$]{}; at (-2,0) [$\L_2$]{}; at (-1,-2) [$\L_3$]{}; (-1,-1) rectangle (10,8); (-1,0)–(10,0); (-1,-1)–(10,10); (-2,6)–(13,-3); (2,-6)–(8,12); at (0,0) ; at (3,3) ; at (6,6) ; at (4,0) ; at (8,0) ; at (4.66,2) ;
at (-2,6) [$\L_1$]{}; at (-2,0) [$\L_2$]{}; at (-1,-2) [$\L_3$]{}; (-1,-1) rectangle (10,8); (-1,0)–(10,0); (-1,-1)–(10,10); (-2,6)–(13,-3); (2,-6)–(8,12); at (0,0) ; at (3,3) ; at (6,6) ; at (4,0) ; at (8,0) ; at (5.5,1.5) ;
at (-2,6) [$\L_1$]{}; at (-2,0) [$\L_2$]{}; at (-1,-2) [$\L_3$]{}; (-1,-1) rectangle (10,8); (-1,0)–(10,0); (-1,-1)–(10,10); (-2,6)–(13,-3); (2,-6)–(8,12); at (0,0) ; at (3,3) ; at (6,6) ; at (4,0) ; at (6.5,0) ; at (5.5,1.5) ;
at (-2,6) [$\L_1$]{}; at (-2,0) [$\L_2$]{}; at (-1,-2) [$\L_3$]{}; (-1,-1) rectangle (10,8); (-1,0)–(10,0); (-1,-1)–(10,10); (-2,6)–(13,-3); (2,-6)–(8,12); at (1,1) ; at (3,3) ; at (6,6) ; at (4,0) ; at (6.5,0) ; at (5.5,1.5) ;
From a geometric point-of-view, these configurations are all qualitatively different in that the number of lines containing $3$ points in each configuration is different (e.g., there are four lines that contain $3$ points of $\X$ in the first configuration, but only one such line in the last configuration). However, from an algebraic point-of-view, because these sets of points are all $\k$-configurations of type $(1,2,3)$, the graded resolutions (and consequently, the Hilbert functions) of these sets of points are all the same. So the algebra does not “see” these lines, and so we cannot distinguish these $\k$-configurations.
Our goal in this paper is to determine how one can distinguish these $\k$-configurations from an algebraic point-of-view. In particular, we wish to distinguish $\k$-configurations by the number of lines that contain $d_s$ points of $\X$. It can be shown (see Remark \[hilbertfcnreducedbnd\]) that the first difference function of the Hilbert function of $\X$ produces only an upper bound on the number of lines. We show that one can obtain an exact value if one instead considers the Hilbert function of the set of fat points supported on the $\k$-configuration. Precisely, we prove:
\[maintheorem\] Let $\X \subseteq \mathbb{P}^2$ be a $\k$-configuration of type $d=(d_1,\dots,d_s) \neq (1)$. Then there exists an integer $m_0$ such that for all $m {\geqslant}m_0$, $$\Delta \H_{m\X}(md_s-1) = \mbox{number
of lines containing exactly $d_s$ points of $\X$,}$$ where $\Delta \H_{m\X}(-)$ is the first difference function of the Hilbert function of fat points of multiplicity $m$ supported on $\X$. Furthermore, if $d_s > s$, then $m_0 = 2$, and if $d_s = s$, then $m_0 = s+1$.
In other words, the number of lines that contain $d_s$ points of $\X$ is encoded in the Hilbert function of fat points supported on $\X$. This provides us with an algebraic method to differentiate $\k$-configurations. Note we exclude the $\k$-configuration of type $d=(1)$ since $\X$ is a single point, and there are an infinite number of lines through this point. Thematically, this paper is similar to works of Bigatti, Geramita, and Migliore [@BGM], and Chiantini and Migliore [@CM] which derived geometric consequences about points from the Hilbert function.
We now give an outline of the paper. In Section 2, we define all the relevant terminology involving $\k$-configurations, and some properties of $\k$-configurations. We also recall a procedure to bound values of the Hilbert function of a set of fat points due to Cooper, Harbourne, and Teitler [@CHT], which will be our main tool. In Section 3, we focus on the case $d_s > s$ and prove Theorem \[maintheorem\] in this case. In Section 4, we focus on the case that $d_s = s$. A more subtle argument is needed to prove Theorem \[maintheorem\] since an extra line may come into play. We will also require a result of Catalisano, Trung, and Valla [@CTV] to complete this case. In the final section, we give a reformulation of Theorem \[maintheorem\], and make a connection to a question of Geramita, Migliore, and Sabourin [@GMS] on the number of Hilbert functions of fat points whose support has a fixed Hilbert function.
Background results {#sec:background}
==================
This section collects the necessary background results. We first review Hilbert functions and ideals of (fat) points in $\mathbb{P}^2$. We then introduce a number of lemmas describing $\k$-configurations. Throughout the remainder of this paper, $R = \k[x_0,x_1,x_2]$ is a polynomial ring over an algebraically closed field $\k$.
Points in P2 and Hilbert functions
----------------------------------
We recall some general facts about (fat) points in $\mathbb{P}^2$ and their Hilbert functions. These results will be used later in our study of $\k$-configurations.
Let $\X = \{P_1,\ldots,P_s\}$ be a set of distinct points in $\mathbb{P}^2$. If $I_{P_i}$ is the ideal associated to $P_i$ in $R =
\k[x_0,x_1,x_2]$, then the homogeneous ideal associated to $\X$ is the ideal $I_\X = I_{P_1} \cap \cdots \cap I_{P_s}$. Given $s$ positive integers $m_1,\ldots,m_s$ (not necessarily distinct), the scheme defined by the ideal [ $I_\Z = I_{P_1}^{m_1} \cap \cdots \cap I_{P_s}^{m_s}$]{} is called a set of [*fat points*]{}. We say that $m_i$ is the [*multiplicity*]{} of the point $P_i$. If $m_1 = \cdots = m_s = m$, then we say $\Z$ is a [*homogeneous set of fat points*]{} of multiplicity $m$. In this case, we normally write $m\X$ for $\Z$, and $I_{m\X}$ for $I_\Z$.
Note that it can be shown that $I_{m\X} = I_{\X}^{(m)}$, the $m$-th symbolic power of the ideal $I_\X$. If $\X = \{P\}$, then we sometimes write $I_{mP}$ for $I_{m\X}$. As well, since $I_P$ is a complete intersection, it follows that $I_{mP} = I_P^{(m)} = I_P^m$ (see Zariski-Samuel [@ZS Appendix 6, Lemma 5]).
An ongoing problem at the intersection of commutative algebra and algebraic geometry is to study and classify the Hilbert functions arising from the homogeneous ideals of sets of fat points. Recall that if $I \subseteq R = \k[x_0,x_1,x_2]$ is any homogeneous ideal, then the [*Hilbert function*]{} of $R/I$, denoted $\H_{R/I}$, is the numerical function $\H_{R/I}:\mathbb{N} \rightarrow \mathbb{N}$ defined by $$\H_{R/I}(t) := \dim_\k R_t - \dim_\k I_t$$ where $R_t$, respectively $I_t$, denotes the $t$-th graded component of $R$, respectively $I$. If $I = I_\Z$ is the defining ideal of a set of (fat) points $\Z$, then we usually write $\H_{\Z}$ for $\H_{R/I_\Z}$. The [*first difference*]{} of the Hilbert function $\H_{R/I}$, is the function $$\Delta \H_{R/I}(t) := \H_{R/I}(t) - \H_{R/I}(t-1) ~~\mbox{for all
$t {\geqslant}0$} ~~\mbox{ where $\H_{R/I}(t) = 0$ for all $t < 0$.}$$
Given a set of points $\Z \subseteq \mathbb{P}^2$, Cooper, Harbourne, and Teitler [@CHT] described a procedure by which one can find both upper and lower bounds on $\H_\Z(t)$ for all $t {\geqslant}0$. This procedure, which we describe below, will be instrumental in the proof of our main results.
Let $\Z=\Z_0$ be a fat point subscheme of $\mathbb{P}^2$. Choose a sequence of lines $\L_1,\dots,\L_r$ and define $\Z_i$ to be the residual of $\Z_{i-1}$ with respect to the line $\L_i$ (i.e. the subscheme of $\mathbb{P}^2$ defined by the ideal $I_{\mathbb{Z}_i} : I_{\L_i}$). Define the associated [*reduction vector*]{} $\mathbf{v} = (v_1,\dots,v_r)$ by taking $v_i = \deg (\L_i \cap \Z_{i-1})$. In particular, $v_i$ is the sum of the multiplicities of the points in $\L_i \cap \Z_{i-1}$. Given $\mathbf{v}=(v_1,\dots,v_r)$, we define functions $$\label{lowerbound}
f_{\mathbf{v}} (t) = \sum_{i=0}^{r-1} \min (t-i+1,v_{i+1})$$ and $$\label{upperbound}
F_{\mathbf{v}} (t) = \min_{0{\leqslant}i{\leqslant}r} \bigg(
\binom{t+2}{2} - \binom{t-i+2}{2} + \sum_{j=i+1}^{r} v_j
\bigg).$$
\[CHT\] Let $\Z=\Z_0$ be a fat point scheme in $\mathbb{P}^2$ with reduction vector $\mathbf{v} = (v_1,\dots,v_r)$ such that $\Z_{r+1} = \varnothing$. Then the Hilbert function $\H_\Z (t)$ of $\Z$ is bounded by $f_{\mathbf{v}} (t) {\leqslant}\H_\Z (t) {\leqslant}F_{\mathbf{v}} (t)$.
\[ex:CHT\] Let $\X$ be the $\k$-configuration of type $(1,3,4,5)$ in Figure \[fig:CHT\]. We illustrate how to use Theorem \[CHT\] to compute $\H_{2\X} (8)$. We take $\Z = 2\X$, i.e., we assume that each point has multiplicity two; this is indicated by the $2$ by each point.
at (-2,6) [$\L_4$]{}; at (-2,0) [$\L_3$]{}; at (-1,-2) [$\L_2$]{}; at (4,-2) [$\L_1$]{}; (-1,-1.2) rectangle (10,8); (-1,0)–(10,0); (-1,-1)–(10,10); (-2,6)–(13,-3); (2,-6)–(8,12); at (8,0) \[pin=[below:2]{}\] ; at (6.33,1) \[pin=[above right:2]{}\] ; at (3,3) \[pin=[above:2]{}\] ; at (1.33,4) \[pin=[above:2]{}\] ; at (-0.33,5) \[pin=[above:2]{}\] ; at (0,0) \[pin=[below:2]{}\] ; at (1.75,0) \[pin=[below:2]{}\] ; at (3.5,0) \[pin=[above:2]{}\] ; at (5.5,0) \[pin=[below:2]{}\] ; at (1.5,1.5) \[pin=[above:2]{}\] ; at (4.5,4.5) \[pin=[above:2]{}\] ; at (6,6) \[pin=[\[pin distance=4pt\]right:2]{}\] ; at (5,3) \[pin=[\[pin distance=4pt\]right:2]{}\] ;
at (-2,6) [$\L_4$]{}; at (-2,0) [$\L_3$]{}; at (-1,-2) [$\L_2$]{}; at (4,-2) [$\L_1$]{}; (-1,-1.2) rectangle (10,8); (-1,0)–(10,0); (-1,-1)–(10,10); (-2,6)–(13,-3); (2,-6)–(8,12); at (8,0) \[pin=[below:1]{}\] ; at (6.33,1) \[pin=[above right:1]{}\] ; at (3,3) \[pin=[above:1]{}\] ; at (1.33,4) \[pin=[above:1]{}\] ; at (-0.33,5) \[pin=[above:1]{}\] ; at (0,0) \[pin=[below:2]{}\] ; at (1.75,0) \[pin=[below:2]{}\] ; at (3.5,0) \[pin=[above:2]{}\] ; at (5.5,0) \[pin=[below:2]{}\] ; at (1.5,1.5) \[pin=[above:2]{}\] ; at (4.5,4.5) \[pin=[above:2]{}\] ; at (6,6) \[pin=[\[pin distance=4pt\]right:2]{}\] ; at (5,3) \[pin=[\[pin distance=4pt\]right:2]{}\] ;
at (-2,6) [$\L_4$]{}; at (-2,0) [$\L_3$]{}; at (-1,-2) [$\L_2$]{}; at (4,-2) [$\L_1$]{}; (-1,-1.2) rectangle (10,8); (-1,0)–(10,0); (-1,-1)–(10,10); (-2,6)–(13,-3); (2,-6)–(8,12); at (8,0) \[pin=[below:0]{}\] ; at (6.33,1) \[pin=[above right:1]{}\] ; at (3,3) \[pin=[above:1]{}\] ; at (1.33,4) \[pin=[above:1]{}\] ; at (-0.33,5) \[pin=[above:1]{}\] ; at (0,0) \[pin=[below:1]{}\] ; at (1.75,0) \[pin=[below:1]{}\] ; at (3.5,0) \[pin=[above:1]{}\] ; at (5.5,0) \[pin=[below:1]{}\] ; at (1.5,1.5) \[pin=[above:2]{}\] ; at (4.5,4.5) \[pin=[above:2]{}\] ; at (6,6) \[pin=[\[pin distance=4pt\]right:2]{}\] ; at (5,3) \[pin=[\[pin distance=4pt\]right:2]{}\] ;
We apply Theorem \[CHT\] using the sequence of lines $\HH_1,\dots,\HH_{8}$, where $\HH_1=\L_4$, $\HH_2=\L_3$, $\HH_3=\L_2$, $\HH_4=\L_1$, $\HH_5=\L_4$, $\HH_6=\L_3$, $\HH_7=\L_2$, and $\HH_8=\L_1$. The reduction vector is ${\bf v}=(10,9,8,3,3,3,2,1)$. To see this, note that $\Z_0 = 2\X$, so $\HH_1 \cap \Z_0$ consists of the five double points on $\L_4$, so $\deg(\HH_1 \cap \Z_0) = 10$. We form $\Z_1$ from $\Z_0$ by reducing the multiplicity of each point on $\HH_1 = \L_4$ by one. Then $\HH_2 \cap \Z_1 = \L_3 \cap \Z_1$ consists of 4 double points and one reduced point, so $\deg(\HH_2 \cap \Z_1) = 2\cdot 4 + 1 = 9$. Figure \[fig:CHT\] illustrates the first two steps of this procedure. Continuing in this fashion allows us to compute ${\bf v}$, ending when we reach $\Z_9 = \varnothing$.
To compute the lower bound $f_{\bf v} (8)$ using Equation , we compare the values of $8-i+1$ and $v_{i+1}$ in the table below (the minimum is in bold).
---------------------------------------------------------------------------------------------------
$i$ 0 1 2 3 4 5 6 7
--------- ------------------------------------------------------- --- --- --- --- --- --- --- -- --
$8-i+1$ **9 & **8 & **7 & 6 & 5 & 4 & 3 & 2\
$v_{i+1}$ & 10 & 9 & 8 & **3 & **3 & **3 & **2 & **1\
****************
---------------------------------------------------------------------------------------------------
Adding up the minimum values gives $
f_{\bf v}(8)=36.
$ To compute an upper bound, we take $i=3$ in Equation ; then we have $$F_{\bf v}(8){\leqslant}\binom{8+2}{2}-\binom{5+2}{2}+\sum_{j=4}^8 v_j=36.$$ This implies that $
f_{\bf v}(8)=F_{\bf v}(8)=\H_{2\X}(8)=36.
$
In Example \[ex:CHT\], we have used the procedure of [@CHT] to find actual values of the Hilbert function. In general, however, one can only expect to find bounds.
Properties of k-configurations
------------------------------
In this section, we record a number of useful facts about $\k$-configurations.
\[bounds\] Suppose that $\X \subseteq \mathbb{P}^2$ is a $\k$-configuration of type $(d_1,\ldots,d_s)$. Then
1. $d_j {\leqslant}d_s - s + j$ for $j=1,\ldots,s$;
2. if $d_s =s$, then $(d_1,\ldots,d_s) = (1,\ldots,s)$;
3. for any line $\L$ in $\mathbb{P}^2$, $|\L \cap \X| {\leqslant}d_s.$
Statements $(i)$ and $(ii)$ follow directly from the definition of $\k$-configurations since $1 {\leqslant}d_1 < d_2 < \cdots < d_s$. Statement $(iii)$ is [@RR Lemma 1.3].
By definition, there is at least one line $\L$ in $\mathbb{P}^2$ that meets a $\k$-configuration $\X$ of type $(d_1,\ldots,d_s)$ at $d_s$ points, namely, the line $\L_s$. As mentioned in the introduction, our goal is to enumerate the lines that meet $\X$ at exactly $d_s$ points. We begin with some useful necessary conditions for a line $\L$ to contain $d_s$ points.
\[necessaryconditions\] Suppose that $\X \subseteq \mathbb{P}^2$ is a $\k$-configuration of type $(d_1,\ldots,d_s)$, and $\L_1,\ldots,\L_s$ are the lines used to define $\X$. Let $\L$ be any line in $\mathbb{P}^2$ such that $|\L \cap \X| = d_s$.
1. If $d_s > s$, then $\L \in \{\L_1,\ldots,\L_s\}$.
2. If $\L = \L_i$, then $d_j = d_s-s+j$ for $j = i,\ldots,s$.
$(i)$ If $\L \not\in \{\L_1,\ldots,\L_s\}$, then $\L \cap \X \subseteq \bigcup_{i=1}^s (\L \cap \L_i)$. So if $s < d_s$, $|\L \cap \X| {\leqslant}\sum_{i=1}^s |\L \cap \L_i| = s < d_s$. In other words, if $|\L \cap \X| = d_s$, then $\L$ must be in $\{\L_1,\ldots,\L_s\}$.
$(ii)$ Suppose $\L = \L_i$ contains $d_s$ points of $\X$. By definition, $\L_i$ contains the $d_i$ points of $\X_i \subseteq \X$. Furthermore, this line cannot contain any of the points in $\X_1,\ldots,\X_{i-1}$. In addition, $\L_i$ can contain at most one point of $\X_{i+1},\ldots,\X_s$. So $d_s = |\L_i \cap \X| {\leqslant}d_i + (s-i)$. But by Lemma \[bounds\], we have $d_i + (s-i) {\leqslant}d_s$, so $d_s = d_i + (s-i)$. To complete the proof, note that $d_i < d_{i+1} < \cdots < d_s$ is a set of $s-i+1$ strictly increasing integers with $d_i = d_s - (s-i)$. This forces $d_j = d_s - (s-j)$ for all $j=i,\ldots,s$.
If $\X$ is a $\k$-configuration of type $(d_1,\ldots,d_s)$ with $d_{s-1} < d_{s}-1$, the above lemma implies that there is exactly one line containing $d_s$ points of $\X$, namely $\L_s$.
If $d_s > s$, Lemma \[necessaryconditions\] implies that the lines we want to count are among the $\L_i$’s, and consequently, there are at most $s$ such lines. The next result shows that when $d_s = s$ (or equivalently, the type is $(1,2,\ldots,s)$) the situation is more subtle. In particular, if there is a line $\L$ that contains $s$ points that is not among the $\L_i$’s, then it must be one of two lines.
\[extraline\] Suppose that $\X \subseteq \mathbb{P}^2$ is a $\k$-configuration of type $(1,2,\ldots,s)$ with $s {\geqslant}2$. Let $\X_1,\ldots,\X_s$ be the subsets of $\X$; let $\L_1,\ldots,\L_s$ be the lines used to define $\X$; let $\X_1 = \{P\}$ be the point on $\L_1$; and let $\X_2 = \{Q_1,Q_2\}$ be the two points on $\L_2$. If $\L$ is a line in $\mathbb{P}^2$ such that $|\L \cap \X| = d_s = s $, and if $\L \not\in \{\L_1,\ldots,\L_s\}$, then $\L$ must either be the line through $P$ and $Q_1$, or the line through $P$ and $Q_2$.
Suppose $|\L \cap \X| = d_s = s$. Since $\L \not\in \{\L_1,\dots,\L_s\}$, we have $s = |\L \cap \X| {\leqslant}|\L \cap \L_1| + \cdots + |\L \cap \L_s| =
s$. In other words, $\L \cap \L_i$ is a point of $\X_i \subseteq \X$ for $i=1,\ldots,s$. So $\L$ must pass through $P$ and either $Q_1$ or $Q_2$.
\[numberoflines\] Suppose that $\X \subseteq \mathbb{P}^2$ is a $\k$-configuration of type $(1,2,\ldots,s)$ with $s {\geqslant}2$. Then there are at most $s+1$ lines that contain $s$ points of $\X$.
The only candidates for the lines that contain $s$ points are the $s$ lines $\L_1,\ldots,\L_s$ that define the $\k$-configuration, and by Lemma \[extraline\], the two lines $\L_{PQ_1}$ and $\L_{PQ_2}$, i.e., the lines that go through the point of $\X_1 = \{P\}$ and one of the two points of $\X_2 = \{Q_1,Q_2\}$. This gives us $s+2$ lines. However, if the lines $\L_{PQ_1}$ and $\L_{PQ_2}$ both contain $s$ points, then either $\L_1$ is one of these two lines, or does not contain $s$ points. Indeed if $\L_1$ contains $s$ points, then $s=|\L_1 \cap \X| = |\X_1| + |\L_1 \cap \L_2| + \cdots +
|\L_1 \cap \L_s| =s$. In particular, $|\L_1 \cap \L_2| =1$, i.e., $\L_1$ must contain one of the two points of $\X_2$, and so $\L_1 = \L_{PQ_1}$ or $\L_{PQ_2}$. So, there are at most $s+1$ lines that contain $s$ points $\X$.
We finish this section with a useful lemma for relabelling a $\k$-configuration. This lemma exploits the fact that the lines and subsets defining a $\k$-configuration need not be unique.
\[relabel1\] Suppose that $\X \subseteq \mathbb{P}^2$ is a $\k$-configuration of type $(d_1,d_2,\ldots,d_s)$ with $s {\geqslant}2$. Let $\X_1,\ldots,\X_s$ be the subsets of $\X$, and $\L_1,\ldots,\L_s$ the lines used to define $\X$. Suppose that
1. $|\L_{s-k} \cap \X| = d_s$ for $k = 0,\ldots,j$.
2. $|\L_{s-k} \cap \X| < d_s$ for $k = j+1,\ldots,i-1$, and
3. $|\L_{s-i} \cap \X| = d_s$.
Set $\mathbb{T} = \L_{s-i} \cap (\X_{s-j-1} \cup \X_{s-j-2} \cup \cdots
\cup \X_{s-i+1})$.
Then the $\k$-configuration $\X$ can also be defined using the subsets $\X'_1,\ldots,\X'_s$ and lines $\L'_1,\ldots,\L'_s$ where
1. $\X'_k = \X_k$ and $\L'_k = \L_k$ for $k=1,\ldots,s-i-1$,
2. $\X'_k = \X_{k+1} \setminus \mathbb{T}$ and $\L'_k = \L_{k+1}$ for $k=s-i,\ldots,s-j-2$,
3. $\X'_{s-j-1} = \X_{s-i} \cup \mathbb{T}$ and $\L'_{s-j-1} = \L_{s-i}$, and
4. $\X'_k = \X_k$ and $\L'_k = \L_k$ for $k=s-j,\ldots,s$.
We need to verify that the subsets $\X'_i$ and lines $\L'_i$ define the same $\k$-configuration, that is, we need to see if they satisfy the conditions (1), (2), and (3) of Definition \[kdefn\].
We first note that condition $(1)$ holds since $$\begin{aligned}
\bigcup_{i=1}^s \X'_k &= & (\X_1 \cup \cdots \cup \X_{s-i-1}) \cup
\left(\bigcup_{k={s-i}}^{s-j-2} (\X_{k+1} \setminus \mathbb{T})\right) \cup ( \X_{s-i} \cup \mathbb{T}) \cup (\X_{s-j} \cup \cdots \cup \X_s) \\
& = & \X_1 \cup \cdots \cup \X_s = \X.\end{aligned}$$
For condition $(2)$, it is clear that $\X'_k \subseteq \L'_k$ for all $k$. We now verify that $|\X'_k| = d_k$ for all $k$. For $k=1,\ldots,s-i-1$ and $k = s-j,\ldots,s$ this is immediate since $\X_k' = \X_k$. Because $|\L_{s-i} \cap \X| = d_s$, it follows by Lemma \[necessaryconditions\] that $d_{s-k} = d_s - s + (s-k) = d_s -k$ for $k=i,\ldots,j+1$. Moreover, as in the proof of Lemma \[necessaryconditions\], $\L_{s-i} \cap \L_{s-k} \in \X_{s-k}$ for all $k=j+1,\ldots,i-1$. So $|\mathbb{T}| = i-1-j$, and thus $$|\X'_{s-j+1}| = |\X_{s-i} \cup \mathbb{T}| = d_s - i + i-1-j = d_s - (j+1) = d_{s-j+1}.$$ Also, again since $\L_{s-i} \cap \L_{s-k} \in \X_{s-k}$ for $k=j+1,\ldots,i-1$, we have $$|\X'_k| = |\X_{k+1} \setminus \mathbb{T}| = d_{k+1} - 1 = d_s-s+k+1 -1 = d_s -s+k
=d_k$$ for $k= s-i,\ldots,s-j-2$.
Finally, for condition $(3)$, we only need to check the line $\L'_{s-j-1}$ since the result is true for the other lines by the construction of $\X$ using the lines $\L_1,\ldots,\L_s$. Now $\L'_{s-j-1} = \L_{s-i}$, and we know that it does not intersect with the points $\X'_k = \X_k$ with $k < s-i$. Also, by construction, $\L'_{s-j-1}$ does not intersect with the points of $\X'_{s-i},\ldots,\X'_{s-j-2}$. So condition $(3)$ holds.
at (7,-2) [$\L_1$]{}; at (-1,-2) [$\L_2$]{}; at (-2,3) [$\L_3$]{}; at (-2,1.5) [$\L_4$]{}; at (-2,0) [$\L_5$]{}; (-1,-1) rectangle (11,8); (-1,0)–(11,0); (-1,-1)–(10,10); (-1,3)–(11,3); (-1,1.5)–(11,1.5); (8,-6)–(2,12); at (0,0) ; at (2,0) ; at (4,0) ; at (6,0) ; at (8,0) ; at (10,0) ; at (3.25,1.5) ; at (5,1.5) ; at (6.75,1.5) ; at (8.75,1.5) ; at (3,3) ; at (5,3) ; at (7,3) ; at (9.25,3) ; at (1.5,1.5) ; at (4,4) ; at (5.5,5.5) ; at (7,7) ; at (3.8,6.6) ;
at (7,-2) [$\L'_1$]{}; at (-1,-2) [$\L'_4$]{}; at (-2,3) [$\L'_2$]{}; at (-2,1.5) [$\L'_3$]{}; at (-2,0) [$\L'_5$]{}; (-1,-1) rectangle (11,8); (-1,0)–(11,0); (-1,-1)–(10,10); (-1,3)–(11,3); (-1,1.5)–(11,1.5); (8,-6)–(2,12); at (0,0) ; at (2,0) ; at (4,0) ; at (6,0) ; at (8,0) ; at (10,0) ; at (3.25,1.5) ; at (5,1.5) ; at (6.75,1.5) ; at (8.75,1.5) ; at (3,3) ; at (5,3) ; at (7,3) ; at (9.25,3) ; at (1.5,1.5) ; at (4,4) ; at (5.5,5.5) ; at (7,7) ; at (3.8,6.6) ;
Figure \[fig:relabel\] gives an example of the relabelling. As before, the shapes denote which points belong to the subsets $\X_i$.
\[relabelcor\] Suppose that $\X \subseteq \mathbb{P}^2$ is a $\k$-configuration of type $(d_1,d_2,\ldots,d_s)$ with $s {\geqslant}2$. Let $\L_1,\ldots,\L_s$ be the lines used to define $\X$. After relabelling, we can assume that there is an $r$ such that $|\L_{s-j} \cap \X| = d_s$ for all $0 {\leqslant}j {\leqslant}r-1$, but $|\L_{s-j} \cap \X| < d_s$ for all $r {\leqslant}j {\leqslant}s-1$.
In the assumptions of Lemma \[relabel1\], we are assuming that $\L_s,\ldots,\L_{s-j}$ all meet $\X$ at $d_s$ points, but $\L_{s-j-1}$ does not. After applying the relabelling of Lemma \[relabel1\], the lines $\L'_s,\ldots,\L'_{s-j-1}$ now meet $\X$ at $d_s$ points. By reiterating Lemma \[relabel1\], we arrive at the conclusion.
\[hilbertfcnreducedbnd\] In the introduction we mentioned that the number of lines that contain $d_s$ points is bounded by a value of the first difference of the Hilbert function. Specifically, the number of lines that contain $d_s$ points is bounded above by $\Delta \H_{\X}(d_s-1) + 1$.
We sketch out how to prove this result. Roberts and Roitman [@RR Theorem 1.2] give a formula for the Hilbert function $\H_{\X}$ of a $\k$-configuration in terms of the type $(d_1,d_2,\ldots,d_s)$. It follows from this formula that $\H_{\X}(d_s-1) = \sum_{i=1}^s d_i$, and $\H_{\X}(d_s-2) = (\sum_{i=1}^s d_i) - t$ where $t$ is the number of consecutive integers at the end of $(d_1,d_2,\ldots,d_s)$. So, $$\Delta \H_\X(d_s-1) = \H_{\X}(d_s-1) - \H_{\X}(d_s-2) = t.$$ Note that if $d_s = s$, then $t=s$. It follows by Lemma \[necessaryconditions\] that if $d_s > s$, then $t$ is an upper bound on the number of lines that contain $d_s$ points, and if $d_s =s$, then by Corollary \[numberoflines\], $t+1 = s+1$ is an upper bound. We can combine this information in the statement that number of lines that contain $d_s$ points is bounded above by $\Delta \H_\X(d_s-1) +1$.
The case ds greater than s {#sec:main}
==========================
In this section, we prove Theorem \[maintheorem\] in the case the $\k$-configuration $\X$ has type $(d_1,\dots,d_s)$ with $d_s > s$.
\[bigmult\] Let $\X \subseteq \mathbb{P}^2$ be a $\k$-configuration of type $d=(d_1,\dots,d_s)$, and assume that there are $r$ lines containing exactly $d_s$ points of $\X$. If $d_s > s$, then $r = \Delta \H_{m\X} (m d_s -1)$ for all $m {\geqslant}2$.
By Lemma \[necessaryconditions\] $(i)$, the lines containing $d_s$ points of $\X$ fall among the lines $\L_1,\dots,\L_s$ defining the $\k$-configuration. By Corollary \[relabelcor\], we may assume that the lines containing exactly $d_s$ points of $\X$ are $\L_s,\dots,\L_{s-r+1}$, while the lines $\L_{s-r},\dots,\L_1$ contain less than $d_s$ points of $\X$.
We will apply Theorem \[CHT\] to compute certain values of $\H_{m\X}$. Towards this goal, we obtain the reduction vector $\bf v$ of $\X$ using the sequence of lines $$\L_s,\dots,\L_1,
\L_s,\dots,\L_1,
\dots,
\L_s,\dots,\L_1,$$ where the subsequence $\L_s,\dots,\L_1$ is repeated $m$ times.
We claim that, for $i=1,\dots,r$, we have $$v_i = m d_s -i+1.$$ Let $\Z_0 = m\X$. Since $|\L_s \cap \X| = d_s$, we have $$v_1 = \deg (\L_s \cap \Z_0) = m d_s.$$ Now let $\Z_1$ be the residual of $\Z_0$ with respect to the line $\L_s$. The line $\L_{s-1}$ contains the $d_{s-1}$ points of $\X_{s-1}$, and the point $\X_s \cap \L_{s-1}$. The multiplicity of the point $\X_s \cap \L_{s-1}$ in $\Z_1$ is $m-1$, while the points of $\X_{s-1}$ have multiplicity $m$ in $\Z_1$. Thus we get $$v_2 = \deg (\L_{s-1} \cap \Z_1) = m d_{s-1} +m-1 =
m (d_s-1) +m-1=m d_s -1,$$ where $d_{s-1} = d_s -1$ by Lemma \[necessaryconditions\] $(ii)$. Continuing in this fashion, for $i=3,\dots,r$, we have a scheme $\Z_{i-1}$. The line $\L_{s-i+1}$ contains the $d_{s-i+1}$ points of $\X_{s-i+1}$, and the points $\X_s \cap \L_{s-i+1}$, $\X_{s-1} \cap \L_{s-i+1}$, $\dots$, $\X_{s-i+2} \cap \L_{s-i+1}$. The former have multiplicity $m$ in $\Z_{i-1}$, while the latter have multiplicity $m-1$ in $\Z_{i-1}$. Thus, for $i=1,\dots,r$, we get $$\begin{split}
v_i &= \deg (\L_{s-i+1} \cap \Z_{i-1})
= m d_{s-i+1} +(m-1)(i-1)\\
&= m (d_s-i+1) +(m-1)(i-1)=m d_s -i+1.
\end{split}$$
Next we claim that, for $i=r+1,\dots,s$, we have $$v_i {\leqslant}m d_s -i.$$ The line $\L_{s-i+1}$ contains the $d_{s-i+1}$ points of $\X_{s-i+1}$, and $e$ points at the intersections $\X_t \cap \L_{s-i+1}$ for $t>s-i+1$. Note that $d_{s-i+1} + e < d_s$ because we assumed that, for $i=r+1,\dots,s$, the line $\L_{s-i+1}$ contains less than $d_s$ points of $\X$. The points of $\X_{s-i+1}$ have multiplicity $m$ in $\Z_{i-1}$, while each point $\X_t \cap \L_{s-i+1}$ has multiplicity $m-1$ in $\Z_{i-1}$. Thus, for $i=r+1,\dots,s$, we get $$\begin{split}
v_i &= \deg (\L_{s-i+1} \cap \Z_{i-1}) = m d_{s-i+1} +(m-1)e\\
& = d_{s-i+1} +(m-1)(d_{s-i+1} +e)\\
&< d_s -i+1 + (m-1) d_s = m d_s -i+1,
\end{split}$$ where the inequality uses Lemma \[bounds\] $(i)$. This proves our claim.
This concludes our first round of removing the lines $\L_s,\dots,\L_1$, corresponding to the entries $v_1,\dots,v_s$ of the reduction vector ${\bf v}$. Now we focus on later passes. We can index later entries of the reduction vector by $v_{js+i}$, where $j=1,\dots,m-1$ keeps track of the current pass (the first pass corresponding to $j=0$), and $i=1,\dots,s$ indicates that we are going to remove the line $\L_{s-i+1}$. We claim that $$v_{js+i} {\leqslant}m d_s - (js+i).$$ We proceed to estimate the multiplicity of points in $\L_{s-i+1} \cap \Z_{js+i-1}$. The line $\L_{s-i+1}$ contains the $d_{s-i+1}$ points of $\X_{s-i+1}$; these have multiplicity at most $m-j$ in $\Z_{js+i-1}$, because the line $\L_{s-i+1}$ was removed $j$ times in previous passes. In addition, the line $\L_{s-i+1}$ contains $e$ points at the intersections $\X_t \cap \L_{s-i+1}$ for $t>s-i+1$, where $d_{s-i+1} + e {\leqslant}d_s$ as before. Each of these points has been removed $j$ times in previous passes and once in the current pass, and therefore, it has multiplicity at most $m-j-1$ in $\Z_{js+i-1}$. Altogether, for $j=1,\dots,m-1$ and $i=1,\dots,s$, we obtain the following estimate: $$\begin{split}
v_{js+i}
&= \deg (\L_{s-i+1} \cap \Z_{js+i-1}) {\leqslant}(m-j) d_{s-i+1} +(m-j-1)e\\
& = d_{s-i+1} +(m-j-1)(d_{s-i+1} +e)\\
& {\leqslant}d_s -i+1 + (m-j-1) d_s = m d_s - j d_s -i+1\\
& < m d_s -j s -i +1,
\end{split}$$ using Lemma \[bounds\] $(i)$ and the hypothesis $d_s > s$. This proves our claim.
Observe that after removing the lines $\L_s,\dots,\L_1$ $m$ times, we have $\Z_{ms+1} = \varnothing$. In other words, ${\bf v} = (v_1,\dots,v_{ms})$ is a complete reduction vector for $m\X$. We can summarize our findings about ${\bf v}$ as follows: $$\begin{aligned}
&v_i = md_s -i +1,& &\mbox{for $i=1,\ldots,r$},\\
&v_i {\leqslant}md_s -i,& &\mbox{for $i=r+1,\ldots,ms$}.
\end{aligned}$$
Now we compute the value $\H_{m\X} (m d_s -2)$ using Theorem \[CHT\]. Recall that a lower bound is given by $$f_{\bf v} (m d_s - 2) = \sum_{i=0}^{ms-1} \min (m d_s -1-i,v_{i+1}).$$ Based on our previous estimates, we have $$\begin{aligned}
&\min (m d_s -1-i,v_{i+1}) = md_s -1-i,&
&\mbox{for $i=0,\ldots,r-1$},\\
&\min (m d_s -1-i,v_{i+1}) = v_{i+1},&
&\mbox{for $i=r,\ldots,ms-1$}.
\end{aligned}$$ Hence we get $$f_{\bf v} (m d_s - 2) = \sum_{i=0}^{r-1} (m d_s -1-i) +
\sum_{i=r}^{ms-1} v_{i+1}.$$ As for the upper bound, it is given by $$F_{\bf v} (m d_s -2) = \min_{0{\leqslant}i{\leqslant}ms} \bigg(
\binom{m d_s}{2} - \binom{m d_s -i}{2} + \sum_{j=i+1}^{ms} v_j
\bigg).$$ Evaluating the right hand side for $i=r$, we get $$\begin{split}
F_{\bf v} (m d_s -2)
&{\leqslant}\binom{m d_s}{2} - \binom{m d_s -r}{2} + \sum_{j=r+1}^{ms} v_j\\
&=\sum_{h=m d_s -r}^{m d_s -1} h + \sum_{j=r+1}^{ms} v_j\\
&=\sum_{i=0}^{r-1} (m d_s -1-i) + \sum_{i=r}^{ms-1} v_{i+1}.
\end{split}$$ Combining these bounds, we obtain $$\H_{m\X} (m d_s -2) =
\sum_{i=0}^{r-1} (m d_s -1-i) + \sum_{i=r}^{ms-1} v_{i+1}.$$ Similarly, we can use Theorem \[CHT\] to compute $\H_{m\X} (m d_s -1)$. In this case, the lower bound is given by $$\begin{split}
f_{\bf v} (m d_s - 1)
&= \sum_{i=0}^{ms-1} \min (m d_s -i, v_{i+1})\\
&= \sum_{i=0}^{r-1} (m d_s -i) +
\sum_{i=r}^{ms-1} v_{i+1} = \sum_{i=0}^{ms-1} v_{i+1}.
\end{split}$$ Note that since $f_{\bf v} (m d_s-1)$ is the sum of all the entries of the reduction vector, $f_{\bf v}(m d_s - 1) = \deg(m\X)$ by [@CHT Remark 1.2.6]. On the other hand, it is well-known that for any zero-dimensional scheme $\Z$, $\H_{\Z}(t) {\leqslant}\deg(\Z)$ for all $t$ (see, e.g. [@CTV]). We thus have $$\H_{m\X}(m d_s -1)=\sum_{i=0}^{r-1} (m d_s -i) +
\sum_{i=r}^{ms-1} v_{i+1}=\deg(m\X).$$ Finally, computing the first difference of the Hilbert function gives the desired result: $$\begin{split}
\Delta \H_{m\X} (md_s -1)
&= \H_{m\X} (md_s -1) - \H_{m\X} (md_s -2) \\
&= \sum_{i=0}^{r-1} (m d_s -i) - \sum_{i=0}^{r-1} (m d_s -1-i)
= \sum_{i=0}^{r-1} 1 = r.
\end{split}$$
Consider the $\k$-configuration $\X$ of type $(1,3,4,5)$ of Example \[ex:CHT\]. There are three lines containing $d_4=5$ points, namely $\L_2$, $\L_3$, and $\L_4$.
Our computation in Example \[ex:CHT\] shows that $\H_{2\X}(2d_4-2) = \H_{2\X} (8) = 36$. In fact, this is an instance of the general computation carried out in the proof of Theorem \[bigmult\]. A similar computation yields $\H_{2\X} (9) = 39$. Therefore we have $$\Delta \H_{2\X} (9) = \H_{2\X}(9)-\H_{2\X}(8)=39-36=3,$$ as desired.
The case ds=s.
==============
In this section we focus on $\k$-configurations of type $d = (d_1,\ldots,d_s)$ with $d_s = s {\geqslant}2$ (as mentioned in the introduction, the case $d=(1)$ is a single point). As noted in Lemma \[bounds\], the $\k$-configuration $\X$ must have type $(1,2,\ldots,s)$. Unlike the case $d_s > s$, the value of $\Delta \H_{2\X}(2d_s-1)$ need not equal the number of lines that contain $d_s = s$ points of $\X$. As a simple example, consider the $\k$-configuration of type $(1,2,3)$ given in Figure \[specialcase\].
at (-2,6) [$\L_1$]{}; at (-2,0) [$\L_2$]{}; at (-1,-2) [$\L_3$]{}; (-1,-1) rectangle (10,8); (-1,0)–(10,0); (-1,-1)–(10,10); (-2,6)–(13,-3); (2,-6)–(8,12); at (1,1) ; at (3,3) ; at (6,6) ; at (4,0) ; at (6.5,0) ; at (5.5,1.5) ;
This $\k$-configuration has exactly one line containing exactly three points (namely, the line $\L_3$). However, when we compute the Hilbert function of $2\X$, we get $$\H_{2\X} : 1~~~3~~~6~~~10~~~~15~~~~18~~~~18~~~~\rightarrow,$$ and consequently, $\Delta \H_{2\X}(2\cdot3-1) = \H_{2\X}(5) - \H_{2\X}(4) = 18-15 = 3$. So, the hypothesis that $d_s > s$ in Theorem \[bigmult\] is necessary.
In this section, we will derive a result similar to Theorem \[bigmult\]. However, in order to find the number of lines that contain $s$ points of $\X$, we need to consider the Hilbert function of $m\X$ with $m {\geqslant}s+1$ instead of $m {\geqslant}2$. We need a more subtle argument, in part, because of Lemma \[extraline\]. That is, unlike the case of $d_s > s$, there may be up to two extra lines $\L$ that contains $s$ points of $\X$, where $\L$ is not among the lines that defines the $\k$-configuration.
We begin with a lemma that allows us to break our argument into three separate cases. This lemma is similar to Lemma \[relabel1\] in that it allows us to make some additional assumptions about the lines $\L_1,\ldots,\L_s$ and points $\X_1,\ldots,\X_s$ used to define the $\k$-configuration.
\[relabel2\] Suppose that $\X \subseteq \mathbb{P}^2$ is a $\k$-configuration of type $(1,2,\ldots,s)$ with $s {\geqslant}2$. Let $\X_1,\ldots,\X_s$ be the subsets of $\X$, and $\L_1,\ldots,\L_s$ the lines used to define $\X$.
Then one of the three disjoint cases must hold:
1. There are exactly $s+1$ lines that contain $s$ points of $\X$, and the points of $\X$ are precisely the pairwise intersections of such lines.
2. There are exactly $s$ lines that contain $s$ points of $\X$, and we can assume that these lines are $\L_1,\ldots,\L_s$. Moreover, for each $i=1,\dots,s$, the set $\X \cap \L_i$ contains $s-1$ points located at the intersection of $\L_i$ and $\L_j$ (with $j\neq i$), and a single point $P_i$ that does not belong to any line $\L_j$ for $j\neq i$.
3. There are $1 {\leqslant}r < s$ lines that contain $s$ points, and furthermore, after a relabelling of the lines $\L_1,\ldots,\L_s$ and subsets $\X_1,\ldots,\X_s$, we can assume that none of these $r$ lines pass through the point of $\X_1 = \{P\}$.
By Corollary \[numberoflines\] there are at most $s+1$ lines that contain $s$ points of $\X$. So, there are three cases: $(i)$ exactly $s+1$ lines that contain $s$ point of $\X$, $(ii)$ exactly $s$ lines that contain $s$ points of $\X$, or $(iii)$ $1{\leqslant}r < s$ lines that contain $s$ points of $\X$. We now show that in each case, we can label the $\X_i$’s and $\L_i$’s as described in the statement.
$(i)$ Suppose that there are exactly $s+1$ lines that contain $s$ points of $\X$. If $s=2$, then the hypothesis that $\X$ is a $\k$-configuration implies that the three points of $\X$ are not colinear. Thus each pair of points of $\X$ uniquely determines a line, and the points of $\X$ are the intersections of such lines. Now suppose that $s>2$, and let $\HH_1,\dots,\HH_s,\HH_{s+1}$ be the $s+1$ lines each passing through $s$ points of $\X$. Define the set of points $\Y := \X \setminus \HH_{s+1}$, so that $\X = \Y \cup (\X \cap \HH_{s+1})$. We have $|\Y| = \binom{s+1}{2} -s = \binom{s}{2}$. Each line $\HH_1,\dots,\HH_s$ passes through $s-1$ points of $\Y$, otherwise we would have $|\Y|>\binom{s}{2}$. By induction on $s$, the points of $\Y$ are the pairwise intersections of the lines $\HH_1,\dots,\HH_s$. By cardinality considerations, the $s$ points of $\X \cap \HH_{s+1}$ must be the intersections of $\HH_{s+1}$ with the lines $\HH_1,\dots,\HH_s$. This shows that the points of $\X$ are precisely the pairwise intersections of the lines $\HH_1,\dots,\HH_s,\HH_{s+1}$.
$(ii)$ Suppose that there are exactly $s$ lines that contain $s$ points of $\X$. There are three subcases: $(a)$ the $s$ lines that contain $s$ points are $\L_1,\ldots,\L_s$; $(b)$ $s-1$ of the lines that contain $s$ points are among $\L_1,\ldots,\L_s$ and there is one more line $\L$; and $(c)$ $s-2$ of the lines that contain $s$ points are among $\L_1,\ldots,\L_s$, and there are two more lines that contain $s$ points. Note that Lemma \[extraline\] implies that there is at most two lines not among the $\L_i$’s that will contain $s$ points, so these are the only three cases. We will first show that if $(c)$ is true, then we can relabel the lines and points so that we can assume case $(b)$ is true. We will then show that in case $(b)$, we can again relabel lines and points so we can assume case $(a)$ is true.
Assume case $(c)$ holds. By Lemma \[extraline\], the two lines that contain $s$ points that are not among the $\L_i$’s are the lines $\L_{PQ_1}$ and $\L_{PQ_2}$ where $\X_1 = \{P\}$ and $\X_2 = \{Q_1,Q_2\}$. As argued in Corollary \[numberoflines\], the line $\L_1$ cannot contain $s$ points. Since $\{P\} = \X_1 \subseteq \L_{PQ_1}$, we then have that the $\k$-configuration can also be defined by the same $\X_i$’s and the lines $\L_{PQ_1},\L_2,\ldots,\L_s$. Note that we are in now case $(b)$.
We now assume case $(b)$, that is, $s-1$ of the lines that contain $s$ points are among $\L_1,\ldots,\L_s$ and there is one additional line $\L$ that contains $s$ points. Suppose that $\L_1$ does not contain $s$ points. By Lemma \[extraline\], the additional line $\L$ contains $\X_1$, so as above, we replace $\L_1$ with $\L$, and the $\k$-configuration is defined by the same $\X_i$’s and the lines $\L,\L_2,\ldots,\L_s$, all of which contain $s$ points. On the other hand, suppose $\L_1$ contains $s$ points. Then there is exactly one line $\L_j \in \{\L_2,\ldots,\L_{s-1}\}$ that does not contain $s$ points (note that $\L_s$ contains $s$ points). Moreover, $\L_1,\ldots,\L_{j-1}$ must all intersect $\L_j$ at distinct points since each such $\L_i$ needs to contain $s$ distinct points.
Set $$\begin{aligned}
\mathbb{T} & = & \L \cap (\X_1 \cup \cdots \cup\X_j).\end{aligned}$$ Since $\L$ contains $s$ points of $\X$, we must have $\L \cap \X_i \neq \varnothing$ for all $i=1,\ldots,s$, and in particular, $|\mathbb{T}| = j$. Then the $\k$-configuration $\X$ can also be defined using the subsets $$\begin{aligned}
\X'_i &=& (\X_i \setminus \L) \cup (\L_i \cap \L_j) ~~\mbox{and}~~
\L'_i = \L_i ~~\mbox{for $i=1,\ldots,j-1$,} \\
\X_j & = & \mathbb{T} ~~ \mbox{and}~~ \L'_j = \L, ~~\mbox{and}\\
\X'_i & =&\X_i ~~\mbox{and}~~ \L'_i = \L_i ~~\mbox{for $i = j+1,\ldots,s$}.\end{aligned}$$ The verification of this fact is similar to the proof of Lemma \[relabel1\]. Note that the line $\L_j$ is no longer used to define the $\k$-configuration; moreover, the $s$ lines that contain the $s$ points are $\L'_1,\ldots,\L'_s$ after this relabelling, i.e., we are now in case $(a)$. We have now verified that we can assume that the lines that contain $s$ points of $\X$ are exactly the lines $\L_1,\ldots,\L_s$. We now verify the second part of $(ii)$.
Now, for each $i=1,\dots,s$, the line $\L_i$ contains exactly $s$ points of $\X$, so at least one of the $s$ points in $\X \cap \L_i$ does not belong to $\L_j$ for $j\neq i$; call this point $P_i$. For $i=1,\dots,s-1$, set $\Y_i := \X_{i+1} \setminus \{P_{i+1}\}$. The set $\Y := \bigcup_{i=1}^{s-1} \Y_i$ is a $\k$-configuration of type $(1,\dots,s-1)$ with supporting lines $\L_{i+1} \supseteq \Y_i$ (this follows from the fact that $\X$ is a $\k$-configuration with supporting lines $\L_i \supseteq \X_i$). Furthermore, there are exactly $s$ lines that contain $s-1$ points of $\Y$, namely the lines $\L_1,\dots,\L_s$. Therefore, by part $(i)$, all points of $\Y$ are precisely the pairwise intersections of the lines $\L_1,\dots,\L_s$. The statement in part $(ii)$ follows.
$(iii)$ Finally, suppose that there are $1 {\leqslant}r < s$ lines that contain $s$ points of $\X$. Like case $(ii)$, there are three subcases: $(a)$ the $r$ lines are among $\L_1,\ldots,\L_s$; $(b)$ $r-1$ of the lines are among $\L_1,\ldots,\L_s$, and there is one additional line $\L$, or $(c)$ $r-2$ of the lines are among $\L_1,\ldots,
\L_s$, and there are two additional lines that contains $s$ points. Like case $(ii)$, we first show that we can relabel case $(c)$ so case $(b)$ is true. We then show that if case $(b)$ is true, we can again relabel so case $(a)$ is true.
If we assume case $(c)$, we first apply Corollary \[relabelcor\] to relabel the lines so that $\L_1$ does not contain $s$ points (since only the last $r-2$ lines will contain $s$ points). Lemma \[extraline\] implies that the two additional lines are $\L_{PQ_1},\L_{PQ_2}$. Since $\X_1 \subseteq \L_{PQ_1}$ we can still define the $\k$-configuration using the same $\X_i$’s, but with the lines $\L_{PQ_1},\L_2,\ldots,\L_s$, i.e., we are in case $(b)$.
In case $(b)$, we again first apply Corollary \[relabelcor\] to relabel our $\k$-configuration so that $\L_1$ does not contain $s$ points. By Lemma \[extraline\], the additional line $\L$ is either $\L_{PQ_1}$ or $\L_{PQ_2}$. In either case, $\X_1 \subseteq \L$, so we again define the $\k$-configuration using the the same $\X_i$’s and the lines $\L,\L_2,\ldots,\L_s$. We have now relabelled the $\k$-configuration so case $(a)$ holds.
Since we can assume that $(a)$ holds, the $1 {\leqslant}r < s$ lines that contains $s$ points are among $\L_1,\ldots,\L_s$. Again, by applying Corollary \[relabelcor\], we can assume that $\L_{s-r+1},\ldots,\L_s$ are the $r$ lines with $s$ points, and in particular, none of these points contain $\X_1 = \{P\}$ by definition of a $\k$-configuration.
Case 1: Exactly s+1 lines
-------------------------
We will now consider the three cases of Lemma \[relabel2\] separately. We first consider the case that there are exactly $s+1$ lines that contain $s$ points of $\X$.
\[case=s+1\] Let $\X \subseteq \mathbb{P}^2$ be a $\k$-configuration of type $d=(d_1,\ldots,d_s) = (1,2,\ldots,s)$ with $s {\geqslant}2$. Assume that there are exactly $s+1$ lines containing $s$ points of $\X$. Then $s+1 = \Delta \H_{m\X} (m d_s -1)$ for all $m {\geqslant}2$.
Let $\HH_1,\dots,\HH_s,\HH_{s+1}$ be the lines containing $s$ points of $\X$; by Lemma \[relabel2\] $(i)$, the points of $\X$ are precisely the intersections of such lines.
To compute bounds on the Hilbert function of $m\X$, we apply Theorem \[CHT\] with the reduction vector $\bf v$ obtained from the sequence of lines $$\HH_{s+1}, \HH_s, \dots, \HH_1,
\HH_{s+1}, \HH_s, \dots, \HH_1,
\dots,
\HH_{s+1}, \HH_s, \dots, \HH_1,$$ where the subsequence $\HH_{s+1}, \HH_s, \dots, \HH_1$ is repeated $\lceil \frac{m}{2}\rceil$ times.
We index the entries of the reduction vector by $v_{j(s+1)+i}$, where $j=0,\dots,\lceil \frac{m}{2}\rceil -1$ is the number of times the subsequence of lines $\HH_{s+1}, \HH_s, \dots, \HH_1$ has been completely removed, and $i=1,\dots,s+1$ indicates that we are going to remove the line $\HH_{s-i+2}$. Note that each time the subsequence $\HH_{s+1}, \HH_s, \dots, \HH_1$ is removed, the multiplicity of each point of $m\X$ decreases by two. If $m$ is even, this process eventually reduces the multiplicity of each point to zero. If $m$ is odd, then the process reduces the multiplicity of each point to one, so removing the sequence of lines $\HH_{s+1}, \HH_s, \dots, \HH_1$ one more time reduces the multiplicity to zero. In particular, $m\X$ will be reduced to $\varnothing$ after removing the subsequence of lines $\HH_{s+1}, \HH_s, \dots, \HH_1$ $\lceil \frac{m}{2}\rceil$ times. At the step corresponding to $v_{j(s+1)+i}$, the line $\HH_{s-i+2}$ contains:
- the points of intersection $\HH_{s-i+2} \cap \HH_k$ for $k>s-i+2$, with multiplicity $m-2j-1$;
- the points of intersection $\HH_{s-i+2} \cap \HH_k$ for $k<s-i+2$, with multiplicity $m-2j$.
This gives $$\label{eq:star_red_vec}
\begin{split}
v_{j(s+1)+i} &= (i-1) (m-2j-1) + (s-i+1) (m-2j)\\
&= (m-2j)s-i+1.
\end{split}$$
When $j=0$, Equation implies $$v_i = ms -i+1,$$ for all $i=1,\dots,s+1$. For $j>0$, we get $$\begin{split}
v_{j(s+1)+i} &= (m-2j)s -i+1 = ms -2js -i+1\\
&< ms -j(s+1) -i+1
\end{split}$$ because $s>1$. This shows that $$v_{j(s+1)+i} {\leqslant}ms -(j(s+1)+i)$$ for all $j=1,\dots,\lceil \frac{m}{2}\rceil -1$ and $i=1,\dots,s+1$.
Since $d_s =s$, we can summarize the results above by writing $$\begin{aligned}
&v_i = m d_s -i +1,& &\mbox{for $i=1,\ldots,s+1$},\\
&v_i {\leqslant}m d_s -i,& &\mbox{for $i=s+2,\ldots,\lceil \frac{m}{2}\rceil (s+1)$}.
\end{aligned}$$ Proceeding as in the proof of Theorem \[bigmult\], we obtain $$\H_{m\X} (m d_s -2) = \sum_{i=0}^s (m d_s -1-i) +
\sum_{i=s+2}^{\lceil \frac{m}{2}\rceil (s+1)-1} v_{i+1}.$$ Also as in the proof of Theorem \[bigmult\], we have $$\H_{m\X} (m d_s -1) = \deg (m\X) =
\sum_{i=0}^{\lceil \frac{m}{2}\rceil (s+1)-1} v_{i+1} =
\sum_{i=0}^s (m d_s -i) +
\sum_{i=s+2}^{\lceil \frac{m}{2}\rceil (s+1)-1} v_{i+1}.$$ We conclude that $$\Delta \H_{m\X} (m d_s -1)
= \H_{m\X} (m d_s -1) - \H_{m\X} (m d_s -2)
= s+1.$$
A $\k$-configuration of type $(1,2,\ldots,s)$ which has exactly $s+1$ lines containing $s$ points is also an example of a star configuration. When $m=2$, Theorem \[case=s+1\] can be deduced from [@GHM Theorem 3.2].
Case 2: Exactly s lines
-----------------------
We next consider the case that there are exactly $s$ lines containing $s$ points. Reasoning as in the previous case, we may compute a reduction vector from these $s$ lines, in order to calculate values of the Hilbert function. However, in this case, the bounds thus obtained may not be tight. The following example illustrates the issue, and a possible workaround.
Consider a $\k$-configuration $\X$ of type $(1,2,3,4)$ with exactly four lines that contain four points of $\X$. By Lemma \[relabel2\] $(ii)$, $\X$ consists of the intersections of the lines $\L_1,\L_2,\L_3,\L_4$ defining the $\k$-configuration, and four non-colinear points $P_1,P_2,P_3,P_4$, with $P_i$ belonging to $\L_i$. We have depicted such an $\X$ in Figure \[fig:issue\_s\_lines\].
at (-2,6) [$\L_4$]{}; at (-2,0) [$\L_3$]{}; at (-1,-2) [$\L_2$]{}; at (4,-2) [$\L_1$]{}; at (6.35,3.65) [$P_1$]{}; at (4.15,5.6) [$P_2$]{}; at (2.65,0.85) [$P_3$]{}; at (7.4,1.6) [$P_4$]{}; (-1,-1) rectangle (10,8); (-1,0)–(10,0); (-1,-1)–(10,10); (-2,6)–(13,-3); (2,-6)–(8,12); at (8,0) ; at (4.66,2) ; at (3,3) ; at (6.5,0.9) ; at (0,0) ; at (4,0) ; at (1.75,0) ; at (6,6) ; at (4.5,4.5) ; at (5.5,4.5) ;
at (-2,6) [$\L_4$]{}; at (-2,0) [$\L_3$]{}; at (-1,-2) [$\L_2$]{}; at (4,-2) [$\L_1$]{}; at (10.75,4.5) [$\HH$]{}; at (1.75,-2) [$\HH_1$]{}; at (6.5,-2) [$\HH_2$]{}; at (6.35,3.65) [$P_1$]{}; at (4.15,5.6) [$P_2$]{}; at (2.65,0.85) [$Q_1$]{}; at (7.4,1.6) [$Q_2$]{}; (-1,-1) rectangle (10,8); (-1,0)–(10,0); (-1,-1)–(10,10); (-2,6)–(13,-3); (2,-6)–(8,12); (3,4.5)–(12,4.5); (1.75,-2)–(1.75,2.5); (6.5,-2)–(6.5,2.5); at (8,0) ; at (4.66,2) ; at (3,3) ; at (6.5,0.9) ; at (0,0) ; at (4,0) ; at (1.75,0) ; at (6,6) ; at (4.5,4.5) ; at (5.5,4.5) ;
We proceed to compute bounds for $\H_{2\X} (2 d_s -2)$ as we did in Example \[ex:CHT\]. First we use the sequence of lines $\L_4,\L_3,\L_2,\L_1,\L_4,\L_3,\L_2,\L_1$. The table below compares the function $(2 d_s -2)-i+1$ with the entries of the reduction vector $\bf v$; the minimum is in bold.
----------------------------------------------------------------------------------------------
$i$ 0 1 2 3 4 5 6 7
--------- -------------------------------------------------- --- --- --- --- --- --- --- -- --
$6-i+1$ **7 & **6 & **5 & **4 & 3 & 2 & 1 & **0\
$v_{i+1}$ & 8 & 7 & 6 & 5 & **1 & **1 & **1 & 1\
****************
----------------------------------------------------------------------------------------------
Summing the minimum values, we obtain the lower bound $\H_{2\X} (6) {\geqslant}f_{\bf v} (6) = 25$.
Now let $\HH$ be the line through $P_1$ and $P_2$. In general, the line $\HH$ could also contain $P_3$ or $P_4$, but not both. However in the $\k$-configuration depicted in Figure \[fig:issue\_s\_lines\] $\HH$ does not contain either $P_3$ or $P_4$. Consider the points of the set $\{P_1,P_2,P_3,P_4\} \setminus \HH$, namely $P_3$ and $P_4$, and relabel them $Q_1$ and $Q_2$. Let $\HH_1$ be a line through $Q_1$ not passing through $Q_2$, and let $\HH_2$ be a line through $Q_2$. We compute a lower bound for $\H_{2\X} (2 d_s -2)$ using the sequence of lines $\L_4,\L_3,\L_2,\L_1,\HH,\HH_1,\HH_2$. The table below summarizes the necessary information.
--------------------------------------------------------------------------------------
$i$ 0 1 2 3 4 5 6
--------- ---------------------------------------------- --- --- --- --- --- --- -- --
$6-i+1$ **7 & **6 & **5 & **4 & 3 & 2 & 1\
$v_{i+1}$ & 8 & 7 & 6 & 5 & **2 & **1 & **1\
**************
--------------------------------------------------------------------------------------
Summing the minimum values, we obtain the lower bound $\H_{2\X} (6) {\geqslant}f_{\bf v} (6) = 26$.
An easy computation with Equation (using either reduction vector) leads to the upper bound $\H_{2\X} (6) {\leqslant}F_{\bf v} (6) {\leqslant}26$. This shows that $\H_{2\X} (6) = 26$. In particular, the lower bound computed from the lines $\L_4,\L_3,\L_2,\L_1$ alone is not tight.
Using the above example as a guide, we prove our main result for the case under consideration.
\[case=s\] Let $\X \subseteq \mathbb{P}^2$ be a $\k$-configuration of type $d=(d_1,\ldots,d_s) = (1,2,\ldots,s)$ with $s{\geqslant}2$. Assume that there are exactly $s$ lines containing $s$ points of $\X$. Then $s = \Delta \H_{m\X} (m d_s -1)$ for all $m {\geqslant}2$.
By Lemma \[relabel2\] $(ii)$, we can assume that the lines containing $s$ points of $\X$ are the lines $\L_1,\ldots,\L_s$ that define the $\k$-configuration. Moreover, for each $i=1,\dots,s$, there is a point $P_i \in \X\cap \L_i$ that does not belong to $\L_j$ for any $j\neq i$. Then the points of $\X$ are the points of intersection of the lines $\L_1,\dots,\L_s$ together with the points $P_1,\dots,P_s$.
To compute bounds on the Hilbert function of $m\X$, we apply Theorem \[CHT\] with the reduction vector $\bf v$ obtained from a sequence of lines $$\L_s,\dots,\L_1,
\L_s,\dots,\L_1,
\dots,
\L_s,\dots,\L_1,
\HH, \HH_1, \dots, \HH_{s-u},$$ where the subsequence $\L_s,\dots,\L_1$ is repeated $m-1$ times and the additional lines $\HH, \HH_1, \dots, \HH_{s-u}$ are constructed as follows.
Let $\HH$ denote the line through $P_1$ and $P_2$. The line $\HH$ contains $u$ points of the set $\{P_1,\dots,P_s\}$, where, by construction, $u{\geqslant}2$. Furthermore, $\HH$ is not one of the lines $\L_1,\dots,\L_s$, and therefore, it cannot contain $s$ points of $\X$, i.e., $u{\leqslant}s-1$. It follows that the set $\{P_1,\dots,P_s\} \setminus \HH$ is not empty, and must in fact contain $s-u$ points, which we denote $Q_1,\dots,Q_{s-u}$. For each $i=1,\dots,s-u$, let $\HH_i$ be a line passing through $Q_i$ that does not contain any point $Q_j$ for $j>i$.
Now we proceed to compute (or bound) the entries of the reduction vector $\bf v$. We claim that, for $i=1,\dots,s$, we have $$v_i = m s -i+1.$$ At the $i$-th step, the line $\L_{s-i+1}$ contains:
- the points of intersection $\L_{s-i+1} \cap \L_k$ for $k>s-i+1$, with multiplicity $m-1$;
- the points of intersection $\L_{s-i+1} \cap \L_k$ for $k<s-i+1$, with multiplicity $m$; and
- the point $P_{s-i+1}$ with multiplicity $m$.
This gives $$v_i = (m-1)(i-1) + m (s-i) + m = ms -i+1,$$ proving the claim.
Next, we claim that, for $l=s+1,\dots,(m-1)s+s-u+1$, we have $$\label{eq:claim}
v_l {\leqslant}m s - l.$$
We first prove this claim for entries $v_{js+i}$, where $j=1,\dots,\lceil \frac{m}{2} \rceil -1$ is the number of times the subsequence of lines $\L_s,\dots,\L_1$ has been completely removed, and $i=1,\dots,s$ indicates that we are going to remove the line $\L_{s-i+1}$. At the step corresponding to $v_{js+i}$, the line $\L_{s-i+1}$ contains:
- the points of intersection $\L_{s-i+1} \cap \L_k$ for $k>s-i+1$, with multiplicity $m-2j-1$;
- the points of intersection $\L_{s-i+1} \cap \L_k$ for $k<s-i+1$, with multiplicity $m-2j$; and
- the point $P_{s-i+1}$ with multiplicity $m-j$.
The $2j$ in the above multiplicities follows from the fact that points located at the intersections of the lines $\L_s,\dots,\L_1$ are removed twice with each full pass along the subsequence $\L_s,\dots,\L_1$. Thus we obtain $$\begin{split}
v_{js+i}
&= (m-2j-1)(i-1)+(m-2j)(s-i) +m-j\\
&= ms -i+j-2js+1 = ms -js-i + j(1-s) +1\\
&< ms -js -i +1,
\end{split}$$ from which the claim of Equation follows for the chosen values of $i$ and $j$.
Next, we prove the claim for $v_{js+i}$, where $j=\lceil \frac{m}{2} \rceil, \dots, m-2$, and $i=1,\dots,s$. Since the lines $\L_s,\dots,\L_1$ have been removed $\lceil \frac{m}{2} \rceil$ times, the multiplicity of the points located at the intersections of the lines $\L_s,\dots,\L_1$ is now zero. Hence, at the step corresponding to $v_{js+i}$, the line $\L_{s-i+1}$ only contains the point $P_{s-i+1}$ with multiplicity $m-j$. We get $$v_{js+i}
= m-j. $$ Since $j{\leqslant}m-2$, we have $m-j{\geqslant}2$ and therefore $$\frac{m-j}{m-j-1} = \frac{m-j-1+1}{m-j-1} =
1 + \frac{1}{m-j-1} {\leqslant}2 {\leqslant}s.$$ This implies $$v_{js+i} {\leqslant}(m -j -1)s = ms -js -s {\leqslant}ms -js-i,$$ thus proving the claim of Equation for the given $i$ and $j$.
At this stage, the multiplicity of the points $P_1,\dots,P_s$ has been reduced to one, because each line $\L_1,\dots,\L_s$ has been removed $m-1$ times. The next step is to find the value of $v_{(m-1)s +1}$, which corresponds to the line $\HH$ defined at the beginning. By construction, $\HH$ contains $u$ points of the set $\{P_1,\dots,P_s\}$, with $u {\leqslant}s-1$. Therefore $$v_{(m-1)s +1} = u {\leqslant}s-1 = ms -((m-1)s+1);$$ this shows that Equation holds for this entry of $\bf v$.
Finally, we evaluate $v_{(m-1)s+h}$, for $h=2,\dots,s-u+1$. For a given value of $h$, we assume that we have already removed $\HH_1,\dots,\HH_{h-2}$ and we are about to remove $\HH_{h-1}$. The line $\HH_{h-1}$ contains a single point of $\X$, namely $Q_{h-1}$. Moreover, $Q_{h-1}$ is by definition one of the points in the set $\{P_1,\dots,P_s\}\setminus \HH$, so its multiplicity is down to one. Thus we have $$v_{(m-1)s+h} = 1 {\leqslant}s-h = ms - ((m-1)s +h).$$ The inequality $1{\leqslant}s-h$ follows from $h{\leqslant}s-u+1$ and $u{\geqslant}2$. Thus we have proved that Equation holds for all the desired values.
To summarize, we showed that $$\begin{aligned}
&v_i = ms -i +1,& &\mbox{for $i=1,\ldots,s$},\\
&v_i {\leqslant}ms -i,& &\mbox{for $i=s+1,\ldots,(m-1)s+s-u+1$}.
\end{aligned}$$ From here on, the proof proceeds as for Theorem \[bigmult\], yielding $$\begin{split}
\Delta \H_{m\X} (ms -1)
&= \H_{m\X} (ms -1) - \H_{m\X} (ms -2) \\
&= \sum_{i=0}^{s-1} (ms -i) - \sum_{i=0}^{s-1} (ms -1-i)
= \sum_{i=0}^{s-1} 1 = s.
\end{split}$$
Case 3: 1=<r<s lines
--------------------------
We consider the final case when there are $1 {\leqslant}r < s$ lines that contain $s$ points of $\X$. Before going forward, we recall a result of Catalisano, Trung, and Valla [@CTV Lemma 3]; we have specialized this result to the case of points in $\mathbb{P}^2$.
\[CTVlemma\] Let $P_1,\ldots,P_k, P$ be distinct points in $\mathbb{P}^2$ and let $I_P$ be the defining prime ideal of $P$. If $m_1,\ldots,m_k,$ and $a$ are positive integers and $I = I_{P_1}^{m_1} \cap \cdots \cap I_{P_k}^{m_k}$, then
1. $\H_{R/(I+I_P^a)}(t) =
\sum_{i=0}^{a-1} \dim_\k [(I+I_P^i)/(I+I_P^{i+1})]_t$ for every $t>0$, with $I^0_{P}=R$.
2. If $P = [1:0:0]$, then $[(I+I_P^i)/(I + I_P^{i+1})]_t = 0$ if and only if $i > t$ or $x_0^{t-i}M \in I + I_P^{i+1}$ for every monomial $M$ of degree $i$ in $x_1,x_2$.
We now prove the remaining open case. Note that unlike Theorems \[case=s+1\] and \[case=s\], we need to assume that $m {\geqslant}s+1$ instead of $m {\geqslant}2$.
\[case<s\] Let $\X \subseteq \mathbb{P}^2$ be a $\k$-configuration of type $d=(d_1,\ldots,d_s) = (1,2,\ldots,s)$ with $s{\geqslant}2$. Assume that there are $1 {\leqslant}r < s$ lines containing $s$ points of $\X$. Then $r = \Delta \H_{m\X} (m d_s -1)$ for all $m {\geqslant}s+1$.
Let $\L_1,\ldots,\L_s$ be the $s$ lines that define the $\k$-configuration. After using Lemma \[relabel2\] $(iii)$ to relabel, we can assume that the $r$ lines that contain $s$ points are among $\L_2,\ldots,\L_s$, and thus the unique point $P$ of $\X_1$ does not lie on any line containing $s$ points of $\X$. So, we can write our $\k$-configuration as $\X = \Y \cup \{P\}$ where $\{P \} = \X_1$ is the point on the line $\L_1$ and $\Y$ is a $\k$-configuration of type $(2,3,\ldots,s) = (d_1',\ldots,d_{s-1}')$. Since there is no line containing $s$ points of $\X$ that passes through the point $P$, the $r$ lines that contain $s$ points of $\X$ must also contain $s$ points of $\Y$. So we can apply Theorem \[bigmult\] to $\Y$.
Suppose that $\X_2 = \{Q_1,Q_2\}$. Since $P,Q_1,Q_2$ do not all lie on the same line, we can make a linear change of coordinates so that $$P = [1:0:0],~~~ Q_1 = [0:1:0],~~~~\mbox{and}~~~~ Q_2 = [0:0:1].$$ We let $L_i$ denote the linear form that defines the line $\L_i$. Note that after we have made our change of coordinates, if $\L$, with defining form $L = ax_0 + bx_1 + cx_2$, is any line that does not pass through $P$, then $L \not\in I_P = \langle x_1,x_2 \rangle$, i.e., $a \neq 0$.
With this setup, we make the following claim:
[*Claim.*]{} For all $m {\geqslant}s+1$, $\H_{R/(I_{m\Y}+I_P^m)}(ms-2) = 0.$
[*Proof of the Claim.*]{} By Lemma \[CTVlemma\], it suffices to show that for each $i=0,\ldots,m-1$, the monomial $x_0^{ms-2-i}M \in \big[I_{m\Y}+I_P^{i+1}\big]_{ms-2}$ where $M$ is any monomial of degree $i$ in $x_1,x_2$. We will treat the cases $i \in \{0,\ldots,m-2\}$ and $i = m-1$ separately.
Fix an $i \in \{0,\ldots,m-2\}$ and let $M$ be any monomial of degree $i$ in $x_1,x_2$. Since none of the lines $\L_2,\ldots,\L_s$ pass through the point $P$, we have $L_k = a_{k,0}x_0 + a_{k,1}x_1 + a_{k,2}x_2$ with $a_{k,0} \neq 0$ for all $k=2,\ldots,s$. Then $$L_2^mL_3^m \cdots L_s^m = ax_0^{ms-m} + \sum_{k=1}^{ms-m} x_0^{ms-m-k}f_k(x_1,x_2)$$ with $a \neq 0$ and where $f_k(x_1,x_2)$ is a homogeneous polynomial of degree $k$ only in $x_1$ and $x_2$. Since $L_2\cdots L_s \in I_\Y$, it follows that $$L_2^m \cdots L_s^m \in \big[(I_\Y)^m\big]_{ms-m} \subseteq \big[I_{m\Y}\big]_{ms-m} \subseteq \big[I_{m\Y} + I_P^{i+1}\big]_{ms-m}$$ and thus $$ax_0^{ms-m}M + \sum_{k=1}^{ms-m} x_0^{ms-m-k}f_k(x_1,x_2)M \in \big[I_{m\Y} + I_P^{i+1}\big]_{ms-m+i}.$$ But $I_P^{i+1} = \langle x_1,x_2 \rangle^{i+1}$, so $f_k(x_1,x_2)M \in I_P^{i+1}$ for each $k=1,\ldots,ms-m$ since $f_k(x_1,x_2)M$ is a homogeneous polynomial only in $x_1,x_2$ of degree $i+k {\geqslant}i+1$. But then this means that $$a^{-1}ax_0^{ms-m}M = x_0^{ms-m}M\in \big[I_{m\Y} + I_P^{i+1}\big]_{ms-m+i}.$$ Since $i {\leqslant}m-2$, we thus have $x_0^{m-2-i}x_0^{ms-m}M = x_0^{ms-2-i}M \in
\big[I_{m\Y} + I_P^{i+1}\big]_{ms-2}$.
Now suppose that $i=m-1$. Consider any monomial $M = x_1^ax_2^b$ with $a+b = m-1$ and $a,b {\geqslant}1$. Since $I_{Q_1} = \langle x_0,x_2 \rangle$ and $I_{Q_2} = \langle x_0,x_1 \rangle$, this means that $x_1^ax_2^b \in
I_{Q_1}^b \cap I_{Q_2}^a$. Because $\L_2$ is the line that passes through $Q_1$ and $Q_2$, we have $L_2^{m-1}M \in (I_{Q_1}^m \cap I_{Q_2}^m)$, and consequently, $$\begin{array}{llllllllll}
L_2^{m-1} L_3^m\cdots L_s^mM
& = & \ds ax_0^{ms-m-1}M + \sum_{k=1}^{ms-m-1} x_0^{ms-m-1-k}f_k(x_1,x_2)M \\[2.5ex]
& \in & \big[I_{m\Y} \big]_{ms-2} \subseteq \big[I_{m\Y} +
I_P^{m}\big]_{ms-2}.
\end{array}$$ Arguing as above, this implies that $x_0^{ms-m-1}M \in \big[I_{m\Y} +
I_P^{m}\big]_{ms-2}.$
It remains to show that $x_0^{ms-m-1}x_1^{m-1}$ and $x_0^{ms-m-1}x_2^{m-1}
\in \big[I_{m\Y} + I_P^{m}\big]_{ms-2}.$ We only verify the second statement since the first statement is similar. Consider the line $\L$ through the point $P$ and $Q_2$. Because $\L$ goes through $P$, it does not contain $s$ points. In particular, there must be some $j \in \{3,\ldots,s\}$ such that $\L \cap \X_j = \varnothing$, i.e., $\L$ does not intersect with any of the points of $\X$ on the line $\L_j$. Let $\X_j = \{S_1,\ldots,S_j\}$ be these $j$ points, and let $\mathbb{H}_\ell$ be the line through $Q_2$ and $S_\ell$ for $\ell = 1,\ldots,j$. Furthermore, let $H_\ell$ denote the associated linear form. Note that none of the lines $\mathbb{H}_\ell$ can pass through the point $P$, so in particular, each $H_\ell$ has the form $H_\ell = a_{\ell}x_0 + b_{\ell}x_1 + c_{\ell}x_2$ with $a_{\ell} \neq 0$.
We now claim that $$F := x_1^{m-1}
H_{1}\cdots H_{j} L_2^{m-j}L_3^{m}\cdots L_{j-1}^mL_{j}^{m-1}
L_{j+1}^m\cdots L_s^m \in I_{m\Y}.$$ Because $j {\leqslant}s$ and $m {\geqslant}s+1$, $m-j {\geqslant}1$. So, in particular, $x_1^{m-1} L_2^{m-j} \in I_{Q_1}^m$. Also, $H_{1}\cdots H_{j} L_2^{m-j} \in I_{Q_2}^m$, so $F$ vanishes at the points of $\{Q_1,Q_2\}$ to the correct multiplicity. Note that $H_{1}\cdots H_{j} L_{j}^{m-1}$ vanishes at all the points on $\L_j$ to multiplicity at least $m$. Furthermore, for any other $k$, $L_k^m$ vanishes at all the points on $\L_k$ to multiplicity at least $m$. So we have $F \in I_{m\Y}$. To finish the proof, we need to note that $$H_{1}\cdots H_{j} L_2^{m-j}L_3^{m}\cdots L_{j-1}^mL_{j}^{m-1}
L_{j+1}^m\cdots L_s^m =
ax_0^{ms-m-1} + \sum_{k=1}^{ms-m-1} x_0^{ms-m-1-k}f_k(x_1,x_2)$$ with $a \neq 0$ and where each $f_k(x_1,x_2)$ is a homogeneous polynomial of degree $k$ only in $x_1,x_2$. The rest of the proof now follows similar to the cases above. This ends the proof of the claim. We now complete the proof. Let $m {\geqslant}s+1$ be any integer, and consider the short exact sequence $$0 \rightarrow (I_{m\Y} \cap I_P^m) \rightarrow
I_{m\Y} \oplus I_P^m \rightarrow I_{m\Y}+I_P^m \rightarrow 0.$$ Note that the ideal of $m\X$ is $I_{m\X} = I_{m\Y} \cap I_P^m$, so the short exact sequence implies $$\H_{m\X}(t) = \H_{m\Y}(t) + \H_{mP}(t) - \H_{R/(I_{m\Y}+I_P^m)}(t)$$ for all $t {\geqslant}0$. Note that $\H_{mP}(t) = \binom{m+1}{2}$ for all $t {\geqslant}m-1$. Using this fact, and the above claim we get $$\begin{aligned}
\Delta \H_{m\X}(ms-1) &= &\H_{m\X}(ms-1) - \H_{m\X}(ms-2) \\
& = &\left(\H_{m\Y}(ms-1) + \H_{mP}(ms-1) - \H_{R/(I_{m\Y}+I_P^m)}(ms-1)\right) - \\
&&
\left(\H_{m\Y}(ms-2) + \H_{mP}(ms-2) - \H_{R/(I_{m\Y}+I_P^m)}(ms-2)\right) \\
& = & \Delta \H_{m\Y}(ms-1) + \left(\binom{m+1}{2}-\binom{m+1}{2}\right)
- (0 -0) \\
& = & r.\end{aligned}$$ The last equality comes from Theorem \[bigmult\] since $\Delta \H_{m\Y}(ms-1) = r$ for all $m {\geqslant}2$.
Notice that in the proof of Theorem \[case<s\], the hypothesis that $m {\geqslant}s+1$ was only used in the proof of the claim to show that a particular monomial belonged to the ideal $I_{m\Y}+I_P^m$. However, there may be some room for improvement on the lower bound $s+1$. For example, for the $\k$-configuration of type $(1,2,3)$ given in Figure \[specialcase\], computer tests have shown that $\Delta \H_{m\X}(m3-1) = 1$ for all $m {\geqslant}s =3$, instead of $s+1 = 4$. Similarly, if we consider [*standard linear configurations*]{} of type $(1,2,\ldots,s)$ (as defined in [@GMS Definition 2.10]), then it can be shown that Theorem \[case<s\] holds for all $m {\geqslant}2$. We omit this proof since it requires the special geometry of standard linear $\k$-configurations.
Concluding Remarks
==================
We conclude this paper with some observations. Following [@CTV], we define the [*regularity index*]{} of a zero-dimensional scheme $\Z \subseteq \mathbb{P}^n$ to be $${\rm ri}(\Z) = \min (t ~|~ \H_{\Z}(t) = \deg(\Z)).$$ Embedded in our proof of Theorem \[maintheorem\], we actually computed the regularity index of multiples of a $\k$-configuration. In particular, we proved that
Let $\X \subseteq \mathbb{P}^2$ be a $\k$-configuration of type $d=(d_1,\dots,d_s)$. Then for all integers $m {\geqslant}s+1$, $${\rm ri}(m\X) = md_s -1.$$
If $d=(1)$, then $\X =\{P\}$ is a single point. It is well-known that $$\H_{m \X} (t) = \min \left(
\binom{t+2}{2}, \binom{m+1}{2}
\right),$$ so the regularity index is $m-1$.
If $d\neq (1)$ and if $m {\geqslant}s+1$, Theorems \[bigmult\], \[case=s+1\], \[case=s\], and \[case<s\], imply $\H_{m\X}(md_s-2) < \H_{m\X}(md_s-1)$. Moreover, as part of our proofs, we argued that $\H_{m\X}(md_s-1) = \deg(m\X)$.
The regularity index ${\rm ri}(\Z)$ can also be defined as the maximal integer $t$ such that $\Delta \H_\Z(t) \ne 0$. So, Theorem \[maintheorem\] can be restated as:
\[restatement\] Let $\X \subseteq \mathbb{P}^2$ be a $\k$-configuration of type $d=(d_1,\dots,d_s) \neq (1)$ and $m {\geqslant}s+1$. Then the number of lines containing exactly $d_s$ points of $\X$ is the last non-zero value of $\Delta \H_{m\X}(t)$.
As a final comment, we turn to a question posed by Geramita, Migliore, and Sabourin [@GMS]:
\[question\] What are all the possible Hilbert functions of fat point schemes in $\mathbb{P}^n$ whose support has a fixed Hilbert function $\H$?
As noted in [@GMS], this question is quite difficult; in fact, [@GMS] focused on the case of double points in $\mathbb{P}^2$. Using the work of this paper, we can give an interesting observation related Question \[question\]:
Fix integers $m {\geqslant}s+1 {\geqslant}3$. Then there are at least $s+1$ possible Hilbert functions of homogeneous fat points of multiplicity $m$ in $\mathbb{P}^2$ whose support has the Hilbert function $$\H(t) = \min\left( \binom{t+2}{2}, \binom{s+1}{2} \right).$$
Any $\k$-configuration $\X$ of type $(1,2,\ldots,s)$ with $s {\geqslant}2$ has Hilbert function $\H_\X(t) = \H(t)$ (see [@RR Theorem 1.2]). By Theorem \[restatement\], $\H_{m\X}(ms-2) = \deg(m\X) - r$ where $r$ is the number of lines that contain $s$ points of $\X$. As shown in Lemma \[relabel2\], $1 {\leqslant}r {\leqslant}s+1$. It suffices to show that each $r$ is possible; this would imply that we have at least $s+1$ different Hilbert functions.
Fix $\L_1,\ldots,\L_{s+1}$ distinct lines. If $r = s+1$, we take all pairwise intersections of these $s+1$ lines to get the desired set of points. So, suppose $1 {\leqslant}r {\leqslant}s$. We construct a $\k$-configuration of type $(1,2,\ldots,s)$ with exactly $r$ lines containing $s$ points as follows:
1. let $\X_1$ be any point in $\L_1 \setminus (\L_2\cup \cdots
\cup \L_{s+1})$.
2. let $\X_2$ be any two points $\L_2 \setminus (\L_1 \cup \L_3
\cup \cdots \cup \L_{s+1})$.
3. 4. let $\X_{s-r}$ be any $s-r$ points $\L_{s-r} \setminus
(\L_1 \cup \cdots \cup \widehat{\L}_{s-r} \cup \cdots \cup \L_s)$.
5. let $\X_{s-r+1}$ be any $s-r+1$ points $\L_{s-r+1} \setminus
(\L_1 \cup \cdots \cup \widehat{\L}_{s-r+1} \cup \cdots \cup \L_s)$.
6. let $\X_{s-r+2}$ any $s-r+1$ points on $\L_{s-r+2} \setminus
(\L_1 \cup \cdots \cup \widehat{\L}_{s-r+2} \cup \cdots \cup \L_s)$ and the point $\L_{s-r+2} \cap \L_{s-r+1}$.
7. let $\X_{s-r+3}$ be any $s-r+1$ points on $\L_{s-r+3} \setminus
(\L_1 \cup \cdots \cup \widehat{\L}_{s-r+3} \cup \cdots \cup \L_s)$ and the two points $\L_{s-r+3} \cap (\L_{s-r+1} \cup \L_{s-r+2})$.
8. 9. let $\X_{s}$ be any $s-r+1$ points on $\L_{s} \setminus
(\L_1 \cup \cdots \cup \widehat{\L}_{s})$ and the $r-1$ points $\L_{s} \cap (\L_{s-r+1} \cup \cdots \cup \L_{s-1})$.
This configuration then gives the desired result.
As mentioned in the introduction, $\k$-configurations of points can be defined in $\mathbb{P}^n$ (see, e.g., [@GHS:1; @GHS:2; @GS; @H]). It is natural to ask if a result similar to Theorem \[maintheorem\] also holds more generally. Based upon some calculations, it appears that this may be the case. For example, let $\X$ be the $\k$-configuration of points in $\mathbb{P}^3$ found in [@GS Example 4.1] (see [@GS] for both the definition and a picture). For this example, one can see that there are three lines that contain four points. The Hilbert function of $2\X$ is given by $$\begin{array}{lllllllllllllllllll}
\H_{2\X} & : & 1 & 4 & 10 & 20 & 35 & 50 & 57 & 60 & \ra.
\end{array}$$ Note that ${\rm ri}(2\X) = 7$. Also, we have $\Delta\H_{2\X}(7)=3$, i.e., the same as the number of lines containing four points, which is similar to our statement in Theorem \[restatement\].
Although we suspect that a more general result holds, our proof relies on techniques developed in [@CHT] that only give precise information when the points are in $\mathbb{P}^2$.
[**Acknowledgements**]{} We would like to thank the referee for their helpful comments and suggestions. Work on this project began when the second and third authors visited Anthony (Tony) V. Geramita in Kingston, Ontario in the summer of 2015. Computer experiments using [CoCoA]{} [@cocoa] inspired our main result. Unfortunately, Tony became quite ill soon after our visit, and he passed away in June 2016. We would like to thank Tony for the input he was able to provide during the very initial stage of this project. Shin’s research was supported by the Basic Science Research Program of the NRF (Korea) under grant (No. 2016R1D1A1B03931683). Van Tuyl’s research was supported in part by NSERC Discovery Grant 2014-03898.
[10]{} J. Abbott, A. Bigatti, G. Lagorio, CoCoA-5: a system for doing Computations in Commutative Algebra. Available at [http://cocoa.dima.unige.it]{}
A. Bigatti, A.V. Geramita, J. Migliore, Geometric consequences of extremal behavior in a theorem of Macaulay. Trans. Amer. Math. Soc. [**346**]{} (1994), 203–235.
M.V. Catalisano, N.V. Trung, G. Valla, A sharp bound for the regularity index of fat points in general position. Proc. Amer. Math. Soc. [**118**]{} (1993), 717–724.
L. Chiantini, J. Migliore, Almost maximal growth of the Hilbert function. J. Algebra [**431**]{} (2015), 38–77.
S. Cooper, B. Harbourne, Z. Teitler, Combinatorial bounds on Hilbert functions of fat points in projective space. J. Pure Appl. Algebra [**215**]{} (2011), 2165–2179.
A.V. Geramita, B. Harbourne, J. Migliore, Star configurations in $\mathbb{P}^n$. J. Algebra [**376**]{} (2013), 279–299.
A.V. Geramita, T. Harima, Y.S. Shin, An alternative to the Hilbert function for the ideal of a finite set of points in $\mathbb{P}^n$. Illinois J. Math. [**45**]{} (2001), 1–23.
A.V. Geramita, T. Harima, Y.S. Shin, Decompositions of the Hilbert function of a set of points in $\P^n$, Canadian J. Math. [**53**]{} (2001), 923–943.
A.V. Geramita, T. Harima, Y.S. Shin, Extremal point sets and Gorenstein ideals. Adv. Math. [**152**]{} (2000), 78–119.
A.V. Geramita, J. Migliore, L. Sabourin, On the first infinitesimal neighborhood of a linear configuration of points in $\mathbb{P}^2$. J. Algebra [**298**]{} (2006), 563–611.
A.V. Geramita, Y.S. Shin, $k$-configurations in $\mathbb{P}^3$ all have extremal resolutions. J. Algebra [**213**]{} (1999), 351–368.
T. Harima, Some examples of unimodal Gorenstein sequences. J. Pure Appl. Algebra [**103**]{} (1995), 313–324.
L.G. Roberts, M. Roitman, On Hilbert functions of reduced and of integral algebras. J. Pure Appl. Algebra [**56**]{} (1989), 85–104.
O. Zariski, P. Samuel, [*Commutative Algebra, vol. II*]{}, Springer, 1960.
[^1]: Last updated:
|
---
abstract: 'Existing algorithms for the optimal control of quantum observables are based on locally optimal steps in the space of control fields, or as in the case of genetic algorithms, operate on the basis of heuristics that do not explicitly take into account details pertaining to the geometry of the search space. We present globally efficient algorithms for quantum observable control that follow direct or close-to-direct paths in the domain of unitary dynamical propagators, based on partial reconstruction of these propagators at successive points along the search trajectory through orthogonal observable measurements. These algorithms can be implemented experimentally and offer an alternative to the adaptive learning control approach to optimal control experiments (OCE). Their performance is compared to that of local gradient-based control optimization.'
author:
- Raj Chakrabarti
- Rebing Wu
- Herschel Rabitz
bibliography:
- 'INTEGRABLE.bib'
date: 24 August 2007
title: 'Orthogonal measurement-assisted quantum control'
---
Introduction {#intro}
============
The optimal control of quantum dynamics is receiving increasing interest due to widespread success in laboratory and computational experiments across a broad scope of systems. With these promising results it becomes imperative to understand the reasons for success and to develop more efficient algorithms that can increase objective yields to higher quality. In the computational setting, the expense of iteratively solving the Schrodinger equation necessitates faster algorithms for the search over control field space if these methods are to be routinely employed for large-scale applications. In the laboratory setting, although closed-loop methodologies have encountered remarkable success, the search algorithms currently used do not typically attain yields as high as those that can be achieved using computational algorithms.
Recently, significant strides have been made towards establishing a foundation for the systematic development of efficient OCT algorithms based on the observation that the landscape traversed by search algorithms in the optimization of quantum controls is not arbitrarily complicated, but rather possesses an analytical structure originating in the geometry of quantum mechanics [@Raj2007]. This structure should allow not only rationalization of the comparative successes of previous quantum control experiments, but also analytical assessment of the comparative efficiencies of new algorithms.
Prior work established important features of these landscapes, in particular, their critical topologies [@RabMik2005; @Mike2006a]. The critical points of a control landscape correspond to locally optimal solutions to the control problem. The most common objective in quantum optimal control is maximization of the expectation value of an observable. The landscape corresponding to this problem was shown to be almost entirely devoid of local traps, i.e., the vast majority of local suboptima are saddles, facilitating the convergence of local search algorithms. The number of local suboptima, as well as the volumes of these critical regions were calculated. However, the relationship between the topology of quantum control landscapes and their geometry, which would dictate the behavior of global search algorithms, was not explored.
Thus far, optimal control algorithms for quantum observables have not exploited the geometry of quantum control landscapes, which may simplify control optimization compared to that for classical systems. Indeed, the majority of quantum optimal control algorithms to date have aimed at optimizing an objective functional, such as the expectation value of an observable operator, directly on the domain of time-dependent control fields $\varepsilon(t)$. Typical approaches to quantum control optimization use the information in the measurement of a single quantum observable to guide the search for optimal controls; the simplest approach is to randomly sample single observable expectation values at various points over the landscape and use genetic algorithms (GA) to update the control field. A recent experimental study [@Roslund2007] demonstrated at least a two-fold improvement in optimization efficiency through the use of local gradient algorithms rather than GA, but the important question remains as to whether global algorithms for quantum control that are not “blind” like GA can be implemented in an experimental setting.
When control optimization seeks to optimize the expectation value of an observable by following, e.g., local gradient information on the domain of controls, the geometry of the underlying space of quantum dynamical propagators ${\mathcal{U}}(N)$ is not explicitly exploited. In particular, optimal control algorithms that are based on locally minimizing an objective function on the domain of control fields do not follow globally optimal paths in ${\mathcal{U}}(N)$. This approach tends to convolute the properties of the map between control fields and associated unitary propagators with the properties of the map between unitary propagators and associated values of the objective function.
An alternative approach to observable maximization is to first solve numerically for the set of unitary matrices $U$ that maximize the expectation value ${{\rm Tr}}(U\rho(0)U^{\dag}\Theta)$ of the observable $\Theta$, and then to determine a control field $\varepsilon(t)$ that produces that $U$ at time $t=T$ [@Khaneja2001; @Khaneja2002a]. For restricted Hamiltonians in low dimensions, analytical solutions for the optimal control field $\varepsilon(t)$ have been shown to exist. However, to date, numerical algorithms for the optimization of unitary propagators in higher dimensions have operated solely on the basis of local gradient information, such that the global geometry of ${\mathcal{U}}(N)$ is again not exploited.
The variational problems of optimal control theory admit two types of minimizers. Denoting the cost functional by $J$, according to the chain rule, $$\frac{\delta J}{\delta \varepsilon(t)} = \frac{{{\rm d}}J}{{{\rm d}}U} \cdot
\frac{\delta U}{\delta \varepsilon(t)}.$$ The first type of minimizer corresponds to those control Hamiltonians that are critical points of the control objective functional, but are not critical points of the map between control fields and associated dynamical propagators (i.e., points at which $\frac{{{\rm d}}J}{{{\rm d}}U}=0$, while the Frechet derivative mapping from the control variation $\delta \varepsilon(t)$ to $\delta U(T)$ at $t=T$ is surjective). The second type corresponds to critical points of the latter map (i.e., points at which the mapping from $\delta \varepsilon(t)$ to $\delta U(T)$ is not locally surjective) [@Wu2007] [^1]. Critical points of the first type, which are referred to as kinematic critical points or normal extremal controls, are either global optima or saddle points, but never local traps[@RabMik2005; @Mike2006a]. Recent work in quantum optimal control theory suggests that the critical points of the map $\varepsilon(t) \rightarrow U(T)$, called abnormal extremal controls, are particularly rare (i.e., there are generally fewer critical points compared to classical control problems) [@Wu2007; @Raj2007].
Because the objective function $J$ is a complete function of $U(T)$, the maximal achievable optimization efficiency is ultimately determined by the properties of the map between control fields and unitary propagators, $\varepsilon(t) \rightarrow U(T)$. Irrespective of the corresponding observable expectation value, updating the control field to produce a unitary propagator that is close to the current propagator will typically be computationally inexpensive. Therefore, following a direct route in the space of unitary propagators is expected to be more efficient in quantum control than following a gradient flow on the space of objective function values that maps to a longer path in ${\mathcal{U}}(N)$. As will be shown, the scarcity of critical points of the map $\varepsilon(t) \rightarrow
U(T)$ in quantum control problems implies that it is surprisingly simple to track arbitrary paths in ${\mathcal{U}}(N)$ during optimization, at least for certain families of Hamiltonians. However, it is not uncommon to encounter regions of ${\mathcal{U}}(N)$ where numerically, the relevant differential equations are ill-conditioned. The ability to selectively avoid such singular regions, which correspond to abnormal extremals, is desirable. One way to achieve this goal is to constrain the search trajectory to only roughly follow a predetermined path in ${\mathcal{U}}(N)$.
In this paper, we develop globally efficient algorithms for the optimization of quantum observables that exploit the geometry of ${\mathcal{U}}(N)$ by *approximately* following a predetermined path in the space of quantum dynamical propagators. This approach to globally efficient quantum control optimization is based on making a partial tomographic set of measurements at various steps along the search trajectory. A complete tomographic set of observations is a set that is adequate for the estimation of all the $N^2$ parameters of the unitary propagator $U(T)$ [@Hradil2003; @Lidar2007]; a partial tomographic set reconstructs only a subset of these parameters. The goal of this approach is to reap the benefits of unitary matrix tracking without encountering the associated singularities. As such, the approach attempts to leverage the methodologies of quantum statistical inference [@Malley1993] in order to reduce the search effort involved in solving quantum control problems.
These experimentally-implementable algorithms for quantum control optimization can be simulated by employing a generalization of the diffeomorphic homotopy tracking methodology D-MORPH (diffeomorphic modulation under observable response-preserving homotopy) [@Rothman2005; @Rothman2006a; @Rothman2006b]. In contrast to observable-preserving diffeomorphic tracking, the orthogonal observable tracking algorithm developed and applied here identifies parametrized paths $\varepsilon(s,t)$ that follow a given predetermined trajectory through ${\mathcal{U}}(N)$. In both cases, a denumerably infinite number of solutions exist to the tracking differential equations; paths $\varepsilon(s,t)$ that optimize desirable physical features of the control field can be tuned through the choice of an auxiliary free function. We will show that a primary difference between scalar and vector observable tracking is that the trajectory followed in ${\mathcal{U}}(N)$ in the former case is highly sensitive to changes in the system Hamiltonian, whereas the $U$-trajectory followed in the latter case can be rendered largely system independent by employing a larger set of orthogonal observables. This suggests that, besides its usefulness as an optimization algorithm, orthogonal observable tracking can reveal universal features underlying the computational effort involved in quantum optimal control searches across diverse systems.
In addition, we compare the trajectories in ${\mathcal{U}}(N)$ followed by standard OCT gradient-following algorithms with those that track optimal paths in the dynamical group, for various Hamiltonians, in order to determine how the geometry of the underlying space affects the convergence of experimental and computational control optimizations that exploit only local gradient information. In so doing, we will show that there exists a special relationship between the gradient flow on $\varepsilon(t)$ and a particular (global) path $U(s)$ in the domain of unitary propagators, namely the gradient flow of the objective on ${\mathcal{U}}(N)$, which offers insight into the convergence properties of the former.
Quantum optimal control gradient flows {#gradientflows}
======================================
Local algorithms for quantum optimal control, whether numerical (OCT) or experimental (OCE), are typically based on the gradient of the objective function. In this section, we review the properties of the gradient for quantum observable expectation value maximization, and dissect these properties into system(Hamiltonian)-dependent and universal system-independent parts.
A generic quantum optimal control cost functional can be written:
$$\begin{gathered}
\label{OCT}
J =\Phi(U(T), T)-\\
\textmd{Re}\left[{{\rm Tr}}\int_{0}^T\left\{\left(\frac{\partial U(t)}{\partial t} +
\frac{i}{\hbar}H(\varepsilon(t))U(t)\right)\beta(t)\right\}dt\right]-\\
\lambda\int_{0}^T \mid \varepsilon(t)\mid^2 {{\rm d}}t\end{gathered}$$
where $H$ is the total Hamiltonian, $\beta(t)$ is a Lagrange multiplier operator constraining the quantum system dynamics to obey the [Schrödinger ]{}equation, $\varepsilon (t)$ is the time-dependent control field, and $\lambda$ weights the importance of the penalty on the total field fluence. Solutions to the optimal control problem correspond to $\frac{\delta J}{\delta \varepsilon(t)} = 0$. The functional $\Phi$, which we refer to as the objective function, can take various forms. The most common form of $\Phi$ is the expectation value of an observable of the system: $$\Phi(U) = {{\rm Tr}}(U(T){\rho(0)}U^{\dag}(T)\Theta)$$ where $\rho(0)$ is the initial density matrix of the system and $\Theta$ is an arbitrary Hermitian observable operator [@Mike2006a].
An infinitesimal functional change in the Hamiltonian $\delta H(t)$ produces an infinitesimal change in the dynamical propagator $U(t,0)$ as follows: $$\delta U(t,0) = - \frac {i}{\hbar} \int_0^t U(t,t') \delta H(t')
U(t',0)dt'$$ where $\delta H(t) = \triangledown_\varepsilon H(t)
\cdot \delta \varepsilon(t)$. The corresponding change in $\Phi$ is then given by $$\delta \Phi = -\frac {i}{\hbar} \int_0^T
{{\rm Tr}}(~[\Theta(T),U^{\dag}(t,0)\delta H(t)U(t,0)~]\rho(0))dt,$$ where $\Theta(T) \equiv U^{\dag}(T){\rho(0)}U(T)$. In the special case of the electric dipole approximation, the Hamiltonian assumes the form $$H(t) = H_0 - \mu \cdot \varepsilon(t)$$ where $H_0$ is the internal Hamiltonian of the system and $\mu$ is its electric dipole operator. $\mu(t)$ is given by $$\mu(t) \equiv U^{\dag}(t,0)\mu U(t,0) = i \hbar M(t)$$ where $M(t) \equiv \frac {i} {\hbar} U^{\dag}(t,0) \triangledown_{\varepsilon}
H(t) U(t,0)$. Within the electric dipole approximation, the gradient of $\Phi$ is [@HoRab2007a]: $$\begin{gathered}
\label{grad1}
\frac{\delta \Phi}{\delta \varepsilon(t)} =
-\frac{i}{\hbar}{{\rm Tr}}\{\left[\Theta(T),\mu(t)\right]\rho(0)\}\\
= \frac{i}{\hbar}\sum_i
\rho(0)\langle i|\Theta(T)\mu(t)-\mu(t)\Theta(T)|i\rangle \\
= \frac {i}{\hbar} \sum_{i=1}^n p_i \sum_{j=1}^N \Big[\langle
i|\Theta(T)|j\rangle \langle j|\mu(t)|i\rangle -\langle
i|\mu(t)|j\rangle \langle j|\Theta(T)|i\rangle \Big]\end{gathered}$$ where the initial density matrix is given as $\rho(0)=\sum_{i=1}^n
p_i |i\rangle\langle i|, p_1 > ...> p_n > 0, \quad \sum_{i=1}^n p_i
= 1.$ The assumption of local surjectivity of $\varepsilon(t)
\rightarrow U(T)$ implies that the functions $\langle
i|\mu(t)|j\rangle $ are $N^2$ linearly independent functions of time. The functions $$\begin{gathered}
\label{gradbasis}
\langle i|U^{\dag}(T) \Theta U(T)|j\rangle \langle j|U^{\dag}(t)\mu U(t)|i\rangle ~ -\\
\langle i|U^{\dag}(t)\mu U(t)|j\rangle \langle j|U^{\dag}(T)\Theta U^{\dag}(T)|i\rangle\end{gathered}$$ therefore constitute natural basis functions for the gradient on the domain $\varepsilon(t)$. We are interested in the global behavior of the flow trajectories followed by these gradients, which are the solutions to the differential equations $$\label{Egrad}\frac{d\varepsilon(s,t)}{ds}= \triangledown \Phi_\varepsilon(\varepsilon(t))
=\alpha \frac {\delta \Phi(s,T) }{\delta \varepsilon(s,t)}$$ where $s > 0$ is a continuous variable parametrizing the algorithmic time evolution of the search trajectory, and $\alpha$ is an arbitrary positive constant that we will set to 1. The existence of the natural basis (\[gradbasis\]) indicates that these flow trajectories evolve on a low-dimensional subspace of $\varepsilon(t)$. However, the gradient flow equations cannot be integrated analytically for arbitrary internal Hamiltonians $H_0$, precluding a deeper understanding of the global dynamics of the search process. In fact, these dynamics do not have universal (Hamiltonian-independent) properties. The explicit path followed by the search algorithm on $\varepsilon(t)$ depends on the solution to the Schrodinger equation for the particular system Hamiltonian and cannot be expressed analytically.
Because the objective functional $\Phi$ is explicitly a function of $U(T)$, any universal properties of the global geometry of the search dynamics must be investigated on this domain. These search dynamics are governed by the gradient flow of $\Phi$ on the domain ${\mathcal{U}}(N)$, given by $$\frac{dU}{ds} = \triangledown \Phi(U).$$ The tangent space of ${\mathcal{U}}(N)$ at any element $U \in
{\mathcal{U}}(N)$ is $$T_U {\mathcal{U}}(N)=\{U \Omega | \Omega^{\dag} = -\Omega, \quad \Omega
\in {\mathbb{C}}^{N\times N}\},$$ where $\Omega$ is an arbitrary skew-Hermitian matrix, and the directional derivative for a function $\Phi$ defined on ${\mathcal{U}}(N)$ is $$\begin{aligned}
D \Phi_U(U \Omega) \equiv {{\rm Tr}}\left((\triangledown\Phi(U))^{\dag}U\Omega\right).\end{aligned}$$ The directional derivative of the objective functional $\Phi$ along an arbitrary direction $U \Omega$ in $T_U {\mathcal{U}}(N)$ can then be written $$\begin{aligned}
D\Phi_{U}(U\Omega) &=& {{\rm Tr}}\left(U^{\dag}\Theta U \Omega \rho(0) + (U
\Omega)^{\dag}\Theta U \rho(0)\right) \\
&=& {{\rm Tr}}\left([\rho(0),U^{\dag}\Theta U]
\Omega\right)\end{aligned}$$ allowing us to identify the gradient of $\Phi$ on ${\mathcal{U}}(N)$ as $$\triangledown \Phi = -U[\rho(0),U^{\dag}\Theta U]=[\Theta,U\rho(0) U^{\dag}]U.$$ Therefore, the equations of motion for the gradient flow lines of objective functional $\Phi$ are $$\label{Uflow}
\frac{dU}{ds} = [\Theta,U\rho(0)U^{\dag}]U = -U \rho(0) U^{\dag}\Theta U + \Theta U\rho(0).$$ In section \[integration\] below, we integrate these equations to obtain the trajectories $U(s)$ followed by gradient algorithms on ${\mathcal{U}}(N)$ over algorithmic time $0\leq s< \infty$.
The essential question arises as to the relationship between the gradient flow on $\varepsilon(t)$ and that on ${\mathcal{U}}(N)$. The gradient on $\varepsilon(t)$ is related to the gradient on ${\mathcal{U}}(N)$ through $$\label{chain}\frac{\delta \Phi}{\delta \varepsilon(t)}=\sum_{i,j}\frac{\delta
U_{ij}}{\delta \varepsilon(t)}\frac{{{\rm d}}\Phi}{{{\rm d}}U_{ij}}.$$ Now suppose that we have the gradient flow of $\varepsilon(s,t)$ that follows (\[Egrad\]) and let $U(s)$ be the projected trajectory on the unitary group ${\mathcal{U}}(N)$ of system propagators at time $T$, driven by $\varepsilon(s,t)$. The algorithmic time derivative of $U(s)$ is then $$\label{Us}
\frac{{{\rm d}}U_{ij}(s)}{{{\rm d}}s}= \int_0^T \frac{\delta U_{ij}(s)}{\delta \varepsilon(s,t)}\frac{\partial \varepsilon(s,t)}{\partial
s} {{\rm d}}t$$ which, combined with (\[Egrad\]) and (\[chain\]), gives $$\label{dot Us}
\frac{{{\rm d}}U_{ij}(s)}{{{\rm d}}s}=\int_0^T \frac{\delta U_{ij}(s)}{\delta \varepsilon(s,t)}\sum_{p,q}\frac{\delta U_{pq}(s)}{\delta \varepsilon(s,t)}\frac{{{\rm d}}\Phi}{{{\rm d}}U_{pq}} {{\rm d}}t.$$ It is convenient to write this equation in vector form, replacing the $N \times N$ matrix $U(s)$ with the $N^2$ dimensional vector $\textbf{u}(s)$: $$\begin{gathered}
\label{Gmat}
\frac{{{\rm d}}\textbf{u}(s)}{{{\rm d}}s} =\left[\int_0^T
\frac{\delta\textbf{u}(s)}{\delta
\varepsilon(s,t)}\frac{\delta{\textbf{u}^T(s)}}{\delta \varepsilon(s,t)}{{\rm d}}t\right]\triangledown \Phi[\textbf{u}(s)] \mathrel{\mathop:}= \\
\textmd{G}[\varepsilon(s,t)]\triangledown \Phi[\textbf{u}(s)]\end{gathered}$$ where the superscript $T$ denotes the transpose. Thus the projected trajectory from the space of control field is different from that driven by the gradient flow in the unitary group: $$\label{gradient_U}
\frac{{{\rm d}}U(s)}{{{\rm d}}s}=\nabla \Phi[U(s)].$$ This relation implies that the variation of the propagator in ${\mathcal{U}}(N)$ caused by tracking the gradient flow in the space of control field is Hamiltonian-dependent, where the influence of the Hamiltonian is all contained in the $N^2$-dimensional symmetric matrix $\textmd{G}[\varepsilon(s,t)]$.
Unitary matrix flow tracking {#unitarytracking}
============================
The matrix $\textmd{G}\left[\varepsilon(s,t)\right]$ in equation (\[Gmat\]) above indicates that the convergence time for local gradient-based OCT algorithms may vary greatly as a function of the Hamiltonian of the system. Given the decomposition of the gradient into Hamiltonian-dependent and Hamiltonian-independent parts, the natural question arises as to whether the Hamiltonian-dependent part can be suppressed to produce an algorithm whose convergence time will be (approximately) dictated by that of the unitary gradient flow, irrespective of the system Hamiltonian.
In order for the projected flow from $\varepsilon(t)$ onto $U(T)$ to match the integrated gradient flow on $U(T)$, the quantity $\frac{\partial{\varepsilon(s,t)}}{\partial s}$ that corresponds to movement in each step must satisfy a generalized differential equation: $$\label{gendiff}
\frac{{{\rm d}}U(s)}{ds} = \int_0^T \frac{\delta U(s)}{\delta
\varepsilon(s,t)}\frac{\partial{\varepsilon(s,t)}}{\partial s}{{\rm d}}t=\triangledown \Phi\left[U(s)\right].$$
In the dipole approximation, this relation becomes the following matrix integral equation: $$\int_0^T \mu (s,t)\frac{\partial
{\varepsilon(s,t)}}{\partial s}{{\rm d}}t = U^{\dag}(s)\triangledown \Phi
\left[U(s)\right],$$ where $\mu(s,t) \equiv U^{\dag}(s,t)\mu
U(s,t).$ When $\Phi$ is the observable expectation value objective function, we have $$\int_0^T \mu(s,t) \frac{\partial \varepsilon(s,t)}{\partial s}{{\rm d}}t =
-\left[\rho,U^{\dag}(s)\Theta U(s)\right].$$ On the basis of eigenstates, the matrix integral equation is written $$\label{matint}
\int_0^T \mu_{ij}(s,t)\frac {\partial \varepsilon(s,t)} {\partial s}
{{\rm d}}t = i\hbar\langle i|U^{\dag}(s,T)\triangledown \Phi
\left[U(s,T)\right]|j\rangle.$$ To solve this equation, we first note that the flexibility in the choice of the representation of the variation in $\varepsilon(s,t)$ allows us to expand it on the basis of functions $\mu_{ij}(s,t)$, as $$\frac {\partial \varepsilon(s,t)}{\partial s} = \sum_{i,j} x_{ij}
\mu_{ij}(s,t).$$ Inserting this expansion into the above equation produces $$\begin{gathered}
\sum_{p,q} x_{pq}(s) \int_0^T \mu_{ij}(s,t) \mu_{pq}(s,t)
{{\rm d}}t = \\
i\hbar \langle i|U^{\dag}(s,T)\triangledown \Phi\left[U(s,T)\right]|j\rangle.\end{gathered}$$ If we denote the correlation matrix $\textmd{G}(s)$ as $$\begin{gathered}
\textmd{G}_{ij,pq}(s) = \int_0^T \mu_{ij}(s,t)\mu_{pq}(s,t) {{\rm d}}t = \\
\int_0^T \langle i |\mu(s,t)|j\rangle \langle p|\mu(s,t)|q\rangle {{\rm d}}t,\end{gathered}$$ (as in eqn (\[Gmat\]) above, but now specifically in the case of the dipole approximation) and define $$\Delta_{ij}(s) \equiv i\hbar \langle i|U^{\dag}(s,T)\triangledown
\Phi\left[U(s,T)\right]|j\rangle,$$ it can be shown [@Dominy2007] that the matrix integral equation (\[matint\]) can be converted into the following nonsingular $N^2$-dimensional algebraic differential equation: $$\label{utrack}
\frac{\partial \varepsilon}{\partial s} = f_s + \Big(v(\Delta) -
\alpha \Big)^T\textmd{G}^{-1}v({\mu(t)})$$ where $f_s = f_s(t)$ is a “free” function resulting from the solution of the homogeneous differential equation, the operator $v$ vectorizes its matrix argument and $\alpha\equiv\int_0^T v(\mu(t))
f_s {{\rm d}}t$.
Solving this set of $N^4$ scalar differential equations requires that the $N^2 \times N^2$ matrix $\textmd{G}$ is invertible. The invertibility of this matrix is equivalent to the claim that the map $\varepsilon(t)\rightarrow U(N)$ between control fields and unitary propagators is surjective, such that it is possible to reach any $U(s+1)$ infinitesimally close to $U(s)$ in a vanishingly small step. Thus, a necessary condition for the existence of a well-determined search direction is the full-rank of the Jacobian, i.e., $$\textmd{rank} \frac {\delta U(s)}{\delta \varepsilon(s,t)}
= \textmd{dim}[{\mathcal{U}}(N)] = N^2$$ which is equivalent to the requirement of local surjectivity of the map $\varepsilon(t) \rightarrow U(T)$.
The problem is undetermined because of the rank of the matrix is lower than the number of variables to be solved, which results from the fact that the optimal control problem itself is undetermined with a multiplicity of solutions. Each “free function” $f_s$ corresponds to a unique algorithmic step in $\varepsilon(t)$; modulating this function allows for systematic exploration of the set of functions $\varepsilon(s,t)$ that are compatible with the gradient step on $U$ [@Dominy2007].
As in the case of the gradient $\triangledown \Phi[\varepsilon(t)]$, the flow on $\varepsilon(t)$ that tracks the $U$-gradient can be expressed in terms of a set of maximally $N^2$ linearly independent functions of time, but whereas the former is a unique functional derivative, the latter is highly degenerate. The former are explicitly determined by the functions $\mu(t)$, while the latter are underdetermined by these functions; only $N^2$ linearly independent components of $\frac{\partial \varepsilon(s,t)}{\partial s}$ are explicitly determined by $\mu(t)$, the rest remaining unspecified.
Although the gradient step $\frac{{{\rm d}}\varepsilon(s,t)}{{{\rm d}}s}=\nabla
\Phi[\varepsilon(s,t)]$ is always locally the direction of fastest decrease in the objective function at $\varepsilon(t)$, the path $\varepsilon(s,t)$ derived from following this gradient has no universal (Hamiltonian-independent) global geometry, since $\Phi$ is not explicitly a function of $\varepsilon(t)$. It is known [@Mike2006a] that this path will not encounter any traps during the search, but beyond this, the geometry can be expected to be rugged and globally suboptimal. Unlike the gradient $\triangledown \Phi[\varepsilon(t)]$, the algorithmic step above follows the gradient flow on ${\mathcal{U}}(N)$ (in the limit of infinitesimally small algorithmic time steps). The $N^2$ functions $\mu(s,t)$ are calculated during the evaluation of $\triangledown
\Phi[\varepsilon(t)]$; hence, the computational overhead incurred by following this flow corresponds to that needed to compute the $N^4$ elements of $\textmd{G}(s)$ and invert the matrix, at each algorithmic time step. This flow respects the geometric formulation of the optimal control objective function in terms of $U(T)$ rather directly in terms of $\varepsilon(t)$. As we shall show below, the global geometry of this path can be completely determined analytically for objective function $\Phi$. The functions $\mu(s,t)$ contain all relevant information about the quantum dynamics, whereas the functions $\triangledown \Phi(U)$ contain complete information about the geometry of the search space.
Incidentally, matrix integral equation (\[matint\]) can be reformulated to provide further insight into the relationship between $\varepsilon(t)$-gradient and $U(T)$-gradient flows. Equation (\[matint\]) can be rewritten $$\begin{gathered}
\label{mu}
\int_0^T\langle \mu(s,t'),\mu(s,t)\rangle \frac{{{\rm d}}\varepsilon(s,t')}{{{\rm d}}s}
{{\rm d}}t' =\\
\langle -i\hbar \left[\Theta(s,T),\rho\right],\mu(s,t)\rangle.\end{gathered}$$ It can be shown that if the time-dependent dipole operator $\mu(s,t)$ displays the Dirac property
$$\langle \mu(s,t'),\mu(s,t) \rangle = \sum_{i=1}^N\sum_{j\geq i}^N \left[\textmd{Re}(\langle i | \mu(s,t') |j \rangle)\textmd{Re}(\langle j | \mu(s,t') | i \rangle)+\textmd{Im}(\langle j | \mu(s,t') | i \rangle)\right]=\delta(t-t'),$$
the corresponding nonsingular initial value problem is $$\frac{{{\rm d}}\varepsilon(s,t)}{{{\rm d}}s} \approx \langle -i\hbar \left[\Theta(s,T),\rho(0)\right],\mu(s,t)\rangle,$$ which is effectively identical to the $\varepsilon(t)$-gradient flow for observable functional $\Phi$. As such, the extent to which condition (\[mu\]) is satisfied for a given Hamiltonian will determine the faithfulness with which this flow tracks the $U(T)$-gradient.
Of course, a multitude of other flows could be substituted for the RHS of equation (\[matint\]). In section \[integration\], we will integrate the $U(T)$-gradient flow and show that it does not follow a globally optimal path. Since we are interested in global optimality, we should choose a flow that follows the shortest possible path from the initial condition to a unitary matrix that maximizes the observable expectation value. It can be shown [@Mike2006a] that a continuous manifold of unitary matrices $W$ maximizes $\Phi(T)$. These $W$s can be determined numerically by standard optimization algorithms on the domain of unitary propagators [@Brockett1991]. The shortest length path in $U(N)$ between $U(0)$ and an optimal $W$ is then the geodesic path that can be parameterized as $U(s)=U(0)\exp(iAs)$ with $A =
-i\log(W^{\dag}U(0))$ where $\log$ denotes the complex matrix logarithm with eigenvalues chosen to lie on the principal branch $-\pi<\theta<\pi$. Thus, if we set $\Delta_{ij}(s) = \langle i|A|j
\rangle = \langle i|-i\log(W^{\dag}U(s)|j \rangle$, the tracking algorithm will attempt to follow this geodesic path. Because this choice of $A$ does not represent the gradient of an objective function, the optimization will not converge exponentially to the solution, but rather will continue past the target matrix $W$ unless stopped [@Dominy2007]. On the other hand, it is in principle possible to choose $A$ that results in the algorithm tracking the same step in ${\mathcal{U}}(N)$, but at a rate that depends on algorithmic time $s$. [^2]
Due to the nonlinearity of the differential equations above, errors in tracking will inevitably occur, increasing the length of the search trajectory beyond that of the minimal geodesic path (see below). These errors will naturally be a function of the system Hamiltonian. It is of interest to examine the dependence of matrix flow tracking errors on the Hamiltonian by continuously morphing the Hamiltonian during the optimization. This efficient approach to Hamiltonian sampling will allow a more systematic comparison of the efficiency of global OCT optimization with that of local gradient-based OCT, which is expected to be much more system-dependent.
Hamiltonian morphing can encompass changes in both the system’s internal Hamiltonian and the dipole operator. We assume these matrices can be written as functions of the algorithmic step $s$ as ${\mathcal{H}}(s) = {\mathcal{H}}_0(s) + \mu(s)\varepsilon(s,t)$. Since we have
$$\frac{\partial {\mathcal{H}}(s,t)}{\partial s} = \frac{{{\rm d}}{\mathcal{H}}_0(s)}{{{\rm d}}s} - \frac{{{\rm d}}\mu(s)}{{{\rm d}}s} \varepsilon(s,t) - \mu(s)\frac{\partial
\varepsilon(s,t)}{\partial s},$$ we can rewrite eqn (\[matint\]) as $$\begin{gathered}
\label{duds2}
\frac{{{\rm d}}U(s)}{{{\rm d}}s} = \int_0^T {{\rm d}}t \big(a_0(s,t,T)\frac{\partial
\varepsilon(s,t)}{\partial s} ~+\\
a_1(s,t,T)\varepsilon(s,t)+a_2(s,t,T)\big) = 0\end{gathered}$$ where $a_0 = \mu(s,t)$, $a_1 =\frac {{{\rm d}}\mu(s,t)}{{{\rm d}}s}$ and $a_2
= \frac {{{\rm d}}{\mathcal{H}}_0(s)}{{{\rm d}}s}$. Thus, if we define $$b(s,T) \equiv - \int_0^T \big( \nu(a_1)\varepsilon(s,t) + \nu(a_2) \big) {{\rm d}}t,$$ where $\nu$ denotes the $N^2$-dimensional vectorized Hermitian matrix as above, we can rewrite the matrix integral equation (\[matint\]) as $$\int_0^T \mu(s,t) \frac{\partial \varepsilon(s,t)}{\partial s}{{\rm d}}t =
-~[\rho,U^{\dag}(s)\Theta U(s)~]+b(s,T).$$ Therefore in the case of combined Hamiltonian morphing and unitary tracking, the D-MORPH differential equation for the control field becomes
$$\label{UHamdiff}
\frac{\partial E}{\partial s} = f_s + \Big(v(\Delta) + b(s,T) -
\alpha \Big)^T\textmd{G}^{-1}v({\mu(t)})$$
Even if $\textmd{G}$ is invertible, it is possible that it is nearly singular, resulting in large numerical errors during the solution to the differential equation. It is convenient to assess the nearness to singularity of $\textmd{G}$ by means of its condition number $C$, namely the ratio of its largest singular value to its smallest singular value, i.e., if $\textmd{G}^{-1}=V\left[\textmd{diag}(1/\omega_j)\right]U^T$, $C=\frac{\max_j\omega_j}{\min_j\omega_j}$.
As mentioned, tracking errors can also originate due to the omission of higher order functional derivatives, such as $\frac{\delta^2
U(T)}{\delta \varepsilon(t)^2}$, in equation (\[duds2\]). These may in principle be large, even if the input-state map is surjective, resulting in large numerical errors for finite step sizes. Since the calculation of these higher derivatives is very expensive, we do not employ them in the calculation of the algorithmic step. The hypothesis that tracking globally optimal paths in ${\mathcal{U}}(N)$ is typically more efficient than local optimization of the expectation value of the target observable is equivalent to the assumption that the second and higher order functional derivatives of the unitary propagator with respect to the control field are relatively small, but not negligible when attempting to traverse a large distance in ${\mathcal{U}}(N)$ in a single control field iteration.
The computational expense of unitary matrix tracking increases fairly steeply with system dimension. Since matrix inversion scales as $N^2$, where $N$ is the dimension of the matrix, the cost of inverting the $\textmd{G}$ matrix scales as $N^4$, where $N$ is the Hilbert space dimension. By contrast, global observable expectation value tracking, discussed in the next section, avoids this overhead, but at the cost of being unable to specify precisely the unitary path followed during optimization.
In order to test the hypothesis that the primary determinant of optimization efficiency is the unitary path length to the target $W$, we compared the optimization efficiencies of algorithms that follow a geodesic on the unitary group versus a faster path on the domain of objective function values that corresponds to a longer path in ${\mathcal{U}}(N)$. These results are presented in section \[numerical\].
Orthogonal observation-assisted quantum control {#orthog}
===============================================
Unitary matrix tracking has the distinct advantage that it can directly follow an optimal path in the space of unitary propagators, assuming the input-state map is surjective and the linear formulation of the tracking equations above is a reasonable approximation. However, it cannot be implemented experimentally without expensive tomography measurements, and carries a computational overhead that scales exponentially with system size. Given the initial state $\rho(0)$ of the system, matrix elements of the unitary operator $U(T)$ can be determined based on knowledge of the final state $\rho(T) = U(T)\rho(0)U^{\dag}(T)$. $\rho(T)$ can be known only if a so-called tomographically complete set of observables has been measured sufficiently many times on identical copies of the system to approximate the expectation value of each observable. Assuming $\rho(0)$ is nondegenerate, if such measurements are made at each step of the control optimization, the unitary matrix tracking described above can be implemented (if $\rho(0)$ is degenerate, the maximum number of $U(T)$ elements that can be reconstructed will be diminished, as shown below). However, the cost of this procedure is very steep for large systems. The natural question arises as to what sort of comparative benefit in optimization efficiency can be accrued from measurement of a limited number $m$ of (orthogonal) operators, where $m < N^2$.
Consider the case where $n \geq m$ distinct observables, denoted $\Theta_1(T),...,\Theta_n(T)$ or $\{\Theta_k\}$, possibly linearly dependent and not necessarily orthogonal, are measured at each step. For simplicity, represent each of the Hermitian matrices as an $N^2$-dimensional vector with real coefficients. Then by Gram-Schmidt orthogonalization, it is always possible to construct an orthogonal basis of $m$ linearly independent $N^2$-dimensional vectors, $\Theta'_1(T),...,\Theta'_m(T)$ that (spans this set) - any element of the set $\{\Theta_k\}$ can be expressed as a linear combination of the basis operators in this set, i.e., for any $k$, $\Theta_k = \sum_{i=1}^m c_{ik} \Theta'_i$. In other words, the information obtained by measuring the expectation values of the set $\{\Theta_k\}$ is equivalent to that obtained by measuring the $\{\Theta'_i\}$, since for each $k$, $\langle \Theta_k \rangle =
\langle \sum_{i=1}^m c_{ik} \Theta'_i \rangle$. orthogonal bases of Hermitian operators are the Pauli (2-d) and Gell-Mann (3-d) matrices.
As above, we restrict ourselves here to coherent quantum dynamics, and additionally assume that a sufficient number of measurements of each observable have been made to accurately estimate its corresponding expectation value. Now consider the $m$ (scalar functions of algorithmic time) $\{\langle \Theta_k(T,s) \rangle\}$ of expectation values for each observable corresponding to a desired unitary track $U(T,s)$. Again, the information about the states of the system $\rho(T,s)$ or equivalently, $U(T,s)$ contained in these measurements is equivalent to that contained in the $m$ functions $\{\langle \Theta'_i(T,s) \rangle\}$. Let us therefore represent this information in the form of the $m$-dimensional vector $\textbf{v}(T,s)$, where $$\textbf{v}_i(T,s) \equiv \langle \sum_k c_{ki}\Theta'_i(T,s)
\rangle.$$ During control optimization, we are interested in tracking these paths $\textbf{v}(T,s)$ in the vector space $\textmd{V}$ that are consistent with the desired path $Q(T,s)$ in $U(N)$.
The generalized differential equation (analogous to eqn (\[gendiff\]) ) that must be satisfied in order to simultaneously track these paths is: $$\begin{gathered}
\frac{{{\rm d}}\textbf{v}(T,s)}{{{\rm d}}s}=\int_0^T \frac{\delta \textbf{v}(T,s)}{\delta \varepsilon(s,t)}\frac{\partial
\varepsilon(s,t)}{\partial s}{{\rm d}}t =\\
\sum_{i=1}^m {{\rm Tr}}\left\{\rho(0)\frac{{{\rm d}}Q^{\dag}(T,s)}{{{\rm d}}s} \left(\sum_k c_{ki}\Theta_i\right)
\frac{{{\rm d}}Q(T,s)}{{{\rm d}}s}\right\}~\textbf{e}_i.\end{gathered}$$ Based on eqn (\[grad1\]), we have $$\frac{\delta \textbf{v}_i(s)}{\delta \varepsilon(s,t)} = \frac{1}{\i\hbar}{{\rm Tr}}\left(
\left[\sum_k c_{ki}\Theta_{i}(T),\rho(0)\right]\mu(t)\right)$$for the gradient of each of the observable expectation values $\langle \Theta_i \rangle$. Following the above derivation, we can convert this generalized differential equation into a vector integral equation: $$\begin{gathered}
\sum_{i=1}^m\int_0^T \frac{1}{\i\hbar}{{\rm Tr}}\left(
\left[\sum_k
c_{ki}\Theta_{i}(T),\rho(0)\right]\mu(t)\right)\textbf{e}_i~
\frac{\partial \varepsilon(s,t)}{\partial s}{{\rm d}}t \\
= \sum_{i=1}^m {{\rm Tr}}\left\{\rho(0) \frac{{{\rm d}}Q^{\dag}(T,s)}{{{\rm d}}s} \left(\sum_k
c_{ki}\Theta_i\right) \frac{{{\rm d}}Q(T,s)}{{{\rm d}}s}\right\}~\textbf{e}_i.\end{gathered}$$ Denoting the vector observable track of interest by $\textbf{w}(s)$, i.e., $$\textbf{w}(s) \equiv \sum_{i=1}^m {{\rm Tr}}\left\{\rho(0) Q^{\dag}(T,s)
\left(\sum_k c_{ki}\Theta_i\right)
Q(T,s)\right\}~\textbf{e}_i,$$ and expanding $\frac{\partial \varepsilon(s,t)}{\partial s}$ on the basis of orthogonal observables, $$\frac{\partial
\varepsilon(s,t)}{\partial s} = \sum_{i=1}^m x_i \frac{\delta
\textbf{v}_i(T,s)}{\delta \varepsilon(s,t)},$$ we have $$\int_0^T
\left(\frac{\delta \textbf{v}(T,s)}{\delta
\varepsilon(s,t)}\right)^T ~\textbf{x}\cdot\frac{\delta
\textbf{v}(T,s)}{\delta \varepsilon(s,t)} {{\rm d}}t= \frac{{{\rm d}}\textbf{w}(s)}{{{\rm d}}s},$$ or equivalently, $$\sum_{j=1}^m \int_0^T\frac{\delta \textbf{v}_i(T,s)}{\delta \varepsilon(s,t)}~~\textbf{x}_j\frac{\delta \textbf{v}_j(T,s)}{\delta \varepsilon(s,t)}{{\rm d}}t= \frac{{{\rm d}}\textbf{w}(s)}{{{\rm d}}s}.$$ Defining the correlation matrix in this case as $$\Gamma_{ij}(s) \equiv \int_0^T \frac{\delta \textbf{v}_i(T,s)}{\delta \varepsilon(s,t)}\frac{\delta \textbf{v}_j(T,s)}{\delta \varepsilon(s,t)}{{\rm d}}t,$$ we obtain the following nonsingular algebraic differential equation for the algorithmic step in the control field: $$\label{vectrack}
\frac {\partial E}{\partial s} = f_s(t) + \left[\frac{{{\rm d}}\textbf{w}}{{{\rm d}}s}-\textbf{a}(s) \right]^T\Gamma^{-1}\frac{\delta \textbf{v}(T,s)}{\delta \varepsilon(s,t)}$$ where $f_s(t)$ is again a free function and we have defined the vector function $\textbf{a}(s)$ by analogy to $\alpha(s)$ above: $$\textbf{a}(s) \equiv \int_0^T\frac{\delta \textbf{v}(T,s)}{\delta \varepsilon(s,t)}f_s(t) {{\rm d}}t.$$ The advantage of orthogonal observable expectation value tracking, compared to unitary matrix tracking, is that the likelihood of the matrix $\Gamma$ being ill-conditioned - even at abnormal extremal control fields $\varepsilon(t)$, where $\textmd{G}$ is singular - diminishes rapidly with $N^2-m$, where $m$ is the number of orthogonal observable operators employed.
In the special case where only the observable of interest $\Theta_1$ is measured at each algorithmic step, this equation reduces to: $$\label{scalarflow}
\frac {\partial \varepsilon(s,t)}{\partial s} = f(s,t)+ \frac{\frac{{{\rm d}}P}{{{\rm d}}s}
-\int_0^T a_0(s,t,T)f(s,t) {{\rm d}}t}{\gamma(s)}a_0(s,t),$$ where $P(s)$ is the desired track for $\langle \Theta_1(T) \rangle$, $a_0(s,t,T)\equiv-\frac{1}{i\hbar}{{\rm Tr}}\left(\rho(0)
\big[U^{\dag}(T,0)\Theta_1U(T,0),U^{\dag}(t,0)\mu(s)U(t,0)\big]\right)$, and $\gamma(s)
\equiv \int_0^T \left[a_0(s,t,T)\right]^2{{\rm d}}t$. Here, it is of course not necessary to carry out any observable operator orthogonalization.
Of course, measuring the expectation values (or gradients) of two or more observable operators is more expensive than following the gradient of a single observable. However, note that the gradients $\frac{\delta \langle \Theta_1(T) \rangle}{\delta \varepsilon(t)}$ and $\frac{\delta \langle \Theta_2(T) \rangle}{\delta
\varepsilon(t)}$ of multiple observables are closely related since $$\frac{\delta \langle \Theta_1 \rangle}{\delta \varepsilon(t)} =
-\frac{i}{\hbar}{{\rm Tr}}\{\left[U^{\dag}(T)\Theta_1U(T),\mu(t)\right]\rho(0)\}$$ while $$\frac{\delta \langle \Theta_2 \rangle}{\delta \varepsilon(t)} =
-\frac{i}{\hbar}{{\rm Tr}}\{\left[U^{\dag}(T)\Theta_2U(T),\mu(t)\right]\rho(0)\}.$$ As such, the information gathered through the estimation of the gradient of $\langle \Theta_1(T) \rangle$ can be used to “inform” the estimation of $\langle \Theta_1(T) \rangle$. In particular, although the norms of these two gradients differ, their time-dependencies - i.e., $\frac{\delta \langle \Theta_1(T)
\rangle}{\delta \varepsilon(t_1)}/\frac{\delta \langle
\Theta_1(T) \rangle}{\delta \varepsilon(t_2)}$ are identical. Hence, only one high-dimensional gradient estimation needs to be carried out.
The above algorithm can be applied to follow an arbitrary set of observable expectation value tracks $\{\langle \Theta_i(s)
\rangle\}$. Here, we are interested in following the observable tracks that correspond to the shortest path between $U_0$ and $W$ on the domain of unitary propagators, namely the geodesic path $U(s)=U(0)\exp(iAs)$ with $A = -i\log(W^{\dag}U(s))$. As mentioned, the matrix $W$ can be determined numerically if $\rho(0)$ and $\Theta$ are known, for minimal computational cost. As shown by Hsieh et al. [@Mike2006a], there exists a continuous submanifold of unitary matrices $W$ that solve the observable maximization problem; if we denote the Hilbert space dimension by $N$, the dimension of this submanifold ranges from $N$ in the case that $\rho(0)$ and $\Theta$ are full rank nondegenerate matrices to $N^2-2N+2$ in the case that $\rho$ and $\Theta$ are both pure state projectors (see section \[phase\]).
The dimension of the subspace $M_T$ of ${\mathcal{U}}(N)$ that is consistent with the observed track $\frac{{{\rm d}}\textbf{v}(T,s)}{{{\rm d}}s}$ displays a complicated dependence on the eigenvalue spectra of $\rho(0)$ and $\{\Theta_i\}$. We demonstrate this explicitly for the case of single observable tracking. In this case, $$\begin{gathered}
M_T \equiv \{V(s) \mid
{{\rm Tr}}\big(V(s)^{\dag}\rho(0) V(s) \Theta\big) = \\
{{\rm Tr}}\big(U(s)^{\dag}\rho(0)U(s) \Theta \big)=\langle \Theta(s)
\rangle\}\end{gathered}$$ where $U(s) = \exp{(i\log{(W^{\dag}U_0)}s)}$. As a first step, we must characterize the degenerate subset $M(s)$ of unitary matrices that are compatible with a given observable expectation value $\langle \Theta \rangle$, as a function of the eigenvalue spectra of $\rho(0)$ and $\Theta$. Let $\rho(0) = Q^{\dag}\epsilon
Q$ and $\Theta = R^{\dag}\lambda R$, where $\epsilon_1,\epsilon_2,...$ and $\lambda_1,...,\lambda_2,...$ are the eigenvalues of $\rho(0)$ and $\Theta$ with associated unitary diagonalization transformations $Q$ and $R$ respectively. Then the observable expectation value corresponding to a given unitary propagator can be written $$\begin{gathered}
J(U)={{\rm Tr}}(U^{\dag}R\hat \rho(0)R^{\dag}US\hat \Theta S^{\dag}) \\
= {{\rm Tr}}\left[(R^{\dag}US)^{\dag}\hat \rho(R^{\dag}US)\hat \Theta\right]={{\rm Tr}}(\hat U^{\dag}\rho(0)\hat U \hat \Theta)\end{gathered}$$ where the isomorphism $\hat U = R^{\dag}US$ also runs over $U(N)$, and $U \equiv U(s)$. Therefore, without loss of generality, we can always assume that both $\rho(0)$ and $\Theta$ are in diagonal form, and determine $M(s)$ in terms of $\hat U(s)$, instead of $U(s)$. Denote by ${\mathcal{U}}(\textbf{n})$ the product group ${\mathcal{U}}(n_1) \times \cdots
\times {\mathcal{U}}(n_r)$, where ${\mathcal{U}}(n_1)$ is the unitary group acting on the $n_i$-dimensional degenerate subspace corresponding to $\lambda_i$, and define ${\mathcal{U}}(\textbf{m})={\mathcal{U}}(m_1)\times \cdots {\mathcal{U}}(m_s)$ in the same manner. Then any transformation $\hat U \rightarrow Q \hat U
T^{\dag}$, where $Q \in {\mathcal{U}}(\textbf{n})$ and $T \in {\mathcal{U}}(\textbf{m})$, leaves $J$ invariant: $$\begin{aligned}
{{\rm Tr}}(T \hat U^{\dag}Q^{\dag}\hat \rho(0)Q \hat U T^{\dag} \hat \Theta) &=& {{\rm Tr}}(\hat U^{\dag}Q^{\dag}\hat \rho(0)Q \hat U T^{\dag} \hat \Theta T)\\
&=& J(\hat U).\end{aligned}$$ This can be seen by observing that since $\hat \rho(0)$ is diagonal, any unitary transformation $\hat \rho(0)\rightarrow Q^{\dag}\hat
\rho(0) Q = \hat \rho(0) Q^{\dag}Q = \hat \rho(0)$, if the unitary blocks of $Q$ are aligned with the degeneracies of $\rho(0)$. By the cyclic invariance of the trace, we also have $\hat \Theta
\rightarrow T^{\dag}\hat \Theta T = \hat \Theta T^{\dag}T=\hat
\Theta.$ Thus, the degenerate manifold can be written $M(s) =
{\mathcal{U}}(\textbf{n})\hat U(s){\mathcal{U}}(\textbf{m})$. Hence, the entire subspace of ${\mathcal{U}}(N)$ that is accessible to the system propagator during global observable tracking is $$M_T = \bigcup_{0\leq s \leq 1}{\mathcal{U}}(\textbf{n})\hat U(s){\mathcal{U}}(\textbf{m}).$$
The manifold $M(s)$ can be expressed as the quotient set $$M(s)=\frac{{\mathcal{U}}(\textbf{n})\times {\mathcal{U}}(\textbf{m})}{{\mathcal{U}}(\textbf{m}) \cap \hat U^{\dag}(s) {\mathcal{U}}(\textbf{n}) \hat U(s)},$$ which can be seen as follows. Define $F_{H}(P,Q): H \rightarrow P H Q$, where $H \in {\mathcal{U}}(N)$ and $(P,Q) \in {\mathcal{U}}(\textbf{n})\times {\mathcal{U}}(\textbf{m})$. Let $stab(H)$ denote the stabilizer of $H$ in ${\mathcal{U}}(\textbf{n})\times {\mathcal{U}}(\textbf{m})$, i.e. the set of matrix pairs $(X,Y) \in {\mathcal{U}}(\textbf{n}) \times {\mathcal{U}}(\textbf{m})$ such that $F_H(X,Y) = X H Y = H$. The stabilizer characterizes the set of points that are equivalent with $H$, hence the manifold $M$ can be identified as the quotient set of ${\mathcal{U}}(\textbf{n})\times {\mathcal{U}}(\textbf{m})$ divided by $stab(H)$. We can specify the stabilizer as follows. First, from $Y=H^{\dag}U^{\dag}H$, we see that $H$ transforms $U \in {\mathcal{U}}(\textbf{n})$ into ${\mathcal{U}}(\textbf{m})$. Hence $Y \in {\mathcal{U}}(\textbf{m}) \cap H^{\dag}{\mathcal{U}}(\textbf{n})H$. Conversely, for any $Y \in {\mathcal{U}}(\textbf{m}) \cap H^{\dag}{\mathcal{U}}(\textbf{n}) H$, the pair $(Y^{\dag}HY,Y)$ must be a member of $stab(H)$. Hence, the stabilizer is isomorphic to ${\mathcal{U}}(\textbf{m}) \cap H^{\dag}{\mathcal{U}}(\textbf{n}) H$. In the present case where $H = U(s)$, we have $$\begin{gathered}
stab(\hat U(s)) = \{(\hat U(s) Y^{\dag}\hat U^{\dag}(s),Y): \\
Y \in {\mathcal{U}}(\textbf{m}) \cap \hat U^{\dag}(s) {\mathcal{U}}(\textbf{n}) \hat U(s)\}.\end{gathered}$$ Thus the dimension of the degenerate manifold $M(s)$ is $$D_0(M(s)) =\textmd{dim}\: {\mathcal{U}}(\textbf{n}) +\textmd{dim}\: {\mathcal{U}}(\textbf{m})-\textmd{dim}\; stab(\hat U(s)).$$
The dimension of this subspace cannot be specified in a simple form for arbitrary $U(s) \in {\mathcal{U}}(N)$, since it is governed by the dimension of the stabilizer. We note a couple of special cases. If $\hat U(s) = R^{\dag} U(s) S$ contains unitary subblocks that fall within the overlapping unitary subblocks in $U(\textbf{n})$ and $U(\textbf{m})$,the dimension of the stabilizer is at least as large as that of the subblocks. If $\hat U(s) = R^{\dag} U(s) S$ is a permutation matrix $\Pi$ or a product of a permutation matrix with a matrix of the aforementioned type, the permutation matrix can act to rearrange the subblocks of $U(\textbf{m})$ so that they overlap with those of $U(\textbf{n})$ and thereby increase the dimension of the stabilizer. The dimension of the subspace (subgroup) of ${\mathcal{U}}(N)$ composed of such matrices $U(s)$ can be shown to increase very rapidly with increasing degeneracies $m$ and $n$ in $\rho(0)$ and $\Theta$, respectively.
The matrix $\hat U(s)$ can only rearrange or diminish the size of existing subgroups ${\mathcal{U}}(n_i)$ within ${\mathcal{U}}(\textbf{n})$, but cannot create larger subgroups. Thus, we can establish a maximal dimension for $stab(\hat U(s))$ for any given $\hat U(s)$ as that which maximizes the overlap between the respective subgroups ${\mathcal{U}}(m_j)$ and ${\mathcal{U}}(n_i)$. This bound is achieved when the conjugation action $U(s)^{\dag}{\mathcal{U}}(\textbf{n}) U(s)$ of $\hat U(s)$ is equivalent to the action $\Pi(s)^{\dag}{\mathcal{U}}(\textbf{n}) \Pi$ of the permutation matrix $\Pi$ that rearranges the subblocks such that they display maximal overlap. Therefore, for fixed $\rho(0)$, $\Theta$, the dimension of the manifold can range from the maximal value of $\sum_in_i^2 +
\sum_jm_j^2 - N$ down to $\sum_{i=1}^r n_i^2 + \sum_{j=1}^s m_j^2 -
\sum_{1\leq i \leq r, 1 \leq j \leq s} k_{ij}^2$ in the case that the conjugation action of $\hat U(s)$ satisfies the above condition, where $k_{ij}$ denotes the number of positions in the diagonal where the eigenvalues $\lambda_i$ and $\epsilon_j$ appear simultaneously after imposition of the permutation matrix $\Pi$.
Thus, we see that the size of the set of unitary matrices $V$ producing the same observable expectation value $\langle
\Theta\rangle$ will change along the trajectory $U(s)$, based on the extent to which the latter reorients the eigenvalues of $\rho(0)$ and $\Theta$ such that degeneracies coincide. In particular, the volume of the subspace $M_T$ of ${\mathcal{U}}(N)$ that is consistent with a track $\langle \Theta(s) \rangle$ derived from a geodesic between $U_0$ and $W$ will display a strong dependence on the choice of $U_0$ and $W$, and will change depending on the matrix $W$ that is chosen from the degenerate submanifold of unitary matrices that solves the observable maximization problem.
Note that the coefficient $a_0$ in the (single) observable expectation value tracking differential equation is in fact equal to the gradient on the domain $\varepsilon(t)$, equation (\[grad1\]). Recall that the gradient flow (\[Egrad\]) is defined by the differential equation $\frac{{{\rm d}}E}{{{\rm d}}s} = \frac{\delta \langle \Theta
\rangle}{\delta E}$. Since the coefficients of $a_0$ are scalars, we see that the algorithmic path for scalar tracking can be expanded on a basis whose dimension is identical to that of the gradient basis (\[gradbasis\]), as expected. We will analyze the dependence of the dimension of this basis on the eigenvalue spectra of $\rho(0)$ and $\Theta$ in section \[integration\].
As a function of the algorithmic step $s$, the coefficient $\frac{\int_0^T a_0(s,t,T)f(s,t) {{\rm d}}t}{\gamma(s)}$ in eqn (\[scalarflow\]) will adjust the step direction so that unitary matrices $V(s)$ at each step are constrained within the subspace $M(s)$. According to the above analysis, this dimension of this subspace will scale more steeply with increasing degeneracies in $\rho(0)$ and $\Theta$. The maximal dimension of $M(s)$ ranges from $N^2-N$ for the problem where $\rho$ and $\Theta$ are both pure state projectors, to $N$ for the case where $\rho$ and $\Theta$ are full rank with completely nondegenerate eigenvalues. Even in the former case, this represents an advantage over the $\varepsilon(t)$-gradient flow, which is free to explore the full $N^2$-dimensional space of dynamical propagators in ${\mathcal{U}}(N)$.
The above analysis assumes that the initial density matrix $\rho(0)$ is known to arbitrary precision. This information is, of course, not required for $\varepsilon(t)$-gradient based optimization, but it is readily acquired in the case that the initial state is at thermal equilibrium. For orthogonal observable tracking, the cost of partial quantum state reconstruction of $\rho(T)$ at each algorithmic step must be weighed against the increase in efficiency obtained by virtue of following a globally optimal path (section \[oce\]).
Error correction and fluence minimization {#errorfluence}
=========================================
### Error correction
In attempting to track paths on ${\mathcal{U}}(N)$, errors will inevitably occur for two reasons. First, the algorithmic step on ${\mathcal{U}}(N)$ will be a linear approximation to the true increment $\delta U(T)$ due to discretization error; this error will increase as a function of the curvature of the integrated flow trajectory at algorithmic time $s$. Second, the D-MORPH integral equation is formulated in terms of only the first-order functional derivative $\frac {\delta U(T)}{\delta \varepsilon(t)}$ (or $\frac {\delta \textbf{v}(T)}{\delta \varepsilon(t)}$, $\frac {\delta \langle \Theta(T) \rangle}{\delta \varepsilon(t)}$ for orthogonal observable and single observable tracking, respectively); the error incurred by neglecting higher order terms in the Taylor expansion will depend on the system Hamiltonian.
In our numerical simulations, we apply error-correction methods to account for these deviations from the track of interest. (These methods can in principle also be implemented in an experimental setting.) For unitary matrix tracking, we correct for these inaccuracies by following the (minimal-length) geodesic from the real point $U(T,s_k)$ to the track point $Q(s_k)$. This correction can be implemented by incorporating the function $C(s_k)=-\frac{i}{s_{k+1}-s_k}\log(Q^{\dag}(s_k)U(T, s_k))$ into the matrix differential equation for the algorithmic time step: $$\frac{\partial E}{\partial
s} = f(s,t) + \Big( v(C(s)) + v(\Delta) -\alpha
\Big)^T\textmd{G}^{-1}v({\mu(t)})$$ In a more efficient approach, we combine error correction and the next gradient step in one iteration [@Dominy2007]. In this case, we define $\Delta(s_k)=-\frac{i}{s_{k+1}-s_k}\log
\left(Q^{\dag}(s_{k+1})U(T,s_k)\right)$ and use $$\frac{\partial E}{\partial s} = f(s,t) + \Big(v(\Delta) -\alpha
\Big)^T\textmd{G}^{-1}v({\mu(t)}).$$
For orthogonal observable expectation value tracking, the vector space within which $\textbf{v}(s)$ resides is not a Lie group, and consequently it is not as straightforward to apply error correction algorithms that exploit the curved geometry of the manifold. We therefore choose the error correction term to be a simple scalar multiple of the difference between the current values of the observable vector and its target value, i.e. $\beta\left[\textbf{w}(s)-\textbf{v}(s)\right]$, such that the tracking differential equation becomes $$\begin{gathered}
\frac {\partial E}{\partial s} = f(s,t) ~+ \\
\left[\beta\left(\textbf{w}(s)-\textbf{v}(s)\right)+\frac{{{\rm d}}\textbf{w}}{{{\rm d}}s}- \textbf{a}(s)\right]^T\Gamma^{-1}\frac{\delta \textbf{v}(T,s)}{\delta \varepsilon(s,t)}.\end{gathered}$$ For the special case of single observable tracking, this reduces to $$\begin{gathered}
\frac {\partial \varepsilon(s,t)}{\partial s} = f(s,t)~+ \\
\frac{\beta(P(s)-\langle \Theta(s) \rangle + \frac{{{\rm d}}P(s)}{{{\rm d}}s} -\int_0^T a_0(s,t,T)f(s,t) {{\rm d}}t}{\gamma(s)}a_0(s,t).\end{gathered}$$
### Fluence minimization
Clearly, the above analysis does not take into account the common physical constraint of penalties on the total field fluence. The effect of the fluence penalty in this scenario is then to decrease the degeneracy in the solutions to the above system of equations for $\frac {\partial \varepsilon(s,t)}{\partial s}$. This is accomplished by choosing the free function $f(s,t)$ in either the unitary or orthogonal observable tracking differential equations to be an explicit function of the electric field. It can be shown [@Rothman2005] that the choice: $$f(s,t) = - \frac{1}{\Delta s} \varepsilon(s,t) W(t),$$ where $W(t)$ is an arbitrary weight function and the $\Delta s$ term controls numerical instabilities, will determine the $\frac
{\partial \varepsilon(s,t)}{\partial s}$ at each algorithmic time step $s$ that minimizes fluence.
Integration of quantum observable expectation value gradient flows {#integration}
==================================================================
We have seen (eqn \[Gmat\]) that in general, the projected path in ${\mathcal{U}}(N)$ that originates from following the local $\varepsilon(t)$-gradient depends on the Hamiltonian of the system through the matrix $\textmd{G}$. Nonetheless, there is still a Hamiltonian independent component to the $\varepsilon(t)$-gradient, which corresponds to the gradient on the domain ${\mathcal{U}}(N)$. Thus, it is of interest to gain some understanding of the behavior of this $U$-gradient flow - whether it follows a direct path toward the target unitary propagator, or whether it biases the $\varepsilon(t)$-gradient flow to follow indirect paths in ${\mathcal{U}}(N)$. Such an analysis may shed light on the comparative optimization efficiencies of the gradient compared to the tracking algorithms described in the previous sections.
It can be shown that the $U(T)$ and $\varepsilon(t)$ gradient flows evolve locally on subspaces of the same dimension, and that this dimension changes predictably as a function of the eigenvalue spectra of $\rho(0)$ and $\Theta$. These gradient flows evolve on a subspace of the homogeneous space of ${\mathcal{U}}(N)$ whose dimension is given by the spectrum of the initial density matrix $\rho(0)$, necessitating the use of a distinct coordinate basis to express the integrated gradient flow trajectories for different classes of $\rho(0)$ that depend on the latter’s number of nonzero and degenerate eigenvalues [@HoRab2007b]. Specifically, let $\rho(0)$ consist of $r$ subsets of degenerate eigenvalues $p_1, \cdots, p_r$, with multiplicities $n_1,\cdots, n_r$. Writing $\rho(0)= \sum_{i=1}^n p_i|i\rangle
\langle i|$, we have $$\begin{gathered}
\frac{\delta \Phi}{\delta \varepsilon(t) } = a_0(t,T) = \\
\frac{i}{\hbar} \sum_{i=1}^n p_i \sum_{j=1}^N \left[ \langle i|\Theta(T)|j \rangle
\langle j|\mu(t)|i \rangle - \langle i|\mu(t)|j \rangle \langle
j|\Theta(T)|i\rangle \right]\end{gathered}$$ Defining $s_k \equiv \sum_{i=1}^{k-1}n_i, \quad k=2,\cdots, r+1$ this can be written
$$\begin{aligned}
\frac{\delta \Phi}{\delta \varepsilon(t)}&=&\frac{i}{\hbar} \sum_{k=1}^r p_k
\sum_{i=s_k+1}^{s_{k+1}} \left[\sum_{j=1}^{s_k} +
\sum_{j=s_k+1}^{s_{k+1}} + \sum_{j=s_{k+1}+1}^N \right] \{\langle
i|\Theta(T)|j \rangle \langle j|\mu(t)|i \rangle -\langle i|\mu(t)|j
\rangle \langle j|\Theta(T)|i \rangle \}\\
&=&\frac{i}{\hbar} \sum_{k=1}^r p_k \sum_{i=s_k+1}^{s_{k+1}}\left[\sum_{j=1}^{s_k} +
\sum_{j=s_{k+1}+1}^N \right] \{\langle i|\Theta(T)|j\rangle \langle
j|\mu(t)|i\rangle - \langle i|\mu(t)|j \rangle \langle j|\Theta(T)|i
\rangle \}\end{aligned}$$
where the second equality follows from the fact that the terms corresponding to $\sum_{j=s_k+1}^{s_{k+1}}$ (i.e., those arising from the same degenerate eigenvalue of $\rho(0))$) are zero. This indicates that the dimension of the subspace of the space of skew-Hermitian matrices upon which the gradient flow evolves is $$D=N^2-(N-n)^2-\sum_{i=1}^r n_i^2 = n(2N-n)-\sum_{i=1}^r n_i^2.$$ This is the dimension of a compact polytope $P$ which is the convex hull of the equilibria of the gradient vector field. The gradient flow involves on the interior of this polytope [@Bloch1992].
Both the $\varepsilon(t)$-gradient flow (\[Egrad\]) and observable tracking (\[scalarflow\]) can be expanded on this basis corresponding to $\frac{\delta \Phi}{\delta \varepsilon(t) }$. It can be shown (see below) that in the case of the gradient, the increased dimension of this basis set for increasing nondegeneracies in $\rho(0)$, $\Theta$ generally results in increased unitary pathlengths; since the entire unitary group is free for exploration, the increased number of locally accessible directions results in the path meandering to more distant regions of the search space. By contrast, in the case of observable tracking, the increased number of locally accessible directions for greater nondegeneracies in $\rho(0)$, $\Theta$ are coupled with a decrease in the dimension of the globally accessible search space in ${\mathcal{U}}(N)$; the greater local freedom is used to follow the target unitary tracks of interest.
In order to shed light on the origin of the aforementioned behavior of the $\varepsilon(t)$-gradient, we consider the global paths followed by $U$-gradient flow of $\Phi$. In a useful analogy, the control optimization process can itself be treated as a dynamical system. The critical manifolds of the objective function then correspond to equilibria of the dynamical system, and the gradient flow trajectories to its phase trajectories. Within this analogy, the gradient flow of $\Phi$ on ${\mathcal{U}}(N)$ can be shown to represent the equations of motion of an integrable dynamical system. The expression (\[Uflow\]) for the gradient flow of $\Phi$ above is cubic in $U$. However, through the change of variables $U(s,T)\rightarrow \rho(s,T) = U(s,T)\rho(0,0)U^{\dag}(s,T)$ we can reexpress it as a quadratic function: $$\begin{gathered}
\dot \rho(s,T) = \\
-U(s,T)\rho(0,0)\dot U^{\dag}(s,T) -\dot U(s,T) \rho(0,0)U^{\dag}(s,T) \\
= \rho^2(s,T) \Theta - 2\rho(s,T) \Theta \rho(s,T) + \Theta \rho^2(s,T)\\
=\left[\rho(s,T),[\rho(s,T),\Theta ] \right]\end{gathered}$$ where $s$ denotes the algorithmic time variable of the gradient flow in ${\mathcal{U}}(N)$ and the dot denotes the $s$-derivative. This quadratic expression for the gradient flow is in so-called double bracket form [@Brockett1991; @Bloch1992; @Helmke1994]. The set of all $U(s,T)\rho(0,0)U^{\dag}(s,T)$ is a homogeneous space $M(\rho)$ for the Lie group ${\mathcal{U}}(N)$, namely the space of all Hermitian matrices with eigenvalues fixed to those of $\rho(0,0)$. Maximizing $\Phi(U)$ over ${\mathcal{U}}(N)$ is equivalent to minimizing the least squares distance $\|
\Theta-U \rho U^{\dag} \|^2$ of $\Theta$ to $U\rho U^{\dag}
\in M(\rho)$: $$\| \Theta-U\rho U^{\dag} \|^2 = \| \Theta \|^2 - 2\Phi(U) +
\| \rho \|^2.$$
Here, we provide an explicit formula pertaining to the analytical solution for the above gradient flow for what is perhaps the most common objective in quantum optimal control theory and experiments, namely the maximization of an arbitrary observable starting from a pure state. In particular, this includes the special case of maximizing the transition probability $P_{if}$ between given initial and final pure states $|i\rangle$ and $|f\rangle$. Whenever $\rho(t=0)$ is a pure state, $n=r=n_1=1$, and $D=2N-2$, and the convex hull of the critical points of the vector field is a $(N-1)$-dimensional simplex. The gradient flow evolves in the interior of this simplex. The analytical solution for the fully general case of a mixed state $\rho(0)$ and nondegenerate $\Theta$ is more complicated and presented in another work of the authors.
Since the objective function is symmetric with respect to $\rho(0)$ and $\Theta$, this formulation applies if either $\rho(0)$ or $\Theta$ is a pure state projection operator, i.e., if at least one of them can be diagonalized by an appropriate change of basis to matrices that have only one nonzero diagonal element, corresponding to $|i\rangle \langle i|$ or $|f \rangle
\langle f|$, respectively. The other operator can have an arbitrary spectrum. The same integrated gradient flow thus applies to the problem of maximizing the transition probability between any generic mixed initial state to any pure state.
Under these conditions, we can execute a change of variables such that the double bracket flow, which evolves on the $\frac{1}{2}
m(m+1)$-dimensional vector space of Hermitian matrices $\rho(0)$ is mapped to a flow on the m-dimensional Hilbert space. Letting $|\psi(s) \rangle =U(s)|i\rangle$, the double bracket flow can be written: $$\begin{aligned}
|\dot \psi(s)\rangle &=&\dot U(s)|i\rangle\\
&=&\left[\Theta U(s)|i\rangle -U(s)|i\rangle\langle i|U^\dagger(s)\Theta U(s)\right]|i\rangle \\
&=& \left[ \Theta - \langle \psi(s)|\Theta|\psi(s) \rangle I\right] ~ |\psi(s)
\rangle.\end{aligned}$$ If we define $x(s) \equiv (|c_1(s)|^2,\cdots,|c_N(s)|^2)$, where $c_1(s),\cdots,c_N(s)$ are the coordinates of $|\psi(s)\rangle$ under the basis that diagonalizes $\Theta$; it can be verified that the integrated gradient flow can be written: $$\begin{aligned}
x(s) &=& \frac{e^{2s\Theta}\cdot(|c_1(0)|^2,\cdots,|c_N(0)|^2)}{\sum_{i=1}^N|c_i(0)|^2e^{2s\lambda_i}} \\
&=&\
\frac{e^{2s\lambda_1}|c_1(0)|^2,\cdots,e^{2s\lambda_N}|c_N(0)|^2}{\sum_{i=1}^N|c_i(0)|^2e^{2s\lambda_i}}\end{aligned}$$ where $\lambda_1,\cdots, \lambda_N$ denote the eigenvalues of $\Theta$.
In the case that $\Theta$ has only one nonzero eigenvalue (pure state), this becomes:
$$\begin{aligned}
x(s)&=& \left(\frac{|c_1(0)|^2}{\sum_{i\neq
j}^m|c_i(0)|^2+e^{2s\lambda_j}|c_j(0)|^2},\cdots,\frac{e^{2s\lambda_j}|c_j(0)|^2}{\sum_{i\neq
j}^m|c_i(0)|^2+e^{2s\lambda_j}|c_j(0)|^2},\cdots \right)\\
&=&
\left(\frac{|c_1(0)|^2}{1+(e^{2s\lambda_j}-1)|c_j(0)|^2},\cdots,\frac{e^{2s\lambda_j}|c_j(0)|^2}{1+(e^{2s\lambda_j}-1)|c_j(0)|^2},\cdots
\right).\end{aligned}$$
In contrast to quantum time evolution of the state vector (which resides on the Hilbert sphere $S^{m-1}$, the matrix $e^{2s\Theta}$ translates the vector $x$, which resides on the $(m-1)$-dimensional simplex, through algorithmic time. Since coherent quantum time evolution cannot change the eigenvalues of the density matrix $\rho$, the optimization of controls simply reorders these eigenvalues. Hence the gradient flow is said to be an “isospectral” flow. We are primarily interested in the gradient flow on the domain of unitary propagators of the quantum dynamics, $U(T)$. In general, there exists a one-to-many map between $\rho(T)$ and $U(T)$. This attests to the existence of a multiplicity of paths through the search space of the quantum optimal control problem that will maximize the objective functional in equivalent dynamical time. Although the analytical solution we have presented is framed on the homogeneous of ${\mathcal{U}}(N)$ rather than on the Lie group itself, we will see in the next section that generic properties of the behavior of the flow trajectories, in particular their phase behavior with respect to local critical points, are identical on these domains.
The above gradient flow for $|\psi\rangle $ only applies when $\Theta$ has no degeneracy in its eigenvalues, including zero eigenvalues. If the maximum eigenvalue of $\Theta$ is degenerate with multiplicity $k$, such that $c_1=c_2=\cdots=c_k$, and $c_k >
c_j, \quad j=k+1,\cdots,n$, then the dynamics converges to the point $\frac{1}{k}(1,\cdots,1,0,\cdots,0).$
A remarkable feature of the gradient flow for objective function $\Phi$ is that it is a Hamiltonian flow [@Faybu1991; @Bloch1990; @Bloch1995], for general $\rho(0)$ and $\Theta$. The eigenvalues can be viewed as the analog of the momenta in the corresponding Hamiltonian system. The “isospectral” character of the flow indicates these momenta are conserved. An alternative proof of the integrability of the flow for $\Phi$ is based on demonstrating that in N dimensions, the flow has N integrals of the motion that are in involution, which is the classical definition of complete integrability for a Hamiltonian system. From the point of view of the modern theory of integrable systems, the double bracket flow can be shown to represent a type of Lax pair, a general form that can be adopted by all completely integrable Hamiltonian systems [@Babelon2003].
Phase behavior of quantum observable maximization gradient flows {#phase}
================================================================
The integrated flow trajectories provided above for the gradient of $\Phi$ on the domain of unitary propagators can be used to provide insight into global behavior of the $\varepsilon(t)$-gradient flow. Just as the global trajectory of the unitary geodesic flow influences, but does not directly determine the unitary path followed by global observable tracking, the integrated $U(T)$-flow trajectories influence but do not completely determine the behavior of the $\varepsilon(t)$-gradient flow. A useful metric for assessing the phase behavior of the $U(T)$-gradient flows is the distance of the search trajectory from the critical manifolds of the objective as a function of algorithmic time.
The distance of the search trajectory to the global optimum of the objective can be expressed:
$$\|x(s)-e_{i*}\|^2 =
\|x(s)\|^2-2\frac{\langle e^{2s\Theta}x(0),e_{i*}\rangle }{\langle
e^{2s\Theta}x(0)\rangle }+1$$ The equilibrium points of this flow are the critical points of the objective function. Hsieh et al. [@Mike2006a] showed that the critical manifolds of $\Phi$ satisfy the condition $$\frac{{{\rm d}}J}{{{\rm d}}s}= i{{\rm Tr}}\left(A\left[\Theta,U^{\dag}\rho(0)U\right]\right)=0.$$ The critical manifolds are then given by matrices of the form $$\hat U_l = QP_lR^{\dag},$$ where $P_l, \quad l=1,\cdots,N!$ is an $N$-fold permutation matrix whose nonzero entries are complex numbers $\exp(i\phi_1),...,\exp(i\phi_N)$ of unit modulus, and $\rho=Q^{\dag}\epsilon Q$ and $\Theta = R^{\dag}\lambda R$. The critical manifolds have a similar topology to the manifolds $M(s)$ shown in section \[orthog\] to map to a given observable expectation value. For example, in the case that $\rho$ and $\Theta$ are fully nondegenerate, they are $N$-torii $T_l^N$. In this case, the number of critical manifolds scales factorially with the Hilbert space dimension $N$. In the case that either $\rho$ or $\Theta$ has only one nonzero eigenvalue, while the other is full rank and nondegenerate, the number of critical manifolds scales linearly with $N$. (In the present case, assuming $\Theta$ is full-rank, the number of these equilibrium points scales exponentially as $2^m$ with the Hilbert space dimension.) The optimal solution to the search problem corresponds to the basis vector where $\Theta$ has its maximal eigenvalue. In the case that the observable operator $\Theta$ has only one nonzero eigenvalue $i$, there are only two critical points, corresponding to $\pm e_i$.
The time derivative of the distance of the search trajectory to a critical manifold (framed on the homogeneous space) is
$$\frac{{{\rm d}}}{{{\rm d}}t}\|x(s)-e_{i*}\|^2
=\frac{e^{2s\lambda_i*}x_{i*}(0)\sum_{j=1}^m(\lambda_{i*}-\lambda_j)e^{2s\lambda_j}x_j(0)}{\left[\sum_{j=1}^me^{2s\lambda_j}x_j(0)\right]^2}.$$
Solving for the zeroes of this time derivative reveals that the distance between the current point on the search trajectory and the solution can alternately increase and decrease with time. The distance of the gradient flow trajectory from the suboptimal critical manifolds can alternately increase and decrease arbitrarily depending on the spectral structure of $\Theta$. Thus the density matrix does not always resemble the target observable operator to a progressively greater extent during the algorithmic time evolution.
Qualitatively, the most obvious and important feature of these trajectories is that the closer the initial condition is to a suboptimal critical manifold, the greater the extent to which the gradient flow follows the boundary of the simplex during its early time evolution. Away from these initial conditions, the behavior of the gradient flow trajectory is considerably more sensitive to the spectral structure of the observable operator $\Theta$ than it is to the initial state $\psi_0$. Indeed, it may be shown [@RajWu2007] that for operators $\Theta$ with two eigenvalues arbitrarily close to each other, the time required for convergence to the global optimum increases without bound, whereas this is not the case for initial states with $c_i$ arbitrarily close to 0.
Of course, the search trajectory slows down in the vicinity of the critical manifolds. Note that at the critical manifolds themselves, the observable tracking equations also encounter singularities, since $a_0 = 0$. In the vicinity of the critical manifolds (the attracting regions) tracking may be less accurate due to small values of $\gamma(s)$ in the denominator of eqn (\[scalarflow\]), corresponding to functions $a_0$ of small modulus.
As the nondegeneracies in $\rho(0)$ and $\Theta$ increase, the number of attractors of the search trajectory increases, indicating an increase in the unitary path length. Therefore, as mentioned above, although the trajectory followed by the gradient locally accesses the same number of directions as observable tracking, its global path appears biased towards being longer, assuming the Hamiltonian is fixed, especially for highly nondegenerate $\rho(0)$, $\Theta$; the local steps in global geodesic observable tracking access the same number of local directions but orient them towards unitary matrices along a shorter path.
Numerical implementation {#numerical}
========================
Simulations comparing the efficiency of unitary or orthogonal observable tracking with gradient-based optimal control algorithms will be presented in a separate paper. However, we provide here a brief summary of numerical methods that can be used to implement the various tracking algorithms described above.
For multiple observation-assisted tracking, a set of $n$ observable operators $\Theta_1,\cdots, \Theta_n$ were either chosen randomly or based on eigenvalue degeneracies. This (possibly linearly dependent) set was orthogonalized by Gram-Schmidt orthogonalization, resulting in a $m$ (where $m \leq N$) dimensional basis set of observable operators. The $m$-dimensional vector $v(s)$ was constructed by tracing each of these observable operators with the density matrix. Numerical solution of the D-MORPH differential equations (\[utrack\], \[vectrack\], \[scalarflow\]) was carried out as follows. The electric field $\varepsilon(s,t)$ was stored as a $p\times q$ matrix, where $p$ and $q$ are the number of discretization steps of the algorithmic time parameter $s$ and the dynamical time $t$, respectively. For each algorithmic step $s_k$, the field was represented as a $q$-vector for the purpose of computations. Starting from an initial guess $\varepsilon(s_0,t)$ for the control field, the Hamiltonian was integrated over the interval $~[0,T~]$ by propagating the Schrodinger equation over each time step $t_k -> t_{k+1}$, producing the local propagator $U(t_{j+1},t_j) = \exp~[-iH(s_i,t_j)T/(q-1)~]$. For this purpose, the propagation toolkit was used. Local propagators were precalculated via diagonalization of the Hamiltonian matrix (at a cost of $N^3$), exponentiation of the diagonal elements, and left/right multiplication of the resulting matrix by the matrix of eigenvectors/transpose of the matrix of eigenvectors. This approach is generally faster than computing the matrix exponential directly. Alternatively, a fourth-order Runge-Kutta integrator, can be employed for the propagation, and is often used for density matrix propagation.
The time propagators $U(t_j,0)=U(t_j,t_{j-1}),\cdots,U(t_1,t_0)$ computed in step 1 were then used to calculate the time-evolved dipole operators $\mu(t_j)=U^{\dag}(t_j,0)\mu U(t_j,0)$, which can be represented as a $q$-dimensional vector of $N\times N$ Hermitian matrices. The $N^2 \times N^2$ matrix $\textmd{G}(s_k)$ and $N^2$ vector $\alpha(s_k)$ (alternatively, the $m \times m$ matrix $\Gamma(s_k)$ and $m$ vector $\textbf{a}(s_k))$) were then computed by time integration of the dipole functions (and appropriate choice of function $f(s,t)$ )described above. For tracking of unitary gradient flows, the next point $Q(s_{k})$ on the target unitary track, necessary for the implementation of error correction, was calculated in one of two ways: i) numerically through $Q(s_{k})=Q(s_{k-1})*\exp(-i\Delta(s_k)
{{\rm d}}s)$, computed using a matrix exponential routine (because of the greater importance of speed vs accuracy for this step) or ii) analytically, through the integrated flow equation above. If error correction was employed, the matrix $C(s_k)=-\frac{i}{s_k - s_{k-1}}
\log (U^{\dag}(s_k)Q(s_k))$ for unitary error correction was calculated by diagonalization of the unitary matrix followed by calculation of the scalar logarithms of the diagonal elements, where the logarithms are restricted to lie of the principal axis.
Next, the control field $\varepsilon(s_k,t)$ was updated to $\varepsilon(s_{k+1},t)$. This step required the inversion of the $N^2 \times N^2$ matrix $\textmd{G}(s_k)$ or $m \times m$ matrix $\Gamma(s_k)$, which was carried out using LU decomposition. The quantities $\textmd{G}^{-1}(s_k)$, $\Delta(s_k)$, $\alpha(s_k)$ (alternatively, $\Gamma^{-1}(s_k), \frac{\textbf{v}(s)}{{{\rm d}}s},
\textbf{a}(s_k)$, for scalar tracking) and $C(s_k)$ were used to compute the $q$-dimensional vector $\frac{\partial \varepsilon(s_{k},t)}{\partial s}$. One of two approaches was used to update the field: (i) a simple linear propagation scheme, i.e. $\varepsilon(s_{k+1},t) = \varepsilon(s_{k},t) + {{\rm d}}s ~
\frac{\partial \varepsilon(s_{k},t)}{\partial s}$, or (ii) a fourth-order Runge-Kutta integrator. Because the accuracy of tracking depended largely on the accuracy of this $s$-propagation step, and because only one $s$-propagation was carried out for each set of $q$ time propagations, a more accurate (but expensive) fifth-order Runge-Kutta integrator was often used in this step. The updated control field $\varepsilon(s_{k+1},t)$ was then again used to the propagate the Schrodinger equation.
In the above numerical tracking scheme, the most computationally intensive step is the propagation of the Schrodinger equation. Although calculation of the matrix elements of $\textmd{G}$ for unitary tracking (respectively, $\Gamma$ for orthogonal observable tracking) and inversion of this matrix scale unfavorably with the dimension of the quantum system, this scaling is polynomial. Since remaining on the target $U$-track can play an important role in the global convergence of the algorithm, especially where the local gradient follows a circuitous route, the additional expense incurred in accurate $s$-propagation may be well-warranted.
Implications for optimal control experiments (OCE) {#oce}
==================================================
The majority of optimal control experiments (OCE) have been implemented with adaptive or genetic search algorithms, which only require measurement of the expectation value of the observable for a given control field. Recently, gradient-following algorithms have been implemented in OCE studies, based on the observation that the control landscape is devoid of suboptimal traps [@Roslund2007]. In computer simulations, it is generally observed that gradient-based algorithms converge more efficiently than genetic algorithms. However, more sophisticated local search algorithms, such as the Krotov or iterative algorithms often used in OCT, are difficult if not impossible to implement experimentally due to the extreme difficulty in measuring second derivatives of the expectation value in the presence of noise and decoherence. Therefore, global search algorithms which can be implemented on the basis of gradient information alone, such as orthogonal observable tracking, are particularly attractive candidates for improving the search efficiency of OCE.
Applying observable tracking experimentally requires a fairly simple extension of gradient methods that have already been applied. The gradient is determined through repeated measurements on identically prepared systems, to account for the impact of noise. Instead of following the path of steepest descent, the laser field is updated in a direction that in the linear approximation would produce the next observable track value $\Theta(s+{{\rm d}}s)$. The assumption is that because this observable step is consistent with a short step on the domain of unitary propagators, the associated error in reaching this expectation value will be smaller than that associated with trying to move over the same fraction of the gradient flow trajectory. Error correction can be implemented using a method identical to that described above for numerical simulations.
In the simplest incarnation of orthogonal observation-assisted quantum control, the full density matrix $\rho(0)$ is estimated at the beginning of the control optimization. This need be done only once and hence adds only a fixed overhead to the experimental effort that does not add substantially to the scaling of algorithmic cost with system dimension. At subsequent steps during the optimization, the experimenter can (possibly adaptively) decide how many orthogonal observations to make in order to estimate the final density matrix $\rho(T)$, with the goal of keeping the unitary path as close as possible to the desired geodesic. The number of distinct observables that must be measured at each step (and hence roughly the total number of measurements) scales linearly with the number of estimated parameters of $\rho(T)$.
In order to properly compare the efficiencies of experimental gradient-following and global observable tracking methods, it is necessary to consider the expense associated with reconstruction of the initial and final density matrices $\rho(0)$ and $\rho(T)$ in the latter case. A variety of different quantum statistical inference methods have been developed over the past several years for the estimation of density matrices on the basis of quantum observations. Like the measurement of the gradient, these methods are based on multiple observations on identically prepared copies of the system. If we consider $M$ measurements on identically prepared copies, each measurement is described by a positive operator-valued measure (POVM). Of interest to us here is the scaling of the number of measurements required to identify the density matrix within a given precision, with respect to the Hilbert space dimension. In most reconstruction techniques, such as quantum tomography [@Ariano1995], a matrix element of the quantum state is obtained by averaging its pattern function over data. In the averaging procedure, the matrix elements are allowed to fluctuate statistically through negative values, resulting in large statistical errors. Recently, the method of maximal likelihood estimation (MLE) of quantum states has received increasing attention due to its greater accuracy [^3]. Denoting by $\hat F_i$ the POVM corresponding to the $i$-th observation, the likelihood functional
$$L(\hat \rho) = \prod_{i=1}^N {{\rm Tr}}(\hat \rho(0)\hat F_i)$$ describes the probability of obtaining the set of observed outcomes for a given density matrix $\hat \rho$. This likelihood functional is maximized over the set of density matrices. An effective parameterization of $\hat \rho$ is $\hat \rho(0) = \hat T^{\dag} \hat
T$, which guarantees positivity and Hermiticity, and the condition of unit trace is imposed via a Lagrange multiplier $\lambda$, to give $$L(\hat T) = \sum_{i=1}^N \ln {{\rm Tr}}(\hat T^{\dag} \hat T \hat F_i) - \lambda {{\rm Tr}}(\hat T^{\dag}\hat T).$$ Standard numerical techniques, such as Newton-Raphson or downhill simplex algorithms, are used to search for the maximum over the $N^2$ parameters of the matrix $\hat T$. The statistical uncertainty in density matrix estimates obtained via MLE can quantified by considering the likelihood to function to represent a probability distribution on the space of density matrix elements - or, in the current parameterization, the $N^2$ parameters constituting the matrix $\hat T$, denoted here by the vector $t$. In the limit of many measurements, this distribution approaches a Gaussian. The Fisher information matrix $I = \frac {\partial^2 L}{\partial t
\partial t'}$, which is the variance of the score function that is set to zero to obtain the MLE estimates, can then be used to quantify the uncertainties in the parameters. Note that the constraint ${{\rm Tr}}(\hat
T^{\dag}\hat T)=1$ implies that the optimization trajectory maintains orthogonality to $u =\frac {\partial {{\rm Tr}}(\hat T^{\dag}\hat
T)}{\partial t}$. Under these conditions it can be shown that the covariance matrix for $t$ is given by $$V =
I^{-1}-I^{-1}uu^TI^{-1}/u^TI^{-1}u.$$ As such, the associated uncertainties on the density matrix elements as a function of the number of measurements can be determined for a given system based on computer simulations.
Banaszek et al. [@Banaszek1999] applied MLE to both discrete and continuous quantum state estimation, comparing to state tomography. In the case of continuous variable states, the density matrix was, of course, truncated. For identical systems, MLE required orders of magnitude fewer measurements $N$ to reconstruct the state with the same accuracy. For example, only $50,000$ homodyne data (compared to $10^8$ for tomography) were required to reconstruct the matrix for a single-mode radiation field. The number of required measurements was not highly sensitive to the truncation dimension, since adaptive techniques can be used to improve efficiency in higher dimensions. Only $500$ measurements were required to reconstruct the density matrix of a discrete quantum system, a pair of spin-$1/2$ particles in the singlet state. Given that accurate estimation of the gradient requires a similar number of measurements [@Roslund2007], if MLE is used for state reconstruction, the additional algorithmic overhead for observable tracking is not limiting. Quantitative calculations of this overhead will be reported in a a forthcoming numerical study.
Discussion
==========
We have presented several global algorithms for the optimization of quantum observables, based on following globally optimal paths in the unitary group of dynamical propagators. The most versatile of these algorithms, orthogonal observable expectation value tracking, aims to simultaneously track a set of observable expectation value paths consistent with the unitary geodesic path to the target propagator. The performance of the latter has been compared theoretically to that of local gradient algorithms. A follow-up paper will report numerical simulations comparing the efficiencies of the algorithms described herein, across various families of Hamiltonians.
Although the $\varepsilon(t)$-gradient flow is always the locally optimal path, its projected path in ${\mathcal{U}}(N)$ is generally much longer than those that can be tracked by global algorithms. The latter often require fewer iterations for convergence. Of course, in order to assess the utility of $U$-flow tracking as a practical alternative to local OCT algorithms, it is necessary to consider the computational overhead incurred in solving the system of tracking differential equations at each algorithmic step, which is on the order of $N^4$-times more costly than solving scalar observable tracking equations.
For the systems studied, the geodesic track in ${\mathcal{U}}(N)$ can typically be followed faithfully by matrix tracking algorithms, assuming the system is controllable on the entire unitary group. However, for many Hamiltonians, unitary matrix tracking can routinely encounter regions of the landscape where the $\textmd{G}$ matrix is ill-conditioned. In contrast, tracking a vector $\textbf{v}(s)$ of $m$ orthogonal observable expectation values corresponding to this unitary matrix track encounters such singularities more rarely, provided $m < N$.
However, the number of observable expectation values tracked is not the only factor that affects the mean pathlength in $U(N)$. The relationship between the basis set of operators (spanning the space of measured observables) and the eigenvalue spectrum of $\rho(0)$ affects the dimension and volume of the subspace of ${\mathcal{U}}(N)$ that is accessible to the search trajectory. Indeed, the comparative advantage of employing global observable tracking algorithms versus local gradient algorithms was found to depend on the spectra of the initial density matrix $\rho(0)$, as predicted based on a geometric analysis of the map between unitary propagators and associated observable expectation values. In particular, measurement of the expectation value of a quantum observable provides more information about the dynamical propagator $U(T)$ when $\rho(0)$ has fewer degenerate eigenvalues, since degeneracies produce symmetries that result in invariant subspaces over which the unitary propagator can vary without altering the observable. Thus, for an identical number of measurements, the global path in ${\mathcal{U}}(N)$ can be tracked with greater precision for systems with nondegenerate $\rho(0)$.
Clearly, a very important question underlying the efficiency of global OCT algorithms based on paths in the unitary group is whether the nonsingularity of $\textmd{G}$ and the assumption of small higher-order functional derivatives remains valid for more general Hamiltonians beyond those considered here. In principle, paths in ${\mathcal{U}}(N)$ that are longer than the geodesic might be significantly easier to track if these assumptions were to break down, for particular systems. For systems where $\textmd{G}$ is almost always close to singular, global tracking algorithms on ${\mathcal{U}}(N)$ may not be viable, even in the presence of error correction. In these cases one would expect global observable tracking to be preferable to matrix tracking algorithms, since the system can “choose” which of the infinite number of degenerate paths in ${\mathcal{U}}(N)$ it follows.
This study, and the associated forthcoming numerical simulations, sets the stage for experimental testing of its prediction that global observable tracking algorithms will display advantages compared to the gradient. As discussed above, an important question for future study is how to implement global observable control algorithms experimentally, and whether nonideal conditions (noise, decoherence) in the laboratory will obscure some of its predicted advantages. In particular, the *effective* degeneracies of $\rho(0)$ and $\Theta$ - e.g., whether the populations of the various levels in a mixed state are sufficiently high to permit accurate determination of the associated unitary propagators in the presence of noise - become very important.
Besides the perceived advantage of global observable control algorithms, they may offer a means of assessing the search effort and search complexity inherent in quantum control problems in a universal manner, since they are more system independent than local gradient-based algorithms. As shown in a separate numerical study comparing these algorithms, over several families of related Hamiltonians, the variance of the convergence time of the $\varepsilon(t)$-gradient flow is significantly greater than that of global observable or unitary matrix tracking. Given that its convergence is also faster, unitary tracking may offer an approach to setting upper bounds on the scaling of the time required for quantum observable control optimizations, as a function of system size.
[^1]: The local surjectivity of $\varepsilon(t)
\rightarrow U(T)$ has important connections to the controllability of the quantum system [@Raj2007]
[^2]: In the case that the control system evolves on a subgroup of $U(N)$, e.g. SU(N), the geodesic on that subgroup can be tracked instead.
[^3]: Other methods for state reconstruction, such as the maximum entropy method or Bayesian quantum state identification [@Buzek1997], can also be employed.
|
---
abstract: 'Following Feigin and Fuchs, we compute the first cohomology of the Lie superalgebra $\mathcal{K}(1)$ of contact vector fields on the (1,1)-dimensional real superspace with coefficients in the superspace of linear differential operators acting on the superspaces of weighted densities. We also compute the same, but $\mathfrak{osp}(1|2)$-relative, cohomology. We explicitly give 1-cocycles spanning these cohomology. We classify generic formal $\mathfrak{osp}(1|2)$-trivial deformations of the $\mathcal{K}(1)$-module structure on the superspaces of symbols of differential operators. We prove that any generic formal $\mathfrak{osp}(1|2)$-trivial deformation of this $\mathcal{K}(1)$-module is equivalent to a polynomial one of degree $\leq4$. This work is the simplest superization of a result by Bouarroudj \[On $\mathfrak{sl}$(2)-relative cohomology of the Lie algebra of vector fields and differential operators, J. Nonlinear Math. Phys., no.1, (2007), 112–127\]. Further superizations correspond to $\mathfrak{osp}(N|2)$-relative cohomology of the Lie superalgebras of contact vector fields on $1|N$-dimensional superspace.'
author:
- |
Imed Basdouri $^1$, Mabrouk Ben Ammar $^1$,\
Nizar Ben Fraj $^2$, Maha Boujelben $^1$ and Kaouthar Kammoun $^1$\
$^1$Département de Mathématiques, Faculté des Sciences de Sfax,\
Route de Soukra 3018 Sfax BP 802, Tunisie\
E-mails: basdourimed@yahoo.fr, mabrouk.benammar@fss.rnu.tn,\
Maha.Boujelben@fss.rnu.tn, lkkaouthar@yahoo.com\
$^2$Institut Supérieur de Sciences Appliquées et Technologie, Sousse, Tunisie\
E-mail: benfraj\_nizar@yahoo.fr\
title: ' Cohomology of the Lie Superalgebra of Contact Vector Fields on $\mathbb{R}^{1|1} $ and Deformations of the Superspace of Symbols '
---
Introduction
============
For motivations, see Bouarroudj’s paper [@b] of which this work is the most natural superization, other possibilities being cohomology of polynomial versions of various infinite dimensional stringy“ Lie superalgebras (for their list, see [@gls]). This list contains several infinite series and several exceptional superalgebras, but to consider cohomology relative a middle” subsuperalgebra similar, in a sense, to $\mathfrak{sl}(2)$ is only possible when such a subsuperalgebra exists which only happens in a few cases. Here we consider the simplest of such cases.
Let $\mathfrak{vect}(1)$ be the Lie algebra of polynomial vector fields on $\mathbb{K}:=\mathbb{R}$ or $\mathbb{C}$. Consider the 1-parameter deformation of the $\mathfrak{vect}(1)$-action on $\mathbb{K}[x]$: $$L_{X\frac{d}{dx}}^\lambda(f)= Xf'+\lambda X'f,$$ where $X, f\in\mathbb{K}[x]$ and $X':=\frac{dX}{dx}$. This deformation shows that on the level of Lie algebras (and similarly below, for Lie superalgebras) it is natural to choose $\mathbb{C}$ as the ground field.
Denote by $\mathcal{F}_\lambda$ the $\mathfrak{vect}(1)$-module structure on $\mathbb{K}[x]$ defined by $L^\lambda$ for a fixed $\lambda$. Geometrically, ${\cal F}_\lambda=\left\{fdx^{\lambda}\mid
f\in \mathbb{K}[x]\right\}$ is the space of polynomial weighted densities of weight $\lambda\in\mathbb{C}$. The space ${\cal
F}_\lambda$ coincides with the space of vector fields, functions and differential 1-forms for $\lambda = -1,\, 0$ and $1$, respectively.
Denote by $\mathrm{D}_{\nu,\mu}:=\mathrm{Hom}_{\rm{diff}}({\cal
F}_\nu, {\cal F}_\mu)$ the $\mathfrak{vect}(1)$-module of linear differential operators with the natural $\mathfrak{vect}(1)$-action denoted $L_X^{\nu,\mu}(A)$. Each module $\mathrm{D}_{\nu,\mu}$ has a natural filtration by the order of differential operators; the graded module ${\cal
S}_{\nu,\mu}:=\mathrm{gr}\mathrm{D}_{\nu,\mu}$ is called the [*space of symbols*]{}. The quotient-module $\mathrm{D}^k_{\nu,\mu}/\mathrm{D}^{k-1}_{\nu,\mu}$ is isomorphic to the module of weighted densities $\mathcal{F}_{\mu-\nu-k}$; the isomorphism is provided by the principal symbol map $\sigma_{\rm
pr}$ defined by: $$A=\sum_{i=0}^ka_i(x)\left(\frac{\partial}{\partial
x}\right)^i\mapsto\sigma_{\rm pr}(A)=a_k(x)(dx)^{\mu-\nu-k},$$ (see, e.g.,[@gmo]). Therefore, as a $\mathfrak{vect}(1)$-module, the space ${\cal S}_{\nu,\mu}$ depends only on the difference $\beta=\mu-\nu$, so that ${\cal S}_{\nu,\mu}$ can be written as ${\cal S}_{\beta}$, and we have $${\cal S}_{\beta} = \bigoplus_{k=0}^\infty \mathcal{F}_{\beta-k}$$ as $\mathfrak{vect}(1)$-modules. The space of symbols of order $\leq n$ is $${\cal S}_\beta^n:=\bigoplus_{k=0}^n{\cal F}_{\beta-k}.$$
In the last two decades, deformations of various types of structures have assumed an ever increasing role in mathematics and physics. For each such deformation problem a goal is to determine if all related deformation obstructions vanish and many beautiful techniques were developed to determine when this is so. Deformations of Lie algebras with base and versal deformations were already considered by Fialowski in 1986 [@f1]. In 1988, Fialowski [@f2] further introduced deformations whose base is a complete local algebra (the algebra is said to be [*local*]{} if it has a unique maximal ideal). Also, in [@f2], the notion of miniversal (or formal versal) deformation was introduced in general, and it was proved that under some cohomology restrictions, a versal deformation exists. Later Fialowski and Fuchs, using this framework, gave a construction for a versal deformation. Formal deformations of the $\mathfrak{vect}(1)$-module ${\cal S}_\beta^n$ were studied in [@aalo; @bbdo]. Moreover, the formal deformations that become trivial once the action is restricted to $\mathrm{\frak {sl}}(2)$ were completely described in [@bb].
According to Nijenhuis-Richardson the space $\mathrm{H}^1\left(\mathfrak{g};\mathrm{End}(V)\right)$ classifies the infinitesimal deformations of a $\mathfrak{g}$-module $V$ and the obstructions to integrability of a given infinitesimal deformation of $V$ are elements of $\mathrm{H}^2\left(\mathfrak{g};\mathrm{End}(V)\right)$. More generally, if $\frak h$ is a subalgebra of $\frak g$, then the $\frak h$-relative cohomology $\mathrm{H}^1\left(\mathfrak{g},\frak h;\mathrm{End}(V)\right)$ measures the infinitesimal deformations that become trivial once the action is restricted to $\frak h$ ($\frak h$-[*trivial deformations*]{}), while the obstructions to extension of any $\frak
h$-trivial infinitesimal deformation to a formal one are related to $\mathrm{H}^2\left(\mathfrak{g},\frak
h;\mathrm{End}(V)\right)$. Similarly, in the infinite dimensional setting, the infinitesimal deformations of the $\mathfrak{vect}(1)$-module ${\mathcal S}^n_\beta$ are classified, from a certain point of view, by the space
$$\label{1}
\mathrm{H}^1_{\rm diff}\left(\mathfrak{vect}(1);
\mathrm{D}\right)=\bigoplus_{0\leq i, j\leq n}\mathrm{H}^1_{\rm
diff}\left(\mathfrak{vect}(1);
\mathrm{D}_{\beta-j,\beta-i}\right),$$
where $\mathrm{D}:=\mathrm{D}(n,\beta)$ is the $\mathfrak{vect}(1)$-module of differential operators in ${\cal
S}_\beta^n$ and where $\mathrm{H}^i_\mathrm{diff}$ denotes the differential cohomology; that is, only cochains given by differential operators are considered. The $\mathrm{\frak{
sl}}(2)$-trivial infinitesimal deformations are classified by the space $$\label{2}
{\rm H}^1_{\rm diff}\left(\mathfrak{vect}(1),\mathrm{\frak{
sl}}(2); \mathrm{D}\right)=\bigoplus_{0\leq i, j\leq n}{\mathrm
H}^1_{\rm diff}\left(\mathfrak{vect}(1),\mathrm{\frak{ sl}}(2);
\mathrm{D}_{\beta-j,\beta-i}\right).$$
Feigin and Fuchs computed $\mathrm{H}^1_{\rm
diff}\left(\mathfrak{vect}(1);
\mathrm{D}_{\lambda,\lambda'}\right)$, see [@ff]. They showed that non-zero cohomology $\mathrm{H}^1_{\rm
diff}\left(\mathfrak{vect}(1);\mathrm{D}_{\lambda,\lambda'}\right)$ only appear for particular values of weights that we call [*resonant*]{} which satisfy $\lambda'-\lambda\in\mathbb{N}$. Therefore, in formulas (\[1\]) and (\[2\]), the summations are only over $i$ and $j$ such that $i\leq j$. Bouarroudj and Ovsienko [@bo] computed ${\mathrm H}^1_{\rm
diff}\left(\mathfrak{vect}(1),\mathrm{{\frak sl}}(2);
\mathrm{D}_{\lambda,\lambda'}\right)$, and Bouarroudj [@b1] solved a multi-dimensional version of the same problem on manifolds.
In this paper we study the simplest super analog of the problem solved in [@ff; @bo; @b1], namely, we consider the superspace $\mathbb{K}^{1|1}$ equipped with the contact structure determined by a 1-form $\alpha$, and the Lie superalgebra $\mathcal{K}(1)$ of contact polynomial vector fields on $\mathbb{K}^{1|1}$. We introduce the $\mathcal{K}(1)$-module $\mathfrak{F}_\lambda$ of $\lambda$-densities on $\mathbb{K}^{1|1}$ and the $\mathcal{K}(1)$-module of linear differential operators, $\frak{D}_{\nu,\mu}
:=\mathrm{Hom}_{\rm{diff}}(\mathfrak{F}_{\nu},\mathfrak{F}_{\mu})$, which are super analogues of the spaces $\mathcal{F}_\lambda$ and $\mathrm{D}_{\nu,\mu}$, respectively. The Lie superalgebra $\mathfrak{osp}(1|2)$, a super analogue of $\mathrm{\frak {sl}}(2)$, can be realized as a subalgebra of $\mathcal{K}(1)$. We compute $\mathrm{H}^1_{\rm
diff}\left(\mathcal{K}(1);\mathfrak{D}_{\lambda,\lambda'}\right)$ and $\mathrm{H}^1_{\rm diff}\left(\mathcal{K}(1), \mathfrak{osp}(1|
2);\mathfrak{D}_{\lambda,\lambda'}\right)$ and we show that, as in the classical setting, non-zero cohomology $\mathrm{H}^1_{\rm
diff}\left(\mathcal{K}(1);\mathfrak{D}_{\lambda,\lambda'}\right)$ only appear for resonant values of weights which satisfy $\lambda'-\lambda\in{1\over2}\mathbb{N}$. So, the super analogue of the space ${\cal S}_\beta^n$ is naturally the superspace (see [@gmo]): $${\frak
S}^n_{\beta}=\bigoplus_{k=0}^{2n}\mathfrak{F}_{\beta-\frac{k}{2}},\quad\text{where}\quad
n\in{1\over2}\mathbb{N}.$$ We use the result to study formal deformations of the $\mathcal{K}(1)$-module structure on ${\frak S}^n_{\beta}$. Denote by $\mathfrak{D}:=\mathfrak{D}(n,\beta)$ the $\mathcal{K}(1)$-module of linear differential operators in ${\frak S_\beta^n}$. The infinitesimal deformations of the $\mathcal{K}(1)$-module structure on ${\frak S}^n_{\beta}$ are classified by the space $$\mathrm{H}^1_{\rm
diff}\left(\mathcal{K}(1);\mathfrak{D}\right)=\bigoplus_{0\leq
i\leq j \leq 2n}\mathrm{H}^1_{\rm
diff}\left(\mathcal{K}(1);\frak{D}_{\beta-{j\over2},\beta-{i\over2}}\right).$$ The $\mathrm{\frak{ osp}}(1|2)$-trivial infinitesimal deformations are classified by the space $$\mathrm{H}^1_{\rm diff}\left(\mathcal{K}(1), \mathfrak{osp}(1|
2);\mathfrak{D}\right)=\bigoplus_{0\leq i\leq j \leq
2n}\mathrm{H}^1_{\rm diff}\left(\mathcal{K}(1), \mathfrak{osp}(1|
2);\frak{D}_{\beta-{j\over2},\beta-{i\over2}}\right).$$
In this work, we study only the generic formal $\mathfrak{osp}(1|2)$-trivial deformations of the action of $\mathcal{K}(1)$ on the space ${\frak S}^n_{\beta}$. In order to study the integrability of a given $\mathrm{\frak{
osp}}(1|2)$-trivial infinitesimal deformation, we need the description of $\mathfrak{osp}(1|2)$-invariant bilinear differential operators $\mathfrak{F}_{\tau}\otimes\mathfrak{F}_\lambda\longrightarrow\mathfrak{F}_{\mu}$.
Definitions and Notations
=========================
The Lie superalgebra of contact vector fields on $\mathbb{K}^{1|n}$
-------------------------------------------------------------------
Let $\mathbb{K}^{1\mid n}$ be the superspace with coordinates $(x,~\theta_1,\ldots,\theta_n),$ where the $\theta_i$ are odd indeterminates equipped with the standard contact structure given by the following $1$-form: $$\alpha_n=dx+\sum_{i=1}^n\theta_id\theta_i.$$ On $\mathbb{K}[x,\theta]:=\mathbb{K}[x,\theta_1,\dots,\theta_n]$, we consider the contact bracket $$\{F,G\}=FG'-F'G-\frac{1}{2}(-1)^{p(F)}\sum_{i=1}^n\overline{\eta}_i(F)\cdot
\overline{\eta}_i(G),$$where $\overline{\eta}_i=\frac{\partial}{\partial
{\theta_i}}-\theta_i\frac{\partial}{\partial x}$ and $p(F)$ is the parity of $F$.
Let $\mathrm{Vect_{Pol}}(\mathbb{K}^{1|n})$ be the superspace of polynomial vector fields on ${\mathbb{K}}^{1|n}$: $$\mathrm{Vect_{Pol}}(\mathbb{K}^{1|n})=\left\{F_0\partial_x
+ \sum_{i=1}^n F_i\partial_i \mid ~F_i\in\mathbb{K}[x,\theta]~
\text{ for all } i \right\},$$ where $\partial_i=\frac{\partial}{\partial\theta_i}$ and $\partial_x=\frac{\partial}{\partial x} $, and consider the superspace $\mathcal{K}(n)$ of contact polynomial vector fields on ${\mathbb{K}}^{1|n}$. That is, $\mathcal{K}(n)$ is the superspace of vector fields on $\mathbb{K}^{1|n}$ preserving the distribution singled out by the $1$-form $\alpha_n$: $$\mathcal{K}(n)=\big\{X\in\mathrm{Vect_{Pol}}(\mathbb{K}^{1|n})~|~\hbox{there
exists}~F\in {\mathbb{K}}[x,\,\theta]~ \hbox{such
that}~{L}_X(\alpha_n)=F\alpha_n\big\}.$$ The Lie superalgebra $\mathcal{K}(n)$ is spanned by the fields of the form: $$X_F=F\partial_x
-\frac{1}{2}\sum_{i=1}^n(-1)^{p(F)}\overline{\eta}_i(F)\overline{\eta}_i,\;\text{where
$F\in \mathbb{K}[x,\theta]$.}$$ In particular, we have $\mathcal{K}(0)=\mathfrak{vect}(1)$. Observe that ${L}_{X_F}(\alpha_n)=X_1(F)\alpha_n$. The bracket in $\mathcal{K}(n)$ can be written as: $ [X_F,\,X_G]=X_{\{F,\,G\}}$.
The subalgebra $\mathfrak{osp}(1|2)$
------------------------------------
In $\mathcal{K}(1)$, there is a subalgebra $\mathfrak{osp}(1|2)$ of projective transformations $$\mathfrak{osp}(1|2)=\text{Span}\left(X_1,\,X_{\theta},\, X_{x},\,X_{x\theta},\,
X_{x^2}\right);\quad(\mathfrak{osp}(1|2))_{\bar{0}}=\text{Span}(X_1,\,X_{x},\,X_{x^2})
\cong\mathfrak{sl}(2) .$$
The space of polynomial weighted densities on $\mathbb{K}^{1|1}$
----------------------------------------------------------------
From now on, $n=1$ and we will denote $\alpha_1$ and $\overline{\eta}_1$ respectively by $\alpha$ and $\overline{\eta}$. We have analogous definition of weighted densities in super setting (see [@ab]) with $dx$ replaced by $\alpha$. The elements of these spaces are indeed (weighted) densities since all spaces of generalized tensor fields have just one parameter relative $\mathcal{K}(1)$ — the value of $X_x$ on the lowest weight vector (the one annihilated by $X_\theta$). From this point of view the volume element (roughly speaking, $\lq\lq
dx\frac{\partial}{\partial\theta}"$) is indistinguishable from $\alpha^{\frac{1}{2}}.$
Consider the $1$-parameter action of $\mathcal{K}(1)$ on $\mathbb{K}[x,\theta]$ given by the rule: $$\label{superaction} \mathfrak{L}^{\lambda}_{X_F}= X_F + \lambda
F',$$ where $F'=\partial_{x}F$, or, in components: $$\label{deriv}
\frak{L}^{\lambda}_{X_F}(G)=L^{\lambda}_{a\partial_x}(g_0)+\frac{1}{2}~bg_1
+\left(L^{\lambda+\frac{1}{2}}_{a\partial_x}(g_1)+\lambda
g_0b'+{1\over2} g'_0 b\right)\theta,$$ where $F=a+b\theta,\,G=g_0+g_1\theta \in\mathbb{K}[x,\theta]$. We denote this $\mathcal{K}(1)$-module by $\mathfrak{F}_{\lambda}$, the space of all polynomial weighted densities on $\mathbb{K}^{1|1}$ of weight $\lambda$: $$\label{densities}
\mathfrak{F}_\lambda=\left\{f(x,\theta)\alpha^{\lambda} \mid
f(x,\theta) \in\mathbb{K}[x,\theta]\right\}.$$ Obviously:
- The adjoint $\mathcal{K}(1)$-module, is isomorphic to $\mathfrak{F}_{-1}.$
- As a $\mathfrak{vect}(1)$-module, $\mathfrak{F}_{\lambda}\simeq\mathcal{F}_\lambda \oplus
\Pi(\mathcal{F}_{\lambda+{1\over2}})$.
Any differential operator $A$ on $\mathbb{K}^{1|1}$ can be viewed as a linear mapping $F\alpha^\lambda\mapsto(AF)\alpha^\mu$ from $\mathfrak{F}_{\lambda}$ to $\mathfrak{F}_\mu$, thus the space of differential operators becomes a $\mathcal{K}(1)$-module denoted $\mathfrak{D}_{\lambda,\mu}$ for the natural action: $$\label{d-action}
\mathfrak{L}^{\lambda,\mu}_{X_F}(A)=\mathfrak{L}^{\mu}_{X_F}\circ
A-(-1)^{p(A)p(F)} A\circ \mathfrak{L}^{\lambda}_{X_F}.$$
\[decom\] As a $\mathfrak{vect}(1)$-module, we have $$(\frak{D}_{\lambda,\mu})_{\bar 0}\simeq \mathrm{D}_{\lambda,\mu}
\oplus \mathrm{D}_{\lambda+\frac{1}{2},\mu+\frac{1}{2}}\\ \hbox{and}\\
(\frak{D}_{\lambda,\mu})_{\bar1}\simeq\Pi(\mathrm{D}_{\lambda+\frac{1}{2},\mu}
\oplus \mathrm{D}_{\lambda,\mu+\frac{1}{2}}).$$
It is clear that the map $$\label{ph}\begin{array}{ll}
\varphi_\lambda:\mathfrak{F}_\lambda&\longrightarrow\mathcal{F}_{\lambda}\oplus
\Pi(\mathcal{F}_{\lambda+{1\over2}})\\
F\alpha^{\lambda}&\mapsto\left((1-\theta\partial_{\theta})(F)
(dx)^{\lambda},~
\Pi(\partial_{\theta}(F)(dx)^{\lambda+{1\over2}})\right)
\end{array}$$ is $\mathfrak{vect}(1)$-isomorphism, see formulae (\[deriv\]). So, we deduce a $\mathfrak{vect}(1)$-isomorphism: $$\label{Phi}
\begin{array}{lcll}\Phi_{\lambda,\mu}:&\frak{D}_{\lambda,\mu}&\longrightarrow&
\mathrm{D}_{\lambda,\mu}\oplus
\mathrm{D}_{\lambda+{1\over2},\mu+{1\over2}}\oplus\Pi(
\mathrm{D}_{\lambda,\mu+{1\over2}})\oplus\Pi(
\mathrm{D}_{\lambda+{1\over2},\mu})\\&A&\mapsto&\varphi_\mu\circ
A\circ\varphi_\lambda^{-1}.
\end{array}$$Here, we identify the $\mathfrak{vect}(1)$-modules via the following isomorphisms: $$\begin{gathered}
\begin{array}{llllllll}
\rm{Hom}_{\rm
diff}\left(\mathcal{F}_\lambda,\Pi(\mathcal{F}_{\mu+\frac{1}{2}})\right)
&\longrightarrow&\Pi\left(\mathrm{D}_{\lambda,\mu+\frac{1}{2}}\right),
\quad &A&\mapsto&\Pi(\Pi\circ A),\\[10pt] \rm{Hom}_{\rm
diff}\left(\Pi(\mathcal{F}_{\lambda+\frac{1}{2}}),\mathcal{F}_{\mu}\right)
&\longrightarrow&\Pi\left(\mathrm{D}_{\lambda+\frac{1}{2},\mu}\right),
\quad &A&\mapsto&\Pi(A\circ\Pi),\\[10pt] \rm{Hom}_{\rm
diff}\left(\Pi(\mathcal{F}_{\lambda+\frac{1}{2}}),\Pi(\mathcal{F}_{\mu+\frac{1}{2}})\right)
&\longrightarrow&\mathrm{D}_{\lambda+\frac{1}{2},\mu+\frac{1}{2}},
\quad &A&\mapsto&\Pi\circ A\circ\Pi.\\[10pt]
\end{array}\end{gathered}$$ Note that the change of parity map $\Pi$ commutes with the $\mathfrak{vect}(1)$-action.
Consider a family of $\mathfrak{vect}(1)$-modules on the space $\mathrm{D}_{(\lambda_1,\dots,\lambda_m);\mu}$ of linear differential operators: $~A: {\cal
F}_{\lambda_1}\otimes\cdots\otimes\mathcal{F}_{\lambda_m}\longrightarrow{\cal
F}_\mu.$ The Lie algebra $\mathfrak{vect}(1)$ naturally acts on $\mathrm{D}_{(\lambda_1,\dots,\lambda_m);\mu}$ (by the Leibniz rule). We similarly consider a family of ${\rm \mathcal{K}}(1)$-modules on the space $\mathfrak{ D}_{(\lambda_1,\dots,\lambda_m);\mu}$ of linear differential operators: $~A: {\frak
F}_{\lambda_1}\otimes\cdots\otimes\frak{F}_{\lambda_m}\longrightarrow{\frak
F}_\mu$.
$\mathfrak{sl}(2)$- and $\mathfrak{osp}(1|2)$-invariant bilinear differential operators
========================================================================================
\[trans2\][(Gordon, [@pg])]{} There exist $\mathfrak{sl}(2)$-invariant bilinear differential operators, called [transvectants]{}, $$J_k^{\tau,\lambda}:
\mathcal{F}_\tau\otimes\mathcal{F}_\lambda\longrightarrow\mathcal{F}_{\tau+\lambda+k},\quad
(\varphi dx^\tau,\phi dx^\lambda)\mapsto
J_k^{\tau,\lambda}(\varphi,\phi)dx^{\tau+\lambda+k}$$ given by $$J_k^{\tau,\lambda}(\varphi,\phi)=\sum_{0\leq i\leq k,
i+j=k}c_{i,j}\varphi^{(i)}\phi^{(j)},$$ where $k\in\mathbb{N}$ and the coefficients $c_{i,j}$ are characterized as follows:
- If $\tau, \lambda\not\in\{0,\,-{1\over2},\,-1,\,\dots,\,-{k-1\over2}\}$, then $ c_{i,j}=\Gamma_{i,j,k-1}^{\tau,\lambda}$, see (\[coe\]).
- If $\tau$ or $\lambda\in\{0,\,-{1\over2},\,-1,\,\dots,\,-{k-1\over2}\}$, the coefficients $c_{i,j}$ satisfy the recurrence relation $$\label{cij}(i+1)(i + 2\tau)c_{i+1,j} + (j+1)(j+2\lambda)c_{i,j+1} = 0.$$ Moreover, the space of solutions of the system (\[cij\]) is two-dimensional if $2\lambda=-s$ and $2\tau=-t$ with $t > k-s-2$, and one-dimensional otherwise.
Gieres and Theisen [@gt] listed the $\mathfrak{osp}(1|2)$-invariant bilinear differential operators, from $\mathfrak{F}_{\tau}\otimes\mathfrak{F}_\lambda$ to $\mathfrak{F}_{\mu}$, called [*supertransvectants*]{}. Gargoubi and Ovsienko [@go] gave an interpretation of these operators. In [@gt], the supertransvectants are expressed in terms of supercovariant derivative. Here, the supertransvectants appear in the context of the $\frak{ osp}(1|2)$-relative cohomology. More precisely, we need to describe the $\mathfrak{osp}(1|2)$-invariant linear differential operators from $\mathcal{K}{(1)}$ to $\frak{D}_{\lambda,\lambda+k-1}$ vanishing on $\mathfrak{osp}(1|2)$. Thus, using the Gordan’s transvectants and the isomorphism (\[ph\]), we give, in the following theorem, another description and other explicit formulas.
\[main\] i) There are only the following $\mathfrak{osp}(1|2)$-invariant bilinear differential operators acting in the spaces $\frak{F}_{\lambda}$: $$\begin{array}{ll}
\frak{J}_{k}^{\tau,\lambda}:
\frak{F}_\tau\otimes\frak{F}_\lambda&\longrightarrow\frak{F}_{\tau+\lambda+k}\\
(F \alpha^\tau, G \alpha^\lambda)&\mapsto
\frak{J}_k^{\tau,\lambda}(F,G)\alpha^{\tau+\lambda+k},\end{array}$$ where $k\in{1\over2}\mathbb{N}$. The operators $\frak{J}_{k}^{\tau,\lambda}$ labeled by semi-integer $k$ are odd; they are given by $$\begin{array}{lllll}
\frak{J}_{k}^{\tau,\lambda}(F,G)&=\displaystyle\sum_{i+j=[k]}
\Gamma_{i,j,k}^{\tau,\lambda}\left((-1)^{p(F)}(2\tau+[k]-j)F^{(i)}\overline{\eta}(G^{(j)})-
(2\lambda+[k]-i)\overline{\eta}(F^{(i)})G^{(j)}\right).
\end{array}$$ The operators $\frak{J}_{k}^{\tau,\lambda}$, where $k\in
\mathbb{N}$, are even; set $\frak{J}_0^{\tau,\lambda}(F,G)=FG$ and $$\begin{array}{lllll}
\frak{J}_k^{\tau,\lambda}(F,G)&=\displaystyle\sum_{i+j=k-1}(-1)^{p(F)}
\Gamma_{i,j,k-1}^{\tau,\lambda}\overline{\eta}(F^{(i)})\overline{\eta}(G^{(j)})
-\displaystyle\sum_{i+j=k}\Gamma_{i,j,k-1}^{\tau,\lambda}F^{(i)}G^{(j)},\end{array}$$where $\big(^x_i\big)=\frac {x(x-1)\cdots (x-i+1)}{i!}$ and $[k]$ denotes the integer part of $k$, $k>0$, and $$\label{coe} \Gamma_{i,j,k}^{\tau,\lambda}=(-1)^{j}
\begin{pmatrix}2\tau+[k]\\j\end{pmatrix}
\begin{pmatrix}2\lambda+[k]\\i\end{pmatrix}.$$ ii) If $\tau,
\lambda\not\in\{0,\,-{1\over2},\,-1,\,\dots,\,-{[k]\over2}\}$, then $\frak{J}_{k}^{\tau,\lambda}$ is the unique (up to a scalar factor) bilinear $\mathfrak{osp}(1|2)$-invariant bilinear differential operator $\frak{F}_\tau\otimes\frak{F}_\lambda
\longrightarrow\frak{F}_{\tau+\lambda+k}$.
iii\) For $k\in{1\over2}(\mathbb{N}+5)$, the space of $\mathfrak{osp}(1|2)$-invariant linear differential operators from $\mathcal{K}{(1)}$ to $\frak{D}_{\lambda,\lambda+k-1}$ vanishing on $\mathfrak{osp}(1|2)$ is one dimensional.
i\) Let $\mathcal{T}:
\frak{F}_\tau\otimes\frak{F}_\lambda\longrightarrow\frak{F}_\mu$ be an $\mathfrak{osp}(1|2)$-invariant differential operator. Using the fact that, as $\mathfrak{vect}(1)$-modules, $$\label{decomposition}\frak{F}_\tau\otimes\frak{F}_\lambda\simeq
\mathcal{F}_\tau\otimes\mathcal{F}_\lambda\oplus\Pi(\mathcal{F}_{\tau+\frac{1}{2}}\otimes
\mathcal{F}_{\lambda+\frac{1}{2}})\oplus
\mathcal{F}_\tau\otimes\Pi(\mathcal{F}_{\lambda+\frac{1}{2}})\oplus\Pi(\mathcal{F}_{\tau+\frac{1}{2}})\otimes
\mathcal{F}_\lambda$$ and $$\frak{F}_\mu\simeq\mathcal{F}_\mu\oplus\Pi(\mathcal{F}_{\mu+\frac{1}{2}}),$$ we can deduce that the restriction of $\mathcal{T}$ to each component of the right hand side of (\[decomposition\]) is a transvectant. So, the parameters $\tau,$ $\lambda$ and $\mu$ must satisfy $$\mu=\lambda+\tau+k,\quad\hbox{ where }\quad
k\in{1\over2}\mathbb{N}.$$ The corresponding operators will be denoted $\frak{J}_{k}^{\tau,\lambda}$. Obviously, if $k$ is integer, then the operator $\frak{J}_{k}^{\tau,\lambda}$ is even and its restriction to each component of the right hand side of (\[decomposition\]) coincides (up to a scalar factor) with the respective transvectants:
$$\label{restric1}
\begin{cases}
& \text{${J}_k^{\tau,\lambda}~~~~~~~:
\mathcal{F}_\tau\otimes\mathcal{F}_\lambda\longrightarrow\mathcal{F}_{\mu},$}\\
& \text{${J}_{k-1}^{\tau+\frac{1}{2},\lambda+\frac{1}{2}}:
\Pi(\mathcal{F}_{\tau+\frac{1}{2}})\otimes\Pi(\mathcal{F}_{\lambda+\frac{1}{2}})\longrightarrow\mathcal{F}_{\mu}$},
\\ & \text{${J}_{k}^{\tau,\lambda+\frac{1}{2}}~~~:
\mathcal{F}_\tau\otimes\Pi(\mathcal{F}_{\lambda+\frac{1}{2}})\longrightarrow\Pi(\mathcal{F}_{\mu+\frac{1}{2}}$}),
\\ & \text{${J}_{k}^{\tau+\frac{1}{2},\lambda}~~~:
\Pi(\mathcal{F}_{\tau+\frac{1}{2}})\otimes\mathcal{F}_\lambda\longrightarrow\Pi(\mathcal{F}_{\mu+\frac{1}{2}}$}).
\\
\end{cases}$$
If $k$ is semi-integer, then the operator $\frak{J}_{k}^{\tau,\lambda}$ is odd and its restriction to each component of the right hand side of (\[decomposition\]) coincides (up to a scalar factor ) with the respective transvectants: $$\label{restric2}
\begin{cases}
& \text{$J_{[k]+1}^{\tau,\lambda}~~~~~~~:
\mathcal{F}_\tau\otimes\mathcal{F}_\lambda\longrightarrow\Pi(\mathcal{F}_{\mu+\frac{1}{2}}),$}\\
& \text{$J_{[k]}^{\tau+\frac{1}{2},\lambda+\frac{1}{2}}:
\Pi(\mathcal{F}_{\tau+\frac{1}{2}})\otimes\Pi(\mathcal{F}_{\lambda+\frac{1}{2}})
\longrightarrow\Pi(\mathcal{F}_{\mu+\frac{1}{2}})$}, \\ &
\text{$J_{[k]}^{\tau,\lambda+\frac{1}{2}}~~~:
\mathcal{F}_\tau\otimes\Pi(\mathcal{F}_{\lambda+\frac{1}{2}})\longrightarrow\mathcal{F}_{\mu}$},
\\ & \text{$J_{[k]}^{\tau+\frac{1}{2},\lambda}~~~:
\Pi(\mathcal{F}_{\tau+\frac{1}{2}})\otimes\mathcal{F}_\lambda\longrightarrow\mathcal{F}_{\mu}$}.
\\
\end{cases}$$
More precisely, let $F\alpha^\tau\otimes G
\alpha^\lambda\in\frak{F}_\tau\otimes\frak{F}_\lambda$, where $F=f_0+\theta f_1$ and $G=g_0+\theta g_1$, with $f_0,\,f_1,\,g_0,\,g_1\in\mathbb{K}[x]$. Then if $k$ is integer, we have $$\label{integer}
\begin{array}{llll}
\frak{J}_{k}^{\tau,\lambda}(\varphi,\psi)=&\Big[a_1J_k^{\tau,\lambda}(f_0,g_0)+
a_2J_{k-1}^{\tau+\frac{1}{2},\lambda+\frac{1}{2}}(f_1,g_1)
+\\[6pt]&\theta\left(a_3J_{k}^{\tau,\lambda+\frac{1}{2}}(f_0,g_1)+a_4J_{k}^{\tau+\frac{1}{2},\lambda}(f_1,g_0)
\right)\Big]\alpha^\mu
\end{array}$$ and if $k$ is semi-integer, we have $$\label{semi}
\begin{array}{llll}
\frak{J}_{k}^{\tau,\lambda}(\varphi,\psi)=&\Big[b_1J_{[k]}^{\tau,\lambda+\frac{1}{2}}(f_0,g_1)+
b_2J_{[k]}^{\tau+\frac{1}{2},\lambda}(f_1,g_0)
+\\[6pt]&\theta\left(b_3J_{[k]+1}^{\tau,\lambda}(f_0,g_0)+b_4J_{[k]}^{\tau+\frac{1}{2},\lambda+\frac{1}{2}}(f_1,g_1)
\right)\Big]\alpha^\mu,
\end{array}$$where the $a_i$ and $b_i$ are constants. The invariance of $\frak{J}_{k}^{\tau,\lambda}$ with respect to $X_{\theta}$ and $X_{x\theta}$ reads: $$\label{T1}
{\frak L}_{X_\theta}^\mu\circ
\frak{J}_{k}^{\tau,\lambda}-(-1)^{2k}\frak{J}_{k}^{\tau,\lambda}\circ
{\frak L}_{X_\theta}^{(\tau,\lambda)}={\frak
L}_{X_{x\theta}}^\mu\circ\frak{J}_{k}^{\tau,\lambda}
-(-1)^{2k}\frak{J}_{k}^{\tau,\lambda}\circ {\frak
L}_{X_{x\theta}}^{(\tau,\lambda)}=0.$$
The formula (\[T1\]) allows us to determine the coefficients $a_i$ and $b_i$. More precisely, the invariance property with respect to $X_{\theta}$ and $X_{x\theta}$ yields [$$a_1=a_2=a_3=a_4,~b_2=-\frac{2\lambda+k-1}{2\tau+k-1}b_1,~ b_3
=\frac{k}{2\tau+k-1}b_1\text{ and }
b_4=-(1+\frac{2\lambda}{2\tau+k-1})b_1.$$]{}
ii\) The uniqueness of supertansvectants follows from the uniqueness of transvectants.
iii\) In the non-super case, according to formulae (\[cij\]), if $2\tau=-1$ and $k\geq2$, the space of $\mathfrak{sl}(2)$-invariant bilinear differential operators ${\cal F}_\tau\otimes{\cal
F}_\lambda\longrightarrow{\mathcal F}_{\tau+\lambda+k}$ is 2-dimensional if and only if $2\lambda=-s$, where $s\in\{k-1,\,k-2\}$. This space is spanned by $J_k^{-{1\over2},-{s\over2}}$ and $I_k^{-{1\over2},-{s\over2}}$, where $$I_k^{-{1\over2},-{s\over2}}(\varphi,\phi)=
\left\{\begin{array}{ll}\varphi\phi^{(k)}\quad&\text{if}\quad
s=k-1\\
\varphi\phi^{(k)}+k\varphi'\phi^{(k-1)}\quad&\text{if}\quad s=k-2
\end{array}\right.$$ and $$J_k^{-{1\over2},-{s\over2}}(\varphi,\phi)=\sum_{i+j=k,\;i\geq
k-s+1 }c_{i,j}\varphi^{(i)}\phi^{(j)},$$ where the coefficients $c_{i,j}$ satisfy (\[cij\]). We see that only the operators $J_k^{-{1\over2},-{s\over2}}$ vanish on the space of affine functions, i.e., of the form $\varphi(x)=ax+b$.
If $k\geq3$, the space of $\mathfrak{sl}(2)$-invariant bilinear differential operators ${\cal F}_{-1}\otimes{\cal
F}_\lambda\longrightarrow\mathcal{F}_{\lambda+k-1}$ is 2-dimensional if and only if $2\lambda=-s$, where $s\in\{k-1,\,k-2,\,k-3\}$. This space is spanned by $J_k^{-1,-{s\over2}}$ and $I_k^{-1,-{s\over2}}$, where $$I_k^{-1,-{s\over2}}(\varphi,\phi)=\begin{cases}
\varphi\phi^{(k)}&\text{if $s=k-1$}\\
\varphi\phi^{(k)}+{k\over2}\varphi'\phi^{(k-1)}&\text{if $
s=k-2$}\\
\varphi\phi^{(k)}+k\varphi'\phi^{(k-1)}+{k(k-1)\over2}\varphi''\phi^{(k-2)}&\text{if
$ s=k-3$}\end{cases}$$ and where $$J_k^{-1,-{s\over2}}(\varphi,\phi)=\sum_{i+j=k,\;i\geq 3
}c_{i,j}\varphi^{(i)}\phi^{(j)}.$$ We see that the operator $I_k^{-1,-{s\over2}}$ does not vanish on $\frak{sl}(2)$, but the operator $J_k^{-1,-{s\over2}}$ vanishes on $\frak{sl}(2)$.
Now, if $\tau=-1, -{1\over2}$ and $2\lambda\notin\{1-k,\,2-k,\,3-k\}$ with $k\geq3$, the space of $\mathfrak{sl}(2)$-invariant bilinear differential operators ${\cal F}_\tau\otimes{\cal F}_\lambda\longrightarrow{\cal
F}_{\tau+\lambda+k}$ is 1-dimensional. But, in this case, we see that the coefficients $c_{i,j}$ satisfying (\[cij\]) are such that $c_{i,j}=0$ if $i\leq2$ for $\tau=-1$ and $c_{i,j}=0$ if $i\leq1$ for $\tau=-{1\over2}$.
Thus, in the super setting, if $2k\geq5$, according to equations (\[integer\]) and (\[semi\]), we see that the space of $\mathfrak{osp}(1|2)$-invariant linear differential operator from $\mathcal{K}{(1)}$ to $\frak{D}_{\lambda,\lambda+k-1}$ vanishing on $\mathfrak{osp}(1|2)$ is one-dimensional.
Cohomology
==========
Let us first recall some fundamental concepts from cohomology theory (see, e.g., [@Fu]). Let $\frak{g}=\frak{g}_{\bar
0}\oplus \frak{g}_{\bar 1}$ be a Lie superalgebra acting on a superspace $V=V_{\bar 0}\oplus V_{\bar 1}$ and let $\mathfrak{h}$ be a subalgebra of $\mathfrak{g}$. (If $\frak{h}$ is omitted it assumed to be $\{0\}$.) The space of $\frak h$-relative $n$-cochains of $\frak{g}$ with values in $V$ is the $\frak{g}$-module $$C^n(\frak{g},\frak{h}; V ) := \mathrm{Hom}_{\frak
h}(\Lambda^n(\frak{g}/\frak{h});V).$$ The [*coboundary operator*]{} $ \delta_n: C^n(\frak{g},\frak{h}; V
)\longrightarrow C^{n+1}(\frak{g},\frak{h}; V )$ is a $\frak{g}$-map satisfying $\delta_n\circ\delta_{n-1}=0$. The kernel of $\delta_n$, denoted $Z^n(\mathfrak{g},\frak{h};V)$, is the space of $\frak h$-relative $n$-[*cocycles*]{}, among them, the elements in the range of $\delta_{n-1}$ are called $\frak
h$-relative $n$-[*coboundaries*]{}. We denote $B^n(\mathfrak{g},\frak{h};V)$ the space of $n$-coboundaries.
By definition, the $n^{th}$ $\frak h$-relative cohomolgy space is the quotient space $$\mathrm{H}^n
(\mathfrak{g},\frak{h};V)=Z^n(\mathfrak{g},\frak{h};V)/B^n(\mathfrak{g},\frak{h};V).$$ We will only need the formula of $\delta_n$ (which will be simply denoted $\delta$) in degrees 0 and 1: for $v \in
C^0(\frak{g},\,\frak{h}; V) =V^{\frak h}$, $\delta v(g) : =
(-1)^{p(g)p(v)}g\cdot v$, where $$V^{\frak h}=\{v\in V~\mid~h\cdot v=0\quad\text{ for all }
h\in\frak h\},$$ and for $ \Upsilon\in C^1(\frak{g}, \frak{h};V )$, $$\delta(\Upsilon)(g,\,h):=
(-1)^{p(g)p(\Upsilon)}g\cdot
\Upsilon(h)-(-1)^{p(h)(p(g)+p(\Upsilon))}h\cdot
\Upsilon(g)-\Upsilon([g,~h])\quad\text{for any}\quad g,h\in
\frak{g}.$$ According to the $\mathbb{Z}_2$-grading (parity) of $\frak g$, for any $\Upsilon\in Z^1(\frak{g}, V)$, we have $$\Upsilon=\Upsilon'+\Upsilon''\in Z^1(\frak{g}_{\bar 0};\, V)\oplus
\mathrm{Hom}(\frak{g}_{\bar 1},\, V)$$ subject to the following three equations: $$\begin{gathered}
\label{coc1} \Upsilon'([g_1,g_2]) -g_1\cdot \Upsilon'(g_2) +
g_2\cdot \Upsilon'(g_1)= 0 \quad\text{for any}\quad
g_1,\,g_2\in\frak{g}_{\bar 0}, \\[10pt] \label{coc2}
\Upsilon''([g,\,h]) - g\cdot \Upsilon''(h) +(-1)^{p(\Upsilon)}
h\cdot \Upsilon'(g)= 0 \quad\text{for any}\quad g\in\frak{g}_{\bar
0},\,h\in\frak{g}_{\bar 1},\\[10pt] \label{coc3}
\Upsilon'([h_1,h_2]) - (-1)^{p(\Upsilon)}\left(h_1\cdot
\Upsilon''(h_2)+ h_2\cdot \Upsilon''(h_1)\right)=0 \quad\text{for
any}\quad h_1,\,h_2\in\frak{g}_{\bar 1}.\end{gathered}$$
Formulas (\[coc1\])–(\[coc3\]) show that $\mathrm{H}^1_{\rm
diff}(\mathcal{K}(1);\frak{D}_{\lambda,\mu})$ and $\mathrm{H}^1_{\rm diff}(\mathfrak{vect}(1);
\mathrm{D}_{\lambda,\mu})$ are closely related. Similarly, $\mathrm{H}^1_{\rm
diff}(\mathcal{K}(1),\frak{osp}(1|2);\frak{D}_{\lambda,\mu})$ is related to $\mathrm{H}^1_{\rm
diff}(\mathfrak{vect}(1),\frak{sl}(2); \mathrm{D}_{\lambda,\mu})$. Therefore, for comparison and to build upon, we recall the description of $\mathrm{H}^1_{\rm diff}(\mathfrak{vect}(1);
\mathrm{D}_{\lambda,\mu})$. Note that $\mathrm{H}^1_{\rm
diff}(\mathcal{K}(1),\frak{osp}(1|2);\frak{D}_{\lambda,\mu})$ is also computed by Conley, see [@c].
Relationship between $\mathrm{H}^1_{\rm
diff}(\mathfrak{vect}(1); \mathrm{D}_{\lambda,\mu})$ and $\mathrm{H}^1_{\rm diff}(\mathcal{K}(1);\frak{D}_{\lambda,\mu})$ {#FirstSect}
-------------------------------------------------------------------------------------------------------------------------
Feigin and Fuchs [@ff] calculated $\mathrm{H}^1_{\rm diff}(\mathfrak{vect}(1);
\mathrm{D}_{\lambda,\mu})$. The result is as follows $$\label{CohSpace2} \mathrm{H}^1_{\rm diff}(\mathfrak{vect}(1);
\mathrm{D}_{\lambda,\mu})\simeq\left\{
\begin{array}{ll}
\mathbb{K}&\hbox{ if }~~ \mu-\lambda=0,2,3,4 \hbox{ for all
}\lambda,\\[2pt] \mathbb{K}^2& \hbox{ if }~~\lambda=0\hbox{ ~and~
}\mu=1 ,\\[2pt] \mathbb{K}&\hbox{ if }~~ \lambda=0 \hbox{ or }
\lambda=-4\hbox{ ~and~ }\mu-\lambda=5, \\[2pt] \mathbb{K}& \hbox{
if }~~ \lambda=-\frac{5\pm \sqrt{19}}{2}\hbox{ ~and~ }\mu-\lambda
=6,\\[2pt] 0 &\hbox{ otherwise. }
\end{array}
\right.$$ For $X\frac{d}{dx}\in\mathfrak{vect}(1)$ and $f{dx}^{\lambda}\in{\cal F}_\lambda$, we write $$\begin{aligned}
\begin{array}{llll}
C_{\lambda,\lambda+k
}(X\frac{d}{dx})(f{dx}^{\lambda})=C_{\lambda,\lambda+k
}(X,f){dx}^{\lambda+k}. \end{array}\end{aligned}$$ The spaces ${\mathrm H}^1_{\rm dif\/f}(\mathfrak{vect}(1),
\mathrm{D}_{\lambda,\lambda+k})$ are generated by the cohomology classes of the following 1-cocycles: $$\label{cocycles}\begin{array}{llllllllll}
C_{\lambda,\lambda}(X,f)&=&X'f \\
C_{0,1}(X,f)&=&X''f \\
{\widetilde C}_{0,1}(X,f)&=&(X'f)' \\
C_{\lambda,\lambda+2}(X,f)&=&X^{(3)}f+2X''f' \\
C_{\lambda,\lambda+3}(X,f)&=&X^{(3)}f'+X''f'' \\
C_{\lambda,\lambda+4}(X,f)&=&-\lambda
X^{(5)}f+X^{(4)}f'-6X^{(3)}f''-4X''f^{(3)}\\
C_{0,5}(X,f)&=&2X^{(5)}f'-5X^{(4)}f''+10X^{(3)}f^{(3)}+5X''f^{(4)}\\
C_{-4,1}(X,f)&=&12X^{(6)}f+22X^{(5)}f'+5X^{(4)}f''-10X^{(3)}f^{(3)}-5X''f^{(4)}\\
C_{a_i,a_i+6}(X,f)&=&\alpha_i X^{(7)}f-\beta_i X^{(6)}f'-\gamma_i
X^{(5)}f''-
5X^{(4)}f^{(3)}+5X^{(3)}f^{(4)}+~&2X''f^{(5)},
\end{array}$$ where$$\begin{array}{llllllll}a_1=-\frac{5+ \sqrt{19}}{2},
&\alpha_1=-\frac{22+ 5\sqrt{19}}{4}, &\beta_1=\frac{31+
7\sqrt{19}}{2}, &\gamma_1=\frac{25+ 7\sqrt{19}}{2}\\[4pt]
a_2=-\frac{5- \sqrt{19}}{2}, &\alpha_2=-\frac{22- 5\sqrt{19}}{4},
& \beta_2=\frac{31- 7\sqrt{19}}{2},&\gamma_2=\frac{25-
7\sqrt{19}}{2}.\end{array}$$
Now, let us study the relationship between any 1-cocycle of ${\mathcal K}(1)$ and its restriction to the subalgebra $\mathfrak{vect}(1)$. More precisely, we study the relationship between $\mathrm{H}_{\rm diff}^1({\mathcal
K}(1);\mathfrak{D}_{\lambda,\mu})$ and $\mathrm{H}_{\rm
diff}^1(\mathfrak{vect}(1); \mathrm{D}_{\lambda,\mu})$. According to Proposition \[decom\], we see that $\mathrm{H}_{\rm
diff}^1(\mathfrak{vect}(1);\mathfrak{D}_{\lambda,\mu})$ can be deduced from the spaces $\mathrm{H}_{\rm
diff}^1(\mathfrak{vect}(1); \mathrm{D}_{\lambda,\mu})$: $$\label{coho}\begin{array}{ll}\mathrm{H}_{\rm
diff}^1\left(\mathfrak{vect}(1);\mathfrak{D}_{\lambda,\mu}\right)&\simeq\mathrm{H}_{\rm
diff}^1\left(\mathfrak{vect}(1);
\mathrm{D}_{\lambda,\mu}\right)\oplus\mathrm{H}_{\rm
diff}^1\left(\mathfrak{vect}(1);
\mathrm{D}_{\lambda+\frac{1}{2},\mu+\frac{1}{2}}\right)\oplus\\[8pt]
&\mathrm{H}_{\rm diff}^1\left(\mathfrak{vect}(1);\Pi(
\mathrm{D}_{\lambda,\mu+\frac{1}{2}})\right) \oplus\mathrm{H}_{\rm
diff}^1\left(\mathfrak{vect}(1);\Pi(
\mathrm{D}_{\lambda+\frac{1}{2},\mu})\right).
\end{array}$$ Moreover, the following lemma shows the close relationship between the cohomolgy spaces $\mathrm{H}^1(\mathcal{K}(1);\mathfrak{D}_{\lambda,\mu})$ and $\mathrm{H}^1(\mathfrak{vect}(1);\mathrm{D}_{\lambda,\mu})$.
\[sa\] The 1-cocycle $\Upsilon$ of $\mathcal{K}(1)$ is a coboundary if and only if its restriction $\Upsilon'$ to $\mathfrak{vect}(1)$ is a coboundary.
It is easy to see that if $\Upsilon$ is a coboundary of $\mathcal{K}(1)$, then $\Upsilon'$ is a coboundary of $\mathfrak{vect}(1)$. Now, assume that $\Upsilon'$ is a coboundary of $\mathfrak{vect}(1)$, that is, there exist ${A}\in\frak{D}_{\lambda,\mu}$ such that $\Upsilon'$ is defined by $$\Upsilon'(X_f)=\mathfrak{L}_{X_f}^{\lambda,\mu}{A}\quad\text{ for
all } f\in\mathbb{K}[x].$$ By replacing $\Upsilon$ by $\Upsilon-\delta{A}$, we can suppose that $\Upsilon'=0$. But, in this case, the map $\Upsilon$ must satisfy the following equations $$\begin{aligned}
\label{sltr1}
&\mathfrak{L}^{\lambda,\mu}_{X_g}\Upsilon(X_{h\theta})-\Upsilon([X_g,X_{h\theta}])=0
\quad\text{ for all } g,\,h\in\mathbb{K}[x].\\[6pt] \label{sltr2}
&\mathfrak{L}^{\lambda,\mu}_{X_{h_1\theta}}\Upsilon(X_{h_2\theta})+
\mathfrak{L}^{\lambda,\mu}_{X_{h_2\theta}}\Upsilon(X_{h_1\theta})=0
\quad\text{ for all } h_1,\,h_2\in\mathbb{K}[x].\end{aligned}$$ The equation (\[sltr1\]) expresses the $\mathfrak{vect}(1)$-invariance of the map $\Upsilon:\Pi(\mathcal{F}_{-{1\over2}})\times
\mathcal{F}_\lambda\longrightarrow\mathcal{F}_\mu$. Therefore, if $\Upsilon$ is an even 1-cocycle, then, according to Proposition \[decom\], we can easily deduce the expression of $\Upsilon$ from the work of P. Grozman [@G4]. More precisely, $\Upsilon$ has, [*a priori*]{}, the following form: $$\Upsilon(X_{h\theta })(F\alpha^\lambda)=\left\{\begin{array}{llll}
(a_1hf\theta)\alpha^{\lambda-1}&\text{if
}\quad\mu=\lambda-1\\[6pt] (a_2hg+a_3({1\over2}hf'+ \lambda
h'f)\theta)\alpha^\lambda &\text{if }\quad\mu=\lambda\\[6pt]
a_4({1\over2}hg'+ \lambda h'g)\alpha^{\lambda+1} &\text{if
}\quad\mu=\lambda+1, \,\lambda\neq0,-{1\over2} \\[6pt]
a_5({1\over2}hg''+ h'g')\alpha^{{3\over2}}&\text{if
}\quad(\lambda,\mu)=(-{1\over2},{3\over2})\\[6pt]
\left(a_6(hg'+h'g)+a_7({1\over2}hf''+ h'f')\theta\right)\alpha
&\text{if }\quad(\lambda,\mu)=(0,1)\\[6pt] a_8(hg''-
h''g)\alpha\quad&\text{if }(\lambda,\mu)=(-1,1)\\[6pt] (a_9hg'+
a_{10}(hf''- h''f)\theta)\alpha^{1\over2}&\text{if
}\quad(\lambda,\mu)=(-{1\over2},{1\over2})\\[6pt] 0
&\text{otherwise},\end{array}\right.$$ where $a_i\in\mathbb{K}$, $f,\,g\in\mathbb{K}[x]$ and $F=f+g\theta$. But, the map $\Upsilon$ must satisfy the equation (\[sltr2\]), so we obtain $a_1=a_4=a_5=a_8=0,~a_3=-2a_2,~a_7=-2a_6$ and $a_{10}=-a_9$. More precisely, up to a scalar factor, $\Upsilon$ is given by: $$\Upsilon=\left\{\begin{array}{lllll}
\delta ((1-\theta\partial_\theta)\partial_x)&\text{ if
}\quad(\lambda,\mu)=(0,1),\\
\delta(\theta\partial_\theta\partial_x)&\text{ if
}\quad(\lambda,\mu)=(-{1\over2},{1\over2}),\\ \delta
(\theta\partial_\theta)&\text{ if }\quad\lambda=\mu,\\ 0 &\text{
otherwise. }
\end{array}\right.$$ Similarly, if $\Upsilon$ is an odd 1-cocycle, then, $\Upsilon$ has, a priori, the following form (see [@G4]): $$\Upsilon(X_{h\theta })(F\alpha^\lambda)=\left\{\begin{array}{llll}
(b_1hf+b_2hg)\alpha^{\lambda-{1\over2}}&\text{if
}\quad\mu=\lambda-{1\over2}\\[6pt] (b_3({1\over2} hf'+\lambda
h'f)+b_4({1\over2}
hg'+(\lambda+{1\over2})h'g)\theta)\alpha^{\lambda+{1\over2}}&\text{if
}\quad\mu=\lambda+{1\over2}\\[6pt] b_5({1\over2}
hf''+h''f)\alpha^{\frac{3}{2}}&\text{if
}\quad(\lambda,\mu)=(0,{3\over2})\\[6pt] (b_6(hf''-h''f)+b_7({1\over2}
hg''+h'g')\theta)\alpha &\text{if
}\quad(\lambda,\mu)=(-{1\over2},1)\\[6pt]
(b_8(hg''-h''g)\theta\alpha^{{1\over2}}&\text{if
}\quad(\lambda,\mu)=(-1,{1\over2})
\\[6pt] 0 &\text{otherwise,}\end{array}\right.$$ where $b_i\in\mathbb{K}$. But, the map $\Upsilon$ must satisfy the equation (\[sltr2\]), so we obtain $b_5=b_8=0,~b_1=b_2,~b_3=b_4$ and $b_{7}=2b_6$. More precisely, up to a scalar factor, $\Upsilon$ is given by: $$\Upsilon=\left\{\begin{array}{lllll}
\delta (\partial_\theta)&\text{ if}\quad\mu=\lambda+{1\over2},\\
\delta (\theta)&\text{ if }\quad\mu=\lambda-{1\over2},\\ \delta
(\partial_\theta\partial_x)&\text{ if
}\quad(\lambda,\mu)=(-{1\over2},1),\\ 0 &\text{ otherwise. }
\end{array}\right.$$ This completes the proof.
The following lemma gives the general form of any 1-cocycle of $\mathcal{K}(1)$.
\[sd\] Let $\Upsilon\in\mathrm{Z}^1_{\rm
diff}(\mathcal{K}(1);\frak{D}_{\lambda,\mu})$. Up to a coboundary, the map $\Upsilon$ has the following general form $$\label{coef}
\Upsilon(X_{F})=\sum_{m,k}(a_{m,k}+\theta
b_{m,k})\overline{\eta}^{k}(F)\overline{\eta}^m,$$where the coefficients $a_{m,k}$ and $b_{m,k}$ are constants.
Since $-\overline{\eta}^2={\partial _x}$, the operator $\Upsilon$ has the form (\[coef\]), where, [*a priori*]{}, the coefficients $a_{m,k}$ and ${b}_{m,k}$ are functions (see [@gmo]), but we will prove that, up to a coboundary, $\Upsilon$ is invariant with respect the vector field $X_1={\partial_x}$. The 1-cocycle condition reads: $$\begin{array}{lll}\label{partial1}
\mathfrak{L}^{\lambda,\mu}_{X_1}(\Upsilon(X_{F}))-
(-1)^{p(F)p(\Upsilon)}\mathfrak{L}^{\lambda,\mu}_{X_{F}}(\Upsilon(X_1))-
\Upsilon([X_1,X_{F}])=0.
\end{array}$$ But, from (\[cocycles\]), up to a coboundary, we have $\Upsilon(X_1)=0$, and therefore the equation (\[partial1\]) becomes $$\begin{array}{lll}\label{}
\mathfrak{L}^{\lambda,\mu}_{X_1}(\Upsilon(X_{F}))-
\Upsilon([X_1,X_{F}])=0
\end{array}$$ which is nothing but the invariance property of $\Upsilon$ with respect the vector field $X_1$.
\[osp\] Any 1-cocycle $\Upsilon\in
Z^1_{\mathrm{diff}}(\mathcal{K}(1);\frak{D}_{\lambda,\mu})$ vanishing on $\frak{osp}(1|2)$ is $\mathfrak{osp}(1|2)$-invariant.
The 1-cocycle relation of $\Upsilon$ reads: $$\label{osp1}
(-1)^{p(F)p(\Upsilon)}\mathfrak{L}_{X_F}^{\lambda,\mu}
\Upsilon(X_G)-(-1)^{p(G)(p(F)+p(\Upsilon))}\mathfrak{L}_{X_G}^{\lambda,\mu}
\Upsilon(X_F)-\Upsilon([X_F,~X_G])=0,$$ where $X_F,\,X_G\in ~\mathcal{K}(1).$ Thus, if $\Upsilon(X_F)=0$ for all $X_F\in\frak{osp}(1|2)$, the equation (\[osp1\]) becomes $$\label{osp2}
(-1)^{p(F)p(\Upsilon)}\mathfrak{L}_{X_F}^{\lambda,\mu}
\Upsilon(X_G)-\Upsilon([X_F,~X_G])=0$$ expressing the $\frak{osp}(1|2)$-invariance of $\Upsilon$.
\[sl2\] ([@bab] Lemma 3.3.) Up to a coboundary, any 1-cocycle $\Upsilon\in
Z_{\mathrm{diff}}^1(\mathcal{K}(1);\frak{D}_{\lambda,\mu})$ vanishing on $\frak{sl}(2)$ is $\mathfrak{osp}(1|2)$-invariant. That is, if $\Upsilon(X_1)=\Upsilon(X_x)=\Upsilon(X_{x^2})=0$, then the restriction of $\Upsilon$ to $\frak{osp}(1|2)$ is trivial.
Recall that, as $\mathfrak{sl}(2)$-module, the subalgebra $\mathfrak{osp}(1|2)$ is isomorphic to $\mathfrak{sl}(2)\oplus\mathfrak{a}$, where $\mathfrak{a}=\text{Span}(X_\theta,\,X_{x\theta})$. Consider a linear operator $A:
\mathfrak{a}\rightarrow\mathrm{D}_{\lambda,\mu} $. By a straightforward computation, we show that if $A$ is $\mathfrak{sl}(2)$-invariant, then $\mu=\lambda-\frac{1}{2}+k,$ where $k\in\mathbb{N} $ and the corresponding operator $A_k$ has the following expression $$\label{inv}
A_k(X_{h\theta})(fdx^{\lambda})=a_k\left(h f^{(k)}
+k(2\lambda+k-1)h'f^{(k-1)}\right)dx^{\lambda-\frac{1}{2}+k},$$ where $$k(k-1)(2\lambda+k-1)(2\lambda+k-2)a_k=0.$$
Now, consider $\Upsilon\in
Z_{\mathrm{diff}}^1(\mathcal{K}(1);\frak{D}_{\lambda,\mu})$ such that $\Upsilon(X_1)=\Upsilon(X_x)=\Upsilon(X_{x^2})=0$. The 1-cocycle relations give, for all $h$, $h_1$, $h_2$ polynomial with degree 0 or 1 and $g$ polynomial with degree 0, 1 or 2, the following equations $$\begin{aligned}
\label{sltriv1} &\mathfrak{L}^{\lambda,\mu}_{X_g}\Upsilon(X_{h\theta})-
\Upsilon([X_g,X_{h\theta}])=0, \\
\label{sltriv2} &\mathfrak{L}^{\lambda,\mu}_{X_{h_1\theta}}\Upsilon(X_{h_2\theta}) +
\mathfrak{L}^{\lambda,\mu}_{X_{h_2\theta}} \Upsilon(X_{h_1\theta})=0.\end{aligned}$$ 1) If $\Upsilon$ is an even 1-cocycle, then, according to Propostion \[decom\], its restriction to $\mathfrak{a}$ is decomposed into two maps: $\mathfrak{a}\rightarrow\Pi(\mathrm{D}_{\lambda+\frac{1}{2},\mu})$ and $\mathfrak{a}\rightarrow\Pi(\mathrm{D}_{\lambda,\mu+\frac{1}{2}})$. The equation (\[sltriv1\]) tell us that these maps are $\mathfrak{sl}(2)$-invariant. Therefore, their expressions are given by (\[inv\]). So, we must have $\mu=\lambda+k=(\lambda+\frac{1}{2})-\frac{1}{2}+k$ (and then $\mu+\frac{1}{2}=\lambda-\frac{1}{2}+k+1$). More precisely, using the equation (\[sltriv2\]), we get (up to a scalar factor): $$\Upsilon_{|\mathfrak{osp}(1|2)}=\left\{\begin{aligned} &0~~\text{
if }~k(k-1)(2\lambda+k)(2\lambda+k-1)\neq0~~\text{ or }~k=1 \text{
and }\lambda\notin\{0,\,-\frac{1}{2}\},\\
&\delta(\theta\partial_\theta\partial_x^k)~~\text{ if
}~(\lambda,\mu)= \left(\frac{-k}{2},\frac{k}{2}\right),\\ &\delta
(\partial_x^k-\theta\partial_\theta\partial_x^k)~~\text{ if
}~(\lambda,\mu)= \left(\frac{1-k}{2},\frac{1+k}{2}\right)~~\text{
or }~\lambda=\mu.
\end{aligned}\right.$$ 2) Similarly, if $\Upsilon$ is an odd 1-cocycle, we get: $$\Upsilon_{|\mathfrak{osp}(1|2)}=\left\{\begin{array}{lll} 0&\text{
if }& k(k-1)(2\lambda+k-1)\neq0,\\ \delta(\theta)&\text{ if
}&\mu=\lambda-\frac{1}{2},\\ \delta(\partial_\theta)&\text{ if
}&\mu=\lambda+\frac{1}{2},\\ \delta(\theta\partial_x^k)&\text{ if
}&(\lambda,\mu)=(\frac{1-k}{2},\frac{k}{2}).
\end{array}\right.$$
Now, we can compute $\mathrm{H}^1_{\rm
diff}(\mathcal{K}(1);\mathfrak{D}_{\lambda,\mu})$ and the $\frak{osp}(1|2)$-relative cohomology\
$\mathrm{H}^1_{\mathrm{diff}}(\mathcal{K}(1),\mathfrak{osp}(1|2);\frak{D}_{\lambda,\mu})$. Let $\Upsilon$ be any 1-cocycle over $\mathcal{K}(1)$. According to Proposition \[decom\], we have $$\Upsilon_{|\mathfrak{vect}(1)}\in\mathrm{H}_{\rm
diff}^1\left(\mathfrak{vect}(1);
\mathrm{D}_{\lambda,\mu}\right)\oplus\mathrm{H}_{\rm
diff}^1\left(\mathfrak{vect}(1);
\mathrm{D}_{\lambda+\frac{1}{2},\mu+\frac{1}{2}}\right)\quad\text{if}\quad\Upsilon\quad\text{is
even}$$ and $$\Upsilon_{|\mathfrak{vect}(1)}\in\mathrm{H}_{\rm
diff}^1\left(\mathfrak{vect}(1);\Pi(
\mathrm{D}_{\lambda,\mu+\frac{1}{2}})\right) \oplus\mathrm{H}_{\rm
diff}^1\left(\mathfrak{vect}(1);\Pi(
\mathrm{D}_{\lambda+\frac{1}{2},\mu})\right)\quad\text{if}\quad\Upsilon\quad\text{is
odd}.$$ We know that non-zero cohomology $\mathrm{H}^1_{\rm
diff}\left(\mathfrak{vect}(1);
\mathrm{D}_{\lambda,\lambda'}\right)$ only appear if $\lambda'-\lambda\in\mathbb{N}$. Thus, according to Lemma \[sa\], the following statements hold:
- If $\mu-\lambda\notin{1\over2}(\mathbb{N}-1)$, then $\mathrm{H}^1_{\mathrm{diff}}(\mathcal{K}(1);\frak{D}_{\lambda,\mu})=0$.
- If $\mu-\lambda$ is integer, then $\mathrm{H}^1_{\mathrm{diff}}(\mathcal{K}(1);\frak{D}_{\lambda,\mu})$ is spanned only by the cohomology classes of even cocycles.
- If $\mu-\lambda$ is semi-integer, then $\mathrm{H}^1_{\mathrm{diff}}(\mathcal{K}(1);\frak{D}_{\lambda,\mu})$ is spanned only by the cohomology classes of odd cocycles.
The space $\mathrm{H}^1_{\mathrm{diff}}(\mathcal{K}(1),\mathfrak{osp}(1|2);\frak{D}_{\lambda,\mu})$
---------------------------------------------------------------------------------------------------
The main result of this subsection is the following:
\[th3\] $\rm{dim}\mathrm{H}^1_{\mathrm{diff}}(\mathcal{K}(1),\frak{
osp}(1|2);\frak{D}_{\lambda,\mu})=1$ if $$\begin{array}{llllll}
\mu-\lambda=\frac{3}{2} &\hbox{ and } \lambda\neq-{1\over2},\\
\mu-\lambda=2 &\hbox{ for all } \lambda,\\ \mu-\lambda=\frac{5}{2}
&\hbox{ and } \lambda\neq-1,\\ \mu-\lambda=3 &\hbox{ and }
\lambda\in\{0,\,-\frac{5}{2}\},\\ \mu-\lambda=4 &\hbox{ and }
\lambda=\frac{-7\pm\sqrt{33}}{4}.
\end{array}$$ Otherwise, $\mathrm{H}^1_{\mathrm{diff}}(\mathcal{K}(1),\frak{
osp}(1|2);\frak{D}_{\lambda,\mu})=0$.
The corresponding spaces ${\mathrm
H}^1_{\mathrm{diff}}(\mathcal{K}(1),\frak{
osp}(1|2);\frak{D}_{\lambda,\lambda+\frac{k}{2}})$ are spanned by the cohomology classes of $\Upsilon_{\lambda,\lambda+\frac{k}{2}}=\frak{J}_{\frac{k}{2}+1}^{-1,\lambda}$, where $k\in\{3,\,4,\,5,\,6,\,8\}$.
Note that, by Lemma \[sa\], the $\frak{
osp}(1|2)$-relative cocycles are related to its homologous in the classical setting, and by Lemma \[osp\], they are supertransvectants. Bouarroudj and Ovsienko [@bo] showed that $$\label{h1}
\mathrm{H}^1_{\rm diff}(\mathfrak{vect}(1),{\rm\frak
sl}(2);\mathrm{D}_{\lambda,\lambda+k})\simeq \left\{
\begin{array}{lll}
\mathbb{K} & \hbox{if} ~~\left\{
\begin{array}{l}
k=2 \hbox{ and } \lambda\neq-{1\over2},\\ k=3 \hbox{ and }
\lambda\neq-1,\\ k=4 \hbox{ and } \lambda\neq-\frac{3}{2},\\ k=5
\hbox{ and } \lambda=0,-4,\\ k=6 \hbox{ and } \lambda=-\frac{5\pm
\sqrt{19}}{2},
\end{array}
\right.
\\[16pt]
0 &\hbox{otherwise}.
\end{array}
\right.$$ These spaces are generated by the cohomology classes of the following non-trivial $\frak{sl}(2)$-relative 1-cocycles, $A_{\lambda,\lambda+k}:$ $$\begin{array}{lllllll}
A_{\lambda,\lambda+2}(X,f)=X^{(3)}f,\qquad
\lambda\neq-\frac{1}{2},\\[2pt]
A_{\lambda,\lambda+3}(X,f)=X^{(3)}f'-\frac{\lambda}{2}X^{(4)}f,\qquad \lambda\neq-1, \\[2pt]
A_{\lambda,\lambda+4}(X,f)=X^{(3)}f''-\frac{2\lambda+1}{2}X^{(4)}f'+\frac{\lambda(2\lambda+1)}{10}X^{(5)}f,
\qquad \lambda\neq-\frac{3}{2},\\[2pt]
A_{0,5}(X,f) =-3X^{(5)}f'+ 15X^{(4)}f'' -10X^{(3)}f^{(3)},\\[2pt]
A_{-4,1}(X,f)=28 X^{(6)}f+63X^{(5)}f'+ 45X^{(4)}f''
+10X^{(3)}f^{(3)}\\[2pt]
A_{a_i,a_i+6}(X,f)=\alpha_iX^{(7)}f-14\beta_iX^{(6)}f'-126\gamma_iX^{(5)}f''
-210\tau_iX^{(4)}f^{(3)}+210X^{(3)}f^{4}\\
\end{array}$$ where $\tau_1=-2+\sqrt{19}\quad\text{and}\quad
\tau_2=-2-\sqrt{19}$. The $a_i$, $\alpha_i$, $\beta_i$ and $\gamma_i$ are those given in (\[cocycles\]).
So, we see first that if $2(\mu-\lambda)\notin\{3,\,\dots,\,13\}$, then by Lemma \[sa\], the corresponding cohomology ${\mathrm
H}^1_{\mathrm{diff}}(\mathcal{K}(1),\frak{
osp}(1|2);\frak{D}_{\lambda,\mu})$ vanish. Indeed, let $\Upsilon$ be any element of\
$Z_{\mathrm{diff}}^1(\mathcal{K}(1),\frak{
osp}(1|2);\mathfrak{D}_{\lambda,\mu})$. Then by (\[h1\]) and (\[coho\]), up to a coboundary, the restriction of $\Upsilon$ to $\mathfrak{vect}(1)$ vanishes, so $\Upsilon=0$ by Lemma \[sa\]. By the same arguments, if $2(\mu-\lambda)>9$, generically, the corresponding cohomology vanish.
For $2(\mu-\lambda)\in\{3,\,\dots,\,13\}$, we study the supertranvectant $\frak{J}_{\mu-\lambda+1}^{-1,\lambda}$. If it is a non-trivial 1-cocycle, then the corresponding cohomology space is one-dimensional, otherwise it is zero. To study any supertranvectant $\frak{J}_{\mu-\lambda+1}^{-1,\lambda}$ satisfying $\delta(\frak{J}_{\mu-\lambda+1}^{-1,\lambda})=0$, we consider the two components of its restriction to $\mathfrak{vect}(1)$ which we compare with $A_{\lambda,\mu}$ and $A_{\lambda+{1\over2},\mu+{1\over2}}$ or $A_{\lambda+{1\over2},\mu}$ and $A_{\lambda,\mu+{1\over2}}$ depending on whether $\lambda-\mu$ is integer or semi-integer. For instance, we show that $\frak{J}_{5\over2}^{-1,\lambda}$ is a 1-cocycle. Moreover, it is non-trivial for $\lambda\neq-{1\over2}$ since, for $g,\,f\in\mathbb{K}[x]$, we have $\frak{J}_{5\over2}^{-1,\lambda}(X_g)(f)=-\theta
A_{\lambda,\lambda+2}(g,f)$. More precisely, we get the following non-trivial 1-cocycles: $$\left\{\begin{array}{lllllllll}
\Upsilon_{\lambda,\lambda+\frac{3}{2}}(X_G)(F\alpha^{\lambda})&=
&\overline{\eta}(G'')F\alpha^{\lambda+\frac{3}{2}}\quad\text{
for } \lambda\neq-{1\over2},\\[4pt]
\Upsilon_{\lambda,\lambda+\frac{5}{2}}(X_G)(F\alpha^{\lambda})&=&\left(
2\lambda\overline{\eta}(G^{(3)})F-{3}\overline{\eta}(G'')F'
-(-1)^{p(G)}G^{(3)}\overline{\eta}(F)
\right)\alpha^{\lambda+\frac{5}{2}}~\text{ for }\lambda\neq-1,\\[4pt]
\Upsilon_{\lambda,\lambda+2}(X_G)(F\alpha^{\lambda})&=&\left({2\over3}\lambda
G^{(3)}F-(-1)^{p(G)}\overline{\eta}(G'') \overline{\eta}(F)
\right)\alpha^{\lambda+2}\quad\text{ for all }\lambda,\\[4pt]
\Upsilon_{\lambda,\lambda+3}(X_G)(F\alpha^{\lambda})&=&\Big((-1)^{p(G)}\overline{\eta}(G'')\overline{\eta}(F')
-{2\lambda+1\over3}\big((-1)^{p(G)}\overline{\eta}(G^{(3)})\overline{\eta}(F)
+G^{(3)}F'\big)\\[2pt]&~&+{\lambda(2\lambda+1)\over6}G^{(4)}F
\Big)\alpha^{\lambda+3}\quad \text{ for
}\lambda=0,\,-\frac{5}{2},\\[4pt]
\Upsilon_{\lambda,\lambda+4}(X_G)(F\alpha^{\lambda})&=&\Big((-1)^{p(G)}\overline{\eta}(G'')\overline{\eta}(F'')
-{2(\lambda+1)\over3}\big(2(-1)^{p(G)}\overline{\eta}(G^{(3)})\overline{\eta}(F')
+G^{(3)}F''\big)\\[2pt]&~&+{(\lambda+1)(2\lambda+1)\over6}\big((-1)^{p(G)}\overline{\eta}
(G^{(4)})\overline{\eta}(F)+2G^{(4)}F'\big)\\[2pt]&~&-
{\lambda(\lambda+1)(2\lambda+1)\over15}G^{(5)}F\Big)\alpha^{\lambda+4}\quad\text{
for } \lambda=\frac{-7\pm\sqrt{33}}{4}.\end{array}\right.$$
The space $\mathrm{H}^1_{\rm
diff}(\mathcal{K}(1);\mathfrak{D}_{\lambda,\mu}$)
-------------------------------------------------
\[th1\] ${\rm dim}\mathrm{H}^1_{\rm
diff}(\mathcal{K}(1);\mathfrak{D}_{\lambda,\mu})=1$ if $$\begin{array}{llll}
\mu-\lambda=0 &\hbox{ for all } \lambda,\\
\mu-\lambda=\frac{3}{2}&\hbox{ for all }~\lambda,\\ \mu-\lambda=2
&\hbox{ for all } \lambda,\\ \mu-\lambda=\frac{5}{2}&\hbox{ for
all }~\lambda,\\ \mu-\lambda=3 &\hbox{ and }
\lambda\in\{0,\,-\frac{5}{2}\},\\ \mu-\lambda=4 &\hbox{ and }
\lambda=\frac{-7\pm\sqrt{33}}{4}.
\end{array}$$ ${\rm dim}\mathrm{H}^1_{\rm
diff}(\mathcal{K}(1);\frak{D}_{0,\frac{1}{2}})=2$. Otherwise, $\mathrm{H}^1_{\rm
diff}(\mathcal{K}(1);\frak{D}_{\lambda,\mu})=0$.
The spaces $\mathrm{H}^1_{\rm
diff}(\mathcal{K}(1);\frak{D}_{\lambda,\mu})$ are spanned by the cohomology classes of the $1$-cocycles $\Upsilon_{\lambda,\mu}$ given in Theorem \[th3\] and by the cohomology classes of the following $1$-cocycles: $$\left\{\begin{array}{llllll}\Upsilon_{\lambda,\lambda}(X_G)(F\alpha^{\lambda})&=&G'F\alpha^{\lambda},\\[4pt]
\Upsilon_{0,\frac{1}{2}}(X_G)(F)&=&
\overline{\eta}(G')F\alpha^{{1\over2}},\\[4pt]
\widetilde{\Upsilon}_{0,\frac{1}{2}}(X_G)(F)&=&
\overline{\eta}(G'F)\alpha^{{1\over2}},\\[4pt]
\Upsilon_{-\frac{1}{2},1}(X_G)(F\alpha^{-{1\over2}})&=&\Big(\overline{\eta}(G'')F
+\overline{\eta}(G')F'+(-1)^{p(G)}G''\overline{\eta}(F)\Big)\alpha\\[4pt]
\Upsilon_{-1,\frac{3}{2}}(X_G)(F\alpha^{-1})&=&
\Big((-1)^{p(G)}(G'''\overline{\eta}(F)+2G''\overline{\eta}(F'))+
2\overline{\eta}(G'')F'+\overline{\eta}(G')F''\Big)\alpha^{\frac{3}{2}}.
\end{array}\right.$$
First, we recall the structure of the space ${\rm H}^1_\mathrm{diff}(\frak
{osp}(1|2);\mathfrak{D}_{\lambda,\mu})$ computed in [@bb]: $$\label{hosp}
{\rm H}^1_\mathrm{diff}({\frak
{osp}}(1|2),{\frak{D}}_{\lambda,\mu})\simeq\left\{
\begin{array}{ll}
\mathbb{K}&\makebox{ if }~~\lambda=\mu, \\[2pt]
\mathbb{K}^2 & \hbox{ if }~~\lambda=\frac{1-k}{2},~\mu=\frac{k}{2},~k\in\mathbb{N}\setminus\{0\},\\[2pt]
0&\makebox { otherwise. }
\end{array}
\right.$$
Note that ${\mathrm H}^1_{\mathrm{diff}}(\mathcal{K}(1),\frak{
osp}(1|2);\frak{D}_{\lambda,\mu})\subset{\mathrm
H}^1_{\mathrm{diff}}(\mathcal{K}(1),\frak{D}_{\lambda,\mu}).$ Moreover, if $\mu\neq\lambda$, then by (\[hosp\]) and Lemma \[sl2\] we can see that ${\mathrm
H}^1_{\mathrm{diff}}(\mathcal{K}(1),\frak{
osp}(1|2);\frak{D}_{\lambda,\mu})={\mathrm
H}^1_{\mathrm{diff}}(\mathcal{K}(1);\frak{D}_{\lambda,\mu})$, except for $$\label{sin} (\lambda,\mu)\in\left\{
\begin{array}{ll}
(0,\frac{1}{2}),(-\frac{1}{2},1), (-1,\frac{3}{2}),
(-\frac{3}{2},2),(-2,\frac{5}{2})
\end{array}
\right\}.$$ Indeed, let $\Upsilon$ be any non-trivial element of $Z_{\mathrm{diff}}^1(\mathcal{K}(1),\mathfrak{D}_{\lambda,\mu})$ where $\mu\neq\lambda$. If $(\lambda,\mu)\neq(\frac{1-k}{2},\frac{k}{2})$ where $k\in\mathbb{N}\setminus\{0\}$ then, by (\[hosp\]), we can see that $\Upsilon_{|\mathfrak{osp}(1|2)}$ is trivial, therefore, we deduce by using Lemma \[osp\] that the 1-cocycle $\Upsilon$ defines a non-trivial cohomology class in ${\mathrm
H}^1_{\mathrm{diff}}({\cal K}(1),\frak{
osp}(1|2);\frak{D}_{\lambda,\mu})$. If $(\lambda,\mu)=(\frac{1-k}{2},\frac{k}{2})$ where $k\in\mathbb{N}\setminus\{0\}$ then, by (\[CohSpace2\]), we can see that, up to a coboundary, generically the 1-cocycle $\Upsilon$ vanishes on $\mathfrak{vect}(1)$ and then we conclude by using Lemma \[sl2\] since $\mathfrak{sl}(2)\subset\mathfrak{vect}(1).$
Thus, we need to study only the case $\mu=\lambda$ together with the singular cases (\[sin\]). According to Proposition \[decom\], if $\mu-\lambda$ is integer, then $$\mathrm{H}^1_{\rm diff}\left(\mathfrak{vect}(1);\mathfrak{D}_{\lambda,\mu}\right)
\simeq\mathrm{H}^1_{\rm diff}\left(\mathfrak{vect}(1);
\mathrm{D}_{\lambda,\mu}\right)\oplus\mathrm{H}^1_{\rm
diff}\left(\mathfrak{vect}(1);
\mathrm{D}_{\lambda+\frac{1}{2},\mu+\frac{1}{2}}\right),$$ and if $\mu-\lambda$ is semi-integer, then $$\mathrm{H}^1_{\rm
diff}\left(\mathfrak{vect}(1);\mathfrak{D}_{\lambda,\mu}\right)
\simeq\mathrm{H}^1_{\rm diff}\left(\mathfrak{vect}(1);\Pi(
\mathrm{D}_{\lambda+\frac{1}{2},\mu})\right)
\oplus\mathrm{H}^1_{\rm diff}\left(\mathfrak{vect}(1);\Pi(
\mathrm{D}_{\lambda,\mu+\frac{1}{2}})\right).$$ Thus, we deduce $\mathrm{H}^1_{\rm
diff}(\mathfrak{vect}(1);\mathfrak{D}_{\lambda,\mu})$ from (\[CohSpace2\]).
Now, let $\Upsilon$ be a $1$-cocycle from $\mathcal{K}(1)$ to $\frak{D}_{\lambda,\lambda},$ that is, $\Upsilon$ is even. The map $\Upsilon_{|\mathfrak{vect}(1)}$ is a 1-cocycle of $\mathfrak{vect}(1)$. So, up to a coboundary, we have (here $\alpha,
\beta\in\mathbb{K}$) $$\label{decomp5}
\Phi_{\lambda,\lambda}\circ\Upsilon_{|\mathfrak{vect}(1)}=\alpha
C_{\lambda,\lambda}+ \beta
C_{\lambda+{1\over2},\lambda+{1\over2}}.$$By Lemma \[sa\], the 1-cocycle $\Upsilon$ is non-trivial if and only if $(\alpha,\beta)\neq(0,0)$. By Lemma \[sd\], we can write $$\Upsilon(X_{h\theta})=\sum_{m,k}b_{m,k}h^{(k)}\theta\partial^m_x+
\sum_{m,k}\widetilde{b}_{m,k}h^{(k)}\partial_{\theta}\partial^m_x,$$ where the coefficients $b_{m,k}$ and $\widetilde{b}_{m,k}$ are constants. Moreover, the map $\Upsilon$ must satisfy the following equations $$\label{decomp6}\left\{\begin{array}{lllll}
\Upsilon([X_g,\,X_{
h\theta}])&=&\mathfrak{L}_{X_g}^{\lambda,\lambda}\Upsilon(X_{h\theta
})
-\mathfrak{L}_{X_{h\theta }}^{\lambda,\lambda}\Upsilon(X_g), \\
\Upsilon([X_{ h_1\theta},\,X_{h_2\theta }])&=&\mathfrak{L}_{X_{
h_1\theta}}^{\lambda,\lambda}\Upsilon(X_{h_2\theta })
+\mathfrak{L}_{X_{
h_2\theta}}^{\lambda,\lambda}\Upsilon(X_{h_1\theta }).
\end{array}\right.$$ We solve the equations (\[decomp5\]) and (\[decomp6\]) for $\alpha,\,\beta,\,b_{k,m},\,\widetilde{b}_{m,k}$. We prove that $\mathrm{H}^1_{\rm diff} (\mathcal{K}(1) ;
\frak{D}_{\lambda,\lambda})$ is spanned by the non-trivial cocycle $\Upsilon_{\lambda,\lambda}$ corresponding to the cocycle $$\Phi_{\lambda,\lambda}^{-1}\circ
\left(C_{\lambda,\lambda}+C_{\lambda+{1\over2},\lambda+{1\over2}}\right)$$ via its restriction to $\mathfrak{vect}(1)$, see (\[cocycles\]).
For the singular cases (\[sin\]), by the same arguments as above, we get:
- $\mathrm{H}^1_{\rm diff} (\mathcal{K}(1) ; \frak{D}_{0,{1\over2}})$ is spanned by the non-trivial cocycles $\Upsilon_{0,{1\over2}}$ and $\widetilde{\Upsilon}_{0,{1\over2}}$ corresponding, respectively, to the cocycles $\Phi_{0,{1\over2}}^{-1}\circ
\Pi\circ (-C_{0,1})$ and $ \Phi_{0,{1\over2}}^{-1}\circ\Pi\circ
(C_{{1\over2},{1\over2}}-\widetilde{C}_{0,1})$, via their restrictions to $\mathfrak{vect}(1)$.
- $\mathrm{H}^1_{\rm diff} (\mathcal{K}(1) ;
\frak{D}_{-{1\over2},1})$ is spanned by the non-trivial cocycle $\Upsilon_{-{1\over2},1}$ corresponding to the cocycle $\Phi_{-{1\over2},1}^{-1}\circ\Pi\circ( C_{0,1}-
C_{-{1\over2},\frac{3}{2}})$ via its restriction to $\mathfrak{vect}(1).$
- $\mathrm{H}^1_{\rm
diff} (\mathcal{K}(1) ; \frak{D}_{-1,\frac{3}{2}})$ is spanned by the non-trivial cocycle $\Upsilon_{-1,\frac{3}{2}}$ corresponding to the cocycle $ \Phi_{-1,\frac{3}{2}}^{-1}\circ\Pi\circ(
C_{-{1\over2},\frac{3}{2}}-3C_{-1,2})$ via its restriction to $\mathfrak{vect}(1)$.
- $\mathrm{H}^1_{\rm
diff} (\mathcal{K}(1) ;
\frak{D}_{-\frac{3}{2},2})=\mathrm{H}^1_{\rm diff} (\mathcal{K}(1)
; \frak{D}_{-2,\frac{5}{2}})=0$.
Deformation Theory and Cohomology
=================================
Deformation theory of Lie algebra homomorphisms was first considered with only one-parameter of deformation [@fi; @nr; @r]. Recently, deformations of Lie (super)algebras with multi-parameters were intensively studied ( see, e.g., [@aalo; @abbo; @bbdo; @bb; @or1; @or2]). Here we give an outline of this theory.
Infinitesimal deformations and the first cohomology
---------------------------------------------------
Let $\rho_0 :\frak g \longrightarrow{\rm End}(V)$ be an action of a Lie superalgebra $\frak g$ on a vector superspace $V$ and let $\frak h$ be a subagebra of $\frak g$. When studying $\frak
h$-trivial deformations of the $\frak g$-action $\rho_0$, one usually starts with [*infinitesimal*]{} deformations: $$\label{infdef}
\rho=\rho_0+t\,\Upsilon,$$ where $\Upsilon:\frak g\to{\rm End}(V)$ is a linear map vanishing on $\frak h$ and $t$ is a formal parameter with $p(t)=p(\Upsilon)$. The homomorphism condition $$\label{homocond}
[\rho(x),\rho(y)]=\rho([x,y]),$$ where $x,y\in\frak g$, is satisfied in order 1 in $t$ if and only if $\Upsilon$ is a $\frak h$-relative 1-cocycle. That is, the map $\Upsilon$ satisfies $$(-1)^{p(x)p(\Upsilon)}[\rho_0(x),\Upsilon(y)]-(-1)^{p(y)(p(x)+p(\Upsilon))}[\rho_0(y),
\Upsilon(x)]-\Upsilon([x,~y])=0.$$ Moreover, two $\frak h$-trivial infinitesimal deformations $
\rho=\rho_0+t\,\Upsilon_1, $ and $ \rho=\rho_0+t\,\Upsilon_2, $ are equivalents if and only if $\Upsilon_1-\Upsilon_2$ is $\frak
h$-relative coboundary: $$(\Upsilon_1-\Upsilon_2)(x)=(-1)^{p(x)p(A)}[\rho_0(x),A]:=\delta
A(x),$$ where $A\in{\rm End}(V)^{\frak h}$ and $\delta$ stands for differential of cochains on $\frak g$ with values in $\mathrm{End}(V)$ (see, e.g., [@Fu; @nr]). So, the space $\mathrm{H}^1(\frak g,\frak h;{\rm End}(V))$ determines and classifies infinitesimal deformations up to equivalence. If $\dim{\mathrm{H}^1(\frak g,\frak h;{\rm End}(V))}=m$, then choose 1-cocycles $\Upsilon_1,\ldots,\Upsilon_m$ representing a basis of $\mathrm{H}^1(\frak g,\frak h;{\rm End(V)})$ and consider the infinitesimal deformation $$\label{InDefGen2} \rho=\rho_0+\sum_{i=1}^m{}t_i\,\Upsilon_i,$$ where $t_1,\ldots,t_m$ are independent parameters with $p(t_i)=p(\Upsilon_i)$.
Since we are interested in the $\mathfrak{osp}(1|2)$- trivial deformations of the $\mathcal{K}(1)$-action on ${\frak
S}^n_{\beta}$, we consider the space ${\mathrm H}^1_{\rm
diff}(\mathcal{K}(1),\mathfrak{osp}(1|2);\mathrm{End}({\frak
S}^n_{\beta}))$ spanned by the classes $\Upsilon_{\lambda,\lambda+\frac{k}{2}}$, where $k=3,\,4,\,5$ and $2(\beta-\lambda)\in\left\{k,\,k+1,\, \dots,\,2n\right\}$ for generic $\beta$. Any infinitesimal $\mathfrak{osp}(1|2)$-trivial deformation of the $\mathcal{K}(1)$-module ${\frak S}^n_{\beta}$ is then of the form $$\label{infdef1}
\widetilde{\frak L}_{X_F}=\frak{L}_{X_F}+{\frak L}^{(1)}_{X_F},$$ where $\frak{L}_{X_F}$ is the Lie derivative of ${\frak
S}^n_{\beta}$ along the vector field $X_F$ defined by (\[superaction\]), and $$\label{infpart}
{\frak L}_{X_F}^{(1)}=\sum_{\lambda}\sum_{k=3,4,5}t_{\lambda,
\lambda+\frac{k}{2}}\,
\Upsilon_{\lambda,\lambda+\frac{k}{2}}(X_F),$$ where the $t_{\lambda,\lambda+\frac{k}{2}}$ are independent parameters with $p(t_{\lambda,\lambda+\frac{k}{2}})=p(\Upsilon_{\lambda,\lambda+\frac{k}{2}})$ and $2(\beta-\lambda)\in\left\{k,\,k+1,\, \dots,\,2n\right\}.$
Integrability conditions and deformations\
over supercommutative algebras
------------------------------------------
Consider the supercommutative associative superalgebra with unity $\mathbb{C}[[t_1,\ldots,t_m]]$ and consider the problem of integrability of infinitesimal deformations. Starting with the infinitesimal deformation (\[InDefGen2\]), we look for a formal series $$\label{BigDef2} \rho= \rho_0+\sum_{i=1}^m{}t_i\,\Upsilon_i+
\sum_{i,j}{}t_it_j\,\rho^{(2)}_{ij}+\cdots,$$ where the higher order terms $\rho^{(2)}_{ij},\rho^{(3)}_{ijk},\ldots$ are linear maps from $\frak g$ to ${\rm End(V)}$ with $p(\rho^{(2)}_{ij})=p(t_it_j),
p(\rho^{(3)}_{ijk})=p(t_it_jt_k),\dots$ such that the map $$\label{map} \rho:\frak g\to
\mathbb{C}[[t_1,\ldots,t_m]]\otimes{\rm End(V)},$$ satisfies the homomorphism condition (\[homocond\]).
Quite often the above problem has no solution. Following [@fi] and [@aalo], we will impose extra algebraic relations on the parameters $t_1,\ldots,t_m$. Let ${\cal R}$ be an ideal in $\mathbb{C}[[t_1,\ldots,t_m]]$ generated by some set of relations, and we can speak about deformations with base ${\cal
A}=\mathbb{C}[[t_1,\ldots,t_m]]/{\cal R}$, (for details, see [@fi]). The map (\[map\]) sends $\frak g$ to ${\cal
A}\otimes{\rm End}(V)$.
Setting $$\varphi_t = {\rho}- \rho_0,\,\,
\rho^{(1)}=\sum {}t_i\,\Upsilon_i,\,\,
\rho^{(2)}=\sum{}t_it_j\,\rho^{(2)}_{ij},\,\dots,$$ we can rewrite the relation (\[homocond\]) in the following way: $$\label{developping} [\varphi_t(x) , \rho_0(y) ] + [\rho_0(x) ,
\varphi_t(y) ] - \varphi_t([x , y]) +\sum_{i,j > 0}
\;[\rho^{(i)}(x) , \rho^{(j)}(y)] = 0.$$ The first three terms are $(\delta\varphi_t) (x,y)$. For arbitrary linear maps $\gamma_1,~ \gamma_2 :\mathfrak{g}
\longrightarrow\mathrm{End}(V)$, consider the standard [*cup-product*]{}: $[\![\gamma_1,\gamma_2]\!]:\frak g \otimes \frak g
\longrightarrow \mathrm{End}(V)$ defined by: $$\label{maurrer cartan1} [\![\gamma_1 , \gamma_2]\!] (x , y) =
(-1)^{p(\gamma_2)(p(\gamma_1)+p(x))}[\gamma_1(x) , \gamma_2(y)] +
(-1)^{p(\gamma_1)p(x)}[\gamma_2(x) , \gamma_1(y)].$$
The relation (\[developping\]) becomes now equivalent to: $$\label{maurrer cartan} \delta\varphi_t +{1\over2} [\![\varphi_t ,
\varphi_t]\!]= 0,$$ Expanding (\[maurrer cartan\]) in power series in $t_1,\dots,t_m
$, we obtain the following equation for $\rho^{(k)}$: $$\label{maurrer cartank} \delta\rho^{(k)} + {1\over2}\sum_{i+j=k}
[\![\rho^{(i)} , \rho^{(j)}]\!] = 0.$$
The first non-trivial relation $\delta{\rho^{(2)}} +{1\over2}
[\![\rho^{(1)} , \rho^{(1)}]\!] = 0 $ gives the first obstruction to integration of an infinitesimal deformation. Thus, considering the coefficient of $t_i\,t_j$, we get $$\label{nizar} \delta{\rho^{(2)}_{ij}}+{1\over2}[\![\Upsilon_{i} ,
\Upsilon_{j}]\!]=0.$$ It is easy to check that for any two $1$-cocycles $\gamma_1$ and $\gamma_2 \in Z^1 (\frak g ,\frak h; \mathrm{End}(V))$, the bilinear map $[\![\gamma_1 , \gamma_2]\!]$ is a $\frak h$-relative $2$-cocycle. The relation (\[nizar\]) is precisely the condition for this cocycle to be a coboundary. Moreover, if one of the cocycles $\gamma_1$ or $\gamma_2$ is a $\frak h$-relative coboundary, then $[\![\gamma_1 , \gamma_2]\!]$ is a $\frak
h$-relative $2$-coboundary. Therefore, we naturally deduce that the operation (\[maurrer cartan1\]) defines a bilinear map: $$\label{cup-product} \mathrm{H}^1 (\frak g ,\frak h;\mathrm{End}(
V))\otimes \mathrm{H}^1 (\frak g ,\frak h; \mathrm{End}(
V))\longrightarrow \mathrm{H}^2 (\frak g ,\frak h; \mathrm{End}(
V)).$$
All the obstructions lie in $\mathrm{H}^2 (\frak g, \frak
h;\mathrm{End}(V))$ and they are in the image of $\mathrm{H}^1
(\frak g,\frak h; \mathrm{End}(V))$ under the cup-product.
Equivalence
------------
Two deformations, $\rho$ and $\rho'$ of a $\frak g$-module $V$ over $\cal A$ are said to be [*equivalent*]{} (see [@fi]) if there exists an inner automorphism $\Psi$ of the associative superalgebra ${\cal A}\otimes{\rm End}(V)$ such that $$\Psi\circ\rho=\rho'\hbox{ and } \Psi(\mathbb{I})=\mathbb{I},$$ where $\mathbb{I}$ is the unity of the superalgebra ${\cal
A}\otimes{\rm End}(V).$
The following notion of miniversal deformation is fundamental. It assigns to a $\frak g$-module $V$ a canonical commutative associative algebra $\mathcal{A}$ and a canonical deformation over $\mathrm{A}$. A deformation (\[BigDef2\]) over $\mathcal{A}$ is said to be [*miniversal*]{} if
- for any other deformation $\rho'$ with base (local) $\cal A'$, there exists a homomorphism $\psi:{\cal A}'\to{\cal
A}$ satisfying $\psi(1)=1$, such that $$\rho=(\psi\otimes\mathrm{Id})\circ\rho'.$$
- under notation of (i), if $\rho$ is infinitesimal, then $\psi$ is unique.
If $\rho$ satisfies only the condition (i), then it is called versal. This definition does not depend on the choice 1-cocycles $\Upsilon_1,\ldots,\Upsilon_m$ representing a basis of $\mathrm{H}^1(\frak g,\frak h;{\rm End(V)})$.
The miniversal deformation corresponds to the smallest ideal $\mathcal{R}$. We refer to [@fi] for a construction of miniversal deformations of Lie algebras and to [@aalo] for miniversal deformations of $\mathfrak{g}$-modules. Superization of these results is immediate: by the Sign Rule.
Integrability Conditions
=========================
In this section we obtain the integrability conditions for the infinitesimal deformation(\[infdef1\]).
\[th2\] The second-order integrability conditions of the infinitesimal deformation (\[infdef1\]) are the following: $$\label{ord2} t_{\lambda,
\lambda+\frac{5}{2}}\,t_{\lambda+\frac{5}{2},
\lambda+5}=0,\quad\text{where}\quad
2(\beta-\lambda)\in\left\{10,\, \dots,\,2n\right\}.$$
To prove Proposition \[th2\], we need the following lemmas
\[trans1\] Consider a linear differential operator $b:\mathcal{K}(1)\longrightarrow\frak{D}_{\lambda,\mu}$. If $b$ satisfies $$\delta(b)(X,Y)=b(X)=0\hbox{ for all }{X\in\frak {osp}}(1|2),$$ then $b$ is a supertransvectant.
For all $X,\,Y\in\mathcal{K}(1)$ we have $$\delta(b)(X,Y):=(-1)^{p(X)p(b)}\mathfrak{L}^{\lambda,\mu}_{X}(b(Y))-
(-1)^{p(Y)(p(X)+p(b))}\mathfrak{L}^{\lambda,\mu}_{Y}(b(X))-b([X,Y]).$$ Since $ \delta(b)(X,Y)=b(X)=0\hbox{ for all }X\in\frak {osp}(1|2)
$ we deduce that $$(-1)^{p(X)p(b)}\mathfrak{L}^{\lambda,\mu}_{X}(b(Y))-b([X,Y])=0.$$ Thus, the map ${b}$ is $\frak{ osp}(1|2)$-invariant.
\[lth2\]The map $B_{\lambda,\lambda+5}=[\![\Upsilon_{\lambda+\frac{5}{2},\lambda+5}
,\, \Upsilon_{\lambda,\lambda+\frac{5}{2}} ]\!]$ is a non-trivial $\frak{osp}(1|2)$-relative 2-cocycle for $\lambda\neq
0,\,-1,\,-\frac{7}{2},\,-\frac{9}{2}$.
First, observe that for $\lambda=-1, -\frac{7}{2}$, the map $B_{\lambda,\lambda+5}$ is not defined (see Theorem \[th3\]). The map $B_{\lambda,\lambda+5}$ is the cup-product of two $\frak{osp}(1|2)$-relative 1-cocycles, so, $B_{\lambda,\lambda+5}$ is a $\frak{osp}(1|2)$-relative 2-cocycle: $B_{\lambda,\lambda+5}\in
\mathrm{Z}^2(\mathcal{K}(1),\frak{osp}(1|2);\frak{D}_{\lambda,\lambda+5})$. This 2-cocycle is trivial if and only if it is the coboundary of a linear differential operator $$b_{\lambda,\lambda+5}:\mathcal{K}(1)\longrightarrow
\frak{D}_{\lambda,\lambda+5}$$ vanishing on $\mathfrak{osp}(1|2)$. Consider $b_{\lambda,\lambda+5}$ as a bilinear map $\mathfrak{F}_{-1}\otimes\mathfrak{F}_\lambda\longrightarrow\mathfrak{F}_{\lambda+5}$. So, according to Lemma \[trans1\] and Theorem \[main\], the operator ${b}_{\lambda,\lambda+5}$ coincides (up to a scalar factor) with the supertransvectant $\frak{J}_{6}^{-1,\lambda}$. But, by a direct computation, we have, up to a multiple $$\begin{array}{llll}
B_{\lambda,\lambda+5}(X_{g_1},\,X_{g_2})(F\alpha^\lambda)&=&
\left(g_1^{(4)}g_2^{(3)}-g_1^{(3)}g_2^{(4)}\right)\left(2\lambda
f_0-(2\lambda+9)f_1\theta\right)\alpha^{\lambda+5},\\[10pt]
\delta(\frak{J}_{6}^{-1,\lambda})(X_{g_1},\,X_{g_2})(F\alpha^\lambda)&=&
\left(g_1^{(3)}g_2^{(4)}-g_1^{(4)}g_2^{(3)}\right)
\Big(\frac{\lambda(2\lambda+3)(\lambda^2+6\lambda+8)}{9}f_0+\\[10pt]
&&+\frac{(2\lambda+1)(\lambda+3)(4\lambda^2+28\lambda+45)}{36}f_1\theta\Big)\alpha^{\lambda+5},
\end{array}$$ where $g_1, g_2\in\mathbb{K}[x]$ and $F=f_0+f_1\theta\in\mathbb{K}[x,\theta]$. Therefore, the restrictions of the maps $B_{\lambda,\lambda+5}$ and $\delta(\frak{J}_{6}^{-1,\lambda})$ to $\mathfrak{vect}(1)\times \mathfrak{vect}(1)$ are linearly dependant if and only if $$\lambda(\lambda+1)(2\lambda+7)(2\lambda+9)(4\lambda+9)=0.$$ Thus, the maps $B_{\lambda,\lambda+5}$ and $\delta(\frak{J}_{6}^{-1,\lambda})$ are linearly independent for $\lambda\neq0,-1,\,-\frac{7}{2},\,-\frac{9}{2},\,-\frac{9}{4}$. Besides, we check that the maps $B_{-\frac{9}{4},\frac{11}{4}}$ and $\delta(\frak{J}_{6}^{-1,-\frac{9}{4}})$ are also linearly independent although their restrictions to $\mathfrak{vect}(1)\times \mathfrak{vect}(1)$ are linearly dependant. Finally, for $\lambda=0,-\frac{9}{2}$, we check that $B_{\lambda,\lambda+5}$ coincides (up to a scalar factor) with $\delta(\frak{J}_{6}^{-1,\lambda})$. This completes the proof.
[The map $B_{\lambda,\lambda+5}:\mathcal{K}(1)\otimes\mathcal{K}(1)
\longrightarrow\frak{D}_{\lambda,\lambda+5}$ is a non-trivial 2-cocycle, so, $\mathrm{H}^2_{\mathrm{diff}}(\mathcal{K}(1),\frak{
osp}(1|2);\frak{D}_{\lambda,\lambda+5})\neq 0$ while ${\mathrm
H}^2_{\rm diff}(\mathfrak{vect}(1),{\rm \frak{sl}}(2);{\rm
D}_{\lambda,\lambda+5})=0$ for generic $\l$ (see [@b]). Hence, for the second cohomology, the analog of Lemma \[sa\] is not true.]{}
of Proposition \[th2\]: Assume that the infinitesimal deformation (\[infdef1\]) can be integrated to a formal deformation: $$\widetilde{\frak L}_{X_F}=\frak{L}_{X_F}+{\frak
L}^{(1)}_{X_F}+{\frak L}^{(2)}_{X_F}+\cdots$$ The homomorphism condition gives, for the term ${\frak
L}^{(2)}_{\lambda,\mu,\lambda',\mu'}$ in $t_{\lambda,\mu}t_{\lambda',\mu'}$, the following equation $$\label{cap}\delta({\frak
L}^{(2)}_{\lambda,\mu,\lambda',\mu'})=-[\![\Upsilon_{\lambda,\mu},\,
\Upsilon_{\lambda',\mu'}]\!].$$ For arbitrary $\lambda$, the right hand side of (\[cap\]) yields the following 2-cocycles: $$\label{2-cocyc}\begin{array}{llllll}
B_{\lambda,\lambda+3}&=&[\![\Upsilon_{\lambda+\frac{3}{2},\lambda+3},\,
\Upsilon_{\lambda,\lambda+\frac{3}{2}}
]\!]&:&\mathcal{K}(1)\otimes\mathcal{K}(1)\rightarrow\frak{D}_{\lambda,\lambda+3},\\
B_{\lambda,\lambda+{7\over2}}&=&[\![\Upsilon_{\lambda+\frac{3}{2},\lambda+{7\over2}},\,
\Upsilon_{\lambda,\lambda+\frac{3}{2}}
]\!]&:&\mathcal{K}(1)\otimes\mathcal{K}(1)\rightarrow\frak{D}_{\lambda,\lambda+{7\over2}},\\
\widetilde{B}_{\lambda,\lambda+{7\over2}}&=&[\![\Upsilon_{\lambda+2,\lambda+{7\over2}},\,
\Upsilon_{\lambda,\lambda+2}
]\!]&:&\mathcal{K}(1)\otimes\mathcal{K}(1)\rightarrow\frak{D}_{\lambda,\lambda+{7\over2}},\\
B_{\lambda,\lambda+4}&=&[\![\Upsilon_{\lambda+\frac{3}{2},\lambda+4}
,\, \Upsilon_{\lambda,\lambda+\frac{3}{2}}
]\!]&:&\mathcal{K}(1)\otimes\mathcal{K}(1)\rightarrow\frak{D}_{\lambda,\lambda+4},\\
\widetilde{B}_{\lambda,\lambda+4}&=&[\![\Upsilon_{\lambda+\frac{5}{2},\lambda+4}
,\, \Upsilon_{\lambda,\lambda+\frac{5}{2}}
]\!]&:&\mathcal{K}(1)\otimes\mathcal{K}(1)\rightarrow\frak{D}_{\lambda,\lambda+4},\\
\overline{B}_{\lambda,\lambda+4}&=&[\![\Upsilon_{\lambda+2,\lambda+4}
,\, \Upsilon_{\lambda,\lambda+2}
]\!]&:&\mathcal{K}(1)\otimes\mathcal{K}(1)\rightarrow\frak{D}_{\lambda,\lambda+4},\\
B_{\lambda,\lambda+{9\over2}}&=&[\![\Upsilon_{\lambda+\frac{5}{2},\lambda+{9\over2}}
,\, \Upsilon_{\lambda,\lambda+\frac{5}{2}}
]\!]&:&\mathcal{K}(1)\otimes\mathcal{K}(1)\rightarrow\frak{D}_{\lambda,\lambda+{9\over2}},\\
\widetilde{B}_{\lambda,\lambda+{9\over2}}&=&[\![\Upsilon_{\lambda+2,\lambda+{9\over2}}
,\, \Upsilon_{\lambda,\lambda+2}
]\!]&:&\mathcal{K}(1)\otimes\mathcal{K}(1)\rightarrow\frak{D}_{\lambda,\lambda+{9\over2}},\\
B_{\lambda,\lambda+5}&=&[\![\Upsilon_{\lambda+\frac{5}{2},\lambda+5}
,\, \Upsilon_{\lambda,\lambda+\frac{5}{2}}
]\!]&:&\mathcal{K}(1)\otimes\mathcal{K}(1)\rightarrow\frak{D}_{\lambda,\lambda+5}.\end{array}$$
The necessary integrability conditions for the second-order terms ${\frak L}^{(2) }$ are that each 2-cocycle $B_{\lambda,\lambda+k}$, where $2k=6,\,7,\,8,\,9,\,10$, must be a coboundary of a linear differential operator $b_{\lambda,\lambda+k}:\mathcal{K}(1)\longrightarrow
\frak{D}_{\lambda,\lambda+k},$ vanishing on $\mathfrak{osp}(1|2)$. More precisely, as in the proof of Lemma \[lth2\], the operator ${b}_{\lambda,\lambda+k}$ coincides (up to a scalar factor) with the supertransvectant $\frak{J}_{k+1}^{-1,\lambda}$. Clearly, $$\widetilde{B}_{\lambda,\lambda+4}=
B_{\lambda,\lambda+4}=3\overline{B}_{\lambda,\lambda+4},\quad
{B}_{\lambda,\lambda+{7\over2}}=-\widetilde{B}_{\lambda,\lambda+{7\over2}},\quad
{B}_{\lambda,\lambda+{9\over2}}=-\widetilde{B}_{\lambda,\lambda+{9\over2}}$$ and, by a direct computation, we check that $$\begin{array}{lll}
B_{\lambda,\lambda+3}(X_{G_1},X_{G_2})(F\alpha^\lambda)&=&\left(-2(-1)^{p(G_1)}\overline{\eta}(G_1'')\overline{\eta}(G_2'')F\right)\alpha^{\lambda+3},\\[10pt]
{B}_{\lambda,\lambda+{7\over2}}(X_{G_1},X_{G_2})(F\alpha^\lambda)&=&
\Big({2\lambda\over3}
\left((-1)^{p(G_1)}G_1^{(3)}\overline{\eta}(G_2'')-\overline{\eta}(G_1'')G_2^{(3)}\right)F+\\
&&2(-1)^{p(G_2)}\overline{\eta}(G_1'')\overline{\eta}(G_2'')\overline{\eta}(F)\Big)
\alpha^{\lambda+{7\over2}},\\[10pt]
B_{\lambda,\lambda+4}(X_{G_1},X_{G_2})(F\alpha^\lambda)&=&
\Big(-2\lambda(-1)^{p(G_1)}\left(\overline{\eta}(G_1^{(3)})\overline{\eta}(G_2'')+
\overline{\eta}(G_1'')\overline{\eta}(G_2^{(3)})\right)F+\\
&&(-1)^{p(G_2)}\left((-1)^{p(G_1)}\overline{\eta}(G_1'')G_2^{(3)}-G_1^{(3)}
\overline{\eta}(G_2'')\right)\overline{\eta}(F)+\\
&&6(-1)^{p(G_1)}\overline{\eta}(G_1'')\overline{\eta}(G_2'')F'\Big)
\alpha^{\lambda+4},\\[10pt]
{B}_{\lambda,\lambda+{9\over2}}(X_{G_1},X_{G_2})(F\alpha^\lambda)&=&\Big(
{2\lambda\over3}(2\lambda+5)\left((-1)^{p(G_1)}G_1^{(3)}\overline{\eta}(G_2^{(3)})-
\overline{\eta}(G_1^{(3)})G_2^{(3)}\right)F+\\ &&
2\lambda\left(\overline{\eta}(G_1'')G_2^{(4)}-(-1)^{p(G_1)p(G_2)}
\overline{\eta}(G_2'')G_1^{(4)}\right)F+\\
&&(2\lambda+1)(-1)^{p(G_2)}\left(\overline{\eta}(G_1'')\overline{\eta}(G_2^{(3)})+
\overline{\eta}(G_1^{(3)})\overline{\eta}(G_2'')\right)\overline{\eta}(F)+\\
&&(2\lambda+1)\left(\overline{\eta}(G_1'')G_2^{(3)}-(-1)^{p(G_1)}G_1^{(3)}
\overline{\eta}(G_2'')\right)F'-\\
&&6(-1)^{p(G_2)}\overline{\eta}(G_1'')\overline{\eta}(G_2'')
\overline{\eta}(F')\Big)\alpha^{\lambda+{9\over2}},
\end{array}$$ where $G_1,G_2,F\in\mathbb{K}[x,\theta].$ Besides, we can see that $$\begin{gathered}
\zeta_\lambda
B_{\lambda,\lambda+3}=\delta(\frak{J}_{4}^{-1,\lambda}),\quad
\text{ where }\quad \zeta_\lambda= {\lambda(2\lambda+5)\over
4}\,\begin{pmatrix}2\lambda+3\\2\end{pmatrix},\\ \alpha_\lambda
B_{\lambda,\lambda+{7\over2}}=\delta(\frak{J}_{9\over2}^{-1,\lambda}),\quad\text{
where }\quad \alpha_\lambda=
{6\lambda+9\over4}\,\begin{pmatrix}2\lambda+4\\3\end{pmatrix},\\
\beta_\lambda
B_{\lambda,\lambda+4}=\delta(\frak{J}_{5}^{-1,\lambda}),\quad\text{
where }\quad \beta_\lambda=
{2\lambda^2+7\lambda+2\over6}\,\begin{pmatrix}2\lambda+4\\2\end{pmatrix},\\
\gamma_\lambda
B_{\lambda,\lambda+{9\over2}}=\delta(\frak{J}_{11\over2}^{-1,\lambda}),\quad\text{
where }\quad \gamma_\lambda= {3\lambda+6\over
2}\,\begin{pmatrix}2\lambda+5\\3\end{pmatrix}.\end{gathered}$$
Now, by Lemma \[lth2\], $B_{\lambda,\lambda+5}$ is a non-trivial $\frak{osp}(1|2)$-relative 2-cocycle, so, its coefficient must vanish, that is, we get the first set of necessary integrability conditions: $$\label{c1}t_{\lambda,
\lambda+\frac{5}{2}}\,t_{\lambda+\frac{5}{2},
\lambda+5}=0,\quad\text{where}\quad
2(\beta-\lambda)\in\left\{10,\, \dots,\,2n\right\}.$$ The equations (\[c1\]) are the unique integrability conditions for the 2nd order term ${\frak L}^{(2) }$. Under these conditions, the second-order term ${\frak L}^{(2) }$ can be given by $$\begin{array}{lll}{\frak L}^{(2) }=
&-\sum_\lambda\zeta_\lambda^{-1}t_{\lambda+\frac{3}{2},\lambda+3}
t_{\lambda,\lambda+\frac{3}{2}} \frak{J}_4^{-1,\lambda}\\[2pt]
&-\sum_\lambda\alpha_\lambda^{-1}(t_{\lambda+\frac{3}{2},\lambda+\frac{7}{2}}
t_{\lambda,\lambda+\frac{3}{2}}-
t_{\lambda+2,\lambda+\frac{7}{2}}t_{\lambda,\lambda+2})
\frak{J}_\frac{9}{2}^{-1,\lambda}\\[2pt]&-\sum_\lambda\beta_\lambda^{-1}(t_{\lambda+\frac{3}{2},\lambda+4}
t_{\lambda,\lambda+\frac{3}{2}}+
t_{\lambda+\frac{5}{2},\lambda+4}t_{\lambda,\lambda+\frac{5}{2}}+{1\over3}t_{\lambda+2,\lambda+4}t_{\lambda,\lambda+2})
\frak{J}_{5}^{-1,\lambda}\\[2pt]
&-\sum_\lambda\gamma_\lambda^{-1}(t_{\lambda+\frac{5}{2},\lambda+\frac{9}{2}}
t_{\lambda,\lambda+\frac{5}{2}}-
t_{\lambda+2,\lambda+\frac{9}{2}}t_{\lambda,\lambda+2})
\frak{J}_\frac{11}{2}^{-1,\lambda}.\end{array}$$
To compute the third term ${\frak L}^{(3) }$, we need the following two lemmas which we can check by a direct computation with the help of [*Maple*]{}.
\[benfraj1\] $$\begin{array}{llllll}
1)~\xi_{\lambda}^{-1}\,\delta(\frak{J}_{11\over2}^{-1,\lambda})&=&
\left(\zeta_\lambda^{-1}[\![\Upsilon_{\lambda+3,\lambda+\frac{9}{2}},\
\frak{J}_4^{-1,\lambda}
]\!]+\zeta_{\lambda+\frac{3}{2}}^{-1}\,[\![\frak{J}_4^{-1,\lambda+\frac{3}{2}},
\,\Upsilon_{\lambda,\lambda+\frac{3}{2}}
]\!]\right),\\[8pt]
2)~
\alpha_{\lambda+\frac{3}{2}}^{-1}[\![\frak{J}_{\frac{9}{2}}^{-1,\lambda+\frac{3}{2}},\
\Upsilon_{\lambda,\lambda+\frac{3}{2}}
]\!]&=&\epsilon_{1,\lambda}\,\alpha_{\lambda}^{-1}\,[\![\Upsilon_{\lambda+\frac{7}{2},\lambda+5},\
\frak{J}_{\frac{9}{2}}^{-1,\lambda}
]\!]+\epsilon_{2,\lambda}\,\zeta_{\lambda}^{-1}\,[\![\Upsilon_{\lambda+3,\lambda+5},\
\frak{J}_{4}^{-1,\lambda}
]\!]+\\&&\epsilon_{3,\lambda}\,\delta(\frak{J}_{6}^{-1,\lambda}),\\[8pt]
3)~\beta_{\lambda+\frac{3}{2}}^{-1}[\![\frak{J}_{5}^{-1,\lambda+\frac{3}{2}},\
\Upsilon_{\lambda,\lambda+\frac{3}{2}}
]\!]&=&\epsilon_{4,\lambda}\beta_{\lambda}^{-1}[\![\Upsilon_{\lambda+4,\lambda+\frac{11}{2}},
\frak{J}_{5}^{-1,\lambda}
]\!]+\epsilon_{5,\lambda}\,\alpha_{\lambda}^{-1}\,[\![\Upsilon_{\lambda+\frac{7}{2},\lambda+\frac{11}{2}},\
\frak{J}_{\frac{9}{2}}^{-1,\lambda}
]\!]+\\&&\epsilon_{6,\lambda}\,\alpha_{\lambda+2}^{-1}\,[\![\frak{J}_{\frac{9}{2}}^{-1,\lambda+2},\
\Upsilon_{\lambda,\lambda+2}]\!],\\[8pt]
4)~
\alpha_\lambda^{-1}[\![\Upsilon_{\lambda+\frac{7}{2},\lambda+6},\
\frak{J}_{\frac{9}{2}}^{-1,\lambda}
]\!]&=&\epsilon_{7,\lambda}\,\beta_{\lambda+2}^{-1}\,[\![\frak{J}_5^{-1,\lambda+2},\
\Upsilon_{\lambda,\lambda+2}
]\!]+\epsilon_{8,\lambda}\,\alpha_{\lambda+\frac{5}{2}}^{-1}\,[\![\frak{J}_{\frac{9}{2}}^{-1,\lambda+\frac{5}{2}},\
\Upsilon_{\lambda,\lambda+\frac{5}{2}}
]\!]+\\&&\epsilon_{9,\lambda}\,\delta(\frak{J}_{7}^{-1,\lambda}),\\[8pt]
5)~\gamma_{\lambda+2}^{-1}[\![\frak{J}_{\frac{11}{2}}^{-1,\lambda+2},\
\Upsilon_{\lambda,\lambda+2}
]\!]&=&\epsilon_{10,\lambda}\,\beta_\lambda^{-1}\,[\![\Upsilon_{\lambda+4,\lambda+\frac{13}{2}},\
\frak{J}_{5}^{-1,\lambda}
]\!]+\epsilon_{11,\lambda}\,\beta_{\lambda+\frac{5}{2}}^{-1}[\![\frak{J}_{5}^{-1,\lambda+\frac{5}{2}},\
\Upsilon_{\lambda,\lambda+\frac{5}{2}} ]\!]+\\&&
\epsilon_{12,\lambda}\,\gamma_{\lambda}^{-1}[\![\Upsilon_{\lambda+\frac{9}{2},\lambda+\frac{13}{2}},\
\frak{J}_{\frac{11}{2}}^{-1,\lambda} ]\!],
\end{array}$$ where $$\small{
\begin{array}{llllll}
\epsilon_{1,\lambda}&=&\frac{(2\lambda+11)(2\lambda+9)(\lambda+2)}{2(\lambda+3)(2\lambda^2+3\lambda-17)},\quad
&\epsilon_{7,\lambda}&=&-\frac{5(6\lambda^2+33\lambda+17)(2\lambda^2+15\lambda+24)}{2(\lambda+5)(\lambda+2)(2\lambda-3)(2\lambda^2+13\lambda+13)},\\[4pt]
\epsilon_{2,\lambda}&=&\frac{15(2\lambda+5)(\lambda+4)}{2(\lambda+3)(2\lambda^2+3\lambda-17)}\quad
&\epsilon_{8,\lambda}&=&\frac{(2\lambda+9)(2\lambda+5)(\lambda+7)(2\lambda^2+11\lambda+4)}{2(\lambda+5)(2\lambda-3)(\lambda+2)(2\lambda^2+13\lambda+13)},\\[4pt]
\epsilon_{3,\lambda}&=&\frac{48}{\lambda(\lambda+3)(2\lambda+3)(2\lambda+5)(2\lambda^2+3\lambda-17)},\quad
&\epsilon_{9,\lambda}&=&\frac{60}{(2\lambda+3)(2\lambda-3)(\lambda+3)(\lambda+4)(2\lambda^2+13\lambda+13)},\\[4pt]
\epsilon_{4,\lambda}&=&-\frac{(2\lambda+9)(2\lambda+3)(2\lambda^2+7\lambda+2)}{(2\lambda+7)(2\lambda+1)(2\lambda^2+13\lambda+17)}\quad
&\epsilon_{10,\lambda}&=&-\frac{(\lambda+5)(\lambda+2)(2\lambda^2+7\lambda+2)}{9(\lambda+4)^2(\lambda+1)},\\[4pt]
\epsilon_{5,\lambda}&=&-\frac{3(2\lambda+9)(2\lambda+3)^2}{2(2\lambda+7)(2\lambda+1)(2\lambda^2+13\lambda+17)}\quad
&\epsilon_{11,\lambda}&=&-\frac{(2\lambda^2+17\lambda+32)}{9(\lambda+4)},\\[4pt]
\epsilon_{6,\lambda}&=&-\frac{3(2\lambda+7)}{2(2\lambda^2+13\lambda+17)}\quad
&\epsilon_{12,\lambda}&=&-\frac{(\lambda+2)^2(\lambda+5)}{(\lambda+4)^2(\lambda+1)},\\[4pt]
\xi_{\lambda}&=& {3\over
16}\lambda(\lambda+4)(2\lambda+3)(2\lambda+5)\,\begin{pmatrix}2\lambda+5\\3\end{pmatrix}.
\end{array}}$$
\[benfraj2\] Each of the following systems is linearly independent $$\begin{array}{llllllllll}
1)\,&\left( [\![\Upsilon_{\lambda+\frac{7}{2},\lambda+5},\
\frak{J}_\frac{9}{2}^{-1,\lambda} ]\!],\,\,
[\![\Upsilon_{\lambda+3,\lambda+5},\ \frak{J}_4^{-1,\lambda}
]\!],\,\, [\![\frak{J}_4^{-1,\lambda+2},
\,\Upsilon_{\lambda,\lambda+2} ]\!],\,\,
\delta(\frak{J}_{6}^{-1,\lambda})\right),\\[2pt]
2)\,&\Big([\![\Upsilon_{\lambda+4,\lambda+\frac{11}{2}},\
\frak{J}_{5}^{-1,\lambda} ]\!],\,\,
[\![\Upsilon_{\lambda+\frac{7}{2},\lambda+\frac{11}{2}},\
\frak{J}_\frac{9}{2}^{-1,\lambda} ]\!],\,\,
[\![\frak{J}_\frac{9}{2}^{-1,\lambda+2},
\,\Upsilon_{\lambda,\lambda+2}
]\!],\,\,[\![\Upsilon_{\lambda+3,\lambda+\frac{11}{2}},\
\frak{J}_{4}^{-1,\lambda} ]\!]\\[2pt] &~~
[\![\frak{J}_{4}^{-1,\lambda+\frac{5}{2}},
\,\Upsilon_{\lambda,\lambda+\frac{5}{2}} ]\!],\,\,
\delta(\frak{J}_{13\over2}^{-1,\lambda})\Big),\\[2pt]
3)\,&\Big([\![\Upsilon_{\lambda+\frac{9}{2},\lambda+6},\
\frak{J}_\frac{11}{2}^{-1,\lambda} ]\!],\,\,[\![
\frak{J}_\frac{11}{2}^{-1,\lambda+\frac{3}{2}},
\,\Upsilon_{\lambda,\lambda+\frac{3}{2}} ]\!],\,\,
[\![\Upsilon_{\lambda+4,\lambda+6},\ \frak{J}_{5}^{-1,\lambda}
]\!],\,\,[\![ \frak{J}_{5}^{-1,\lambda+2},
\,\Upsilon_{\lambda,\lambda+2}
]\!],\,\,\\[2pt]&~~
[\![\frak{J}_\frac{9}{2}^{-1,\lambda+\frac{5}{2}},
\,\Upsilon_{\lambda,\lambda+\frac{5}{2}} ]\!],\,\,
\delta(\frak{J}_{7}^{-1,\lambda})\Big),\\[2pt]
4)\,&\left([\![\Upsilon_{\lambda+4,\lambda+\frac{13}{2}},\
\frak{J}_{5}^{-1,\lambda} ]\!],\, \,[\![
\frak{J}_{5}^{-1,\lambda+\frac{5}{2}},
\,\Upsilon_{\lambda,\lambda+\frac{5}{2}}
]\!],\,\,[\![\Upsilon_{\lambda+{9\over2},\lambda+\frac{13}{2}},\
\frak{J}_{11\over2}^{-1,\lambda} ]\!],\,\,
\delta(\frak{J}_{15\over2}^{-1,\lambda})\right),\\[2pt]
5)\,&\left([\![\Upsilon_{\lambda+{9\over2},\lambda+7},\
\frak{J}_{11\over2}^{-1,\lambda} ]\!],\,\, [\![
\frak{J}_{11\over2}^{-1,\lambda+{5\over2}},
\,\Upsilon_{\lambda,\lambda+{5\over2}} ]\!],\,\,
\delta(\frak{J}_{8}^{-1,\lambda})\right).
\end{array}$$
Now, we are in position to exhibit the 3rd order integrability conditions.
\[pr3\] The 3rd order integrability conditions of the infinitesimal deformation (\[infdef1\]) are the following:
- For $2(\beta-\lambda)\in\left\{10,\,
\dots,\,2n\right\}:$ $$\begin{gathered}
t_{\lambda,\lambda+\frac{3}{2}}\left(\epsilon_{1,\lambda}\,
t_{\lambda+3,\lambda+5}
t_{\lambda+\frac{3}{2},\lambda+3}+(1-\epsilon_{1,\lambda})\,
t_{\lambda+\frac{7}{2},\lambda+5}
t_{\lambda+\frac{3}{2},\lambda+\frac{7}{2}}
\right)=0,\notag\\
t_{\lambda,\lambda+\frac{3}{2}}\left(\epsilon_{2,\lambda}\,t_{\lambda+\frac{7}{2},\lambda+5}
t_{\lambda+\frac{3}{2},\lambda+\frac{7}{2}}-(1+\epsilon_{2,\lambda})\,
t_{\lambda+3,\lambda+5}
t_{\lambda+\frac{3}{2},\lambda+3}\right)=0,\notag\\
t_{\lambda+\frac{7}{2},\lambda+5}t_{\lambda+2,\lambda+\frac{7}{2}}
t_{\lambda,\lambda+2}=0.\end{gathered}$$
- For $2(\beta-\lambda)\in\left\{11,\,
\dots,\,2n\right\}:$ $$\begin{gathered}
t_{\lambda+4,\lambda+\frac{11}{2}}\,t_{\lambda+\frac{5}{2},\lambda+4}
t_{\lambda,\lambda+\frac{5}{2}}= t_{\lambda,\lambda+\frac{3}{2}}
t_{\lambda+3,\lambda+\frac{11}{2}}
t_{\lambda+\frac{3}{2},\lambda+3}=0,\\
t_{\lambda+4,\lambda+\frac{11}{2}}\left(3(1+\epsilon_{4,\lambda})\,t_{\lambda+\frac{3}{2},\lambda+4}
t_{\lambda,\lambda+\frac{3}{2}}+
t_{\lambda+2,\lambda+4}t_{\lambda,\lambda+2}\right)+\\
\quad+\,\epsilon_{4,\lambda}\, t_{\lambda,\lambda+\frac{3}{2}}
t_{\lambda+\frac{7}{2},\lambda+\frac{11}{2}}t_{\lambda+\frac{3}{2},\lambda+\frac{7}{2}}
=0,\\
t_{\lambda+\frac{7}{2},\lambda+\frac{11}{2}}
\left((1+\frac{\epsilon_{5,\lambda}}{3})\,t_{\lambda+\frac{3}{2},\lambda+\frac{7}{2}}
t_{\lambda,\lambda+\frac{3}{2}}-
t_{\lambda+2,\lambda+\frac{7}{2}}t_{\lambda,\lambda+2}\right)+\\
\quad+\,\epsilon_{5,\lambda}\,t_{\lambda+4,\lambda+\frac{11}{2}}t_{\lambda+\frac{3}{2},\lambda+4}
t_{\lambda,\lambda+\frac{3}{2}}=0,\\
\epsilon_{6,\lambda}\,t_{\lambda,\lambda+\frac{3}{2}}\left(
t_{\lambda+4,\lambda+\frac{11}{2}}t_{\lambda+\frac{3}{2},\lambda+4}+\frac{1}{3}\,
t_{\lambda+\frac{7}{2},\lambda+\frac{11}{2}}t_{\lambda+\frac{3}{2},\lambda+\frac{7}{2}}\right)+\\
\quad
+t_{\lambda,\lambda+2}\left(t_{\lambda+\frac{7}{2},\lambda+\frac{11}{2}}
t_{\lambda+2,\lambda+\frac{7}{2}}-
t_{\lambda+4,\lambda+\frac{11}{2}}t_{\lambda+2,\lambda+4}\right)=0.\end{gathered}$$
- For $2(\beta-\lambda)\in\left\{12,\,
\dots,\,2n\right\}:$$$\begin{gathered}
t_{\lambda+\frac{9}{2},\lambda+6}
\left(t_{\lambda+\frac{5}{2},\lambda+\frac{9}{2}}
t_{\lambda,\lambda+\frac{5}{2}}-
t_{\lambda+2,\lambda+\frac{9}{2}}t_{\lambda,\lambda+2}\right)=0,\\
t_{\lambda,\lambda+\frac{3}{2}} \left(t_{\lambda+4,\lambda+6}
t_{\lambda+\frac{3}{2},\lambda+4}-
t_{\lambda+\frac{7}{2},\lambda+6}t_{\lambda+\frac{3}{2},\lambda+\frac{7}{2}}\right)=0,\\
t_{\lambda+4,\lambda+6}\left(3\,t_{\lambda+\frac{3}{2},\lambda+4}
t_{\lambda,\lambda+\frac{3}{2}}+
3\,t_{\lambda+\frac{5}{2},\lambda+4}t_{\lambda,\lambda+\frac{5}{2}}+
t_{\lambda+2,\lambda+4}t_{\lambda,\lambda+2}\right)=0,\\
t_{\lambda,\lambda+2}\left((1-\epsilon_{7,\lambda})\,t_{\lambda+\frac{7}{2},\lambda+6}
t_{\lambda+2,\lambda+\frac{7}{2}}+
t_{\lambda+\frac{9}{2},\lambda+6}t_{\lambda+2,\lambda+\frac{9}{2}}+{1\over3}
t_{\lambda+4,\lambda+6}t_{\lambda+2,\lambda+4}\right)+\\
\quad +\,\epsilon_{7,\lambda}\,t_{\lambda+\frac{7}{2},\lambda+6}
t_{\lambda+\frac{3}{2},\lambda+\frac{7}{2}}
t_{\lambda,\lambda+\frac{3}{2}}=0,\\
\epsilon_{8,\lambda}\,t_{\lambda+\frac{7}{2},\lambda+6}
\left(t_{\lambda+\frac{3}{2},\lambda+\frac{7}{2}}
t_{\lambda,\lambda+\frac{3}{2}}-
t_{\lambda+2,\lambda+\frac{7}{2}}t_{\lambda,\lambda+2}\right)+\\
\quad +\left(t_{\lambda+4,\lambda+6}
t_{\lambda+\frac{5}{2},\lambda+4}-
t_{\lambda+\frac{9}{2},\lambda+6}t_{\lambda+\frac{5}{2},\lambda+\frac{9}{2}}\right)
t_{\lambda,\lambda+\frac{5}{2}}=0.\end{gathered}$$
- For $2(\beta-\lambda)\in\left\{13,\,
\dots,\,2n\right\}:$$$\begin{gathered}
t_{\lambda+4,\lambda+\frac{13}{2}}\left(t_{\lambda+\frac{3}{2},\lambda+4}
t_{\lambda,\lambda+\frac{3}{2}}+
t_{\lambda+\frac{5}{2},\lambda+4}t_{\lambda,\lambda+\frac{5}{2}}+(\frac{1}{3}-\epsilon_{10,\lambda})
t_{\lambda+2,\lambda+4}t_{\lambda,\lambda+2}\right)+\\\quad
+\,\epsilon_{10,\lambda} t_{\lambda,\lambda+2}
t_{\lambda+\frac{9}{2},\lambda+\frac{13}{2}}
t_{\lambda+2,\lambda+\frac{9}{2}}=0,\\[0.5pt]
t_{\lambda,\lambda+\frac{5}{2}}\left(t_{\lambda+4,\lambda+\frac{13}{2}}
t_{\lambda+\frac{5}{2},\lambda+4}+\frac{1}{3}
\,t_{\lambda+\frac{9}{2},\lambda+\frac{13}{2}}t_{\lambda+\frac{5}{2},\lambda+\frac{9}{2}}\right)+
\\ \quad+\,\epsilon_{11,\lambda}t_{\lambda,\lambda+2}
\left(t_{\lambda+\frac{9}{2},\lambda+\frac{13}{2}}
t_{\lambda+2,\lambda+\frac{9}{2}}-
t_{\lambda+4,\lambda+\frac{13}{2}}t_{\lambda+2,\lambda+4}\right)=0,\\
t_{\lambda+\frac{9}{2},\lambda+\frac{13}{2}}
\left(t_{\lambda+\frac{5}{2},\lambda+\frac{9}{2}}
t_{\lambda,\lambda+\frac{5}{2}}+(\epsilon_{12,\lambda}-1)\,
t_{\lambda+2,\lambda+\frac{9}{2}}t_{\lambda,\lambda+2}\right)-\epsilon_{12,\lambda}
t_{\lambda,\lambda+2}t_{\lambda+4,\lambda+\frac{13}{2}}t_{\lambda+2,\lambda+4}=0.\end{gathered}$$
- For $2(\beta-\lambda)\in\left\{14,\,
\dots,\,2n\right\}:$$$\begin{gathered}
t_{\lambda+\frac{9}{2},\lambda+7}
t_{\lambda+2,\lambda+\frac{9}{2}}t_{\lambda,\lambda+2}
=t_{\lambda,\lambda+\frac{5}{2}}
t_{\lambda+\frac{9}{2},\lambda+7}t_{\lambda+\frac{5}{2},\lambda+\frac{9}{2}}
=0.\end{gathered}$$
Considering again the homomorphism condition, we compute the 3rd order term ${\frak L}^{(3)}$ which is a solution of the Maurer-Cartan equation: $$\label{cap3}\delta({\frak
L}^{(3)})=-{1\over2}\left([{\frak L}^{(1)},\, {\frak
L}^{(2)}]\!]+[\![{\frak L}^{(2)},\, {\frak
L}^{(1)}]\!]\right).$$ The right hand side of (\[cap3\]) together with equation (\[ord2\]) yield the following maps: $$\label{2-cocyc}\begin{array}{llllll}
\Omega_{\lambda,\lambda+\frac{9}{2}}&=& \varphi_1(t)\,
[\![\Upsilon_{\lambda+3,\lambda+\frac{9}{2}},\
\frak{J}_4^{-1,\lambda}
]\!]+\psi_1(t)\,[\![\frak{J}_4^{-1,\lambda+\frac{3}{2}},
\,\Upsilon_{\lambda,\lambda+\frac{3}{2}} ]\!]
&:&\mathcal{K}(1)\otimes\mathcal{K}(1)\rightarrow\frak{D}_{\lambda,\lambda+\frac{9}{2}},\\
\Omega_{\lambda,\lambda+5}&=&\varphi_2(t)\,[\![\Upsilon_{\lambda+\frac{7}{2},\lambda+5},\
\frak{J}_\frac{9}{2}^{-1,\lambda} ]\!]+\psi_2(t)\,
[\![\frak{J}_\frac{9}{2}^{-1,\lambda+\frac{3}{2}},
\,\Upsilon_{\lambda,\lambda+\frac{3}{2}}
]\!]&:&\mathcal{K}(1)\otimes\mathcal{K}(1)\rightarrow\frak{D}_{\lambda,\lambda+5},\\
\widetilde{\Omega}_{\lambda,\lambda+5}&=&\widetilde{\varphi}_2(t)\,[\![\Upsilon_{\lambda+3,\lambda+5},\
\frak{J}_4^{-1,\lambda} ]\!]+\widetilde{\psi}_2(t)\,
[\![\frak{J}_4^{-1,\lambda+2}, \,\Upsilon_{\lambda,\lambda+2}
]\!]&:&\mathcal{K}(1)\otimes\mathcal{K}(1)\rightarrow\frak{D}_{\lambda,\lambda+5},\\
\Omega_{\lambda,\lambda+{11\over2}}&=&\varphi_3(t)\,[\![\Upsilon_{\lambda+4,\lambda+\frac{11}{2}},\
\frak{J}_{5}^{-1,\lambda} ]\!]+\psi_3(t)\,
[\![\frak{J}_{5}^{-1,\lambda+\frac{3}{2}},
\,\Upsilon_{\lambda,\lambda+\frac{3}{2}}
]\!]&:&\mathcal{K}(1)\otimes\mathcal{K}(1)\rightarrow\frak{D}_{\lambda,\lambda+\frac{11}{2}},\\
\widetilde{\Omega}_{\lambda,\lambda+{11\over2}}&=&\widetilde{\varphi}_3(t)\,[\![\Upsilon_{\lambda+\frac{7}{2},\lambda+\frac{11}{2}},\
\frak{J}_\frac{9}{2}^{-1,\lambda} ]\!]+\widetilde{\psi}_3(t)\,
[\![\frak{J}_\frac{9}{2}^{-1,\lambda+2},
\,\Upsilon_{\lambda,\lambda+2}
]\!]&:&\mathcal{K}(1)\otimes\mathcal{K}(1)\rightarrow\frak{D}_{\lambda,\lambda+\frac{11}{2}},\\
\overline{\Omega}_{\lambda,\lambda+{11\over2}}&=&\overline{\varphi}_3(t)
\,[\![\Upsilon_{\lambda+3,\lambda+\frac{11}{2}},\
\frak{J}_4^{-1,\lambda} ]\!]+\overline{\psi}_3(t)\,
[\![\frak{J}_4^{-1,\lambda+\frac{5}{2}},
\,\Upsilon_{\lambda,\lambda+\frac{5}{2}}
]\!]&:&\mathcal{K}(1)\otimes\mathcal{K}(1)\rightarrow\frak{D}_{\lambda,\lambda+\frac{11}{2}},\\
\Omega_{\lambda,\lambda+6}&=&{\varphi}_4(t)\,[\![\Upsilon_{\lambda+\frac{9}{2},\lambda+6},\
\frak{J}_\frac{11}{2}^{-1,\lambda} ]\!]+{\psi}_4(t) \,[\![
\frak{J}_\frac{11}{2}^{-1,\lambda+\frac{3}{2}},
\,\Upsilon_{\lambda,\lambda+\frac{3}{2}}
]\!]&:&\mathcal{K}(1)\otimes\mathcal{K}(1)
\rightarrow\frak{D}_{\lambda,\lambda+6},\\
\widetilde{\Omega}_{\lambda,\lambda+6}&=&\widetilde{\varphi}_4(t)\,[\![\Upsilon_{\lambda+4,\lambda+6},\
\frak{J}_{5}^{-1,\lambda} ]\!]+\widetilde{\psi}_4(t) \,[\![
\frak{J}_{5}^{-1,\lambda+2}, \,\Upsilon_{\lambda,\lambda+2}
]\!]&:&\mathcal{K}(1)\otimes\mathcal{K}(1)
\rightarrow\frak{D}_{\lambda,\lambda+6},\\
\overline{\Omega}_{\lambda,\lambda+6}&=&\overline{\varphi}_4(t)\,[\![\Upsilon_{\lambda+\frac{7}{2},\lambda+6},\
\frak{J}_\frac{9}{2}^{-1,\lambda} ]\!]+\overline{\psi}_4(t)\,[\![
\frak{J}_\frac{9}{2}^{-1,\lambda+\frac{5}{2}},
\,\Upsilon_{\lambda,\lambda+\frac{5}{2}}
]\!]&:&\mathcal{K}(1)\otimes\mathcal{K}(1)
\rightarrow\frak{D}_{\lambda,\lambda+6},\\
\Omega_{\lambda,\lambda+{13\over2}}&=&{\varphi}_5(t)\,[\![\Upsilon_{\lambda+4,\lambda+\frac{13}{2}},\
\frak{J}_{5}^{-1,\lambda} ]\!]+{\psi}_5(t) \,[\![
\frak{J}_{5}^{-1,\lambda+\frac{5}{2}},
\,\Upsilon_{\lambda,\lambda+\frac{5}{2}}
]\!]&:&\mathcal{K}(1)\otimes\mathcal{K}(1)
\rightarrow\frak{D}_{\lambda,\lambda+\frac{13}{2}},\\
\widetilde{\Omega}_{\lambda,\lambda+{13\over2}}&=&\widetilde{\varphi}_5(t)\,[\![\Upsilon_{\lambda+{9\over2},\lambda+\frac{13}{2}},\
\frak{J}_{11\over2}^{-1,\lambda} ]\!]+\widetilde{\psi}_5(t)\, [\![
\frak{J}_{11\over2}^{-1,\lambda+2}, \,\Upsilon_{\lambda,\lambda+2}
]\!]&:&\mathcal{K}(1)\otimes\mathcal{K}(1)
\rightarrow\frak{D}_{\lambda,\lambda+\frac{13}{2}},\\
\Omega_{\lambda,\lambda+7}&=&\varphi_6(t)\,[\![\Upsilon_{\lambda+{9\over2},\lambda+7},\
\frak{J}_{11\over2}^{-1,\lambda} ]\!]+\psi_6(t)\, [\![
\frak{J}_{11\over2}^{-1,\lambda+{5\over2}},
\,\Upsilon_{\lambda,\lambda+{5\over2}}
]\!]&:&\mathcal{K}(1)\otimes\mathcal{K}(1)
\rightarrow\frak{D}_{\lambda,\lambda+7},\end{array}$$ where $$\begin{array}{lll}
\varphi_1(t)=\zeta_\lambda^{-1}t_{\lambda+3,\lambda+\frac{9}{2}}
t_{\lambda+\frac{3}{2},\lambda+3}
t_{\lambda,\lambda+\frac{3}{2}},\\
\psi_1(t)=\zeta_{\lambda+\frac{3}{2}}^{-1}t_{\lambda+3,\lambda+\frac{9}{2}}
t_{\lambda+\frac{3}{2},\lambda+3}
t_{\lambda,\lambda+\frac{3}{2}},\\
\varphi_2(t)=\alpha_\lambda^{-1}t_{\lambda+\frac{7}{2},\lambda+5}
\left(t_{\lambda+\frac{3}{2},\lambda+\frac{7}{2}}
t_{\lambda,\lambda+\frac{3}{2}}-
t_{\lambda+2,\lambda+\frac{7}{2}}t_{\lambda,\lambda+2}\right),\\
\psi_2(t)=\alpha_{\lambda+\frac{3}{2}}^{-1}
\left(t_{\lambda+3,\lambda+5} t_{\lambda+\frac{3}{2},\lambda+3}-
t_{\lambda+\frac{7}{2},\lambda+5}t_{\lambda+\frac{3}{2},\lambda+\frac{7}{2}}\right)
t_{\lambda,\lambda+\frac{3}{2}},\\
\widetilde{\varphi}_2(t)=\zeta_\lambda^{-1}
t_{\lambda+3,\lambda+5}
t_{\lambda+\frac{3}{2},\lambda+3}t_{\lambda,\lambda+\frac{3}{2}},\\
\widetilde{\psi}_2(t)=\zeta_{\lambda+2}^{-1}
t_{\lambda+\frac{7}{2},\lambda+5}t_{\lambda+2,\lambda+\frac{7}{2}}
t_{\lambda,\lambda+2},\\
\varphi_3(t)=\beta_\lambda^{-1}t_{\lambda+4,\lambda+\frac{11}{2}}\left(t_{\lambda+\frac{3}{2},\lambda+4}
t_{\lambda,\lambda+\frac{3}{2}}+
t_{\lambda+\frac{5}{2},\lambda+4}t_{\lambda,\lambda+\frac{5}{2}}+
{1\over3}t_{\lambda+2,\lambda+4}t_{\lambda,\lambda+2}\right),\\
\psi_3(t)=\beta_{\lambda+\frac{3}{2}}^{-1}t_{\lambda,\lambda+\frac{3}{2}}\left(t_{\lambda+3,\lambda+\frac{11}{2}}
t_{\lambda+\frac{3}{2},\lambda+3}+
t_{\lambda+4,\lambda+\frac{11}{2}}t_{\lambda+\frac{3}{2},\lambda+4}+
{1\over3}t_{\lambda+\frac{7}{2},\lambda+\frac{11}{2}}
t_{\lambda+\frac{3}{2},\lambda+\frac{7}{2}}\right),\\
\widetilde{\varphi}_3(t)=\alpha_\lambda^{-1}t_{\lambda+\frac{7}{2},\lambda+\frac{11}{2}}
\left(t_{\lambda+\frac{3}{2},\lambda+\frac{7}{2}}
t_{\lambda,\lambda+\frac{3}{2}}-
t_{\lambda+2,\lambda+\frac{7}{2}}t_{\lambda,\lambda+2}\right),\\
\widetilde{\psi}_3(t)=\alpha_{\lambda+2}^{-1}t_{\lambda,\lambda+2}
\left(t_{\lambda+\frac{7}{2},\lambda+\frac{11}{2}}
t_{\lambda+2,\lambda+\frac{7}{2}}-
t_{\lambda+4,\lambda+\frac{11}{2}}t_{\lambda+2,\lambda+4}\right),\\
\overline{\varphi}_3(t)=\zeta_\lambda^{-1}t_{\lambda+3,\lambda+\frac{11}{2}}
t_{\lambda+\frac{3}{2},\lambda+3}
t_{\lambda,\lambda+\frac{3}{2}},\\
\overline{\psi}_3(t)=\zeta_{\lambda+\frac{5}{2}}^{-1}t_{\lambda+4,\lambda+\frac{11}{2}}
t_{\lambda+\frac{5}{2},\lambda+4}
t_{\lambda,\lambda+\frac{5}{2}},\\
\varphi_4(t)=\gamma_\lambda^{-1}t_{\lambda+\frac{9}{2},\lambda+6}
\left(t_{\lambda+\frac{5}{2},\lambda+\frac{9}{2}}
t_{\lambda,\lambda+\frac{5}{2}}-
t_{\lambda+2,\lambda+\frac{9}{2}}t_{\lambda,\lambda+2}\right),\\\end{array}$$$$\begin{array}{llllllllllllllll}
\psi_4(t)=\gamma_{\lambda+\frac{3}{2}}^{-1}
\left(t_{\lambda+4,\lambda+6} t_{\lambda+\frac{3}{2},\lambda+4}-
t_{\lambda+\frac{7}{2},\lambda+6}t_{\lambda+\frac{3}{2},\lambda+\frac{7}{2}}\right)
t_{\lambda,\lambda+\frac{3}{2}},\\
\widetilde{\varphi}_4(t)=\beta_\lambda^{-1}t_{\lambda+4,\lambda+6}\left(t_{\lambda+\frac{3}{2},\lambda+4}
t_{\lambda,\lambda+\frac{3}{2}}+
t_{\lambda+\frac{5}{2},\lambda+4}t_{\lambda,\lambda+\frac{5}{2}}+
{1\over3}t_{\lambda+2,\lambda+4}t_{\lambda,\lambda+2}\right),\\
\widetilde{\psi}_4(t)=\beta_{\lambda+2}^{-1}t_{\lambda,\lambda+2}\left(t_{\lambda+\frac{7}{2},\lambda+6}
t_{\lambda+2,\lambda+\frac{7}{2}}+
t_{\lambda+\frac{9}{2},\lambda+6}t_{\lambda+2,\lambda+\frac{9}{2}}+
{1\over3}t_{\lambda+4,\lambda+6}t_{\lambda+2,\lambda+4}\right),\\
\overline{\varphi}_4(t)=\alpha_\lambda^{-1}t_{\lambda+\frac{7}{2},\lambda+6}
\left(t_{\lambda+\frac{3}{2},\lambda+\frac{7}{2}}
t_{\lambda,\lambda+\frac{3}{2}}-
t_{\lambda+2,\lambda+\frac{7}{2}}t_{\lambda,\lambda+2}\right),\\
\overline{\psi}_4(t)=\alpha_{\lambda+\frac{5}{2}}^{-1}
\left(t_{\lambda+4,\lambda+6} t_{\lambda+\frac{5}{2},\lambda+4}-
t_{\lambda+\frac{9}{2},\lambda+6}t_{\lambda+\frac{5}{2},\lambda+\frac{9}{2}}\right)
t_{\lambda,\lambda+\frac{5}{2}},\\
{\varphi}_5(t)=\beta_\lambda^{-1}t_{\lambda+4,\lambda+\frac{13}{2}}\left(t_{\lambda+\frac{3}{2},\lambda+4}
t_{\lambda,\lambda+\frac{3}{2}}+
t_{\lambda+\frac{5}{2},\lambda+4}t_{\lambda,\lambda+\frac{5}{2}}+
{1\over3}t_{\lambda+2,\lambda+4}t_{\lambda,\lambda+2}\right),
\\
{\psi}_5(t)=\beta_{\lambda+\frac{5}{2}}^{-1}t_{\lambda,\lambda+\frac{5}{2}}
\left(t_{\lambda+4,\lambda+\frac{13}{2}}
t_{\lambda+\frac{5}{2},\lambda+4}+
{1\over3}t_{\lambda+\frac{9}{2},\lambda+\frac{13}{2}}t_{\lambda+\frac{5}{2},\lambda+\frac{9}{2}}\right),\\
\widetilde{\varphi}_5(t)=\gamma_\lambda^{-1}t_{\lambda+\frac{9}{2},\lambda+\frac{13}{2}}
\left(t_{\lambda+\frac{5}{2},\lambda+\frac{9}{2}}
t_{\lambda,\lambda+\frac{5}{2}}-
t_{\lambda+2,\lambda+\frac{9}{2}}t_{\lambda,\lambda+2}\right),
\\\widetilde{\psi}_5(t)={\gamma}_{\lambda+2}^{-1}t_{\lambda,\lambda+2}
\left(t_{\lambda+\frac{9}{2},\lambda+\frac{13}{2}}
t_{\lambda+2,\lambda+\frac{9}{2}}-
t_{\lambda+4,\lambda+\frac{13}{2}}t_{\lambda+2,\lambda+4}\right),\\
\varphi_6(t)=\gamma_\lambda^{-1}t_{\lambda+\frac{9}{2},\lambda+7}
\left(t_{\lambda+\frac{5}{2},\lambda+\frac{9}{2}}
t_{\lambda,\lambda+\frac{5}{2}}-
t_{\lambda+2,\lambda+\frac{9}{2}}t_{\lambda,\lambda+2}\right),
\\\psi_6(t)=\gamma_{\lambda+\frac{5}{2}}^{-1}t_{\lambda,\lambda+\frac{5}{2}}
t_{\lambda+\frac{9}{2},\lambda+7}t_{\lambda+\frac{5}{2},\lambda+\frac{9}{2}}.
\end{array}$$ Now, the same arguments, as in the proof of Proposition \[th2\], show that we must have: $$\label{E}
\begin{array}{llllll}
\Omega_{\lambda,\lambda+{9\over2}}&=&
\omega_1(t)\delta\left(\frak{J}_{{11\over2}}^{-1,\lambda}\right),\\
\Omega_{\lambda,\lambda+5}+\widetilde{\Omega}_{\lambda,\lambda+5}&=&
\omega_2(t)\delta\left(\frak{J}_{6}^{-1,\lambda}\right),\\
\Omega_{\lambda,\lambda+{11\over2}}+\widetilde{\Omega}_{\lambda,\lambda+{11\over2}}+
\overline{\Omega}_{\lambda,\lambda+{11\over2}}&=&
\omega_3(t)\delta\left(\frak{J}_{13\over2}^{-1,\lambda}\right),\\
\Omega_{\lambda,\lambda+6}+\widetilde{\Omega}_{\lambda,\lambda+6}+\overline{\Omega}_{\lambda,\lambda+6}&=&
\omega_4(t)\delta\left(\frak{J}_{7}^{-1,\lambda}\right),\\
\Omega_{\lambda,\lambda+{13\over2}}+\widetilde{\Omega}_{\lambda,\lambda+{13\over2}}&=&
\omega_5(t)\delta\left(\frak{J}_{15\over2}^{-1,\lambda}\right),\\
\Omega_{\lambda,\lambda+7}&=&
\omega_6(t)\delta\left(\frak{J}_{8}^{-1,\lambda}\right),\end{array}$$ where $\omega_1,\,\dots,\,\omega_5$ are some functions. So, by Lemma \[benfraj1\] and Lemma \[benfraj2\], we obtain for the nonzero $\varphi_i(t), \widetilde{\varphi}_i(t),
\overline{\varphi}_i, \psi_i(t), \widetilde{\psi}_i(t),
\overline{\psi}_i(t)$ and $\omega_i(t)$ the following relation: $$\begin{array}{llllll}
\alpha_{\lambda}\,\varphi_2(t)+\epsilon_{1,\lambda}\,
\alpha_{\lambda+\frac{3}{2}}\,\psi_2(t)=0,\quad
&\zeta_\lambda\,\widetilde{\varphi}_2(t)+\epsilon_{2,\lambda}\,
\alpha_{\lambda+\frac{3}{2}}\,\psi_2(t)=0,\\[1pt]
\beta_{\lambda}\,\varphi_3(t)+\epsilon_{4,\lambda}\,
\beta_{\lambda+\frac{3}{2}}\,\psi_3(t)=0,\quad
&\alpha_{\lambda}\,\widetilde{\varphi}_3(t)+\epsilon_{5,\lambda}\,
\beta_{\lambda+\frac{3}{2}}\,\psi_3(t)=0,\\[1pt]
\alpha_{\lambda+2}\,\widetilde{\psi}_3(t)+\epsilon_{6,\lambda}\,
\beta_{\lambda+\frac{3}{2}}\,\psi_3(t)=0,\quad
&\beta_{\lambda+2}\,\widetilde{\psi}_4(t)+\epsilon_{7,\lambda}\,
\alpha_{\lambda}\,\overline{\varphi}_4(t)=0,\\[1pt]
\alpha_{\lambda+\frac{5}{2}}\,\overline{\psi}_4(t)+\epsilon_{8,\lambda}\,
\alpha_{\lambda}\,\overline{\varphi}_4(t)=0,\quad &
\beta_{\lambda}\,\varphi_5(t)+\epsilon_{10,\lambda}\,\gamma_{\lambda+2}\widetilde{\psi}_5(t)=0,\\[1pt]
\beta_{\lambda+\frac{5}{2}}\,\psi_5(t)+\epsilon_{11,\lambda}\,\gamma_{\lambda+2}\widetilde{\psi}_5(t)=0,\quad
&\gamma_{\lambda}\,\widetilde{\varphi}_5(t)+\epsilon_{12,\lambda}
\,\gamma_{\lambda+2}\widetilde{\psi}_5(t)=0,\\[1pt]
\omega_4(t)=\epsilon_{9,\lambda}\,\alpha_{\lambda}\,\overline{\varphi}_4(t),\quad
&\omega_2(t)=\epsilon_{3,\lambda}\,
\alpha_{\lambda+\frac{3}{2}}\,\psi_2(t),\\[1pt]
\omega_1(t)=\xi_\lambda^{-1}\,
\zeta_\lambda\,\varphi_1(t)=\xi_\lambda^{-1}\,\zeta_{\lambda+\frac{3}{2}}\,\psi_1(t).
\end{array}$$ Therefore, we get the necessary integrability conditions for ${\frak L}^{(3)}$. Under these conditions, the third-order term ${\frak L}^{(3)}$ can be given by: $$\begin{array}{llll}
{\frak L}^{(3) }=&
\sum_\lambda\xi_\lambda^{-1}t_{\lambda+3,\lambda+\frac{9}{2}}
t_{\lambda+\frac{3}{2},\lambda+3}
t_{\lambda,\lambda+\frac{3}{2}}\,
\frak{J}_{11\over2}^{-1,\lambda}+\\
&\sum_\lambda\epsilon_{3,\lambda} \left(t_{\lambda+3,\lambda+5}
t_{\lambda+\frac{3}{2},\lambda+3}-
t_{\lambda+\frac{7}{2},\lambda+5}t_{\lambda+\frac{3}{2},\lambda+\frac{7}{2}}\right)
t_{\lambda,\lambda+\frac{3}{2}}\,
\frak{J}_{6}^{-1,\lambda}+\\
&\sum_\lambda\epsilon_{9,\lambda}\,t_{\lambda+\frac{7}{2},\lambda+6}
\left(t_{\lambda+\frac{3}{2},\lambda+\frac{7}{2}}
t_{\lambda,\lambda+\frac{3}{2}}-
t_{\lambda+2,\lambda+\frac{7}{2}}t_{\lambda,\lambda+2}\right)\frak{J}_{7}^{-1,\lambda}.
\end{array}$$
\[th4\] The 4th order integrability conditions of the infinitesimal deformation (\[infdef1\]) are the following:
- For $2(\beta-\lambda)\in\left\{12,\,
\dots,\,2n\right\}:$ $$\begin{gathered}
t_{\lambda+\frac{9}{2},\lambda+6}
t_{\lambda+3,\lambda+\frac{9}{2}}t_{\lambda+\frac{3}{2},\lambda+3}
t_{\lambda,\lambda+\frac{3}{2}} =0.\end{gathered}$$
- For $2(\beta-\lambda)\in\left\{13,\,
\dots,\,2n\right\}:$$$\begin{gathered}
t_{\lambda+5,\lambda+\frac{13}{2}}
t_{\lambda+\frac{7}{2},\lambda+5}t_{\lambda+\frac{3}{2},\lambda+\frac{7}{2}}
t_{\lambda,\lambda+\frac{3}{2}}=0,\\
t_{\lambda+\frac{3}{2},\lambda+3} t_{\lambda,\lambda+\frac{3}{2}}
t_{\lambda+5,\lambda+\frac{13}{2}}t_{\lambda+3,\lambda+5}=0,\\
t_{\lambda+\frac{9}{2},\lambda+\frac{13}{2}}
t_{\lambda+3,\lambda+\frac{9}{2}}
t_{\lambda+\frac{3}{2},\lambda+3}
t_{\lambda,\lambda+\frac{3}{2}}=0.\end{gathered}$$
- For $2(\beta-\lambda)\in\left\{14,\,
\dots,\,2n\right\}:$ $$\begin{gathered}
t_{\lambda+\frac{9}{2},\lambda+7}
t_{\lambda+3,\lambda+\frac{9}{2}}
t_{\lambda+\frac{3}{2},\lambda+3}
t_{\lambda,\lambda+\frac{3}{2}}=0,\\
t_{\lambda,\lambda+\frac{3}{2}}t_{\lambda+\frac{3}{2},\lambda+3}
t_{\lambda+5,\lambda+7}t_{\lambda+3,\lambda+5}
=0,\\
t_{\lambda,\lambda+2}t_{\lambda+2,\lambda+\frac{7}{2}}
t_{\lambda+\frac{11}{2},\lambda+7}t_{\lambda+\frac{7}{2},\lambda+\frac{11}{2}}
=0,\\
t_{\lambda,\lambda+\frac{3}{2}}t_{\lambda+\frac{3}{2},\lambda+\frac{7}{2}}
t_{\lambda+\frac{11}{2},\lambda+7}t_{\lambda+\frac{7}{2},\lambda+\frac{11}{2}}
=0,\\
t_{\lambda,\lambda+\frac{3}{2}}t_{\lambda+5,\lambda+7}
t_{\lambda+\frac{7}{2},\lambda+5}t_{\lambda+\frac{3}{2},\lambda+\frac{7}{2}}
=0,\\
t_{\lambda+\frac{11}{2},\lambda+7}
t_{\lambda+4,\lambda+\frac{11}{2}}\left(3t_{\lambda+\frac{3}{2},\lambda+4}
t_{\lambda,\lambda+\frac{3}{2}}+
t_{\lambda+2,\lambda+4}t_{\lambda,\lambda+2}\right)=0.\end{gathered}$$
- For $2(\beta-\lambda)\in\left\{15,\,
\dots,\,2n\right\}:$$$\begin{gathered}
t_{\lambda,\lambda+\frac{3}{2}}
t_{\lambda+\frac{3}{2},\lambda+3}t_{\lambda+5,\lambda+\frac{15}{2}}
t_{\lambda+3,\lambda+5}=0,\\
t_{\lambda,\lambda+\frac{3}{2}}
t_{\lambda+5,\lambda+\frac{15}{2}}t_{\lambda+\frac{7}{2},\lambda+5}
t_{\lambda+\frac{3}{2},\lambda+\frac{7}{2}}=0,\\
t_{\lambda,\lambda+\frac{5}{2}} t_{\lambda+\frac{5}{2},\lambda+4}
t_{\lambda+6,\lambda+\frac{15}{2}}t_{\lambda+4,\lambda+6}=0,\\
t_{\lambda+\frac{11}{2},\lambda+\frac{15}{2}}
t_{\lambda+\frac{7}{2},\lambda+\frac{11}{2}}\left(t_{\lambda+\frac{3}{2},\lambda+\frac{7}{2}}
t_{\lambda,\lambda+\frac{3}{2}}-
t_{\lambda+2,\lambda+\frac{7}{2}}t_{\lambda,\lambda+2}\right)=0,\\
t_{\lambda+\frac{11}{2},\lambda+\frac{15}{2}}
t_{\lambda+4,\lambda+\frac{11}{2}}\left(3t_{\lambda+\frac{3}{2},\lambda+4}
t_{\lambda,\lambda+\frac{3}{2}}+
t_{\lambda+2,\lambda+4}t_{\lambda,\lambda+2}\right)=0,\\
t_{\lambda+6,\lambda+\frac{15}{2}}
t_{\lambda+\frac{7}{2},\lambda+6}\left(t_{\lambda+\frac{3}{2},\lambda+\frac{7}{2}}
t_{\lambda,\lambda+\frac{3}{2}}-
t_{\lambda+2,\lambda+\frac{7}{2}}t_{\lambda,\lambda+2}\right)=0.\end{gathered}$$
- For $2(\beta-\lambda)\in\left\{16,\,
\dots,\,2n\right\}:$$$\begin{array}{lllllll}
\epsilon_{14,\lambda}\,t_{\lambda+6,\lambda+8}t_{\lambda+\frac{7}{2},\lambda+6}
\left(t_{\lambda+\frac{3}{2},\lambda+\frac{7}{2}}
t_{\lambda,\lambda+\frac{3}{2}}-
t_{\lambda+2,\lambda+\frac{7}{2}}t_{\lambda,\lambda+2}\right)+\\
\quad~+
t_{\lambda+\frac{13}{2},\lambda+8}t_{\lambda+\frac{9}{2},\lambda+\frac{13}{2}}
\left( t_{\lambda+\frac{5}{2},\lambda+\frac{9}{2}}
t_{\lambda,\lambda+\frac{5}{2}}-
t_{\lambda+2,\lambda+\frac{9}{2}}t_{\lambda,\lambda+2}\right)=0
,\\
\epsilon_{16,\lambda}\,t_{\lambda+6,\lambda+8}t_{\lambda+\frac{7}{2},\lambda+6}
\left(t_{\lambda+\frac{3}{2},\lambda+\frac{7}{2}}
t_{\lambda,\lambda+\frac{3}{2}}-
t_{\lambda+2,\lambda+\frac{7}{2}}t_{\lambda,\lambda+2}\right)+\\
\quad~+ t_{\lambda,\lambda+2}t_{\lambda+\frac{11}{2},\lambda+8}
\left(t_{\lambda+\frac{7}{2},\lambda+\frac{11}{2}}
t_{\lambda+2,\lambda+\frac{7}{2}}-
t_{\lambda+4,\lambda+\frac{11}{2}}t_{\lambda+2,\lambda+4}\right)=0,\\
\left((1+\epsilon_{15,\lambda})\,t_{\lambda+6,\lambda+8}t_{\lambda+\frac{7}{2},\lambda+6}-
t_{\lambda+\frac{11}{2},\lambda+8}t_{\lambda+\frac{7}{2},\lambda+\frac{11}{2}}
\right)\times\\
\quad~\times \left(t_{\lambda+\frac{3}{2},\lambda+\frac{7}{2}}
t_{\lambda,\lambda+\frac{3}{2}}-
t_{\lambda+2,\lambda+\frac{7}{2}}t_{\lambda,\lambda+2}\right)=0,\\\end{array}$$ $$\begin{array}{llllll}
\epsilon_{13,\lambda}\,t_{\lambda+6,\lambda+8}t_{\lambda+\frac{7}{2},\lambda+6}
\left(t_{\lambda+\frac{3}{2},\lambda+\frac{7}{2}}
t_{\lambda,\lambda+\frac{3}{2}}-
t_{\lambda+2,\lambda+\frac{7}{2}}t_{\lambda,\lambda+2}\right)+\\
\quad~+t_{\lambda+\frac{11}{2},\lambda+8}
t_{\lambda+4,\lambda+\frac{11}{2}}
\left(t_{\lambda+\frac{3}{2},\lambda+4}
t_{\lambda,\lambda+\frac{3}{2}}+{1\over3}
t_{\lambda+2,\lambda+4}t_{\lambda,\lambda+2}\right)+\\
\quad~+ t_{\lambda+\frac{13}{2},\lambda+8}
t_{\lambda+4,\lambda+\frac{13}{2}}
\left(t_{\lambda+\frac{3}{2},\lambda+4}
t_{\lambda,\lambda+\frac{3}{2}}+
t_{\lambda+\frac{5}{2},\lambda+4}t_{\lambda,\lambda+\frac{5}{2}}+
\frac{1}{3}t_{\lambda+2,\lambda+4}t_{\lambda,\lambda+2}\right)=0.\end{array}$$
- For $2(\beta-\lambda)\in\left\{17,\,
\dots,\,2n\right\}:$$$\begin{gathered}
t_{\lambda,\lambda+\frac{5}{2}} t_{\lambda+6,\lambda+\frac{17}{2}}
t_{\lambda+4,\lambda+6} t_{\lambda+\frac{5}{2},\lambda+4}=0,\\
t_{\lambda+\frac{13}{2},\lambda+\frac{17}{2}}
t_{\lambda+\frac{9}{2},\lambda+\frac{13}{2}}
\left(t_{\lambda+\frac{5}{2},\lambda+\frac{9}{2}}
t_{\lambda,\lambda+\frac{5}{2}}-
t_{\lambda+2,\lambda+\frac{9}{2}}t_{\lambda,\lambda+2}\right)=0,\\
t_{\lambda+6,\lambda+\frac{17}{2}}
t_{\lambda+\frac{7}{2},\lambda+6}
\left(t_{\lambda+\frac{3}{2},\lambda+\frac{7}{2}}
t_{\lambda,\lambda+\frac{3}{2}}-
t_{\lambda+2,\lambda+\frac{7}{2}}t_{\lambda,\lambda+2}\right)=0,\\
t_{\lambda+\frac{13}{2},\lambda+\frac{17}{2}}
t_{\lambda+4,\lambda+\frac{13}{2}}
\left(3\,t_{\lambda+\frac{3}{2},\lambda+4}
t_{\lambda,\lambda+\frac{3}{2}}+
3\,t_{\lambda+\frac{5}{2},\lambda+4}t_{\lambda,\lambda+\frac{5}{2}}+
t_{\lambda+2,\lambda+4}t_{\lambda,\lambda+2}\right)0.\end{gathered}$$
- For $2(\beta-\lambda)\in\left\{18,\,
\dots,\,2n\right\}:$$$\begin{gathered}
t_{\lambda+\frac{13}{2},\lambda+9}t_{\lambda+\frac{9}{2},\lambda+\frac{13}{2}}
\left(t_{\lambda+\frac{5}{2},\lambda+\frac{9}{2}}
t_{\lambda,\lambda+\frac{5}{2}}-
t_{\lambda+2,\lambda+\frac{9}{2}}t_{\lambda,\lambda+2}\right)=0.\end{gathered}$$
To prove Proposition \[th4\], we need the following two lemmas which we can check by a direct computation with the help of [*Maple*]{}.
\[benfraj3\] We have $$\begin{array}{llllll}
\epsilon_{9,\lambda}\,[\![\Upsilon_{\lambda+6,\lambda+8},\
\frak{J}_{7}^{-1,\lambda}
]\!]&=&\epsilon_{13,\lambda}\,\beta_{\lambda+4}^{-1}\,\beta_{\lambda}^{-1}\,
[\![\frak{J}_5^{-1,\lambda+4},\ \frak{J}_5^{-1,\lambda}
]\!]+\epsilon_{14,\lambda}\,\alpha_{\lambda+\frac{9}{2}}^{-1}\,\gamma_{\lambda}^{-1}
[\![\frak{J}_{\frac{9}{2}}^{-1,\lambda+\frac{9}{2}},\
\frak{J}_{\frac{11}{2}}^{-1,\lambda}
]\!]+\\&&\epsilon_{15,\lambda}\,\alpha_{\lambda}^{-1}\,\gamma_{\lambda+\frac{7}{2}}^{-1}
[\![\frak{J}_{\frac{11}{2}}^{-1,\lambda+\frac{7}{2}},\
\frak{J}_{\frac{9}{2}}^{-1,\lambda} ]\!]+
\epsilon_{16,\lambda}\,\epsilon_{9,\lambda+2}\,[\![\frak{J}_{7}^{-1,\lambda+2},\
\Upsilon_{\lambda,\lambda+2}]\!]+\\&&\epsilon_{17,\lambda}\,
\delta(\frak{J}_{9}^{-1,\lambda}),
\end{array}$$ where $$\small{
\begin{array}{llllll}
\epsilon_{13,\lambda}&=&\frac{\epsilon_{9,\lambda}\,(2\lambda+3)
(2\lambda+5)(2\lambda+9)(\lambda+2)(\lambda+3)(\lambda+5)(2\lambda^2+7\lambda+2)
(2\lambda^2+23\lambda+62)(16\lambda^4+240\lambda^3+1034\lambda^2+1005\lambda+300)}{36(\lambda+4)(2\lambda+7)
(32\lambda^6+784\lambda^5+7156\lambda^4+29576\lambda^3+53961\lambda^2+40281\lambda+11760)},\\[4pt]
\epsilon_{14,\lambda}&=&
\frac{\epsilon_{9,\lambda}\,(2\lambda+3)(2\lambda+5)(2\lambda-5)(2\lambda+9)(\lambda-4)(\lambda+2)(\lambda+7)(\lambda+9)}
{(\lambda+4)(32\lambda^6+784\lambda^5+7156\lambda^4+29576\lambda^3+53961\lambda^2+40281\lambda+11760)},\\[4pt]
\epsilon_{15,\lambda}&=&
-\frac{\epsilon_{9,\lambda}\,(2\lambda-3)(2\lambda+1)(2\lambda+3)(2\lambda+6)
(2\lambda+23)(\lambda+2)(\lambda+5)(\lambda+10)}
{(2\lambda+7)(32\lambda^6+784\lambda^5+7156\lambda^4+29576\lambda^3+53961\lambda^2+40281\lambda+11760)},\\[4pt]
\epsilon_{16,\lambda}&=&-\frac{\epsilon_{9,\lambda}\,(2\lambda+3)(\lambda+2)
(32\lambda^6+656\lambda^5+4756\lambda^4+14104\lambda^3+14901\lambda^2+7059\lambda+240)}
{\epsilon_{9,\lambda+2}(2\lambda+11)(\lambda+6)(32\lambda^6+784\lambda^5+
7156\lambda^4+29576\lambda^3+53961\lambda^2+40281\lambda+11760)},\\[4pt]
\epsilon_{17,\lambda}&=&-\frac{\epsilon_{9,\lambda}\,(2\lambda+5)(2\lambda+9)(2\lambda+6)
(\lambda+5)(16\lambda^4+240\lambda^3+1034\lambda^2+1005\lambda+420)}
{(2\lambda+11)(2\lambda+7)(\lambda+4)(\lambda+6)(32\lambda^6+784\lambda^5+7156\lambda^4+29576\lambda^3+
53961\lambda^2+40281\lambda+11760)}
\end{array}}$$
\[benfraj4\] Each of the following systems is linearly independent $$\begin{array}{lll}
1)\,&\left(\delta(\frak{J}_{7}^{-1,\lambda}),\,\,
\zeta_{\lambda+3}^{-1}\zeta_{\lambda}^{-1}[\![\frak{J}_{4}^{-1,\lambda+3},\,
\frak{J}_{4}^{-1,\lambda}]\!]+\xi_{\lambda}^{-1}[\![\Upsilon_{\lambda+\frac{9}{2},\lambda+6}
,\,\frak{J}_{\frac{11}{2}}^{-1,\lambda}]\!]+\xi_{\lambda+\frac{3}{2}}^{-1}
[\![\frak{J}_{\frac{11}{2}}^{-1,\lambda+\frac{3}{2}},\,
\Upsilon_{\lambda,\lambda+\frac{3}{2}}]\!]\right),\\[2pt]
2)\,&\Big(\delta(\frak{J}_{\frac{15}{2}}^{-1,\lambda}),\,\,
[\![\frak{J}_{4}^{-1,\lambda+\frac{7}{2}},\,
\frak{J}_{\frac{9}{2}}^{-1,\lambda}]\!],\,\,
[\![\Upsilon_{\lambda+5,\lambda+\frac{13}{2}},\,
\frak{J}_{6}^{-1,\lambda}]\!],\,\,[\![\Upsilon_{\lambda+\frac{9}{2},\lambda+\frac{13}{2}},\,
\frak{J}_{\frac{11}{2}}^{-1,\lambda}]\!],\,\,\\&~~~
\zeta^{-1}_\lambda\alpha_{\lambda+3}^{-1}
[\![\frak{J}_{\frac{9}{2}}^{-1,\lambda+3},\
\frak{J}_{4}^{-1,\lambda}
]\!]+\epsilon_{3,\lambda+\frac{3}{2}}[\![\frak{J}_{6}^{-1,\lambda+\frac{3}{2}},\
\Upsilon_{\lambda,\lambda+\frac{3}{2}} ]\!]\Big),\\[2pt]
3)\,&\Big(\delta(\frak{J}_{8}^{-1,\lambda}),\,\,
[\![\frak{J}_{\frac{9}{2}}^{-1,\lambda+\frac{7}{2}},\,
\frak{J}_{\frac{9}{2}}^{-1,\lambda}]\!],\,\,
[\![\frak{J}_{4}^{-1,\lambda+4},\,
\frak{J}_{5}^{-1,\lambda}]\!],\,\,[\![\frak{J}_{5}^{-1,\lambda+3},\,
\frak{J}_{4}^{-1,\lambda}]\!],\,\,[\![\frak{J}_{6}^{-1,\lambda+2},\,
\Upsilon_{\lambda,\lambda+2}]\!],\\&~~~[\![\Upsilon_{\lambda+\frac{9}{2},\lambda+7},\,
\frak{J}_{\frac{11}{2}}^{-1,\lambda}]\!],\,\,[\![\Upsilon_{\lambda+5,\lambda+7},\,
\frak{J}_{6}^{-1,\lambda}]\!]
\Big),\\[2pt]
4)\,&\Big(\delta(\frak{J}_{\frac{17}{2}}^{-1,\lambda}),\,\,
[\![\frak{J}_{5}^{-1,\lambda+\frac{7}{2}},\,
\frak{J}_{\frac{9}{2}}^{-1,\lambda}]\!],\,\,
[\![\frak{J}_{\frac{9}{2}}^{-1,\lambda+4},\,
\frak{J}_{5}^{-1,\lambda}]\!],\,\,
[\![\frak{J}_{\frac{11}{2}}^{-1,\lambda+3},\,
\frak{J}_{4}^{-1,\lambda}]\!],\,\,
[\![\Upsilon_{\lambda+6,\lambda+\frac{15}{2}},\,
\frak{J}_{7}^{-1,\lambda}]\!],\\&~~~
[\![\frak{J}_{6}^{-1,\lambda+\frac{5}{2}},\,
\Upsilon_{\lambda,\lambda+\frac{5}{2}}]\!],\,\,
\epsilon_{3,\lambda} [\![\Upsilon_{\lambda+5,\lambda+\frac{15}{2}}
,\,\frak{J}_{6}^{-1,\lambda}]\!]+\epsilon_{9,\lambda+\frac{3}{2}}
[\![\frak{J}_{7}^{-1,\lambda+\frac{3}{2}},\
\Upsilon_{\lambda,\lambda+\frac{3}{2}} ]\!] \Big),
\\[2pt]
5)\,&\left(\delta(\frak{J}_{9}^{-1,\lambda}),\,\,
[\![\frak{J}_{5}^{-1,\lambda+4},\,
\frak{J}_{5}^{-1,\lambda}]\!],\,\,[\![\frak{J}_{\frac{9}{2}}^{-1,\lambda+\frac{9}{2}},\,
\frak{J}_{\frac{11}{2}}^{-1,\lambda}]\!],\,\,
[\![\frak{J}_{\frac{11}{2}}^{-1,\lambda+\frac{7}{2}},\,
\frak{J}_{\frac{9}{2}}^{-1,\lambda}]\!],\,\,[\![\frak{J}_{7}^{-1,\lambda+2},\,
\Upsilon_{\lambda,\lambda+2}]\!]\right),
\\[2pt]
6)\,&\left(\delta(\frak{J}_{\frac{19}{2}}^{-1,\lambda}),\,\,[\![\frak{J}_{5}^{-1,\lambda+\frac{9}{2}},\,
\frak{J}_{\frac{11}{2}}^{-1,\lambda}]\!],\,\,
[\![\frak{J}_{\frac{11}{2}}^{-1,\lambda+4},\,
\frak{J}_{5}^{-1,\lambda}]\!],\,\,[\![\frak{J}_{7}^{-1,\lambda+\frac{5}{2}},\,
\Upsilon_{\lambda,\lambda+\frac{5}{2}}]\!],\,\,[\![\Upsilon_{\lambda+6,\lambda+\frac{17}{2}},\,
\frak{J}_{7}^{-1,\lambda}]\!]\right),\\[2pt]
7)\,&\left(\delta(\frak{J}_{10}^{-1,\lambda}),\,\,[\![\frak{J}_{\frac{11}{2}}^{-1,\lambda+\frac{9}{2}},\,
\frak{J}_{\frac{11}{2}}^{-1,\lambda}]\!]\right).
\end{array}$$
(Proposition \[th4\]) The fourth order integrability conditions of the infinitesimal deformation (\[infdef1\]) follow from Lemma \[benfraj3\] and Lemma \[benfraj4\] together with Proposition \[th2\] and Proposition \[pr3\] and arguments similar to those from the proof of proposition \[pr3\]. Under these conditions, the fourth-order term ${\frak L}^{(4)}$ can be given by: $${\frak L}^{(4)
}=-\epsilon_{17,\lambda}\,t_{\lambda+6,\lambda+8}t_{\lambda+\frac{7}{2},\lambda+6}
\left(t_{\lambda+\frac{3}{2},\lambda+\frac{7}{2}}
t_{\lambda,\lambda+\frac{3}{2}}-
t_{\lambda+2,\lambda+\frac{7}{2}}t_{\lambda,\lambda+2}\right)\frak{J}_{9}^{-1,\lambda}.$$
\[th5\] The 5th order integrability conditions of the infinitesimal deformation (\[infdef1\]) are the following:
- For $2(\beta-\lambda)\in\left\{19,\,
\dots,\,2n\right\}:$ $$\begin{gathered}
t_{\lambda+8,\lambda+\frac{19}{2}}
t_{\lambda+6,\lambda+8}t_{\lambda+\frac{7}{2},\lambda+6}
\left(t_{\lambda+\frac{3}{2},\lambda+\frac{7}{2}}
t_{\lambda,\lambda+\frac{3}{2}}-
t_{\lambda+2,\lambda+\frac{7}{2}}t_{\lambda,\lambda+2}\right) =0,\end{gathered}$$
- For $2(\beta-\lambda)\in\left\{20,\,
\dots,\,2n\right\}:$ $$\begin{gathered}
t_{\lambda+8,\lambda+10}
t_{\lambda+6,\lambda+8}t_{\lambda+\frac{7}{2},\lambda+6}
\left(t_{\lambda+\frac{3}{2},\lambda+\frac{7}{2}}
t_{\lambda,\lambda+\frac{3}{2}}-
t_{\lambda+2,\lambda+\frac{7}{2}}t_{\lambda,\lambda+2}\right)=0,\\
t_{\lambda,\lambda+2}t_{\lambda+8,\lambda+10}
t_{\lambda+\frac{11}{2},\lambda+8}
\left(t_{\lambda+\frac{7}{2},\lambda+\frac{11}{2}}
t_{\lambda+2,\lambda+\frac{7}{2}}-
t_{\lambda+4,\lambda+\frac{11}{2}}t_{\lambda+2,\lambda+4}\right)=0,\end{gathered}$$
- For $2(\beta-\lambda)\in\left\{21,\,
\dots,\,2n\right\}:$ $$\begin{gathered}
t_{\lambda+8,\lambda+\frac{21}{2}}
t_{\lambda+6,\lambda+8}t_{\lambda+\frac{7}{2},\lambda+6}
\left(t_{\lambda+\frac{3}{2},\lambda+\frac{7}{2}}
t_{\lambda,\lambda+\frac{3}{2}}-
t_{\lambda+2,\lambda+\frac{7}{2}}t_{\lambda,\lambda+2}\right)=0,\\
t_{\lambda+8,\lambda+\frac{21}{2}}t_{\lambda+\frac{13}{2},\lambda+8}
t_{\lambda+\frac{9}{2},\lambda+\frac{13}{2}}
\left(t_{\lambda+\frac{5}{2},\lambda+\frac{9}{2}}
t_{\lambda,\lambda+\frac{5}{2}}-
t_{\lambda+2,\lambda+\frac{9}{2}}t_{\lambda,\lambda+2}\right)=0,\\
t_{\lambda,\lambda+\frac{5}{2}}t_{\lambda+\frac{17}{2},\lambda+\frac{21}{2}}
t_{\lambda+6,\lambda+\frac{17}{2}}
t_{\lambda+\frac{9}{2},\lambda+6}t_{\lambda+\frac{5}{2},\lambda+\frac{9}{2}}=0.\end{gathered}$$
To prove Proposition \[th5\], we need the following lemma which we can check by a direct computation.
\[benfraj5\] Each of the following systems is linearly independent $$\begin{array}{lll}
1)\,&\left(\delta(\frak{J}_{\frac{21}{2}}^{-1,\lambda}),\,\,
\alpha_{\lambda+6}^{-1}\epsilon_{9,\lambda}
[\![\frak{J}_{\frac{9}{2}}^{-1,\lambda+6},\,
\frak{J}_{7}^{-1,\lambda}]\!]-\epsilon_{17,\lambda}
[\![\Upsilon_{\lambda+8,\lambda+\frac{19}{2}}
,\,\frak{J}_{9}^{-1,\lambda}]\!]\right),\\[2pt]
2)\,&\left(\delta(\frak{J}_{11}^{-1,\lambda}),\,\,
[\![\frak{J}_{9}^{-1,\lambda+2},\,
\Upsilon_{\lambda,\lambda+2}]\!],\,\, \epsilon_{17,\lambda}
[\![\Upsilon_{\lambda+8,\lambda+10},\,
\frak{J}_{9}^{-1,\lambda}]\!]+\frac{1}{3}\beta^{-1}_{\lambda+6}\epsilon_{9,\lambda}
[\![\frak{J}_{5}^{-1,\lambda+6},\,
\frak{J}_{7}^{-1,\lambda}]\!]\right),\\[2pt]
3)\,&\Big(\delta(\frak{J}_{\frac{23}{2}}^{-1,\lambda}),\,\,
[\![\frak{J}_{7}^{-1,\lambda+\frac{9}{2}},\,
\frak{J}_{\frac{11}{2}}^{-1,\lambda}]\!],\,\,
[\![\frak{J}_{9}^{-1,\lambda+\frac{5}{2}},\,
\Upsilon_{\lambda,\lambda+\frac{5}{2}}]\!],\,\,\\
&\quad~~~~ \epsilon_{9,\lambda}\gamma_{\lambda+6}^{-1}
[\![\frak{J}_{\frac{11}{2}}^{-1,\lambda+6},\,
\frak{J}_{7}^{-1,\lambda}]\!]-
\epsilon_{17,\lambda}[\![\Upsilon_{\lambda+8,\lambda+\frac{21}{2}},\,
\frak{J}_{9}^{-1,\lambda}]\!] \Big).
\end{array}$$
(Proposition \[th5\]) Using the same arguments as in proof of proposition \[pr3\] together with Lemma \[benfraj5\], Proposition \[pr3\] and Proposition \[th4\], we get the necessary integrability conditions for ${\frak L}^{(5)}$. Under these conditions, it can be easily checked that $\delta({\frak L}^{(m)})=0$ for $m=5,6,7,8.$
The main result in this section is the following theorem.
\[threc\] The conditions given in Propositions \[th2\], \[pr3\], \[th4\], \[th5\] are necessary and sufficient for the integrability of the infinitesimal deformation (\[infdef1\]). Moreover, any formal $\mathfrak{osp}(1|2)$-trivial deformation of the $\mathcal{K}(1)$-module ${\frak S}^n_{\beta}$ is equivalent to a polynomial one of degree $\leq4$.
Of course these conditions are necessary. Now, we show that these conditions are sufficient. The solution $\frak{L}^{(m)}$ of the Maurer-Cartan equation is defined up to a 1-cocycle and it has been shown in [@fi; @aalo] that different choices of solutions of the Maurer-Cartan equation correspond to equivalent deformations. Thus, we can always reduce $\frak{L}^{(m)},$ for $m=5,6,7,8,$ to zero by equivalence. Then, by recurrence, the terms $\frak{L}^{(m)}$, for $m\geq9$, satisfy the equation $\delta(\frak{L}^{(m)})=0$ and can also be reduced to the identically zero map.
Examples
========
We study formal $\mathfrak{osp}(1|2)$-trivial deformations of $\mathcal{K}(1)$-modules ${\frak S}^{n}_{\lambda+n}$ for some $n\in{1\over2}\mathbb{N}$ and for arbitrary generic $\lambda\in\mathbb{K}.$ For $n<5$, each of these deformations is equivalent to its infinitesimal one, without any integrability condition.
[\[Example1\][The $\mathcal{K}(1)$-module ${\frak
S}^{5}_{\lambda+5}$.]{}]{}
The $\mathcal{K}(1)$-module ${\frak S}^{5}_{\lambda+5}$ admits six formal $\mathfrak{osp}(1|2)$-trivial deformations with 18 independent parameters. These deformations are polynomial of degree 3.
In this case, any $\mathfrak{osp}(1|2)$-trivial deformation is given by $$\widetilde{\frak L}_{X_F}=\frak{L}_{X_F}+{\frak
L}^{(1)}_{X_F}+{\frak L}^{(2)}_{X_F}+{\frak L}^{(3)}_{X_F},$$ where $$\begin{array}{ll}
{\frak L}^{(1)}=&t_{\lambda, \lambda+\frac{3}{2}}\,
\Upsilon_{\lambda,\lambda+\frac{3}{2}}+t_{\lambda, \lambda+2}\,
\Upsilon_{\lambda,\lambda+2}+t_{\lambda, \lambda+\frac{5}{2}}\,
\Upsilon_{\lambda,\lambda+\frac{5}{2}}+t_{\lambda+{1\over2},
\lambda+2}\,
\Upsilon_{\lambda+{1\over2},\lambda+2}\\&+t_{\lambda+{1\over2},
\lambda+\frac{5}{2}}\,\Upsilon_{\lambda+{1\over2},\lambda+\frac{5}{2}}+t_{\lambda+{1\over2},
\lambda+3}\,\Upsilon_{\lambda+{1\over2},\lambda+3}+t_{\lambda+1,
\lambda+\frac{5}{2}}\,
\Upsilon_{\lambda+1,\lambda+\frac{5}{2}}+t_{\lambda+1,
\lambda+3}\, \Upsilon_{\lambda+1,\lambda+3}\\&+t_{\lambda+1,
\lambda+\frac{7}{2}}\,
\Upsilon_{\lambda+1,\lambda+\frac{7}{2}}+t_{\lambda+\frac{3}{2},
\lambda+3}\,
\Upsilon_{\lambda+\frac{3}{2},\lambda+3}+t_{\lambda+\frac{3}{2},
\lambda+\frac{7}{2}}\,
\Upsilon_{\lambda+\frac{3}{2},\lambda+\frac{7}{2}}+t_{\lambda+\frac{3}{2},
\lambda+4}\,
\Upsilon_{\lambda+\frac{3}{2},\lambda+4}\\&+t_{\lambda+2,
\lambda+\frac{7}{2}}\,
\Upsilon_{\lambda+2,\lambda+\frac{7}{2}}+t_{\lambda+2,
\lambda+4}\, \Upsilon_{\lambda+2, \lambda+4}+t_{\lambda+2,
\lambda+\frac{9}{2}}\, \Upsilon_{\lambda+2,
\lambda+\frac{9}{2}}+t_{\lambda+\frac{5}{2}, \lambda+4}\,
\Upsilon_{\lambda+\frac{5}{2},
\lambda+4}\\&+t_{\lambda+\frac{5}{2}, \lambda+\frac{9}{2}}\,
\Upsilon_{\lambda+\frac{5}{2},
\lambda+\frac{9}{2}}+t_{\lambda+\frac{5}{2}, \lambda+5}\,
\Upsilon_{\lambda+\frac{5}{2}, \lambda+5}+t_{\lambda+3,
\lambda+\frac{9}{2}}\, \Upsilon_{\lambda+3,
\lambda+\frac{9}{2}}+t_{\lambda+3, \lambda+5}\,
\Upsilon_{\lambda+3, \lambda+5}\\&+t_{\lambda+\frac{7}{2},
\lambda+5}\, \Upsilon_{\lambda+\frac{7}{2}, \lambda+5},\\[10pt]
{\frak L}^{(2) }=
&-\sum_\mu\zeta_\mu^{-1}t_{\mu+\frac{3}{2},\mu+3}
t_{\mu,\mu+\frac{3}{2}}\frak{J}_4^{-1,\mu}\\[2pt]
&-\sum_\nu\alpha_\nu^{-1}(t_{\nu+\frac{3}{2},\nu+\frac{7}{2}}
t_{\nu,\nu+\frac{3}{2}}- t_{\nu+2,\nu+\frac{7}{2}}t_{\nu,\nu+2})
\frak{J}_\frac{9}{2}^{-1,\nu}\\[2pt]&-\sum_\varepsilon\beta_\varepsilon^{-1}
(t_{\varepsilon+\frac{3}{2},\varepsilon+4}
t_{\varepsilon,\varepsilon+\frac{3}{2}}+
t_{\varepsilon+\frac{5}{2},\varepsilon+4}
t_{\varepsilon,\varepsilon+\frac{5}{2}}+{1\over3}t_{\varepsilon+2,\varepsilon+4}t_{\varepsilon,\varepsilon+2})
\frak{J}_{5}^{-1,\varepsilon}\\[2pt]
&-\sum_\ell\gamma_\ell^{-1}(t_{\ell+\frac{5}{2},\ell+\frac{9}{2}}
t_{\ell,\ell+\frac{5}{2}}-
t_{\ell+2,\ell+\frac{9}{2}}t_{\ell,\ell+2})
\frak{J}_\frac{11}{2}^{-1,\ell},\\[10pt]
{\frak L}^{(3) }=
&\sum_\ell\xi_\ell^{-1}t_{\ell+3,\ell+\frac{9}{2}}
t_{\ell+\frac{3}{2},\ell+3} t_{\ell,\ell+\frac{3}{2}}\,
\frak{J}_{11\over2}^{-1,\ell}\\[2pt]&+\,\epsilon_{3,\lambda}\,
\left(t_{\lambda+3,\lambda+5} t_{\lambda+\frac{3}{2},\lambda+3}-
t_{\lambda+\frac{7}{2},\lambda+5}t_{\lambda+\frac{3}{2},\lambda+\frac{7}{2}}\right)
t_{\lambda,\lambda+\frac{3}{2}}
\frak{J}_{6}^{-1,\lambda}\end{array}$$ with $\mu\in\{\lambda,\,\lambda+{1\over2},\,\lambda+1,\,\lambda+\frac{3}{2},\lambda+2\}$, $\nu\in\{\lambda,\,\lambda+{1\over2},\,\lambda+1,\,\lambda+\frac{3}{2}\}$, $\varepsilon\in\{\lambda,\,\lambda+{1\over2},\,\lambda+1\}$ and $\ell\in\{\lambda,\,\lambda+{1\over2}\}$ . The following equations $$\begin{gathered}
\label{co1}t_{\lambda,
\lambda+\frac{5}{2}}\,t_{\lambda+\frac{5}{2}, \lambda+5}=0,\\
t_{\lambda,\lambda+\frac{3}{2}}\left(\epsilon_{1,\lambda}\,
t_{\lambda+3,\lambda+5}
t_{\lambda+\frac{3}{2},\lambda+3}+(1-\epsilon_{1,\lambda})\,
t_{\lambda+\frac{7}{2},\lambda+5}
t_{\lambda+\frac{3}{2},\lambda+\frac{7}{2}}
\right)=0,\\
t_{\lambda,\lambda+\frac{3}{2}}\left(\epsilon_{2,\lambda}\,t_{\lambda+\frac{7}{2},\lambda+5}
t_{\lambda+\frac{3}{2},\lambda+\frac{7}{2}}-(1+\epsilon_{2,\lambda})\,
t_{\lambda+3,\lambda+5}
t_{\lambda+\frac{3}{2},\lambda+3}\right)=0,\\
t_{\lambda+\frac{7}{2},\lambda+5}t_{\lambda+2,\lambda+\frac{7}{2}}
t_{\lambda,\lambda+2}=0.\label{co2}\end{gathered}$$ are the integrability conditions of the infinitesimal deformation. The formal deformations with the greatest number of independent parameters are those corresponding to $t_{\lambda,\lambda+\frac{5}{2}}t_{\lambda+\frac{5}{2},\lambda+5}=t_{\lambda+\frac{7}{2},\lambda+5}t_{\lambda+2,\lambda+\frac{7}{2}}
t_{\lambda,\lambda+2}=t_{\lambda,\lambda+\frac{3}{2}}=0$. So, we must kill at least three parameters and there are six choices. Thus, there are only six deformations with eighteen independent parameters. Of course, there are many formal deformations with less then eighteen independent parameters. The deformation $
\widetilde{\frak L}_{X_F}=\frak{L}_{X_F}+{\frak
L}^{(1)}_{X_F}+{\frak L}^{(2) }_{X_F}+{\frak L}^{(3) }_{X_F}, $ is the miniversal $\mathfrak{osp}(1|2)$-trivial deformation of ${\frak S}^{5}_{\lambda+5}$ with base $\mathcal{A}=
\mathbb{C}[t]/\mathcal{R}$, where $t=(t_{\lambda,
\lambda+\frac{3}{2}},\dots)$ is the family of all parameters given in the expression of ${\frak L}^{(1)}$ and $\mathcal{R}$ is the ideal generated by the left hand sides of (\[co1\])–(\[co2\]).
[\[Example2\] [The $\mathcal{K}(1)$-module ${\frak S}^\frac{11}{2}_{\lambda+\frac{11}{2}}$.]{}]{}
The $\mathcal{K}(1)$-module ${\frak S}^\frac{11}{2}_{\lambda+\frac{11}{2}}$ admits 36 $\mathfrak{osp}(1|2)$-trivial deformations with 17 independent parameters. These deformations are polynomial of degree 3.
Any $\mathfrak{osp}(1|2)$-trivial deformation of ${\frak
S}^\frac{11}{2}_{\lambda+\frac{11}{2}}$ is given by $$\widetilde{\frak L}_{X_F}=\frak{L}_{X_F}+{\frak
L}^{(1)}_{X_F}+{\frak L}^{(2)}_{X_F}+{\frak L}^{(3)}_{X_F},$$ where $$\begin{array}{lll}
{\frak L}^{(1)}=&t_{\lambda, \lambda+\frac{3}{2}}\,
\Upsilon_{\lambda,\lambda+\frac{3}{2}}+t_{\lambda, \lambda+2}\,
\Upsilon_{\lambda,\lambda+2}+t_{\lambda, \lambda+\frac{5}{2}}\,
\Upsilon_{\lambda,\lambda+\frac{5}{2}}+t_{\lambda+{1\over2},
\lambda+2}\,
\Upsilon_{\lambda+{1\over2},\lambda+2}\\&+t_{\lambda+{1\over2},
\lambda+\frac{5}{2}}\,\Upsilon_{\lambda+{1\over2},\lambda+\frac{5}{2}}+t_{\lambda+{1\over2},
\lambda+3}\,\Upsilon_{\lambda+{1\over2},\lambda+3}+t_{\lambda+1,
\lambda+\frac{5}{2}}\,
\Upsilon_{\lambda+1,\lambda+\frac{5}{2}}+t_{\lambda+1,
\lambda+3}\, \Upsilon_{\lambda+1,\lambda+3}\\&+t_{\lambda+1,
\lambda+\frac{7}{2}}\,
\Upsilon_{\lambda+1,\lambda+\frac{7}{2}}+t_{\lambda+\frac{3}{2},
\lambda+3}\,
\Upsilon_{\lambda+\frac{3}{2},\lambda+3}+t_{\lambda+\frac{3}{2},
\lambda+\frac{7}{2}}\,
\Upsilon_{\lambda+\frac{3}{2},\lambda+\frac{7}{2}}+t_{\lambda+\frac{3}{2},
\lambda+4}\,
\Upsilon_{\lambda+\frac{3}{2},\lambda+4}\\&+t_{\lambda+2,
\lambda+\frac{7}{2}}\,
\Upsilon_{\lambda+2,\lambda+\frac{7}{2}}+t_{\lambda+2,
\lambda+4}\, \Upsilon_{\lambda+2, \lambda+4}+t_{\lambda+2,
\lambda+\frac{9}{2}}\, \Upsilon_{\lambda+2,
\lambda+\frac{9}{2}}+t_{\lambda+\frac{5}{2}, \lambda+4}\,
\Upsilon_{\lambda+\frac{5}{2},
\lambda+4}\\&+t_{\lambda+\frac{5}{2}, \lambda+\frac{9}{2}}\,
\Upsilon_{\lambda+\frac{5}{2},
\lambda+\frac{9}{2}}+t_{\lambda+\frac{5}{2}, \lambda+5}\,
\Upsilon_{\lambda+\frac{5}{2}, \lambda+5}+t_{\lambda+3,
\lambda+\frac{9}{2}}\, \Upsilon_{\lambda+3,
\lambda+\frac{9}{2}}+t_{\lambda+3, \lambda+5}\,
\Upsilon_{\lambda+3, \lambda+5}\\&+t_{\lambda+3,
\lambda+\frac{11}{2}}\, \Upsilon_{\lambda+3,
\lambda+\frac{11}{2}}+t_{\lambda+{7\over2}, \lambda+5}\,
\Upsilon_{\lambda+{7\over2}, \lambda+5}+t_{\lambda+\frac{7}{2},
\lambda+\frac{11}{2}}\, \Upsilon_{\lambda+\frac{7}{2},
\lambda+\frac{11}{2}}\\&+t_{\lambda+4, \lambda+\frac{11}{2}}\,
\Upsilon_{\lambda+4, \lambda+\frac{11}{2}},\\[10pt]
{\frak L}^{(2) }=
&-\sum_\mu\zeta_\mu^{-1}t_{\mu+\frac{3}{2},\mu+3}
t_{\mu,\mu+\frac{3}{2}}\frak{J}_4^{-1,\mu}\\[2pt]
&-\sum_\nu\alpha_\nu^{-1}(t_{\nu+\frac{3}{2},\nu+\frac{7}{2}}
t_{\nu,\nu+\frac{3}{2}}- t_{\nu+2,\nu+\frac{7}{2}}t_{\nu,\nu+2})
\frak{J}_\frac{9}{2}^{-1,\nu}\\[2pt]&-\sum_\varepsilon\beta_\varepsilon^{-1}
(t_{\varepsilon+\frac{3}{2},\varepsilon+4}
t_{\varepsilon,\varepsilon+\frac{3}{2}}+
t_{\varepsilon+\frac{5}{2},\varepsilon+4}
t_{\varepsilon,\varepsilon+\frac{5}{2}}+{1\over3}t_{\varepsilon+2,\varepsilon+4}t_{\varepsilon,\varepsilon+2})
\frak{J}_{5}^{-1,\varepsilon}\\[2pt]
&-\sum_\ell\gamma_\ell^{-1}(t_{\ell+\frac{5}{2},\ell+\frac{9}{2}}
t_{\ell,\ell+\frac{5}{2}}-
t_{\ell+2,\ell+\frac{9}{2}}t_{\ell,\ell+2})
\frak{J}_\frac{11}{2}^{-1,\ell},\\[10pt]
{\frak L}^{(3) }=
&\sum_\ell\xi_\ell^{-1}t_{\ell+3,\ell+\frac{9}{2}}
t_{\ell+\frac{3}{2},\ell+3} t_{\ell,\ell+\frac{3}{2}}\,
\frak{J}_{11\over2}^{-1,\ell}+\\&\sum_\iota\epsilon_{3,\iota}\,
\left(t_{\iota+3,\iota+5} t_{\iota+\frac{3}{2},\iota+3}-
t_{\iota+\frac{7}{2},\iota+5}t_{\iota+\frac{3}{2},\iota+\frac{7}{2}}\right)
t_{\iota,\iota+\frac{3}{2}}
\frak{J}_{6}^{-1,\iota}\\\end{array}$$ with $\mu\in\{\lambda,\,\lambda+{1\over2},\,\lambda+1,\,\lambda+\frac{3}{2},\lambda+2,\,\lambda+\frac{5}{2}\}$, $\nu\in\{\lambda,\,\lambda+{1\over2},\,\lambda+1,\,\lambda+\frac{3}{2},\,\lambda+2\}$, $\varepsilon\in\{\lambda,\,\lambda+{1\over2},\,\lambda+1,\,\lambda+\frac{3}{2}\}$, $\ell\in\{\lambda,\,\lambda+{1\over2},\,\lambda+1\}$ and $\iota\in\{\lambda,\,\lambda+{1\over2}\}$. The integrability conditions of this infinitesimal deformation vanishing of the following polynomials, where in the first four lines $\mu\in\left\{\lambda,\, \lambda+{1\over2}\right\}:$ $$\label{2-cocy}\begin{array}{llllllll}
t_{\mu,\mu+\frac{5}{2}}\,t_{\mu+\frac{5}{2}, \mu+5},\\
t_{\mu+\frac{7}{2},\mu+5}t_{\mu+2,\mu+\frac{7}{2}}
t_{\mu,\mu+2},\\
t_{\mu,\mu+\frac{3}{2}}\left(\epsilon_{1,\mu}\, t_{\mu+3,\mu+5}
t_{\mu+\frac{3}{2},\mu+3}+(1-\epsilon_{1,\mu})\,
t_{\mu+\frac{7}{2},\mu+5} t_{\mu+\frac{3}{2},\mu+\frac{7}{2}}
\right),\\
t_{\mu,\mu+\frac{3}{2}}\left(\epsilon_{2,\mu}\,t_{\mu+\frac{7}{2},\mu+5}
t_{\mu+\frac{3}{2},\mu+\frac{7}{2}}-(1+\epsilon_{2,\mu})\,
t_{\mu+3,\mu+5}
t_{\mu+\frac{3}{2},\mu+3}\right),\\
t_{\lambda,\lambda+\frac{3}{2}} t_{\lambda+3,\lambda+\frac{11}{2}}
t_{\lambda+\frac{3}{2},\lambda+3},\\
t_{\lambda+4,\lambda+\frac{11}{2}}t_{\lambda+\frac{5}{2},\lambda+4}
t_{\lambda,\lambda+\frac{5}{2}},\\
t_{\lambda+4,\lambda+\frac{11}{2}}\left(3(1+\epsilon_{4,\lambda})\,t_{\lambda+\frac{3}{2},\lambda+4}
t_{\lambda,\lambda+\frac{3}{2}}+
t_{\lambda+2,\lambda+4}t_{\lambda,\lambda+2}\right)+
\epsilon_{4,\lambda}\, t_{\lambda,\lambda+\frac{3}{2}}
t_{\lambda+\frac{7}{2},\lambda+\frac{11}{2}}t_{\lambda+\frac{3}{2},\lambda+\frac{7}{2}},\\
t_{\lambda+\frac{7}{2},\lambda+\frac{11}{2}}
\left((1+\frac{\epsilon_{5,\lambda}}{3})\,t_{\lambda+\frac{3}{2},\lambda+\frac{7}{2}}
t_{\lambda,\lambda+\frac{3}{2}}-
t_{\lambda+2,\lambda+\frac{7}{2}}t_{\lambda,\lambda+2}\right)+\epsilon_{5,\lambda}\,t_{\lambda+4,\lambda+\frac{11}{2}}t_{\lambda+\frac{3}{2},\lambda+4}
t_{\lambda,\lambda+\frac{3}{2}},\\
\epsilon_{6,\lambda}\,t_{\lambda,\lambda+\frac{3}{2}}\left(
t_{\lambda+4,\lambda+\frac{11}{2}}t_{\lambda+\frac{3}{2},\lambda+4}+\frac{1}{3}\,
t_{\lambda+\frac{7}{2},\lambda+\frac{11}{2}}t_{\lambda+\frac{3}{2},\lambda+\frac{7}{2}}\right)+\\
\quad
t_{\lambda,\lambda+2}\left(t_{\lambda+\frac{7}{2},\lambda+\frac{11}{2}}
t_{\lambda+2,\lambda+\frac{7}{2}}-
t_{\lambda+4,\lambda+\frac{11}{2}}t_{\lambda+2,\lambda+4}\right).
\end{array}$$ These deformations are with 24 parameters $t_{\mu,\nu}$ which are subject to conditions (\[2-cocy\]). Obviously, we can construct many $\mathfrak{osp}(1|2)$-trivial deformation of $\mathfrak{S}^\frac{11}{2}_{\lambda+\frac{11}{2}}$ with independent parameters. But, to have the greatest number of independent parameters, we see that we must kill at least seven parameters, that is , we put $$t_{\lambda,\lambda+\frac{3}{2}}=t_{\lambda+{1\over2},\lambda+2}=t_{\lambda,\lambda+2}=0
\quad\text{ and }\quad
t_{\mu,\mu+\frac{5}{2}}\,t_{\mu+\frac{5}{2},
\mu+5}=t_{\mu+\frac{7}{2},\mu+5}t_{\mu+2,\mu+\frac{7}{2}}
t_{\mu,\mu+2}=0$$ where $\mu=\lambda$ or $\lambda+{1\over2}$. So, there are 36 possible choices of such parameters.
Acknowledgments {#acknowledgments .unnumbered}
===============
We would like to thank Dimitry Leites and Valentin Ovsienko for helpful discussions. We are also grateful to the referee for his comments and suggestion.
[00]{} Agrebaoui B, Ammar F, Lecomte P, Ovsienko V, Multi-parameter deformations of the module of symbols of differential operators, [*Internat. Mathem. Research Notices*]{} [**16**]{} (2002), 847–869.
Agrebaoui B and Ben Fraj N, On the cohomology of the Lie superalgebra of contact vector fields on $S^{1|1}$, [*Bell. Soc. Roy. Sci. Liège*]{} [**72**]{}, 6, 2004, 365–375.
Agrebaoui A, Ben Fraj N, Ben Ammar M, Ovsienko V, Deformations of modules of differential forms, [*J. Nonlinear Math. Phys.*]{} [**10**]{} (2003), 148–156.
Basdouri I, Ben Ammar M, Cohomology of $\mathfrak {osp}(1|2)$ acting on linear differential operators on the supercircle $S^{1|1}$. Letters in Mathematical Physics (2007) 81:239–251
Basdouri I, Ben Ammar M, Dali B, Omri S, Deformation of ${\rm
Vect_P}(\mathbb{R})$-modules of symbols. `arXiv: math.RT/0702664`.
Ben Ammar M, Boujelbene M, $\mathrm{sl}(2)$-Trivial deformations of ${\rm Vect_P}(\mathbb{R})$-modules of symbols. SIGMA 4 (2008), 065.
Bouarroudj S, On sl(2)-relative cohomology of the Lie algebra of vector fields and differential operators, J. Nonlinear Math. Phys., no.1, (2007), 112–127.
Bouarroudj S, Projective and conformal Schwarzian derivatives and cohomology of Lie algebras vector fields related to differential operators, [*Int. Jour. Geom. Methods. Mod. Phys.*]{} [**3**]{} (2006), 667–696.
Bouarroudj S, Ovsienko V, Three cocycles on Diff($S^1$) generalizing the Schwarzian derivative, [*Internat. Math. Res. Notices*]{} [**1**]{} (1998), 25–39.
Conley C H, Conformal symbols and the action of contact vector fields over the superline, `arXiv: 0712.1780v2 [math.RT]`.
Feigin B L, Fuchs D B, Homology of the Lie algebras of vector fields on the line, [*Func. Anal. Appl*]{}., 14 (1980) 201–212.
Fialowski A, Deformations of Lie algebras, Math. USSR-Sb. [**55**]{} (1986), 467-473.
Fialowski A, An example of formal deformations of Lie algebras , Deformation Theory of Algebras and Structures and Appl., Kluwer (1988), 375-401.
Fialowski A, Fuchs D B, Construction of miniversal deformations of Lie algebras, [*J. Func. Anal.*]{} [**161**]{} (1999) 76–110.
Fuchs D B, [*Cohomology of infinite-dimensional Lie algebras*]{}, Plenum Publ. New York, 1986.
Gargoubi H, Mellouli N and Ovsienko V, Differential operators on supercircle: conformally equivariant quantization and symbol calculus, Letters in Mathematical Physics (2007) [**79**]{}: 51–65.
Gargoubi H and Ovsienko V, Supertransvectants and symplectic geometry, `arXiv: 0705.1411v1 [math-ph]`.
Gieres F, Theisen S, Superconformally covariant operators and super W-algebras, J. Math. Phys. 34 (1993) 5964-5985.
Gordan P, [*Invariantentheorie*]{}, Teubner, Leipzig, 1887.
Grozman P., Invariant bilinear differential operators. `arXiv: math.RT/0509562`.
Grozman P, Leites D and Shchepochkina I, Lie superalgebras of string theories, Acta Mathematica Vietnamica, v. 26, 2001, no. 1, 27–63; `arXiv: hep-th/9702120`
Grozman P, Leites D and Shchepochkina I, Invariant differential operators on supermanifolds and Standard Models, In: Olshanetski, M., Vainstein, A. (eds.) Multiple Facets Quantization and Supersymmetry, pp. 508–555. World Scientific, Singapore (2002). `arXiv: math.RT/0202193`
Nijenhuis A, Richardson R W Jr., Deformations of homomorphisms of Lie groups and Lie algebras, [*Bull. Amer. Math. Soc.*]{} [**73**]{} (1967), 175–179.
Ovsienko V, Roger C, Deforming the Lie algebra of vector fields on $S^1$ inside the Lie algebra of pseudodifferential operators on $S^1$, [*AMS Transl. Ser. 2,*]{} (Adv. Math. Sci.) [**194**]{} (1999), 211–227.
Ovsienko V, Roger C, Deforming the Lie algebra of vector fields on $S^1$ inside the Poisson algebra on $\dot T^*S^1$, [*Comm. Math. Phys.*]{} [**198**]{} (1998), 97–110.
Richardson R W, Deformations of subalgebras of Lie algebras, [*J. Diff. Geom.*]{} [**3**]{} (1969), 289–308.
|
---
abstract: 'We show a Lichnerowicz-Obata type estimate for the first eigenvalue of the Laplacian of $n$-dimensional closed Riemannian manifolds with an almost parallel $p$-form ($2\leq p \leq n/2$) in $L^2$-sense, and give an almost decomposition result of the manifold under some pinching conditions when $2\leq p<n/2$.'
address: 'Graduate School of Mathematics, Nagoya University, Chikusa-Ku Nagoya, 464-8602, Japan'
author:
- Masayuki Aino
title: 'Lichnerowicz-Obata Estimate, Almost Parallel $p$-form and Almost Product Manifolds'
---
Introduction
============
In this paper we give an estimate for the first eigenvalue of the Laplacian of closed Riemannian manifolds with positive Ricci curvature and an almost parallel form, and give a pinching result about the almost equality case.
One of the most famous theorem about the estimate of the first eigenvalue of the Laplacian is the Lichnerowicz-Obata theorem. Lichnerowicz showed the optimal comparison result for the first eigenvalue when the Riemannian manifold has positive Ricci curvature, and Obata showed that the equality of the Lichnerowicz estimate implies that the Riemannian manifold is isometric to the standard sphere. In the following, $\lambda_k(g)$ denotes the $k$-th eigenvalue of the Laplacian $\Delta:=-{\mathop{\mathrm{tr}}\nolimits}_g {\mathop{\mathrm{Hess}}\nolimits}$ acting on functions.
Take an integer $n\geq 2$. Let $(M,g)$ be an $n$-dimensional closed Riemannian manifold. If ${\mathop{\mathrm{Ric}}\nolimits}\geq (n-1) g$, then $\lambda_1(g)\geq n$. The equality holds if and only if $(M,g)$ is isometric to the standard sphere of radius $1$.
Petersen [@Pe1], Aubry [@Au] and Honda [@Ho] showed the stability result of the Lichnerowicz-Obata theorem. In the following, $d_{GH}$ denotes the Gromov-Hausdorff distance and $S^n$ denotes the $n$-dimensional standard sphere of radius $1$. (see Definition \[DGH\] for the definition of the Gromov-Hausdorff distance).
\[PA\] For given an integer $n\geq 2$ and a positive real number $\epsilon>0$, there exists $\delta(n,\epsilon)>0$ such that if $(M,g)$ is an $n$-dimensional closed Riemannian manifold with ${\mathop{\mathrm{Ric}}\nolimits}\geq (n-1) g$ and $\lambda_n(g)\leq n+\delta$, then $d_{GH}(M,S^n)\leq \epsilon$.
Note that Petersen considered the pinching condition on $\lambda_{n+1}(g)$, and Aubry and Honda improved it independently.
We mention some improvements of the Lichnerowicz estimate when the Riemannian manifold has a special structure. If $(M,g)$ is a real $n$-dimensional Kähler manifold with ${\mathop{\mathrm{Ric}}\nolimits}\geq (n-1)g$, then the Lichnerowicz estimate is improved as follows: $$\label{kae}
\lambda_1(g)\geq 2(n-1).$$ See [@Be Theorem 11.49] for the proof. If $(M,g)$ is a real $n$-dimensional quaternionic Kähler manifold with ${\mathop{\mathrm{Ric}}\nolimits}\geq (n-1)g$, then we have $$\label{qk}
\lambda_1(g)\geq \frac{2n+8}{n+8}(n-1).$$ See [@AM] for the proof. For these cases, the Riemannian manifold $(M,g)$ has a non-trivial parallel $2$ and $4$-form, respectively. When $(M,g)$ is an $n$-dimensional product Riemannian manifold $(N_1\times N_2,g_1+g_2)$ with ${\mathop{\mathrm{Ric}}\nolimits}\geq (n-1)g$, then we have $$\lambda_1(g)\geq \min_{i\in\{1,2\}}\left\{\frac{\dim N_i}{\dim N_i-1}\right\}(n-1),$$ and $M$ has a non-trivial parallel form if either $N_1$ or $N_2$ is orientable.
Grosjean [@gr] gave a unified proof of the improvements of the Lichnerowicz estimate when the Riemannian manifold has a non-trivial parallel form.
\[grosjean\] Let $(M,g)$ be an $n$-dimensional closed Riemannian manifold. Assume that ${\mathop{\mathrm{Ric}}\nolimits}\geq (n-p-1)g$ and that there exists a nontrivial parallel $p$-form on $M$ $(2\leq p\leq n/2)$. Then, we have $$\label{grs}
\lambda_1(g)\geq n-p.$$
Moreover, if $p<n/2$ and if in addition $M$ is simply connected, then the equality in $(\ref{grs})$ implies that $(M,g)$ is isometric to a product $S^{n-p}\times (X,g')$, where $(X,g')$ is some $p$-dimensional closed Riemannian manifold.
We give several remarks on this theorem.
- Grosjean also showed this type theorem when $M$ has a convex smooth boundary.
- Though Grosjean originally assumed the manifold is orientable, the assumption can be easily removed by taking the orientable double covering.
- If $M$ is simply connected, $p=n/2$ and $n\geq 6$, then it is not difficult to show that the equality in (\[grs\]) also implies that $M$ is isometric to a product $S^{n/2}\times X$ (see Corollary \[p3d\]).
- If $(M,g)$ is either a Kähler manifold or a quaternionic Kähler manifold, then the estimate (\[kae\]) or (\[qk\]) is better.
- If there exists a non-trivial parallel $p$-form $\omega$ ($1\leq p\leq n-1$) on an $n$-dimensional Riemannian manifold $(M,g)$, then $\omega(x)\in \bigwedge^p T^\ast_x M$ ($x\in M$) is invariant under the Holonomy action, and so the Holonomy group coincides with neither $\mathrm{SO}(n)$ nor $\mathrm{O}(n)$.
The main aim of this paper is to show the almost version of Grosjean’s result. We also give the almost version of the estimate (\[kae\]) in Appendix B.
We first note that, for a closed Riemannian manifold $(M,g)$, there exists a non zero $p$-form $\omega$ with $\|\nabla \omega\|_2^2\leq \delta\|\omega\|_2^2$ for some $\delta>0$ if and only if $\lambda_1(\Delta_{C,p})\leq \delta$ holds, where $\lambda_1(\Delta_{C,p})$ is defined by $$\lambda_1(\Delta_{C,p}):=\inf\left\{\frac{\|\nabla \omega\|_2^2}{\|\omega\|_2^2}: \omega\in\Gamma(\bigwedge^p T^\ast M) \text{ with }\omega\neq 0\right\}.$$
Let us state our eigenvalue estimate.
For given integers $n\geq 4$ and $2\leq p \leq n/2$, there exists a constant $C(n,p)>0$ such that if $(M,g)$ is an $n$-dimensional closed Riemannian manifold with ${\mathop{\mathrm{Ric}}\nolimits}_g\geq (n-p-1)g$, then we have $$\lambda_1(g)\geq n-p-C(n,p)\lambda_1(\Delta_{C,p})^{1/2}.$$
We immediately have the following corollary:
For given integers $n\geq 4$ and $2\leq p \leq n/2$, there exists a constant $C(n,p)>0$ such that if $(M,g)$ is an $n$-dimensional closed Riemannian manifold with ${\mathop{\mathrm{Ric}}\nolimits}_g\geq (n-p-1)g$ and $$\frac{n(n-p-1)}{n-1}\leq \lambda_1(g)\leq n-p,$$ then we have $$\lambda_1(\Delta_{C,p})\geq \left(\frac{n-p-\lambda_1(g)}{C(n,p)}\right)^2.$$
Note that we always have the lower bound on the eigenvalue of the Laplacian $\lambda_1(g)\geq n(n-p-1)/(n-1)$ if ${\mathop{\mathrm{Ric}}\nolimits}_g\geq (n-p-1)g$ by the Lichnerowicz estimate. An upper bound on $C(n,p)$ is computable. However, we do not know the optimal value of it.
We next state the eigenvalue pinching result.
For given integers $n\geq 5$ and $2\leq p < n/2$ and a positive real number $\epsilon>0$, there exists $\delta=\delta(n,p,\epsilon)>0$ such that if $(M,g)$ is an $n$-dimensional closed Riemannian manifold with ${\mathop{\mathrm{Ric}}\nolimits}_g\geq (n-p-1)g$, $$\lambda_{n-p+1}(g)\leq n-p+\delta$$ and $$\lambda_1(\Delta_{C,p})\leq \delta,$$ then $M$ is orientable and $$d_{GH}(M,S^{n-p}\times X)\leq \epsilon,$$ where $X$ is some compact metric space.
In fact, we prove that there exist constants $C(n,p)>0$ and $\alpha(n)>0$ such that $$d_{GH}(M,S^{n-p}\times X)\leq C(n,p)\delta^{\alpha(n)}$$ under the assumption of Main Theorem 2. One can easily find the explicit value of $\alpha(n)$ (see Notation \[order\] and Theorem \[MT2\]). However, it might be far from the optimal value. By the Gromov’s pre-compactness theorem, we can take $X$ to be a geodesic space. However, we lose the information about the convergence rate in that case.
Based on Theorem \[PA\], one might expect that we can replace the assumption “$\lambda_{n-p+1}(g)\leq n-p+\delta$” in Main Theorem 2 to the weaker assumption “$\lambda_{n-p}(g)\leq n-p+\delta$”. However, an example shows that we cannot do it even if $\delta=0$ (see Proposition \[p3e\]). Instead of that, we have the following theorems:
For given integers $n\geq 4$ and $2\leq p \leq n/2$, there exists a constant $C(n,p)>0$ such that if $(M,g)$ is an $n$-dimensional closed Riemannian manifold with ${\mathop{\mathrm{Ric}}\nolimits}_g\geq (n-p-1)g$, then we have $$\lambda_1(g)\geq n-p-C(n,p)\lambda_1(\Delta_{C,n-p})^{1/2}.$$
For given integers $n\geq 5$ and $2\leq p < n/2$ and a positive real number $\epsilon>0$, there exists $\delta=\delta(n,p,\epsilon)>0$ such that if $(M,g)$ is an $n$-dimensional closed Riemannian manifold with ${\mathop{\mathrm{Ric}}\nolimits}_g\geq (n-p-1)g$, $$\lambda_{n-p}(g)\leq n-p+\delta$$ and $$\lambda_1(\Delta_{C,n-p})\leq \delta,$$ then we have $$d_{GH}(M,S^{n-p}\times X)\leq \epsilon,$$ where $X$ is some compact metric space.
Note that the assumption “$\lambda_1(\Delta_{C,n-p})\leq \delta$” is equivalent to the assumption “$\lambda_1(\Delta_{C,p})\leq \delta$” if the manifold is orientable. In particular, we have the following corollary:
For given integers $n\geq 5$ and $2\leq p < n/2$ and a positive real number $\epsilon>0$, there exists $\delta=\delta(n,p,\epsilon)>0$ such that if $(M,g)$ is an $n$-dimensional orientable closed Riemannian manifold with ${\mathop{\mathrm{Ric}}\nolimits}_g\geq (n-p-1)g$, $$\lambda_{n-p}(g)\leq n-p+\delta$$ and $$\lambda_1(\Delta_{C,p})\leq \delta,$$ then we have $$d_{GH}(M,S^{n-p}\times X)\leq \epsilon,$$ where $X$ is some compact metric space.
We would like to point out that our work was motivated by Honda’s spectral convergence theorem [@Ho2], which asserts the continuity of the eigenvalues of the connection Laplacian $\Delta_{C,p}$ acting on $p$-forms with respect to the non-collapsing Gromov-Hausdorff convergence assuming the two-sided bound on the Ricci curvature. By virtue of his theorem, we can generalize our main theorems to Ricci limit spaces under such assumptions. See Appendix A for detail. Note that we show our main theorems without the non-collapsing assumption, i.e., without assuming the lower bound on the volume of the Riemannian manifold.
Our work was also motivated by the Cheeger-Colding almost splitting theorem (see [@Ch Theorem 9.25]), whose conclusion is the Gromov-Hausdorff approximation to a product $\mathbb{R}\times X$. As the almost splitting theorem, we need to show the almost Pythagorean theorem under the assumption of Main Theorem 2. One step of the proof (Lemma \[p54i\]) is similar to the final step of the almost splitting theorem [@Ch Lemma 9.16].
The structure of this paper is as follows.
In section 2, we recall some basic definitions and facts, and give calculations of differential forms.
In section 3, we assume that the Riemannian manifold has a non-trivial parallel $p$-form. We give an easy proof of the formula used by Grosjean to prove Theorem \[grosjean\].
In section 4, we estimate the error terms of the Grosjean’s formula when the Riemannian manifold has a non-trivial almost parallel $p$-form. As a consequence, we prove Main Theorem 1 and Main Theorem 3.
In section 5, we prove Main Theorem 2 and Main Theorem 4. In subsection 5.1, we list some useful techniques for pinching problems. In subsection 5.2, we show some pinching conditions on the eigenfunctions along geodesics under the assumption $\lambda_k(g)\leq n-p+\delta$ and $\lambda_1(\Delta_{C,p})\leq \delta$. In subsection 5.3, we show that similar results hold under the assumption $\lambda_k(g)\leq n-p+\delta$ and $\lambda_1(\Delta_{C,n-p})\leq \delta$. In subsection 5.4, we show that the eigenfunctions are almost cosine functions in some sense under our pinching condition. In subsection 5.5, we construct an approximation map and show Main Theorem 2 except for the orientability. In subsection 5.6, we give some lemmas to prove the remaining part of main theorems. In subsection 5.7, we show the orientability of the manifold under the assumption of Main Theorem 2, and complete the proof of it. In subsection 5.8, we show that the assumption of Main Theorem 4 implies that $\lambda_{n-p+1}(g)$ is close to $n-p$, and complete the proof of Main Theorem 4.
In Appendix A, we discuss Ricci limit spaces. We show a gap theorem of the first eigenvalue of the Laplacian acting on $n$-forms for $n$-dimensional unorientable closed Riemannian manifolds. As a consequence, we show the stability of unorientability under the non-collapsing Gromov-Hausdorff convergence assuming the two-sided bound on the Ricci curvature and the upper bound on the diameter. This enable us to generalize our main theorems to Ricci limit spaces under such assumptions.
In Appendix B, we give the almost version of the estimate (\[kae\]) assuming that there exists a $2$-form $\omega$ which satisfies that $\|\nabla \omega\|_2$ and $\|J_\omega^2+{\mathop{\mathrm{Id}}\nolimits}\|_2$ are small, where $J_\omega\in\Gamma(T^\ast M\otimes T M)$ is defined so that $\omega=g(J_\omega\cdot,\cdot)$.
[**Acknowledgments**]{}. I am grateful to my supervisor, Professor Shinichiroh Matsuo for his advice. I also thank Professor Shouhei Honda for helpful discussions about the orientability of Ricci limit spaces. I thank Shunsuke Kano for the discussions about the examples. The works in section 3 were done during my stay at the University of Côte d’Azur. I would like to thank Professor Erwann Aubry for his warm hospitality. This work was supported by JSPS Overseas Challenge Program for Young Researchers and by JSPS Research Fellowships for Young Scientists (JSPS KAKENHI Grant Number JP18J11842).
Preliminaries
=============
Basic Definitions
-----------------
We first recall some basic definitions and fix our convention.
\[Dhau\] Let $(X,d)$ be a metric space. For each point $x_0\in X$, subsets $A,B\subset X$ and $r>0$, define $$\begin{aligned}
d(x_0,A):=&\inf\{d(x_0,a):a\in A\},\\
B_{r}(x_0):=&\{x\in X: d(x,x_0)<r\},\\
B_{r}(A):=&\{x\in X:d(x,A)<r\},\\
d_{H,d}(A,B):=&\inf\{\epsilon>0:A\subset B_{\epsilon}(B) \text{ and } B\subset B_{\epsilon}(A)\}\end{aligned}$$ We call $d_{H,d}$ the Hausdorff distance.
The Hausdorff distance defines a metric on the collection of compact subsets of $X$.
\[DGH\] Let $(X,d_X),(Y,d_Y)$ be metric spaces. Define $$\begin{aligned}
d_{GH}(X,Y):=\inf\Big\{d_{H,d}(X,Y): &\text{ $d$ is a metric on $X\coprod Y$ such that}\\
&\qquad\qquad\quad\text{$d|_X=d_X$ and $d|_Y=d_Y$}\Big\}.\end{aligned}$$
The Gromov-Hausdorff distance defines a metric on the set of isometry classes of compact metric spaces (see [@Pe3 Proposition 11.1.3]).
\[hap\] Let $(X,d_X),(Y,d_Y)$ be metric spaces. We say that a map $f\colon X\to Y$ is an $\epsilon$-Hausdorff approximation map for $\epsilon>0$ if the following two conditions hold.
- For all $a,b\in X$, we have $|d_X(a,b)-d_Y(f(a),f(b))|< \epsilon$,
- $f(X)$ is $\epsilon$-dense in $Y$, i.e., for all $y\in Y$, there exists $x\in X$ with $d_Y(f(x),y)< \epsilon$.
If there exists an $\epsilon$-Hausdorff approximation map $f\colon X\to Y$, then we can show that $d_{GH}(X,Y)\leq 3\epsilon/2$ by considering the following metric $d$ on $X\coprod Y$:
[align\*]{} &d\_X(a,b)&& (a,bX),\
& +\_[xX]{}(d\_X(a,x)+d\_Y(f(x),b))&&(aX,bY),\
&d\_Y(a,b)&&(a,bY).
If $d_{GH}(X,Y)< \epsilon$, then there exists a $2\epsilon$-Hausdorff approximation map from $X$ to $Y$.
Let $C(u_1,\ldots,u_l)>0$ denotes a positive function depending only on the numbers $u_1,\ldots,u_l$. For a set $X$, ${\mathop{\mathrm{Card}}\nolimits}X$ denotes a cardinal number of $X$.
Let $(M,g)$ be a closed Riemannian manifold. For any $p\geq 1$, we use the normalized $L^p$-norm: $$\|f\|_p^p:=\frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)}\int_M |f|^p\,d\mu_g,$$ and $\|f\|_{\infty}:=\mathop{\mathrm{sup~ess}}\limits_{x\in M}|f(x)|$ for a measurable function $f$ on $M$. We also use this notation for tensors. We have $\|f\|_p\leq \|f\|_q$ for any $p\leq q \leq \infty$.
Let $\nabla$ denotes the Levi-Civita connection. Throughout in this paper, $0=\lambda_0(g)< \lambda_1(g) \leq \lambda_2(g) \leq\cdots \to \infty$ denotes the eigenvalues of the Laplacian $\Delta=-\sum_{i,j}g^{ij}\nabla_i \nabla_j$ acting on functions. We sometimes identify $TM$ and $T^\ast M$ using the metric $g$. Given points $x,y\in M$, let $\gamma_{x,y}$ denotes one of minimal geodesics with unit speed such that $\gamma_{x,y}(0)=x$ and $\gamma_{x,y}(d(x,y))=y$. For given $x\in M$ and $u\in T_x M$ with $|u|=1$, let $\gamma_{u}\colon \mathbb{R}\to M$ denotes the geodesic with unit speed such that $\gamma_u(0)=x$ and $\dot{\gamma}_u(0)=u$.
For any $x\in M$ and $u\in T_x M$ with $|u|=1$, put $$t(u):=\sup\{t\in\mathbb{R}_{>0}: d(x,\gamma_u(t))=t\},$$ and define the interior set $I_x\subset M$ at $x$ (see also [@Sa p.104]) by $$I_x:=\{\gamma_u (t): u\in T_x M \text{ with $|u|=1$ and } 0\leq t< t(u)\}.$$ Then, $I_x$ is open and ${\mathop{\mathrm{Vol}}\nolimits}(M\setminus I_x)=0$ [@Sa III Lemma 4.4]. For any $y\in I_x\setminus \{x\}$, the minimal geodesic $\gamma_{x,y}$ is uniquely determined. The function $d(x,\cdot)\colon M\to \mathbb{R}$ is differentiable in $I_x\setminus\{x\}$ and $\nabla d(x,\cdot)(y)=\dot{\gamma}_{x,y}(d(x,y))$ holds for any $y\in I_x\setminus \{x\}$ [@Sa III Proposition 4.8].
Let $V$ be an $n$-dimensional real vector space with an inner product $\langle,\rangle$. We define inner products on $\bigwedge^k V$ and $V\otimes \bigwedge^k V$ as follows: $$\begin{split}
&\langle v_1\wedge\ldots\wedge v_k,w_1\wedge \ldots\wedge w_k\rangle=\det \{\langle v_i,w_j\rangle\}_{i,j},\\
&\langle v_0\otimes v_1\wedge\ldots\wedge v_k,w_0\otimes w_1\wedge \ldots\wedge w_k\rangle=\langle v_0,w_0\rangle \det \{\langle v_i,w_j\rangle \}_{i,j},
\end{split}$$ for $v_0,\ldots,v_k,w_0,\ldots,w_k\in V$. For $\alpha\in V$ and $\omega\in \bigwedge^k V$, there exists unique $\iota(\alpha)\omega\in \bigwedge^{k-1} V$ such that $\langle\iota(\alpha)\omega,\eta\rangle=\langle\omega,\alpha \wedge \eta\rangle$ holds for any $\eta\in \bigwedge^{k-1} V $. If $k=0$, we define $\iota(\alpha)\omega=0$ and $\bigwedge^{-1}V=\{0\}$. Then, $\iota$ defines a bi-linear map: $$\iota\colon V\times \bigwedge^k V\to \bigwedge^{k-1} V.$$ By identifying $V$ and $V^\ast$ using $\langle,\rangle$, we also use the notation $\iota$ for the bi-linear map: $$\iota\colon V^\ast \times \bigwedge^k V\to \bigwedge^{k-1} V.$$
For any Riemannian manifold $(M,g)$, we define operators $\nabla^\ast \colon \Gamma(T^\ast M\otimes \bigwedge^k T^\ast M)\to \Gamma(\bigwedge^k T^\ast M)$ and $d^\ast \colon \Gamma(\bigwedge^k T^\ast M)\to \Gamma(\bigwedge^{k-1}T^\ast M)$ by $$\begin{aligned}
\nabla^\ast(\alpha\otimes \beta):&=-{\mathop{\mathrm{tr}}\nolimits}_{T^\ast M} \nabla(\alpha\otimes \beta)
=-\sum_{i=1}^n \left(\nabla_{e_i}\alpha\right)(e_i)\cdot \beta-\sum_{i=1}^n\alpha(e_i)\cdot\nabla_{e_i}\beta.\\
d^\ast \omega:&=-\sum_{i=1}^n\iota(e_i)\nabla_{e_i}\omega\end{aligned}$$ for all $\alpha\otimes\beta\in \Gamma(T^\ast M\otimes\bigwedge^k T^\ast M)$ and $\omega\in\Gamma(\bigwedge^k T^\ast M)$, where $n=\dim M$ and $\{e_1,\ldots,e_n\}$ is an orthonormal basis of $TM$. If $M$ is closed, then we have $$\begin{aligned}
\int_M \langle T,\nabla \alpha\rangle\,d\mu_g&=\int_M \langle \nabla^\ast T, \alpha\rangle\,d\mu_g,\\
\int_M \langle \omega,d\eta \rangle\,d\mu_g&=\int_M \langle d^\ast \omega, \eta \rangle\,d\mu_g\end{aligned}$$ for all $T\in\Gamma(T^\ast M\otimes\bigwedge^k T^\ast M)$, $\alpha\in\Gamma(\bigwedge^k T^\ast M)$, $\omega\in\Gamma(\bigwedge^k T^\ast M)$ and $\eta\in\Gamma(\bigwedge^{k-1} T^\ast M)$ by the divergence theorem. The Hodge Laplacian $\Delta\colon \Gamma(\bigwedge^k T^\ast M)\to\Gamma(\bigwedge^k T^\ast M)$ is defined by $$\Delta:=d d^\ast +d^\ast d.$$
For an $n$-dimensional Riemannian manifold $(M,g)$, we can take orthonormal basis of $TM$ only locally in general. However, for example, the tensor $$\sum_{i=1}^n e^i\otimes \iota(\nabla_{e_i} \nabla f)\omega\in \Gamma(T^\ast M\otimes \bigwedge^{k-1} T^\ast M)\quad (f\in C^\infty(M),\,\omega\in \Gamma(\bigwedge^k T^\ast M))$$ is defined independently of the choice of the orthonormal basis $\{e_1,\ldots,e_n\}$ of $TM$, where $\{e^1,\ldots,e^n\}$ denotes its dual. Thus, we sometimes use such notation without taking a particular orthonormal basis.
Finally, we list some important notation. Let $(M,g)$ be a closed Riemannian manifold.
- $d$ denotes the Riemannian distance function.
- ${\mathop{\mathrm{Ric}}\nolimits}$ denotes the Ricci curvature tensor.
- ${\mathop{\mathrm{diam}}\nolimits}$ denotes the diameter.
- ${\mathop{\mathrm{Vol}}\nolimits}$ or $\mu_g$ denotes the Riemannian volume measure.
- $\|\cdot\|_p$ denotes the normalized $L^p$-norm for each $p\geq 1$, which is defined by $$\|f\|_p^p:=\frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)}\int_M |f|^p\,d\mu_g$$ for any measurable function $f$ on $M$.
- $\|f\|_{\infty}$ denotes the essential sup of $|f|$ for any measurable function $f$ on $M$.
- $\nabla$ denotes the Levi-Civita connection.
- $\nabla^2$ denotes the Hessian for functions.
- $\Delta\colon \Gamma(\bigwedge^k T^\ast M)\to\Gamma(\bigwedge^k T^\ast M)$ denotes the Hodge Laplacian defined by $\Delta:=d d^\ast +d^\ast d$. We frequently use the Laplacian acting on functions. Note that $\Delta=-{\mathop{\mathrm{tr}}\nolimits}_g \nabla^2$ holds for functions under our sign convention.
- $0=\lambda_0(g)< \lambda_1(g) \leq \lambda_2(g) \leq\cdots \to \infty$ denotes the eigenvalues of the Laplacian acting on functions.
- $\gamma_{x,y}\colon [0,d(x,y)]\to M$ denotes one of minimal geodesics with unit speed such that $\gamma_{x,y}(0)=x$ and $\gamma_{x,y}(d(x,y))=y$ for any $x,y\in M$.
- $\gamma_{u}\colon \mathbb{R}\to M$ denotes the geodesic with unit speed such that $\gamma_u(0)=x$ and $\dot{\gamma}(0)=u$ for any $x\in M$ and $u\in T_x M$ with $|u|=1$.
- $I_x$ denotes the interior set at $x\in M$. We have ${\mathop{\mathrm{Vol}}\nolimits}(M\setminus I_x)=0$. We have that $\gamma_{x,y}$ is uniquely determined and $\nabla d(x,\cdot)=\dot{\gamma}_{x,y}(d(x,y))$ holds for any $y\in I_x\setminus\{x\}$.
- $\Delta_{C,k}=\nabla^\ast \nabla\colon \Gamma(\bigwedge^k T^\ast M)\to \Gamma(\bigwedge^k T^\ast M)$ denotes the connection Laplacian acting on $k$-forms.
- $0\leq \lambda_1(\Delta_{C,k}) \leq \lambda_2(\Delta_{C,k}) \leq\cdots \to \infty$ denotes the eigenvalues of the connection Laplacian $\Delta_{C,k}$ acting on $k$-forms.
- $S^n(r)$ denotes the $n$-dimensional standard sphere of radius $r$.
- $S^n:=S^n(1)$.
Note that the lowest eigenvalue of the Laplacian $\Delta$ acting on function is always equal to $0$, and so we start counting the eigenvalues of it from $i=0$. This is not the case with the connection Laplacian $\Delta_{C,k}$ acting on $k$-forms, and so we start counting the eigenvalues of it from $i=1$. For any $i\in\mathbb{Z}_{>0}$, we have $$\lambda_i(\Delta_{C,0})=\lambda_{i-1}(g).$$
Calculus of Differential Forms
------------------------------
In this subsection, we recall some facts about differential forms, and do some calculations.
We first recall the decomposition: $$T^\ast M\otimes \bigwedge^k T^\ast M=T^{k,1}M\oplus\bigwedge^{k+1} T^\ast M\oplus \bigwedge^{k-1} T^\ast M.$$ See also [@Se Section 2].
Let $V$ be an $n$-dimensional real vector space with an inner product $\langle,\rangle$. We put $$\begin{split}
&P_1\colon V\otimes \bigwedge^k V\to \bigwedge^{k+1} V,\quad P_1(\alpha\otimes \omega):=\left(\frac{1}{k+1}\right)^\frac{1}{2}\alpha\wedge\omega,\\
&P_2\colon V\otimes \bigwedge^k V\to \bigwedge^{k-1} V,\quad P_2(\alpha\otimes \omega):=\left(\frac{1}{n-k+1}\right)^\frac{1}{2}\iota(\alpha)\omega,\\
&Q_1\colon \bigwedge^{k+1} V\to V\otimes \bigwedge^k V,\quad Q_1(\zeta):=\left(\frac{1}{k+1}\right)^\frac{1}{2}\sum_{i=1}^n e^i\otimes\iota(e^i)\zeta,\\
&Q_2\colon \bigwedge^{k-1} V\to V\otimes \bigwedge^k V,\quad Q_2(\eta):=\left(\frac{1}{n-k+1}\right)^\frac{1}{2}\sum_{i=1}^n e^i\otimes e^i\wedge\eta,
\end{split}$$ where $\{e^1,\ldots,e^n\}$ is orthonormal basis of $V$. Then, we have
- ${\mathop{\mathrm{Im}}\nolimits}Q_1\bot {\mathop{\mathrm{Im}}\nolimits}Q_2$,
- $P_i\circ Q_i={\mathop{\mathrm{Id}}\nolimits}$ for each $i=1,2$,
- $Q_1$ and $Q_2$ preserve the norms,
- $Q_i\circ P_i\colon V\otimes \bigwedge^k V\to V\otimes \bigwedge^k V$ is symmetric and $(Q_i\circ P_i)^2=Q_i\circ P_i$ for each $i=1,2$.
Therefore, $Q_i\circ P_i$ is the orthogonal projection $V\otimes \bigwedge^k V\to {\mathop{\mathrm{Im}}\nolimits}Q_i$. Since $\bigwedge^{k+1} V\cong {\mathop{\mathrm{Im}}\nolimits}Q_1$ and $\bigwedge^{k-1} V \cong{\mathop{\mathrm{Im}}\nolimits}Q_2$, we can regard $\bigwedge^{k+1} V$ and $\bigwedge^{k-1} V$ as subspaces of $V\otimes \bigwedge^k V$.
Take an $n$-dimensional Riemannian manifold $(M,g)$ and consider the case when $V=T^\ast_x M$ ($x\in M$). We can take a sub-bundle $T^{k,1}M$ of $T^\ast M\otimes \bigwedge^k T^\ast M$ such that $$T^\ast M\otimes \bigwedge^k T^\ast M=T^{k,1}M\oplus\bigwedge^{k+1} T^\ast M\oplus \bigwedge^{k-1} T^\ast M$$ is an orthogonal decomposition. Then, for $\omega\in\Gamma(\bigwedge^k T^\ast M)$, we can decompose $\nabla \omega\in \Gamma(T^\ast M\otimes\bigwedge^k T^\ast M)$, the $\bigwedge^{k+1} T^\ast M$-component is equal to $\left(1/(k+1)\right)^\frac{1}{2}d\omega$ and the $\bigwedge^{k-1} T^\ast M$-component is equal to $-\left(1/(n-k+1)\right)^{1/2} d^\ast \omega$. Let $T(\omega)$ denotes the remaining part ($T\colon \Gamma(\bigwedge^k T^\ast M)\to \Gamma(T^{k,1}M)$). Then, we have $$\nabla \omega=T(\omega)+ \left(\frac{1}{k+1}\right)^\frac{1}{2} Q_1(d\omega)-\left(\frac{1}{n-k+1}\right)^\frac{1}{2}Q_2(d^\ast w).$$ Therefore, we get $$\label{2b}
|\nabla\omega|^2=|T(\omega)|^2+\frac{1}{k+1} |d\omega|^2+\frac{1}{n-k+1}|d^\ast \omega|^2.$$ If $d^\ast \omega=0$ and $T(\omega)=0$, then $\omega$ is called a Killing k-form (see also [@Se Definition 2.1]).
We next recall the Bochner-Weitzenböck formula.
\[p2a\] Let $(M,g)$ be an $n$-dimensional Riemannian manifold. We define a homomorphism $\mathcal{R}_k\colon \bigwedge^k T^\ast M\to \bigwedge^k T^\ast M$ as $$\mathcal{R}_k \omega=-\sum_{i,j}e^i\wedge \iota(e_j)\left(R(e_i,e_j)\omega\right)$$ for any $\omega\in\bigwedge^k T^\ast M$, where $\{e_1,\ldots,e_n\}$ is an orthonormal basis of $TM$, $\{e^1,\ldots,e^n \}$ is its dual and $R(e_i,e_j)\omega$ is defined by $$R(e_i,e_j)\omega=\nabla_{e_i}\nabla_{e_j}\omega-\nabla_{e_j}\nabla_{e_i}\omega-\nabla_{[e_i,e_j]}\omega\in \Gamma(\bigwedge^k T^\ast M).$$
Note that if $k=1$, then we have $\mathcal{R}_1 \omega={\mathop{\mathrm{Ric}}\nolimits}(\omega,\cdot)$ for any $\omega\in\Gamma(T^\ast M)$.
The Bochner-Weitzenböck formula is stated as follows:
\[p2b\] For any $\omega\in\Gamma (\bigwedge^k T^\ast M)$, we have $$\Delta\omega=\nabla^\ast \nabla \omega+\mathcal{R}_k \omega.$$
In particular, we have the following theorem when $k=1$:
For any $\omega\in\Gamma(T^\ast M)$, we have $$\Delta \omega =\nabla^\ast \nabla \omega + {\mathop{\mathrm{Ric}}\nolimits}(\omega,\cdot).$$
Let us do some calculations of differential forms.
\[p2c\] Let $(M,g)$ be a Riemannian manifold of dimension $n$. Take a vector field $X\in\Gamma(TM)$, a $p$-form $\omega\in\Gamma(\bigwedge^p T^\ast M)$ $(p\geq 1)$ and a local orthonormal bases $\{e_1,\ldots,e_n\}$ of $TM$.
- We have $$\mathcal{R}_{p-1}(\iota(X)\omega)=\iota(X) \mathcal{R}_p \omega+\iota({\mathop{\mathrm{Ric}}\nolimits}(X))\omega+2\sum_{i=1}^n\iota(e_i)(R(X,e_i)\omega).$$
- We have $$\Delta (\iota(X)\omega)=\iota(\Delta X)\omega+\iota(X)\Delta \omega
+2\sum_{i=1}^n\iota(e_i) (R(X,e_i)\omega)-2\sum_{i=1}^n\iota(\nabla_{e_i}X) (\nabla_{e_i}\omega).$$
- We have $$\sum_{i=1}^n\iota(e_i) (R(X,e_i)\omega)
=-\nabla_X d^\ast \omega +d^\ast \nabla_X \omega+\sum_{i,j=1}^n \langle \nabla_{e_j} X, e_i\rangle\iota(e_j)\nabla_{e_i}\omega.$$
Let $\{e^1,\ldots,e^n\}$ be the dual basis of $\{e_1,\ldots,e_n\}$.
We first show (i). If $p=1$, both sides are equal to $0$. Let us assume $p\geq 2$. We have $$\label{2c}
\begin{split}
&\iota({\mathop{\mathrm{Ric}}\nolimits}(X))\omega\\
=&\frac{1}{(p-1)!}\sum_{i,i_1,\ldots,i_{p-1}}\omega(R(X,e_i)e_i,e_{i_1},\cdots,e_{i_{p-1}})e^{i_1}\wedge\cdots\wedge e^{i_{p-1}}\\
=&\frac{-1}{(p-1)!}\sum_{i,i_1,\ldots,i_{p-1}} (R(X,e_i)\omega)(e_i,e_{i_1},\ldots,e_{i_n})e^{i_1}\wedge\cdots\wedge e^{i_{p-1}}\\
&-\frac{1}{(p-1)!}\sum_{i,i_1,\ldots,i_{p-1}} \sum_{l=1}^{p-1} \omega(e_i,e_{i_1},\cdots,R(X,e_i)e_{i_l},\ldots,e_{i_{p-1}})e^{i_1}\wedge\cdots\wedge e^{i_{p-1}}\\
=&-\sum_{i=1}^n\iota(e_i)(R(X,e_i)\omega)\\
&-\frac{1}{(p-1)!}\sum_{i,i_1,\ldots,i_{p-1}} \sum_{l=1}^{p-1} \omega(e_i,e_{i_1},\cdots,R(X,e_i)e_{i_l},\ldots,e_{i_{p-1}})e^{i_1}\wedge\cdots\wedge e^{i_{p-1}}
\end{split}$$ We calculate the second term. $$\begin{split}
-&\frac{1}{(p-1)!}\sum_{i,i_1,\ldots,i_{p-1}} \sum_{l=1}^{p-1} \omega(e_i,e_{i_1},\cdots,R(X,e_i)e_{i_l},\ldots,e_{i_{p-1}})e^{i_1}\wedge\cdots\wedge e^{i_{p-1}}\\
=&\frac{1}{(p-1)!}\sum_{l=1}^{p-1} \sum_{i,j,i_1,\ldots,i_{p-1}}\langle R(e_j,e_{l_l})X,e_i\rangle\omega(e_i,e_j,e_{i_1},\cdots,\widehat{e_{i_l}},\ldots,e_{i_{p-1}})\\
&\qquad\qquad\qquad \qquad\qquad\qquad\qquad\qquad\qquad\qquad
e^{i_l}\wedge e^{i_1}\wedge\cdots\wedge\widehat{e^{i_l}}\wedge\cdots\wedge e^{i_{p-1}}\\
=&\sum_{j,k} e^k\wedge\iota(e_j)\iota(R(e_j,e_{k})X)\omega\\
=&\sum_{j,k} e^k\wedge\iota(e_j)R(e_j,e_{k})(\iota(X)\omega)-\sum_{j,k} e^k\wedge\iota(e_j)\iota(X)R(e_j,e_{k})\omega\\
=&\mathcal{R}_{p-1}(\iota(X)\omega)-\iota(X)\mathcal{R}_{p}\omega
-\sum_{j=1}^n \iota(e_j)(R(X,e_j)\omega)
\end{split}$$ Combining this and (\[2c\]), we get (i).
Let us show (ii). We have $$\nabla^\ast \nabla \iota(X)\omega
=\iota(\nabla^\ast \nabla X)\omega
-2\sum_{i} \iota(\nabla_{e_i}X)\nabla_{e_i}\omega+\iota(X)\nabla^\ast \nabla\omega.$$ Thus, by (i), we get $$\begin{split}
\Delta ( \iota(X)\omega)
=&\nabla^\ast \nabla \iota(X)\omega+\mathcal{R}_{p-1}\iota(X)\omega\\
=&\iota(\nabla^\ast \nabla X)\omega-2\sum_{i} \iota(\nabla_{e_i}X)\nabla_{e_i}\omega+\iota(X)\nabla^\ast \nabla\omega+\mathcal{R}_{p-1}\iota(X)\omega\\
=&\iota(\Delta X)\omega+\iota(X)\Delta \omega
+2\sum_{i=1}^n\iota(e_i) (R(X,e_i)\omega)-2\sum_{i=1}^n\iota(\nabla_{e_i}X) (\nabla_{e_i}\omega).
\end{split}$$ This gives (ii).
Finally, we show (iii). We have $$\begin{split}
\sum_{i=1}^n \iota(e_i)(R(X,e_i)\omega)
=&\sum_{i=1}^n \iota(e_i)\left(\nabla_{X}\nabla_{e_i}\omega-\nabla_{e_i}\nabla_X\omega-\nabla_{\nabla_{X} e_i}\omega+\nabla_{\nabla_{e_i}X}\omega\right)\\
=&-\nabla_{X}d^\ast \omega+d^\ast \nabla_X\omega+\sum_{i,j=1}^n \langle \nabla_{e_j} X, e_i\rangle\iota(e_j)\nabla_{e_i}\omega.
\end{split}$$ This gives (iii).
When $\omega$ is parallel, we have the following corollary.
\[p2d\] Let $(M,g)$ be a Riemannian manifold of dimension $n$. Take a vector field $X\in\Gamma(TM)$ and a parallel $p$-form $\omega\in\Gamma(\bigwedge^p T^\ast M)$ $(p\geq 1)$.
- We have $$\mathcal{R}_{p-1}(\iota(X)\omega)=\iota({\mathop{\mathrm{Ric}}\nolimits}(X))\omega.$$
- We have $$\Delta (\iota(X)\omega)=\iota(\Delta X)\omega.$$
Finally, we give some easy equations for later use. Let $(M,g)$ be a Riemannian manifold of dimension $n$. Take a local orthonormal basis $\{e_1,\ldots,e_n\}$ of $TM$. Let $\{e^1,\ldots,e^n\}$ be its dual. For any $\omega,\eta\in\Gamma(\bigwedge^k T^\ast M)$, we have $$\sum_{i=1}^n \langle e^i\wedge \omega, e^i\wedge \eta \rangle=(n-k)\langle\omega,\eta\rangle,
\quad \sum_{i=1}^n \langle \iota(e_i)\omega, \iota(e_i) \eta \rangle=k\langle\omega,\eta\rangle.$$ For any $\alpha_1,\ldots,\alpha_k\in \Gamma(T^\ast M)$, we have $$Q_1(\alpha_1\wedge\cdots\wedge\alpha_k)=\left(\frac{1}{k}\right)^{1/2}\sum_{i=1}^k(-1)^{i-1}\alpha_i\otimes\alpha_1\wedge\cdots\wedge\widehat{\alpha_i}\wedge\cdots\wedge\alpha_k.$$ Since $Q_1$ preserves the norms, we have $$\label{q1k}
\begin{split}
&k\left|\alpha_1\wedge\cdots\wedge\alpha_k\right|^2\\
=&\left|\sum_{i=1}^k(-1)^{i-1}\alpha_i\otimes\alpha_1\wedge\cdots\wedge\widehat{\alpha_i}\wedge\cdots\wedge\alpha_k\right|^2
\end{split}$$ for any $\alpha_1,\ldots,\alpha_k\in \Gamma(T^\ast M)$.
Suppose that $M$ is oriented. For any $k$, the Hodge star operator $\ast\colon \bigwedge^k T^\ast M\to \bigwedge^{n-k} T^\ast M$ is defined so that $$\langle\ast\omega,\eta \rangle V_g=\omega\wedge\eta$$ for all $\omega\in\Gamma(\bigwedge^k T^\ast M)$ and $\eta\in\Gamma(\bigwedge^{n-k} T^\ast M)$, where $V_g$ denotes the volume form on $(M,g)$. For any $\alpha\in\Gamma(T^\ast M)$, $\omega\in\Gamma(\bigwedge^k T^\ast M)$ and $\eta\in\Gamma(\bigwedge^{k-1} T^\ast M)$, we have $$\begin{aligned}
\langle\ast(\omega \wedge \alpha),\eta\rangle V_g&=\omega \wedge \alpha \wedge \eta,\\
\langle\iota(\alpha)\ast \omega,\eta\rangle V_g=\langle\ast \omega,\alpha\wedge \eta\rangle V_g&=\omega\wedge \alpha \wedge \eta.\end{aligned}$$ Thus, we get $$\label{hstar2}
\ast(\omega \wedge \alpha)=\iota(\alpha)\ast \omega.$$ Therefore, for any $\alpha,\beta\in\Gamma(T^\ast M)$ and $\omega,\eta\in\Gamma(\bigwedge^k T^\ast M)$, we have $$\begin{split}
&\langle\iota (\alpha)\omega,\iota(\beta)\eta\rangle
=\langle\omega,\alpha \wedge \iota(\beta)\eta\rangle\\
=&-\langle\beta \wedge \omega,\alpha \wedge \eta\rangle+\langle\alpha,\beta\rangle\langle\omega,\eta\rangle
=-\langle\iota(\beta)\ast \omega,\iota(\alpha)\ast \eta\rangle+\langle\alpha,\beta\rangle\langle\omega,\eta\rangle,
\end{split}$$ and so $$\label{hstar}
\langle\iota (\alpha)\omega,\iota(\beta)\eta\rangle+\langle\iota(\beta)\ast \omega,\iota(\alpha)\ast \eta\rangle=\langle\alpha,\beta\rangle\langle\omega,\eta\rangle.$$
Parallel $p$-form
=================
In this section, we consider Riemannian manifolds with a non-trivial parallel differential form. The reader who is interested only in the proof of the main theorems can skip this section.
Bochner-Reilly-Grosjean Formula
-------------------------------
The aim of this subsection is to give an easy proof of what Grosjean called a new Bochner-Reilly formula [@gr Proposition 3.1] when the Riemannian manifold has a non-trivial parallel $p$-form $\omega$. In section 4, we estimate the error terms when the manifold has no boundary and $\omega$ is not parallel.
\[p3a\] Let $(M,g)$ be a compact $n$-dimensional Riemannian manifold possibly with a smooth boundary $(\partial M,g')$, and let $\nu$ be the outward unit normal vector field. For any $f\in C^\infty(M)$ and any parallel $p$-form $\omega$ $(1\leq p \leq n-1)$ on $M$, we have $$\begin{split}
&\int_M |T (\iota(\nabla f)\omega)|^2\,d\mu_g\\
=&\frac{p-1}{p}\int_M\langle\iota(\nabla f)\omega, \iota(\nabla\Delta f)\omega\rangle \,d\mu_g-\int_M \langle\iota({\mathop{\mathrm{Ric}}\nolimits}(\nabla f))\omega,\iota(\nabla f)\omega\rangle\,d\mu_g\\
&-\frac{1}{p}\int_{\partial M} \langle \iota(\nu)d(\iota(\nabla f)\omega),\iota(\nabla f)\omega\rangle\,d\mu_{g'}
+\int_{\partial M} \langle\nabla_{\nu}(\iota(\nabla f)\omega),\iota(\nabla f)\omega\rangle\,d\mu_{g'}.
\end{split}$$
Since $d^\ast \iota(\nabla f) \omega=-d^\ast d^\ast(f\omega)=0$, we have $$\label{3a}
\begin{split}
&\int_M \langle\iota({\mathop{\mathrm{Ric}}\nolimits}(\nabla f))\omega,\iota(\nabla f)\omega\rangle\,d\mu_g\\
=&\int_M \langle\mathcal{R}_{p-1}(\iota(\nabla f)\omega),\iota(\nabla f)\omega\rangle\,d\mu_g\\
=&\int_M \langle\Delta(\iota(\nabla f)\omega),\iota(\nabla f)\omega\rangle\,d\mu_g
-\int_M \langle\nabla^\ast\nabla(\iota(\nabla f)\omega),\iota(\nabla f)\omega\rangle\,d\mu_g\\
=&\int_M \langle d(\iota(\nabla f)\omega),d(\iota(\nabla f)\omega)\rangle\,d\mu_g
-\int_M \langle\nabla(\iota(\nabla f)\omega),\nabla(\iota(\nabla f)\omega)\rangle\,d\mu_g\\
-&\int_{\partial M} \langle \iota(\nu)d(\iota(\nabla f)\omega),\iota(\nabla f)\omega\rangle\,d\mu_{g'}
+\int_{\partial M} \langle\nabla_{\nu}(\iota(\nabla f)\omega),\iota(\nabla f)\omega\rangle\,d\mu_{g'}
\end{split}$$ by Corollary \[p2d\] (i), Bochner-Weitzenböck formula and the divergence theorem. By (\[2b\]) and Corollary \[p2d\] (ii), we have $$\label{3b}
\begin{split}
&\int_M \langle d(\iota(\nabla f)\omega),d(\iota(\nabla f)\omega)\rangle\,d\mu_g
-\int_M \langle\nabla(\iota(\nabla f)\omega),\nabla(\iota(\nabla f)\omega)\rangle\,d\mu_g\\
=&\frac{p-1}{p}\int_M \langle d(\iota(\nabla f)\omega),d(\iota(\nabla f)\omega)\rangle\,d\mu_g-\int_M |T(\iota(\nabla f)\omega)|^2\,d\mu_g\\
=&\frac{p-1}{p}\int_M \langle \iota(\nabla\Delta f)\omega),\iota(\nabla f)\omega\rangle\,d\mu_g
\\
&\quad +\frac{p-1}{p}\int_{\partial M} \langle \iota(\nu)d(\iota(\nabla f)\omega),\iota(\nabla f)\omega\rangle\,d\mu_{g'}-\int_M |T(\iota(\nabla f)\omega)|^2\,d\mu_g
\end{split}$$ By (\[3a\]) and (\[3b\]), we get the proposition.
Estimate and Equality Case
--------------------------
In this subsection, we give more general result than Theorem \[grosjean\] without assuming positive Ricci curvature.
For any closed Riemannian manifold $(M,g)$, we define $$\Omega_1(g)=\sup\left\{ \frac{\int_M {\mathop{\mathrm{Ric}}\nolimits}(\nabla f, \nabla f)\,d\mu_g}{\int_M (\Delta f)^2\,d\mu_g}: \text{ $f$ is a non-constant function on $M$}\right\}.$$
By the Bochner formula, we always have $$\Omega_1(g)\leq\frac{n-1}{n},$$ where $n=\dim M$, and $$\|\Delta f\|_2^2\leq \frac{1}{1-\Omega_1(g)}\|\nabla^2 f\|_2^2$$ for all $f\in C^\infty(M)$. Since $$\|\nabla^2 f+\frac{\Delta f}{n}g\|_2^2=\|\nabla^2 f\|_2^2-\frac{1}{n}\|\Delta f\|_2^2,$$ we have $$\|\nabla^2 f\|_2^2\leq \frac{n}{n-\frac{1}{1-\Omega_1(g)}}\|\nabla^2 f+\frac{\Delta f}{n}g\|_2^2$$ for all $f\in C^\infty(M)$ if $\Omega_1(g)< (n-1)/n$. If $\Omega_1(g)=(n-1)/n$, then $\Omega_1(g)$ is attained by a non-constant function $f\in C^\infty(M)$ such that $\nabla^2 f+(\Delta f/n)g=0$, and so $(M,g)$ is isometric to $S^n$ with a rotationally symmetric metric by the Tashiro theorem [@T1] (see also [@Ai Property A]). If ${\mathop{\mathrm{Ric}}\nolimits}_g\geq k g$ ($k>0$), we easily get $\Omega_1(g)>0$ and $$\label{3c}
\lambda_1(g)\geq\frac{k}{\Omega_1(g)}.$$ If ${\mathop{\mathrm{Ric}}\nolimits}_g= k g$ ($k>0$), then we have $$\label{3ca}
\lambda_1(g)=\frac{k}{\Omega_1(g)}.$$ The following proposition is the main result of this subsection.
\[p3c\] Let $(M,g)$ be an $n$-dimensional closed Riemannian manifold. Assume that there exists a non-trivial parallel $p$-form $\omega$ on $M$ $(1\leq p\leq n/2)$. Then, we have $$\label{3d}
\Omega_1(g)\leq \frac{n-p-1}{n-p}.$$
Moreover, if either $p\neq \frac{n}{2}$ or $n\geq 6$ and if in addition $M$ is simply connected, then the equality in $(\ref{3d})$ implies that $(M,g)$ is isometric to a product $(S^{n-p},g_r)\times (X,g')$, where $g_r$ is some rotationally symmetric metric on $S^{n-p}$ and $(X,g')$ is a $p$-dimensional closed Riemannian manifold.
We first show (\[3d\]). By taking the two-sheeted orientable Riemannian covering of $(M,g)$ if necessary, we can assume that $(M,g)$ is oriented. Then, we have $$\begin{split}
\int_M \langle\iota({\mathop{\mathrm{Ric}}\nolimits}(\nabla f))\omega,\iota(\nabla f)\omega\rangle\,d\mu_g
\leq &\frac{p-1}{p}\int_M\langle\iota(\nabla\Delta f)\omega, \iota(\nabla f)\omega\rangle \,d\mu_g,\\
\int_M \langle\iota({\mathop{\mathrm{Ric}}\nolimits}(\nabla f))\ast\omega,\iota(\nabla f)\ast\omega\rangle\,d\mu_g
\leq &\frac{n-p-1}{n-p}\int_M\langle\iota(\nabla\Delta f)\ast\omega, \iota(\nabla f)\ast\omega\rangle \,d\mu_g.
\end{split}$$ Thus, we get $$\begin{split}
\int_M {\mathop{\mathrm{Ric}}\nolimits}(\nabla f,\nabla f)\,d\mu_g
\leq &\frac{n-p-1}{n-p}\int_M(\Delta f)^2 \,d\mu_g
\end{split}$$ by (\[hstar\]). This implies the estimate (\[3d\]).
We next consider the equality case. Suppose that $M$ is simply connected. Let $$TM=\bigoplus_{i=1}^k E_i$$ be the irreducible decomposition of the holonomy representation, and let $$(M,g)=(M_1,g_1)\times\cdots\times (M_k,g_k)$$ be the corresponding de Rham decomposition. There exist non-negative integers $p_1,\ldots,p_k\in\mathbb{Z}_{\geq 0}$ such that $p_1+\cdots+p_k=p$ and the $\bigwedge^{p_1} E^1\otimes\cdots\otimes\bigwedge^{p_k} E^k$-component of $\omega$ is non-zero and parallel, where $E^i$ is the sub-bundle of $T^\ast M$ that corresponds to $E_i$. Thus, we can assume $\omega\in\Gamma(\bigwedge^{p_1} E^1\otimes\cdots\otimes\bigwedge^{p_k} E^k)$.
Take $i$ with $p_i\neq 0$. Let us show that there exists a non-trivial parallel $p_i$-form on $M_i$. Take some $x\in M$ and decompose $\omega_x$ as $$\omega_x=\sum_{j=1}^l \eta_j\wedge\gamma_j,$$ where $\eta_j\in \bigwedge^{p_i} E^i_x$ and $$\gamma_j\in\bigwedge^{p_1} E^1_x\otimes\cdots\otimes\bigwedge^{p_{i-1}} E^{i-1}_x\otimes\bigwedge^{p_{i+1}} E^{i+1}_x\otimes\cdots\otimes\bigwedge^{p_k} E^k_x$$ with $\langle\gamma_j,\gamma_k\rangle=\delta_{jk}$ for all $j,k\in\{1,\ldots,l\}$. Then, $\eta_j$ is invariant under the holonomy representation of $M_i$ for each $j$. Thus, $\eta_j$ defines a parallel $p_i$-form on $M_i$. Therefore, there exists a non-trivial parallel $p_i$-form $\omega_i$ on $M_i$. Then, the eigenspace of the symmetric form on $T M_i$ $$\langle\iota(\cdot)\omega_i, \iota(\cdot)\omega_i\rangle$$ is invariant under the the holonomy representation. Since $T M_i$ is irreducible, there exists a positive number $\mu_i>0$ such that $$\label{wnondeg}
\langle\iota(\cdot)\omega_i, \iota(\cdot)\omega_i\rangle=\mu_i \langle\cdot,\cdot\rangle.$$ Thus, we get $$\begin{split}
\int_{M_i}{\mathop{\mathrm{Ric}}\nolimits}(\nabla f,\nabla f)\,d\mu_{g_i}\leq \frac{p_i-1}{p_i}\int_{M_i} (\Delta f)^2 \,d\mu_{g_i}
\end{split}$$ for all $f\in C^\infty (M_i)$ by Proposition \[p3a\], and so $$\label{3e}
\Omega_1(g_i)\leq\frac{p_i-1}{p_i}.$$ By considering $\ast \omega$, we also have $$\label{3f}
\Omega_1(g_i)\leq\frac{\dim M_i-p_i-1}{\dim M_i-p_i}$$ if $p_i\neq \dim M_i$.
By (\[3e\]), (\[3f\]) and [@Ai Proposition 2.4], we get $$\label{3g}
\Omega_1(g)=\max\{\Omega_1(g_1),\ldots,\Omega_1(g_k)\}\leq \max_i \left\{\frac{\overline{p}_i-1}{\overline{p}_i}\right\},$$ where we put
[align\*]{} &{p\_i,M\_i-p\_i}&& (p\_i0,M\_i),\
&M\_i &&(p\_i=0,M\_i).
Suppose that $$\label{3h}
\Omega_1(g)= \frac{n-p-1}{n-p}$$ and either $p\neq \frac{n}{2}$ or $n\geq 6$. Without loss of generality, we can assume that $\dim M_1=\max_i\{\dim M_i\}$. If $\dim M_1< n-p$, then we get $$\Omega_1(g)\leq\frac{n-p-2}{n-p-1} < \frac{n-p-1}{n-p}$$ by (\[3g\]). This contradicts to (\[3h\]), and so we have $\dim M_1\geq n-p$.
We consider the following three cases:
- $n-p<\dim M_1<n$,
- $\dim M_1=n-p$,
- $\dim M_1=n$.
We first suppose that $n-p<\dim M_1<n$. Then, $p_2+\cdots+p_k\leq n-\dim M_1<p$, and so $p_1\neq 0$. Moreover, we have $p_1\leq p<\dim M_1$. Thus, we have $$\overline{p}_1=\min\{p_1,\dim M_1- p_1\}\leq \frac{\dim M_1}{2}< \frac{n}{2}\leq n-p.$$ Since $\dim M_i<n-p$ for all $i=\{2,\ldots,k\}$, we get $$\Omega_1(g)< \frac{n-p-1}{n-p}$$ by (\[3g\]). This contradicts to (\[3h\]).
We next suppose that $\dim M_1=n$. Then, we have $M=M_1$. Since we have $\Omega_1(g)\leq (p-1)/p$ and $p\leq n-p$, we get $$\label{3i}
p=n-p=n/2\geq 3$$ by (\[3h\]). Since there exists a non-trivial parallel $p$-form, the holonomy group of $(M,g)$ is not equal to $\mathrm{SO}(n)$. If ${\mathop{\mathrm{Ric}}\nolimits}\leq 0$, then $\Omega_1(g)=0$, and so we have one of the following by the Berger classification theorem:
- $(M,g)$ is a Kähler manifold,
- $(M,g)$ is a quaternionic Kähler manifold,
- $(M,g)$ is a symmetric space.
If $(M,g)$ is Kähler manifold, then there exists a Kähler form. Thus, we get $\Omega_1(g)\leq \frac{1}{2}<\frac{n-p-1}{n-p}$ by (\[3g\]) and (\[3i\]). This contradicts to (\[3h\]). If $(M,g)$ is quaternionic Kähler manifold of dimension $n=4d$ ($d\geq 2$), then $(M,g)$ is a positive Einstein manifold $${\mathop{\mathrm{Ric}}\nolimits}=c g \quad (c>0)$$ by $\Omega_1(g)>0$ and [@Be Theorem 14.39]. Thus, we get $$\Omega_1(g)=\frac{c}{\lambda_1(g)}\leq \frac{1}{2}\frac{d+2}{d+1}\leq\frac{2}{3}<\frac{3}{4}$$ by (\[qk\]) and (\[3ca\]). This contradicts to (\[3h\]) and $n-p=p=2d\geq 4$. Finally, we suppose that $(M,g)$ is a symmetric space. Since $(M,g)$ has a non-trivial parallel $p$-form, we have $M\neq S^n$. Thus, by [@bms Theorem 1.1], there exists no non-parallel Killing $(p-1)$-form on $(M,g)$, and so $T(\iota(\nabla f)\omega)\neq 0$ for any non-constant function $f\in C^\infty(M)$. Since $\Omega_1(g)$ is attained by some smooth function [@Ai Lemma 2.1], we get $\Omega_1(g)<(p-1)/p$ by (\[wnondeg\]) and Proposition \[p3a\]. This contradicts to (\[3h\]).
Therefore, we get $\dim M_1=n-p$. Put $(X,g')=(M_2,g_2)\times\cdots\times (M_k,g_k)$. Then, we have $$\Omega_1(g)=\max\{\Omega_1(g_1),\Omega_1(g')\}$$ by [@Ai Proposition 2.4], and so either $
\Omega_1(g_1)=(n-p-1)/(n-p)
$ or $
\Omega_1(g')=(n-p-1)/(n-p).
$ If $
\Omega_1(g_1)=(n-p-1)/(n-p)
$ (resp. $
\Omega_1(g')=(n-p-1)/(n-p)
$), then $(M_1,g_1)$ (resp. $(X,g')$) is isometric to $(S^{n-p},g_r)$, where $g_r$ is rotationally symmetric metric on $S^{n-p}$.
As a corollary, we get the following:
\[p3d\] Let $(M,g)$ be an $n$-dimensional closed Riemannian manifold. Assume that ${\mathop{\mathrm{Ric}}\nolimits}\geq (n-p-1)g$ and there exists a non-trivial parallel $p$-form on $M$ $(2\leq p\leq n/2)$. Then, we have $$\label{3j}
\lambda_1(g)\geq n-p.$$
Moreover, if either $p\neq n/2$ or $n\geq 6$ and if in addition $M$ is simply connected, the equality in (\[3j\]) implies that $(M,g)$ is isometric to a product $S^{n-p}\times X$, where $X$ is a $p$-dimensional closed Riemannian manifold.
By (\[3c\]) and (\[3d\]), we get $$\label{3k}
\lambda_1(g)\geq\frac{1}{\Omega_1(g)}(n-p-1)\geq n-p.$$ This implies (\[3j\]).
Suppose that $M$ is simply connected, $\lambda_1(g)=n-p$ and either $p\neq \frac{n}{2}$ or $n\geq6$. Then, we have $\Omega_1(g)=(n-p-1)/(n-p)$ by (\[3k\]), and so $(M,g)$ is isometric to a product $(S^{n-p},g_r)\times (X,g')$, where $g_r$ is some rotationally symmetric metric on $S^{n-p}$ and $(X,g')$ is a $p$-dimensional closed Riemannian manifold by Proposition \[p3c\]. Since we have ${\mathop{\mathrm{Ric}}\nolimits}_{g_r}\geq (n-p-1)g_r$, ${\mathop{\mathrm{Ric}}\nolimits}_{g'}\geq (n-p-1)g'$ and $n-p=\lambda_1(g)=\min\{\lambda_1(g_r),\lambda_1(g')\}$, we get that either $(S^{n-p},g_r)$ or $(X,g')$ is isometric to $S^{n-p}(1)$ by the Lichnerowicz-Obata theorem (Theorem 1).
If we assume more strong condition on eigenvalues, then the assumption that the manifold is simply connected can be removed.
\[p3d2\] Let $(M,g)$ be an $n$-dimensional closed Riemannian manifold. Assume that ${\mathop{\mathrm{Ric}}\nolimits}\geq (n-p-1)g$ and there exists a non-trivial parallel $p$-form on $M$ $(2\leq p< n/2)$. If $$\lambda_{n-p+1}(g)= n-p,$$ then $(M,g)$ is isometric to a product $S^{n-p}\times (X,g')$, where $(X,g')$ is a $p$-dimensional closed Riemannian manifold.
Let $f_k$ be the $k$-th eigenfunction of the Laplacian on $S^{n-p}$. Note that the functions $f_1,\ldots,f_{n-p+1}$ are height functions.
By Corollary \[p3d\], the universal cover $(\widetilde{M},\tilde{g})$ of $(M,g)$ is isometric to a product $S^{n-p}\times (X,g')$, where $(X,g')$ is a $p$-dimensional closed Riemannian manifold. We regard the function $f_i$ as a function on $\widetilde{M}$. Since $\lambda_{n-p+1}(g)= n-p$, each $f_i\in C^\infty(\widetilde{M})$ ($i=1,\ldots,n-p+1$) is a pull back of some function on $M$. Thus, the covering transformation preserves $f_1,\ldots,f_{n-p+1}$. Therefore, the covering transformation does not act on $S^{n-p}$, and so we get the corollary.
The almost version of this corollary is Main Theorem 2.
Examples
--------
In this subsection, we show that the assumption of Corollary \[p3d2\] is optimal in some sense by giving examples.
Take a positive odd integer $p$ with $p\geq 3$ and a positive integer $n$ with $n> 2p$. Put $a:=\sqrt{(p-1)/(n-p-1)}$. We define an equivalence relation $\sim$ on $S^{n-p}\times S^p(a)$ as follows: $$\begin{split}
&((x_0,\ldots,x_{n-p}),(y_0,\ldots,y_p))\sim ((x'_0,\ldots,x'_{n-p}),(y'_0,\ldots,y'_p))\\
\Leftrightarrow &\text{ there exists $k\in \mathbb{Z}$ such that}\\
&((x'_0,\ldots,x'_{n-p}),(y'_0,\ldots,y'_p))=(((-1)^k x_0, x_1,\ldots,x_{n-p}),(-1)^k(y_0,\ldots,y_p))
\end{split}$$ for any $((x_0,\ldots,x_{n-p}),(y_0,\ldots,y_p)), ((x'_0,\ldots,x'_{n-p}),(y'_0,\ldots,y'_p))\in S^{n-p}\times S^p(a)$. Then, we have the following:
\[p3e\] We have the following properties:
- $(M,g)=(S^{n-p}\times S^p(a))/\sim$ is an $n$-dimensional closed Riemannian manifold with a non-trivial parallel $p$-form.
- ${\mathop{\mathrm{Ric}}\nolimits}= (n-p-1)g$.
- $\lambda_{n-p}(g)=n-p$.
- $(M,g)$ is not isometric to any product Riemannian manifolds.
Let $\omega$ be the volume form on $S^p(a)$. Since the action on $S^{n-p}\times S^p(a)$ preserves $\omega$, there exists a non-trivial parallel $p$-form on $(M,g)$. We also denote it by $\omega$. Since the action on $S^{n-p}\times S^p(a)$ preserves the function $$x_i \colon S^{n-p}\times S^p(a)\to \mathbb{R},\,((x_0,\ldots,x_{n-p}),(y_0,\ldots,y_p))\mapsto x_i$$ for each $i=1,\ldots,n-p$, we have $\lambda_{n-p}(g)=n-p$.
Suppose that $(M,g)$ is isometric to a product $(M^{n-k}_1,g_1)\times (M^{k}_2,g_2)$ ($k\leq n-k$) for some $(n-k)$ and $k$-dimensional closed Riemannian manifolds $(M_1,g_1)$ and $(M_2,g_2)$. Since we have the irreducible decomposition $T_{(x,y)} M\cong T_x S^{n-p}\oplus T_y S^p(a)$ of the restricted holonomy action, we get $k=p$. Since $\lambda_1(g)=n-p$, we have that $(M_1,g_1)$ is isometric to $S^{n-p}$. Thus, we get $\lambda_{n-p+1}(g)=n-p$. However the action on $S^{n-p}\times S^p(a)$ does not preserve the function $$x_0\colon S^{n-p}\times S^p(a)\to \mathbb{R},\,((x_0,\ldots,x_{n-p}),(y_0,\ldots,y_p))\mapsto x_0,$$ and so $\lambda_{n-p+1}(g)\neq n-p$. This is a contradiction.
We next define an equivalence relation $\sim'$ on $S^{n-p}\times S^p(a)$ as follows: $$\begin{split}
&((x_0,\ldots,x_{n-p}),(y_0,\ldots,y_p))\sim' ((x'_0,\ldots,x'_{n-p}),(y'_0,\ldots,y'_p))\\
\Leftrightarrow &\text{ there exists $k\in \mathbb{Z}$ such that}\\
&((x'_0,\ldots,x'_{n-p}),(y'_0,\ldots,y'_p))=(((-1)^k x_0, (-1)^k x_1,x_2,\ldots,x_{n-p}),(-1)^k(y_0,\ldots,y_p))
\end{split}$$ for any $((x_0,\ldots,x_{n-p}),(y_0,\ldots,y_p)), ((x'_0,\ldots,x'_{n-p}),(y'_0,\ldots,y'_p))\in S^{n-p}\times S^p(a)$. Similarly to Proposition \[p3e\], we have the following proposition:
\[p3f\] We have the following properties:
- $(M',g')=(S^{n-p}\times S^p(a))/\sim'$ is an $n$-dimensional closed orientable Riemannian manifold with a non-trivial parallel $p$-form.
- ${\mathop{\mathrm{Ric}}\nolimits}= (n-p-1)g'$.
- $\lambda_{n-p-1}(g')=n-p$.
- $(M',g')$ is not isometric to any product Riemannian manifolds.
Almost Parallel $p$-form
========================
In this section, we show Main Theorem 1 and Main Theorem 3. Recall that $\lambda_1(\Delta_{C,p})$ denotes the first eigenvalue of the connection Laplacian acting on $p$-forms, and $$\Delta_{C,p}:=\nabla^\ast\nabla \colon \Gamma(\bigwedge^p T^\ast M)\to\Gamma(\bigwedge^p T^\ast M).$$ It is enough to show Main Theorem 1 when $\lambda_1(\Delta_{C,p})\leq 1$. Note that we always have $$\lambda_1(\Delta_{C,1})\geq 1$$ if ${\mathop{\mathrm{Ric}}\nolimits}_g\geq (n-1)g$.
Error Estimates
---------------
In this subsection, we give error estimates about Proposition \[p3a\]. Lemma \[p4e\] (vii) corresponds to Proposition \[p3a\]. We list the assumptions of this subsection. We mention that most techniques in this paper can be used under the assumption ${\mathop{\mathrm{Ric}}\nolimits}_g\geq -Kg$ and ${\mathop{\mathrm{diam}}\nolimits}(M)\leq D$.
In this subsection, we assume the following:
- $(M,g)$ is an $n$-dimensional closed Riemannian manifold with ${\mathop{\mathrm{Ric}}\nolimits}_g\geq -Kg$ and ${\mathop{\mathrm{diam}}\nolimits}(M)\leq D$ for some positive real numbers $K>0$ and $D>0$.
- $1\leq k \leq n-1$.
- A $k$-form $\omega\in \Gamma(\bigwedge^k T^\ast M)$ satisfies $\|\omega\|_2=1$, $\|\omega\|_\infty\leq L_1$ and $\|\nabla \omega\|_2^2\leq \lambda$ for some $L_1>0$ and $0\leq \lambda\leq 1$.
- A function $f\in C^\infty(M)$ satisfies $\|f\|_{\infty}\leq L_2\|f\|_2$, $\|\nabla f\|_{\infty}\leq L_2\|f\|_2$ and $\|\Delta f\|_2\leq L_2\|f\|_2$ for some $L_2>0$.
Note that we have $$\label{4a0}
\|\nabla^2 f\|_2^2=\|\Delta f\|_2^2-\frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)}\int_M {\mathop{\mathrm{Ric}}\nolimits}(\nabla f,\nabla f)\,d\mu_g\leq (1+K)L^2_2\|f\|_2^2$$ by the Bochner formula.
We first show the following:
\[p4c\] There exists a positive constant $C(n,K,D)>0$ such that $\||\omega|-1\|_2\leq C \lambda^{1/2}$ holds.
Put $$\overline{\omega}:=\frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)}\int_M |\omega| \,d\mu_g.$$ Since we have $|\omega|\in W^{1,2}(M)$, we get $$\||\omega|-\overline{\omega}\|_2^2\leq \frac{1}{\lambda_1(g)}\|\nabla|\omega|\|_2^2\leq \frac{1}{\lambda_1(g)}\|\nabla\omega\|_2^2\leq\frac{\lambda}{\lambda_1(g)}$$ by the Kato inequality. Thus, by the Li-Yau estimate [@SY p.116], we have $$\||\omega|-\overline{\omega}\|_2\leq C\lambda^{1/2},$$ and so $$|1-\overline{\omega}|=\left|\|\omega\|_2-\|\overline{\omega}\|_2\right|\leq \||\omega|-\overline{\omega}\|_2\leq C\lambda^{1/2}.$$ Therefore, we get $
\||\omega|-1\|_2\leq C\lambda^{1/2}.
$
Let us give error estimates about Proposition \[p3a\].
\[p4d\] There exists a positive constant $C=C(n,k,K,D,L_1,L_2)>0$ such that the following properties hold:
- We have $$\frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)} \int_M |d^{\ast}(\iota(\nabla f)\omega)|^2\,d\mu_g
\leq C\|f\|_2^2\lambda.$$
- We have $$\left|\frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)}\int_M \Big(\langle \iota({\mathop{\mathrm{Ric}}\nolimits}(\nabla f))\omega,\iota(\nabla f)\omega\rangle -\langle \mathcal{R}_{k-1}(\iota(\nabla f)\omega),\iota(\nabla f)\omega\rangle \Big)\,d\mu_g\right|
\leq C\|f\|_2^2\lambda^{1/2}.$$
- We have $$\left|\frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)} \int_M \Big(\langle\Delta(\iota(\nabla f)\omega),\iota(\nabla f)\omega\rangle
-\langle
\iota(\nabla \Delta f)\omega,\iota(\nabla f)\omega\rangle\Big) \,d\mu_g\right|
\leq C\|f\|_2^2\lambda^{1/2}.$$
- We have $$\frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)}\int_M \left|\nabla (\iota(\nabla f)\omega)-\sum_{i=1}^n e^i\otimes \iota(\nabla_{e_i}\nabla f)\omega\right|^2 \,d\mu_g\leq C\|f\|_2^2\lambda.$$
- We have $$\frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)}\int_M \left|d (\iota(\nabla f)\omega)-\sum_{i=1}^n e^i\wedge \iota(\nabla_{e_i}\nabla f)\omega\right|^2 \,d\mu_g\leq C\|f\|_2^2\lambda.$$
- We have $$\frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)}\int_M |\nabla (\iota(\nabla f)\omega) |^2\,d\mu_g\leq C\|f\|_2^2.$$
- We have $$\begin{aligned}
&\Bigg|\frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)}\int_M \langle \iota({\mathop{\mathrm{Ric}}\nolimits}(\nabla f))\omega,\iota(\nabla f)\omega\rangle
\,d\mu_g\\
&\quad-
\frac{k-1}{k} \frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)}\int_M \langle \iota(\nabla\Delta f)\omega,\iota(\nabla f)\omega\rangle \,d\mu_g+\|T(\iota(\nabla f)\omega)\|_2^2\Bigg|
\leq C\|f\|_2^2\lambda^{1/2}.\end{aligned}$$
- If $M$ is oriented and $1\leq k\leq n/2$, then we have $$\begin{aligned}
&\frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)}\int_M {\mathop{\mathrm{Ric}}\nolimits}(\nabla f,\nabla f)|\omega|^2\,d\mu_g\\
\leq &
\frac{n-k-1}{n-k} \frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)}\int_M \langle \nabla\Delta f,\nabla f\rangle|\omega|^2 \,d\mu_g
-\|T(\iota(\nabla f)\omega)\|_2^2
-\|T(\iota(\nabla f)\ast\omega)\|_2^2\\
&\qquad\qquad -\left(\frac{n-k-1}{n-k} -\frac{k-1}{k} \right)\|d(\iota(\nabla f)\omega)\|^2_2
+C\|f\|_2^2\lambda^{1/2}.\end{aligned}$$
Although an orthonormal basis $\{e_1,\ldots,e_n\}$ of $TM$ is defined only locally, $\sum_{i=1}^n e^i\otimes \iota(\nabla_{e_i}\nabla f)\omega$ and $\sum_{i=1}^n e^i\wedge \iota(\nabla_{e_i}\nabla f)\omega$ are well-defined as tensors.
We first prove (i). Since $$d^\ast (f\omega)=-\iota(\nabla f)\omega +f d^\ast \omega$$ and $d^\ast\circ d^\ast=0$, we have $$d^\ast (\iota(\nabla f)\omega)=-\iota(\nabla f)d^\ast \omega.$$ Thus, we get $$\frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)} \int_M |d^{\ast}(\iota(\nabla f)\omega|^2\,d\mu_g
\leq C\|\nabla f\|_{\infty}^2 \|\nabla \omega\|_2^2
\leq C\|f\|_2^2 \lambda.$$
To prove (ii) and (iii), we estimate following terms: $$\begin{aligned}
&\frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)} \int_M \langle\iota(\nabla f)\Delta \omega,\iota(\nabla f) \omega\rangle\,d\mu_g,\\
&\frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)} \int_M \langle\iota(\nabla f)\nabla^\ast \nabla \omega,\iota(\nabla f) \omega\rangle\,d\mu_g,\\
&\frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)} \int_M \langle \iota(\nabla f) \mathcal{R}_k \omega, \iota(\nabla f)\omega\rangle\,d\mu_g,\\
&\frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)} \int_M \langle \sum_{i=1}^n\iota(\nabla_{e_i}\nabla f) (\nabla_{e_i}\omega),\iota(\nabla f)\omega\rangle\,d\mu_g,\\
&\frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)} \int_M \langle\sum_{i=1}^n\iota(e_i)(R(\nabla f,e_i)\omega),\iota(\nabla f) \omega\rangle\,d\mu_g.\end{aligned}$$
We have $$\begin{aligned}
&\int_M \langle\iota(\nabla f)\Delta \omega,\iota(\nabla f) \omega\rangle\,d\mu_g\\
=&\int_M \langle d \omega,d (d f \wedge\iota(\nabla f) \omega)\rangle\,d\mu_g+\int_M \langle d^\ast\omega,d^\ast(d f \wedge\iota(\nabla f) \omega)\rangle\,d\mu_g\end{aligned}$$ and $$\begin{aligned}
&|\langle d \omega,d (d f \wedge\iota(\nabla f) \omega)\rangle|\\
=&|\langle d \omega, \sum_{i=1}^n d f\wedge e^i \wedge\left(\iota(\nabla_{e_i}\nabla f) \omega+\iota(\nabla f) \nabla_{e_i}\omega \right)\rangle|\\
\leq& C|\nabla \omega||\nabla f|(|\nabla^2 f||\omega|+|\nabla f||\nabla \omega|),\\
&|\langle d^\ast \omega,d^\ast (d f \wedge\iota(\nabla f) \omega)\rangle|\\
=&|\langle d^\ast \omega, \sum_{i=1}^n \iota(e_i)\left( \nabla_{e_i} d f\wedge \iota(\nabla f) \omega+ d f\wedge \iota(\nabla_{e_i} \nabla f) \omega+
d f\wedge\iota(\nabla f) \nabla_{e_i}\omega \right)\rangle|\\
\leq& C|\nabla \omega||\nabla f|(|\nabla^2 f||\omega|+|\nabla f||\nabla \omega|).\end{aligned}$$ Thus, we get $$\label{4a}
\left|\frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)} \int_M \langle\iota(\nabla f)\Delta \omega,\iota(\nabla f) \omega\rangle\,d\mu_g\right|
\leq C\|f\|_2^2\lambda^{1/2}.$$
We have $$\begin{aligned}
&\int_M \langle\iota(\nabla f)\nabla^\ast\nabla \omega,\iota(\nabla f) \omega\rangle\,d\mu_g\\
=&\int_M \langle \nabla \omega,\nabla (d f \wedge\iota(\nabla f) \omega)\rangle\,d\mu_g\end{aligned}$$ and $$\begin{aligned}
&|\langle \nabla \omega,\nabla (d f \wedge\iota(\nabla f) \omega)\rangle|\\
=& C|\nabla \omega||\nabla f|(|\nabla^2 f||\omega|+|\nabla f||\nabla \omega|).\end{aligned}$$ Thus, we get $$\label{4aa}
\left|\frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)} \int_M \langle\iota(\nabla f)\nabla^\ast\nabla \omega,\iota(\nabla f) \omega\rangle\,d\mu_g\right|
\leq C\|f\|_2^2\lambda^{1/2}.$$
By Theorem \[p2b\], (\[4a\]) and (\[4aa\]), we have $$\label{4b}
\begin{split}
&\left|\frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)} \int_M \langle \iota(\nabla f) \mathcal{R}_k \omega, \iota(\nabla f)\omega\rangle\,d\mu_g\right|\\
\leq & \frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)}\left(\left|\int_M \langle \iota(\nabla f)\Delta \omega, \iota(\nabla f) \omega\rangle\,d\mu_g\right|+ \left|\int_M \langle \iota(\nabla f)\nabla^\ast\nabla \omega, \iota(\nabla f)\omega\rangle\,d\mu_g\right|\right)\\
\leq & C\|f\|_2^2\lambda^{1/2}.
\end{split}$$
Since $$|\langle \sum_{i=1}^n\iota(\nabla_{e_i}\nabla f) (\nabla_{e_i}\omega),\iota(\nabla f)\omega\rangle|
\leq C|\omega||\nabla f| |\nabla \omega||\nabla^2 f|,$$ we have $$\label{4c}
\left|\frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)} \int_M \langle \sum_{i=1}^n\iota(\nabla_{e_i}\nabla f) (\nabla_{e_i}\omega),\iota(\nabla f)\omega\rangle\,d\mu_g\right|
\leq C \|f\|_2^2\lambda^{1/2}.$$
To estimate $$\frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)} \int_M \langle\sum_{i=1}^n\iota(e_i)(R(\nabla f,e_i)\omega),\iota(\nabla f) \omega\rangle\,d\mu_g,$$ we estimate the following terms: $$\begin{aligned}
&\frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)} \int_M \langle \nabla_{\nabla f} d^\ast \omega,\iota(\nabla f) \omega\rangle\,d\mu_g,\\
&\frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)} \int_M \langle d^\ast \nabla_{\nabla f} \omega, \iota(\nabla f)\omega\rangle\,d\mu_g,\\
&\frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)} \int_M \langle \sum_{i,j=1}^n \langle \nabla_{e_j} \nabla f, e_i\rangle\iota(e_j)\nabla_{e_i}\omega,\iota(\nabla f)\omega\rangle\,d\mu_g.\end{aligned}$$ We have $$\begin{split}
\left|\frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)} \int_M \langle \nabla_{\nabla f} d^\ast \omega,\iota(\nabla f) \omega\rangle\,d\mu_g\right|&
=\left|\frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)} \int_M \langle d^\ast \omega,\nabla^\ast (d f\otimes\iota(\nabla f) \omega)\rangle\,d\mu_g\right|\\
&\leq C\|f\|_2^2\lambda^{1/2},\\
\left|\frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)} \int_M \langle d^\ast \nabla_{\nabla f}\omega,\iota(\nabla f) \omega\rangle\,d\mu_g\right|&
=\left|\frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)} \int_M \langle \nabla \omega, d f \otimes d (\iota(\nabla f) \omega)\rangle\,d\mu_g\right|\\
&\leq C\|f\|_2^2\lambda^{1/2}
\end{split}$$ and $$\left|\frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)} \int_M \langle \sum_{i,j=1}^n \langle \nabla_{e_j} \nabla f, e_i\rangle\iota(e_j)\nabla_{e_i}\omega,\iota(\nabla f)\omega\rangle\,d\mu_g\right|\\
\leq C\|f\|_2^2\lambda^{1/2}.$$ Thus, by Lemma \[p2c\] (iii), we get $$\label{4d}
\left|\frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)} \int_M \langle\sum_{i=1}^n\iota(e_i)(R(\nabla f,e_i)\omega),\iota(\nabla f) \omega\rangle\,d\mu_g\right|
\leq C\|f\|_2^2\lambda^{1/2}.$$
By (\[4a\]), (\[4b\]), (\[4c\]), (\[4d\]) and Lemma \[p2c\], we get (ii) and (iii).
Since $$\nabla (\iota(\nabla f)\omega)-\sum_{i=1}^n e^i\otimes \iota(\nabla_{e_i}\nabla f)\omega
=\sum_{i=1}^n e^i\otimes\iota(\nabla f)\nabla_{e_i}\omega,$$ we get (iv).
Since $$d (\iota(\nabla f)\omega)-\sum_{i=1}^n e^i\wedge \iota(\nabla_{e_i}\nabla f)\omega
=\sum_{i=1}^n e^i\wedge\iota(\nabla f)\nabla_{e_i}\omega,$$ we get (v).
Since $$\nabla (\iota(\nabla f)\omega)
=\sum_{i=1}^n e^i\otimes \iota(\nabla_{e_i}\nabla f)\omega+\sum_{i=1}^n e^i\otimes\iota(\nabla f)\nabla_{e_i}\omega,$$ we get (vi).
By Theorem \[p2b\] and (\[2b\]), we have $$\begin{split}
&\frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)}\int_M \langle \mathcal{R}_{k-1}(\iota(\nabla f)\omega),\iota(\nabla f)\omega\rangle \,d\mu_g\\
=&\frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)}\int_M \langle (\Delta-\nabla^\ast\nabla)(\iota(\nabla f)\omega),\iota(\nabla f)\omega\rangle \,d\mu_g\\
=&\frac{k-1}{k} \frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)}\int_M |d(\iota(\nabla f)\omega)|^2\,d\mu_g\\
&+\frac{n-k+1}{n-k+2} \frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)}\int_M |d^\ast(\iota(\nabla f)\omega)|^2\,d\mu_g-\|T(\iota(\nabla f)\omega)\|_2^2\\
=& \frac{k-1}{k} \frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)}\int_M \langle \Delta (\iota(\nabla f)\omega), \iota(\nabla f)\omega\rangle\,d\mu_g\\
&+\left(\frac{n-k+1}{n-k+2}-\frac{k-1}{k} \right)\|d^\ast(\iota(\nabla f)\omega)\|_2^2-\|T(\iota(\nabla f)\omega)\|_2^2.
\end{split}$$ Thus, by (i), (ii) and (iii), we get (vii) Finally, we prove (viii). Suppose that $M$ is oriented and $1\leq k\leq n/2$. Since $\nabla (\ast \omega)=\ast\nabla \omega$, we have $$\begin{aligned}
&\frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)}\int_M \langle \iota({\mathop{\mathrm{Ric}}\nolimits}(\nabla f))\ast\omega,\iota(\nabla f)\ast\omega\rangle
\,d\mu_g\\
\leq &
\frac{n-k-1}{n-k} \frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)}\int_M \langle \iota(\nabla\Delta f)\ast\omega,\iota(\nabla f)\ast\omega\rangle \,d\mu_g-\|T(\iota(\nabla f)\ast\omega)\|_2^2+C\|f\|_2^2\lambda^{1/2}\end{aligned}$$ by (vii). Thus, by (\[hstar\]), (i), (iii) and (vii), we get $$\begin{aligned}
&\frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)}\int_M {\mathop{\mathrm{Ric}}\nolimits}(\nabla f,\nabla f)|\omega|^2\,d\mu_g\\
\leq &
\frac{n-k-1}{n-k} \frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)}\int_M \langle \nabla\Delta f,\nabla f\rangle|\omega|^2 \,d\mu_g
-\|T(\iota(\nabla f)\omega)\|_2^2
-\|T(\iota(\nabla f)\ast\omega)\|_2^2\\
&\qquad -\left(\frac{n-k-1}{n-k} -\frac{k-1}{k} \right)
\frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)}\int_M \langle \iota(\nabla\Delta f)\omega,\iota(\nabla f)\omega\rangle \,d\mu_g
+C\|f\|_2^2\lambda^{1/2}\\
\leq &
\frac{n-k-1}{n-k} \frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)}\int_M \langle \nabla\Delta f,\nabla f\rangle|\omega|^2 \,d\mu_g
-\|T(\iota(\nabla f)\omega)\|_2^2
-\|T(\iota(\nabla f)\ast\omega)\|_2^2\\
&\qquad\qquad -\left(\frac{n-k-1}{n-k} -\frac{k-1}{k} \right)\|d(\iota(\nabla f)\omega)\|^2_2
+C\|f\|_2^2\lambda^{1/2}.\end{aligned}$$ This gives (viii).
Eigenvalue Estimate
-------------------
In this subsection, we complete the proofs of Main Theorem 1 and Main Theorem 3.
We need the following $L^\infty$ estimates.
\[Linfes\] Take an integer $n\geq 2$ and positive real numbers $K>0$, $D>0$, $\Lambda>0$. Let $(M,g)$ be an $n$-dimensional closed Riemannian manifold with ${\mathop{\mathrm{Ric}}\nolimits}\geq-Kg$ and ${\mathop{\mathrm{diam}}\nolimits}(M)\leq D$. Then, we have the following:
- For any function $f\in C^\infty(M)$ and any $\lambda\geq 0$ with $\Delta f=\lambda f$ and $\lambda\leq \Lambda$, then we have $\|\nabla f\|_\infty\leq C(n,K,D,\Lambda)\|f\|_2$ and $\|f\|_\infty\leq C(n,K,D,\Lambda)\|f\|_2$.
- For any $p$-form $\omega\in \Gamma\left(\bigwedge^p T^\ast M\right)$ and any $\lambda\geq 0$ with $\Delta_{C,p} \omega=\lambda \omega$ and $\lambda\leq \Lambda$, then we have $\|\omega\|_\infty\leq C(n,K,D,\Lambda)\|\omega\|_2$.
By the gradient estimate for eigenfunctions [@Pe1 Theorem 7.3], we get (i).
Let us show (ii). Since we have $$\Delta |\omega|^2=2\langle \Delta_{C,p} \omega, \omega \rangle-2|\nabla \omega|^2\leq 2 \Lambda |\omega|^2,$$ we get $\|\omega\|_\infty\leq C$ by [@Pe3 Proposition 9.2.7] (see also Proposition 7.1.13 and Proposition 7.1.17 in [@Pe3]). Note that our sign convention of the Laplacian is different from [@Pe3].
We use the following proposition not only for the proofs of Main Theorem 1 and Main Theorem 3 but also for other main theorems.
\[p4e\] For given integers $n\geq 4$ and $2\leq p \leq n/2$, there exists a constant $C(n,p)>0$ such that the following property holds. Let $(M,g)$ be an $n$-dimensional closed oriented Riemannian manifold with ${\mathop{\mathrm{Ric}}\nolimits}_g\geq (n-p-1)g$. Suppose that an integer $i\in\mathbb{Z}_{>0}$ satisfies $\lambda_i(g)\leq n-p+1$, and there exists an eigenform $\omega$ of the connection Laplacian $\Delta_{C,p}$ acting on $p$-forms with $\|\omega\|_2=1$ corresponding to the eigenvalue $\lambda$ with $0\leq \lambda\leq 1$. Then, we have $$\begin{aligned}
&\frac{n-p-1}{n-p}\lambda_i(g)\left(\lambda_i(g)-(n-p)\right)\|f_i\|^2\\
\geq&\|T(\iota(\nabla f_i)\omega)\|_2^2
+\|T(\iota(\nabla f_i)\ast\omega)\|_2^2\\
&+\left(\frac{n-p-1}{n-p} -\frac{p-1}{p} \right)\|d(\iota(\nabla f_i)\omega)\|^2_2
-C\lambda^{1/2}\|f_i\|_2^2,\end{aligned}$$ where $f_i$ denotes the $i$-th eigenfunction of the Laplacian acting on functions.
By Lemma \[p4d\] (viii), we have $$\begin{aligned}
&\frac{n-p-1}{{\mathop{\mathrm{Vol}}\nolimits}(M)}\int_M \langle\nabla f_i,\nabla f_i\rangle|\omega|^2\,d\mu_g\\
\leq&\frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)}\int_M {\mathop{\mathrm{Ric}}\nolimits}(\nabla f_i,\nabla f_i)|\omega|^2\,d\mu_g\\
\leq&\frac{n-p-1}{n-p}\frac{\lambda_i(g)}{{\mathop{\mathrm{Vol}}\nolimits}(M)}\int_M \langle\nabla f_i,\nabla f_i\rangle|\omega|^2\,d\mu_g
-\|T(\iota(\nabla f_i)\omega)\|_2^2
-\|T(\iota(\nabla f_i)\ast\omega)\|_2^2\\
&\qquad\qquad -\left(\frac{n-p-1}{n-p} -\frac{p-1}{p} \right)\|d(\iota(\nabla f_i)\omega)\|^2_2
+C\lambda^{1/2}\|f_i\|_2^2.\end{aligned}$$ Thus, we get the proposition by Lemma \[p4c\].
If $M$ is orientable, we get the theorem immediately by Proposition \[p4e\]. If $M$ is not orientable, we get the theorem by considering the two-sheeted orientable Riemannian covering $\pi\colon (\widetilde{M},\tilde{g})\to (M,g)$ because we have $
\lambda_1(g)\geq\lambda_1(\tilde{g})
$ and $
\lambda_1(\Delta_{C,p},g)\geq \lambda_1(\Delta_{C,p},\tilde{g}).
$
Similarly, we get Main Theorem 3 because $\lambda_1(\Delta_{C,p},g)=\lambda_1(\Delta_{C,n-p},g)$ holds if the manifold is orientable.
Pinching
========
In this section, we show the remaining main theorems. Main Theorem 2 is proved in subsection 5.5 except for the orientability, and the orientability is proved in subsection 5.7. Main Theorem 4 is proved in subsection 5.8. We list assumptions of this section.
\[asu1\] Throughout in this section except for subsection 5.1, we assume the following:
- $n\geq 5$ and $2\leq p < n/2$.
- $(M,g)$ is an $n$-dimensional closed Riemannian manifold with ${\mathop{\mathrm{Ric}}\nolimits}_g\geq (n-p-1)g$.
- $C=C(n,p)>0$ denotes a positive constant depending only on $n$ and $p$.
- $\delta>0$ satisfies $\delta\leq \delta_0$ for sufficiently small $\delta_0=\delta_0(n,p)>0$.
Note that, for given real numbers $a,b$ with $0<b<a$ and a positive constant $C>0$, we can assume that $$C \delta^a\leq\delta^b.$$ For most subsections, we list additional assumptions at the beginning of them.
Useful Techniques
-----------------
In this subsection, we list some useful techniques for the pinching problems.
The following lemma is a variation of the Cheng-Yau estimate. See [@Ai2 Lemma 2.10] for the proof (see also [@Ch Theorem 7.1]).
\[chya\] Take an integer $n\geq 2$ and positive real numbers $K>0$, $D>0$, $\Lambda>0$ and $0<\epsilon_1 \leq1$. Then, there exists a positive constant $ C(n,K,D,\Lambda)>0$ such that the following property holds. Let $(M,g)$ be an $n$-dimensional closed Riemannian manifold with ${\mathop{\mathrm{Ric}}\nolimits}\geq-Kg$ and ${\mathop{\mathrm{diam}}\nolimits}(M)\leq D$. Take a function $$f\in \bigoplus_{\lambda_j(g)\leq \Lambda} \mathbb{R} f_j,$$ where $f_j$ denotes the $j$-th eigenfunction of the Laplacian acting on functions. Let $p\in M$ be a maximum point of $f$. Then, we have $$|\nabla f|^2(x)\leq \frac{C}{\epsilon_1}\left(f(p)-f(x)+\epsilon_1\|f\|_2\right)^2$$ for all $x\in M$.
The following theorem is an easy consequence of the Bishop-Gromov inequality.
\[bigr\] Given a positive integer $n\geq2$ and positive real numbers $K>0$ and $D>0$, there exists a positive constant $ C(n,K,D)>0$ such that the following property holds. Let $(M,g)$ be an $n$-dimensional closed Riemannian manifold with ${\mathop{\mathrm{Ric}}\nolimits}\geq -Kg$ and ${\mathop{\mathrm{diam}}\nolimits}(M)\leq D$. Then, for any $p\in M$ and $0<r\leq D+1$, we have $r^n {\mathop{\mathrm{Vol}}\nolimits}(M)\leq C{\mathop{\mathrm{Vol}}\nolimits}(B_r(p))$.
The following theorem is due to Cheeger-Colding [@CC2] (see also [@Pe3 Theorem 7.1.10]). By this theorem, we get integral pinching conditions along the geodesics under the integral pinching condition for a function on $M$.
\[seg\] Given an integer $n\geq 2$ and positive real numbers $K>0$ and $D>0$, there exists a positive constant $ C(n,K,D)>0$ such that the following property holds. Let $(M,g)$ be an $n$-dimensional closed Riemannian manifold with ${\mathop{\mathrm{Ric}}\nolimits}\geq -Kg$ and ${\mathop{\mathrm{diam}}\nolimits}(M)\leq D$. For any non-negative measurable function $h\colon M\to \mathbb{R}_{\geq 0}$, we have $$\frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)^2}\int_{M\times M} \frac{1}{d(y_1,y_2)}\int_0^{d(y_1,y_2)} h\circ \gamma_{y_1,y_2}(s) \,dsdy_1dy_2\leq \frac{C}{{\mathop{\mathrm{Vol}}\nolimits}(M)}\int_M h\,d\mu_g.$$
The book [@Pe3] deals with the segment $c_{y_1,y_2}\colon[0,1]\to M$ for each $y_1,y_2\in M$, defined to be $c_{y_1,y_2}(0)=y_1$, $c_{x,y}(1)=y_2$ and $\nabla_{\partial /\partial t} \dot{c}=0$. We have $c_{x,y}(t)=\gamma_{x,y}(t d(x,y))$ for all $t\in[0,1]$ and $$d(y_1,y_2)\int_0^1 h\circ c_{y_1,y_2}(t) \,d t=\int_0^{d(y_1,y_2)} h\circ \gamma_{y_1,y_2}(s) \,d s.$$
After getting integral pinching conditions along the geodesics, we use the following lemma to get $L^\infty$ error estimate along them. The proof is standard (c.f. [@CC2 Lemma 2.41]).
\[trif\] Take positive real numbers $l,\epsilon>0$ and a non-negative real number $r\geq 0$. Suppose that a smooth function $u\colon [0,l]\to \mathbb{R}$ satisfies $$\int_0^l |u''(t)+r^2 u(t)| \,dt\leq\epsilon.$$ Then, we have $$\begin{split}
\left|u(t)-u(0) \cos r t- \frac{u'(0)}{r} \sin r t\right|&\leq \epsilon\frac{\sinh rt}{r},\\
\left|u'(t)+ r u(0)\sin r t- u'(0)\cos r t\right|&\leq \epsilon+\int_0^t\left|u(s)-u(0)\cos r s-\frac{u'(0)}{r}\sin r s\right|\,ds,
\end{split}$$ for all $t\in [0,l]$, where we defined $$\begin{aligned}
\frac{1}{r}\sin r t:=t,\quad
\frac{1}{r}\sinh r t:=t\end{aligned}$$ if $r=0$.
The following lemma is standard.
\[cosi\] For all $t\in \mathbb{R}$, we have $$1-\frac{1}{2}t^2\leq \cos t\leq 1-\frac{1}{2}t^2+\frac{1}{24}t^4.$$ For any $t\in [-\pi,\pi]$, we have $\cos t\leq 1-\frac{1}{9}t^2$, and so $|t|\leq3(1-\cos t)^{1/2}$. For any $t_1,t_2 \in [0,\pi]$, we have $|t_1-t_2|\leq3|\cos t_1-\cos t_2|^{1/2}$.
Finally, we recall some facts about the geodesic flow. Let $(M,g)$ be a closed Riemannian manifold and let $U M$ denotes the sphere bundle defined by $$U M:=\{u\in TM:|u|=1\}.$$ There exists a natural Riemannian metric $G$ on $UM$, which is the restriction of the Sasaki metric on $TM$ (see [@Sa p.55]). The Riemannian volume measure $\mu_G$ satisfies $$\int_{UM} F\,d\mu_G=\int_M \int_{U_p M} F(u)\, d\mu_0(u) \,d\mu_g(p)$$ for any $F\in C^\infty(U M)$, where $\mu_0$ denotes the standard measure on $U_p M\cong S^{n-1}$. The geodesic flow $\phi_t\colon U M\to U M$ ($t\in\mathbb{R}$) is defined by $$\phi_t(u):=\left.\frac{\partial}{\partial s}\right|_{s=t}\gamma_u (s)\in U_{\gamma_u(t)} M$$ for any $u\in U M$. Though $\phi_t$ does not preserve the metric $G$ in general, it preserves the measure $\mu_G$. This is an easy consequence of [@Sa Lemma 4.4], which asserts that the geodesic flow on $T M$ preserve the natural symplectic structure on $T M$. We use the following lemma.
\[geofl\] Let $(M,g)$ be a closed Riemannian manifold. For any $f\in C^\infty (M)$ and $l>0$, we have $$\frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)}\int_M f \,d\mu_g=\frac{1}{l{\mathop{\mathrm{Vol}}\nolimits}(UM)}\int_{UM}\int_0^l f\circ\gamma_u(t)\,d t\,d\mu_G(u).$$
Let $\pi\colon UM\to M$ be the projection. Since the geodesic flow $\phi_t$ preserves the measure $\mu_G$, we have $$\begin{aligned}
&\frac{1}{l{\mathop{\mathrm{Vol}}\nolimits}(UM)}\int_{UM}\int_0^l f\circ\gamma_u(t)\,d t\,d\mu_G(u)\\
=&\frac{1}{l{\mathop{\mathrm{Vol}}\nolimits}(M){\mathop{\mathrm{Vol}}\nolimits}(S^{n-1})}\int_{UM}\int_0^l f\circ\pi\circ\phi_t(u)\,d t\,d\mu_G(u)\\
=&\frac{1}{l{\mathop{\mathrm{Vol}}\nolimits}(M){\mathop{\mathrm{Vol}}\nolimits}(S^{n-1})}\int_0^l\int_{UM} f\circ\pi(u)\,d\mu_G(u)\,d t\\
=&\frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)}\int_M f \,d\mu_g.\end{aligned}$$ This gives the lemma.
This kind of lemma was used by Colding [@Co1] to prove that the almost equality of the Bishop comparison theorem implies the Gromov-Hausdorff closeness to the standard sphere.
Estimates for the Segments
--------------------------
The goal of this subsection is to give error estimates along the geodesics.
In this subsection, we assume the following in addition to Assumption \[asu1\].
- $1\leq k\leq n-p+1$
- $f_i\in C^\infty(M)$ ($i\in\{1,\ldots,k\}$) is an eigenfunction of the Laplacian acting on functions with $\|f_i\|_2^2=1/(n-p+1)$ corresponding to the eigenvalue $\lambda_i$ with $0<\lambda_i\leq n-p+\delta$ such that $$\int_M f_i f_j\,d\mu_g=0$$ holds for any $i\neq j$.
- $\omega\in\Gamma(\bigwedge^p T^\ast M)$ is an eigenform of the connection Laplacian $\Delta_{C,p}$ with $\|\omega\|_2=1$ corresponding to the eigenvalue $\lambda$ with $0\leq \lambda \leq \delta$.
Note that we have $\|\omega\|_\infty\leq C$, $\|f_i\|_\infty \leq C $ and $\|\nabla f_i\|_\infty \leq C$ for all $i\in\{1,\ldots,k\}$ (see Lemma \[Linfes\]). By Main Theorem 1, we have $$\lambda_i\geq n-p-C(n,p)\delta^{1/2}$$ for all $i$. If the manifold is orientable, one can regard $f_i$ as the $i$-th eigenfunction and $\lambda_i$ as $\lambda_i(g)$. However, for the unorientable case, our assumption is convenient when we consider $f_i\circ \pi$, where $\pi: (\widetilde{M},\tilde{g})\to (M,g)$ denotes the two-sheeted orientable Riemannian covering.
We first list some basic consequences of our pinching condition.
\[p5c\] For any $f\in {\mathop{\mathrm{Span}}\nolimits}_{\mathbb{R}}\{f_1,\ldots,f_{k}\}$, we have
- $\|\iota(\nabla f)\omega\|_2^2\leq C\delta^{1/2}\|f\|_2^2$,
- $\|\nabla(\iota(\nabla f)\omega)\|_2^2\leq C\delta^{1/2}\|f\|_2^2$,
- $\|(|\nabla^2 f|^2-\frac{1}{n-p}|\Delta f|^2)|\omega|^2\|_1\leq C\delta^{1/4}\|f\|_2^2$.
It is enough to consider the case when $M$ is orientable.
We first assume that $f=f_i$ for some $i=1,\ldots,k$. Then, we have $$\label{5a0}
\begin{split}
&\|d(\iota(\nabla f)\omega)\|^2_2\leq C\delta^{1/2}\|f\|_2^2,\\
&\|d^\ast (\iota(\nabla f)\omega)\|^2_2 \leq C\delta^{1/2}\|f\|_2^2,\quad
\|T(\iota(\nabla f)\omega)\|_2^2\leq C\delta^{1/2}\|f\|_2^2,\\
&\|d^\ast (\iota(\nabla f)\ast \omega)\|^2_2 \leq C\delta^{1/2}\|f\|_2^2,\quad
\|T(\iota(\nabla f)\ast\omega)\|_2^2 \leq C\delta^{1/2}\|f\|_2^2
\end{split}$$ by Lemma \[p4d\] (i) and Proposition \[p4e\]. Thus, by (\[2b\]), we get $$\label{5a}
\|\nabla (\iota(\nabla f)\omega)\|^2_2\leq C\delta^{1/2}\|f\|_2^2$$ and $$\label{5b}
\|\nabla (\iota(\nabla f)\ast\omega)\|^2_2\leq \frac{1}{n-p} \|d (\iota(\nabla f)\ast \omega)\|^2_2+C\delta^{1/2}\|f\|_2^2.$$ Moreover, by Lemma \[p4d\] (iii), we have $$\label{5c}
\begin{split}
\|\iota(\nabla f)\omega\|_2^2
=&\frac{1}{\lambda_i}\frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)}\int_M \langle \iota(\nabla \Delta f)\omega, \iota(\nabla f)\omega\rangle\,d\mu_g\\
\leq& C\|d(\iota(\nabla f)\omega)\|^2_2+C\|d^\ast (\iota(\nabla f)\omega)\|^2_2+C\delta^{1/2}\|f\|_2^2\\
\leq& C\delta^{1/2}\|f\|_2^2.
\end{split}$$
For any $f=a_1 f_1+\cdots + a_k f_k\in {\mathop{\mathrm{Span}}\nolimits}_{\mathbb{R}}\{f_1,\ldots,f_{k}\}$, we have (\[5a0\]), (\[5a\]), (\[5b\]), (\[5c\]). For example, we have $$\begin{split}
\|\nabla (\iota(\nabla f)\omega)\|_2\leq&\sum_{i=1}^k |a_k|\|\nabla (\iota(\nabla f_i)\omega)\|_2\leq C\delta^{1/4}\sum_{i=1}^k |a_k|\|f_i\|_2\leq C\delta^{1/4}\|f\|_2,\\
\|\nabla (\iota(\nabla f)\ast\omega)\|_2^2=&\frac{1}{n-p}\|d(\iota(\nabla f)\ast\omega)\|_2^2+\frac{1}{p+2}\|d^\ast(\iota(\nabla f)\ast\omega)\|_2^2+\|T(\iota(\nabla f)\ast\omega)\|_2^2\\
\leq& \frac{1}{n-p}\|d(\iota(\nabla f)\ast\omega)\|_2^2+C\delta^{1/2}\|f\|_2^2.
\end{split}$$ Thus, we get (i) and (ii) by (\[5a\]) and (\[5c\]).
Finally, we prove (iii). Take arbitrary $f\in {\mathop{\mathrm{Span}}\nolimits}_{\mathbb{R}}\{f_1,\ldots,f_{k}\}$. We have $$\label{5ca}
\begin{split}
&\left|\sum_{i=1}^n e^i\otimes \iota(\nabla_{e_i}\nabla f)\ast\omega\right|^2\\
=&\sum_{i=1}^n \langle \iota(\nabla_{e_i} \nabla f)\ast\omega,\iota(\nabla_{e_i} \nabla f)\ast\omega\rangle
=|\nabla^2 f|^2|\omega|^2-\left|\sum_{i=1}^n e^i\otimes \iota(\nabla_{e_i}\nabla f)\omega\right|^2.
\end{split}$$ Thus, we have $$\begin{split}
&\frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)}\int_M \left||\nabla(\iota(\nabla f)\ast \omega)|^2-|\nabla^2 f|^2|\omega|^2\right|\,d\mu_g\\
\leq &\frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)}\int_M \left||\nabla(\iota(\nabla f)\ast \omega)|^2-\left|\sum_{i=1}^n e^i\otimes \iota(\nabla_{e_i}\nabla f)\ast\omega\right|^2\right| \,d\mu_g\\
&\qquad+\frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)}\int_M \left|\sum_{i=1}^n e^i\otimes \iota(\nabla_{e_i}\nabla f)\omega\right|^2\,d\mu_g,
\end{split}$$ and so we get $$\label{5d}
\frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)}\int_M \left||\nabla(\iota(\nabla f)\ast \omega)|^2-|\nabla^2 f|^2|\omega|^2\right|\,d\mu_g
\leq C\delta^{1/2}\|f\|_2^2$$ by (ii) and Lemma \[p4d\] (iv) and (vi). We have $$\label{5e}
\begin{split}
&\left|\sum_{i=1}^n e^i\wedge \iota(\nabla_{e_i}\nabla f)\ast\omega\right|^2\\
=&\sum_{i=1}^n |\iota(\nabla_{e_i} \nabla f)\ast\omega|^2-\sum_{i,j=1}^n \langle \iota(e_i)\iota(\nabla_{e_j}\nabla f)\ast \omega, \iota(e_j)\iota(\nabla_{e_i}\nabla f)\ast \omega \rangle\\
=&|\nabla^2 f|^2|\omega|^2-\left|\sum_{i=1}^n e^i\otimes \iota(\nabla_{e_i}\nabla f)\omega\right|^2\\
&\qquad -\sum_{i,j,k,l=1}^n \nabla^2 f(e_i,e_k)\nabla^2 f(e_j,e_l)\langle e^i\wedge e^l \wedge \omega,
e^j\wedge e^k \wedge \omega \rangle
\end{split}$$ by (\[5ca\]) and (\[hstar2\]). Since $$\begin{aligned}
\langle e^i\wedge e^l \wedge \omega,
e^j\wedge e^k \wedge \omega \rangle
=&(\delta_{i j}\delta_{k l}-\delta_{i k}\delta_{j l})|\omega|^2
-\delta_{i j}\langle \iota(e_k)\omega,\iota(e_l)\omega\rangle\\
&+\delta_{i k}\langle \iota(e_j)\omega,\iota(e_l)\omega\rangle
+\langle e^l\wedge \omega,e^j\wedge e^k\wedge\iota(e_i)\omega\rangle,\end{aligned}$$ we have $$\label{5f}
\begin{split}
&\sum_{i,j,k,l=1}^n \nabla^2 f(e_i,e_k)\nabla^2 f(e_j,e_l)\langle e^i\wedge e^l \wedge \omega,
e^j\wedge e^k \wedge \omega \rangle\\
=&|\nabla^2 f|^2|\omega|^2-(\Delta f)^2|\omega|^2
-\sum_{i=1}^n | \iota(\nabla_{e_i}\nabla f)\omega|^2
-\sum_{i=1}^n\Delta f \langle \iota(\nabla_{e_i}\nabla f)\omega,\iota(e_i)\omega\rangle\\
&\qquad+\sum_{j,k,l=1}^n \nabla^2 f(e_j,e_l)\langle e^l\wedge \omega,e^j\wedge e^k\wedge\iota(\nabla_{e_k} \nabla f)\omega\rangle.
\end{split}$$ By (\[5e\]) and (\[5f\]), we get $$\begin{split}
\left|\sum_{i=1}^n e^i\wedge \iota(\nabla_{e_i}\nabla f)\ast\omega\right|^2
=&(\Delta f)^2|\omega|^2+\sum_{i=1}^n\Delta f \langle \iota(\nabla_{e_i}\nabla f)\omega,\iota(e_i)\omega\rangle\\
-&\sum_{j,k,l=1}^n \nabla^2 f(e_j,e_l)\langle e^l\wedge \omega,e^j\wedge e^k\wedge\iota(\nabla_{e_k} \nabla f)\omega\rangle,
\end{split}$$ and so $$\label{5g}
\left|\left|\sum_{i=1}^n e^i\wedge \iota(\nabla_{e_i}\nabla f)\ast\omega\right|^2-(\Delta f)^2|\omega|^2\right|
\leq C|\nabla^2 f| |\omega|\left|\sum_{i=1}^n e^i\otimes \iota(\nabla_{e_i}\nabla f)\omega\right|$$ By (\[5g\]), (ii) and Lemma \[p4d\], we get $$\label{5h}
\frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)}\int_M \left|
|d (\iota(\nabla f)\ast\omega)|^2-(\Delta f)^2|\omega|^2\right|\,d\mu_g
\leq C\delta^{1/4}\|f\|_2^2.$$ Since we have $$|\nabla (\iota(\nabla f)\ast\omega)|^2\geq \frac{1}{n-p} |d (\iota(\nabla f)\ast \omega)|^2$$ at each point by (\[2b\]), we get (iii) by (\[5b\]), (\[5d\]) and (\[5h\]).
We use the following notation.
\[np5d\] Take $f\in {\mathop{\mathrm{Span}}\nolimits}_{\mathbb{R}}\{f_1,\ldots,f_{k}\}$ with $\|f\|_2^2=1/(n-p+1)$ and put $$\begin{aligned}
h_0&:=|\nabla^2 f|^2, \quad h_1:=||\omega|^2-1|, \quad h_2:=|\nabla \omega|^2,\\
h_3&:=|\iota(\nabla f)\omega |^2, \quad h_4:=|\nabla (\iota(\nabla f)\omega)|^2,\quad h_5:=\left|\sum_{i=1}^n e^i\otimes\iota(\nabla_{e_i}\nabla f)\omega\right|^2\\
h_6&:=\left||\nabla^2 f|^2-\frac{1}{n-p}(\Delta f)^2\right||\omega|^2.\end{aligned}$$ For each $y_1\in M$, we define $$\begin{aligned}
D_f(y_1):=&\Big\{y_2\in I_{y_1}\setminus\{y_1\}:\frac{1}{d(y_1,y_2)}\int_{0}^{d(y_1,y_2)} h_0\circ \gamma_{y_1,y_2}(s)\,d s\leq \delta^{-1/50} \text{ and}\\
&\qquad \quad\frac{1}{d(y_1,y_2)}\int_{0}^{d(y_1,y_2)} h_i\circ \gamma_{y_1,y_2}(s)\,d s\leq \delta^{1/5} \text{ for all $i=1,\ldots,6$}
\Big\},\\
Q_f:=&\{y_1\in M: {\mathop{\mathrm{Vol}}\nolimits}(M\setminus D_f(y_1))\leq\delta^{1/100}{\mathop{\mathrm{Vol}}\nolimits}(M)\},\\
E_f(y_1):=&\Big\{u\in U_{y_1} M: \frac{1}{\pi }\int_{0}^{\pi} h_0\circ \gamma_u(s)\,d s\leq \delta^{-1/50} \text{ and }\frac{1}{\pi}\int_{0}^{\pi} h_i\circ \gamma_u (s)\,d s\leq \delta^{1/5} \\
&\qquad \quad\qquad \quad\qquad \quad\qquad \quad\qquad \quad\qquad \quad\qquad \quad\text{ for all $i=1,\ldots,6$}
\Big\},\\
R_f:=&\{y_1\in M: {\mathop{\mathrm{Vol}}\nolimits}(U_{y_1} M\setminus E_f(y_1))\leq\delta^{1/100}{\mathop{\mathrm{Vol}}\nolimits}(U_{y_1}M)\}.\end{aligned}$$
Now, we use the segment inequality and Lemma \[geofl\]. We show that we have the integral pinching condition along most geodesics.
\[p5d\] Take $f\in {\mathop{\mathrm{Span}}\nolimits}_{\mathbb{R}}\{f_1,\ldots,f_{k}\}$ with $\|f\|_2^2=1/(n-p+1)$. Then, we have the following properties:
- ${\mathop{\mathrm{Vol}}\nolimits}(M\setminus Q_f)\leq C\delta^{1/100}{\mathop{\mathrm{Vol}}\nolimits}(M).$
- ${\mathop{\mathrm{Vol}}\nolimits}(M\setminus R_f)\leq C\delta^{1/100}{\mathop{\mathrm{Vol}}\nolimits}(M).$
We have $\|h_i\|_1\leq C\delta^{1/4}$ for all $i=1,\ldots,6$ by the assumption, Lemma \[p4c\], Lemma \[p4d\] (iv) and Lemma \[p5c\], and we have $\|h_0\|_1\leq C$ by (\[4a0\]).
For any $y_1\in M\setminus Q_f$, we have ${\mathop{\mathrm{Vol}}\nolimits}(M\setminus D_f(y_1))>\delta^{1/100}{\mathop{\mathrm{Vol}}\nolimits}(M)$, and so we have either $$\frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)}\int_M\frac{1}{d(y_1,y_2)}\int_0^{d(y_1,y_2)}h_0\circ \gamma_{y_1,y_2}(s)\,d s \,d y_2\geq \frac{1}{7}\delta^{-1/100}$$ or $$\frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)}\int_M\frac{1}{d(y_1,y_2)}\int_0^{d(y_1,y_2)}h_i\circ \gamma_{y_1,y_2}(s)\,d s \,d y_2\geq\frac{1}{7}\delta^{21/100}$$ for some $i=1,\ldots,6$. Thus, we get either $$\frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)}\int_M \int_M\frac{1}{d(y_1,y_2)}\int_0^{d(y_1,y_2)}h_0\circ \gamma_{y_1,y_2}(s)\,d s \,d y_1\,d y_2\geq \frac{1}{49}\delta^{-1/100}{\mathop{\mathrm{Vol}}\nolimits}(M\setminus Q_f)$$ or $$\frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)}\int_M \int_M\frac{1}{d(y_1,y_2)}\int_0^{d(y_1,y_2)}h_i\circ \gamma_{y_1,y_2}(s)\,d s \,d y_1\,d y_2\geq \frac{1}{49}\delta^{21/100}{\mathop{\mathrm{Vol}}\nolimits}(M\setminus Q_f)$$ for some $i=1,\ldots,6$. Therefore, we get (i) by the segment inequality (Theorem \[seg\]).
Similarly, we get (ii) by Lemma \[geofl\].
Under the pinching condition along the geodesic, we get the following:
\[p5e\] Take $f\in {\mathop{\mathrm{Span}}\nolimits}_{\mathbb{R}}\{f_1,\ldots,f_{k}\}$ with $\|f\|_2^2=1/(n-p+1)$. Suppose that a geodesic $\gamma\colon [0,l]\to M$ satisfies one of the following:
- There exist $x\in M$ and $y\in D_f(x)$ such that $l=d(x,y)$ and $\gamma=\gamma_{x,y}$,
- There exist $x\in M$ and $u\in E_f(x)$ such that $l=\pi$ and $\gamma=\gamma_u$.
Then, we have $$||\omega|^2(s)-1|\leq C\delta^{1/10},\quad |\iota(\nabla f)\omega|(s)\leq C\delta^{1/10}$$ for all $s\in [0,l]$, and at least one of the following:
- $\frac{1}{l}\int_0^l|\nabla^2 f|\circ \gamma(s)\,d s\leq C\delta^{1/250}$,
- There exists a parallel orthonormal basis $\{E^1(s),\ldots,E^n(s)\}$ of $T_{\gamma(s)}^\ast M$ along $\gamma$ such that $$|\omega-E^{n-p+1}\wedge\cdots\wedge E^n|(s)\leq C\delta^{1/25}$$ for all $s\in[0,l]$, and $$\frac{1}{l}\int_0^l|\nabla^2 f+f\sum_{i=1}^{n-p}E^i\otimes E^i|(s)\, d s\leq C\delta^{1/200},$$ where we write $|\cdot|(s)$ instead of $|\cdot|\circ\gamma(s)$.
In particular, for both cases, there exists a parallel orthonormal basis $\{E^1(s),\ldots,E^n(s)\}$ of $T_{\gamma(s)}^\ast M$ along $\gamma$ such that $$\frac{1}{l}\int_0^l|\nabla^2 f+f\sum_{i=1}^{n-p}E^i\otimes E^i|(s)\, d s\leq C\delta^{1/250}.$$
Moreover, if we put $$\dot{\gamma}^E:=\sum_{i=1}^{n-p} \langle\dot{\gamma},E_i\rangle E_i,$$ where $\{E_1,\ldots,E_n\}$ denotes the dual basis of $\{E^1,\ldots,E^n\}$, then $|\dot{\gamma}^E|$ is constant along $\gamma$, and $$\begin{split}
\left|f\circ \gamma(s)-f(\gamma(s_0))\cos (|\dot{\gamma}^E|(s-s_0))-\frac{1}{|\dot{\gamma}^E|}\langle\nabla f,\dot{\gamma}(s_0)\rangle\sin (|\dot{\gamma}^{E}|(s-s_0))\right|&\leq C\delta^{1/250},\\
\left| \langle \nabla f, \dot{\gamma}(s)\rangle+f(\gamma(s_0))|\dot{\gamma}^{E}|\sin (|\dot{\gamma}^{E}|(s-s_0))-\langle\nabla f,\dot{\gamma}(s_0)\rangle\cos (|\dot{\gamma}^{E}|(s-s_0))\right|&\leq C\delta^{1/250}
\end{split}$$ for all $s,s_0\in[0,l]$.
We first show the first assertion. Since $\frac{d}{d s}|\omega|^2(s)=2\langle\nabla_{\dot{\gamma}}\omega,\omega\rangle$, we have $$\begin{aligned}
\left||\omega|^2(s)-|\omega|^2(0)\right|
=&\left|\int_0^s \frac{d}{d s}|\omega|^2(t)\,d t\right|\\
\leq& 2 \left(\int_0^s |\nabla \omega|^2 (t)\,d t\right)^{1/2} \left(\int_0^s |\omega|^2 (t)\, d t\right)^{1/2}
\leq C\delta^{1/10}\end{aligned}$$ for all $s\in[0,l]$. Since we have $\int_0^l||\omega|^2-1|\, d t \leq \delta^{1/5}$, we get $||\omega|^2(s)-1|\leq C\delta^{1/10}$. In particular, $|\omega|(s)\geq 1/2$, and so $$\label{5i}
\frac{1}{l}\int_0^l\left||\nabla^2 f|^2-\frac{1}{n-p}(\Delta f)^2\right|(s)\,d s\leq 2\delta^{1/5}.$$ Similarly, we have $|\iota(\nabla f)\omega|(s)\leq C\delta^{1/10}$ for all $s\in [0,l]$.
We show the remaining assertions. Put $$\begin{aligned}
A_1:=&\left\{s\in [0,l]:\left|\sum_{i=1}^n e^i\otimes\iota(\nabla_{e_i}\nabla f)\omega\right|^2(s)>\delta^{1/10}\right\},\\
A_2:=&\left\{s\in [0,l]:\left||\nabla^2 f|^2-\frac{1}{n-p}(\Delta f)^2\right|(s)>\delta^{1/10}\right\},\\
A_3:=&\left\{s\in [0,l]:|\nabla^2 f|(s)<\delta^{1/250}\right\}.\end{aligned}$$ Then, we have $H^1(A_1)\leq \delta^{1/10}l$ and $H^1(A_2)\leq 2\delta^{1/10} l$, where $H^1$ denotes the one dimensional Hausdorff measure. We consider the following two cases:
- $[0,l]=A_1\cup A_2\cup A_3$,
- $[0,l]\neq A_1\cup A_2\cup A_3$.
We first consider the case (a). Since $$H^1([0,l]\setminus A_3)\leq 3 \delta^{1/10} l,$$ we have $$\begin{aligned}
\int_{[0,l]\setminus A_3}|\nabla^2 f|(s)\,d s
\leq&
\left(\int_{[0,l]\setminus A_3}|\nabla^2 f|^2(s)\,d s\right)^{1/2}H^1 ([0,l]\setminus A_3)^{1/2}\\
\leq &C \delta^{-1/100}\delta^{1/20} l=C\delta^{1/25}l.\end{aligned}$$ On the other hand, we have $$\int_{A_3} |\nabla^2 f|(s)\,d s\leq \delta^{1/250} l.$$ Therefore, we get $$\frac{1}{l}\int_0^l|\nabla^2 f|(s)\,d s\leq C\delta^{1/250}.$$ This implies (i). Moreover, since $|\Delta f|\leq \sqrt{n}|\nabla^2 f|$ and $\left\|\Delta f-(n-p)f\right\|_{\infty}\leq C\delta$, we get $$\frac{1}{l}\int_0^l|\nabla^2 f+f\sum_{i=1}^{n-p}E^i\otimes E^i|(s)\, d s\leq C\delta^{1/250},$$ where $\{E^1(s),\ldots,E^n(s)\}$ is any parallel orthonormal basis of $T_{\gamma(s)}^\ast M$ along $\gamma$.
We next consider the case (b). There exists $t\in[0,l]$ such that $$\begin{aligned}
\left|\sum_{i=1}^n e^i\otimes\iota(\nabla_{e_i}\nabla f)\omega\right|^2(t)&\leq\delta^{1/10},\\
\left||\nabla^2 f|^2-\frac{1}{n-p}(\Delta f)^2\right|(t)&\leq\delta^{1/10},\\
|\nabla^2 f|(t)&\geq\delta^{1/250}.\end{aligned}$$ Take an orthonormal basis $\{e_1,\ldots,e_n\}$ of $T_{\gamma(t)}M$ such that $$\nabla^2 f(e_i,e_j)=\mu_i\delta_{i j}\quad (\mu_i\in\mathbb{R})$$ for all $i,j=1,\ldots,n$. Let $\{e^1,\ldots,e^n\}$ be the dual basis of $T_{\gamma(t)}^\ast M$. Then, we have $$\delta^{1/10}\geq \left|\sum_{i=1}^n e^i\otimes\iota(\nabla_{e_i}\nabla f)\omega\right|^2(t)
=\sum_{i=1}^n\mu_i^2 |\iota(e_i)\omega|^2(t).$$ Thus, for each $i=1,\ldots,n$, we have at least one of the following:
- $|\mu_i|\leq \delta^{1/100}$,
- $|\iota(e_i)\omega|(t)\leq \delta^{1/25}$.
Since $|\omega|(t)\geq 1/2$, we have $${\mathop{\mathrm{Card}}\nolimits}\{i: |\iota(e_i)\omega|(t)\leq \delta^{1/25}\}\leq n-p,$$ and so $${\mathop{\mathrm{Card}}\nolimits}\{i: |\mu_i|\leq \delta^{1/100}\}\geq p.$$ Therefore, we can assume $|\mu_i|\leq \delta^{1/100}$ for all $i=n-p+1,\ldots, n$. Then, we get $$\begin{aligned}
\left|
\nabla^2 f+\frac{\Delta f}{n-p}\sum_{i=1}^{n-p} e^i\otimes e^i
\right|^2(t)
=&|\nabla^2 f|^2(t)+\frac{2}{n-p}(\Delta f)(t)\sum_{i=1}^{n-p}\mu_i
+\frac{(\Delta f)^2(t)}{n-p}\\
=&|\nabla^2 f|^2(t)-\frac{(\Delta f)^2(t)}{n-p}-\frac{2}{n-p}(\Delta f)(t)\sum_{i=n-p+1}^{n}\mu_i\\
\leq& C\delta^{1/100}. \end{aligned}$$ Putting $e_i\otimes e_i$ into the inside of the left hand side, we get $$\left|\mu_i+\frac{\Delta f(t)}{n-p}\right|^2\leq C\delta^{1/100}$$ for all $i=1,\ldots, n-p$, and so $$\begin{aligned}
|\mu_i|\geq \frac{|\Delta f(t)|}{n-p}-C\delta^{1/200}
\geq &\left(\frac{|\nabla^2 f|^2(t)-\delta^{1/10}}{n-p}\right)^{1/2}-C\delta^{1/200}\\
\geq &\left(\frac{\delta^{1/125}-\delta^{1/10}}{n-p}\right)^{1/2}-C\delta^{1/200}
>\delta^{1/100}.\end{aligned}$$ Thus, we have $$|\iota(e_i)\omega|(t)\leq \delta^{1/25}$$ for all $i=1,\ldots,n-p$. Thus, we get either $|\omega(t)-e^{n-p+1}\wedge\cdots\wedge e^n|\leq C\delta^{1/25}$ or $|\omega(t)+e^{n-p+1}\wedge\cdots\wedge e^n|\leq C\delta^{1/25}$ by $||\omega|^2(t)-1|\leq C\delta^{1/10}$. We can assume that $|\omega(t)-e^{n-p+1}\wedge\cdots\wedge e^n|\leq C\delta^{1/25}$.
Let $\{E_1,\ldots,E_n\}$ be the parallel orthonormal basis of $TM$ along $\gamma$ such that $E_i(t)=e_i$, and let $\{E^1,\ldots,E^n\}$ be its dual. Because $$\begin{aligned}
&\int_0^l \left|\frac{d}{d s}|\omega-E^{n-p+1}\wedge\cdots \wedge E^n|^2(s)\right|\,d s\\
\leq&
2 \int_0^l |\nabla \omega|(s)|\omega-E^{n-p+1}\wedge\cdots \wedge E^n|(s)\,d s
\leq C\delta^{1/10},\end{aligned}$$ we get $$|\omega-E^{n-p+1}\wedge\cdots\wedge E^n|(s)\leq C\delta^{1/25}$$ for all $s\in [0,l]$. Thus, we get $$|\langle\iota(E_i)\omega,\iota(E_j)\omega\rangle|\leq C\delta^{1/25}$$ for all $i=1,\cdots,n$ and $j=1,\ldots,n-p$, and $$|\langle\iota(E_i)\omega,\iota(E_j)\omega\rangle-\delta_{i j}|\leq C\delta^{1/25}$$ for all $i,j=n-p+1,\cdots,n$. Therefore, we get $$\begin{aligned}
&\left|
\left|\sum_{i=1}^n E^i\otimes\iota(\nabla_{E_i}\nabla f)\omega\right|^2-\sum_{i=1}^n\sum_{j=n-p+1}^n(\nabla^2 f(E_i,E_j))^2
\right|\\
=&\left|
\sum_{i,j,k=1}^n \nabla^2 f(E_i,E_j)\nabla^2 f(E_i,E_k)\langle\iota(E_j)\omega,\iota(E_k)\omega\rangle-\sum_{i=1}^n\sum_{j=n-p+1}^n(\nabla^2 f (E_i,E_j))^2
\right|\\
\leq&C |\nabla^2 f|^2 \delta^{1/25}.\end{aligned}$$ Thus, for all $i=1,\cdots,n$ and $j=1,\ldots,n-p$, we get $$|\nabla^2 f(E_i,E_j)|^2\leq \left|\sum_{k=1}^n E^k\otimes\iota(\nabla_{E_k}\nabla f)\omega\right|^2+C |\nabla^2 f|^2 \delta^{1/25},$$ and so $$\frac{1}{l}\int_0^l|\nabla^2 f (E_i,E_j)|^2(s)\,d s
\leq 2\delta^{1/5}+C\delta^{-1/50}\delta^{1/25}
\leq C\delta^{1/50}.$$ This gives $$\frac{1}{l}\int_0^l|\nabla^2 f (E_i,E_j)|(s)\,d s
\leq C\delta^{1/100}$$ for all $i=1,\cdots,n$ and $j=1,\ldots,n-p$. Because $$\begin{aligned}
\left|
\nabla^2 f+\frac{\Delta f}{n-p}\sum_{i=1}^{n-p}E^i\otimes E^i
\right|^2
=&|\nabla^2 f|^2+2\frac{\Delta f}{n-p}\sum_{i=1}^{n-p}\nabla^2 f (E_i,E_i)+\frac{(\Delta f)^2}{n-p}\\
=&|\nabla^2 f|^2-\frac{(\Delta f)^2}{n-p}-2\frac{\Delta f}{n-p}\sum_{i=n-p+1}^{n}\nabla^2 f (E_i,E_i),\end{aligned}$$ we have $$\frac{1}{l}\int_0^l\left|
\nabla^2 f+\frac{\Delta f}{n-p}\sum_{i=1}^{n-p}E^i\otimes E^i
\right|^2\,d s
\leq 2\delta^{1/5}+C\delta^{1/100}\leq C\delta^{1/100}.$$ Since $$\left\|\frac{\Delta f}{n-p}-f\right\|_{\infty}\leq C\delta$$ we get $$\frac{1}{l}\int_0^l|\nabla^2 f+f\sum_{i=1}^{n-p}E^i\otimes E^i|(s)\, d s\leq C\delta^{1/200}.$$ This implies (ii).
Let us show the final assertion. It is trivial that $|\dot{\gamma}^E|$ is constant along $\gamma$. Since we have $$\left(\nabla^2 f+f \sum_{i=1}^{n-p}E^i\otimes E^i\right)(\dot{\gamma},\dot{\gamma})
=\frac{d^2}{d s^2} f\circ \gamma + |\dot{\gamma}^E|^2 f\circ \gamma,$$ we get $$\int_0^l\left|\frac{d^2}{d s^2} f\circ \gamma(s) + |\dot{\gamma}^E|^2 f\circ \gamma(s)\right|\,d s\leq C\delta^{1/250}.$$ Thus, we get the lemma by Lemma \[trif\].
Almost Parallel $(n-p)$-form I
------------------------------
In this subsection, we consider the pinching condition on $\lambda_1(\Delta_{C,n-p})$ for $2\leq p< n/2$. If $M$ is oriented, then this is coincide with the pinching condition on $\lambda_1(\Delta_{C,p})$. Thus, we only consider the case when $M$ is not orientable.
\[anori\] In this subsection, we assume the following in addition to Assumption \[asu1\].
- $M$ is not orientable.
- $1\leq k\leq n-p+1$.
- $f_i\in C^\infty(M)$ ($i\in\{1,\ldots,k\}$) is an eigenfunction of the Laplacian acting on functions with $\|f_i\|_2^2=1/(n-p+1)$ corresponding to the eigenvalue $\lambda_i$ with $0<\lambda_i\leq n-p+\delta$ such that $$\int_M f_i f_j\,d\mu_g=0$$ holds for any $i\neq j$.
- $\xi\in\Gamma(\bigwedge^{n-p} T^\ast M)$ is an eigenform of the connection Laplacian $\Delta_{C,n-p}$ with $\|\xi\|_2=1$ corresponding to the eigenvalue $\lambda$ with $0\leq \lambda \leq \delta$.
Under these assumptions, we use the following notation.
\[np5f\] Take $f\in{\mathop{\mathrm{Span}}\nolimits}_{\mathbb{R}}\{f_1,\ldots, f_k\}$ with $\|f\|_2^2=1/(n-p+1)$. Let $\pi\colon (\widetilde{M},\tilde{g})\to (M,g)$ be the two-sheeted oriented Riemannian covering. Put $
\tilde{f}:=f\circ \pi\in C^\infty(\widetilde{M})$, $\widetilde{\xi}:=\pi^\ast \xi\in\Gamma(\bigwedge^{n-p}T^\ast M)$ and $\omega:=\ast \widetilde{\xi}\in\Gamma(\bigwedge^{p}T^\ast \widetilde{M})$. Define $h_0,\ldots,h_6$, $Q_{\tilde{f}}$, $D_{\tilde{f}}(\tilde{y}_1)$, $R_{\tilde{f}}$ and $E_{\tilde{f}}(\tilde{y_1})$ as Notation \[np5d\] for $\tilde{f}$, $\omega$ and $\tilde{y}_1\in \widetilde{M}$. Put $$\begin{aligned}
Q_f:=&M\setminus \pi\left(\widetilde{M}\setminus Q_{\tilde{f}}\right),\quad D_f(y_1):=&&M\setminus \pi\left(\widetilde{M}\setminus\bigcap_{\tilde{y}\in\pi^{-1}(y_1)} D_{\tilde{f}}(\tilde{y})\right),\\
R_f:=&M\setminus \pi\left(\widetilde{M}\setminus R_{\tilde{f}}\right),\quad E_f(y_1):=&&U_{y_1} M\setminus \bigcup_{\tilde{y}\in\pi^{-1}(y_1)}\pi_\ast\left(U_{\tilde{y}}\widetilde{M}\setminus E_{\tilde{f}}(\tilde{y})\right)\end{aligned}$$ for each $y_1\in M$.
We immediately have the following lemmas by Lemma \[p5d\] and Lemma \[p5e\].
\[p5f\] We have the following:
- ${\mathop{\mathrm{Vol}}\nolimits}(M\setminus Q_f)\leq C\delta^{1/100}{\mathop{\mathrm{Vol}}\nolimits}(M)$, and ${\mathop{\mathrm{Vol}}\nolimits}(M\setminus D_f(y_1))\leq2\delta^{1/100}{\mathop{\mathrm{Vol}}\nolimits}(\widetilde{M})=4\delta^{1/100}{\mathop{\mathrm{Vol}}\nolimits}(M)$ for each $y_1\in Q_f$.
- ${\mathop{\mathrm{Vol}}\nolimits}(M\setminus R_f)\leq C\delta^{1/100}{\mathop{\mathrm{Vol}}\nolimits}(M)$, and ${\mathop{\mathrm{Vol}}\nolimits}(U_{y_1} M\setminus E_f(y_1))\leq2\delta^{1/100}{\mathop{\mathrm{Vol}}\nolimits}(U_{y_1}M)$ for each $y_1\in R_f$.
- Take $y_1\in M$ and $y_2\in D_f(y_1)$ and one of the lift of $\gamma_{y_1,y_2}$: $$\tilde{\gamma}_{y_1,y_2}\colon[0,d(y_1,y_2)]\to \widetilde{M}.$$ Put $\tilde{y}_1:=\tilde{\gamma}_{y_1,y_2}(0)\in \widetilde{M}$ and $\tilde{y}_2:=\tilde{\gamma}_{y_1,y_2}(d(y_1,y_2))\in \widetilde{M}$. Then, we have $\tilde{y}_2\in D_{\tilde{f}}(\tilde{y}_1)$.
- Take $y_1\in M$ and $u\in E_f(y_1)$ and one of the lift of $\gamma_u$: $$\tilde{\gamma}_{u}\colon[0,\pi]\to \widetilde{M}.$$ Put $\tilde{y}_1:=\tilde{\gamma}_{u}(0)\in \widetilde{M}$ and $\tilde{u}:=\dot{\tilde{\gamma}}_{u}(0)\in U_{\tilde{y}_1}\widetilde{M}$. Then, we have $\tilde{u}\in E_{\tilde{f}}(\tilde{y}_1)$.
\[p5g\] Suppose that a geodesic $\gamma\colon [0,l]\to M$ satisfies one of the following:
- There exist $x\in M$ and $y\in D_f(x)$ such that $l=d(x,y)$ and $\gamma=\gamma_{x,y}$,
- There exist $x\in M$ and $u\in E_f(x)$ such that $l=\pi$ and $\gamma=\gamma_u$.
Let $\tilde{\gamma}\colon [0,l]\to\widetilde{M}$ be one of the lift of $\gamma$. Then, we have $$||\omega|^2(\tilde{\gamma}(s))-1|\leq C\delta^{1/10},\quad |\iota(\nabla \tilde{f})(\omega)|\circ\tilde{\gamma}(s)\leq C\delta^{1/10}$$ for all $s\in [0,l]$, and at least one of the following:
- $\frac{1}{l}\int_0^l|\nabla^2 f|\circ \gamma(s)\,d s\leq C\delta^{1/250}$,
- There exists a parallel orthonormal basis $\{E^1(s),\ldots,E^n(s)\}$ of $T_{\gamma(s)}^\ast M$ along $\gamma$ such that $$|\xi-E^{1}\wedge\cdots\wedge E^{n-p}|(s)\leq C\delta^{1/25}$$ for all $s\in[0,s]$, and $$\frac{1}{l}\int_0^l|\nabla^2 f+f\sum_{i=1}^{n-p}E^i\otimes E^i|(s)\, d s\leq C\delta^{1/200}.$$
In particular, for both cases, there exists a parallel orthonormal basis $\{E^1(s),\ldots,E^n(s)\}$ of $T_{\gamma(s)}^\ast M$ along $\gamma$ such that $$\frac{1}{l}\int_0^l|\nabla^2 f+f\sum_{i=1}^{n-p}E^i\otimes E^i|(s)\, d s\leq C\delta^{1/250}.$$ Moreover, if we put $$\dot{\gamma}^E:=\sum_{i=1}^{n-p} \langle\dot{\gamma},E_i\rangle E_i,$$ where $\{E_1,\ldots,E_n\}$ denotes the dual basis of $\{E^1,\ldots,E^n\}$, then $|\dot{\gamma}^E|$ is constant along $\gamma$, and $$\begin{split}
\left|f\circ \gamma(s)-f(\gamma(s_0))\cos (|\dot{\gamma}^E|(s-s_0))-\frac{1}{|\dot{\gamma}^E|}\langle\nabla f,\dot{\gamma}(s_0)\rangle\sin (|\dot{\gamma}^{E}|(s-s_0))\right|&\leq C\delta^{1/250},\\
\left| \langle \nabla f, \dot{\gamma}(s)\rangle+f(\gamma(s_0))|\dot{\gamma}^{E}|\sin (|\dot{\gamma}^{E}|(s-s_0))-\langle\nabla f,\dot{\gamma}(s_0)\rangle\cos (|\dot{\gamma}^{E}|(s-s_0))\right|&\leq C\delta^{1/250}
\end{split}$$ for all $s,s_0\in[0,l]$.
Eigenfunction and Distance
--------------------------
In this subsection, we show that the function is an almost cosine function in some sense under our pinching condition.
In this subsection, we assume the following in addition to Assumption \[asu1\].
- $1\leq k\leq n-p+1$.
- $f_i\in C^\infty(M)$ ($i\in\{1,\ldots,k\}$) is an eigenfunction of the Laplacian acting on functions with $\|f_i\|_2^2=1/(n-p+1)$ corresponding to the eigenvalue $\lambda_i$ with $0<\lambda_i\leq n-p+\delta$ such that $$\int_M f_i f_j\,d\mu_g=0$$ holds for any $i\neq j$.
- Either $\lambda_1(\Delta_{C,p})\leq \delta$ or $\lambda_1(\Delta_{C,n-p})\leq \delta.$
The following proposition is the goal of this subsection. See Notation \[np5d\] and Notation \[np5f\] for the definitions of $D_f$, $Q_f$, $E_f$ and $R_f$.
\[p53a\] Take $f\in {\mathop{\mathrm{Span}}\nolimits}_{\mathbb{R}}\{f_1,\ldots,f_{k}\}$ with $\|f\|_2^2=1/(n-p+1)$. There exists a point $p_f\in Q_f$ such that the following properties hold:
- $\sup_M f\leq f(p_f)+C\delta^{1/100n}$ and $|f(p_f)-1|\leq C\delta^{1/800n}$,
- For any $x\in D_f(p_f)$ with $|\nabla f|(x)\leq \delta^{1/800n}$, we have $$||f(x)|-1|\leq C\delta^{1/800n}.$$
- For any $x\in D_f(p_f)\cap Q_f\cap R_f$, we have $$|f(x)^2+|\nabla f|^2(x)-1|\leq C \delta^{1/800n}.$$
- Put $$A_f:=\{x\in M: |f(x)-1|\leq \delta^{1/900n}\}.$$ Then, we have $$|f(x)-\cos d(x,A_f)|\leq C\delta^{1/2000n}$$ for all $x\in M$, and $$\sup_{x\in M}d(x,A_f)\leq \pi+ C\delta^{1/100n}.$$
Take a maximum point $\tilde{p}\in M$ of $f$. Then, by the Bishop-Gromov theorem and Lemma \[p5d\] (or Lemma \[p5f\]), there exists a point $p_f\in Q_f$ with $d(\tilde{p},p_f)\leq C \delta^{1/100n}$. By Lemma \[chya\], we have $$|\nabla f|^2(p_f)\leq \frac{C}{\delta^{1/100n}}(f(\tilde{p})-f(p_f)+\delta^{1/100n}\|f\|_2)^2\leq C\delta^{1/100n},$$ and so $$\label{54b}
|\nabla f|(p_f)\leq C\delta^{1/200n}.$$
\[c0\] For any $x\in D_f(p_f)$ with $|\nabla f|(x)\leq C\delta^{1/800n}$, we have $$||f(x)|-|f(p_f)||\leq C\delta^{1/800n}.$$
Since $$|\nabla f|(p_f)\leq C\delta^{1/200n}, \quad |\nabla f|(x)\leq C\delta^{1/800n},$$ we get $$\begin{aligned}
|f\circ \gamma_{p_f,x}(s)-f(p_f)\cos ( |\dot{\gamma}_{p_f,x}^E| s)|&\leq C\delta^{1/200n},\\
|f\circ \gamma_{p_f,x}(d(p_f,x)-s)-f(x)\cos ( |\dot{\gamma}_{p_f,x}^E|s)|&\leq C\delta^{1/800n}\end{aligned}$$ for all $s\in[0,d(p_f,x)]$ by Lemma \[p5e\] or Lemma \[p5g\]. Thus, we have $$\begin{aligned}
|f(x)-f(p_f)\cos ( |\dot{\gamma}_{p_f,x}^E| d(p_f,x))|&\leq C\delta^{1/200n},\\
|f(p_f)-f(x)\cos ( |\dot{\gamma}_{p_f,x}^E|d(p_f,x))|&\leq C\delta^{1/800n},\end{aligned}$$ and so we get $||f(x)|-|f(p_f)||\leq C\delta^{1/800n}$.
Similarly to $p_f$, we take a point $q_f\in Q_{f}(x)$ with $d(\tilde{q},q_f)\leq C\delta^{1/100n}$, where $\tilde{q}\in M$ is minimum point of $f$. By $\|f\|_{\infty}\geq\|f\|_2=1/\sqrt{n-p+1}$, we have $\max\{|f(p_f)|,|f(q_f)|\}\geq 1/\sqrt{n-p+1}-C\delta^{1/100n}$. Since $|\nabla f|(q_f)\leq C\delta^{1/200n}$, we have $|f(p_f)|\geq |f(q_f)|-C\delta^{1/800n}$ by Claim \[c0\]. Therefore, we get $$\label{54c0}
f(p_f)\geq \frac{1}{\sqrt{n-p+1}}-C\delta^{1/800n}\geq\frac{1}{2\sqrt{n-p+1}}.$$
\[c1\] Take $x\in M$ and $y\in D_f(x)$. Let $\{E^1,\ldots,E^n\}$ be a parallel orthonormal basis along $\gamma_{x,y}$ in Lemma \[p5e\] or Lemma \[p5g\]. If $(i)$ holds in the lemmas, we can assume that $E_1=\dot{\gamma}_{x,y}$. Then, we have $$\begin{aligned}
\label{54ba}|\langle\nabla f(x),\dot{\gamma}_{x,y}(0)\rangle-\langle
\nabla f(x),\dot{\gamma}_{x,y}^{E}(0)\rangle|&\leq C\delta^{1/25},\\
\label{54c} |\langle\nabla f(x),\dot{\gamma}_{x,y}(0)\rangle|&\leq |\nabla f(x)||\dot{\gamma}_{x,y}^{E}|+C\delta^{1/25}\end{aligned}$$ and $$\begin{split}
\left|f\circ \gamma_{x,y}(s)-f(x)\cos (|\dot{\gamma}_{x,y}^{E}|s)-\frac{1}{|\dot{\gamma}_{x,y}^{E}|}\langle\nabla f(x),\dot{\gamma}_{x,y}(0)\rangle\sin (|\dot{\gamma}_{x,y}^{E}|s)\right|&\leq C\delta^{1/250},\\
\left| \langle \nabla f, \dot{\gamma}_{x,y}(s)\rangle+f(x)|\dot{\gamma}_{x,y}^{E}|\sin (|\dot{\gamma}_{x,y}^{E}|s)-\langle\nabla f(x),\dot{\gamma}_{x,y}(0)\rangle\cos (|\dot{\gamma}_{x,y}^{E}|s)\right|&\leq C\delta^{1/250}
\end{split}$$ for all $s\in[0,d(x,y)]$.
If (i) holds in the lemmas, $\dot{\gamma}_{x,y}=\dot{\gamma}_{x,y}^E$, and so (\[54ba\]) and (\[54c\]) are trivial. If (ii) in the lemma holds, we have $|\iota(\nabla f)(E^{n-p+1}\wedge\cdots\wedge E^n)|\leq C\delta^{1/25}$, and so $|\langle\nabla f(x),E_i\rangle|\leq C\delta^{1/25}$ for all $i=n-p+1,\ldots,n$. This gives (\[54ba\]) and (\[54c\]). We get the remaining part of the claim by Lemma \[p5e\] or Lemma \[p5g\] putting $s_0=0$.
\[c2\] For any $x\in D_f(p_f)\cap Q_f\cap R_f$ with $|\nabla f|(x)\geq \delta^{1/800n}$, we have $$|f(x)^2+|\nabla f|^2(x)-f(p_f)^2|\leq C\delta^{1/800n}.$$ Moreover, there exists a point $y\in D_f(p_f)\cap D_f(x)$ such that the following properties hold.
- $d(x,y)< \pi$,
- $|f(p_f)-f(y)|\leq C \delta^{1/800n}$,
- $|f(x)-f(p_f)\cos d(x,y)|\leq C \delta^{1/800n},$
- For any $z\in M$ with $d(x,z)\leq d(x,y)-\delta^{1/2000n}$, we have $$f(p_f)-f(z)\geq \frac{1}{C}\delta^{1/1000n}.$$
Take $x\in D_f(p_f)\cap Q_f\cap R_f$ with $|\nabla f|(x)\geq \delta^{1/800n}$. By the definition of $R_f$, there exists a vector $u\in E_f(x)$ with $$\left|
\frac{\nabla f}{|\nabla f|}(x)-u
\right|\leq C \delta^{1/100n}.$$ Thus, we have $$\label{54d}
\Big|\langle\nabla f(x),\dot{\gamma}_u(0)\rangle-|\nabla f|(x)\Big|=|\nabla f|(x)-\langle\nabla f(x), u\rangle\leq C\delta^{1/100n}.$$ Let $\{E^1,\ldots,E^n\}$ be a parallel orthonormal basis along $\gamma_{u}$ in Lemma \[p5e\] or Lemma \[p5g\]. We first suppose that (ii) holds in the lemmas. Then, for all $i=n-p+1,\ldots, n$, we have $|\langle\nabla f, E_i\rangle|\leq C\delta^{1/25}$, and so $$|\langle u,E_i\rangle|\leq \left| u-\frac{\nabla f}{|\nabla f|}(x)\right|+\left|\langle \frac{\nabla f}{|\nabla f|}(x), E_i\rangle\right|\leq C\delta^{1/100n}+C\delta^{1/25}\delta^{-1/800n}\leq C\delta^{1/100n}.$$ Thus, we get $$|\dot{\gamma}_u^E|^2=|u^E|^2=1-\sum_{i=n-p+1}^n\langle u, E_i\rangle^2\geq 1-C\delta^{1/100n}.$$ If (i) holds in the lemmas, we can assume $u=E_1$, and so $|\dot{\gamma}_u^E|=|u^E|=1$. For both cases, we get we get $$\label{54e}
\begin{split}
|f\circ \gamma_u(s)-f(x)\cos s-|\nabla f|(x)\sin s|\leq& C\delta^{1/100n}\\
|\langle\nabla f,\dot{\gamma}_u(s)\rangle+f(x)\sin s-|\nabla f|(x)\cos s|\leq& C\delta^{1/100n}
\end{split}$$ for all $s\in [0,\pi]$ by (\[54d\]). Take $s_0\in[0,\pi]$ such that $$\begin{aligned}
\frac{f(x)}{(f(x)^2+|\nabla f|^2(x))^{1/2}}=&\cos s_0,\\
\frac{|\nabla f|(x)}{(f(x)^2+|\nabla f|^2(x))^{1/2}}=&\sin s_0.\end{aligned}$$ Since $\sin s_0\geq \frac{1}{C}\delta^{1/800n}$ by the assumption, we have $$\label{54ea}
\frac{1}{C}\delta^{1/800n}\leq s_0\leq \pi-\frac{1}{C}\delta^{1/800n}.$$ By the addition theorem, we have $$\begin{split}
(f(x)^2+|\nabla f|^2(x))^{1/2}\cos (s-s_0)=&f(x)\cos s+|\nabla f|(x)\sin s,\\
(f(x)^2+|\nabla f|^2(x))^{1/2}\sin (s-s_0)=&f(x)\sin s-|\nabla f|(x)\cos s,
\end{split}$$ and so we get $$\begin{split}
|f\circ \gamma_u(s)-(f(x)^2+|\nabla f|^2(x))^{1/2}\cos (s-s_0)|\leq& C\delta^{1/100n},\\
|\langle\nabla f,\dot{\gamma}_u(s)\rangle+(f(x)^2+|\nabla f|^2(x))^{1/2}\sin (s-s_0)|\leq& C\delta^{1/100n}
\end{split}$$ for all $s\in [0,\pi]$ by (\[54e\]). In particular, we get $$\label{54f}
\begin{split}
|f\circ \gamma_u(s_0)-(f(x)^2+|\nabla f|^2(x))^{1/2}|\leq& C\delta^{1/100n},\\
|\langle\nabla f,\dot{\gamma}_u(s_0)\rangle|\leq& C\delta^{1/100n}.
\end{split}$$ Take $y\in D_f(p_f)\cap D_f(x)$ with $d(\gamma_u(s_0),y)\leq C\delta^{1/100n}$. We have $$\label{54fa}
d(x,y)\leq d(x,\gamma_u(s_0))+d(\gamma_u(s_0),y)\leq s_0+C\delta^{1/100n}.$$ By (\[54f\]), we get $$\label{54fb}
|f(y)-(f(x)^2+|\nabla f|^2(x))^{1/2}|\leq C\delta^{1/100n}$$ Take a parallel orthonormal basis $\{\widetilde{E^1},\ldots,\widetilde{E^n}\}$ of $T^\ast M$ along $\gamma_{x,y}$ in Lemma \[p5e\] or Lemma \[p5g\]. By (\[54ea\]) and (\[54fa\]), we get (a) and $$\frac{1}{C}\delta^{1/800n}\leq
|\dot{\gamma}_{x,y}^{\widetilde{E}}|d(x,y)+s_0
\leq 2\pi-\frac{1}{C}\delta^{1/800n},$$ and so $$\label{54g}
\cos (|\dot{\gamma}_{x,y}^{\widetilde{E}}|d(x,y)+s_0)\leq 1-\frac{1}{C}\delta^{1/400n}.$$ If $|\dot{\gamma}_{x,y}^{\widetilde{E}}|\leq \delta^{1/100}$, we have $|f(y)-f(x)|\leq C\delta^{1/250}$ by Claim \[c1\], and so $$(f(x)^2+|\nabla f|^2(x))^{1/2}-f(x)\leq C\delta^{1/100n}$$ by (\[54f\]). This contradicts to $
|\nabla f|(x)\geq \delta^{1/800n}.
$ Thus, we get $|\dot{\gamma}_{x,y}^{\widetilde{E}}|\geq \delta^{1/100}$. Then, we have $$\label{54g1}
\frac{1}{|\dot{\gamma}_{x,y}^{\widetilde{E}}|}\langle\nabla f(x),\dot{\gamma}_{x,y}(0)\rangle\leq |\nabla f|(x)+C\delta^{3/100}$$ and $$\begin{aligned}
&(f(x)^2+|\nabla f|^2(x))^{1/2}\\
\leq& f(y)+C\delta^{1/100n}\\
\leq& f(x)\cos (|\dot{\gamma}_{x,y}^{E}|d(x,y))+\frac{1}{|\dot{\gamma}_{x,y}^{E}|}\langle\nabla f(x),\dot{\gamma}_{x,y}(0)\rangle\sin (|\dot{\gamma}_{x,y}^{E}|d(x,y))+C\delta^{1/100n}\\
\leq &\left(f(x)^2+\frac{1}{|\dot{\gamma}_{x,y}^{\widetilde{E}}|^2}\langle\nabla f(x),\dot{\gamma}_{x,y}(0)\rangle^2\right)^{1/2}+C\delta^{1/100n}\end{aligned}$$ by Claim \[c1\] and (\[54fb\]). Thus, $$\label{54g2}
|\nabla f|^2(x)
\leq
\frac{1}{|\dot{\gamma}_{x,y}^{\widetilde{E}}|^2}\langle\nabla f(x),\dot{\gamma}_{x,y}(0)\rangle^2 +C\delta^{1/100n}.$$ By (\[54g1\]) and (\[54g2\]), we get $$\label{54h0}
\left|\frac{1}{|\dot{\gamma}_{x,y}^{\widetilde{E}}|^2}\langle\nabla f(x),\dot{\gamma}_{x,y}(0)\rangle^2-|\nabla f|^2(x)\right|\leq C\delta^{1/100n}.$$ This gives $$\label{54h}
\begin{split}
&\left|\frac{1}{|\dot{\gamma}_{x,y}^{\widetilde{E}}|}|\langle\nabla f(x),\dot{\gamma}_{x,y}(0)\rangle|-|\nabla f|(x)\right|\\
\leq& \left|\frac{1}{|\dot{\gamma}_{x,y}^{\widetilde{E}}|^2}\langle\nabla f(x),\dot{\gamma}_{x,y}(0)\rangle^2-|\nabla f|^2(x)\right|\delta^{-1/800n}\leq
C\delta^{7/800n}.
\end{split}$$ We show that $\langle\nabla f(x),\dot{\gamma}_{x,y}(0)\rangle> 0$. If $\langle\nabla f(x),\dot{\gamma}_{x,y}(0)\rangle\leq 0$, we get $$\left|f(y)-f(x)\cos (|\dot{\gamma}_{x,y}^{\widetilde{E}}|d(x,y))+ |\nabla f|\sin (|\dot{\gamma}_{x,y}^{\widetilde{E}}|d(x,y))\right|\leq C\delta^{7/800n}$$ by (\[54h\]) and Claim \[c1\], and so $$\left|f(y)-(f(x)^2+|\nabla f|^2(x))^{1/2}\cos (|\dot{\gamma}_{x,y}^{\widetilde{E}}|d(x,y)+s_0)\right|\leq C\delta^{7/800n}.$$ Thus, we get $$\begin{split}
&(f(x)^2+|\nabla f|^2(x))^{1/2}\\
\leq &f(y)+C\delta^{1/100n}\\
\leq &(f(x)^2+|\nabla f|^2(x))^{1/2} \cos (|\dot{\gamma}_{x,y}^{\widetilde{E}}|d(x,y)+s_0)+C\delta^{7/800n}\\
\leq &(f(x)^2+|\nabla f|^2(x))^{1/2} -\frac{1}{C}\delta^{3/800n}
\end{split}$$ by (\[54fb\]), (\[54g\]) and $|\nabla f|(x)\geq \delta^{1/800n}$. This is a contradiction. Therefore, we get $\langle\nabla f(x),\dot{\gamma}_{x,y}(0)\rangle>0$. Thus, $$\label{54ha}
\begin{split}
\left|f(y)-(f(x)^2+|\nabla f|^2(x))^{1/2}\cos (|\dot{\gamma}_{x,y}^{\widetilde{E}}|d(x,y)-s_0)\right|&\leq C\delta^{7/800n},\\
\left| \langle \nabla f(y), \dot{\gamma}_{x,y}\rangle+|\dot{\gamma}_{x,y}^{\widetilde{E}}|(f(x)^2+|\nabla f|^2(x))^{1/2}\sin (|\dot{\gamma}_{x,y}^{\widetilde{E}}|d(x,y)-s_0)\right|&\leq C\delta^{7/800n}
\end{split}$$ by (\[54h\]) and Claim \[c1\]. Then, we have $$\begin{aligned}
(f(x)^2+|\nabla f|^2(x))^{1/2} (1-\cos(|\dot{\gamma}_{x,y}^{\widetilde{E}}|d(x,y)-s_0))
\leq C\delta^{7/800n}\end{aligned}$$ by (\[54fb\]), and so $$1-\cos(|\dot{\gamma}_{x,y}^{\widetilde{E}}|d(x,y)-s_0)\leq C\delta^{3/400n}.$$ by $|\nabla f|(x)\geq\delta^{1/800n}$. Since $$-\pi<|\dot{\gamma}_{x,y}^{\widetilde{E}}|d(x,y)-s_0<\pi,$$ we get $$\label{54i}
\left||\dot{\gamma}_{x,y}^{\widetilde{E}}|d(x,y)-s_0\right|\leq C\delta^{3/800n}.$$ Thus, we have $$s_0\leq |\dot{\gamma}_{x,y}^{\widetilde{E}}|s_0+ C\delta^{3/800n}$$ by (\[54fa\]), and so $$\label{54j}
1-|\dot{\gamma}_{x,y}^{\widetilde{E}}| \leq C\delta^{1/400n}$$ by (\[54ea\]). Thus, we get $$\label{54k}
|d(x,y)-s_0|\leq C\delta^{1/400n}.$$ By (\[54ha\]) and (\[54i\]), we have $$\label{54l}
|\langle\nabla f(y), \dot{\gamma}_{x,y}(d(x,y))\rangle|\leq C\delta^{3/800n}.$$
We have $$\label{54m}
\begin{split}
&\frac{d}{d s}\left(|\nabla f|^2(s)-\langle\nabla f,\dot{\gamma}_{x,y}(s)\rangle^2\right)\\
=&2\left(\langle\nabla_{\dot{\gamma}_{x,y}}\nabla f,\nabla f\rangle(s)-\langle\nabla_{\dot{\gamma}_{x,y}}\nabla f,\dot{\gamma}_{x,y}(s)\rangle\langle\nabla f,\dot{\gamma}_{x,y}(s)\rangle\right)\\
=&2\langle \nabla^2 f+ f\sum_{i=1}^{n-p}\widetilde{E}^i\otimes \widetilde{E}^i,\dot{\gamma}_{x,y}\otimes\nabla f\rangle(s)-2f\langle\nabla f,\dot{\gamma}_{x,y}^{\widetilde{E}}\rangle\\
&-2\langle \nabla^2 f+ f\sum_{i=1}^{n-p}\widetilde{E}^i\otimes \widetilde{E}^i,\dot{\gamma}_{x,y}\otimes\dot{\gamma}_{x,y}
\rangle(s)\langle\nabla f,\dot{\gamma}_{x,y}(s)\rangle\\
&+2f|\dot{\gamma}_{x,y}^{\widetilde{E}}|^2\langle\nabla f,\dot{\gamma}_{x,y}(s)\rangle.
\end{split}$$ Thus, we get $$\label{54n}
\begin{split}
&\left|\frac{d}{d s}\left(|\nabla f|^2(s)-\langle\nabla f,\dot{\gamma}_{x,y}(s)\rangle^2\right)\right|\\
\leq&C\left|\nabla^2 f+ f\sum_{i=1}^{n-p}\widetilde{E}^i\otimes \widetilde{E}^i\right|
+C\left|\langle\nabla f,\dot{\gamma}_{x,y}^{\widetilde{E}}\rangle-|\dot{\gamma}_{x,y}^{\widetilde{E}}|^2\langle\nabla f,\dot{\gamma}_{x,y}(s)\rangle\right|.\\
\leq &C\left|\nabla^2 f+ f\sum_{i=1}^{n-p}\widetilde{E}^i\otimes \widetilde{E}^i\right|+
C\delta^{1/400n}
\end{split}$$ by (\[54ba\]) and (\[54j\]). By integration, we get $$\int_0^{d(x,y)} \left|\frac{d}{d s}\left(|\nabla f|^2(s)-\langle\nabla f,\dot{\gamma}_{x,y}(s)\rangle^2\right)\right|\,d s
\leq C\delta^{1/400n},$$ and so $$\Big||\nabla f|^2(y)-\langle\nabla f(y),\dot{\gamma}_{x,y}(d(x,y))\rangle^2
-|\nabla f|^2(x)+\langle\nabla f(x),\dot{\gamma}_{x,y}(0)\rangle^2
\Big|\leq C\delta^{1/400n}.$$ Thus, we get $$|\nabla f|(y)\leq C\delta^{1/800n}.$$ by (\[54h0\]), (\[54j\]) and (\[54l\]). By Claim \[c0\] and (\[54c0\]), we get $$\left||f(y)|-f(p_f)\right|\leq C\delta^{1/800n}.$$ Since $$f(y)\geq (f(x)^2+|\nabla f|^2(x))^{1/2}-C\delta^{1/100n}\geq \delta^{1/800n}-C\delta^{1/100n}>0$$ by (\[54fb\]), we get $$\left|f(y)-f(p_f)\right|\leq C\delta^{1/800n}.$$ This gives (b). We get $$\label{54o}
|(f(x)^2+|\nabla f|^2(x))^{1/2}-f(p_f)|
\leq C\delta^{1/800n}$$ by (\[54ha\]), (\[54i\]) and (b), and so we get (c) by the definition of $s_0$ and (\[54k\]). (\[54o\]) implies the first assertion.
Finally, we show (d). Suppose that a point $z\in M$ satisfies $d(x,z)\leq d(x,y)-\delta^{1/2000n}$. Then, $d(x,y)\geq \delta^{1/2000n}$, and so $$f(x)\leq f(p_f)\cos d(x,y)+C\delta^{1/800n}\leq f(p_f)-\frac{1}{C}\delta^{1/1000n}$$ by (\[54c0\]). There exists $w\in D_f(x)$ with $d(z,w)\leq C\delta^{1/100n}$. Let $\{\overline{E}^1,\ldots,\overline{E}^n\}$ be a parallel orthonormal basis along $\gamma_{x,w}$ in Lemma \[p5e\] or Lemma \[p5g\]. If (i) holds in the lemmas, we assume that $\overline{E}_1=\dot{\gamma}_{x,w}$. If $|\dot{\gamma}_{x,w}^{\overline{E}}|\leq \delta^{1/100}$, we have $$f(z)\leq f(w)+C\delta^{1/100n}
\leq f(x)+ C\delta^{1/100n}
\leq f(p_f)-\frac{1}{C}\delta^{1/1000n}$$ by Claim \[c1\]. If $|\dot{\gamma}_{x,w}^{\overline{E}}|\geq \delta^{1/100}$, we have $$\begin{split}
f(z)\leq& f(w)+C\delta^{1/100n}\\
\leq& f(x)\cos (|\dot{\gamma}_{x,w}^{\overline{E}}|d(x,z))+|\nabla f|(x)\sin (|\dot{\gamma}_{x,w}^{\overline{E}}|d(x,z))+C\delta^{1/100n}\\
\leq& f(p_f)\cos (|\dot{\gamma}_{x,w}^{\overline{E}}|d(x,z)-d(x,y))+\delta^{1/800n}
\leq f(p_f)-\frac{1}{C}\delta^{1/1000n}
\end{split}$$ by Claim \[c1\], (\[54k\]), (\[54o\]) and $-\pi\leq|\dot{\gamma}_{x,w}^{\overline{E}}|d(x,z)-d(x,y)\leq -\delta^{1/2000n}$. For both cases, we get (d).
By Claim \[c0\] and Claim \[c2\], we get $$\label{54p}
|f(x)^2+|\nabla f|^2(x)-f(p_f)^2|\leq C\delta^{1/800n}$$ for all $x\in D_f(p_f)\cap Q_f\cap R_f$.
\[c3\] We have $$|f(p_f)-1|\leq C\delta^{1/800n}.$$
Since $$\|f^2+|\nabla f|^2-f(p_f)^2\|_{\infty}\leq C$$ and $${\mathop{\mathrm{Vol}}\nolimits}(M\setminus (D_f(p_f)\cap Q_f\cap R_f) )\leq C\delta^{1/100},$$ we get $$\frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)}\int_M|f(x)^2+|\nabla f|^2(x)-f(p_f)^2| \,d\mu_g\leq C \delta^{1/800n}$$ by (\[54p\]). By the assumption, we have $$\frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)}\left|\int_M (f(x)^2+|\nabla f|^2(x)-1) \,d\mu_g\right|\leq C \delta^{1/2}$$ Thus, we get $$|f(p_f)^2-1|\leq C\delta^{1/800n}.$$ Since $f(p_f)>0$, we get the claim.
By Claim \[c0\], Claim \[c3\] and (\[54p\]), we get (i), (ii) and (iii).
Finally, we prove (iv). Put $$A_f:=\{x\in M: |f(x)-1|\leq \delta^{1/900n}\}.$$ Since we have $$|f(w)-\cos d(w,A_f)|\leq \delta^{1/900n}$$ for all $w\in A_f$, we get (iv) on $A_f$.
Let us show (iv) on $M\setminus A_f$. Take $w\notin A_f$ and $x\in D_f(p_f)\cap Q_f\cap R_f $ with $d(w,x)\leq C\delta^{1/100n}$.
We first suppose that $|\nabla f|(x)\geq \delta^{1/800n}$. Take $y\in D_f(p_f)\cap D_f(x)$ of Claim \[c2\]. Then, $|f(y)-1|\leq C\delta^{1/800n}$, and so $y\in A_f$. Thus, $$\label{54q}
d(x, A_f)\leq d(x,y)<\pi.$$ For all $z\in A_f$, we have $|f(p_f)-f(z)|\leq C\delta^{1/900n}$, and so $d(x,z)> d(x,y)-\delta^{1/2000n}$ by Claim \[c2\] (d). Thus, $$\label{54r}
d(x,A_f)\geq d(x,y)-\delta^{1/2000n}.$$ By (\[54q\]) and (\[54r\]), we get $$|d(x,A_f)- d(x,y)|\leq \delta^{1/2000n}.$$ Therefore, we have $|f(x)-\cos d(x,A_f)|\leq C\delta^{1/2000n}$ by Claim \[c2\] (c), and so $|f(w)-\cos d(w,A_f)|\leq C\delta^{1/2000n}$. By (\[54q\]), we have $d(w,A_f)\leq \pi+C\delta^{1/100n}$.
We next suppose that $|\nabla f|(x)\leq \delta^{1/800n}$. Then, $||f|(x)-1|\leq C\delta^{1/800n}$ by Claim \[c0\]. If $f(x)\geq 0$, then $w \in A_f$. This contradicts to $w\notin A_f$. Thus, we have $|f(x)+1|\leq C\delta^{1/800n}$. We see that (i) in Lemma \[p5e\] or Lemma \[p5g\] cannot occur for $\gamma_{p_f,x}$ because we have $$|\nabla^2 f|\geq \frac{1}{\sqrt{n}}|\Delta f|\geq\frac{n-p}{\sqrt{n}}|f|-C\delta^{1/2}.$$ Thus, there exists an orthonormal basis $\{e^1,\ldots,e^n\}$ of $T_x^\ast M$ such that $|\omega(x)-e^{n-p+1}\wedge\cdots\wedge e^n|\leq C\delta^{1/25}$ if $\lambda_1(\Delta_{C,p})\leq \delta$, where $\omega$ denotes the first eigenform of $\Delta_{C,p}$ with $\|\omega\|_2=1$, and $|\xi(x)-e^{1}\wedge\cdots\wedge e^{n-p}|\leq C\delta^{1/25}$ if $\lambda_1(\Delta_{C,n-p})\leq \delta$, where $\xi$ denotes the first eigenform of $\Delta_{C,n-p}$ with $\|\xi\|_2=1$. Take $u\in E_f(x)$ with $|u-e_1|\leq C\delta^{1/100n}$. Then, we get $$|f\circ\gamma_u(s)+\cos s|\leq C\delta^{1/800n}$$ for all $s\in [0,\pi]$ by Lemma \[p5e\] or Lemma \[p5g\]. Thus, we get $\gamma_u(\pi)\in A_f$, and so $$\label{54s}
d(w,A_f)\leq \pi+C\delta^{1/100n}.$$ For any $y\in A_f$, there exists $z\in D_f(x)$ with $d(y,z)\leq C\delta^{1/100n}$. Let $\{E^1,\ldots,E^n\}$ be a parallel orthonormal basis of $T^\ast M$ along $\gamma_{x,z}$ of Claim\[c1\]. Then, $$|1+\cos ( |\dot{\gamma}_{x,z}^{E}| d(x,z))|\leq C\delta^{1/900n}$$ by Claim \[c1\]. Thus, we get $d(x,z)\geq\pi-C\delta^{1/1800n}$, and so $$\label{54t}
d(w,A_f)\geq \pi- C\delta^{1/1800n}.$$ By (\[54s\]) and (\[54t\]), we get $|d(w,A_f)-\pi|\leq C\delta^{1/1800n}$, and so $|f(w)-\cos d(w,A_f)|\leq C\delta^{1/1800n}$.
For both cases, we get (iv).
Gromov-Hausdorff Approximation
------------------------------
In this subsection, we construct a Hausdorff approximation map, and show that the Riemannian manifold is close to the product metric space $S^{n-p}\times X$ in the Gromov-Hausdorff topology under our pinching condition.
In this subsection, we assume the following in addition to Assumption \[asu1\].
- $\lambda_{n-p+1}(g)\leq n-p+\delta$.
- $f_i$ is the $i$-th eigenfunction of the Laplacian acting on functions with $\|f_i\|_2^2=1/(n-p+1)$.
- Either $\lambda_1(\Delta_{C,p})\leq \delta$ or $\lambda_1(\Delta_{C,n-p})\leq \delta.$
The following proposition is based on [@Pe1 Lemma 5.2].
\[p54a\] Define $\widetilde{\Psi}:=(f_1,\dots,f_{n-p+1})\colon M\to \mathbb{R}^{n-p+1}$. Then, we have $$\||\widetilde{\Psi}|^2-1\|_{\infty}\leq C\delta^{1/800n^2}.$$
We first prove the following claim:
\[p54b\] For any $x\in M$, we have $|\widetilde{\Psi}|(x)\leq 1+C\delta^{1/800n}$
If $|\widetilde{\Psi}|(x)=0$, the claim is trivial. Thus, we assume that $|\widetilde{\Psi}|(x)\neq 0$. Put $$f_x:=\frac{1}{|\widetilde{\Psi}|(x)}\sum_{i=1}^{n-p+1} f_i(x)f_i.$$ Then, we have $$\|f_x\|_2^2=\frac{1}{n-p+1}.$$ Thus, we get $$|\widetilde{\Psi}|(x)=f_x(x)\leq 1+ C\delta^{1/800n}$$ by Proposition \[p53a\] (i). This gives the claim.
We need the following claim [@Pe1 Theorem 7.1]. Note that our sign convention of the Laplacian is different from [@Pe3].
\[claimpe\] For a smooth functions $u\in C^\infty(M)$ and a non-negative continuous function $F$ with $\Delta u\leq F$, we have $$u\leq C\|F\|_n+\frac{1}{{\mathop{\mathrm{Vol}}\nolimits}M}\int_M u\,d\mu_g.$$
To apply Claim \[claimpe\] to $-|\widetilde{\Psi}|^2$, we compute $\Delta|\widetilde{\Psi}|^2$. $$\label{55a}
\begin{split}
\Delta|\widetilde{\Psi}|^2
=&\Delta \sum_{i=1}^{n-p+1} f_i^2\\
=&2\sum_{i=1}^{n-p+1} f_i \Delta f_i-2\sum_{i=1}^{n-p+1} |\nabla f_i|^2\\
=&2\sum_{i=1}^{n-p+1} (\Delta f_i-(n-p) f_i)f_i+2(n-p+1)(|\widetilde{\Psi}|^2-1)\\
&\quad-2\sum_{i=1}^{n-p+1} (f_i^2+|\nabla f_i|^2-1).
\end{split}$$ We estimate each component.
By the assumption, we have $$\label{55b}
\|(\Delta f_i-(n-p) f_i)f_i\|_{\infty}\leq C \delta^{1/2}$$ for each $i$.
We next estimate $\||\widetilde{\Psi}|^2-1\|_n$. For $x\in M$ with $|\widetilde{\Psi}(x)|^2-1< 0$, we have $||\widetilde{\Psi}(x)|^2-1|=1-|\widetilde{\Psi}(x)|^2$. For $x\in M$ with $|\widetilde{\Psi}(x)|^2-1\geq 0$, we have $||\widetilde{\Psi}(x)|^2-1|=|\widetilde{\Psi}(x)|^2-1 \leq 1-|\widetilde{\Psi}(x)|^2+C\delta^{1/800n}$ by Claim \[p54b\]. For both cases, we have $||\widetilde{\Psi}(x)|^2-1|\leq 1-|\widetilde{\Psi}(x)|^2+C\delta^{1/800n}$. Combining this and $\|\widetilde{\Psi}\|_2=1$, we get $$\||\widetilde{\Psi}|^2-1\|_1 \leq C \delta^{1/800n}.$$ Since $||\widetilde{\Psi}(x)|^2-1|\leq 1$ for all $x\in M$, we get $$\||\widetilde{\Psi}|^2-1\|_n^n\leq \||\widetilde{\Psi}|^2-1\|_1\leq C \delta^{1/800n}.$$ Thus, we get $$\label{55c}
\||\widetilde{\Psi}|^2-1\|_n\leq C\delta^{1/800n^2}.$$
Finally, we estimate $\|f_i^2+|\nabla f_i|^2-1\|_n$. Since we have $$\|f_i^2 +|\nabla f_i|^2-1\|_\infty\leq C,$$ we get $$\begin{split}
\|f_i^2+|\nabla f_i|^2 -1\|_n^n
\leq &C \|f_i^2 +|\nabla f_i|^2-1\|_1
\leq C\delta^{1/800n}
\end{split}$$ by Proposition \[p53a\] (iii). Therefore, we get $$\label{55d}
\|f_i^2+|\nabla f_i|^2 -1\|_n\leq C\delta^{1/800n^2}.$$
By (\[55a\]), (\[55b\]), (\[55c\]) and (\[55d\]), we get $$\|\Delta|\widetilde{\Psi}|^2\|_n\leq C \delta^{1/800n^2}.$$ Since we have $$\frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)}\int_M |\widetilde{\Psi}|^2\,d\mu_g=1,$$ we get the lemma by Claim \[claimpe\].
In the remaining part of this subsection, we use the following notation.
- Let $d_S$ denotes the intrinsic distance function on $S^{n-p}(1)$. Note that we have $\cos d_S(x,y)=x\cdot y$ and $$d_{\mathbb{R}^{n-p+1}}(x,y)\leq d_{S}(x,y)\leq 3 d_{\mathbb{R}^{n-p+1}}(x,y)$$ for all $x,y\in S^{n-p}\subset\mathbb{R}^{n-p+1}$.
- For each $f\in{\mathop{\mathrm{Span}}\nolimits}_{\mathbb{R}}\{f_1,\ldots, f_{n-p+1}\}$, we use the notation $p_f$ and $A_f$ of Proposition \[p53a\]. Recall that we defined $$A_f:=\{x\in M: |f(x)-1|\leq \delta^{1/900n}\}.$$
- Define $\widetilde{\Psi}:=(f_1,\dots,f_{n-p+1})\colon M\to \mathbb{R}^{n-p+1}$ and $$\Psi:=\frac{\widetilde{\Psi}}{|\widetilde{\Psi}|}\colon M\to S^{n-p}.$$
- For each $x\in M$, put $$f_x:=\frac{1}{|\widetilde{\Psi}|(x)}\sum_{i=1}^{n-p+1} f_i(x)f_i=\sum_{i=1}^{n-p+1} \Psi_i(x)f_i,$$ $p_x:=p_{f_x}$ and $A_x:=A_{f_x}$.
- For each $x\in M$ and $f\in{\mathop{\mathrm{Span}}\nolimits}_{\mathbb{R}}\{f_1,\ldots, f_{n-p+1}\}$ with $\|f\|_2^2=1/(n-p+1)$, choose $a_f(x)\in A_f$ such that $$d(x,A_f)=d(x,a_f(x)).$$
The goal of this subsection is to show that $$\Phi_f \colon M\to S^{n-p}\times A_f,\,x\mapsto (\Psi(x),a_f(x))$$ is an approximation map.
\[p54c00\] For all $x,y\in M$, we have $$|\Psi(x)-\Psi(y)|\leq Cd(x,y).$$
Since we have $\|\nabla f_i\|_{\infty}\leq C$ for all $i\in\{1,\ldots,n-p+1\}$, we get $|\widetilde{\Psi}(x)-\widetilde{\Psi}(y)|\leq Cd(x,y)$ for all $x,y\in M$. Thus, we get the lemma by Lemma \[p54a\].
\[p54c0\] Take $u\in S^{n-p}$ and put $f=\sum_{i=1}^{n-p+1}u_i f_i$. Then, we have $$|d_S(\Psi(y),u)-d(y,A_{f})|\leq C\delta^{1/1600n^2}$$ for all $y\in M$.
Since $$f(y)=u\cdot\widetilde{\Psi}(y),$$ we have $$|u \cdot\widetilde{\Psi}(y)-\cos d(y,A_{f})|\leq C\delta^{1/2000n}$$ by Proposition \[p53a\], and so $$|u\cdot \Psi(y)-\cos d(y,A_{f})|\leq C\delta^{1/800n^2}$$ by Lemma \[p54a\]. Thus, we get $$|\cos d_S(\Psi(y),u)-\cos d(y,A_{f})|\leq C\delta^{1/800n^2}.$$ This and $d(y,A_{f})\leq \pi+C\delta^{1/100n}$ give the lemma.
By the definition of $A_{y}$, we immediately get the following corollaries:
\[p54c01\] Take $u\in S^{n-p}$ and put $f=\sum_{i=1}^{n-p+1}u_i f_i$. Then, we have $$d_S(\Psi(p_f),u)\leq C\delta^{1/1600n^2}.$$
\[p54c\] For each $y_1,y_2\in M$, we have $$|d_S(\Psi(y_1),\Psi(y_2))-d(y_2,A_{y_1})|\leq C\delta^{1/1600n^2}.$$
\[p54c1\] For each $y\in M$, we have $$d(y,A_{y})\leq C\delta^{1/1600n^2}.$$
We need to show the almost Pythagorean theorem for our purpose. To do this, we regard $|\dot{\gamma}^E| s$ in Lemma \[p5e\] or Lemma \[p5g\] as a moving distance in $S^{n-p}$. We first approximate their cosine.
\[p54d\] Take $y_1\in M$, $\tilde{y}_1\in D_{f_{y_1}}(p_{y_1})\cap R_{f_{y_1}}\cap Q_{f_{y_1}}$ with $d(y_1,\tilde{y}_1)\leq C\delta^{1/100n}$ and $y_2\in D_{f_{y_1}}(\tilde{y}_1)$ $($note that we can take such $\tilde{y}_1$ for any $y_1$ by the Bishop-Gromov theorem$)$. Let $\{E^1,\ldots,E^n\}$ be a parallel orthonormal basis of $T^\ast M$ along $\gamma_{\tilde{y}_1,y_2}$ in Lemma \[p5e\] or Lemma \[p5g\] for $f_{y_1}$. Then, $(ii)$ holds in the lemmas, and $$|\cos(|\dot{\gamma}_{\tilde{y}_1,y_2}^E|s)-\cos d_S(\Psi(y_1),\Psi(\gamma_{\tilde{y}_1,y_2}(s)))|\leq C\delta^{1/1600n^2}$$ for all $s\in[0,d(\tilde{y}_1,y_2)]$. In particular, we have $$|\cos(|\dot{\gamma}_{\tilde{y}_1,y_2}^E|d(\tilde{y}_1,y_2))-\cos d_S(\Psi(y_1),\Psi(y_2))|\leq C\delta^{1/1600n^2}.$$
By Corollary \[p54c1\], we have $$d(\tilde{y}_1,A_{y_1})\leq C\delta^{1/1600n^2},$$ and so we get $$f\circ \gamma_{\tilde{y}_1,y_2}(s)
\geq \cos d(\gamma_{\tilde{y}_1,y_2}(s),A_{y_1})- C\delta^{1/2000n}
\geq \cos s- C\delta^{1/1600n^2}
\geq \frac{1}{\sqrt{2}}- C\delta^{1/1600n^2}$$ for all $s\leq\min\{\pi/4,d(\tilde{y}_1,y_2)\}$. Therefore, we have $$|\nabla^2 f|(\gamma_{\tilde{y}_1,y_2}(s))
\geq \frac{1}{\sqrt{n}}|\Delta f|(\gamma_{\tilde{y}_1,y_2}(s))
\geq \frac{n-p}{\sqrt{2n}}- C\delta^{1/1600n^2}$$ for all $s\leq\min\{\pi/4,d(\tilde{y}_1,y_2)\}$. Thus, (i) in Lemma \[p5e\] or Lemma \[p5g\] cannot occur, and so (ii) holds in the lemmas.
Since we have $f_{y_1}(y_1)=|\widetilde{\Psi}(y_1)|$, we get $$\label{ad1}
|f_{y_1}(\tilde{y}_1)-1|\leq C\delta^{1/800n^2}$$ by Lemma \[p54a\] and $d(y_1,\tilde{y}_1)\leq C\delta^{1/100n}$. By (\[ad1\]) and Proposition \[p53a\] (iii), we have $$|\nabla f_{y_1}|(\tilde{y}_1)\leq C\delta^{1/1600n^2}.$$ Thus, we get $$|f_{y_1}(\gamma_{\tilde{y}_1,y_2}(s))-\cos(|\dot{\gamma}_{\tilde{y}_1,y_2}^E|s)|\leq C\delta^{1/1600n^2}$$ for all $s\in[0,d(\tilde{y}_1,y_2)]$ by Lemma \[p5e\] or Lemma \[p5g\]. On the other hand, we have $$|f_{y_1}(\gamma_{\tilde{y}_1,y_2}(s))-\cos d_S(\Psi(y_1),\Psi(\gamma_{\tilde{y}_1,y_2}(s)))|\leq C\delta^{1/1600n^2}$$ for all $s\in[0,d(\tilde{y}_1,y_2)]$ by Proposition \[p53a\] (iv) and Corollary \[p54c\]. Thus, we get the lemma.
We use the following notation:
- For any $y_1,y_2\in M$ and $f\in {\mathop{\mathrm{Span}}\nolimits}_{\mathbb{R}}\{f_1,\ldots, f_{n-p+1}\}$ with $\|f\|_2^2=1/(n-p+1)$, define $$\begin{aligned}
&G_f^{y_1}(y_2)\\
:=&\langle\dot{\gamma}_{y_2,y_1}(0),\nabla f(y_2)\rangle d(y_1,y_2)\sin d_S(\Psi(y_1),\Psi(y_2))\\
&\quad +\Big(\cos d(y_2, A_f)\cos d_S(\Psi(y_1),\Psi(y_2))-\cos d(y_1,A_f)\Big)
d_S(\Psi(y_1),\Psi(y_2)).\end{aligned}$$
- For any $y_1,y_2\in M$, define
[align\*]{} &1 d(y\_1,y\_2),\
&0 d(y\_1,y\_2)>.
- For any $y_1,y_2\in M$ and $f\in {\mathop{\mathrm{Span}}\nolimits}_{\mathbb{R}}\{f_1,\ldots, f_{n-p+1}\}$ with $\|f\|_2^2=1/(n-p+1)$, define $$\begin{aligned}
C_f^{y_1}(y_2):=&\Big\{y_3\in M : \gamma_{y_2,y_3}(s)\in I_{y_1}\setminus\{y_1\} \text{ for almost all $s\in[0,d(y_2,y_3)]$, and}\\
&\qquad \qquad\qquad \qquad \int_{0}^{d(y_2,y_3)} |G_f^{y_1}H^{y_1}|(\gamma_{y_2,y_3}(s))\,d s\leq \delta^{1/9600n^2}\Big\},\\
P_f^{y_1}:=&\{y_2\in M: {\mathop{\mathrm{Vol}}\nolimits}(M\setminus C_f^{y_1}(y_2))\leq\delta^{1/9600n^2}{\mathop{\mathrm{Vol}}\nolimits}(M)\}.\end{aligned}$$
Pinching condition on $G_f^{y_1}$ plays a crucial role for our purpose. Let us estimate $G_f^{y_1}$.
\[p54e\] Take $\eta>0$ with $\eta\geq \delta^{1/2000n}$, $f\in{\mathop{\mathrm{Span}}\nolimits}_{\mathbb{R}}\{f_1,\ldots, f_{n-p+1}\}$ with $\|f\|_2^2=1/(n-p+1)$, $y_1\in Q_f$ and $y_2\in D_f(y_1)$. Let $\{E^1,\ldots,E^n\}$ be a parallel orthonormal basis of $T^\ast M$ along $\gamma_{y_1,y_2}$ in Lemma \[p5e\] or Lemma \[p5g\] for $f$. If $$||\dot{\gamma}_{y_1,y_2}^E|d(y_1,y_2)-d_S(\Psi(y_1),\Psi(y_2))|\leq \eta,$$ then $$|G_f^{y_1}(y_2)|\leq C\eta.$$
We have $$\begin{aligned}
\Big|f(y_1)-f(y_2)\cos &(|\dot{\gamma}_{y_1,y_2}^E|d(y_1,y_2))\\
&-\frac{1}{|\dot{\gamma}_{y_1,y_2}^E|}\langle\nabla f(y_2),\dot{\gamma}_{y_2,y_1}(0)\rangle\sin (|\dot{\gamma}_{y_1,y_2}^E|d(y_1,y_2))
\Big|
\leq C\delta^{1/250}\end{aligned}$$ by Lemma \[p5e\] or Lemma \[p5g\]. Thus, by Proposition \[p53a\] (iv), we get $$\begin{aligned}
\Big||\dot{\gamma}_{y_1,y_2}^E|\cos d(y_1,A_f)&-|\dot{\gamma}_{y_1,y_2}^E|\cos d(y_2, A_f)\cos (|\dot{\gamma}_{y_1,y_2}^E|d(y_1,y_2))\\
&-\langle\nabla f(y_2),\dot{\gamma}_{y_2,y_1}(0)\rangle\sin (|\dot{\gamma}_{y_1,y_2}^E|d(y_1,y_2))
\Big|
\leq C\delta^{1/2000n},\end{aligned}$$ and so we get the lemma.
The quantity $|\dot{\gamma}_{y_1,y_2}^E|$ in the above lemma is slightly different from that of Lemma \[p54d\]. Comparing these two quantity, we get the following:
\[p54f0\] Take $\eta>0$ with $\eta\geq \delta^{1/2000n}$, $f\in{\mathop{\mathrm{Span}}\nolimits}_{\mathbb{R}}\{f_1,\ldots, f_{n-p+1}\}$ with $\|f\|_2^2=1/(n-p+1)$, $y_1\in M$, $\tilde{y}_1\in D_{f_{y_1}}(p_{y_1})\cap R_{f_{y_1}}\cap Q_{f_{y_1}}\cap Q_f$ with $d(y_1,\tilde{y}_1)\leq C\delta^{1/100n}$ and $y_2\in D_{f_{y_1}}(\tilde{y}_1)\cap D_f(\tilde{y}_1)$. Let $\{E^1,\ldots,E^n\}$ be a parallel orthonormal basis of $T^\ast M$ along $\gamma_{\tilde{y}_1,y_2}$ in Lemma \[p5e\] or Lemma \[p5g\] for $f_{y_1}$. If $$||\dot{\gamma}_{\tilde{y}_1,y_2}^E|d(\tilde{y}_1,y_2)-d_S(\Psi(\tilde{y}_1),\Psi(y_2))|\leq \eta,$$ then $$|G_f^{\tilde{y}_1}(y_2)|\leq C\eta.$$
Let $\{\widetilde{E}^1,\ldots,\widetilde{E}^n\}$ be a parallel orthonormal basis of $T^\ast M$ along $\gamma_{\tilde{y}_1,y_2}$ in Lemma \[p5e\] or Lemma \[p5g\] for $f$ (if (i) holds, then we can assume that $\widetilde{E}^i=E^i$ for all $i$). We show that $$\left||\dot{\gamma}_{\tilde{y}_1,y_2}^E|-|\dot{\gamma}_{\tilde{y}_1,y_2}^{\widetilde{E}}|\right|\leq C\delta^{1/50}.$$ Then, we immediately get the corollary by Lemma \[p54e\].
We first suppose that $\lambda_1(\Delta_{C,p})\leq \delta$. Let $\omega$ be the first eigenform of $\Delta_{C,p}$ with $\|\omega\|_2=1$. We have $|\omega(y_2)-E^{n-p+1}\wedge \cdots\wedge E^n|\leq C\delta^{1/25}$ by Lemma \[p5e\] and Lemma \[p54d\]. Since $|\dot{\gamma}_{\tilde{y}_1,y_2}^E|^2=1-|\iota(\dot{\gamma}_{\tilde{y}_1,y_2})(E^{n-p+1}\wedge \cdots\wedge E^n)|^2$, we get $$\label{55e}
\left||\dot{\gamma}_{\tilde{y}_1,y_2}^E|^2-\left(1-|\iota(\dot{\gamma}_{\tilde{y}_1,y_2})\omega|^2(y_2)\right)\right|\leq C\delta^{1/25}.$$ Similarly, we get $$\label{55f}
\left||\dot{\gamma}_{\tilde{y}_1,y_2}^{\widetilde{E}}|^2-\left(1-|\iota(\dot{\gamma}_{\tilde{y}_1,y_2})\omega|^2(y_2)\right)\right|\leq C\delta^{1/25}.$$ By (\[55e\]) and (\[55f\]), we get $$\left||\dot{\gamma}_{\tilde{y}_1,y_2}^E|-|\dot{\gamma}_{\tilde{y}_1,y_2}^{\widetilde{E}}|\right|\leq C\delta^{1/50}.$$
We next suppose that $\lambda_1(\Delta_{C,n-p})\leq \delta$. Let $\xi$ be the first eigenform of $\Delta_{C,n-p}$ with $\|\xi\|_2=1$. Similarly to the case when $\lambda_1(\Delta_{C,p})\leq \delta$, we have $$\begin{aligned}
\left||\dot{\gamma}_{\tilde{y}_1,y_2}^E|^2-|\iota(\dot{\gamma}_{\tilde{y}_1,y_2})\xi|^2(y_2)\right|\leq& C\delta^{1/25},\\
\left||\dot{\gamma}_{\tilde{y}_1,y_2}^{\widetilde{E}}|^2-|\iota(\dot{\gamma}_{\tilde{y}_1,y_2})\xi|^2(y_2)\right|\leq& C\delta^{1/25},\end{aligned}$$ and so $$\left||\dot{\gamma}_{\tilde{y}_1,y_2}^E|-|\dot{\gamma}_{\tilde{y}_1,y_2}^{\widetilde{E}}|\right|\leq C\delta^{1/50}.$$
By the above two cases, we get the corollary.
Let us show the integral pinching condition.
\[p54f\] Take $f\in{\mathop{\mathrm{Span}}\nolimits}_{\mathbb{R}}\{f_1,\ldots, f_{n-p+1}\}$ with $\|f\|_2^2=1/(n-p+1)$, $y_1\in M$ and $\tilde{y}_1\in D_{f_{y_1}}(p_{y_1})\cap R_{f_{y_1}}\cap Q_{f_{y_1}}\cap Q_f$ with $d(y_1,\tilde{y}_1)\leq C\delta^{1/100n}$. Then, $$\|G_f^{\tilde{y}_1} H_{\tilde{y}_1}\|_1\leq C\delta^{1/3200n^2},$$ and $${\mathop{\mathrm{Vol}}\nolimits}(M\setminus P_f^{\tilde{y}_1})\leq C\delta^{1/9600n^2}.$$
Take arbitrary $y_2\in D_f(\tilde{y_1})\cap D_{f_{y_1}}(\tilde{y}_1)$. Let $\{E^1,\ldots,E^n\}$ be a parallel orthonormal basis of $T^\ast M$ along $\gamma_{y_1,y_2}$ in Lemma \[p5e\] or Lemma \[p5g\] for $f_{y_1}$. Then, we have $$||\dot{\gamma}_{\tilde{y}_1,y_2}^E|d(\tilde{y}_1,y_2)-d_S(\Psi(\tilde{y}_1),\Psi(y_2))|\leq C\delta^{1/3200n^2},$$ if $d(\tilde{y}_1,y_2)\leq \pi$ by Lemma \[p54c00\] and Lemma \[p54d\]. Thus, by Corollary \[p54f0\], we have $$\sup_{D_f(\tilde{y_1})\cap D_{f_{y_1}}(\tilde{y}_1)}|G_f^{\tilde{y}_1} H^{\tilde{y}_1}|\leq C\delta^{1/3200n^2}.$$ Since ${\mathop{\mathrm{Vol}}\nolimits}(M\setminus (D_f(\tilde{y}_1)\cap D_{f_{y_1}}(\tilde{y}_1)))\leq C\delta^{1/100}{\mathop{\mathrm{Vol}}\nolimits}(M)$ and $\|G_f^{\tilde{y}_1} H^{\tilde{y}_1}\|_\infty\leq C$, we get $$\|G_f^{\tilde{y}_1} H^{\tilde{y}_1}\|_1\leq C\delta^{1/3200n^2}.$$ By the segment inequality (Theorem \[seg\]), we get the lemma.
\[order\] We use the following notation.
- $\eta_0=\delta^{1/9600n^3}$.
- $\eta_1=\eta_0^{1/26}$.
- $\eta_2=\eta_1^{1/78}$.
- $L=\eta_2^{1/150}$.
We use Lemma \[p54f\] to give the almost Pythagorean theorem for the special case (see Lemma \[p54l\]). For the general case, we need to estimate $\|G_f^{\tilde{y}_1}\|_1$. To do this, we show that $|\dot{\gamma}_{\tilde{y}_1,y_2}^E|d(\tilde{y}_1,y_2)\leq \pi+L$ under the assumption of Lemma \[p54d\] in Lemma \[p54n\]. Then, we can estimate $\|G_f^{\tilde{y}_1}\|_1$ similarly to Lemma \[p54f\]. After proving that, we use Lemma \[p54i\] again to give the almost Pythagorean theorem for the general case. The following lemma, which guarantees that an almost shortest pass from a point in $M$ to $A_f$ almost corresponds to a geodesic in $S^{n-p}$ through $\Psi$ under some assumptions, is the first step to achieve these objectives.
\[p54g\] Take
- $f\in{\mathop{\mathrm{Span}}\nolimits}_{\mathbb{R}}\{f_1,\ldots, f_{n-p+1}\}$ with $\|f\|_2^2=1/(n-p+1)$,
- $u\in S^{n-p}$ with $f=\sum_{i=1}^{n-p+1}u_i f_i$,
- $x,y\in M$,
- $\eta>0$ with $\eta_0\leq\eta\leq L^{1/3n}$.
Suppose
- $d(y,A_f)\leq C \eta$,
- $|d(x,A_f)-d(x,y)|\leq C\eta$.
Then, we have the following for all $s,s'\in[0,d(x,y)]$:
- $|d(\gamma_{y,x}(s),A_f)-s|\leq C\eta$,
- $\left||s-s'|-d_S\left(\Psi(\gamma_{y,x}(s)),\Psi(\gamma_{y,x}(s'))\right)\right|\leq C\eta$,
- If in addition $d(x,A_f)\geq \frac{1}{C}\eta^{1/26}$, there exists $v\in S^{n-p}$ such that $u\cdot v=0$ and $$d_S(\Psi(\gamma_{y,x}(s)),\gamma_v(s))\leq C\eta^{3/13}$$ for all $s\in[0,d(x,y)]$, where we define $\gamma_v(s):=\cos s u+\sin s v\in S^{n-p}$.
We first prove (i). We have $$d(\gamma_{y,x}(s),A_f)\leq d(\gamma_{y,x}(s),y)+d(y,A_f)\leq s+ C\eta,$$ and $$\begin{aligned}
d(x,y)-C\eta\leq d(x,A_f)\leq &d(\gamma_{y,x}(s),A_f)+d(\gamma_{y,x}(s),x)\\
=&d(\gamma_{y,x}(s),A_f)+d(x,y)-s.\end{aligned}$$ Thus, we get (i).
We next prove (ii). Since $d(y,A_f)\leq C \eta$, we have $\cos d(y,A_f)\geq 1-C\eta^2$, and so $$\label{55g}
|f(y)-1|\leq C\eta^2$$ by Proposition \[p53a\] (iv). On the other hand, we have $f(y)=u\cdot \widetilde{\Psi}(y)$, and so $$\label{55h}
|f(y)-\cos d_S(u,\Psi(y))|\leq C\delta^{1/800n^2}$$ by Lemma \[p54a\]. By (\[55g\]) and (\[55h\]), we get $|1-\cos d_S(u,\Psi(y))|\leq C\eta^2$. This gives $d_S(u,\Psi(y))\leq C\eta$. Thus, we get $$\label{55i}
|s-d_S(\Psi(\gamma_{y,x}(s)),\Psi(y))|\leq C \eta$$ for all $s\in[0,d(x,y)]$ by (i) and Lemma \[p54c0\]. Take arbitrary $s,s'\in[0,d(x,y)]$ with $s<s'$. Then, $$\label{55j}
\begin{split}
s'-s=d(\gamma_{y,x}(s),\gamma_{y,x}(s'))&\geq d(\gamma_{y,x}(s),A_{\gamma_{y,x}(s')})-d(\gamma_{y,x}(s'),A_{\gamma_{y,x}(s')})\\
&\geq d_S(\Psi(\gamma_{y,x}(s)),\Psi(\gamma_{y,x}(s')))-C\delta^{1/1600n^2}
\end{split}$$ by Corollary \[p54c\] and Corollary \[p54c1\]. On the other hand, we have $$\begin{split}
s'-C\eta\leq& d_S(\Psi(\gamma_{y,x}(s')),\Psi(y))\\
\leq &d_S(\Psi(\gamma_{y,x}(s)),\Psi(\gamma_{y,x}(s')))+d_S(\Psi(\gamma_{y,x}(s)),\Psi(y))\\
\leq &d_S(\Psi(\gamma_{y,x}(s)),\Psi(\gamma_{y,x}(s'))) +s+C\eta
\end{split}$$ by (\[55i\]), and so $$\label{55k}
s'-s\leq d_S(\Psi(\gamma_{y,x}(s)),\Psi(\gamma_{y,x}(s'))) +C\eta.$$ By (\[55j\]) and (\[55k\]), we get (ii).
Finally, we prove (iii). Since $d(x,A_f)\geq\frac{1}{C}\eta^{1/26}$, there exists $s_0\in[0,d(x,y)]$ such that $\frac{1}{C}\eta^{1/26}\leq d(z,y)\leq \pi- \frac{1}{C}\eta^{1/26}$, where we put $z=\gamma_{y,x}(s_0)$. Then, there exists $v\in S^{n-p}$ with $u\cdot v=0$ and $t_1\in[0,\pi]$ such that $$\Psi(z)=\cos t_1 u+\sin t_1 v.$$ We have $$\begin{aligned}
|\cos t_1-\cos d(z,y)|=&|\cos d_S(\Psi(z),u)-\cos s_0|\\
\leq& |\cos d(z,A_f)-\cos s_0|+C\delta^{1/1600n^2}
\leq C\eta\end{aligned}$$ by Lemma \[p54c0\] and (i). This gives $$\label{55l}
|t_1-d(z,y)|\leq C\eta^{1/2}.$$ Take arbitrary $s\in [0,d(x,y)]$. Then, there exist $w\in S^{n-p}$ and $x_1,x_2,x_3\in \mathbb{R}$ such that $w\perp {\mathop{\mathrm{Span}}\nolimits}_{\mathbb{R}}\{u,v\}$, $x_1^2+x_2^2+x_3^2=1$ and $$\Psi(\gamma_{y,x}(s))=x_1 u+x_2 v+ x_3 w.$$ Since we have $$|s-d_S(\Psi(\gamma_{y,x}(s)),u)|\leq C\eta$$ by (i) and Lemma \[p54c0\], and $\cos d_S(\Psi(\gamma_{y,x}(s)),u)=x_1$, we get $$\label{55m}
|\cos s- x_1|\leq C\eta.$$ We have $$\left||d(z,y)-s|-d_S(\Psi(\gamma_{y,x}(s)),\Psi(z))\right|\leq C\eta$$ by (ii). Since $\cos d_S(\Psi(\gamma_{y,x}(s)),\Psi(z))=x_1 \cos t_1+x_2\sin t_1$, we get $$\label{55n}
|\cos(d(z,y)-s)- x_1 \cos d(z,y)-x_2\sin d(z,y)|\leq C\eta^{1/2}$$ by (\[55l\]). By (\[55m\]) and (\[55n\]), we have $$\sin d(z,y)|\sin s- x_2|\leq C\eta^{1/2}.$$ By the assumption, we have $$\sin d(z,y)\geq \frac{1}{C}\eta^{1/26},$$ and so we get $$\label{55o}
|\sin s- x_2|\leq C\eta^{6/13}.$$ By (\[55m\]) and (\[55o\]), we get $$|\cos d_S(\Psi(\gamma_{y,x}(s)),\gamma_v(s))-1|
=|x_1 \cos s+x_2\sin s-1|\leq C\eta^{6/13}.$$ Thus, we get (iii).
The following lemma asserts that the differential of an almost shortest pass from a point in $M$ to $A_f$ is in the direction of $\nabla f$ under some assumptions.
\[p54h\] Take
- $f\in{\mathop{\mathrm{Span}}\nolimits}_{\mathbb{R}}\{f_1,\ldots, f_{n-p+1}\}$ with $\|f\|_2^2=1/(n-p+1)$,
- $x\in D_f(p_f)\cap Q_f \cap R_f$,
- $y\in D_f(x)\cap D_f(p_f)\cap Q_f\cap R_f$,
- $\eta>0$ with $\eta_0\leq\eta\leq L^{1/3n}$.
Suppose
- $d(x,A_f)\geq\frac{1}{C}\eta^{1/26}$,
- $d(y,A_f)\leq C \eta$,
- $|d(x,A_f)-d(x,y)|\leq C\eta$.
Let $\{E^1,\ldots,E^n\}$ be a parallel orthonormal basis of $T^\ast M$ along $\gamma_{y,x}$ in Lemma \[p5e\] or Lemma \[p5g\] for $f$. Then, we have the following for all $s\in[0,d(x,y)]$:
- $||\dot{\gamma}^E_{y,x}|-1|\leq C \eta^{6/13}$,
- $|\nabla f (\gamma_{y,x}(s))+\sin s \dot{\gamma}_{y,x}(s)|\leq C\eta^{3/26}$.
We first note that we have $$\label{55p}
d(x,y)\leq \pi+C\eta$$ by the assumption and Proposition \[p53a\] (iv).
Let us prove (i). By $d(y,A_f)\leq C \eta$, we have $\cos d(y,A_f)\geq 1- C\eta^2$. Thus, we have $$\label{55q}
|1-f(y)|\leq C\eta^2$$ by Proposition \[p53a\] (iv). By Proposition \[p53a\] (iii), we get $$|\nabla f|(y)\leq C\eta.$$ Thus, we have $$\label{55r}
|f(x)-\cos(|\dot{\gamma}_{y,x}^E|d(x,y))|\leq C\eta$$ by Lemma \[p5e\] or Lemma \[p5g\], and so $$||\dot{\gamma}_{y,x}^E|d(x,y)-d(x,A_f)|\leq C\eta^{1/2}$$ by Proposition \[p53a\] (iv) and (\[55p\]). By the assumption, we get $$||\dot{\gamma}_{y,x}^E|-1|d(x,A_f)\leq C\eta^{1/2}.$$ This gives (i).
We next prove (ii). By Proposition \[p53a\], we have $$||\nabla f|^2(x)-\sin^2 d(x,A_f)|\leq C\delta^{1/2000n}.$$ Thus, we get $$||\nabla f|(x)-|\sin d(x,A_f)||\leq C\delta^{1/4000n}.$$ Since $\sin d(x, A_f)\geq -C\delta^{1/100n}$ by Proposition \[p53a\] (iv), we have $$||\nabla f|(x)-\sin d(x,A_f)|\leq C\delta^{1/4000n}.$$ Thus, we get $$\label{55s}
||\nabla f|(x)-\sin d(x,y)|\leq C\eta$$ by the assumption. On the other hand, by (i) and Lemma \[p5e\] or Lemma \[p5g\], we have $$|f(y)-f(x)\cos d(x,y)-\langle\nabla f(x),\dot{\gamma}_{x,y}(0)\rangle\sin d(x,y)|\leq C\eta^{6/13},$$ and so $$\label{55t}
|\sin^2 d(x,y)-\langle\nabla f(x),\dot{\gamma}_{x,y}(0)\rangle\sin d(x,y)|\leq C\eta^{6/13}$$ by (\[55q\]) and (\[55r\]).
We consider the following two cases:
- $d(x,y)\leq \pi-\eta^{3/13}$,
- $d(x,y)> \pi-\eta^{3/13}$.
We first suppose that $d(x,y)\leq \pi-\eta^{3/13}$. We get $$|\sin d(x,y)-\langle\nabla f(x),\dot{\gamma}_{x,y}(0)\rangle|\leq C\eta^{3/13}$$ by the assumption and (\[55t\]). By (\[55s\]), we get $$\label{55u}
|\nabla f|(x)-\langle\nabla f(x),\dot{\gamma}_{x,y}(0)\rangle \leq C\eta^{3/13}.$$
We next suppose that $d(x,y)> \pi-\eta^{3/13}$. Then, we have $\cos d(x,A_f)\leq -1+C\eta^{6/13}$, and so $|\nabla f|(x)\leq C\eta^{3/13}$ by Proposition \[p53a\] (iii) and (iv). Thus, we also get (\[55u\]) for this case.
By (i), (\[54m\]) and Lemma \[p5e\] or \[p5g\], we have $$\int_0^{d(x,y)} \left|\frac{d}{d s}\left(|\nabla f|^2(\gamma_{x,y}(s))-\langle\nabla f(\gamma_{x,y}(s)), \dot{\gamma}_{x,y}(s)\rangle^2\right)\right|\,d s\leq C\eta^{6/13}.$$ Thus, we get $$\label{55v}
|\nabla f|^2(\gamma_{x,y}(s))-\langle\nabla f(\gamma_{x,y}(s)), \dot{\gamma}_{x,y}(s)\rangle^2\leq C\eta^{3/13}$$ for all $s\in[0,d(x,y)]$ by (\[55u\]). Since $$|\nabla f (\gamma_{x,y}(s))-\langle\nabla f(\gamma_{x,y}(s)), \dot{\gamma}_{x,y}(s)\rangle\dot{\gamma}_{x,y}(s)|^2=|\nabla f|^2(\gamma_{x,y}(s))-\langle\nabla f(\gamma_{x,y}(s)), \dot{\gamma}_{x,y}(s)\rangle^2,$$ we get $$|\nabla f (\gamma_{x,y}(s))-\langle\nabla f(\gamma_{x,y}(s)), \dot{\gamma}_{x,y}(s)\rangle\dot{\gamma}_{x,y}(s)|\leq
C\eta^{3/26}$$ by (\[55v\]). Since we have $$|\langle\nabla f(\gamma_{x,y}(s)), \dot{\gamma}_{x,y}(s)\rangle+\cos d(x,y)\sin s-\sin d(x,y) \cos s|\leq C\eta^{3/13}$$ by (\[55r\]), (\[55s\]), (\[55u\]), (i) and Lemma \[p5e\] or Lemma \[p5g\], we get $$|\nabla f (\gamma_{x,y}(s))-\sin (d(x,y)-s)\dot{\gamma}_{x,y}(s)|\leq
C\eta^{3/26}$$ This gives (ii).
The following lemma is crucial to show the almost Pythagorean theorem.
\[p54i\] Take
- $f\in{\mathop{\mathrm{Span}}\nolimits}_{\mathbb{R}}\{f_1,\ldots, f_{n-p+1}\}$ with $\|f\|_2^2=1/(n-p+1)$,
- $x\in D_f(p_f)\cap Q_f \cap R_f$,
- $y\in D_f(x)\cap D_f(p_f)\cap Q_f\cap R_f$,
- $z\in M$,
- $\eta>0$ with $\eta_0\leq\eta\leq L^{1/3n}$ and $T\in [0, d(x,y)]$.
Suppose
- $d(y,A_f)\leq C\eta$,
- $|d(x,A_f)-d(x,y)|\leq C\eta$,
- $\gamma_{y,x}(s)\in I_z\setminus\{z\}$ for almost all $s\in [T,d(x,y)]$,
- $\int_T^{d(x,y)} |G_f^z(\gamma_{y,x}(s))|\,d s\leq C\eta^{3/26}$.
Then, we have $$\left| d(z,x)^2-d_S(\Psi(z),\Psi(x))^2- d(z,\gamma_{y,x}(T))^2+d_S(\Psi(z),\Psi(\gamma_{y,x}(T)))^2
\right|\leq C\eta^{1/26}.$$
If $d(x,A_f)\leq\eta^{1/26}$, then $d(x,y)\leq C\eta^{1/26}$, and so $d(x,\gamma_{y,x}(T))\leq C\eta^{1/26}$. Thus, we immediately get the lemma by Lemma \[p54c00\] if $d(x,A_f)\leq\eta^{1/26}$. In the following, we assume that $d(x,A_f)\geq\eta^{1/26}$. Take $u\in S^{n-p}$ with $f=\sum_{i=1}^{n-p}u_i f_i$, and $v\in S^{n-p}$ of Lemma \[p54g\] (iii). Define $$r(s):=d_S(\Psi (z),\gamma_v(s)).$$ Then, by the triangle inequality and Lemma \[p54g\] (iii), we have $$\label{55w}
|r(s)-d_S(\Psi (z),\Psi(\gamma_{y,x}(s)))|\leq C\eta^{3/13}.$$
There exist $w\in S^{n-p}$ and $x_1,x_2,x_3\in \mathbb{R}$ such that $w\perp {\mathop{\mathrm{Span}}\nolimits}_{\mathbb{R}}\{u,v\}$, $x_1^2+x_2^2+x_3^2=1$ and $$\Psi(z)=x_1 u+x_2 v+ x_3 w.$$ Then, $$\label{55x}
\cos r(s)=x_1\cos s+x_2\sin s$$ by the definition of $\gamma_v$ in Lemma \[p54g\] (iii), and so $$-x_1\sin s+x_2\cos s
=\frac{d}{d s} \cos r(s)
=-r'(s)\sin r(s)$$ Thus, we get $$\label{55y}
\begin{split}
-r'(s)\sin r(s) \sin s=-x_1\sin^2 s+x_2\sin s\cos s=\cos r(s)\cos s-x_1
\end{split}$$ by (\[55x\]). Since $x_1=\Psi(z)\cdot u$ and $f(z)=\widetilde{\Psi}(z)\cdot u$, we have $$\label{55z}
|x_1-\cos d(z,A_f)|\leq C\delta^{1/800n^2}$$ by Proposition \[p53a\] (iv) and Lemma \[p54a\]. By Lemma \[p54g\], (\[55w\]), (\[55y\]) and (\[55z\]), we get $$\label{56a}
\begin{split}
&\Big|\Big(\cos d(\gamma_{y,x}(s),A_f)\cos d_S(\Psi(z),\Psi(\gamma_{y,x}(s)))-\cos d(z,A_f)\Big)d_S(\Psi(z),\Psi(\gamma_{y,x}(s)))\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad +r'(s)r(s)\sin r(s) \sin s\Big|\leq C\eta^{3/13}.
\end{split}$$
Define $$l(s):=d(z,\gamma_{y,x}(s)).$$ Then, for all $s\in [0,d(x,y)]$ with $\gamma_{y,x}(s)\in I_z\setminus\{z\}$, we have $$l'(s)=\langle\dot{\gamma}_{z,\gamma_{y,x}(s)}(l(s)),\dot{\gamma}_{y,x}(s)\rangle$$ by the first variation formula, and so $$|l'(s)\sin s+\langle\dot{\gamma}_{z,\gamma_{y,x}(s)}(l(s)),\nabla f(\gamma_{y,x}(s))\rangle|\leq C\eta^{3/26}$$ by Lemma \[p54h\] (ii). Thus, for almost all $s\in [0,d(x,y)]$, we have $$\label{56b}
\begin{split}
\Big|\langle\dot{\gamma}_{\gamma_{y,x}(s),z}(0),&\nabla f(\gamma_{y,x}(s))\rangle l(s)\sin d_S(\Psi(z),\Psi(\gamma_{y,x}(s))\\
&-l'(s)l(s)\sin r(s)\sin s \Big|\leq C\eta^{3/26}
\end{split}$$ by (\[55w\]). By the definition of $G_f^z$, (\[56a\]) and (\[56b\]), for almost all $s\in [0,d(x,y)]$, we have $$\begin{aligned}
\Big|
G_f^z(\gamma_{y,x}(s))-l'(s)l(s)\sin r(s)\sin s+r'(s)r(s)\sin r(s) \sin s
\Big|\leq C\eta^{3/26}.\end{aligned}$$ Thus, by the assumption, we get $$\label{56c}
\int_T^{d(x,y)}\left|\left(\frac{d}{d s}(l(s)^2-r(s)^2)\right)\sin r(s)\sin s\right|\,d s
\leq C\eta^{3/26}.$$ Define $$\begin{aligned}
I&:=\{s\in [T,d(x,y)]: \eta^{1/26}\leq s\leq \pi -\eta^{1/26}\text{ and }\eta^{1/26}\leq r(s) \leq\pi -\eta^{1/26}
\}\\
II&:=[T,d(x,y)]\setminus I.\end{aligned}$$ Then, we have $$\label{56ca}
\int_I \left|\frac{d}{d s}(l(s)^2-r(s)^2)\right|\,d s
\leq C\eta^{1/26}$$ by (\[56c\]). Let us estimate $H^1(II)$, where $H^1$ denotes the $1$-dimensional Hausdorff measure. Suppose that $$\{s\in [T,d(x,y)]: r(s)<\eta^{1/26} \text{ or } r(s)>\pi-\eta^{1/26}\}\neq \emptyset,$$ and take arbitrary $s\in[T,d(x,y)]$ such that $r(s)<\eta^{1/26}$ or $r(s)>\pi-\eta^{1/26}$. Then, we have $$\label{56d}
||\cos r(s)|-1|\leq C\eta^{1/13}.$$ Note that we have $r(s)\leq \pi$ by ${\mathop{\mathrm{diam}}\nolimits}(S^{n-p})=\pi$. By (\[55x\]), we get $$\label{56e}
1-C\eta^{1/13}\leq (x_1^2+x_2^2)^{1/2}\leq 1.$$ Take $s_1\in[0,2\pi]$ such that $$\begin{aligned}
\cos s_1=&\frac{x_1}{(x_1^2+x_2^2)^{1/2}},\\
\sin s_1=&\frac{x_2}{(x_1^2+x_2^2)^{1/2}}.\end{aligned}$$ Then, we get $||\cos (s-s_1)|-1|\leq C\eta^{1/13}$ by (\[55x\]), (\[56d\]) and (\[56e\]). Thus, there exists $n\in \mathbb{Z}$ such that $$|s-s_1-n\pi|\leq C\eta^{1/26}.$$ Then, we have $|n|\leq 2$, and so $$H^1\left(\{s\in [T,d(x,y)]: r(s)<\eta^{1/26} \text{ or } r(s)>\pi-\eta^{1/26}\}\right)\leq C\eta^{1/26}.$$ Note that we have $d(x,y)\leq d(x,A_f)+C\eta\leq \pi+C\eta$ by the assumption and Proposition \[p53a\] (iv). Since we have $$H^1\left(\{s\in [T,d(x,y)]: s<\eta^{1/26} \text{ or } s>\pi-\eta^{1/26}\}\right)\leq C\eta^{1/26},$$ we get $H^1(II)\leq C\eta^{1/26}$. Since $\left|\frac{d}{d s}(l(s)^2-r(s)^2)\right|\leq C$ for almost all $s\in[L,d(x,y)]$, we get $$\label{56f}
\int_{II} \left|\frac{d}{d s}(l(s)^2-r(s)^2)\right|\,d s
\leq C\eta^{1/26}.$$
By (\[56ca\]) and (\[56f\]), we get $$\int_T^{d(x,y)}\left|\frac{d}{d s}(l(s)^2-r(s)^2)\right|\,d s
\leq C\eta^{1/26}.$$ Thus, we have $$|l(d(x,y))^2-r(d(x,y))^2-l(T)^2+r(T)^2|\leq C\eta^{1/26}.$$ By (\[55w\]) and the definition of $l$, we get the lemma.
In the following, the term “$\eta^{1/26}$” frequently appears. Since it appears only due to technical reasons, we put $\tau:=1/26$.
Take $f\in{\mathop{\mathrm{Span}}\nolimits}_{\mathbb{R}}\{f_1,\ldots, f_{n-p+1}\}$ with $\|f\|_2^2=1/(n-p+1)$. By Lemma \[p54f\] and the Bishop-Gromov inequality, for any triple $(x_1,x_2,x_3)\in M\times M\times M$, we can take points $\tilde{x}_1\in D_{f_{x_1}}(p_{x_1})\cap Q_{f_{x_1}} \cap R_{f_{x_1}}\cap Q_f$, $\tilde{x}_2\in D_f(p_f)\cap Q_f \cap R_f\cap P_f^{\tilde{x}_1}$ and $\tilde{x}_3\in D_f(\tilde{x}_2)\cap D_f(p_f)\cap Q_f\cap R_f\cap C_f^{\tilde{x}_1}(\tilde{x}_2)$ such that $d(x_1,\tilde{x}_1)\leq C\delta^{1/100n}$, $d(x_2,\tilde{x}_2)\leq C\eta_0$, $d(x_3,\tilde{x}_3)\leq C\eta_0$. We call the triple $(\tilde{x}_1,\tilde{x}_2,\tilde{x}_3)$ a “[*$\Pi$-triple for $(x_1,x_2,x_3,f)$*]{}”.
\[ptrp\] Take
- $f\in{\mathop{\mathrm{Span}}\nolimits}_{\mathbb{R}}\{f_1,\ldots, f_{n-p+1}\}$ with $\|f\|_2^2=1/(n-p+1)$,
- $x,y,z\in M$,
- $\eta>0$ with $\eta_0\leq\eta\leq L^{1/3n}$ and $T\in [0, d(x,y)]$.
Take a $\Pi$-triple $(\tilde{z},\tilde{x},\tilde{y})$ for $(z,x,y,f)$. Suppose
- $d(y,A_f)\leq C\eta$,
- $|d(x,A_f)-d(x,y)|\leq C\eta$,
- $d(\tilde{z},\gamma_{\tilde{y},\tilde{x}}(s))\leq \pi$ for all $s\in[T,d(\tilde{x},\tilde{y})]$.
Then, we have $$\left| d(\tilde{z},\tilde{x})^2-d_S(\Psi(\tilde{z}),\Psi(\tilde{x}))^2- d(\tilde{z},\gamma_{\tilde{y},\tilde{x}}(T))^2+d_S(\Psi(\tilde{z}),\Psi(\gamma_{\tilde{y},\tilde{x}}(T)))^2
\right|\leq C\eta^{\tau}.$$
We have $(G^{\tilde{z}}_f H^{\tilde{z}})(\gamma_{\tilde{y},\tilde{x}}(s))=G^{\tilde{z}}_f(\gamma_{\tilde{y},\tilde{x}}(s))$ for all $s\in[T,d(\tilde{x},\tilde{y})]$. Thus, we get the lemma immediately by the definition of $C_f^{\tilde{z}}(\tilde{x})$ and Lemma \[p54i\].
The following lemma guarantees that if the images of two points in $M$ under $\Phi_f$ are close to each other in $S^{n-p}\times A_f$, then their distance in $M$ are close to each other under some assumptions.
\[p54j\] Take
- $f\in{\mathop{\mathrm{Span}}\nolimits}_{\mathbb{R}}\{f_1,\ldots, f_{n-p+1}\}$ with $\|f\|_2^2=1/(n-p+1)$,
- $x,y,z,w\in M$,
- $\eta>0$ with $\eta_0\leq\eta\leq L^{1/3n}$.
Suppose
- $d(x,A_f)\leq \pi- \frac{1}{C}\eta^{\tau/3}$ and $d(z,A_f)\leq \pi- \frac{1}{C}\eta^{\tau/3}$,
- $d(y,A_f)\leq C\eta$ and $d(w,A_f)\leq C\eta$,
- $|d(x,A_f)-d(x,y)|\leq C\eta$ and $|d(z,A_f)-d(z,w)|\leq C\eta$
- $d(y,w)\leq C\eta$,
- $d_S(\Psi(x),\Psi(z))\leq C\eta$.
Then, we have $$d(x,z)\leq C\eta^{\tau/2}.$$
We first show the following claim.
\[p54k\] If $x,y,z,w\in M$ satisfies:
- $d(x,A_f)\leq \frac{1}{2}\pi- \frac{1}{C}\eta^{1/2}$ and $d(z,A_f)\leq \frac{1}{2}\pi- \frac{1}{C}\eta^{1/2}$,
- $d(y,A_f)\leq C\eta$ and $d(w,A_f)\leq C\eta$,
- $|d(x,A_f)-d(x,y)|\leq C\eta$ and $|d(z,A_f)-d(z,w)|\leq C\eta$
- $d(y,w)\leq C\eta$,
- $d_S(\Psi(x),\Psi(z))\leq C\eta^{\tau/2}$.
Then, we have $$d(x,z)\leq C\eta^{\tau/2}.$$
Take $u\in S^{n-p}$ with $f=\sum_{i=1}^{n-p+1} u_i f_i$. By the assumptions and Lemma \[p54c0\], we have $$\begin{aligned}
d_S(u,\Psi(y))\leq& C\eta,\\
|d_S(\Psi(z),u)-d(z,A_f)|\leq &C\delta^{1/1600n^2}.\end{aligned}$$ Since we have $|d(z,A_f)-d(z,y)|\leq C\eta$ by the assumptions, we get $$\label{57a0}
|d_S(\Psi(z),\Psi(y))-d(z,y)|\leq C\eta.$$ Take a $\Pi$-triple $(\tilde{z},\tilde{x},\tilde{y})$ for $(z,x,y,f)$. Then, we have $$\begin{aligned}
d(\tilde{z},\gamma_{\tilde{y},\tilde{x}}(s))\leq d(\tilde{z},\tilde{y})+d(\tilde{y},\tilde{x})
\leq &d(z,w)+d(y,w)+d(x,y)+C\eta_0\\
\leq &\pi-\frac{1}{C}\eta^{1/2}+C\eta\leq \pi\end{aligned}$$ for all $s\in[0,d(\tilde{x},\tilde{y})]$, and so $$\left| d(z,x)^2-d_S(\Psi(z),\Psi(x))^2- d(z,y)^2+d_S(\Psi(z),\Psi(y))^2
\right|\leq C\eta^{\tau}$$ by Lemma \[p54c00\] and Lemma \[ptrp\]. Thus, we get $d(x,z)\leq C\eta^{\tau/2}$ by (\[57a0\]).
Let us suppose that $x,y,z,w\in M$ satisfies the assumptions of the lemma. Take $u\in S^{n-p}$ with $f=\sum_{i=1}^{n-p+1}u_i f_i$. By the assumptions and Lemma \[p54c0\], we have $$\label{57a1}
|d(x,A_f)-d(z,A_f)|
\leq |d_S(\Psi(x),u)-d(\Psi(z),u)|+C\delta^{1/1600n^2}
\leq C\eta$$ Thus, if either $d(x,A_f)\leq \eta^{\tau}$ or $d(z,A_f)\leq \eta^{\tau}$ holds, then the lemma is trivial. In the following, we assume $d(x,A_f)\geq \eta^{\tau}$ and $d(z,A_f)\geq \eta^{\tau}$. Take a $\Pi$-triple $(\tilde{z},\tilde{x},\tilde{y})$ for $(z,x,y,f)$. By Lemma \[p54g\] (iii), we can take $v_1,v_2\in S^{n-p}$ such that $u\cdot v_i=0$ ($i=1,2$), $$\label{57a}
d_S(\Psi(\gamma_{\tilde{y},\tilde{x}}(s)),\gamma_{v_1}(s))\leq C\eta^{3/13}$$ for all $s\in [0,d(\tilde{y},\tilde{x})]$ and $$\label{57b}
d_S(\Psi(\gamma_{w,\tilde{z}}(s)),\gamma_{v_2}(s))\leq C\eta^{3/13}$$ for all $s\in [0,d(w,\tilde{z})]$, where $\gamma_{v_i}(s):=\cos s u+\sin s v_i \in S^{n-p}$ ($i=1,2$). By the assumptions and (\[57a1\]), we get $$\label{57b1}
|d(\tilde{y},\tilde{x})-d(w,\tilde{z})|\leq C\eta,$$ and so $$\begin{aligned}
\sin d(\tilde{y},\tilde{x}) |v_1- v_2|
\leq &C d_S(\gamma_{v_1}(d(\tilde{y},\tilde{x})),\gamma_{v_2}(d(\tilde{y},\tilde{x})))\\
\leq &Cd_S(\Psi(\tilde{x}),\Psi(\tilde{z}))+C\eta^{3/13}
\leq C\eta^{3/13}\end{aligned}$$ by (\[57a\]) and (\[57b\]). By $\eta^{\tau}\leq d(x,A_f)\leq \pi-\frac{1}{C}\eta^{\tau/3}$, we have $\sin d(\tilde{y},\tilde{x})\geq \frac{1}{C}\eta^{\tau}$. Thus, we get $$|v_1-v_2|\leq C\eta^{\tau}.$$ This gives $$\label{57c}
d_S(\gamma_{v_1}(s),\gamma_{v_2}(s))\leq C\eta^{\tau}.$$ for all $s\in \mathbb{R}$.
Put $$a=\gamma_{\tilde{y},\tilde{x}}\left(\frac{1}{2}d(\tilde{y},\tilde{x})\right),
\quad b=\gamma_{w,\tilde{z}}\left(\frac{1}{2}d(w,\tilde{z})\right).$$ By (\[57a\]), (\[57b\]), (\[57b1\]) and (\[57c\]), we have $$d_S(\Psi(a),\Psi(b))\leq C\eta^{\tau}.$$ Moreover, other assumptions of Claim \[p54k\] hold for the pair $(a,y,b,w)$ by Lemma \[p54g\] (i). Thus, we get $$d(a,b)\leq C\eta^{\tau/2}$$ by Claim \[p54k\]. Therefore, we have $$d(\tilde{z}, \gamma_{\tilde{y},\tilde{x}}(s))\leq d(\tilde{z},b)+d(a,b)+d(\gamma_{\tilde{y},\tilde{x}}(s),a)\leq \frac{1}{2}d(\tilde{x},\tilde{y})+\frac{1}{2}d(\tilde{z},w)+C\eta^{\tau/2}\leq \pi$$ for all $s\in[0,d(\tilde{y},\tilde{x})]$, and so $d(\tilde{x},\tilde{z})\leq C\eta^{\tau/2}$ similarly to Claim \[p54k\]. Thus, we get the lemma.
Let us show the almost Pythagorean theorem for the special case. Recall that we defined $\eta_1:=\eta_0^\tau$.
\[p54l\] Take
- $f\in{\mathop{\mathrm{Span}}\nolimits}_{\mathbb{R}}\{f_1,\ldots, f_{n-p+1}\}$ with $\|f\|_2^2=1/(n-p+1)$,
- $x,y,z,w\in M$,
- $\eta>0$ with $\eta_1\leq \eta\leq L^{1/3n}$.
Suppose
- $d(x,z)\leq C\eta$,
- $d(x,A_f)\leq \pi- \frac{1}{C}\eta^{1/2}$ and $d(z,A_f)\leq \pi- \frac{1}{C}\eta^{1/2}$,
- $d(y,A_f)\leq C\eta_0$ and $d(w,A_f)\leq C\eta_0$,
- $|d(x,A_f)-d(x,y)|\leq C\eta_0$ and $|d(z,A_f)-d(z,w)|\leq C\eta_0$.
Then, we have $$|d(x,z)^2-d_S(\Psi(x),\Psi(z))^2-d(y,w)^2|\leq C\eta_1.$$
By Lemma \[p54c0\], we have $$\label{57d0}
d_S(\Psi(y),\Psi(w))
\leq d(y,A_f)+d(w,A_f)+C\delta^{1/1600n^2}
\leq C\eta_0.$$
Put $a_0:=x$ and $b_0:=z$. In the following, we define $a_{i},b_{i}\in M$ ($i=1,2,3$) so that
- $d(a_{i},b_{i})\leq C\eta^{1/2}$,
- $|d(a_{i},A_f)-d(a_{i},y)|\leq C\eta_0$ and $|d(b_{i},A_f)-d(b_{i},w)|\leq C\eta_0$,
- $d(a_{i},A_f)\leq \frac{3-i}{3}\pi+C\eta_0$ and $d(b_{i},A_f)\leq \frac{3-i}{3}\pi+C\eta_0$,
- $|d(a_{i+1},b_{i+1})^2-d_S(\Psi(a_{i+1}),\Psi(b_{i+1}))^2-d(a_{i},b_{i})^2+d_S(\Psi(a_{i}),\Psi(b_{i}))^2|\leq C\eta_0^{\tau}$ ($i=0,1,2$),
- $d(y,a_3)\leq C\eta_0$ and $d(w,b_3)\leq C\eta_0$.
If we succeed in defining such $a_i$ and $b_i$, we have $$|d(x,z)^2-d_S(\Psi(x),\Psi(z))^2-d(y,w)^2+d_S(\Psi(y),\Psi(w))^2|\leq C\eta_0^{\tau}=C\eta_1$$ by (iv) and (v), and so we get the lemma by (\[57d0\]).
Take arbitrary $i\in\{0,1,2\}$ and suppose that we have chosen $a_i,b_i\in M$ such that (i), (ii) and (iii) hold if $i\geq 1$. Let us define $a_{i+1},b_{i+1}\in M$ that satisfy our properties. Take a $\Pi$-triple $(\tilde{b}_i,\tilde{a}_i, \tilde{y}_i)$ for $(b_i,a_i,y,f)$. Define $$a_{i+1}:=\gamma_{\tilde{y}_i,\tilde{a}_i}\left(\frac{2-i}{3-i}d(\tilde{y}_i,\tilde{a}_i)\right).$$ Since $$d(\tilde{b}_i, \gamma_{\tilde{y}_i,\tilde{a}_i}(s))\leq d(\tilde{a}_i,\tilde{b}_i)
+d(\tilde{a}_i,\gamma_{\tilde{y}_i,\tilde{a}_i}(s))\leq \frac{\pi}{3}+C\eta^{1/2}$$ for all $s\in\left[\frac{2-i}{3-i}d(\tilde{y}_i,\tilde{a}_i),d(\tilde{y}_i,\tilde{a}_i)\right]$ by the assumptions, we get $$\label{57d}
|d(a_{i+1},b_{i})^2-d_S(\Psi(a_{i+1}),\Psi(b_{i}))^2-d(a_{i},b_{i})^2+d_S(\Psi(a_{i}),\Psi(b_{i}))^2|\leq C\eta_0^{\tau}$$ by Lemma \[p54c00\] and Lemma \[ptrp\]. Take a $\Pi$-triple $(\overline{a}_{i+1},\overline{b}_i,\overline{w}_i)$ for $(a_{i+1},b_i,w,f)$. Define $$b_{i+1}:=\gamma_{\overline{w}_i,\overline{b}_i}\left(\frac{2-i}{3-i}d(\overline{w}_i,\overline{b}_i)\right).$$ Since $$d(\overline{a}_{i+1},\gamma_{\overline{w}_i,\overline{b}_i}(s))\leq d(\overline{a}_{i+1},a_i)+d(a_i,\overline{b}_i)+d(\overline{b_i},\gamma_{\overline{w}_i,\overline{b}_i}(s))\leq \frac{2}{3}\pi +C\eta^{1/2}$$ for all $s\in\left[\frac{2-i}{3-i}d(\overline{w}_i,\overline{b}_i),d(\overline{w}_i,\overline{b}_i)\right]$ by the assumptions, we get $$\label{57e}
|d(a_{i+1},b_{i+1})^2-d_S(\Psi(a_{i+1}),\Psi(b_{i+1}))^2-d(a_{i+1},b_{i})^2+d_S(\Psi(a_{i+1}),\Psi(b_{i}))^2|\leq C\eta_0^{\tau}$$ by Lemma \[p54c00\] and Lemma \[ptrp\]. By (\[57d\]) and (\[57e\]), we get (iv).
By the assumptions and Lemma \[p54g\], we get (ii) for $a_{i+1}$ and $b_{i+1}$.
By the assumptions, we have $$\begin{aligned}
d(a_{i+1},A_f)
\leq &d(a_{i+1},\tilde{y}_i)+d(y,A_f)+C\eta_0\\
=&\frac{2-i}{3-i}d(\tilde{a}_{i},\tilde{y}_i)+C\eta_0
\leq \frac{2-i}{3}\pi+C\eta_0.\end{aligned}$$ Similarly, we have $d(b_{i+1},A_f)\leq \frac{2-i}{3}\pi+C\eta_0$. Thus, we get (iii) for $a_{i+1}$ and $b_{i+1}$.
By definition, we have $$a_3=\tilde{y}_3,\quad b_3=\overline{w}_3.$$ Thus, we get (v).
In the following, we prove (i) for $a_{i+1}$ and $b_{i+1}$. If $d(a_i,y)\leq \eta_0^{\tau}$, then we have $$\begin{aligned}
d(b_i,w)\leq d(b_i,A_f)+C\eta_0
\leq d(a_i,A_f)+C\eta^{1/2}
\leq C\eta^{1/2},\end{aligned}$$ and so $$\begin{aligned}
d(y,w)\leq&C\eta^{1/2},\\
d(a_{i+1},y)\leq& C\eta^{1/2},\\
d(b_{i+1},w)\leq & C\eta^{1/2}.\end{aligned}$$ Then, we have $d(a_{i+1},b_{i+1})\leq C\eta^{1/2}$. Similarly, if $d(b_i,w)\leq \eta_0^{\tau}$, then $d(a_{i+1},b_{i+1})\leq C\eta^{1/2}$. Thus, in the following, we assume that $d(a_i,y)\geq \eta_0^{\tau}$ and $d(b_i,w)\geq \eta_0^{\tau}$. By Lemma \[p54g\], we can take $u,v_1,v_2\in S^{n-p}$ such that $f=\sum_{j=1}^{n-p+1}u_j f_j$, $ u\cdot v_k=0$ ($k=1,2$), $$\label{57f}
d_S(\Psi(\gamma_{\tilde{y}_i,\tilde{a}_i}(s)),\gamma_{v_1}(s))\leq C\eta_0^{3/13}$$ for all $s\in [0,d(\tilde{a}_i,\tilde{y}_i)]$ and $$\label{57g}
d_S(\Psi(\gamma_{\overline{w}_i,\overline{b}_i}(s)),\gamma_{v_2}(s))\leq C\eta_0^{3/13}$$ for all $s\in [0,d(\overline{b}_i,\overline{w}_i)]$, where $\gamma_{v_k}(s):=\cos s u+\sin s v_k\in S^{n-p}$ ($k=1,2$). Since $$|d(\tilde{a}_i,\tilde{y}_i)-d(\overline{b}_i,\overline{w}_i)|\leq
|d(a_i,A_f)-d(b_i,A_f)|+C\eta_0\leq d(a_i,b_i)+C\eta_0,$$ we have $$\label{57h}
\left|d_S(\Psi(\tilde{a}_i),\Psi(\overline{b}_i))
-d_S\left(\gamma_{v_1}(l_i),\gamma_{v_2} (l_i)\right)
\right|\leq d(a_i,b_i)+C\eta_0^{3/13}$$ and $$\label{57i}
\left|d_S(\Psi(a_{i+1}),\Psi(b_{i+1}))
-d_S\left(\gamma_{v_1}\left(\frac{2-i}{3-i}l_i\right),\gamma_{v_2} \left(\frac{2-i}{3-i}l_i\right)\right)
\right|\leq d(a_i,b_i)+C\eta_0^{3/13}$$ by (\[57f\]) and (\[57g\]), where we put $l_i:=d(\tilde{a}_i,\tilde{y}_i)$. By (\[57h\]) and Lemma \[p54c00\], we get $$\label{57i1}
|v_1-v_2|\sin l_i
\leq C d_S\left(\gamma_{v_1}(l_i),\gamma_{v_2} (l_i)\right)
\leq Cd(a_i,b_i)+C\eta_0^{3/13}.$$ Note that we assumed $$\label{57j}
d(a_i,b_i)\leq C\eta^{1/2}$$ and $$\label{57k}
d(a_0,b_0)\leq C\eta.$$
We first suppose that $d(a_i,y)\leq \pi/6$. By (\[57h\]), (\[57i\]) and (\[57j\]), we get $$\label{57l}
\left|d_S(\Psi(\tilde{a}_i),\Psi(\overline{b}_i))
-d_S\left(\gamma_{v_1}(l_i),\gamma_{v_2} (l_i)\right)
\right|\leq C\eta^{1/2}$$ and $$\label{57m}
\left|d_S(\Psi(a_{i+1}),\Psi(b_{i+1}))
-d_S\left(\gamma_{v_1}\left(\frac{2-i}{3-i}l_i\right),\gamma_{v_2} \left(\frac{2-i}{3-i}l_i\right)\right)
\right|\leq C\eta^{1/2},$$ Since $l_i\leq \pi/2$, we have $$\sin \left(\frac{2-i}{3-i}l_i\right)
\leq \sin l_i,$$ and so $$\begin{aligned}
d_S(\Psi(a_{i+1}),\Psi(b_{i+1}))
\leq& d_S\left(\gamma_{v_1}\left(\frac{2-i}{3-i}l_i\right),\gamma_{v_2} \left(\frac{2-i}{3-i}l_i\right)\right)+C\eta^{1/2}\\
\leq& C|v_1-v_2|\sin \left(\frac{2-i}{3-i}l_i\right)+C\eta^{1/2}\\
\leq& C|v_1-v_2|\sin l_i+C\eta^{1/2}\\
\leq& C d_S(\Psi(\tilde{a}_i),\Psi(\overline{b}_i))+C\eta^{1/2}
\leq C\eta^{1/2}\end{aligned}$$ by (\[57l\]) and (\[57m\]). Thus, we get $d(a_{i+1},b_{i+1})\leq C\eta^{1/2}$ by (iv).
We next suppose that $\pi/6\leq d(a_i,y)\leq 5\pi/6$. By (\[57i1\]) and (\[57j\]), we have $|v_1-v_2|\leq C\eta^{1/2}$. Thus, we get $$\begin{aligned}
d_S(\Psi(a_{i+1}),\Psi(b_{i+1}))
\leq C\eta^{1/2}\end{aligned}$$ by (\[57i\]). Thus, we get $d(a_{i+1},b_{i+1})\leq C\eta^{1/2}$ by (iv).
If $i\geq 1$, we have $d(a_i,y)\leq 5\pi/6$, and so we get $d(a_{i+1},b_{i+1})\leq C\eta^{1/2}$ by the above two cases.
Finally, we suppose that $i=0$ and $d(x,y)\geq 5\pi/6$. By (\[57i1\]) and (\[57k\]), we have $|v_1-v_2|\sin l_0\leq C\eta$. By the definition of $l_0$, we have $|l_0-d(x,y)|\leq C\eta_0.$ Thus, we have $\sin l_0\geq \frac{1}{C}(\pi- l_0)\geq \frac{1}{C}\eta^{1/2}$, and so we get $|v_1-v_2|\leq C\eta^{1/2}$, and so we have $$\begin{aligned}
d_S(\Psi(a_{i+1}),\Psi(b_{i+1}))
\leq C\eta^{1/2}\end{aligned}$$ by (\[57i\]). Thus, $d(a_{i+1},b_{i+1})\leq C\eta^{1/2}$ by (iv).
Therefore, we have (i) for all cases, and we get the lemma.
Let us show that the map $\Phi_f\colon M\to S^{n-p}\times A_f,\,x\mapsto (\Psi(x), a_f(x))$ is almost surjective.
\[p54m\] Take $f\in {\mathop{\mathrm{Span}}\nolimits}_{\mathbb{R}}\{f_1,\ldots, f_{n-p+1}\}$ with $\|f\|_2^2=1/(n-p+1)$. For any $(v,a)\in S^{n-p}\times A_f$, there exists $x\in M$ such that $d(\Phi_f(x),(v,a))\leq C\eta_1^{1/2}$ holds.
Take arbitrary $(v,a)\in S^{n-p}\times A_f$. Take $u\in S^{n-p}$ with $f=\sum_{i=1}^{n-p+1} u_i f_i$. Since there exists $\tilde{v}\in S^{n-p}$ such that $d_S(u,\tilde{v})\leq \pi-\eta_1^{1/2}$ and $d_S(v,\tilde{v})\leq \eta_1^{1/2}$, it is enough to prove the proposition assuming $d_S(u,v)\leq \pi-\eta_1^{1/2}$.
Put $F_v:=\sum_{i=1}^{n-p+1}v_i f_i$. Then, $|F_v(p_{F_v})-1|\leq C\delta^{1/800n}$ and $A_{F_v}=\{x\in M:|F_v(x)-1|\leq \delta^{1/900n}\}$ by Proposition \[p53a\]. In the following, we show that $a_v:=a_{F_v}(a)\in A_{F_v}$ has the desired property. By Lemma \[p54c0\], we get $$\begin{aligned}
\notag d_S(\Psi(a),u)\leq &C\delta^{1/1600n^2},\\
\label{57n}
d_S(\Psi(a_v),v)\leq &C\delta^{1/1600n^2}.\end{aligned}$$ Thus, by Lemma \[p54c0\], we get $$\begin{aligned}
|d(a,a_v)-d(a_f(a_v),a_v)|=&|d(a,A_{F_v})-d(a_v,A_f)|\\
\leq& |d_S(\Psi(a),v)-d_S(\Psi(a_v),u)|+C\delta^{1/1600n^2}\\
\leq& C\delta^{1/1600n^2}\leq \eta_0\end{aligned}$$ and $$d(a_v,A_f)\leq d_S(\Psi(a_v),u)+C\delta^{1/1600n^2}
\leq d_S(u,v)+C\delta^{1/1600n^2}\leq \pi-\frac{1}{2}\eta_1^{1/2}.$$ Since we have $d(a_v,A_f)=d(a_v,a_f(a_v))$, we get $$\begin{aligned}
|d(a_v,A_f)-d(a_v,a)|\leq |d(a_v,A_f)-d(a_v,a_f(a_v))|+\eta_0=\eta_0,\end{aligned}$$ and so we get $$\label{57o}
d(a,a_f(a_v))\leq C\eta_0^{\tau/2}=C\eta_1^{1/2}$$ by Lemma \[p54l\] putting $x=z=a_v$, $y=a$ and $w=a_f(a_v)$.
By (\[57n\]) and (\[57o\]), putting $x=a_v$, we get the proposition.
Now, we are in position to show $|\dot{\gamma}_{\tilde{y}_1,y_2}^E|d(\tilde{y}_1,y_2)\leq \pi+L$ under the assumption of Lemma \[p54d\]. Note that we defined $\tau=1/26$, $\eta_2=\eta_1^{\tau/3}$ and $L=\eta_2^{1/150}$.
\[p54n\] Take $y_1\in M$, $\tilde{y}_1\in D_{f_{y_1}}(p_{y_1})\cap R_{f_{y_1}}\cap Q_{f_{y_1}}$ with $d(y_1,\tilde{y}_1)\leq C\delta^{1/100n}$ and $y_2\in D_{f_{y_1}}(\tilde{y}_1)$. Let $\{E_1,\ldots,E_n\}$ be a parallel orthonormal basis of $TM$ along $\gamma_{\tilde{y}_1,y_2}$ in Lemma \[p5e\] or Lemma \[p5g\] for $f_{y_1}$. Then, $$|\dot{\gamma}_{\tilde{y}_1,y_2}^E|d(\tilde{y}_1,y_2)\leq \pi+ L$$ and $$||\dot{\gamma}_{\tilde{y}_1,y_2}^E|d(\tilde{y}_1,y_2)-d_S(\Psi(y_1),\Psi(y_2))|\leq CL.$$
We immediately get the second assertion by the first assertion and Lemma \[p54d\].
Let us show the first assertion. Suppose that $$|\dot{\gamma}_{\tilde{y}_1,y_2}^E|d(\tilde{y}_1,y_2)>\pi+ L.$$ Put $$\begin{aligned}
f:=&-f_{y_1},\\
\gamma:=&\gamma_{\tilde{y}_1,y_2},\\
s_0:=&\frac{1}{|\dot{\gamma}^E|}\eta_2^{\tau/4},\\
s_1:=&\frac{1}{|\dot{\gamma}^E|}(\pi+L).\end{aligned}$$ Take $k\in \mathbb{N}$ to be $$\frac{1}{\eta_2}(s_1-s_0)<k\leq \frac{1}{\eta_2}(s_1-s_0)+1,$$ and put $$t_j:=\frac{j}{k}(s_1-s_0)+s_0$$ for each $j\in\{0,\ldots,k\}$. Note that we have $t_0=s_0$, $t_k=s_1$ and $$\begin{aligned}
\label{57p0}
k\leq& C\eta_2^{-1},\\
\label{57p1}\frac{1}{k}\leq& C\eta_2.\end{aligned}$$
For all $s\in[s_0,s_1]$, we have $$\begin{aligned}
\cos d_S(\Psi(y_1),\Psi(\gamma(s)))
\leq \cos (|\dot{\gamma}^E| s)+C\delta^{1/1600n^2}
\leq 1-\frac{1}{C}\eta_2^{\tau/2}\end{aligned}$$ for all $s\in[s_0,s_1]$ by Lemma \[p54d\]. Since $$f(\gamma(s))=-|\widetilde{\Psi}|(\gamma(s))\cos d_S(\Psi(y_1),\Psi(\gamma(s)))$$ by the definitions of $f_{y_1}$ and $f$, we get $$f(\gamma(s))\geq -1+\frac{1}{C}\eta_2^{\tau/2}$$ for all $s\in[s_0,s_1]$ by Lemma \[p54a\]. This gives $$\label{57p}
d(\gamma(s),A_f)\leq \pi-\frac{1}{C}\eta_2^{\tau/4}$$ $s\in[s_0,s_1]$ by Proposition \[p53a\]. By the definition of $t_j$ and (\[57p\]), we have $$\begin{aligned}
\label{57p11} d(\gamma(t_j),\gamma(t_{j+1}))\leq& \eta_2,\\
\notag d(\gamma(t_j),A_f)\leq & \pi-\frac{1}{C}\eta_2^{\tau/4}\leq \pi-\eta_2^{1/2},\\
\notag d(\gamma(t_{j+1}),A_f)\leq &\pi-\frac{1}{C}\eta_2^{\tau/4}\leq\pi-\eta_2^{1/2}\end{aligned}$$ for all $j\in\{0,\ldots,k-1\}$, and so we get $$\label{57q}
|d(\gamma(t_j),\gamma(t_{j+1}))^2-d_S(\Psi(\gamma(t_j)),\Psi(\gamma(t_{j+1})))^2-d(a_f(\gamma(t_j)),a_f(\gamma(t_{j+1})))^2|\leq C\eta_1$$ by Lemma \[p54l\]. In particular, we get $$\label{57r}
d(a_f(\gamma(t_j)),a_f(\gamma(t_{j+1})))\leq C\eta_2$$ by (\[57p11\]).
Take $j_0\in\{1,\ldots, k-1\}$ to be $$|\dot{\gamma}^E|t_{j_0}< \pi \leq |\dot{\gamma}^E|t_{j_0+1}.$$ Since $$||\dot{\gamma}^E|s-d_S(\Psi(y_1),\Psi(\gamma(s)))|\leq C\delta^{1/3200n^2}$$ for all $s\in\left[0,\frac{1}{|\dot{\gamma}^E|}\pi\right]$ by Lemma \[p54d\], we get $$\label{57s}
\begin{split}
d_S(\Psi(\gamma(t_j)),\Psi(\gamma(t_{j+1})))
\geq &d_S(\Psi(y_1),\Psi(\gamma(t_{j+1})))- d_S(\Psi(y_1),\Psi(\gamma(t_{j})))\\
\geq &|\dot{\gamma}^E|(t_{j+1}-t_j)-C\delta^{1/3200n^2}
\end{split}$$ for all $j\in \{0,\ldots,j_0-1\}$. Since $$|2\pi-|\dot{\gamma}^E|s-d_S(\Psi(y_1),\Psi(\gamma(s)))|\leq C\delta^{1/3200n^2}$$ for all $s\in\left[\frac{1}{|\dot{\gamma}^E|}\pi,s_1\right]$ by Lemma \[p54d\], we get $$\label{57t}
d_S(\Psi(\gamma(t_j)),\Psi(\gamma(t_{j+1})))
\geq |\dot{\gamma}^E|(t_{j+1}-t_j)-C\delta^{1/3200n^2}$$ for all $j\in \{j_0+1,\ldots,k-1\}$. By (\[57q\]), (\[57s\]) and (\[57t\]), we get $$\label{57u}
d(a_f(\gamma(t_j)),a_f(\gamma(t_{j+1})))^2
\leq d(\gamma(t_j),\gamma(t_{j+1}))^2-|\dot{\gamma}^E|^2(t_{j+1}-t_j)^2+C\eta_1$$ for all $j\in\{0,\ldots,k-1\}\setminus \{j_0\}$.
Since we have $$\begin{aligned}
d_S(\Psi(\gamma(s_l)),\Psi(p_f))
\leq d(\gamma(s_l),A_f)+C\delta^{1/1600n^2}
\leq \pi-\frac{1}{C}\eta_2^{\tau/4}\end{aligned}$$ for each $l=0,1$ by Lemma \[p54c0\], Corollary \[p54c01\] and (\[57p\]), we can take a curve $\beta\colon[0,K]\to S^{n-p}$ in $S^{n-p}$ with unit speed ($K$ is some constant) such that $$\begin{aligned}
\beta(0)=&\Psi(\gamma(s_0)),\\
\beta(K)=&\Psi(\gamma(s_1)),\\
|d_S(\Psi(\gamma(s_0)),\Psi(\gamma(s_1)))-K|\leq &C\eta_2^{\tau/4},\\
d_S(\beta(s),\Psi(p_f))\leq &\pi-\frac{1}{C}\eta_2^{\tau/4}\end{aligned}$$ for all $s\in[0,K]$. Note that we can find such $\beta$ by taking an almost shortest pass in $$\{u\in S^{n-p}: d(u,\Psi(p_f))\leq \pi-\frac{1}{C}\eta_2^{\tau/4}\}.$$ By Proposition \[p54m\], there exists $x_j\in M$ such that $$\label{57v}
d\left(\Phi_f(x_j),\left(\beta\left(\frac{j}{k}K\right),a_f(\gamma(t_j))\right)\right)\leq C\eta_1^{1/2}$$ for each $j\in\{0,\ldots,k\}$. By (\[57p1\]), (\[57r\]), (\[57v\]), Lemma \[p54c0\] and Corollary \[p54c01\], we have $$\begin{aligned}
\notag d(a_f(x_j),a_f(x_{j+1}))\leq &C\eta_2,\\
\label{57v1}d_S(\Psi(x_j),\Psi(x_{j+1}))\leq &\frac{1}{k}K+C\eta_1^{1/2}\leq C\eta_2,\\
\label{57v2}d(x_j,A_f)\leq &d_S(\Psi(x_j),\Psi(p_f))+C\delta^{1/1600n^2}\\
\notag \leq &d_S\left(\beta\left(\frac{j}{k}K\right),\Psi(p_f)\right)+C\eta_1^{1/2}
\leq\pi-\frac{1}{C}\eta_2^{\tau/4}\end{aligned}$$ for all $j$, and so $$\label{57v3}
d(x_j,x_{j+1})\leq C\eta_2^{\tau/2}$$ by Lemma \[p54j\] putting $x=x_j, y=a_f(x_j), z=x_{j+1}, w=a_f(x_{j+1})$ and $\eta=\eta_2$. By (\[57v2\]), (\[57v3\]) and Lemma \[p54l\] putting $x=x_j, y=a_f(x_j), z=x_{j+1}, w=a_f(x_{j+1})$ and $\eta=\eta_2^{\tau/2}$, we get $$\label{57w}
|d(x_j,x_{j+1})^2-d_S(\Psi(x_j),\Psi(x_{j+1}))^2-d(a_f(x_j),a_f(x_{j+1}))^2|\leq C\eta_1$$ for all $j\in\{0,\ldots,k-1\}$. By (\[57u\]), (\[57v1\]) and (\[57w\]), we have $$\label{57x}
\begin{split}
d(x_j,x_{j+1})^2
\leq &\frac{1}{k^2}K^2+d(a_f(x_j),a_f(x_{j+1}))^2+C\eta_1^{1/2}\\
\leq &\frac{1}{k^2}K^2+d(\gamma(t_j),\gamma(t_{j+1}))^2-|\dot{\gamma}^E|^2(t_{j+1}-t_j)^2+C\eta_1^{1/2}
\end{split}$$ for all $j\in\{0,\ldots,k-1\}\setminus\{j_0\}$. Since $K\leq \pi+C\eta_2^{\tau/4}$, we have $$\label{57y}
\frac{1}{k^2}K^2\leq \frac{\pi^2}{k^2}+\frac{C}{k^2}\eta_2^{\tau/4}.$$ Since $$|\dot{\gamma}^E|(t_{j+1}-t_j)=\frac{|\dot{\gamma}^E|}{k}(s_1-s_0)
=\frac{1}{k}(\pi+L-\eta_2^{\tau/4})
\geq\frac{1}{k}\left(\pi+\frac{1}{2}L\right),$$ we have $$\label{57z}
|\dot{\gamma}^E|^2(t_{j+1}-t_j)^2\geq \frac{\pi^2}{k^2}+\frac{1}{k^2}L$$ for all $j\in\{0,\ldots,k-1\}$. By (\[57y\]) and (\[57z\]), we get $$|\dot{\gamma}^E|^2(t_{j+1}-t_j)^2-\frac{1}{k^2}K^2\geq \frac{1}{k^2}L-\frac{C}{k^2}\eta_2^{\tau/4} \geq
\frac{1}{2k^2}L$$ for all $j\in\{0,\ldots,k-1\}$. Thus, by (\[57x\]), we have $$d(x_j,x_{j+1})^2
\leq d(\gamma(t_j),\gamma(t_{j+1}))^2-\frac{1}{2k^2}L+C\eta_1^{1/2}
\leq d(\gamma(t_j),\gamma(t_{j+1}))^2-\frac{1}{4k^2}L$$ for all $j\in\{0,\ldots,k-1\}\setminus\{j_0\}$. Since $d(\gamma(t_j),\gamma(t_{j+1}))+d(x_j,x_{j+1})\leq 1$, we get $$\label{58a}
\frac{1}{4k^2}L\leq d(\gamma(t_j),\gamma(t_{j+1}))^2-d(x_j,x_{j+1})^2
\leq d(\gamma(t_j),\gamma(t_{j+1}))-d(x_j,x_{j+1})$$ $j\in\{0,\ldots,k-1\}\setminus\{j_0\}$. By (\[57p0\]), (\[57v3\]) and (\[58a\]), we get $$\label{58b}
\begin{split}
d(x_0,x_k)\leq &\sum_{i=0}^{k-1}d(x_j,x_{j+1})
\leq \sum_{i=0}^{k-1}d(\gamma(t_j),\gamma(t_{j+1}))-\frac{k-1}{4k^2}L+d(x_{j_0},x_{j_0+1})\\
\leq& d(\gamma(s_0),\gamma(s_1))-\frac{1}{8k}L\leq d(\gamma(s_0),\gamma(s_1))-\frac{1}{C}\eta_2 L.
\end{split}$$ By (\[57v\]), we have $$\begin{aligned}
d_S(\Psi(x_0),\Psi(\gamma(s_0)))\leq C\eta_1,\quad
d(a_f(x_0),a_f(\gamma(s_0)))\leq C\eta_1.\end{aligned}$$ Thus, by (\[57p\]), (\[57v2\]) and Lemma \[p54j\], we get $$\label{58c}
d(x_0,\gamma(s_0))\leq C\eta_1^{\tau/2}.$$ Similarly, we get $$\label{58d}
d(x_k,\gamma(s_1))\leq C\eta_1^{\tau/2}.$$ By (\[58b\]), (\[58c\]) and (\[58d\]), we get $$\eta_2 L\leq C\eta_1^{\tau/2}.$$ This contradicts to the definitions of $\eta_2$ and $L$. Thus, we get the lemma.
For all $y_1,y_2\in M$, define $$\begin{aligned}
\overline{C}_f^{y_1}(y_2)=&\Big\{y_3\in M : \gamma_{y_2,y_3}(s)\in I_{y_1}\setminus\{y_1\} \text{ for almost all $s\in[0,d(y_2,y_3)]$, and}\\
&\qquad \qquad\qquad \qquad \int_{0}^{d(y_2,y_3)} |G_f^{y_1}|(\gamma_{y_2,y_3}(s))\,d s\leq L^{1/3}\Big\},\\
\overline{P}_f^{y_1}=&\{y_2\in M: {\mathop{\mathrm{Vol}}\nolimits}(M\setminus C_f^{y_1}(y_2))\leq L^{1/3}{\mathop{\mathrm{Vol}}\nolimits}(M)\}.\end{aligned}$$
Let us complete the Gromov-Hausdorff approximation.
\[MT2\] Take $f\in{\mathop{\mathrm{Span}}\nolimits}_{\mathbb{R}}\{f_1,\ldots, f_{n-p+1}\}$ with $\|f\|_2^2=1/(n-p+1)$. Then, the map $\Phi_f\colon M\to S^{n-p}\times A_f$ is a $CL^{1/156n}$-Hausdorff approximation map. In particular, we have $d_{GH}(M, S^{n-p}\times A_f)\leq CL^{1/156n}$.
Take arbitrary $y_1\in M$ and $\tilde{y}_1\in D_{f_{y_1}}(p_{y_1})\cap R_{f_{y_1}}\cap Q_{f_{y_1}}\cap Q_f$ with $d(y_1,\tilde{y}_1)\leq C\delta^{1/100n}$. By Lemma \[p54c00\], Corollary \[p54f0\] and Lemma \[p54n\], we have $$|G_f^{\tilde{y}_1}|(y_2)\leq CL$$ for all $y\in D_f(\tilde{y_1})\cap D_{f_{y_1}}(\tilde{y}_1)$. Since ${\mathop{\mathrm{Vol}}\nolimits}(M\setminus (D_f(\tilde{y_1})\cap D_{f_{y_1}}(\tilde{y}_1)))\leq C\delta^{1/100}{\mathop{\mathrm{Vol}}\nolimits}(M)$ and $\|G_f^{\tilde{y}_1}\|_\infty\leq C$, we get $$\|G_f^{\tilde{y}_1}\|_1\leq CL.$$ Thus, by the segment inequality, we get $${\mathop{\mathrm{Vol}}\nolimits}(M\setminus \overline{P}^{\tilde{y}_1}_f)\leq CL^{1/3}.$$
Take arbitrary $x,z\in M$. By the Bishop-Gromov inequality, there exist $\tilde{z}\in D_{f_{z}}(p_{z})\cap Q_{f_{z}} \cap R_{f_{z}}\cap Q_f$, $\tilde{x}\in D_f(p_f)\cap Q_f \cap R_f\cap\overline{P}_f^{\tilde{z}}$ and $\tilde{y}\in D_f(\tilde{x})\cap D_f(p_f)\cap Q_f\cap R_f\cap \overline{C}_f^{\tilde{x}}(\tilde{z})$ such that $d(z,\tilde{z})\leq C \delta^{1/100n}$, $d(x,\tilde{x})\leq CL^{1/3n}$ and $d(a_f(x),\tilde{y})\leq CL^{1/3n}$. Here, we used the estimate ${\mathop{\mathrm{Vol}}\nolimits}(M\setminus \overline{P}^{\tilde{z}}_f)\leq CL^{1/3}$. Then, we get $$\left| d(\tilde{z},\tilde{x})^2-d_S(\Psi(\tilde{z}),\Psi(\tilde{x}))^2- d(\tilde{z},\tilde{y})^2+d_S(\Psi(\tilde{z}),\Psi(\tilde{y}))^2
\right|\leq CL^{1/78n}$$ by Lemma \[p54i\]. Thus, we get $$\label{59a}
\left| d(z,x)^2-d_S(\Psi(z),\Psi(x))^2- d(z,a_f(x))^2+d_S(\Psi(z),\Psi(a_f(x)))^2
\right|\leq CL^{1/78n}$$ by Lemma \[p54c00\]. Similarly, we have $$\label{59b}
\begin{split}
&\left| d(a_f(x),z)^2-d_S(\Psi(a_f(x)),\Psi(z))^2- d(a_f(x),a_f(z))^2+ d_S(\Psi(a_f(x)),\Psi(a_f(z)))^2\right|\\
\leq &CL^{1/78n}.
\end{split}$$ Since we have $d_S(\Psi(a_f(x)),\Psi(a_f(z)))\leq C\delta^{1/1600n^2}$ by Lemma \[p54c0\], we get $$\left| d(z,x)^2-d_S(\Psi(z),\Psi(x))^2- d(a_f(x),a_f(z))^2\right|\leq CL^{1/78n}.$$ by (\[59a\]) and (\[59b\]). This gives $$\begin{split}
&\left| d(z,x)-d(\Phi_f(z),\Phi_f(x))\right|\\
=&\left| d(z,x)-\left(d_S(\Psi(z),\Psi(x))^2+ d(a_f(x),a_f(z))^2\right)^{1/2}\right|\leq CL^{1/156n}.
\end{split}$$ Combining this and Proposition \[p54l\], we get the theorem.
By the above theorem, we get Main Theorem 2 except for the orientability, which is proved in subsection 5.7.
Further Inequalities
--------------------
In this subsection, we show two lemmas to prove the remaining part of main theorems.
In this subsection, we assume the following in addition to Assumption \[asu1\].
- $1\leq k\leq n-p+1$.
- $f_i\in C^\infty(M)$ ($i\in\{1,\ldots,k\}$) is an eigenfunction of the Laplacian acting on functions with $\|f_i\|_2^2=1/(n-p+1)$ corresponding to the eigenvalue $\lambda_i$ with $0<\lambda_i\leq n-p+\delta$ such that $$\int_M f_i f_j\,d\mu_g=0$$ holds for any $i\neq j$.
- $\omega\in\Gamma(\bigwedge^p T^\ast M)$ is an eigenform of the connection Laplacian $\Delta_{C,p}$ with $\|\omega\|_2=1$ corresponding to the eigenvalue $\lambda$ with $0\leq \lambda \leq \delta$.
Note that we have $\|\omega\|_\infty\leq C$, $\|f_i\|_\infty \leq C $ and $\|\nabla f_i\|_\infty \leq C$ for all $i\in\{1,\ldots,k\}$ by Lemma \[Linfes\], and $\lambda_i\geq n-p-C\delta^{1/2}$ by Main Theorem 1.
\[pfua\] For any $f\in {\mathop{\mathrm{Span}}\nolimits}_{\mathbb{R}}\{f_1,\ldots,f_{k}\}$, we have $$\left\|\sum_{i=1}^n e^i\otimes (\nabla_{e_i}d f+f e^i)\wedge \omega \right\|_2\leq C\delta^{1/8}\|f\|_2.$$
We have $$\label{fua}
\begin{split}
&\left|\sum_{i=1}^n e^i\otimes (\nabla_{e_i}d f+f e^i)\wedge \omega\right|^2\\
=&\sum_{i=1}^n \langle (\nabla_{e_i}d f) \wedge \omega, (\nabla_{e_i}d f) \wedge \omega\rangle+ 2\sum_{i=1}^n \langle(\nabla_{e_i}d f) \wedge \omega, f e^i \wedge \omega\rangle\\
&\qquad\qquad\qquad\qquad \qquad\qquad\qquad \qquad+\sum_{i=1}^n f^2 \langle e^i \wedge \omega ,e^i \wedge \omega\rangle\\
=&|\nabla^2 f|^2|\omega|^2 -\sum_{i=1}^n|\iota(\nabla_{e_i}\nabla f)\omega|^2
-2 f\Delta f|\omega|^2\\
&\qquad \qquad\qquad \qquad\qquad -2\sum_{i=1}^n f \langle\omega , e^i\wedge \iota(\nabla_{e_i}\nabla f)\omega\rangle
+(n-p)f^2|\omega|^2\\
=& |\nabla^2 f|^2|\omega|^2-\frac{1}{n-p}(\Delta f)^2|\omega|^2
+2\Delta f \left(\frac{1}{n-p}\Delta f-f\right)|\omega|^2\\
-&(n-p)\left(\left(\frac{\Delta f}{n-p}\right)^2-f^2\right)|\omega|^2
-\left|\sum_{i=1}^n e^i \otimes\iota(\nabla_{e_i}\nabla f)\omega\right|^2-2\sum_{i=1}^n f \langle\omega , e^i\wedge \iota(\nabla_{e_i}\nabla f)\omega\rangle.
\end{split}$$ By the assumption, we have $$\begin{aligned}
\label{fub}\left\|\Delta f \left(\frac{1}{n-p}\Delta f-f\right)|\omega|^2\right\|_1\leq &C\delta^{1/2}\|f\|_2^2,\\
\label{fuc}\left\|\left(\left(\frac{\Delta f}{n-p}\right)^2-f^2\right)|\omega|^2\right\|_1\leq &C\delta^{1/2}\|f\|_2^2.\end{aligned}$$ By Lemma \[p4d\] (iv) and Lemma \[p5c\] (ii), we have $$\label{fud}
\left\|\sum_{i=1}^n e^i \otimes\iota(\nabla_{e_i}\nabla f)\omega\right\|_2
\leq \|\nabla (\iota(\nabla f)\omega)\|_2+ C\delta^{1/2}\|f\|_2\leq C\delta^{1/4}\|f\|_2,$$ and so $$\label{fue}
\left\|\sum_{i=1}^n f \langle\omega , e^i\wedge \iota(\nabla_{e_i}\nabla f)\omega\rangle\right\|_1\leq C\|f\|_2\left\|\sum_{i=1}^n e^i \otimes\iota(\nabla_{e_i}\nabla f)\omega\right\|_2
\leq C\delta^{1/4}\|f\|_2^2.$$ By Lemma \[p5c\], (\[fua\]), (\[fub\]), (\[fuc\]), (\[fud\]) and (\[fue\]), we get the lemma.
\[pfub\] Define $G=G(f_1,\ldots,f_k)$ by $$\begin{split}
G:=\Big\{x\in M: & |f_i^2+|\nabla f_i|^2-1|(x)\leq\delta^{1/1600n}\text{ for all $i=1,\ldots,k$, and}\\
&\left|\frac{1}{2}(f_i+f_j)^2+\frac{1}{2}|\nabla f_i+\nabla f_j|^2-1\right|(x)\leq \delta^{1/1600n},\\
&\left|\frac{1}{2}(f_i-f_j)^2+\frac{1}{2}|\nabla f_i-\nabla f_j|^2-1\right|(x)\leq \delta^{1/1600n}\text{ for all $i\neq j$}
\Big\}.
\end{split}$$ Then, we have the following properties.
- We have ${\mathop{\mathrm{Vol}}\nolimits}(M\setminus G)\leq C\delta^{1/1600n}{\mathop{\mathrm{Vol}}\nolimits}(M)$.
- For all $x\in G$ and $i,j$ with $i\neq j$, we have $\left|f_i f_j+\langle\nabla f_i,\nabla f_j\rangle\right|(x)\leq\delta^{1/1600n}$.
By Proposition \[p53a\] (iii), we have $$\begin{split}
\|f_i^2+|\nabla f_i|^2-1\|_1\leq&C\delta^{1/800n},\\
\left\|\frac{1}{2}(f_i+f_j)^2+\frac{1}{2}|\nabla f_i+\nabla f_j|^2-1\right\|_1\leq &C\delta^{1/800n},\\
\left\|\frac{1}{2}(f_i-f_j)^2+\frac{1}{2}|\nabla f_i-\nabla f_j|^2-1\right\|_1\leq &C\delta^{1/800n}
\end{split}$$ for all $i\neq j$. Therefore, we get $$\begin{aligned}
&{\mathop{\mathrm{Vol}}\nolimits}\left(\left\{x\in M: \left|f_i^2+|\nabla f_i|^2-1\right|(x)>\delta^{1/1600n}\right\}\right)\\
\leq& \delta^{-1/1600n}\int_M \left|f_i^2+|\nabla f_i|^2-1\right|\,d\mu_g\\
=& \delta^{-1/1600n}\|f_i^2+|\nabla f_i|^2-1\|_1{\mathop{\mathrm{Vol}}\nolimits}(M)\leq C\delta^{1/1600n}{\mathop{\mathrm{Vol}}\nolimits}(M)\end{aligned}$$ for all $i$. Similarly, we have $$\begin{aligned}
{\mathop{\mathrm{Vol}}\nolimits}\left(\left\{x\in M: \left|\frac{1}{2}(f_i+f_j)^2+\frac{1}{2}|\nabla f_i+\nabla f_j|^2-1\right|(x)>\delta^{1/1600n}\right\}\right)&\leq C\delta^{1/1600n}{\mathop{\mathrm{Vol}}\nolimits}(M),\\
{\mathop{\mathrm{Vol}}\nolimits}\left(\left\{x\in M: \left|\frac{1}{2}(f_i-f_j)^2+\frac{1}{2}|\nabla f_i-\nabla f_j|^2-1\right|(x)>\delta^{1/1600n}\right\}\right)&\leq C\delta^{1/1600n}{\mathop{\mathrm{Vol}}\nolimits}(M)\end{aligned}$$ for all $i\neq j$. Thus, we get (i).
For all $x\in G$ and $i,j$ with $i\neq j$, we have $$\begin{aligned}
&\left|f_i f_j+\langle\nabla f_i,\nabla f_j\rangle\right|(x)\\
=&\frac{1}{2}\left|
\frac{1}{2}(f_i+f_j)^2+\frac{1}{2}|\nabla f_i+\nabla f_j|^2
-\frac{1}{2}(f_i-f_j)^2-\frac{1}{2}|\nabla f_i-\nabla f_j|^2\right|(x)
\leq\delta^{1/1600n}.\end{aligned}$$ Thus, we get (ii).
Orientability
-------------
The goal of this subsection is to show the orientability of the manifold under the assumption of Main Theorem 2.
Note that our assumptions are Assumption \[asu1\].
\[pora\] If $$\lambda_{n-p+1}(g)\leq n-p+\delta$$ and $$\lambda_1(\Delta_{C,p})\leq \delta$$ hold, then $M$ is orientable.
To prove the theorem, we use the following claim:
\[porb\] Define $$\lambda_1(\Delta_{C,n}):=\inf \left\{\frac{\|\nabla \eta\|_2^2}{\|\eta\|_2^2}: \eta\in \Gamma(\bigwedge^n T^\ast M)\text{ with } \eta\neq 0\right\}.$$ If $$\lambda_1(\Delta_{C,n})< \frac{n}{n-1}(n-p-1)$$ holds, then $M$ is orientable.
See Corollary \[papeb\] and Remark \[papec\] in the appendix for the proof of Claim \[porb\].
Let $f_i$ be the $i$-th eigenfunction of the Laplacian with $\|f_i\|_2^2=1/(n-p+1)$ for each $i$, and $\omega$ be the first eigenform of the connection Laplacian $\Delta_{C,p}$ acting on $\Gamma(\bigwedge^p T^\ast M)$ with $\|\omega\|_2=1$. Put $$V:=\sum_{i=1}^{n-p+1} (-1)^{i-1} f_i d f_1\wedge\cdots \wedge \widehat{d f_i}\wedge\cdots \wedge d f_{n-p+1}\wedge \omega\in \Gamma(\bigwedge^n T^\ast M).$$ In the following, we show that $\|\nabla V\|_2^2/\|V\|_2^2< n(n-p+1)/(n-1)$.
Define a vector bundle $E:=T^\ast M\oplus \mathbb{R}e$, where $\mathbb{R}e$ denotes the trivial bundle of rank $1$ with a global non-vanishing section $e$. We consider an inner product $\langle\cdot,\cdot\rangle$ on $E$ defined by $$\langle \alpha+ f e,\beta +h e\rangle:=\langle\alpha,\beta\rangle+ fh$$ for all $\alpha,\beta\in\Gamma(T^\ast M)$ and $f,h\in C^\infty(M)$. Put $$S_i=d f_i +f_i e\in \Gamma(E)$$ for each $i$, and $$\alpha:=S_1\wedge\cdots \wedge S_{n-p+1}\in \Gamma(\bigwedge^{n-p+1} E).$$ Then, we have $\alpha\wedge \omega=e\wedge V$, and so $$\label{ora}
|\alpha\wedge \omega|=|V|.$$
For each $k=1,\ldots,n-p+1$, we have $$\begin{split}
&\Big\|
\big\langle
S_k\wedge\cdots \wedge S_{n-p+1}\wedge \omega,
\left(\iota(S_{k-1})\cdots\iota(S_1)\alpha\right)\wedge \omega
\big\rangle\\
&\qquad-\big\langle S_{k+1} \wedge\cdots \wedge S_{n-p+1}\wedge \omega,
\left(\iota(S_k)\cdots\iota(S_1)\alpha\right)\wedge \omega
\big\rangle
\Big\|_1\\
=&\left\|
\big\langle S_{k+1} \wedge\cdots \wedge S_{n-p+1}\wedge \omega,
\left(\iota(S_{k-1})\cdots\iota(S_1)\alpha\right)\wedge \iota(d f_k)\omega
\big\rangle
\right\|_1\\
\leq& C\|\iota(d f_k)\omega\|_2\leq C\delta^{1/4}
\end{split}$$ by Lemma \[p5c\] (i). By induction, we get $$\label{orb}
\||\alpha\wedge \omega|^2-|\alpha|^2|\omega|^2\|_1\leq C\delta^{1/4}.$$ In particular, we have $$\label{orc}
\left|\|\alpha\wedge \omega\|_2^2-\||\alpha|^2|\omega|^2\|_1\right|
\leq C\delta^{1/4}.$$
Since we have $$\left|\langle S_i(x), S_j(x)\rangle -\delta_{i j}\right|\leq \delta^{1/1600n}$$ for all $x\in G=G(f_1,\ldots,f_{n-p+1})$ and $i,j$ by Lemma \[pfub\] (ii), we get $$||\alpha|^2(x)-1|\leq C\delta^{1/1600n}$$ for all $x\in G$. Thus, we get $$\label{ord}
\begin{split}
&\left|
\frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)}\int_M(|\alpha|^2|\omega|^2-1) \,d\mu_g
\right|\\
=&\Bigg|
\frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)}\int_G(|\alpha|^2-1)|\omega|^2 \,d\mu_g\\
&\qquad+\frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)}\int_{M\setminus G}(|\alpha|^2-1)|\omega|^2 \,d\mu_g+\frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)}\int_M(|\omega|^2-1) \,d\mu_g
\Bigg|\\
\leq &C\delta^{1/1600n}
\end{split}$$ by Lemma \[p4c\] and Lemma \[pfub\] (i). By (\[ora\]), (\[orc\]) and (\[ord\]), we get $$\label{ore}
|\|V\|_2^2-1|\leq C\delta^{1/1600n}.$$
We next estimate $\|\nabla V\|_2^2$. We have $$\begin{split}
&\nabla V\\
=& \sum_{i=1}^{n-p+1} (-1)^{i-1} d f_i\otimes d f_1\wedge\cdots \wedge \widehat{d f_i}\wedge\cdots \wedge d f_{n-p+1}\wedge \omega\\
+&\sum_{j<i}\sum_{k=1}^n (-1)^{i-1}(-1)^{j-1} f_i e^k\otimes (\nabla_{e_k} d f_j)\wedge d f_1\wedge\cdots\wedge\widehat{d f_j} \wedge\cdots\wedge \widehat{d f_i}\wedge\cdots \wedge d f_{n-p+1}\wedge \omega\\
+&\sum_{i<j}\sum_{k=1}^n (-1)^{i-1}(-1)^{j} f_i e^k\otimes (\nabla_{e_k} d f_j)\wedge d f_1\wedge\cdots\wedge\widehat{d f_i} \wedge\cdots\wedge\widehat{d f_j}\wedge\cdots \wedge d f_{n-p+1}\wedge \omega\\
+&\sum_{i=1}^{n-p+1} \sum_{k=1}^n (-1)^{i-1} f_i e^k \otimes d f_1\wedge\cdots \wedge \widehat{d f_i}\wedge\cdots \wedge d f_{n-p+1}\wedge \nabla_{e_k}\omega.
\end{split}$$ Thus, we get $$\label{orf}
\begin{split}
&\left\|
\nabla V
- \sum_{i=1}^{n-p+1} (-1)^{i-1} d f_i\otimes d f_1\wedge\cdots \wedge \widehat{d f_i}\wedge\cdots \wedge d f_{n-p+1}\wedge \omega
\right\|_2\\
\leq&
\Bigg\|\sum_{j<i}\sum_{k=1}^n (-1)^{i-1}(-1)^{j-1} f_i f_j e^k\otimes e^k\wedge d f_1\wedge\cdots\wedge\widehat{d f_j} \wedge\cdots\wedge \widehat{d f_i}\wedge\cdots \wedge d f_{n-p+1}\wedge \omega\\
&+\sum_{i<j}\sum_{k=1}^n (-1)^{i-1}(-1)^{j} f_i f_j e^k\otimes e^k\wedge d f_1\wedge\cdots\wedge\widehat{d f_i} \wedge\cdots\wedge\widehat{d f_j}\wedge\cdots \wedge d f_{n-p+1}\wedge \omega\Bigg\|_2\\
&+C\sum_{i=1}^{n-p+1}\left\|\sum_{k=1}^n e^k\otimes (\nabla_{e_k}d f_i+f_i e^k)\wedge\omega\right\|_2
+ C\|\nabla\omega\|_2\\
\leq &C\delta^{1/8}
\end{split}$$ by Lemma \[pfua\].
Similarly to (\[orb\]), we have $$\label{org}
\begin{split}
&\Bigg\|\left|\sum_{i=1}^{n-p+1} (-1)^{i-1} d f_i\otimes d f_1\wedge\cdots \wedge \widehat{d f_i}\wedge\cdots \wedge d f_{n-p+1}\wedge \omega\right|^2\\
&\qquad-\left|\sum_{i=1}^{n-p+1} (-1)^{i-1} d f_i\otimes d f_1\wedge\cdots \wedge \widehat{d f_i}\wedge\cdots \wedge d f_{n-p+1}\right|^2|\omega|^2\Bigg\|_1\\
&\leq C\delta^{1/4}.
\end{split}$$
Since we have $$d f_1\wedge\cdots\wedge d f_{n-p+1}\wedge\omega=0,$$ we get $$\label{orh}
\begin{split}
&\|
|d f_1\wedge\cdots\wedge d f_{n-p+1}|^2|\omega|^2
\|_1\\
=&
\||d f_1\wedge\cdots\wedge d f_{n-p+1}|^2|\omega|^2-
|d f_1\wedge\cdots\wedge d f_{n-p+1}\wedge\omega|^2
\|_1
\leq C\delta^{1/4}
\end{split}$$ similarly to (\[orb\]). By (\[q1k\]), we get $$\label{ori}
\begin{split}
&\left|
\sum_{i=1}^{n-p+1}(-1)^{i-1}d f_i\otimes d f_1\wedge\cdots\wedge \widehat{d f_i}\wedge \cdots\wedge d f_{n-p+1}
\right|^2\\
=&(n-p+1) |d f_1\wedge \cdots\wedge d f_{n-p+1}|^2.
\end{split}$$ By (\[orh\]) and (\[ori\]), we get $$\label{orj}
\left\|
\left|\sum_{i=1}^{n-p+1}(-1)^{i-1}d f_i\otimes d f_1\wedge\cdots\wedge \widehat{d f_i}\wedge \cdots\wedge d f_{n-p+1}\right|^2|\omega|^2
\right\|_1\leq C\delta^{1/4}.$$ By (\[org\]) and (\[orj\]), we have $$\label{ork}
\begin{split}
&\left\|
\sum_{i=1}^{n-p+1}(-1)^{i-1}d f_i\otimes d f_1\wedge\cdots\wedge \widehat{d f_i}\wedge \cdots\wedge d f_{n-p+1}\wedge\omega
\right\|_2^2\\
=&\left\|
\left|\sum_{i=1}^{n-p+1}(-1)^{i-1}d f_i\otimes d f_1\wedge\cdots\wedge \widehat{d f_i}\wedge \cdots\wedge d f_{n-p+1}\wedge\omega\right|^2
\right\|_1\leq C\delta^{1/4}.
\end{split}$$ By (\[orf\]) and (\[ork\]), we get $$\label{orl}
\|\nabla V\|_2\leq C\delta^{1/8}.$$ By (\[ore\]) and (\[orl\]), we get $$\lambda_1(\Delta_{C,n})\leq C\delta^{1/4},$$ and so we get the theorem by Claim \[porb\].
Combining Theorem \[MT2\] and Theorem \[pora\], we get Main Theorem 2.
Almost Parallel $(n-p)$-form II
-------------------------------
In this subsection, we show that the assumption “$\lambda_{n-p}(g)$ is close to $n-p$” implies the condition “$\lambda_{n-p+1}(g)$ is close to $n-p$” under the assumption $\lambda_1(\Delta_{C,n-p})\leq \delta$.
Note that our assumptions are Assumption \[asu1\].
\[pala\] Suppose that
- $f_i\in C^\infty(M)$ $(i\in\{1,\ldots,n-p\})$ is an eigenfunction of the Laplacian acting on functions with $\|f_i\|_2^2=1/(n-p+1)$ corresponding to the eigenvalue $\lambda_i$ with $0<\lambda_i\leq n-p+\delta$ such that $$\int_M f_i f_j\,d\mu_g=0$$ holds for any $i\neq j$,
- $\xi\in\Gamma(\bigwedge^{n-p} T^\ast M)$ is an eigenform of the connection Laplacian $\Delta_{C,n-p}$ with $\|\xi\|_2=1$ corresponding to the eigenvalue $\lambda$ with $0\leq \lambda \leq \delta$.
Put $$F:= \langle d f_1\wedge\ldots\wedge d f_{n-p}, \xi \rangle\in C^\infty(M).$$ Then, we have $$\left|\|F\|_2^2-\frac{1}{n-p+1}\right|\leq C\delta^{1/1600n},\quad \left|\|\nabla F\|_2^2-\frac{n-p}{n-p+1}\right|\leq C\delta^{1/1600n}$$ and $$\left|\frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)}\int_M f_i F\,d\mu_g\right|\leq C\delta^{1/2}$$ for all $i=1,\ldots, n-p$.
If $M$ is not orientable, we take the two-sheeted oriented Riemannian covering $\pi\colon (\widetilde{M},\tilde{g})\to (M,g)$, and put $$\widetilde{F}:=F\circ \pi,\quad \tilde{f}_i:=f_i\circ \pi.$$ Then, we have $
\|F\|_2=\|\widetilde{F}\|_2$, $\|\nabla F\|_2=\|\nabla \widetilde{F}\|_2,$ $$\frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(\widetilde{M})}\int_{\widetilde{M}} \tilde{f}_i \widetilde{F} \,d\mu_{\tilde{g}}=
\frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)}\int_M f_i F \,d\mu_g$$ and $$\widetilde{F}=\langle d \tilde{f}_1\wedge\ldots\wedge d \tilde{f}_{n-p}, \pi^\ast \xi\rangle.$$ Thus, it is enough to consider the case when $M$ is orientable. In the following, we assume that $M$ is orientable, and we fix an orientation of $M$.
Put $$\omega:=\ast \xi\in \Gamma(\bigwedge^p T^\ast M).$$ Let $V_g\in \Gamma(\bigwedge^n T^\ast M)$ be the volume form of $(M,g)$. Then, we have $$\label{ala}
F V_g= d f_1\wedge\cdots \wedge d f_{n-p}\wedge \omega.$$ Define a vector bundle $E:=T^\ast M\oplus \mathbb{R}e$, where $\mathbb{R}e$ denotes the trivial bundle of rank $1$ with a global non-vanishing section $e$. We consider an inner product $\langle\cdot,\cdot\rangle$ on $E$ defined by $$\langle \alpha+ f e,\beta +h e\rangle:=\langle\alpha,\beta\rangle+ fh$$ for all $\alpha,\beta\in\Gamma(T^\ast M)$ and $f,h\in C^\infty(M)$. Put $$S_i:=d f_i +f_i e\in \Gamma(E)$$ for each $i$, and $$\beta:=S_1\wedge\cdots \wedge S_{n-p}\in \Gamma(\bigwedge^{n-p} E).$$
Since we have $|F|=|F V_g|$, we get $$\||F|^2-|d f_1\wedge\cdots \wedge d f_{n-p} |^2|\omega|^2\|_1\leq C\delta^{1/4}$$ similarly to (\[orb\]) by (\[ala\]), and so $$\label{alb}
\left|\|F\|_2^2-\left\||d f_1\wedge\cdots \wedge d f_{n-p} |^2|\omega|^2\right\|_1\right|\leq C\delta^{1/4}$$
By Lemma \[pfua\] and (\[ala\]), we have $$\left\|
\nabla (F V_g)+\sum_{i=1}^{n-p} \sum_{k=1}^n(-1)^{i-1} f_i e^k\otimes e^k\wedge d f_1\wedge \cdots\wedge \widehat{d f_i}\wedge \cdots\wedge d f_{n-p}\wedge \omega
\right\|_2\leq C\delta^{1/8}.$$ Since $|\nabla(F V_g)|=|\nabla F|$, we get $$\label{alc}
\left|\|\nabla F\|_2^2-\left\|\left|\sum_{i=1}^{n-p} \sum_{k=1}^n(-1)^{i-1} f_i e^k\otimes e^k\wedge d f_1\wedge \cdots\wedge \widehat{d f_i}\wedge \cdots\wedge d f_{n-p}\wedge \omega\right|^2\right\|_1\right|\leq C\delta^{1/8}.$$ We have $$\label{ald}
\begin{split}
&\left|\sum_{i=1}^{n-p} \sum_{k=1}^n(-1)^{i-1} f_i e^k\otimes e^k\wedge d f_1\wedge \cdots\wedge \widehat{d f_i}\wedge \cdots\wedge d f_{n-p}\wedge \omega
\right|^2\\
=&\sum_{k=1}^n \left|e^k\wedge\left(\sum_{i=1}^{n-p} (-1)^{i-1} f_i d f_1\wedge \cdots\wedge \widehat{d f_i}\wedge \cdots\wedge d f_{n-p}\right)\wedge \omega\right|^2\\
=&\left|\sum_{i=1}^{n-p} (-1)^{i-1} f_i d f_1\wedge \cdots \wedge \widehat{d f_i}\wedge \cdots\wedge d f_{n-p}\wedge \omega\right|^2.
\end{split}$$ Similarly to (\[orb\]), we have $$\begin{split}
&\Bigg\|\left|\sum_{i=1}^{n-p} (-1)^{i-1} f_i d f_1\wedge \cdots \wedge\widehat{d f_i}\wedge \cdots\wedge d f_{n-p}\wedge \omega\right|^2\\
&\qquad -\left|\sum_{i=1}^{n-p} (-1)^{i-1} f_i d f_1\wedge \cdots\wedge \widehat{d f_i}\wedge \cdots\wedge d f_{n-p}\right|^2|\omega|^2\Bigg\|_1\leq C\delta^{1/4}.
\end{split}$$ Since we have $$\iota(e)\beta=\sum_{i=1}^{n-p} (-1)^{i-1} f_i d f_1\wedge \cdots\wedge \widehat{d f_i}\wedge \cdots\wedge d f_{n-p},$$ we get $$\label{alf}
\left\|\left|\sum_{i=1}^{n-p} (-1)^{i-1} f_i d f_1\wedge \cdots\wedge \widehat{d f_i}\wedge \cdots\wedge d f_{n-p}\wedge \omega\right|^2 -|\iota(e)\beta|^2|\omega|^2\right\|_1\leq C\delta^{1/4}.$$ By (\[alc\]), (\[ald\]) and (\[alf\]), we get $$\label{alf1}
\left|\|\nabla F\|_2^2-\left\||\iota(e)\beta|^2|\omega|^2\right\|_1\right|\leq C\delta^{1/8}.$$
We have $$\label{alg}
|\beta|^2=|d f_1\wedge\cdots\wedge d f_{n-p}|^2+|\iota(e)\beta|^2.$$ We calculate $\sum_{k=1}^n\left|e^k\wedge \beta\right|^2$ in two ways. We have $$\label{alh}
\begin{split}
\sum_{k=1}^n|e^k\wedge \beta|^2=&(p+1)|\beta|^2-|e\wedge\beta|^2\\
=&(p+1)|\beta|^2-|d f_1\wedge\cdots\wedge d f_{n-p}|^2= p|\beta|^2+|\iota(e)\beta|^2
\end{split}$$ by (\[alg\]). For all $\eta\in \Gamma(T^\ast M)$, we have $$\begin{aligned}
&|\eta\wedge\beta|^2\\
=&|\eta|^2|\beta|^2-\langle\iota(\eta)\beta,\iota(\eta)\beta\rangle\\
=&|\eta|^2|\beta|^2-\sum_{i,j=1}^{n-p}(-1)^{i+j}\langle \eta, d f_i\rangle\langle\eta, d f_j\rangle
\langle S_1\wedge\cdots\wedge \widehat{S_i}\wedge\cdots \wedge S_{n-p},S_1\wedge\cdots\wedge \widehat{S_j}\wedge\cdots \wedge S_{n-p}
\rangle,\end{aligned}$$ and so we get $$\label{ali}
\begin{split}
&\sum_{k=1}^n|e^k\wedge \beta|^2\\
=&n|\beta|^2-\sum_{i,j=1}^{n-p}(-1)^{i+j}\langle d f_i,d f_j\rangle
\langle S_1\wedge\cdots\wedge \widehat{S_i}\wedge\cdots \wedge S_{n-p},S_1\wedge\cdots\wedge \widehat{S_j}\wedge\cdots \wedge S_{n-p}\rangle.
\end{split}$$ By (\[alh\]) and (\[ali\]), we get $$\label{alj}
\begin{split}
&|\iota(e)\beta|^2\\
=&(n-p)|\beta|^2-\sum_{i,j=1}^{n-p}(-1)^{i+j}\langle d f_i,d f_j\rangle
\langle S_1\wedge\cdots\wedge \widehat{S_i}\wedge\cdots \wedge S_{n-p},S_1\wedge\cdots\wedge \widehat{S_j}\wedge\cdots \wedge S_{n-p}\rangle
\end{split}$$
Since we have $|\langle S_i,S_j\rangle(x)-\delta_{i j}|\leq C\delta^{1/1600n}$ for all $x\in G=G(f_1,\ldots, f_{n-p})$ by Lemma \[pfub\] (ii), we have $$\label{alk}
\begin{split}
&\Bigg\|\sum_{i=1}^{n-p}|d f_i|^2\\
&-\sum_{i,j=1}^{n-p}(-1)^{i+j}\langle d f_i,d f_j\rangle
\langle S_1\wedge\cdots\wedge \widehat{S_i}\wedge\cdots \wedge S_{n-p},S_1\wedge\cdots\wedge \widehat{S_j}\wedge\cdots \wedge S_{n-p}\rangle|\omega|^2
\Bigg\|_1
\leq C\delta^{1/1600n}
\end{split}$$ and $$\label{all}
\left|\left\||\beta|^2|\omega|^2\right\|_1-1\right|\leq C\delta^{1/1600n}$$ by Lemma \[p4c\] and Lemma \[pfub\] (i). By the assumption, we have $$\label{alm}
\left|\sum_{i=1}^{n-p}\|d f_i\|_2^2-\frac{(n-p)^2}{n-p+1}\right|\leq C\delta^{1/2}.$$ By (\[alj\]), (\[alk\]), (\[all\]) and (\[alm\]), we get $$\label{aln}
\left|\left\||\iota(e)\beta|^2|\omega|^2\right\|_1-\frac{n-p}{n-p+1}\right|\leq C\delta^{1/1600n},$$ and so $$\label{alo}
\left|\left\||d f_1\wedge\cdots\wedge d f_{n-p}|^2|\omega|^2\right\|_1-\frac{1}{n-p+1}\right|\leq C\delta^{1/1600n}$$ by (\[alg\]) and (\[all\]). By (\[alb\]) and (\[alo\]), we get $$\left|\|F\|_2^2-\frac{1}{n-p+1}\right|\leq C\delta^{1/1600n}.$$ By (\[alf1\]) and (\[aln\]), we get $$\left|\|\nabla F\|_2^2- \frac{n-p}{n-p+1}\right| \leq C\delta^{1/1600n}.$$
Let us show the remaining assertion. Since we have $$\begin{aligned}
f_i F V_g=&\frac{1}{2}(-1)^{i-1} d \left(f_i^2 d f_1\wedge\cdots\wedge\widehat{d f_i}\wedge \cdots \wedge d f_{n-p}\wedge\omega\right)\\
-&\frac{1}{2}(-1)^{i-1} (-1)^{n-p-1}f_i^2 d f_1\wedge\cdots\wedge\widehat{d f_i}\wedge\cdots \wedge d f_{n-p}\wedge d \omega,\end{aligned}$$ we get $$\left|\frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)}\int_M f_i F\,d\mu_g\right|\leq
C\|\nabla \omega\|_2 \leq C\delta^{1/2}$$ by the Stokes theorem.
By applying the Rayleigh principle $$\begin{aligned}
&\lambda_{n-p+1}(g)\\
=&\inf\left\{\sup_{f\in V\setminus\{0\}}\frac{\|\nabla f\|_2^2}{\|f\|_2^2}: V\text{ is an $(n-p+1)$-dimensional subspace of } C^\infty (M)
\right\}\end{aligned}$$ to the subspace ${\mathop{\mathrm{Span}}\nolimits}_{\mathbb{R}}\{f_1,\ldots, f_{n-p}, F\}$, we immediately get the following corollary:
\[palb\] If $$\lambda_{n-p}(g)\leq n-p+\delta$$ and $$\lambda_1(\Delta_{C,n-p})\leq \delta$$ hold, then we have $$\lambda_{n-p+1}(g)\leq n-p+C\delta^{1/1600n}.$$
Combining Theorem \[MT2\] and Corollary \[palb\], we get Main Theorem 4.
Finally, we investigate the Gromov-Hausdorff limit of the sequence of the Riemannian manifolds that satisfy our pinching condition.
Take $n\geq 5$ and $2\leq p < n/2$. Let $\{(M_i,g_i)\}_{i\in\mathbb{N}}$ be a sequence of $n$-dimensional closed Riemannian manifolds with ${\mathop{\mathrm{Ric}}\nolimits}_{g_i}\geq (n-p-1)g_i$ that satisfies one of the following:
- $\lim_{i\to\infty}\lambda_{n-p+1}(g_i)=n-p$ and $\lim_{i\to \infty}\lambda_1(\Delta_{C,p},g_i)=0$,
- $M_i$ is orientable for each $i$, $\lim_{i\to\infty}\lambda_{n-p}(g_i)=n-p$ and $\lim_{i\to \infty}\lambda_1(\Delta_{C,p},g_i)=0$,
- $\lim_{i\to\infty}\lambda_{n-p}(g_i)=n-p$ and $\lim_{i\to \infty}\lambda_1(\Delta_{C,n-p},g_i)=0$.
If $\{(M_i,g_i)\}_{i\in\mathbb{N}}$ converges to a geodesic space $X$, then there exists a geodesic space $Y$ such that $X$ is isometric to $S^{n-p}\times Y$.
By Main Theorem 2 and Main Theorem 4, we get that there exist a sequence of positive real numbers $\{\epsilon_i\}$ and compact metric spaces $\{Y_i\}$ such that $\lim_{i\to \infty}\epsilon_i=0$ and $d_{GH}(M_i,S^{n-p}\times Y_i)\leq \epsilon_i$. Then, $\{S^{n-p}\times Y_i\}$ converges to $X$ in the Gromov-Hausdorff topology, and so $\{Y_i\}$ is pre-compact in the Gromov-Hausdorff topology by [@Pe3 Theorem 11.1.10]. Thus, there exists a subsequence that converges to some compact metric space $Y$. Therefore, we get that $X$ is isometric to $S^{n-p}\times Y$. Since $X$ is a geodesic space, $Y$ is also a geodesic space.
Limit Spaces and Unorientability
================================
In this appendix, we generalize our main theorems to certain limit spaces. We first give the proof of Claim \[porb\] in the following form.
\[papea\] Let $(M,g)$ be an $n$-dimensional unorientable closed Riemannian manifold. Define $$\lambda_1(\Delta_{C,n},g):=\inf \left\{\frac{\|\nabla \eta\|_2^2}{\|\eta\|_2^2}: \eta\in \Gamma(\bigwedge^n T^\ast M)\text{ with } \eta\neq 0\right\}.$$ Then, we have the following:
- If ${\mathop{\mathrm{Ric}}\nolimits}\geq (n-1)g$, then we have $$\lambda_1(\Delta_{C,n},g)\geq n.$$
- If ${\mathop{\mathrm{Ric}}\nolimits}\geq -K g$ and ${\mathop{\mathrm{diam}}\nolimits}(M)\leq D$ $(K,D>0)$, then we have $$\lambda_1(\Delta_{C,n},g)\geq C_1(n,K,2D),$$ where $C_1(n,K,D)$ is defined by $$C_1(n,K,D):=\frac{1}{(n-1)D^2\exp\left(1+\sqrt{1+4(n-1)KD^2}\right)}.$$
Take the two-sheeted oriented Riemannian covering $\pi\colon (\widetilde{M},\tilde{g})\to (M,g)$. Let us show that $$\lambda_1(\Delta_{C,n},g)\geq \lambda_1(\tilde{g}).$$ If we succeed in proving this inequality, we get (i) by the Lichnerowicz inequality, and we get (ii) by ${\mathop{\mathrm{diam}}\nolimits}(\widetilde{M})\leq 2 {\mathop{\mathrm{diam}}\nolimits}(M)$ and the Li-Yau estimate [@SY p.116], which asserts that $$\lambda_1(g_1)\geq C_1(n,K,D)$$ holds for any $n$-dimensional closed Riemannian manifold $(N_1,g_1)$ with ${\mathop{\mathrm{Ric}}\nolimits}_{g_1}\geq -K g_1$ and ${\mathop{\mathrm{diam}}\nolimits}(N_1)\leq D$.
Take the first eigenform $\omega\in\Gamma(\bigwedge^n T^\ast M)$ of the connection Laplacian $\Delta_{C,n}$. Put $$\widetilde{\omega}:=\pi^\ast \omega \in\Gamma(\bigwedge^n T^\ast \widetilde{M}).$$ Then, we have $$\Delta_{C,n}\widetilde{\omega}=\lambda_1(\Delta_{C,n},g)\widetilde{\omega}.$$ By the correspondence through the Hodge star operator, we have that $\lambda_1(\Delta_{C,n},g)$ is the eigenvalue of the Laplacian acting on functions on $\widetilde{M}$. If $\lambda_1(\Delta_{C,n},g)=0$, then $\omega$ defines an orientation of $M$, and so we have $$\lambda_1(\Delta_{C,n},g)\neq 0=\lambda_0(\tilde{g}).$$ Thus, we get $$\lambda_1(\Delta_{C,n},g)\geq\lambda_1(\tilde{g}).$$ Therefore, we get the lemma.
We immediately get the following corollary.
\[papeb\] Let $(M,g)$ be an $n$-dimensional closed Riemannian manifold. If one of the following properties holds, then $M$ is orientable.
- ${\mathop{\mathrm{Ric}}\nolimits}\geq (n-1)g$ and $$\lambda_1(\Delta_{C,n},g)< n.$$
- ${\mathop{\mathrm{Ric}}\nolimits}\geq -K g$, ${\mathop{\mathrm{diam}}\nolimits}(M)\leq D$ $(K,D>0)$ and $$\lambda_1(\Delta_{C,n},g)< C_1(n,K,2D).$$
\[papec\] For any closed Riemannian manifold $(M,g)$ and a positive real number $a>0$, we have ${\mathop{\mathrm{Ric}}\nolimits}_{g}={\mathop{\mathrm{Ric}}\nolimits}_{a g}$ and $$\lambda_k(a g)=\frac{1}{a}\lambda_k(g)$$ for all $k$. Thus, we get Claim \[porb\] by Corollary \[papeb\].
As an application of Lemma \[papea\], we show the stability of unorientability under the non-collapsing Gromov-Hausdorff convergence assuming the two-sided bound on the Ricci curvature.
\[paped\] Take real numbers $K_1,K_2\in\mathbb{R}$ and positive real numbers $D>0$ and $v>0$. Let $\{(M_i,g_i)\}$ be a sequence of $n$-dimensional unorientable closed Riemannian manifolds with $K_1 g_i\leq {\mathop{\mathrm{Ric}}\nolimits}_{g_i}\leq K_2 g_i$, ${\mathop{\mathrm{diam}}\nolimits}(M)\leq D$ and ${\mathop{\mathrm{Vol}}\nolimits}(M)\geq v$. Suppose that $\{(M_i,g_i)\}$ converges to a limit space $X$ in the Gromov-Hausdorff sense. Then, $X$ is not orientable in the sense of Honda [@Hoor] $($see also the definition below$)$.
Note that Honda [@Hoor Theorem 1.3] showed the stability of orientability without assuming the upper bound on the Ricci curvature.
Before proving Theorem \[paped\], we fix our notation and recall definitions about limit spaces.
Take real numbers $K_1,K_2\in\mathbb{R}$ and positive real numbers $D>0$ and $v>0$.
- Let $\mathcal{M}_1=\mathcal{M}_1(n,K_1,K_2,v)$ be the set of isometry classes of $n$-dimensional closed Riemannian manifolds $(M,g)$ with $K_1g\leq {\mathop{\mathrm{Ric}}\nolimits}_g \leq K_2 g$ and ${\mathop{\mathrm{Vol}}\nolimits}(M)\geq v$. Let $\overline{\mathcal{M}}_1=\overline{\mathcal{M}}_1(n,K_1,K_2,v)$ be the closure of $\mathcal{M}$ in the Gromov-Hausdorff topology.
- Let $\mathcal{M}_2=\mathcal{M}_2(n,K_1,K_2,D,v)$ be the set of isometry classes of $n$-dimensional closed Riemannian manifolds $(M,g)$ with $K_1g\leq {\mathop{\mathrm{Ric}}\nolimits}_g \leq K_2 g$, ${\mathop{\mathrm{diam}}\nolimits}(M)\leq D$ and ${\mathop{\mathrm{Vol}}\nolimits}(M)\geq v$. Let $\overline{\mathcal{M}}_2=\overline{\mathcal{M}}_2(n,K_1,K_2,D,v)$ be the closure of $\mathcal{M}$ in the Gromov-Hausdorff topology.
By the Myers theorem, we have $$\begin{aligned}
\mathcal{M}_1(n,K_1,K_2,v)\subset &\mathcal{M}_2(n,K_1,K_2,\sqrt{(n-1)/K_1}\pi,v),\\
\overline{\mathcal{M}}_1(n,K_1,K_2,v)\subset&\overline{\mathcal{M}}_2(n,K_1,K_2,\sqrt{(n-1)/K_1}\pi,v)\end{aligned}$$ if $K_1>0$.
If $X_i\in\overline{\mathcal{M}}_2$ ($i\in \mathbb{N}$) converges to $X\in\overline{\mathcal{M}}_2$ in the Gromov-Hausdorff topology, then there exist a sequence of positive real numbers $\{\epsilon_i\}_{i\in \mathbb{N}}$ with $\lim_{i\to \infty}\epsilon_i=0$, and a sequence of $\epsilon_i$-Hausdorff approximation maps $\phi_i \colon X_i\to X$. Fix such a sequence. We say a sequence $x_i\in X_i$ converges to $x\in X$ if $\lim_{i\to \infty}\phi_i(x_i)=x$ (denote it by $x_i\stackrel{GH}{\to} x$). By the volume convergence theorem [@CC1 Theorem 5.9], $(X_i,H^n)$ converges to $(X,H^n)$ in the measured Gromov-Hausdorff sense, i.e., for all $r>0$ and all sequence $x_i\in X_i$ that converges to $x\in X$, we have $\lim_{i\to \infty}H^n(B_r (x_i))=H^n(B_r(x))$, where $H^n$ denotes the $n$-dimensional Hausdorff measure.
For all $X\in \overline{\mathcal{M}}_2$, we can consider the cotangent bundle $\pi \colon T^\ast X \to X$ with a canonical inner product by [@Ch0] and [@CC3] (see also [@Ho1 Section 2] for a short review). We have $H^n(X\setminus \pi(T^\ast X))=0$ and $T^\ast_x X:=\pi^{-1}(x)$ is an $n$-dimensional vector space for all $x\in \pi(T^\ast X)$. For all Lipschitz function $f$ on $X$, we can define $d f(x)\in T_x^\ast X$ for almost all $x\in X$, and we have $d f\in L^\infty(T^\ast X)$.
Let us recall definitions of functional spaces on limit spaces. Note that we can define such functional spaces on more general spaces than our assumption. Some of the following functional spaces are first introduced by Gigli [@Gig].
Let $X\in \overline{\mathcal{M}}_2$.
- Let ${\mathop{\mathrm{LIP}}\nolimits}(X)$ be the set of the Lipschitz functions on $X$. For all $f\in {\mathop{\mathrm{LIP}}\nolimits}(X)$, we define $\|f\|_{H^{1,2}}^2=\|f\|_2^2+\|d f\|_2^2$. Let $H^{1,2}(X)$ be the completion of ${\mathop{\mathrm{LIP}}\nolimits}(X)$ with respect to this norm.
- Define $$\begin{split}
\mathcal{D}^2(\Delta,X):=\Big\{f\in H^{1,2}(X)& : \text{there exists $F\in L^2(X)$ such that}\\
&\int_X \langle df, dh \rangle\,d H^n=\int_X F h\,d H^n \text{ for all $h\in H^{1,2}(X)$} \Big\}.
\end{split}$$ For any $f\in\mathcal{D}^2(\Delta,X)$, the function $F\in L^2(X)$ is uniquely determined. Thus, we define $\Delta f:=F$.
- Define $$\begin{split}
{\mathop{\mathrm{Test}}\nolimits}F(X):=&\left\{f\in\mathcal{D}^2(\Delta,X)\cap {\mathop{\mathrm{LIP}}\nolimits}(X):\Delta f\in H^{1,2}(X)\right\},\\
{\mathop{\mathrm{TestForm}}\nolimits}_p(X):=&\left\{\sum_{i=1}^N f_{0,i} d f_{1,i}\wedge\ldots\wedge d f_{p,i}: N\in \mathbb{N},\, f_{j,i}\in {\mathop{\mathrm{Test}}\nolimits}F(X)\right\}
\end{split}$$ for all $p\in\{1,\ldots,n\}$.
- The operator $$\nabla\colon {\mathop{\mathrm{TestForm}}\nolimits}_p(X)\to L^2(T^\ast X \otimes \bigwedge^p T^\ast X)$$ is defined by $$\begin{aligned}
&\nabla\sum_{i=1}^N f_{0,i} d f_{1,i}\wedge\ldots\wedge d f_{p,i}\\
:=&\sum_{i=1}^N \left(d f_{0,i}\otimes d f_{1,i}\wedge\ldots\wedge d f_{p,i}+ \sum_{j=1}^p f_{0,i} d f_{1,i}\wedge\ldots\wedge\nabla^2 f_{j,i}\wedge\ldots\wedge d f_{p,i}\right),\end{aligned}$$ where $\nabla^2$ denotes the Hessian ${\mathop{\mathrm{Hess}}\nolimits}$ defined in [@Gig Definition 3.3.1] or [@Ho0]. Gigli defined the functional space $W^{2,2}(X)\subset H^{1,2}(X)$, on which we can define the Hessian as an $L^2$-tensor, and showed that $\mathcal{D}(\Delta,X)\subset W^{2,2}(X)$. Honda showed that for any $f\in \mathcal{D}^2(\Delta,X)$, $f$ is weakly twice differentiable [@Ho3 Theorem 1.9] in the sense of [@Ho0], that we can define the Hessian ${\mathop{\mathrm{Hess}}\nolimits}f$ using the Levi-Civita connection defined in [@Ho0], and that ${\mathop{\mathrm{Hess}}\nolimits}f\in L^2(T^\ast X\otimes T^\ast X)$ [@Ho3 Theorem 4.11]. Moreover, Honda showed that his definition of the Hessian coincides with Gigli’s one [@Ho3 Theorem 1.9].
- For any $\omega\in {\mathop{\mathrm{TestForm}}\nolimits}_p(X)$, we define $\|\omega\|_{H_C^{1,2}}^2:=\|\omega\|_2^2+\|\nabla \omega\|_2^2$. Let $H^{1,2}_C(\bigwedge^p T^\ast X)$ be the completion of ${\mathop{\mathrm{TestForm}}\nolimits}_p (X)$ with respect to this norm.
- Define $$\begin{split}
\mathcal{D}^2(\Delta_{C,p},X):=\Big\{\omega \in& H^{1,2}_C(\bigwedge^p T^\ast X) : \text{there exists $\hat{\omega}\in L^2(\bigwedge^p T^\ast X)$ such that}\\
&\int_X \langle \nabla \omega, \nabla \eta \rangle\,d H^n=\int_X \langle\hat{\omega}, \eta\rangle \,d H^n \text{ for all $\eta\in H_C^{1,2}(\bigwedge^p T^\ast X)$} \Big\}.
\end{split}$$ For any $\omega\in\mathcal{D}^2(\Delta_{C,p},X)$, the form $\hat{\omega}\in L^2(\bigwedge^p T^\ast X)$ is uniquely determined. Thus, we put $\Delta_{C,p} \omega:=\hat{\omega}$.
- For all $k\in \mathbb{Z}_{>0}$, we define $$\begin{split}
\lambda_k(\Delta_{C,p},X):=\inf\left\{\sup_{\omega\in \mathcal{E}_k\setminus \{0\}}\frac{\|\nabla \omega\|^2_2}{\|\omega\|^2_2}: \mathcal{E}_k\subset H^{1,2}_C(\bigwedge^p T^\ast X)\text{ is a $k$-dimensional subspace}\right\}.
\end{split}$$
Similarly to the smooth case, there exists a complete orthonormal system of eigenforms of the connection Laplacian $\Delta_{C,p}$ in $L^2(\bigwedge^p T^\ast M)$, and each eigenform is an element of $\mathcal{D}^2(\Delta_{C,p},X)$ (see [@Ho2 Theorem 4.17]).
Honda [@Ho2] showed the following theorem:
\[papee\] Let $\{X_i\}_{i\in \mathbb{N}}$ be a sequence in $\overline{\mathcal{M}}_2$ and let $X\in\overline{\mathcal{M}}_2$ be its Gromov-Hausdorff limit. Then, we have $$\lim_{i\to \infty}\lambda_k(\Delta_{C,p},X_i)=\lambda_k(\Delta_{C,p},X)$$ for all $p\in\{0,\ldots,n\}$ and $k\in\mathbb{Z}_{>0}$.
Let $X\in \overline{\mathcal{M}}_2$. We say that $X$ is orientable if there exists $\omega\in L^\infty(\bigwedge^n T^\ast X)$ such that $|\omega|(z)=1$ for almost all $z\in X$ and that $$\langle\omega,\eta\rangle\in H^{1,2}(X)$$ for any $\eta \in {\mathop{\mathrm{TestForm}}\nolimits}_n(X)$. Then, we call $\omega$ an orientation of $X$.
\[papef\] Let $X\in \overline{\mathcal{M}}_2$. Then, $X$ is orientable if and only if $\lambda_1(\Delta_{C,n},X)=0$.
We first suppose that $X$ is orientable and show $\lambda_1(\Delta_{C,n},X)=0$. Let $\omega\in L^\infty(\bigwedge^n T^\ast X)$ be the orientation of $X$. By [@Hoor Proposition 6.5], for almost all $z\in X$, $\omega$ is differentiable at $z$ and $\nabla^{g_X}\omega(z)=0$, where $\nabla^{g_X}$ denotes the Levi-Civita connection defined in [@Ho0]. By Proposition 4.5 and Remark 4.7 in [@Ho2], we have $\omega\in H^{1,2}_C(\bigwedge^p T^\ast X)$. By [@Ho3 Corollary 7.10], we have $\nabla \omega(z)=\nabla^{g_X}\omega(z)=0$ for almost all $z\in X$. Thus, we get $$\lambda_1(\Delta_{C,n},X)=0$$ by the definition of $\lambda_1(\Delta_{C,n},X)$.
We next suppose $\lambda_1(\Delta_{C,n},X)=0$ and show that $X$ is orientable. Let $\{(M_i,g_i)\}_{i\in \mathbb{N}}$ be a sequence in $\mathcal{M}_2$ that converges to $X$ in the Gromov-Hausdorff topology. Then, we have $\lim_{i\to \infty}\lambda_1(\Delta_{C,n},g_i)=0$ by Theorem \[papee\]. Thus, by Corollary \[papeb\], we get that $M_i$ is orientable for sufficiently large $i$, and so $X$ is orientable by the stability of orientability [@Hoor Theorem 1.3].
Let $\{(M_i,g_i)\}_{i\in \mathbb{N}}$ be a sequence in $\mathcal{M}_2$ and let $X$ be its Gromov-Hausdorff limit. Suppose that each $M_i$ is not orientable. Then, we have $$\lambda_1(\Delta_{C,n},g_i)\geq C_1(n,K_1,2D)$$ by Lemma \[papea\]. By Theorem \[papee\], we get $$\lambda_1(\Delta_{C,n},X)\geq C_1(n,K_1,2D)>0.$$ Thus, by Lemma \[papef\], we get the theorem.
\[papeg\] Let $X\in \overline{\mathcal{M}}_2$. If $X$ is not orientable, then we have $$\lambda_1(\Delta_{C,n},X)\geq C_1(n,K_1,2D).$$
Let $\{(M_i,g_i)\}_{i\in \mathbb{N}}$ be a sequence in $\mathcal{M}_2$ that converges to $X$ in the Gromov-Hausdorff topology. By Lemma \[papef\], we have $\lambda_1(\Delta_{C,n},X)>0$, and so we get $$\lambda_1(\Delta_{C,n},g_i)>0$$ for sufficiently large $i$ by Theorem \[papee\]. Thus, $M_i$ is not orientable and $$\lambda_1(\Delta_{C,n},g_i)\geq C_1(n,K_1,2D)$$ for sufficiently large $i$ by Lemma \[papea\]. By Theorem \[papee\], we get the theorem.
We immediately get the following corollaries:
\[papeh\] Let $\{X_i\}_{i\in \mathbb{N}}$ be a sequence in $\overline{\mathcal{M}}_2$ and let $X\in\overline{\mathcal{M}}_2$ be its Gromov-Hausdorff limit. If $X_i$ is not orientable for each $i$, then $X$ is not orientable.
\[papei\] Let $\{X_i\}_{i\in \mathbb{N}}$ be a sequence in $\overline{\mathcal{M}}_2$ and let $X\in\overline{\mathcal{M}}_2$ be its Gromov-Hausdorff limit. Then, the following two conditions are mutually equivalent.
- $X_i$ is orientable for sufficiently large $i$.
- $X$ is orientable.
By Corollary \[papei\], we have that if $X_1\in\overline{\mathcal{M}}_2$ is orientable and $X_2\in\overline{\mathcal{M}}_2$ is unorientable, then $X_1$ and $X_2$ belong to different connected components in $\overline{\mathcal{M}}_2$ with respect to the Gromov-Hausdorff topology.
Now, we generalize our main theorems to Ricci limit spaces. We get the following theorem by Main Theorem 1 and Theorem \[papee\].
\[papej\] For given integers $n\geq 4$ and $2\leq p \leq n/2$ and positive real numbers $K>n-p-1$ and $v>0$, there exists a constant $C(n,p)>0$ such that $$\lambda_1(X)\geq n-p-C(n,p)\lambda_1(\Delta_{C,p},X)^{1/2}$$ holds for all $X\in \overline{\mathcal{M}}_1(n,n-p-1,K,v)$.
We get the following theorem by Main Theorem 2, Main Theorem 4, Theorem \[papee\] and Corollary \[papei\].
\[papek\] For given integers $n\geq 5$ and $2\leq p < n/2$ and positive real numbers $\epsilon>0$, $K>n-p-1$ and $v>0$, there exists a constant $\delta=\delta(n,p,\epsilon)>0$ such that if $X\in \overline{\mathcal{M}}_1(n,n-p-1,K,v)$ satisfies one of
- $\lambda_{n-p+1}(X)\leq n-p+\delta$ and $\lambda_1(\Delta_{C,p},X)\leq \delta$,
- $X$ is orientable, $\lambda_{n-p}(X)\leq n-p+\delta$ and $\lambda_1(\Delta_{C,p},X)\leq \delta$,
- $\lambda_{n-p}(X)\leq n-p+\delta$ and $\lambda_1(\Delta_{C,n-p},X)\leq \delta$,
then there exists a geodesic space $Y$ such that $d_{GH}(X,S^{n-p}\times Y)\leq \epsilon$ holds. Moreover, if $(i)$ holds, then $X$ is orientable.
Eigenvalue Estimate for $L^2$ Almost Kähler Manifolds
=====================================================
In this section, we consider $L^2$ almost Kähler manifolds, i.e., we assume that there exists a $2$-form $\omega$ which satisfies that $\|\nabla \omega\|_2$ and $\|J_\omega^2+{\mathop{\mathrm{Id}}\nolimits}\|_1$ are small, where $J_\omega\in\Gamma(T^\ast M\otimes T M)$ is defined so that $\omega=g(J_\omega\cdot,\cdot)$. The main goal is to give the almost version of (\[kae\]).
Let $(M,g)$ be a Riemannian manifold. For each $2$-form $\omega\in \Gamma(\bigwedge^2 T^\ast M)$, let $J_\omega\in\Gamma(T^\ast M\otimes T M)$ denotes the anti-symmetric tensor that satisfies $\omega=g(J_\omega\cdot,\cdot)$.
We first show the following easy lemmas.
\[pB2\] Let $(M,g)$ be an $n$-dimensional closed Riemannian manifold. If there exists a $2$-form $\omega$ such that $\|J_\omega^2+{\mathop{\mathrm{Id}}\nolimits}\|_1<1$ holds, then $n$ is an even integer.
There exists a point $x\in M$ such that $|J_\omega^2(x)+{\mathop{\mathrm{Id}}\nolimits}_{T_x M}|<1$. For any $v\in T_x M$ with $|v|=1$, we have $|J_\omega^2(x)(v)+v|<1$, and so $|J_\omega^2(x)(v)|>0$. Thus, $J_\omega(x)$ is non-degenerate. Therefore, $(T_x M,\omega_x)$ is a symplectic vector space. This implies the lemma.
\[pB3\] Given integers $n\geq 2$, $1\leq p\leq n-1$, and positive real numbers $K>0$, $D>0$, there exists $\delta_0(n,p,K,D)>0$ such that if $(M,g)$ is an $n$-dimensional closed Riemannian manifold with ${\mathop{\mathrm{Ric}}\nolimits}\geq-K g$ and ${\mathop{\mathrm{diam}}\nolimits}(M)\leq D$, then we have $$\lambda_{\alpha(n,p)+1}(\Delta_{C,p})\geq \delta_0(n,p,K,D),$$ where we defined $$\alpha(n,p):=\binom{n}{p}=\frac{n!}{p!(n-p)!}.$$
Put $\delta:=\lambda_{\alpha(n,p)+1}(\Delta_{C,p})$. If $\delta\geq 1$, we get the lemma. Thus, we assume that $\delta<1$. Let $\omega_i\in\Gamma(\bigwedge^p T^\ast M)$ denotes the $i$-th eigenform of the connection Laplacian $\Delta_{C,p}$ acting on $p$-forms with $\|\omega_i\|_2=1$.
We have $$\label{l2es}
\|\langle \omega_i,\omega_j\rangle\|_2^2\leq \frac{1}{\lambda_1(g)}\|\nabla\langle \omega_i,\omega_j\rangle\|_2^2\leq C(n,p,K,D)\delta$$ for each $i,j=1,\ldots, \alpha(n,p)+1$ with $i\neq j$ by the Li-Yau estimate [@SY p.116] and Lemma \[Linfes\]. By Lemma \[p4c\] and (\[l2es\]), we have $$\begin{aligned}
\|\langle \omega_i,\omega_j\rangle\|_1&\leq C(n,p,K,D)\delta^{1/2} \quad (i,j=1,\ldots, \alpha(n,p)+1 \text{ with } i\neq j),\\
\||\omega_i|^2-1\|_1&\leq C(n,p,K,D)\delta^{1/2} \quad (i=1,\ldots, \alpha(n,p)+1).\end{aligned}$$ Put $$\begin{aligned}
G:=\Big\{x\in M: & ||\omega_i|^2-1|(x)\leq\delta^{1/4}\text{ for all $i=1,\ldots, \alpha(n,p)+1$, and}\\
&\left|\langle \omega_i,\omega_j\rangle\right|(x)\leq \delta^{1/4}\text{ for all $i,j=1,\ldots, \alpha(n,p)+1$ with $i\neq j$}
\Big\}.\end{aligned}$$ Then, we have ${\mathop{\mathrm{Vol}}\nolimits}(M\backslash G)\leq C_1(n,p,K,D)\delta^{1/4}{\mathop{\mathrm{Vol}}\nolimits}(M)$ for some positive constant $C_1(n,p,K,D)$ depending only on $n,p,K$ and $D$ similarly to Lemma \[pfub\].
Let us show $\delta\geq \min\left\{1/C_1(n,p,K,D)^4,1/(\alpha(n,p)+1)^4\right\}$ by contradiction. Suppose that that $\delta< \min\left\{1/C_1(n,p,K,D)^4,1/(\alpha(n,p)+1)^4\right\}$. Then, we have $G\neq \emptyset$, and so we can take a point $x_0\in G$. We show that $\omega_{1}(x_0),\ldots, \omega_{\alpha(n,p)+1}(x_0)\in \bigwedge^p T_{x_0}^\ast M$ are linearly independent. Take arbitrary $a_1,\ldots, a_k\in \mathbb{R}$ with $a_1 \omega_1(x_0)+\cdots+a_k \omega_{\alpha(n,p)+1}(x_0)=0$. Take $i$ with $|a_i|=\max\{|a_1|,\ldots,|a_k|\}$. Since we have $\langle a_1 \omega_1(x_0)+\cdots+a_k \omega_{\alpha(n,p)+1}(x_0),\omega_i(x_0)\rangle=0$, we get $$\begin{split}
0\geq |a_i||\omega_i(x_0)|^2-\sum_{i\neq j}\left|a_j\langle \omega_i(x_0), \omega_j(x_0)\rangle\right|
\geq& |a_i|(1-\delta^{1/4})-\sum_{i\neq j}|a_i|\delta^{1/4}\\
\geq& |a_i|\left(1-(\alpha(n,p)+1)\delta^{1/4}\right).
\end{split}$$ Thus, $|a_i|=0$, and so $a_1=\cdots=a_k=0$. This implies the linearly independence of $\omega_{1}(x_0),\ldots, \omega_{\alpha(n,p)+1}(x_0)$. This contradicts to $\dim\left(\bigwedge^p T^\ast_{x_0} M\right)=\alpha(n,p)$. Thus, we get $\lambda_{\alpha(n,p)+1}(\Delta_{C,p})=\delta\geq \min\left\{1/C_1(n,p,K,D)^4,1/(\alpha(n,p)+1)^4\right\}$.
\[pB4\] Let $(M,g)$ be an $n$-dimensional closed Riemannian manifold. Suppose that a $2$-form $\omega$ satisfies
- $\|\nabla \omega\|_2^2\leq \delta\|\omega\|_2^2$,
- $\|J_\omega^2+{\mathop{\mathrm{Id}}\nolimits}\|_1\leq \delta^{1/4}\|\omega\|_2^2$
for some $0<\delta \leq 1/4$. Let $\omega_\alpha$ be its image of the orthogonal projection $$P_{\delta}:L^2\left(\bigwedge^2 T^\ast M \right)\to \bigoplus_{\lambda_i(\Delta_{C,2})\leq \delta^{1/2}} \mathbb{R}\omega_i,$$ where $\omega_i$ denotes the $i$-th eigenform of the connection Laplacian $\Delta_{C,2}$ with $\|\omega_i\|_2=1$ $(\omega_\alpha :=P_{\delta} (\omega))$. Then, we have
- $\|\nabla \omega_\alpha\|_2^2\leq 2\delta\|\omega_\alpha\|_2^2$,
- $\|J_{\omega_\alpha}^2+{\mathop{\mathrm{Id}}\nolimits}\|_1\leq 10\delta^{1/4}\|\omega_\alpha\|_2^2$.
Put $\omega_\beta:=\omega-\omega_\alpha$. Then, we have $\|\omega\|^2_2=\|\omega_\alpha\|^2_2+\|\omega_\beta\|^2_2$. By the assumption (i), we have $$\begin{aligned}
\delta \|\omega\|_2^2
\geq \|\nabla\omega\|_2^2
=\|\nabla \omega_\alpha\|_2^2+\|\nabla \omega_\beta\|_2^2
\geq \|\nabla \omega_\alpha\|_2^2+\delta^{1/2}\|\omega_\beta\|_2^2.\end{aligned}$$ Thus, we get $$\begin{aligned}
\label{AB1}\|\nabla \omega_\alpha\|_2^2&\leq \delta \|\omega\|_2^2,\\
\label{AB2}\|\omega_\beta\|_2^2&\leq \delta^{1/2}\|\omega\|_2^2,\end{aligned}$$ and so $$\label{AB3}
\|\omega_\alpha\|_2^2=\|\omega\|^2_2 -\|\omega_\beta\|^2_2\geq (1-\delta^{1/2})\|\omega\|_2^2\geq \frac{1}{2}\|\omega\|_2^2.$$ By the definitions of the norms, we have $|J_\omega|^2=2|\omega|^2$ and $|J_{\omega_\alpha}|^2=2|\omega_\alpha|^2$. Since we have $$J_\omega^2-J_{\omega_\alpha}^2
=J_\omega(J_\omega-J_{\omega_\alpha})+(J_\omega-J_{\omega_\alpha})J_{\omega_\alpha},$$ we get $$|J_\omega^2-J_{\omega_\alpha}^2|
\leq 2(|\omega|+|\omega_\alpha|)|\omega_\beta|.$$ Therefore, we have $$\|J_\omega^2-J_{\omega_\alpha}^2\|_1
\leq 4\|\omega\|_2\|\omega_\beta\|_2\leq 4\delta^{1/4}\|\omega\|_2^2$$ by (\[AB2\]), and so $$\label{AB4}
\|J_{\omega_\alpha}^2+{\mathop{\mathrm{Id}}\nolimits}\|_1
\leq \|J_{\omega}^2+{\mathop{\mathrm{Id}}\nolimits}\|_1+\|J_\omega^2-J_{\omega_\alpha}^2\|_1\leq 5 \delta^{1/4}\|\omega\|_2^2\leq 10 \delta^{1/4}\|\omega_\alpha\|_2^2$$ by (\[AB3\]). By (\[AB1\]) and (\[AB4\]), we get the lemma.
Let us show the orientability for $L^2$ almost Kähler manifolds.
For any integer $n\geq 2$ and positive real numbers $K>0$, $D>0$, there exists a constant $\delta_1(n,K,D)>0$ such that the following property holds. Let $(M,g)$ be an $n$-dimensional closed Riemannian manifold with ${\mathop{\mathrm{Ric}}\nolimits}\geq-K g$ and ${\mathop{\mathrm{diam}}\nolimits}(M)\leq D$. If there exists a $2$-form $\omega$ such that
- $\|\nabla \omega\|_2^2\leq \delta_1\|\omega\|_2^2$,
- $\|J_\omega^2+{\mathop{\mathrm{Id}}\nolimits}\|_1\leq \delta_1^{1/4}\|\omega\|_2^2$,
then $M$ is orientable.
By Lemma \[pB2\], we have that $n=2m$ is an even integer. We first assume that $\delta_1< \min\{1/4m^2,\delta_0(n,2,K,D)^2\}$. Since $J_{\omega}$ is anti-symmetric, we have $|J_{\omega}|^2\leq \sqrt{2m}|J_{\omega}^2|$. Thus, we get $$\sqrt{\frac{2}{m}}\|\omega\|_2^2=
\frac{1}{\sqrt{2m}}\|J_{\omega}\|_2^2\leq \sqrt{2m}+\delta_1^{1/4} \|\omega\|_2^2$$ by $|{\mathop{\mathrm{Id}}\nolimits}|=\sqrt{2m}$. This and $\delta_1^{1/4}\leq \frac{1}{2}\sqrt{\frac{2}{m}}$ imply that $\|\omega\|_2\leq \sqrt{2m}$. Put $\omega_\alpha:=P_{\delta_1}(\omega)$. Note that we have that $\|\omega_\alpha\|_2\leq\|\omega\|_2\leq \sqrt{2m}$ and that $\|\omega_\alpha\|_\infty\leq C(n,K,D)$ by Lemma \[Linfes\] and Lemma \[pB3\].
We first fix $x\in M$, and consider the $\mathbb{C}$-linear map $$J_{\omega_\alpha}(x)\colon T_x M\otimes_\mathbb{R} \mathbb{C}\to T_x M\otimes_\mathbb{R} \mathbb{C}.$$ Let us extend the Riemannian metric $\langle\cdot,\cdot\rangle
$ to $T_x M\otimes_\mathbb{R} \mathbb{C}$ so that $$\langle u_1 + i v_1, u_2+iv_2 \rangle=(\langle u_1,u_2\rangle+\langle v_1,v_2\rangle)+i(\langle v_1, u_2\rangle-\langle u_1, v_2\rangle)$$ for all $u_1,u_2,v_1,v_2\in T_x M$. Since $J_{\omega_\alpha}(x)$ is anti-symmetric, there exist eigenvalues $\{\lambda_1,\overline{\lambda_1},\ldots,\lambda_m,\overline{\lambda_m}\}$ of $J_{\omega_\alpha}(x)$ and an orthogonal basis $\{E_1,\overline{E_1},\ldots,E_m,\overline{E_m}\}$ of $T_{x} M\otimes_\mathbb{R} \mathbb{C}$ such that $J_{\omega_\alpha}(x) E_i=\lambda_i E_i$, where the overline denotes the complex conjugate. Note that each $\lambda_i$ is a pure imaginary number. Let $\{E^1,\overline{E^1},\ldots,E^m,\overline{E^m}\}\subset T_x^\ast M\otimes_{\mathbb{R}} \mathbb{C}\cong(T_x M\otimes_{\mathbb{R}} \mathbb{C} )^\ast$ be the dual basis of $\{E_1,\overline{E_1},\ldots,E_m,\overline{E_m}\}$. If we extend $\omega_\alpha(x)$ to a complex bilinear form, then we have $$\omega_\alpha(x)=\sum_{i=1}^m \lambda_i E^i\wedge \overline{E^i}.$$ Thus, we get $$\omega_{\alpha}^m(x)=m! \lambda_1\cdots \lambda_m E^1\wedge\overline{E^1}\wedge E^m\wedge\overline{E^m},$$ and so $$|\omega_{\alpha}^m(x)|=m!|\lambda_1|\cdots |\lambda_m|.$$ Since we have $$|\lambda_i|^2=|J_{\omega_\alpha}^2 E_i|=|(J_{\omega_\alpha}^2+{\mathop{\mathrm{Id}}\nolimits})E_i-E_i|,$$ we get $$\left|1-|\lambda_i|^2\right|\leq|J_{\omega_\alpha}^2+{\mathop{\mathrm{Id}}\nolimits}|(x)$$ and $$|\lambda_i|\leq |J_{\omega_\alpha}|^2=2|\omega_\alpha|^2\leq C(n,K,D).$$ Therefore, we get $$\left||\omega_\alpha^m|^2-(m!)^2\right|\leq C|J_{\omega_\alpha}^2+{\mathop{\mathrm{Id}}\nolimits}|,$$ and so $$\left|\|\omega_\alpha^m\|_2^2-(m!)^2\right|\leq C\delta_1^{1/4}$$ by Lemma \[pB4\]. Since we have $$\|\nabla (\omega_\alpha^m)\|_2^2\leq \delta_1$$ by Lemma \[pB4\], we get the proposition taking $\delta_1$ sufficiently small by Corollary \[papeb\] (ii).
The following theorem is the goal of this section.
For any integer $n\geq 2$, there exists a constant $C(n)>0$ such that the following property holds. Let $(M,g)$ be an $n$-dimensional closed Riemannian manifold with ${\mathop{\mathrm{Ric}}\nolimits}\geq (n-1) g$. If there exists a $2$-form $\omega$ such that
- $\|\nabla \omega\|_2^2\leq \delta\|\omega\|_2^2$,
- $\|J_\omega^2+{\mathop{\mathrm{Id}}\nolimits}\|_1\leq \delta^{1/4}\|\omega\|_2^2$,
for some $\delta>0$, then we have $$\lambda_1(g)\geq 2(n-1)- C(n)\delta^{1/2}.$$
It is enough to prove the theorem when $\delta$ is small. Thus, we can assume that $n=2m$ is an even integer by Lemma \[pB2\]. If $n=2$, then $\lambda_1(g)\geq 2(n-1)$ is the original Lichnerowicz estimate. If $n=4$, the conclusion of the theorem can also be deduced from Main Theorem 1.
We first assume that $\delta< \min\{1/4m^2,\delta_0(n,2,K,D)^2\}$. Put $\omega_\alpha:=P_{\delta}(\omega)=\sum_{i=1}^k a_i \omega_i$. Here, $\omega_i$ is the $i$-th eigenform of the connection Laplacian $\Delta_{C,2}$ with $\|\omega_i\|_2=1$ corresponding to the eigenvalue $\lambda_i(\Delta_{C,2})\leq \delta^{1/2}$ for each $i=1,\ldots, k$. We have $k\leq \alpha(n,2)$ by Lemma \[pB3\], and $\|\omega_\alpha\|_\infty\leq C$ by Lemma \[Linfes\].
Let $f\in C^\infty(M)$ be the first eigenfunction of the Laplacian with $\|f\|_2=1$. If $\lambda_1(g)\geq 2(n-1)+1$, we get the theorem. Thus, we assume that $\lambda_1(g)\leq 2(n-1)+1$. Then, we have $\|f\|_\infty\leq C$ and $\|\nabla f\|_\infty \leq C$ by Lemma \[Linfes\]. By Lemma \[p4d\] (i) and (iii), we have $$\label{AB5}
\begin{split}
&\frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)}\left|\int_M \langle\Delta (\iota(\nabla f)\omega_\alpha),\iota(\nabla f)\omega_\alpha\rangle-\lambda_1(g)\langle\iota(\nabla f)\omega_\alpha,\iota(\nabla f)\omega_\alpha\rangle\,d\mu_g
\right|\\
\leq &C\delta^{1/2}\|\omega_\alpha\|_2^2
\end{split}$$ and $$\label{AB6}
\|d^\ast (\iota(\nabla f)\omega_\alpha)\|_2^2\leq C\delta\|\omega_\alpha\|_2^2.$$ By (\[2b\]), (\[AB5\]), (\[AB6\]) and the Bochner formula, we get $$\label{AB7}
\begin{split}
&\frac{n-1}{{\mathop{\mathrm{Vol}}\nolimits}(M)}\int_M
\langle J_{\omega_\alpha}\nabla f,J_{\omega_\alpha}\nabla f\rangle\,d\mu_g\\
\leq&\frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)}\int_M{\mathop{\mathrm{Ric}}\nolimits}(J_{\omega_\alpha}\nabla f,J_{\omega_\alpha}\nabla f) \,d\mu_g\\
=&\frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)}\int_M \langle\Delta (\iota(\nabla f)\omega_\alpha),\iota(\nabla f)\omega_\alpha\rangle\,d\mu_g-\frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)}\int_M |\nabla (\iota(\nabla f)\omega_\alpha)|^2\,d\mu_g\\
\leq& \frac{\lambda_1(g)}{2{\mathop{\mathrm{Vol}}\nolimits}(M)}\int_M \langle\iota(\nabla f)\omega_\alpha,\iota(\nabla f)\omega_\alpha\rangle\,d\mu_g+C\delta^{1/2}\\
=&\frac{\lambda_1(g)}{2{\mathop{\mathrm{Vol}}\nolimits}(M)}\int_M \langle J_{\omega_\alpha}\nabla f,J_{\omega_\alpha}\nabla f \rangle\,d\mu_g+C\delta^{1/2}.
\end{split}$$
Since $J_{\omega_\alpha}$ is anti-symmetric, we have $$\begin{split}
&\frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)}\left|\int_M \langle J_{\omega_\alpha}\nabla f,J_{\omega_\alpha}\nabla f\rangle\,d\mu_g- \int_M \langle \nabla f,\nabla f\rangle\,d\mu_g\right|\\
\leq&
\frac{1}{{\mathop{\mathrm{Vol}}\nolimits}(M)}\left|\int_M \langle (J_{\omega_\alpha}^2+{\mathop{\mathrm{Id}}\nolimits})\nabla f,\nabla f\rangle\,d\mu_g\right|\\
\leq &C\|J_{\omega_\alpha}^2+{\mathop{\mathrm{Id}}\nolimits}\|_1\leq C\delta^{1/4}
\end{split}$$ by Lemma \[pB4\]. Thus, taking $\delta$ sufficiently small, we get $$\label{AB8}
\|J_{\omega_\alpha}\nabla f\|_2^2
\geq \|\nabla f\|_2^2-C\delta^{1/4}= \lambda_1(g) -C\delta^{1/4}\geq n-C\delta^{1/4}\geq \frac{n}{2}$$ by the Lichnerowicz estimate. By (\[AB7\]) and (\[AB8\]), we get the theorem.
[99]{} M. Aino, [Riemannian invariants that characterize rotational symmetries of the standard sphere,]{} Manuscripta Math. 156 (2018), 241–272. M. Aino, [Sphere theorems and eigenvalue pinching without positive Ricci curvature assumption,]{} Calc. Var. Partial Differential Equations, to appear. D. V. Alekseevsky, S. Marchiafava, [Transformations of a quaternionic Kähler manifold,]{} C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), no. 6, 703–708. E. Aubry, [Pincement sur le spectre et le volume en courbure de Ricci positive,]{} Ann. Sci. École Norm. Sup. (4) 38 (2005), 387–405. F. Belgun, A. Moroianu, U Semmelmann, [Killing forms on symmetric spaces,]{} Differential Geom. Appl. 24 (2006), no. 3, 215–222. A. Besse, [Einstein manifolds,]{} Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer-Verlag, Berlin (1987), xii+510 pp. J. Cheeger, [Differentiability of Lipschitz functions on metric measure spaces,]{} Geom. Funct. Anal. 9 (1999), 428–517. J. Cheeger, [Degeneration of Riemannian metrics under Ricci curvature bounds,]{} Lezioni Fermiane, Scuola Normale Superiore, Pisa, (2001). J. Cheeger, T. Colding, [Lower bounds on Ricci curvature and the almost rigidity of warped products,]{} Ann. of Math. (2) 144 (1996), 189–237. J. Cheeger, T. Colding, [On the structure of spaces with Ricci curvature bounded below. I,]{} J. Differential Geom. 46 (1997), 406–480. J. Cheeger, R. Colding, [On the structure of spaces with Ricci curvature bounded below. III,]{} J. Differential Geom. 54 (2000), no. 1, 37–74. T. Colding, [Shape of manifolds with positive Ricci curvature,]{} Invent. Math. 124 (1996), 175–191. N. Gigli, [Nonsmooth differential geometry–an approach tailored for spaces with Ricci curvature bounded from below,]{} Mem. Amer. Math. Soc. 251 (2018), no. 1196, v+161 pp. J. F. Grosjean, [A new Lichnerowicz-Obata estimate in the presence of a parallel p-form,]{} Manuscripta Math. 107 (2002), no. 4, 503–520. S. Honda, [Ricci curvature and almost spherical multi-suspension,]{} Tohoku Math. J. (2) 61 (2009), 499–522. S. Honda, [A weakly second-order differential structure on rectifiable metric measure spaces,]{} Geom. Topol. 18 (2014), 633–668. S. Honda, [Ricci curvature and $L^p$-convergence,]{} J. Reine Angew. Math. 705 (2015), 85–154. S. Honda, [Ricci curvature and orientability,]{} Calc. Var. Partial Differential Equations 56 (2017), no. 6, Art. 174, 47 pp. S. Honda, [Spectral convergence under bounded Ricci curvature,]{} J. Funct. Anal. 273 (2017), 1577–1662. S. Honda, [Elliptic PDEs on compact Ricci limit spaces and applications,]{} Mem. Amer. Math. Soc. 253 (2018), v+92 pp. P. Petersen, [On eigenvalue pinching in positive Ricci curvature,]{} Invent. Math. 138 (1999), 1–21. P. Petersen, [Riemannian geometry. Third edition,]{} Springer, Cham, 2016. xviii+499 pp. T. Sakai, [Riemannian geometry,]{} Translated from the 1992 Japanese original by the author. Translations of Mathematical Monographs, 149. American Mathematical Society, Providence, RI, (1996), xiv+358 pp. R. Schoen, S.T. Yau, [Lectures on differential geometry,]{} Conference Proceedings and Lecture Notes in Geometry and Topology, I. International Press, Cambridge, MA, (1994) v+235 pp. U. Semmelmann, [Conformal Killing forms on Riemannian manifolds,]{} Math. Z. 245 (2003), no. 3, 503–-527. Y. Tashiro, [Complete Riemannian manifolds and some vector fields,]{} Trans. Amer. Math. Soc. 117, (1965), 251–275.
|
---
author:
- 'Wen-Yuan Liu'
- Shaojun Dong
- Chao Wang
- Yongjian Han
- Hong An
- 'Guang-Can Guo'
- Lixin He
date:
-
-
title: 'Reply to comments on “Gapless spin liquid ground state of spin-1/2 $J_1$-$J_2$ Heisenberg model on square lattices”'
---
=5000
=1000
In a recent comments[@zhao2019], Zhao et al. argue that the definition of dimer orders used in our paper (Ref. ) may not rule out valence bond solid (VBS) orders of $J_1$-$J_2$ model on the open boundary conditions (OBC). In this reply, we show that their argument does not apply to our case.
The definition of dimer order parameter in our paper [@liu2018] is: $$m^2_{d\alpha}=\frac{1}{N_b^2}\sum_{{\bf i},{\bf j}}(\langle B^{\alpha}_{\bf i}B^{\alpha}_{\bf j}\rangle-\langle B^{\alpha}_{\bf i}\rangle \langle B^{\alpha}_{\bf j} \rangle) {\rm e}^{i{\bf k \cdot (i-j)}},
\label{eq:OriginDef1}$$ where $B^{\alpha}_{\bf i}=\bf{S_i}\cdot \bf{S_{i+e_{\alpha}}}$ is the bond operator defined on a pair of nearest neighbour sites ${\bf i}$ and ${\bf i+e_{\alpha}}$ along the $\alpha$ direction with $\alpha=x$ or $y$. Horizontal dimer values $m^2_{dx}$ and vertical ones $m^2_{dy}$ are obtained with ${\bf k}_x=(\pi,0)$ and ${\bf k}_y=(0,\pi)$, respectively. The Eq. \[eq:OriginDef1\] can be also expressed as, $$m^2_{d\alpha}=\langle D^{2}_{\alpha} \rangle-\langle D_{\alpha} \rangle^2.
\label{eq:OriginDef2}$$ where $D_{\alpha}=\frac{1}{N_b}\sum_{\bf i}(-1)^{i_\alpha}B^{\alpha}_{\bf i}$. It has been analytically proven that the dimer order parameter is nonzero in the thermodynamic limit for typical VBS states[@mambrini2006], and it has been used to detect VBS orders on periodic boundary conditions (PBC)[@mambrini2006], as well as cylindrical geometryies[@jiang2012; @gong2014] in numerical simulations.
![Inverse system size dependence of different definitions of the VBS order parameter at $J_2/J_1$=0.55. []{data-label="fig:VBSorder"}](./VBSorderCompare.eps){width="3.2in"}
In the comments of Ref., Zhao et al. argue that $m^2_{d\alpha}$ can not be used to rule out VBS orders. They demonstrated their argument on the so-called $J-Q_3$ (with $J=0$) model on open square lattices, whose ground state is a strong VBS state. They propose that one should use other VBS order parameters, such as $\langle D^2_{\alpha}\rangle$, $\langle D_\alpha\rangle^2$, or a symmetrized form $\langle D_{x}^2\rangle+ \langle D_{y}^2\rangle-{1\over 2}(\langle D_{x}\rangle+\langle D_{y}\rangle)^2$. However, their arguments have a prerequisite, i.e. the ground state has symmetry breaking. In our paper [@liu2018], we have checked very carefully that there is no symmetry breaking in the ground state. In fact, we have pointed out in our paper [@liu2018] that “We find that $m^2_{dx}$ and $m^2_{dy}$ are almost the same within numerical precision at each lattice size, reflecting the isotropy of horizontal and vertical directions, which is expected for the true ground states and exclude the CVB phases.”
To directly address Zhao et al.’s concerns, we plot the VBS order parameters of various definitions including $m^2_{dx}$+$m^2_{dy}$, $\langle D^2_{x}\rangle$+$\langle D^2_{y}\rangle$, and $\langle D_{x}^2\rangle+\langle D_{y}^2\rangle-{1\over 2}(\langle D_{x}\rangle+\langle D_{y}\rangle)^2$ etc. at $J_2/J_1$=0.55 up to a $16 \times 16$ lattice [@liu2018] in Fig. \[fig:VBSorder\]. As one can see that all the different definitions of VBS order parameters approach zero in the thermodynamic limit by a second order polynomial fitting. Especially, $\langle D_{x}\rangle^2$ and $\langle D_{y}\rangle^2$ are almost identical, and as a consequence the symmetrized order parameters $\langle D_{x}^2\rangle+
\langle D_{y}^2\rangle-{1\over 2}(\langle D_{x}\rangle+\langle D_{y}\rangle)^2$ are identical to $m^2_{dx}$+$m^2_{dy}$. These are strong evidences to rule out the VBS states based on our current calculations.
In addition, our conclusion that the intermediate nonmagnetic phase of $J_1$-$J_2$ model is a gapless spin liquid state is based on a series of consistent evidences. Besides the dimer order parameters, we have also showed that the spin correlation functions have power law decays, suggesting the states have zero ${\bf S}$=1 gap. These results are contradict to those of the VBS states in which the gap is non-zero, and exponential decays of the spin correlation functions are expected. Importantly, the behaviours at $J_2/J_1$=0.55 are almost the same as those of $J_2/J_1$=0.5, suggesting that no phase transition happens in this region.
We note some progress on tensor network algorithms was reported in a very recent paper[@liu2019]. It is expected that the $J_1$-$J_2$ model up to 24$\times$24 will be further investigated in the near future, to reexamine all different scenarios among the calculations by tensor network methods[@wang2016; @reza2017; @liu2018], the density matrix renormailization group (DMRG) method[@jiang2012; @gong2014; @wang2018], as well as variational quantum Monte Carlo methods[@hu2013].
[10]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{}
, , , ().
, , , , , , , ****, ().
, , , , ****, ().
, , , ****, ().
, , , , , ****, ().
, , , , ().
, , , , ****, ().
, ****, ().
, ****, ().
, , , , ****, ().
|
---
abstract: 'We present an analysis of the effect of dephasing on the single channel charge relaxation resistance of a mesoscopic capacitor in the linear low frequency regime. The capacitor consists of a cavity which is via a quantum point contact connected to an electron reservoir and Coulomb coupled to a gate. The capacitor is in a perpendicular high magnetic field such that only one (spin polarized) edge state is (partially) transmitted through the contact. In the coherent limit the charge relaxation resistance for a single channel contact is independent of the transmission probability of the contact and given by half a resistance quantum. The loss of coherence in the conductor is modeled by attaching to it a fictitious probe, which draws no net current. In the incoherent limit one could expect a charge relaxation resistance that is inversely proportional to the transmission probability of the quantum point contact. However, such a two terminal result requires that scattering is between two electron reservoirs which provide full inelastic relaxation. We find that dephasing of a single edge state in the cavity is not sufficient to generate an interface resistance. As a consequence the charge relaxation resistance is given by the sum of one constant interface resistance and the (original) Landauer resistance. The same result is obtained in the high temperature regime due to energy averaging over many occupied states in the cavity. Only for a large number of open dephasing channels, describing spatially homogenous dephasing in the cavity, do we recover the two terminal resistance, which is inversely proportional to the transmission probability of the QPC. We compare different dephasing models and discuss the relation of our results to a recent experiment.'
author:
- 'Simon E. Nigg'
- Markus Büttiker
bibliography:
- 'biblio.bib'
title: |
Quantum to Classical Transition of the Charge Relaxation Resistance\
of a Mesoscopic Capacitor
---
Introduction\[sec:introduction\]
================================
Interest in quantum coherent electron transport in the AC regime has been revived recently thanks to progress made in controlling and manipulating small high mobility mesoscopic structures driven by high frequency periodic voltages at ultra low temperatures. The state of the art includes the realization of high frequency single electron sources, which might be important for metrology. In Ref. this was achieved by applying large amplitude periodic voltage pulses of a few hundred MHz on the gate of a mesoscopic capacitor. The accuracy of this single electron emitter was analyzed theoretically in Ref. . In Ref. , pulses of surface acoustic waves were used to transport electrons one by one on a piezoelectric GaAs substrate. Two parameter quantized pumping with localized electrical potentials has been demonstrated in Ref. and one parameter non-adiabatic quantized charge pumping in Ref. . These experiments use frequencies in the GHz range to control the population and depopulation of one (or several) localized level(s). Thus the dynamics of charge relaxation is of central importance for these experiments.
Of particular interest to us here is the work of Gabelli et al. [@Gabelli:06], who succeeded in measuring both the in and out of phase parts of the linear AC conductance $G(\omega)=I(\omega)/V(\omega)$ of a mesoscopic capacitor at the driving frequency $\omega\approx 1\, {\rm GHz}$. The capacitor consists of a sub-micrometer Quantum Dot (QD) connected to an electron reservoir via a tunable Quantum Point Contact (QPC) and capacitively coupled to a metallic back or top gate (see Fig. 1).
The question “What is the RC-time of a quantum coherent capacitor?” has been theoretically addressed by Büttiker, Thomas and Prêtre [@Buttiker:93b]. In the low frequency regime $\omega \ll 1/\tau_{RC}$, where $\tau_{RC}$ is the RC-time of the system, the response is determined by an [*electrochemical capacitance*]{} $C_{\mu}$ and a [*charge relaxation resistance*]{} $R_q$.
[\[\]\[\]\[1.2\][$V_{QPC}$]{} \[\]\[l\]\[1.2\][$V(\omega)$]{} \[l\]\[\]\[1.2\][$V_{\phi}(\omega)$]{} \[\]\[r\][$N_{\phi}$]{} \[\]\[r\][$N_{\phi}-1$]{} \[\]\[\]\[1.2\][$V_g$]{} \[\]\[\]\[1.2\][$U(\omega)$]{} \[\]\[\]\[1.4\][$\color{blue}\mathcal{T}$]{} ![(Color online) Mesoscopic Capacitor. The full blue curve represents the current carrying channel connected to the reservoir via the QPC while the dashed dotted black curves represent additional localized states, disconnected from the reservoir. As an example, the innermost edge state (highest Landau Level) is here split into three localized states illustrating the possibility of having more than one localized state per Landau Level. The red dashed lines represent incoherent scattering between states in the cavity. $\mathcal{T}$ is the transmission probability of the QPC and $\varepsilon$ is the inter-channel coupling strength. $U(\omega)$ is the Fourier transform of the time-dependent electric potential $U(t)$ inside the cavity. The functions of the various voltages are discussed in the text. \[fig:1a\]](fig1.eps "fig:"){width="40.00000%"} ]{}
Together these determine the RC-time in complete analogy to the classical case: $\tau_{RC}=R_qC_{\mu}$ . These two quantities however differ fundamentally from their classical counterparts. In particular the quantum RC-time obtained from their product, is sensitive to the quantum coherence of the system and consequently displays typical mesoscopic fluctuations [@Gopar:96a; @Brouwer:97a; @Brouwer:97c; @Buttiker:07a]. For a system, with many conducting channels [@Brouwer:97a], these fluctuations are present separately in both the capacitance and the resistance. Surprisingly, for a coherent capacitor with a single channel, only the capacitance fluctuates and the resistance is found to be constant and given by half a resistance quantum [@Buttiker:93a] $$R_q = \frac{h}{2e^2} \,.
\label{eq1}$$ This quantization has indeed been observed experimentally [@Gabelli:06] thus establishing a novel manifestation of quantum coherence in the AC regime.
The claim that the quantization of $R_q$ requires quantum coherence is perhaps not so astonishing. The interesting question is the length scale on which coherence is necessary. For the integer quantized Hall effect [@vKlitzing:80] coherence is necessary only over a cyclotron radius which is sufficient to establish a Landau level structure. In fact as discussed in Ref. inelastic scattering (the destruction of long range coherence) can even help to establish quantization of the Hall resistance. Similarly in quantum point contacts [@vWees:88; @Wharam:88] coherence over the width of the conduction channel is in principle sufficient to establish a step-like structure of the conductance. In contrast, as we will show, the quantization of the charge relaxation resistance requires coherence over the entire capacitance plate (the quantum dot) and not only over the contact region. Thus the quantized charge relaxation resistance in Eq. (\[eq1\]) is indeed very sensitive to dephasing.
There is a second important aspect in which the quantized charge relaxation resistance $R_q=h/(2e^2)$ differs from quantization of a Hall resistance [@vKlitzing:80] or of a ballistic conductance [@vWees:88; @Wharam:88]. In both of these latter cases quantization is associated with perfect transmission channels which permit unidirectional electron motion through the sample. In contrast, the quantization of $R_q$ is independent of the transmission probability of the contact! For a coherent capacitor plate connected via a single spin polarized channel to a reservoir the quantization of $R_q$ is truly universal and holds even in the Coulomb blockade regime [@Nigg:06].
Of course, no matter how pure the samples are, a spurious interaction of the system with environmental degrees of freedom, is unavoidable. This introduces dephasing into the system.
It is thus of interest to ask how dephasing affects the quantization of the single channel charge relaxation resistance and to investigate the crossover from the coherent to the incoherent regime. Furthermore, in typical measurements the temperature, though low compared with the level spacing of the sample, is still comparable to other relevant energy scales such as the driving frequency or the coupling strength between cavity and lead. From a theoretical point of view it is thus desirable to be able to distinguish between thermal averaging and effects due to pure dephasing and to understand the interplay between these two fundamental mechanisms. Intuitively, one would expect that in the presence of strong enough dephasing, the QD starts behaving like an electron reservoir and thus that a fully incoherent single channel capacitor should exhibit the two terminal resistance $$R_q = \frac{h}{e^2} \frac{1}{\mathcal{T}},
\label{eq2}$$ where $\mathcal{T}$ is the transmission probability of the channel through the QPC connecting the system to the electron reservoir. Interestingly, neither dephasing nor energy averaging (high-temperature limit) lead directly to Eq. (\[eq2\]). We find that for the QD to become a true electron reservoir it is necessary that many channels participate in the inelastic relaxation process which a true reservoir must provide.
In the present work we employ a description of dephasing provided by the voltage and dephasing probe models [@Buttiker:86; @Buttiker:88; @Datta:91; @DeJong:96; @Pilgram:06], where one attaches a fictitious probe to the system which can absorb and re-emit electrons from or into the conductor. If the probe supports only one channel, we find that the charge relaxation resistance of the [*fully incoherent*]{} mesoscopic capacitor is given by $$R_q = \frac{h}{2e^2}+\frac{h}{e^2}\frac{1-\mathcal{T}}{\mathcal{T}}\,,$$ Hence, the charge relaxation resistance is given by the sum of the resistance as found from the (original) Landauer [@Landauer:70] formula $h/e^2(1-\mathcal{T})/\mathcal{T}$ and [ *one*]{} interface resistance [@Imry:86; @Landauer:87] $h/(2e^2)$. Incidentally, as we show below, this is also the value of $R_q$ obtained in the high temperature limit for the coherent system, illustrating an interesting relation between single channel dephasing and temperature induced phase averaging. A hybrid superconducting-normal conductor provides another geometry with only one normal narrow-wide interface [@Sols:99].
In the next two sections, we introduce the physical system and the dephasing models. Then in section \[sec:interf-edge-state\], we specialize our model to a specific form of the scattering matrix appropriate for transport along edge states of the integer quantum Hall regime and discuss the main results. Finally, our conclusions are given in section \[sec:conclusion\].
The mesoscopic capacitor\[sec:mesoscopic-capacitor\]
====================================================
The system we consider can be viewed as the mesoscopic equivalent of the ubiquitous classical series RC circuit. One of the macroscopic “plates” of the classical capacitor is replaced by a QD and the role of the resistor is played by a QPC connecting this QD to an electron reservoir. This system is represented schematically in Fig. \[fig:1a\]. The curves with arrows represent the transport channels of the system corresponding physically to edge states of the integer (spin polarized) quantum Hall regime, in which the experiment of Ref. was performed. By varying the gate voltage $V_{QPC}$, one changes both the transparency of the QPC and the electrostatic potential in the cavity. In the present work we take the gate voltage $V_g$, applied to the macroscopic “plate” of the capacitor, as a fixed voltage reference and set it to zero. A sinusoidal AC voltage $V(\omega)$, applied to the electron reservoir, drives an AC current through the system.
The low frequency linear AC response of the mesoscopic capacitor can be characterized [@Buttiker:93b] by an electrochemical capacitance $C_{\mu}$ and a charge relaxation resistance $R_q$, defined via the AC conductance as $$\label{eq:2}
G(\omega) = -i\omega C_{\mu}+\omega^2C_{\mu}^2R_q+O(\omega^3)\,.$$ The linear low frequency regime is given by $eV_{ac}\ll \hbar\omega\ll\Delta$, where $V_{ac}$ is the amplitude of the AC voltage and $\Delta$ is the mean level spacing in the QD.
Even in very clean samples some coupling of the current carrying edge channel to some environmental states is unavoidable. For example, we can expect that an electron entering the QD in the current carrying edge channel (full blue curve in Fig. \[fig:1a\]) may be scattered (red dashed lines in Fig. \[fig:1a\]) by phonons or other electrons into localized states belonging to other (higher) Landau levels not directly connected to the lead, before being scattered back into the open edge channel and returning to the electron reservoir. If on the one hand, this inter-edge state scattering is purely [*elastic*]{}, the presence of these closed states is known to lead to a periodic modulation of the conductance as a function of gate voltage, the period of which is proportional to the number of closed states [@Staring:92; @Heinzel:94]. Such modulations, with a period corresponding to about $10$ to $15$ closed states, have indeed been observed in the experiment of Ref. at low temperatures for a magnetic field strength of $1.3\,{\rm T}$. If on the other hand the scattering is [*inelastic*]{}, such processes will in general be incoherent, i.e. they will destroy the information carried by the phase of the electronic wave and hence lead to dephasing.
The idea of the present work is to mimic the latter processes using the voltage and dephasing probe models as illustrated in Fig. \[fig:1b\]. The extension of these models to the AC regime is presented in the next section. For simplicity, we will here neglect the contribution of the elastic processes and focus solely on the inelastic ones.
Voltage and dephasing probe models in the AC regime\[sec:volt-deph-probe\]
==========================================================================
[\[\]\[\]\[1.2\][$V_{QPC}$]{} \[\]\[l\]\[1.2\][$V(\omega)$]{} \[l\]\[\]\[1.2\][$V_{\phi}(\omega)$]{} \[\]\[r\][$N_{\phi}$]{} \[\]\[r\][$N_{\phi}-1$]{} \[\]\[\]\[1.2\][$V_g$]{} \[l\]\[l\]\[1\][$U(\omega)$]{} \[\]\[\][DP, VP]{} \[\]\[\]\[0.9\][$\color{blue}\mathcal{T}$]{} \[\]\[\]\[1\][$I_{\phi}=0$]{} \[\]\[\][DP or VP, $I_{\phi}=0$]{} ![(Color online) Schematic representation of the voltage and dephasing probe models. The incoherent inter-channel coupling depicted in Fig. \[fig:1a\], is mediated by a Voltage Probe (VP) or a Dephasing Probe (DP) represented as a shaded plane. The entire system including the fictitious probe, which draws no net current, is again described as a [ *coherent*]{} multi-terminal scatterer. \[fig:1b\]](fig2.eps "fig:"){width="38.00000%"}]{}
To simulate the loss of phase coherence of electrons inside the cavity, we attach to the quantum dot a fictitious probe [@Buttiker:86; @Buttiker:88; @Datta:91; @DeJong:96], which draws no net current. An electron entering this probe is immediately replaced by an electron re-injected incoherently into the conductor. The main advantage of this approach is that the entire system consisting of the conductor and the probe can be treated as a coherent multi-terminal conductor within the scattering matrix approach. Some recent applications of this approach include investigations on the effect of dephasing on quantum pumping [@Moskalets:01; @Chung:07], on quantum limited detection [@Clerk:04] and on photon assisted shot noise [@Polianski:05]. The effect of dephasing on shot noise and higher moments (counting statistics) has been investigated in Refs. and . A probe which dephases spin states has been introduced in Ref. .
In terms of the spectral current density $i_{\alpha}(E,\omega)$, the current at the driving frequency $\omega$ into probe $\alpha$ is expressed as $$\label{eq:3}
I_{\alpha}(\omega) = \int dE\, i_{\alpha}(E,\omega)\,.$$ The gauge invariant spectral current in turn is given by $$\label{eq:4}
i_{\alpha}(E,\omega) = \sum_{\beta}g_{\alpha\beta}(E,\omega)(V_{\beta}(\omega)-U(\omega))\,,$$ where $$\label{eq:5}
g_{\alpha\beta}(E,\omega) =\frac{e^2}{h}F_{\beta}(E,\omega){\ensuremath{{\rm tr}}}[\openone_{\alpha}\delta_{\alpha\beta}-S^{\dagger}_{\alpha\beta}(E)S_{\alpha\beta}(E+\hbar\omega)]$$ is the (unscreened) spectral AC conductance from probe $\beta$ to probe $\alpha$ and $F_{\beta}(E,\omega)=[f_{\beta}(E)-f_{\beta}(E+\hbar\omega)]/\hbar\omega$, $f_{\beta}$ being the electron distribution function in probe $\beta$. $V_{\beta}(\omega)$ is the voltage applied to reservoir $\beta$ and $U(\omega)$ is the Fourier transform of the electric potential inside the QD, which is assumed to be homogeneous. The inclusion of this potential, which accounts for the screening interaction between charges on the conductor and charges on the gate electrode, is essential to ensure gauge invariance in the dynamical regime [@Buttiker:93b]. Finally, $S_{\alpha\beta}(E)$ is the $(N_{\alpha}+N_{\beta})\times (N_{\alpha}+N_{\beta})$ scattering matrix for electrons with energy $E$ scattered from the $N_{\beta}$ channels of probe $\beta$ to the $N_{\alpha}$ channels of probe $\alpha$.
In the following we will be interested in the situation where only one current carrying channel (full blue curve in Fig. \[fig:1b\]) connects the QD to the electron reservoir ($N_{1} = 1$), while the number of channels coupling to the fictitious probe $N_{\phi}$ is arbitrary.
For the voltage probe, we require that the current into the fictitious probe vanishes at each instant of time or equivalently at all frequencies, i.e. $I_{\phi}(\omega)=0$. For the dephasing probe, we require in addition that the current into the probe vanishes in any infinitesimal energy interval $dE$ and thus that the spectral current $i_{\phi}(E,\omega)=0$. This latter condition simulates quasi-elastic scattering where the energy exchanged is small compared to all other energy scales. Clearly, with these definitions, a dephasing probe is also a voltage probe but a voltage probe need not be a dephasing probe. In both cases however, current conservation implies that $I(\omega)\equiv I_1(\omega) =
-i\omega CU(\omega)$, where $C$ is the geometrical capacitance of the QD. This relation, together with Eqs. (\[eq:3\]) and (\[eq:4\]), allows us to self-consistently eliminate the internal potential $U(\omega)$ in the usual fashion [@Buttiker:93b].
Voltage probe\[sec:voltage-probe\]
----------------------------------
From the condition $I_{\phi}(\omega) =0$, we find the AC conductance $$\label{eq:6}
G(\omega) =\frac{I(\omega)}{V(\omega)}=\frac{-i\omega C\chi(\omega)}{\chi(\omega)-i\omega C}\,,$$ where $$\label{eq:7}
\chi(\omega) = g_{11}(\omega)-\frac{g_{1\phi}(\omega)g_{\phi 1}(\omega)}{g_{\phi\phi}(\omega)}\,.$$ Here and for all of the following, we have introduced the notation $g_{\alpha\beta}(\omega)=\int dE g_{\alpha\beta}(E,\omega)$. Upon expanding to second order in $\omega$ and comparing coefficients with (\[eq:2\]), we find $$\label{eq:16}
C_{\mu} = \frac{C\chi_1}{-iC+\chi_1}\quad\text{and}\quad R_q = -\frac{\chi_2}{{\chi_1}^2}\,,$$ with $$\label{eq:14}
\chi_1 =
\sum_{\alpha,\beta}g^{1}_{\alpha\beta}\,\,\, \text{and}\,\,\,\chi_2 =
\sum_{\alpha\beta}\Big(g^{2}_{\alpha\beta}-g^{1}_{\alpha\phi}g^{1}_{\phi\beta}/g^{0}_{\phi\phi}\Big)\,,$$ where $g_{\alpha\beta}(\omega)=g^{0}_{\alpha\beta}+g^{1}_{\alpha\beta}\omega+g^{2}_{\alpha\beta}\omega^2+O(\omega^3)$ and $\chi(\omega)=\chi_1\omega+\chi_2\omega^2+O(\omega^3)$. The conductance expansion coefficients are given in terms of the scattering matrix and its energy derivatives as $$\label{eq:19}
g_{\alpha\beta}^i = \int dE
f_{\beta}'(E)A_{\alpha\beta}^i(E)\,,\quad (i=1,2,3)$$ with $$\begin{aligned}
\label{eq:9}
A_{\alpha\beta}^0&=-\frac{e^2}{h}{\ensuremath{{\rm tr}}}[\openone_{\alpha}\delta_{\alpha\beta}-S^{\dagger}_{\alpha\beta}S_{\alpha\beta}]\,,\nonumber\\
A_{\alpha\beta}^1&=\frac{e^2}{4\pi}{\ensuremath{{\rm tr}}}[S_{\alpha\beta}^{\dagger}S_{\alpha\beta}'-{(S_{\alpha\beta}')^{\dagger}}S_{\alpha\beta}]\,,\\
A_{\alpha\beta}^2&=-\frac{e^2h}{8\pi^2}{\ensuremath{{\rm tr}}}[{{S_{\alpha\beta}'}^{\dagger}}S_{\alpha\beta}'-\frac{1}{3}\left(S_{\alpha\beta}^{\dagger}S_{\alpha\beta}\right)'']\,,\nonumber\end{aligned}$$ where $'$ denotes differentiation with respect to $E$ and for compactness we have suppressed the energy arguments. In the voltage probe model the electrons in the fictitious lead are allowed to relax towards equilibrium arbitrarily fast and we thus have $f_{\phi}(E)=f_{1}(E)=1/[1+\exp(\beta(E-E_F))]\equiv f(E)$.
Dephasing probe\[sec:dephasing-probe\]
--------------------------------------
In contrast to the voltage probe, the distribution function $f_{\phi}(E)$ of the dephasing probe is a priori not known. The requirement $i_{\phi}(E,\omega)=0$, together with Eq. (\[eq:4\]) yields $$\label{eq:8}
G(\omega)=\frac{-i\omega
C\tilde\chi(\omega)}{\tilde\chi(\omega)-i\omega C}\,,$$ where $$\label{eq:11}
\tilde\chi(\omega)\equiv g_{11}(\omega)-\int
dE\frac{g_{1\phi}(E,\omega)g_{\phi 1}(E,\omega)}{g_{\phi\phi}(E,\omega)}\,.$$ The electrochemical capacitance and the charge relaxation resistance are given in terms of the first and second order frequency expansion coefficients $\tilde\chi_1$ and $\tilde\chi_2$ as $$\label{eq:12}
C_{\mu}=\frac{C\tilde\chi_1}{-iC+\tilde\chi_1}\quad\text{and}\quad R_q
= -\frac{\tilde\chi_2}{{\tilde\chi_1}^{\ 2}}\,.$$ Making use of the unitarity of the scattering matrix, we find explicitly $$\tilde\chi_1 = \sum_{\alpha\beta}\int dE f'(E)A_{\alpha\beta}^1(E)=\chi_1\,,$$ and $$\label{eq:20}
\tilde\chi_2 = \sum_{\alpha\beta}\left(g_{\alpha\beta}^2
-\int dEf'(E)\frac{A_{\alpha\phi}^1(E)A_{\phi\beta}^1(E)}{A_{\phi\phi}^0(E)}\right)\,.$$
Comparing with Eqs. (\[eq:14\]), we see that at zero temperature, voltage and dephasing probes equally affect the AC conductance. At finite temperature, the electrochemical capacitance of the mesoscopic capacitor does still not distinguish between dephasing and voltage probes, while the charge relaxation resistance is in principle sensitive to whether the dephasing mechanism is quasi-elastic (dephasing probe) or inelastic (voltage probe).
Interfering edge state model\[sec:interf-edge-state\]
=====================================================
Scattering matrix for independent channels\[sec:scatt-matr-indep\]
------------------------------------------------------------------
We next apply the two dephasing models described in the previous section to a model for the scattering matrix of the mesoscopic capacitor in the integer quantum Hall regime introduced in Refs. and , which is here extended to include a voltage (dephasing) probe. The special form of the scattering matrix arises due to multiple reflections of the electronic wavefunction within the cavity in close analogy with a Fabry-Perot interferometer.
The additional probe, with $N_{\phi}$ channels is coupled to the single edge channel propagating through the QPC. Clearly $N_{\phi}-1$ channels of the probe are perfectly reflected at the QPC from within the cavity as depicted in Figs. \[fig:1b\] and \[fig:2\]. For simplicity, we shall assume the channels to be independent, which means that we consider the physical edge channels to coincide with the eigen-channels of the transmission matrix. Furthermore, we consider a symmetric QPC and assume that each channel couples identically to the fictitious probe with strength $\varepsilon$. Then the $(N_{\phi}+1)\times(N_{\phi}+1)$ scattering matrix $S_1$ of the QPC and the $2N_{\phi}\times 2N_{\phi}$ scattering matrix $S_{\varepsilon}$ of the fictitious probe have block diagonal form and may be parameterized as follows $$S_1=\begin{pmatrix}r_1&t'_1\\t_1&r'_1\end{pmatrix}\quad\text{and}\quad S_{\varepsilon}=\begin{pmatrix}r_{\varepsilon}&t'_{\varepsilon}\\t_{\varepsilon}&r'_{\varepsilon}\end{pmatrix}$$ with $r_1=ir$ and $r'_1={\rm diag}(ir{\ensuremath{e}}^{i\phi_1(E)},{\ensuremath{e}}^{i\phi_2(E)},\dots$ $\dots,{\ensuremath{e}}^{i\phi_{N_{\phi}}(E)})$ where we take $r$ to be real and $\phi_l(E)$ is the phase accumulated by an electron during one round trip along the $l$-th edge state through the QD. $t_1=(\sqrt{1-r^2},0,\dots,0)^T$ and $t^{\prime}_1=(\sqrt{1-r^2}{\ensuremath{e}}^{i\phi_1(E)},0,\dots,0)$. Finally $r^{(\prime)}_{\varepsilon}={\rm
diag}(i\sqrt{1-\varepsilon},\dots,i\sqrt{1-\varepsilon})$ and $t^{(\prime)}_{\varepsilon}={\rm
diag}(\sqrt{\varepsilon},\dots,\sqrt{\varepsilon})$.
The total $(N_{\phi}+1)\times (N_{\phi}+1)$ scattering matrix, which is obtained from the series combination of the two scattering matrices $S_1$ and $S_{\varepsilon}$, takes the form $$\label{eq:13}
S=\begin{pmatrix}S_{11}&S_{1\phi}\\S_{\phi1}&S_{\phi\phi}\end{pmatrix}\,,$$ with $$\begin{aligned}
\label{eq:15}
S_{11}&=\frac{i(r+{\ensuremath{e}}^{i\phi_1}\sqrt{1-\varepsilon})}{1+r{\ensuremath{e}}^{i\phi_1}\sqrt{1-\varepsilon}}\,,\nonumber\\
S_{1\phi}&=(\frac{\sqrt{\varepsilon(1-r^2)}{\ensuremath{e}}^{i\phi_1}}{1+r{\ensuremath{e}}^{i\phi_1}\sqrt{1-\varepsilon}},\overbrace{0,\dots,0}^{N_{\phi}-1})\,,\\
S_{\phi 1}&=(\frac{\sqrt{\varepsilon(1-r^2)}}{1+r{\ensuremath{e}}^{i\phi_1}\sqrt{1-\varepsilon}},\overbrace{0,\dots,0}^{N_{\phi}-1})^T\,,\nonumber\\
S_{\phi\phi}&={\rm
diag}(\frac{i(\sqrt{1-\varepsilon}+r{\ensuremath{e}}^{i\phi_1})}{1+r{\ensuremath{e}}^{i\phi_1}\sqrt{1-\varepsilon}},\frac{i\sqrt{1-\varepsilon}+{\ensuremath{e}}^{i\phi_2}}{1-i{\ensuremath{e}}^{i\phi_2}\sqrt{1-\varepsilon}},\dots\nonumber\\
&\phantom{{\rm
diag}(r'_{\varepsilon_1}}\dots,\frac{i\sqrt{1-\varepsilon}+{\ensuremath{e}}^{i\phi_{N_{\phi}}}}{1-i{\ensuremath{e}}^{i\phi_{N_{\phi}}}\sqrt{1-\varepsilon}})\,.\nonumber\end{aligned}$$
[ \[\]\[\]\[0.7\][$\phi_1$]{} \[\]\[\]\[0.7\][$\phi_2$]{} \[\]\[\]\[0.7\][$\phi_3$]{} \[\]\[\]\[0.7\][$\phi_4\dots$]{} ![(Color online) $S_1$ is the $(N_{\phi}+1)\times
(N_{\phi}+1)$ scattering matrices of the QPC relating the incoming channel to the $N_{\phi}$ channels inside the cavity and $S_{\varepsilon}$ is the $2N_{\phi}\times 2N_{\phi}$ scattering matrix relating the $N_{\phi}$ channels in the cavity with the $N_{\phi}$ channels in the probe. From these two matrices one can derive $S$, the total scattering matrix relating the incoming channel to the $N_{\phi}$ channels in the probe. An electron in the $i$-th channel accumulates a phase $\phi_i$ inside the cavity.\[fig:2\]](fig3.eps "fig:"){width="40.00000%"}]{}
Using these expressions together with (\[eq:16\]) for the voltage probe, respectively (\[eq:12\]) for the dephasing probe, we can express the electrochemical capacitance and the charge relaxation resistance as a function of the transparency $\mathcal{T}=1-r^2$ of the current carrying channel, the phases $\phi_1,\dots,\phi_{N_{\phi}}$ and the coupling strength $\varepsilon$. In order to investigate the crossover from the coherent to the incoherent regime, we will later on make a specific physically motivated choice for the energy dependence of the phases. However, even without specifying the form of the energy-phase relation, we can already draw some general conclusions by looking at the incoherent limit $\varepsilon\rightarrow 1$. This we do next after briefly reviewing the coherent case $\varepsilon=0$.
Results and discussion\[sec:results-discussion\]
------------------------------------------------
### General results at zero temperature\[sec:general-results-at\]
We first consider the zero temperature limit. In this case voltage and dephasing probe models are equivalent as shown in section \[sec:volt-deph-probe\]. The capacitance and the resistance are given by $$\label{eq:23}
C_{\mu}=\frac{-C\sum_{\alpha\beta}A_{\alpha\beta}^1}{-iC-\sum_{\alpha\beta}A_{\alpha\beta}^1}\,,$$ and $$\label{eq:22}
R_q = \sum_{\alpha\beta}\left(A_{\alpha\beta}^2-\frac{A^1_{\alpha\phi}A^1_{\phi\beta}}{A^0_{\phi\phi}}\right)\Big/\left(\sum_{\alpha\beta}A_{\alpha\beta}^1\right)^2\,,$$ where $A^i_{\alpha\beta}\equiv A^i_{\alpha\beta}(E_F)$ are given in Eq. (\[eq:9\]). In the coherent regime ($\varepsilon=0$), we recover the universal result $R_q=h/(2e^2)$ for the resistance while the capacitance is given by $C_{\mu}=Ce^2\nu/(C+e^2\nu)$, with the density of states of the cavity $\nu(E) =1/(2\pi i)S^{\dagger}(E)dS(E)/dE$, where $S(E)=\lim_{\varepsilon\rightarrow 0}S_{11}(E)$, for $S_{11}$ given in Eq. (\[eq:15\]). In the opposite, fully incoherent regime ($\varepsilon=1$), we find $$\label{eq:21}
C_{\mu}=\frac{C\frac{e^2}{2\pi}\sum_{i=1}^{N_{\phi}}\phi_i^{\prime}}{C+\frac{e^2}{2\pi}\sum_{i=1}^{N_{\phi}}\phi_i^{\prime}}\,,$$ which is independent of $\mathcal{T}$, and $$\label{eq:17}
R_q =
\frac{h}{2e^2}\frac{\sum_{i=1}^{N_{\phi}}(\phi_i^{\prime})^2}{\left(\sum_{i=1}^{N_{\phi}}\phi_i^{\prime}\right)^2}+\frac{h}{e^2}\left(\frac{1}{\mathcal{T}}-\frac{\phi_1^{\prime}}{\sum_{i=1}^{N_{\phi}}\phi_i^{\prime}}\right)\,.$$ For a single open dephasing channel ($N_{\phi}=1$), Eq. (\[eq:17\]) reduces to $$\label{eq:24}
R_q = \frac{h}{2e^2}+\frac{h}{e^2} \frac{1-\mathcal{T}}{\mathcal{T}}\,.$$ Thus, as mentioned in the introduction, if only the current carrying channel is dephased, the charge relaxation resistance is the sum of a constant interface resistance [@Imry:86; @Landauer:87; @Sols:99] $R_c=h/(2e^2)$ and the original Landauer resistance $R_L=h/e^2(1-\mathcal{T})/\mathcal{T}$ of the QPC.
### Smooth potential approximation\[sec:smooth-potent-appr\]
In the following, we will assume that the potential in the cavity is sufficiently smooth so that the energy dependent part of the accumulated phase is the same for all channels. Then $\phi_1^{\prime}\approx\phi_2^{\prime}\approx\dots\phi_{N_{\phi}}^{\prime}\equiv\phi^{\prime}$. Within this approximation, Eqs. (\[eq:21\]) and (\[eq:17\]) reduce to $$\label{eq:26}
C_{\mu}=\frac{C\frac{e^2}{2\pi}N_{\phi}\phi^{\prime}}{C+\frac{e^2}{2\pi}N_{\phi}\phi^{\prime}}\,,$$ and $$\label{eq:25}
R_q =\frac{h}{e^2}\frac{1-\mathcal{T}}{\mathcal{T}}+\frac{h}{2e^2}+\frac{h}{2e^2}\frac{N_{\phi}-1}{N_{\phi}}\,.$$ Written in this way, this expression for $R_q$ again lends itself to a simple interpretation. The first term on the righthand side of Eq. (\[eq:25\]), is the original Landauer resistance of the QPC. The second term $R_c\equiv\frac{h}{2e^2}$ is the interface resistance of the real reservoir-conductor interface and the third term $R_{\phi}=\frac{h}{2e^2}\frac{N_{\phi}-1}{N_{\phi}}$ is the resistance contributed to by the dephasing. In the limit of a very large number of open dephasing channels ($\varepsilon=1,\ N_{\phi}\gg 1$), which corresponds physically to spatially homogeneous dephasing, $R_{\phi}\rightarrow R_c$ and so $R_q\rightarrow h/e^2(1/\mathcal{T})$ as well as $C_{\mu}\rightarrow C$. Thus, in this limit the fictitious probe contributes half a resistance quantum and the mesoscopic capacitor behaves like a classical RC circuit with a two terminal resistance in series with the geometrical gate capacitance.
Next we investigate the crossover from the coherent to the incoherent regime. For this purpose, we assume that the accumulated phase depends linearly on energy in the vicinity of the Fermi energy; explicitly we take $\phi_1(E) =\phi_2(E)=\dots=\phi_{N_{\phi}}(E)=2\pi E/\Delta$, where $\Delta$ is the mean level spacing in the cavity. Then, the fictitious probe is characterized by only two parameters; the number of channels $N_{\phi}$ and the coupling strength $\epsilon$. Following Ref. , the latter can be related to the dephasing time $\tau_{\phi}$. The scattering amplitudes have poles at the complex energies $E=E_n-i\Gamma_e/2-i\Gamma_{\phi}/2$, where $E_n = (2n+1)\Delta/2$ with $n=0,1,\dots$ is a resonant energy and $\Gamma_e=-(\Delta/\pi)\ln[r]$ and $\Gamma_{\phi}=-(\Delta/\pi)\ln[\sqrt{1-\varepsilon}]$ are respectively the elastic and inelastic widths. The dephasing time $\tau_{\phi}=\hbar/\Gamma_{\phi}$ is then related to $\varepsilon$ by $\varepsilon=1-\exp[-h/(\tau_{\phi}\Delta)]$.
In Fig. \[fig:3\] we show the behavior of $R_q$ as a function of the dephasing strength $\varepsilon$, if the probe is weakly coupled so that only one channel ($N_{\phi}=1$) with transmission probability $\varepsilon$ connects the cavity to the fictitious reservoir.
[ ![(Color online) $R_q$ as a function of the dephasing strength $\varepsilon$ of a single channel probe ($N_{\phi}=1$) at zero temperature, for different values of the channel transmission probability $\mathcal{T}$. $(1)$: $\mathcal{T}=0.6$, $(2)$: $\mathcal{T}=0.8$ and $(3)$: $\mathcal{T}=1$. The dashed curves $(a)$ show the off-resonant case $\Delta E\equiv \min_n(E_F-n\Delta)=0.5\Delta$, while the full curves $(b)$ show the on-resonant case $\Delta E=0$. The horizontal dotted lines represent the value of $\frac{h}{2e^2}+\frac{h}{e^2}\frac{1-\mathcal{T}}{\mathcal{T}}$ for the different transmission probabilities.\[fig:3\]](fig4.eps "fig:"){width="45.00000%"}]{}
We see that if the current carrying channel is perfectly transmitted through the QPC, i.e. for $\mathcal{T}=1$, the resistance is insensitive to dephasing and is fixed to its coherent value of half a resistance quantum (curve $3$). This is reasonable since for perfect transmission the electronic wavefunction is not split at the QPC and hence an electron cannot interfere with itself whether or not it evolves coherently along the edge channel. We emphasize however, that this simple argument holds only if the dephasing probe couples to a single channel. If the probe is coupled more strongly, such that it couples to additional (closed) channels inside the cavity ($N_{\phi}>1$), the ensuing effective incoherent coupling between channels will affect $R_q$ in an $\varepsilon$-dependent manner. Turning our attention back to the single channel case of Fig. \[fig:3\], we see that as the transparency of the channel is reduced, the charge relaxation resistance increases with $\varepsilon$. Also evident is that dephasing affects the resistance non-monotonically in the off-resonant case (curves a), where the energy of the electron lies between two Fabry-Perot-like resonances, and monotonically in the on-resonant case (curves b). This can be related to the fact that dephasing induces both a decrease of the peak value and a broadening of the density of states (DOS) in the cavity. On resonance the net result is thus always a monotonous decrease of the amplitude of the DOS. Off-resonance however, the amplitude may first increase due to the widening of the closest resonance. Finally, as expected dephasing is seen to affect the resistance the stronger, the weaker the coupling to the reservoir is, i.e. the longer an electron dwells inside the cavity, where it can undergo dephasing.
### Dephasing vs Temperature induced phase averaging\[sec:deph-vs-temp\]
It is instructive to compare the results obtained in the previous section in the incoherent limit $\varepsilon=1$ at zero temperature with finite temperature effects in the coherent regime $\varepsilon=0$. At finite temperature and for a perfectly coherent single channel system, the charge relaxation resistance is given by [@Buttiker:93b] $$\label{eq:1}
R_q = \frac{h}{2e^2}\frac{\int dE(-f'(E))\nu(E)^2}{\left(\int dE(-f'(E))\nu(E)\right)^2}\,,$$ where $\nu(E)$ is the density of states of the channel which was defined above and is here explicitly given by [@Gabelli:06] $\nu(E)=(1/\Delta)\left[(1-r^2)/(1+r^2+2r\cos(2\pi E/\Delta))\right]$. At low temperature $k_BT\ll\Delta$, an expansion around the Fermi energy yields $R_q=\frac{h}{2e^2}\left(1+\frac{\pi^2}{3\beta^2}\left(\ln'[\nu(E_F)]\right)^2\right)$ with $\beta=1/(k_BT)$. Finite temperature effects thus arise at order $T^2$ and lead to the appearance of pairs of peaks in the resistance as a function of Fermi energy around each resonance, where the square of the derivative of the density of states is maximal [@Nigg:06] (see thin red curve in Fig. \[fig:5\], top). This behavior has indeed been observed experimentally [@GabelliThesis:06] in the weakly coupled regime, where $\Delta\gg k_BT\gg\mathcal{T}\Delta$. At very high temperature $k_BT\gg\Delta$, the integrals in (\[eq:1\]) may be evaluated asymptotically as shown in the appendix and we obtain $R_q=h/(2e^2)+h/e^2(1-\mathcal{T})/\mathcal{T}$. Thus phase averaging in the high temperature coherent regime ($\varepsilon=0$) is equivalent to dephasing via a fictitious probe with a single open channel ($\varepsilon=1$) at zero temperature (see Eq. (\[eq:24\])). This fact and the crossover from the low to the high temperature regime are illustrated in the upper panel of Fig. \[fig:6\]. There we show the charge relaxation resistance as a function of the inverse temperature $\beta$ for different dephasing strengths $\varepsilon$ for $N_{\phi}=1$. For complete dephasing (curve a), $R_q$ is temperature independent and given by Eq. (\[eq:17\]) with $N_{\phi}=1$. Interestingly, we find that for a single channel probe, voltage and dephasing probes equally affect the resistance even at finite temperature. Technically this is due to the fact that for a linear energy-phase relation such as assumed in this work, the energy dependent parts of each factor in the integrands of Eq. (\[eq:18\]) below are identical.
[ \[\]\[l\][$\frac{h}{e^2}\frac{1-\mathcal{T}}{\mathcal{T}}+\frac{h}{2e^2}+\frac{h}{2e^2}\frac{N_{\phi}-1}{N_{\phi}}$]{} ![(Color online) [*Upper panel*]{}: $R_q$ as a function of the inverse temperature $\beta$ for $N_{\phi}=1$ and dephasing strengths: ($a$) $\varepsilon=1$, ($b$) $\varepsilon=0.9$, ($c$) $\varepsilon=0.5$ and ($d$) $\varepsilon=0$. The dashed-dotted red line gives the value $h/e^2(1-\mathcal{T})/\mathcal{T}+h/(2e^2)$ and the dotted line the value $h/(2e^2)$. As discussed in the text, dephasing and voltage probes are indistinguishable in this case. [*Lower panel*]{}: $R_q$ as a function of the inverse temperature $\beta$ for $N_{\phi}=2$ and dephasing strengths: ($a$) $\varepsilon=1$, ($b$) $\varepsilon=0.7$ and ($c$) $\varepsilon=0$. The dashed-dotted red line gives the value $h/e^2(1-\mathcal{T})/\mathcal{T}+\frac{h}{2e^2}+\frac{h}{2e^2}(N_{\phi}-1)/(N_{\phi})$, the dotted line the value $h/e^2(1-\mathcal{T})/\mathcal{T}+h/(2e^2)]$ and the dashed line the value of $h/(2e^2)$. As discussed in the text, dephasing and voltage probes differ for finite dephasing if $\varepsilon\not=1$, (curves $b$). $\beta$ is given in units of the inverse level spacing $\Delta^{-1}$. We show here the resonant cases $\Delta E\equiv \min_n(E_F-n\Delta)=0$; the off-resonant behavior is similar.\[fig:6\]](fig5.eps "fig:"){width="45.00000%"}\
![(Color online) [*Upper panel*]{}: $R_q$ as a function of the inverse temperature $\beta$ for $N_{\phi}=1$ and dephasing strengths: ($a$) $\varepsilon=1$, ($b$) $\varepsilon=0.9$, ($c$) $\varepsilon=0.5$ and ($d$) $\varepsilon=0$. The dashed-dotted red line gives the value $h/e^2(1-\mathcal{T})/\mathcal{T}+h/(2e^2)$ and the dotted line the value $h/(2e^2)$. As discussed in the text, dephasing and voltage probes are indistinguishable in this case. [*Lower panel*]{}: $R_q$ as a function of the inverse temperature $\beta$ for $N_{\phi}=2$ and dephasing strengths: ($a$) $\varepsilon=1$, ($b$) $\varepsilon=0.7$ and ($c$) $\varepsilon=0$. The dashed-dotted red line gives the value $h/e^2(1-\mathcal{T})/\mathcal{T}+\frac{h}{2e^2}+\frac{h}{2e^2}(N_{\phi}-1)/(N_{\phi})$, the dotted line the value $h/e^2(1-\mathcal{T})/\mathcal{T}+h/(2e^2)]$ and the dashed line the value of $h/(2e^2)$. As discussed in the text, dephasing and voltage probes differ for finite dephasing if $\varepsilon\not=1$, (curves $b$). $\beta$ is given in units of the inverse level spacing $\Delta^{-1}$. We show here the resonant cases $\Delta E\equiv \min_n(E_F-n\Delta)=0$; the off-resonant behavior is similar.\[fig:6\]](fig6.eps "fig:"){width="45.00000%"}]{}
[*Dephasing vs Voltage probe*]{}. At the end of the last paragraph, we concluded that dephasing and voltage probes are indistinguishable for a single channel probe as long as the accumulated phase is linear in energy. When $N_{\phi}\geq 2$, the two dephasing models differ at finite temperature. Introducing the [*emittances*]{} [@Pretre:96] $\mathcal{N}_{\alpha}^{{\rm
em}}\equiv 1/(2\pi
i)\sum_{\beta}{\ensuremath{{\rm tr}}}[S_{\alpha\beta}^{\dagger}S^{\prime}_{\alpha\beta}]=1/(ie^2)\sum_{\beta}A^1_{\alpha\beta}$, which represent the DOS of carriers emitted into probe $\alpha$ and the [*injectances*]{} $\mathcal{N}_{\beta}^{{\rm in}}\equiv1/(2\pi
i)\sum_{\alpha}{\ensuremath{{\rm tr}}}[S_{\alpha\beta}^{\dagger}S^{\prime}_{\alpha\beta}]=1/(ie^2)\sum_{\alpha}A^1_{\alpha\beta}$ representing the DOS of carriers injected from probe $\beta$, we may write the difference of resistance between the two models $\Delta R_q\equiv R_q^{{\rm DP}}-R_q^{{\rm VP}}$ as $$\label{eq:18}
\Delta R_q = \frac{h}{e^2}\frac{\frac{\int dE
f'\mathcal{N}_{\phi}^{{\rm em}}\int dEf'\mathcal{N}_{\phi}^{{\rm in}}}{\int dEf'{\ensuremath{{\rm tr}}}[\openone_{\phi}-S^{\dagger}_{\phi\phi}S_{\phi\phi}]}-\int dE
f'\frac{\mathcal{N}_{\phi}^{{\rm em}}\mathcal{N}_{\phi}^{{\rm in}}}{{\ensuremath{{\rm tr}}}[\openone_{\phi}-S^{\dagger}_{\phi\phi}S_{\phi\phi}]}}{\left(\int
dEf'\mathcal{N}\right)^2}\,,$$ where $\mathcal{N}=\sum_{\alpha}\mathcal{N}_{\alpha}^{{\rm
in}}=\sum_{\alpha}\mathcal{N}_{\alpha}^{{\rm em}}$ is the total DOS and for compactness we have suppressed all the energy arguments. As illustrated in the lower panel of Fig. \[fig:6\], we find somewhat counter-intuitively, that the resistance is larger in the presence of a dephasing probe than in the presence of a voltage probe. Indeed, one would have expected that since the voltage probe is dissipative and the dephasing probe is not, the former should lead to a larger resistance than the latter. This intuition fails when applied to $R_q$. Finally, for complete dephasing (curve $a)$), the two models coincide again. This is due to the fact that for $\varepsilon=1$ the coefficients given in (\[eq:9\]) become energy independent as a consequence of the linear energy-phase relation.
### Comparison with experiment\[sec:comp-with-exper\]
Comparison with the experiment [@Gabelli:06], leads us to some important conclusions. In this experiment, the real and imaginary parts of the AC conductance (\[eq:2\]) were measured at $mK$ temperatures $k_BT\ll\Delta$ while varying the transmission of the QPC, giving access to the charge relaxation resistance over a wide range of channel transparencies. In the highly transmissive regime the quantization of the charge relaxation resistance of a single channel mesoscopic capacitor could thereby be verified. As the coupling to the lead was reduced an oscillating increase in resistance was measured and excellent agreement with a theoretical model including only temperature broadening effects was obtained in the regime $\Delta\gg k_BT\gg\mathcal{T}\Delta$. For higher temperatures $T\sim 4K>\Delta/k_B$ the resistance was found to approach $\frac{h}{e^2}$ for a single perfectly open channel [@GabelliThesis:06] indicating that in this regime, the cavity truly acts like an additional reservoir. Indeed, according to our discussion in section \[sec:deph-vs-temp\], pure phase averaging due to temperature broadening would instead lead to $R_q\xrightarrow{k_BT\gg\Delta}\frac{h}{2e^2}$ for $\mathcal{T}=1$. Thus the observed value of the resistance hints at the presence of a spatially homogenous dephasing mechanism effective at high temperatures, which is suppressed at low temperatures. One such mechanism is the thermally activated tunneling from the current carrying edge channel to nearby localized states, which together act as a many channel voltage (dephasing) probe depending on the energetics of the scattering process. In Fig. \[fig:5\], we show the charge relaxation resistance as a function of the QPC voltage. The dependence of the transmission probability on $V_{QPC}$ is modeled assuming that the constriction is well described by a saddle-like potential [@Buttiker:90; @Gabelli:06] $\mathcal{T}=[1+\exp\{-a_0(V_{QPC}-E_0)\}]^{-1}$ and a change in $V_{QPC}$ is assumed to induce a proportional shift in the electrostatic potential of the QD. In the upper panel we show the coherent case $\varepsilon=0$ for three different temperatures $k_BT/\Delta = 0.8,\ 0.1$ and $0.01$. At low temperature and small transmission we recognize the resistance oscillations discussed in section \[sec:deph-vs-temp\]. As the temperature is increased, $R_q$ goes towards $h/e^2(1-\mathcal{T})/\mathcal{T} +h/(2e^2)$. This is to be contrasted with the situation shown in the lower panel where we display the incoherent case for a voltage probe with $\varepsilon=0.9$, $N_{\phi}=10$ and the same set of temperatures. For an open constriction ($\mathcal{T}\approx 1$), $R_q$ is now close to $h/e^2$ in the high temperature regime in agreement with the experimental observation.
[![(Color online) $R_q$ as a function of $V_{QPC}$ for different temperatures for a coherent system $\varepsilon=0$ (upper panel) and for a strongly incoherent system $\varepsilon=0.9$ and $N_{\phi}=10$ (lower panel). The inverse temperature $\beta$ is given in units of the inverse level spacing $\Delta^{-1}$. \[fig:5\]](fig7.eps "fig:"){width="45.00000%"}]{}\
Conclusion\[sec:conclusion\]
============================
In this work, we investigate the effect of decoherence on the dynamic electron transport in a mesoscopic capacitor. Extending the voltage and dephasing probe models to the AC regime, we calculate the charge relaxation resistance and the electrochemical capacitance, which together determine the RC-time of the system. Dephasing breaks the universality of the single channel, zero temperature charge relaxation resistance and introduces a dependency on the transparency of the QPC. We find that complete [*intra-channel*]{} relaxation alone is not sufficient to recover the two terminal resistance formula but rather yields a resistance which is the sum of the original Landauer formula and the interface resistance to the reservoir. This is also the resistance obtained in the high temperature limit of the coherent single channel system. Only in the presence of perfect [*inter-channel*]{} relaxation with a large number of channels, does the QD act as an additional reservoir and we recover the classically expected two terminal resistance.
We thank Heidi F[ö]{}rster and Mikhail Polianski for helpful comments on the manuscript. This work was supported by the Swiss NSF, the STREP project SUBTLE and the Swiss National Center of Competence in Research MaNEP.
High temperature regime integrals\[sec:high-temp-regime\]
=========================================================
In this appendix we compute the integrals appearing in the high temperature limit of Eq. (\[eq:1\]). Asymptotically we have $$\label{eq:10}
\lim_{\beta\rightarrow 0}R_q = \frac{h}{2e^2}\frac{I_2}{(I_1)^2}\,,$$ where $$I_1=\int_0^{\Delta}dE\nu(E),\quad\text{and}\quad I_2=\Delta\int_0^{\Delta}dE\nu(E)^2\,$$ with $\nu(E)=\frac{1}{\Delta}\frac{1-r^2}{1+2r\cos\frac{2\pi
E}{\Delta}+r^2}$. Following a change of variables $x=\frac{2\pi E}{\Delta}$ we get $$I_1=\frac{1}{2\pi}\int_0^{2\pi}dx\frac{1-r^2}{1+2r\cos
x+r^2}\,,$$ and $$I_2=\frac{1}{2\pi}\int_0^{2\pi}dx\frac{(1-r^2)^2}{(1+2r\cos x+r^2)^2}\,.$$ A simple way of computing these integrals is to use the (Poisson kernel) identity $$\frac{1-r^2}{1+2r\cos x+r^2}=\sum_{k=-\infty}^{\infty}(-r)^{|k|}{\ensuremath{e}}^{ikx}\,,$$ which can easily be verified by splitting the sum as $\sum_{k=-\infty}^{\infty}x^k=\sum_{k=0}^{\infty}x^k+\sum_{k=-\infty}^{0}x^k-1$ and utilizing the fact that for $|r|<1$, the geometric series converge. Integrating the sum in $I_1$ term by term and using the identity $\int_0^{2\pi}{\ensuremath{e}}^{ikx}dx=2\pi\delta_{k0}$ for $k\in\mathbb{Z}$, we immediately find $I_1 = 1$. Similarly we have $$\begin{aligned}
I_2 &=
\frac{1}{2\pi}\int_0^{2\pi}dx\left(\sum_{k=-\infty}^{\infty}(-r)^{|k|}{\ensuremath{e}}^{ikx}\right)^2\nonumber\\
&=\frac{1}{2\pi}\sum_{k,k'=-\infty}^{\infty}(-r)^{|k|+|k'|}\int_0^{2\pi}dx{\ensuremath{e}}^{i(k+k')x}\nonumber\nonumber\\
&=\sum_{k,k'=-\infty}^{\infty}(-r)^{|k|+|k'|}\delta_{k+k',0}=\frac{1}{\Delta}\sum_{k=-\infty}^{\infty}(-r)^{2|k|}\nonumber\\
&=\sum_{k=0}^{\infty}(-r)^{2k}+\sum_{k=-\infty}^{0}(-r)^{-2k}-1\nonumber\\
&=\frac{2}{1-r^2}-1=\frac{1+r^2}{1-r^2}\,.\end{aligned}$$ Substituting back into (\[eq:10\]) yields the desired result.
|
---
abstract: 'We study various properties of the soliton solutions of the modified regularized long-wave equation. This model possesses exact one- and two-soliton solutions but no other solutions are known. We show that numerical three-soliton configurations, for which the initial conditions were taken in the form of a linear superposition of three single-soliton solutions, evolve in time as three-soliton solutions of the model and in their scatterings each individual soliton experiences a total phase-shift that is the sum of pairwise phase-shifts. We also investigate the soliton resolution conjecture for this equation, and find that individual soliton-like lumps initially evolve very much like lumps for integrable models but eventually (at least) some blow-up, suggesting basic instability of the model.'
---
[**Aspects of the modified regularized long-wave equation**]{}
F. ter Braak$^{\dagger}$ and W. J. Zakrzewski$^{\dagger \dagger}$
.2in Department of Mathematical Sciences,\
University of Durham, Durham DH1 3LE, U.K.\
${}$ $^{(\dagger)}$ floris.ter-braak@durham.ac.uk\
$^{(\dagger \dagger)}$ w.j.zakrzewski@durham.ac.uk
Introduction {#Introduction}
============
Integrable partial differential equations in $(1+1)$ dimensions, such as the KdV and the non-linear Schrödinger equation, possess infinitely many conservation laws which implies that such systems possess soliton solutions; these solitons are localised waves which preserve their shape and velocity before and after the scattering, but they do experience a phase-shift as a result of the scattering [@Hietarinta0]. However, integrable models are quite rare, and some physical events can be described by models which possess soliton-like structures (such as general vortices, skyrmions and baby skyrmions) but are not integrable. Furthermore, the scattering properties of their soliton-like solutions are often not that different, in the sense that the amount of emitted radiation is not very large. Therefore, these models can loosely be described as ‘almost-integrable’. This has lead to various attempts to define the concept of *quasi-integrability* (see, [*e.g.*]{}, paper [@Ferreira] and references therein). Attempts have also been made to relate this concept to the extra (very special) symmetries satisfied by the two-soliton solutions [@Ferreira1].
In fact, the whole concept of integrability can be defined in various ways [@LMS]. One of such definitions is *Hirota integrability*, which is based on Hirota’s method for obtaining multi-soliton solutions of non-linear models (and we will, when it is important to stress this fact, refer to them as Hirota solutions). In addition, it is often claimed that if this method leads to the construction of the exact one-, two- and three-soliton Hirota solutions, the model is considered to be Hirota integrable [@Hietarinta]. On the other hand, partial differential equations which only possess one- and two-soliton solutions, also known as *partial integrable* models, are not Hirota integrable [@Hietarinta0]. (Note that this does not necessarily mean that solutions describing three or more solitons do not exist for these models; it only shows that they cannot be obtained using the Hirota method.) One of such models is the modified regularized long-wave (mRLW) equation,[^1] which we discuss in detail in this paper.
It has also been observed that if a set of equations is Hirota integrable, they satisfy the more conventional definitions of integrability [@Hietarinta1]. Thus it is interesting to try to better understand why, whereas models with only one- and two-soliton Hirota solutions do not appear to satisfy the conventional definitions of integrability, systems which also possesses three-soliton Hirota solutions do satisfy the conventional definitions of integrability. This raises the question: what is so special about three-soliton Hirota solutions?
In this paper we look in detail at various (numerical) properties of the mRLW equation. In the next section we recall this equation and the exact form of its one- and two-soliton solutions. Then we compare the analytical two-soliton scattering with our numerical approximation in order to test the accuracy and stability of our finite difference scheme. We also check whether the numerically evolved linear superposition of two single-soliton solutions is a good approximation to the corresponding analytical two-soliton solution, because we want to use a linear superposition of three single-soliton solutions as the initial conditions for our numerical three-soliton simulations. We find that this approach gives us essentially the same results as the ones obtained by using the exact two-soliton solutions. In section \[Numerical\_three\_soliton\_solution\] we investigate the time evolution of three-soliton systems obtained in such a way and look at the phase-shifts experienced by these solitons during their interactions. Finally, in section \[soliton\_resolution\_conjecture\] we look at some simulations using lumps ([*i.e.*]{}, functions which do not solve the mRLW equation) in order to test the soliton resolution conjecture. The last section of the paper presents our conclusions and plans for future work.
A large part of our results is based on numerical simulations of the mRLW equation. Since we used a numerical procedure which combined explicit and implicit finite difference methods, the work involved the discretisation in both time and space, and so had large memory requirements. Most of the simulations involved grid-spacing of $h=0.1$ and time-steps of $\tau = 0.001$. To assess the reliability of our procedures we have altered these values and we are confident that the results presented in the paper are correct. Since our procedure is second order in time, we require an analytic expressions of the fields at the first $2$ time levels. We took them from the expressions of $q(x,t)$ given in the paper and refer to as ‘initial conditions’. The spatial boundary conditions were fixed. More details on the numerical procedure are presented in appendix \[Numerical\_approx\], and the particular values of various parameters used in the reported simulations are gathered in a table in appendix \[summary\_of\_variables\].
The modified regularized long-wave equation {#Modified_regularized_long-wave_equation_equation}
===========================================
The mRLW equation, introduced by J. D. Gibbon, J. C. Eilbeck and R. K. Dodd [@Gibbon], is defined by $$q_{xxtt} + 2q_{xx} q_{tt} + 4q_{xt}^2 - q_{xt} - q_{tt} = 0 \,, \label{MRL}$$ where $q = q(x, t)$ is a real-valued function, and the subscripts $x$ and $t$ denote partial differentiation with respect to these variables. It is known that this equation possesses analytical one- and two-soliton Hirota solutions [@Gibbon], where the one-soliton solutions are given by $$q = \ln \left(1 + e^{\eta_1} \right) \label{1.1}$$ and the two-soliton solutions take the form $$q = \ln \left( 1 + e^{\eta_1} + e^{\eta_2} + A_{12} e^{\eta_1 + \eta_2} \right) \,, \label{1.2}$$ where $$\eta_i = k_i x - \omega_i t + \delta_i \,,\quad i = 1,2 \,,$$ and $$A_{12} = - \frac{(\omega_1 - \omega_2)^2(k_1 - k_2)^2 + (\omega_1 - \omega_2)(k_1 - k_2) - (\omega_1 - \omega_2)^2}{(\omega_1 + \omega_2)^2(k_1 + k_2)^2 + (\omega_1 + \omega_2)(k_1 + k_2) - (\omega_1 + \omega_2)^2} \,. \label{1.2.1}$$ In these expressions, the parameters $k_i$ and $\omega_i$ are constrained by the following dispersion relation $$\omega_i = \frac{k_i}{1-k_i^2} \,, \quad i = 1,2 \,. \label{1.3}$$ The actual soliton fields of Gibbon *et al. *are defined by $u \equiv - q_{xt}$.
In what follows, for definiteness, we consider only $1 > k_1 > k_2 > 0$. This implies that both the amplitude and velocity of each soliton will be positive. Furthermore, the soliton corresponding to $\eta_1$ will have a larger amplitude and velocity than the soliton corresponding to $\eta_2$. To illustrate this, in figure \[plot0\_2to0\_7\] the red lines present the plots of the spatial dependence of $u$ at various values of $t$ of an analytical two-soliton simulation. The other curves will be discussed below.
[.34]{} ![This figure shows two solitons interacting with each other at different points in time for three simulations. The red line is obtained using the analytic expression given by equation (\[1.2\]), the green line shows the numerical time evolution of equation (\[1.2\]) as initial conditions, and the blue line shows the numerical time evolution of equation (\[1.4\]) as initial conditions. (Note that the three simulations are so close that their plots are barely distinguishable with the naked eye.)[]{data-label="plot0_2to0_7"}](10_t0_U_field_2sol_second_order_num "fig:")
[.34]{} ![This figure shows two solitons interacting with each other at different points in time for three simulations. The red line is obtained using the analytic expression given by equation (\[1.2\]), the green line shows the numerical time evolution of equation (\[1.2\]) as initial conditions, and the blue line shows the numerical time evolution of equation (\[1.4\]) as initial conditions. (Note that the three simulations are so close that their plots are barely distinguishable with the naked eye.)[]{data-label="plot0_2to0_7"}](10_t200_U_field_2sol_second_order_num "fig:")
[.34]{} ![This figure shows two solitons interacting with each other at different points in time for three simulations. The red line is obtained using the analytic expression given by equation (\[1.2\]), the green line shows the numerical time evolution of equation (\[1.2\]) as initial conditions, and the blue line shows the numerical time evolution of equation (\[1.4\]) as initial conditions. (Note that the three simulations are so close that their plots are barely distinguishable with the naked eye.)[]{data-label="plot0_2to0_7"}](10_t225_U_field_2sol_second_order_num "fig:")
[.34]{} ![This figure shows two solitons interacting with each other at different points in time for three simulations. The red line is obtained using the analytic expression given by equation (\[1.2\]), the green line shows the numerical time evolution of equation (\[1.2\]) as initial conditions, and the blue line shows the numerical time evolution of equation (\[1.4\]) as initial conditions. (Note that the three simulations are so close that their plots are barely distinguishable with the naked eye.)[]{data-label="plot0_2to0_7"}](10_t250_U_field_2sol_second_order_num "fig:")
[.34]{} ![This figure shows two solitons interacting with each other at different points in time for three simulations. The red line is obtained using the analytic expression given by equation (\[1.2\]), the green line shows the numerical time evolution of equation (\[1.2\]) as initial conditions, and the blue line shows the numerical time evolution of equation (\[1.4\]) as initial conditions. (Note that the three simulations are so close that their plots are barely distinguishable with the naked eye.)[]{data-label="plot0_2to0_7"}](10_t275_U_field_2sol_second_order_num "fig:")
[.34]{} ![This figure shows two solitons interacting with each other at different points in time for three simulations. The red line is obtained using the analytic expression given by equation (\[1.2\]), the green line shows the numerical time evolution of equation (\[1.2\]) as initial conditions, and the blue line shows the numerical time evolution of equation (\[1.4\]) as initial conditions. (Note that the three simulations are so close that their plots are barely distinguishable with the naked eye.)[]{data-label="plot0_2to0_7"}](10_t375_U_field_2sol_second_order_num "fig:")
The Hirota method does not generate three-soliton solutions for the mRLW equation, which implies that the model is not Hirota integrable. Furthermore, to our knowledge, nobody has found an analytic expression describing three or more solitons (using any method). Thus, we need to use numerical methods (see the appendix for more details) in order to study the time evolution of three-soliton configurations. To test this scheme empirically, we have used the initial field configurations ([*i.e.*]{}, the initial conditions) expressed by equation (\[1.2\]) and evolved the configuration in time using our procedures. The green lines in figure \[plot0\_2to0\_7\] shows the plots for such a simulation which is produced using the same values for the angular frequencies $\omega_i$ and phase constants $\delta_i$ as for the aforementioned analytical simulation (see appendix \[summary\_of\_variables\] for more information). This allows us to compare the numerical solution with the analytical results. Since these two lines in figure \[plot0\_2to0\_7\] are so close to each other that one can hardly distinguish them, the numerical solution is a good approximation of the analytical values.
Furthermore, it will be useful for the simulations discussed in the next section to also consider the time evolution of the following initial conditions $$q = \ln \left(1 + e^{\eta_1} \right) + \ln \left(1 + e^{\eta_2} \right) \,. \label{1.4}$$ The blue lines in figure \[plot0\_2to0\_7\] present the results of this evolution, where again the parameters governing $\eta_1$ and $\eta_2$ have exactly the same values as for the previously two discussed simulations. All together, figure \[plot0\_2to0\_7\] demonstrates that the numerical time evolution of a linear superposition of two exact one-soliton solutions produces results that are also almost indistinguishable from those of the analytical and numerical two-soliton simulations (provided that the two solitons are initially placed far apart from each other).
The above described three different simulations are so close that with the naked eye they are indistinguishable on the scale used in figure \[plot0\_2to0\_7\]. Therefore, in order to illustrate how close the three lines are, in figure \[plot0\_7\] we have added an insert of the region near the amplitude of the smallest soliton on a much smaller scale. This insert demonstrates very clearly that there is indeed a very small discrepancy between the analytical result and both the numerical simulations, but even on this scale we cannot distinguish between the numerical time evolution of equation (\[1.2\]) and equation (\[1.4\]) (and upon zooming in on the region of the larger soliton, we find that the discrepancies between the simulations are of similarily small magnitudes). This is expected since looking at the time evolution of the two solitons, we see that most of the interaction takes place when the solitons are close together. We see that in this case each soliton has a size of about $L\sim 20$, and since they are initially placed further apart than $L$ (see figure \[plot0\_2\]), the errors of such an approximation are only in the interaction of their ‘tails’.
When the two solitons scatter, the only result of their interaction after the collision is the phase-shift they experience. In order to determine the analytical expression of this phase-shift, let us introduce two new variables $y \equiv x - \frac{\omega_1}{k_1}t$ and $z \equiv x - \frac{\omega_2}{k_2}t$. Then, substituting $y$ into $\eta_1$ and $\eta_2$ ([*i.e.*]{}, $\eta_1 = k_1 y + \delta_1$ and $\eta_2 = k_2(y + \frac{\omega_1}{k_1} t) - \omega_2t + \delta_2$ ) the exact two-soliton solution can be asymptotically approximated as $$\lim_{t \to - \infty} u(y,t) \approx \frac{k_1 \omega_1 e^{\eta_1}}{\left( 1 + e^{\eta_1} \right)^2} \quad \text{and} \quad \lim_{t \to \infty} u(y,t) \approx \frac{k_1 \omega_1 e^{\eta_1 + \ln A_{12}}}{\left( 1 + e^{\eta_1 + \ln A_{12}} \right)^2} \,.$$ Similarly, introducing the variable $z$ gives $$\lim_{t \to - \infty} u(z,t) \approx \frac{k_2 \omega_2 e^{\eta_2 + \ln A_{12}}}{\left( 1 + e^{\eta_2 + \ln A_{12}} \right)^2} \quad \text{and} \quad \lim_{t \to \infty} u(z,t) \approx \frac{k_2 \omega_2 e^{\eta_2}}{\left( 1 + e^{\eta_2} \right)^2} \,.$$ Thus, after the collision the solitary wave corresponding to $\eta_1$ is phase-shifted forward by $\ln A_{12}$ and the wave corresponding to $\eta_2$ is phase-shifted by $\ln A_{12}$ in the opposite direction.
In the following section we analyse the phase-shift that solitons experience during three-soliton scattering. Since no three-soliton solutions are known, we investigate them numerically, and so it will be useful to test the reliability of our numerical method for determining the phase-shifts of the analytical and numerical two-soliton simulations.[^2] We estimate this error by dividing the computationally observed phase-shift (from either the analytical or numerical simulation) by $\ln A_{12}$ ([*i.e.*]{}, the analytical expression of the phase shift-shift). With this definition, we have found that the error is always small but it is somewhat sensitive to the details of our procedure. In fact we have found that in all our (analytical and numerical) simulations the error had always been smaller than $5.0 \%$. Since we also observed errors close to $5.0 \%$ for the analytical simulations, we can conclude that these errors are mainly a result of the algorithm of determining the phase-shift rather than resulting from the finite difference scheme approximating the mRLW equation.
Finally, let us briefly discuss the conserved charges for the mRLW equation. As far as we are aware, its only known conservation laws are given by $$\partial_t \int\limits^\infty_{-\infty } \mathrm{d} x \; (-q_{xt}) = \partial_t \int\limits^\infty_{-\infty } \mathrm{d} x \; u = 0$$ and $$\partial_t \int\limits^\infty_{-\infty } \mathrm{d} x \; (-q_{xx}) = 0 \,,$$ which can be easily verified by taking the derivative of equation (\[MRL\]) with respect to $x$. Let us add that one can approximate the values of these conserved charges for the analytical one- and two-soliton solutions as follows $$Q_1 \equiv \int\limits^\infty_{-\infty } \mathrm{d} x \; (-q_{xt}) = [-q_t]^\infty_{x=-\infty} \approx \sum\limits_{i=1}^N \omega_i \,,\quad N = 1,2 \,, \label{1.5}$$ and $$Q_2 \equiv \int\limits^\infty_{-\infty } \mathrm{d} x \; (-q_{xx}) = [-q_x]^\infty_{x=-\infty} \approx -\sum\limits_{i=1}^N k_i \,,\quad N = 1,2 \,. \label{1.6}$$ In the next section, we check whether these quantities are also conserved for the numerically evolved three-soliton configurations.
Numerical three-soliton solutions {#Numerical_three_soliton_solution}
=================================
So far we discussed mainly two-soliton configurations. In this section we look at systems involving three solitons. As we stated before, the analytical three-soliton solutions (or solutions involving even more solitons) of the mRLW equation are not known, and they cannot be found by the Hirota method. This has been stated in literature and we have verified this claim for three and four solitons. So, we do not really know whether such solutions exist or not; all we know is that if they exist, their forms cannot be found by the Hirota method.
As we have shown in the previous section, a linear superposition of two single-soliton solutions is almost indistinguishable from the analytical two-soliton solution when, initially, these two solitons are far enough apart. Armed with this observation, we have numerically simulated the time evolution of a three-soliton system by using the linear superposition of three well-separated one-soliton solutions as the initial conditions for our simulations. In other words, we use $$q = \ln \left(1 + e^{\eta_1} \right) + \ln \left(1 + e^{\eta_2} \right) + \ln \left(1 + e^{\eta_3} \right) \label{sup3}$$ to construct the initial conditions of a three-soliton system. We have performed many such simulations using a range of variables describing the frequencies of individual solitons and, in the next subsection, we discuss the results of two of such simulations for illustrative purposes.
Three-soliton interactions
--------------------------
We are primarily interested in three-soliton interactions, and so for the simulations discussed in this section the phase constants $\delta_i$ have always been chosen in such a way that all three solitons scatter with each other at more or less the same time. To illustrate this, figures \[plot1t200to1t675\]
[.34]{} ![This figure shows three solitons interacting with each other at various points in time. Note that at all times during the interaction, there are three distinct maxima present.[]{data-label="plot1t200to1t675"}](11t200_U_field_matrix_3sol_second_order_num "fig:")
[.34]{} ![This figure shows three solitons interacting with each other at various points in time. Note that at all times during the interaction, there are three distinct maxima present.[]{data-label="plot1t200to1t675"}](11t400_U_field_matrix_3sol_second_order_num "fig:")
[.34]{} ![This figure shows three solitons interacting with each other at various points in time. Note that at all times during the interaction, there are three distinct maxima present.[]{data-label="plot1t200to1t675"}](11t425_U_field_matrix_3sol_second_order_num "fig:")
[.34]{} ![This figure shows three solitons interacting with each other at various points in time. Note that at all times during the interaction, there are three distinct maxima present.[]{data-label="plot1t200to1t675"}](11t450_U_field_matrix_3sol_second_order_num "fig:")
[.34]{} ![This figure shows three solitons interacting with each other at various points in time. Note that at all times during the interaction, there are three distinct maxima present.[]{data-label="plot1t200to1t675"}](11t475_U_field_matrix_3sol_second_order_num "fig:")
[.34]{} ![This figure shows three solitons interacting with each other at various points in time. Note that at all times during the interaction, there are three distinct maxima present.[]{data-label="plot1t200to1t675"}](11t675_U_field_matrix_3sol_second_order_num "fig:")
and \[plot3t200to1t775\] show two of such interactions, where both simulations start at $t=0$ with the solitons placed far apart of each other. For these two simulations, figures \[1amplitude\_vs\_time\_and\_1location\_vs\_time\]
[.34]{} ![For figures \[1amplitude\_vs\_time\] and \[1location\_vs\_time\], the red dots present the time dependence of the amplitude and the location of the left-soliton of the simulation shown in figure \[plot1t200to1t675\]. Similarly, the green and blue dots correspond to the middle- and right-soliton, respectively. Finally, figure \[1charge\_vs\_time\] shows how the conserved charges of the corresponding simulation vary with time.[]{data-label="1amplitude_vs_time_and_1location_vs_time"}](11amplitude_vs_time "fig:")
[.34]{} ![For figures \[1amplitude\_vs\_time\] and \[1location\_vs\_time\], the red dots present the time dependence of the amplitude and the location of the left-soliton of the simulation shown in figure \[plot1t200to1t675\]. Similarly, the green and blue dots correspond to the middle- and right-soliton, respectively. Finally, figure \[1charge\_vs\_time\] shows how the conserved charges of the corresponding simulation vary with time.[]{data-label="1amplitude_vs_time_and_1location_vs_time"}](11location_of_soliton_vs_time "fig:")
[.34]{} ![For figures \[1amplitude\_vs\_time\] and \[1location\_vs\_time\], the red dots present the time dependence of the amplitude and the location of the left-soliton of the simulation shown in figure \[plot1t200to1t675\]. Similarly, the green and blue dots correspond to the middle- and right-soliton, respectively. Finally, figure \[1charge\_vs\_time\] shows how the conserved charges of the corresponding simulation vary with time.[]{data-label="1amplitude_vs_time_and_1location_vs_time"}](11conserved_charge "fig:")
and \[3amplitude\_vs\_time\_and\_1location\_vs\_time\]
[.34]{} ![This figure shows three solitons interacting with each other at various points in time. Note that there is a small period of time ($17 \lesssim t \lesssim 23$) during the collision where the smallest wave is ‘absorbed’ and there are only two distinct maxima present (see figure \[3t425\] for example).[]{data-label="plot3t200to1t775"}](12t200_U_field_matrix_3sol_second_order_num "fig:")
[.34]{} ![This figure shows three solitons interacting with each other at various points in time. Note that there is a small period of time ($17 \lesssim t \lesssim 23$) during the collision where the smallest wave is ‘absorbed’ and there are only two distinct maxima present (see figure \[3t425\] for example).[]{data-label="plot3t200to1t775"}](12t400_U_field_matrix_3sol_second_order_num "fig:")
[.34]{} ![This figure shows three solitons interacting with each other at various points in time. Note that there is a small period of time ($17 \lesssim t \lesssim 23$) during the collision where the smallest wave is ‘absorbed’ and there are only two distinct maxima present (see figure \[3t425\] for example).[]{data-label="plot3t200to1t775"}](12t425_U_field_matrix_3sol_second_order_num "fig:")
[.34]{} ![This figure shows three solitons interacting with each other at various points in time. Note that there is a small period of time ($17 \lesssim t \lesssim 23$) during the collision where the smallest wave is ‘absorbed’ and there are only two distinct maxima present (see figure \[3t425\] for example).[]{data-label="plot3t200to1t775"}](12t550_U_field_matrix_3sol_second_order_num "fig:")
[.34]{} ![This figure shows three solitons interacting with each other at various points in time. Note that there is a small period of time ($17 \lesssim t \lesssim 23$) during the collision where the smallest wave is ‘absorbed’ and there are only two distinct maxima present (see figure \[3t425\] for example).[]{data-label="plot3t200to1t775"}](12t575_U_field_matrix_3sol_second_order_num "fig:")
[.34]{} ![This figure shows three solitons interacting with each other at various points in time. Note that there is a small period of time ($17 \lesssim t \lesssim 23$) during the collision where the smallest wave is ‘absorbed’ and there are only two distinct maxima present (see figure \[3t425\] for example).[]{data-label="plot3t200to1t775"}](12t775_U_field_matrix_3sol_second_order_num "fig:")
[.34]{} ![For figures \[3amplitude\_vs\_time\] and \[3location\_vs\_time\], the red dots present the time dependence of the amplitude and the location of the left-soliton of the simulation shown in figure \[plot3t200to1t775\]. Similarly, the green and blue dots correspond to the middle- and right-soliton, respectively. Finally, figure \[3charge\_vs\_time\] shows how the conserved charges of the corresponding simulation vary with time.[]{data-label="3amplitude_vs_time_and_1location_vs_time"}](12amplitude_vs_time "fig:")
[.34]{} ![For figures \[3amplitude\_vs\_time\] and \[3location\_vs\_time\], the red dots present the time dependence of the amplitude and the location of the left-soliton of the simulation shown in figure \[plot3t200to1t775\]. Similarly, the green and blue dots correspond to the middle- and right-soliton, respectively. Finally, figure \[3charge\_vs\_time\] shows how the conserved charges of the corresponding simulation vary with time.[]{data-label="3amplitude_vs_time_and_1location_vs_time"}](12location_of_soliton_vs_time "fig:")
[.34]{} ![For figures \[3amplitude\_vs\_time\] and \[3location\_vs\_time\], the red dots present the time dependence of the amplitude and the location of the left-soliton of the simulation shown in figure \[plot3t200to1t775\]. Similarly, the green and blue dots correspond to the middle- and right-soliton, respectively. Finally, figure \[3charge\_vs\_time\] shows how the conserved charges of the corresponding simulation vary with time.[]{data-label="3amplitude_vs_time_and_1location_vs_time"}](12conserved_charge "fig:")
show, respectively, the time dependence of the amplitude and the location (defined as the location of the amplitude of the wave) of each soliton, and the time dependence of the charges $Q_1$ and $Q_2$. Using these results, we have observed that for all our simulations, after the three-soliton interaction, the solitons recovered their original amplitudes and velocities with a numerical error of less than $0.1 \%$. Furthermore, the quantities $Q_1$ and $Q_2$ have been incredibly well conserved for all our simulations with a numerical error of less than $0.001 \%$ compared to the approximations given by equations (\[1.5\]) and (\[1.6\]). These results demonstrate that the solitons, in these configurations, behave as solitons in integrable models.
Assuming that the total phase-shift each soliton experiences when three solitons collide is the sum of the pairwise phase-shifts, then, analytically, we expect the location of the largest soliton (related to $\eta_1$) to experience a phase-shift forward by $\ln(A_{12})+\ln(A_{13})$, the soliton corresponding to $\eta_2$ to experience a phase-shift given by $-\ln(A_{12})+\ln(A_{23})$ and the smallest soliton to be phase-shifted by $-\ln(A_{13})-\ln(A_{23})$, where $A_{13}$ and $A_{23}$ are defined in a similar way as $A_{12}$ (see equation (\[1.2.1\])). We found that the phase-shifts for all our simulations have always been within $5.0\%$ of the expected analytic value. Since these errors are consistent with the errors discussed near the end of section \[Modified\_regularized\_long-wave\_equation\_equation\], we conclude that the phase-shifts experienced by each soliton during a numerical three-soliton interaction is additive, which suggest the non-existence of additional ‘three-body’ forces, ([*i.e.*]{}, the phase shifts can be explained by the additivity of ‘two-body’ forces). This is another indicator that the solitons of our model behave like solitons in an integrable model.
Soliton resolution conjecture {#soliton_resolution_conjecture}
=============================
In this section we investigate the time evolution of soliton-like lumps which are not analytical solutions of the mRLW equation. This is a test of the idea that such initial field configurations may eventually decouple into soliton-like and radiation-like components. For many non-linear dispersive equations there is evidence suggesting that such arbitrary finite-energy initial configurations always decouple in such a manner and this expectation is also referred to as the soliton resolution conjecture [@tao].[^3]
Of course the choice of the initial lump-like configuration is very arbitrary - one could take a Gaussian field or any other more complicated initial condition field for which the $u=-q_{xt}$ resembles a soliton-like structure. However, since our numerical scheme requires an analytical expression for $q$ and $q_t$ as initial conditions, and the field we are interested in is described by $u=-q_{xt}$, we are somewhat limited in our choices.
Figure \[plot8\_000to6\_250\]
[.34]{} ![Plots of $x$-dependence of $u(x,t)$ (at various values of $t$) of the numerical evolution of $q$ field with the initial condition taken from equation (\[2.2\]).[]{data-label="plot8_000to6_250"}](13_t0_U_field_2sol_second_order_num "fig:")
[.34]{} ![Plots of $x$-dependence of $u(x,t)$ (at various values of $t$) of the numerical evolution of $q$ field with the initial condition taken from equation (\[2.2\]).[]{data-label="plot8_000to6_250"}](13_t2_U_field_2sol_second_order_num "fig:")
[.34]{} ![Plots of $x$-dependence of $u(x,t)$ (at various values of $t$) of the numerical evolution of $q$ field with the initial condition taken from equation (\[2.2\]).[]{data-label="plot8_000to6_250"}](13_t8_U_field_2sol_second_order_num "fig:")
[.34]{} ![Plots of $x$-dependence of $u(x,t)$ (at various values of $t$) of the numerical evolution of $q$ field with the initial condition taken from equation (\[2.2\]).[]{data-label="plot8_000to6_250"}](13_t36_U_field_2sol_second_order_num "fig:")
[.34]{} ![Plots of $x$-dependence of $u(x,t)$ (at various values of $t$) of the numerical evolution of $q$ field with the initial condition taken from equation (\[2.2\]).[]{data-label="plot8_000to6_250"}](13_t72_U_field_2sol_second_order_num "fig:")
[.34]{} ![Plots of $x$-dependence of $u(x,t)$ (at various values of $t$) of the numerical evolution of $q$ field with the initial condition taken from equation (\[2.2\]).[]{data-label="plot8_000to6_250"}](13_t130_U_field_2sol_second_order_num "fig:")
shows the time evolution of our first choice of the lump given by $$q = -t \tanh(x) \implies u = \operatorname{sech}^2(x) \,. \label{2.2}$$ We started this simulation at $t=1$ and, as fig \[8\_000\] shows, the initial conditions describe a lump with a positive amplitude located at $x=0$. Then figures \[8\_050\] and \[8\_100\] show that this initial configuration evolved into a lump and an anti-lump configuration. The anti-lump traveled to the left leaving behind some ‘radiation’, whereas the positive lump moved to the right without emitting any (visible) ‘radiation’ (see figures \[8\_150\], \[8\_200\], and \[8\_250\]). The ‘radiation’ left behind by the anti-lump was also slowly moving to the left and (although it is difficult to see this by looking at figure \[plot8\_000to6\_250\]) it also emitted some further ‘radiation’ which started to travel to the right (but not fast enough to catch up with the positive lump). We ran this simulation until $t = 241$ without the system blowing up. The simulation might blow up if we run it for a longer time, but we lacked the computational power for studying this further. Let us add that when we repeated this simulation with a negative amplitude ([*i.e.*]{}, using $u = -\operatorname{sech}^2(x)$ as initial conditions), then the simulation blew up at approximately $t = 28.9$. From this we see that at least the ‘lump-like’ described by equation (\[2.2\]) was reasonably robust and long-lived.
However, this was not the case for the evolution of another ‘lump-like’ configuration, this time described by the following Gaussian function $$q = -t \operatorname{erf}(0.5 x) \implies u = \frac{1}{\sqrt{\pi}} \exp\left(-x^2\right) \,. \label{2.1}$$ Its time evolution is presented in figure \[plot6\_000to6\_200\].
[.34]{} ![Plots of $x$-dependence of $u(x,t)$ (at various values of $t$) of the numerical evolution of $q$ field with the initial condition taken from equation (\[2.1\]).[]{data-label="plot6_000to6_200"}](14_t0_U_field_2sol_second_order_num "fig:")
[.34]{} ![Plots of $x$-dependence of $u(x,t)$ (at various values of $t$) of the numerical evolution of $q$ field with the initial condition taken from equation (\[2.1\]).[]{data-label="plot6_000to6_200"}](14_t25_U_field_2sol_second_order_num "fig:")
[.34]{} ![Plots of $x$-dependence of $u(x,t)$ (at various values of $t$) of the numerical evolution of $q$ field with the initial condition taken from equation (\[2.1\]).[]{data-label="plot6_000to6_200"}](14_t50_U_field_2sol_second_order_num "fig:")
[.34]{} ![Plots of $x$-dependence of $u(x,t)$ (at various values of $t$) of the numerical evolution of $q$ field with the initial condition taken from equation (\[2.1\]).[]{data-label="plot6_000to6_200"}](14_t225_U_field_2sol_second_order_num "fig:")
[.34]{} ![Plots of $x$-dependence of $u(x,t)$ (at various values of $t$) of the numerical evolution of $q$ field with the initial condition taken from equation (\[2.1\]).[]{data-label="plot6_000to6_200"}](14_t275_U_field_2sol_second_order_num "fig:")
[.34]{} ![Plots of $x$-dependence of $u(x,t)$ (at various values of $t$) of the numerical evolution of $q$ field with the initial condition taken from equation (\[2.1\]).[]{data-label="plot6_000to6_200"}](14_t290_U_field_2sol_second_order_num "fig:")
This time we saw that the lump, which at the start of the simulation ($t=1$) was located at $x=0$ (see figure \[6\_000\]), started to move to the right, also leaving some ‘radiation’ behind. However, our plots show that the ‘radiation’ started to develop larger amplitudes as the time progressed. This can be seen by comparing figure \[6\_160\] with figure \[6\_200\] which show that the amplitudes of the ‘radiation’ had grown significantly during a short period of time. In fact, at approximately $t=67.2$ the whole system blew up. The same happened when we repeated the simulation with a negative amplitude ([*i.e.*]{}, using $u = -\pi^{-1/2} \exp\left(-x^2\right)$ as initial conditions). In this case the simulation blew up at approximately $t =11.9$ ([*i.e.*]{} again, smaller than for a positive initial amplitude of the lump).
Let us add that we believe that the blow-ups described in this section are not numerical artefacts, since we checked this by changing the parameters of the grid-spacing and time-steps, and they have always happened at roughly the same values of $t$.
So, to sum up, we feel that the blow-ups described in this section imply that the soliton resolution conjecture does not hold for the mRLW equation, whereas it does hold for many integrable systems such as the KdV and the non-linear Schrödinger equation. Numerically, this is the only observation we have found that distinguishes the non-integrable mRLW equation from many other integrable systems and this may be related to the fact that this model does not possess a conserved quantity which controls and limits the growth of the amplitude (such as the energy in many systems).
Conclusions and further comments
================================
In this paper we have investigated the (numerical) time evolution of two- and three-soliton configurations of the mRLW equation. When we numerically evolved the initial conditions described by the analytic two-soliton solution and compared the results with the corresponding analytical simulation, we have found that they were essentially indistinguishable. This provided a good test of our numerical procedure and reassured us that the results of our simulations could be trusted. Furthermore, our results agreed with the results presented in the original paper by Gibbon *et al.* [@Gibbon], where they overlapped. In addition to the investigation of the numerical two-soliton solutions, we have also studied the numerical time evolution of systems constructed by the superposition of two one-soliton solutions, and have found that these configurations approximated the analytical two-soliton solutions very closely. This has led us to consider three-soliton configurations constructed by taking a superposition of three one-soliton solutions. The numerical time evolution of these three-soliton configurations behaved very much like those seen in integrable models; the field evolved well, there did not seem to be any breaking or changes to the individual solitons whenever they were far from each other, and after the scattering the solitons emerged with their original shapes and velocities. Furthermore, the phase-shift experienced by each soliton was additive, and the known conservation laws were also obeyed for such configurations. This suggests to us that analytical three-soliton solutions may exist (though their analytic forms cannot be found by Hirota’s method).
We also looked at the time evolutions of various lumps - [*i.e.*]{}, fields that crudely resemble a single soliton field but are not exact solutions of the mRLW equation. We have found that the system blows up for some lumps, which is most likely a consequence of instabilities of the model which lead to the development of very steep gradients causing our numerical procedures to break down. We have checked that these blow-ups were genuine properties of the evolution of these field configurations rather than numerical artefacts. Our results imply that the soliton resolution conjecture does not hold for the mRLW equation. Numerically, this is the only property of this non-integrable model with a behaviour that differs from many other integrable models. It would therefore be interesting to study other systems which are not Hirota integrable but do possess one- and two-soliton Hirota solutions and check if such systems show similar properties to the mRLW equation ([*i.e.*]{}, whether they possess numerical three-soliton solutions in which the solitons behave exactly as one would expect from integrable solitons, but the soliton resolution conjecture does not hold). This could shine some new light on the connection between (Hirota) integrability and the soliton resolution conjecture.
So overall, our results show that the mRLW model has many interesting properties and in many ways behaves like an integrable model. This has led us to consider whether we can think of this model as a finite perturbation of an integrable model and an obvious suggestion here is to think of this model as a perturbation of the KdV equation.[^4] So together with L. A. Ferreira we are now trying to carry out such a procedure and we hope that it will help us to understand the integrability/quasi-integrability properties of this model better. We hope to be able to say more on this topic soon.
[**Acknowledgements:**]{} [We would like to thank L. A. Ferreira for many constructive and helpful comments and for working with us on some problems discussed in this paper. WJZ would like to thank the Royal Society for its grant to collaborate with L. A. Ferreira. Both authors thank the FAPESP/Durham University for their grant to facilitate their visits to the USP at São Carlos, and the Department of Physics in São Carlos for its hospitality.]{}
Numerical methods {#Numerical_method}
=================
Our numerical procedure for solving the mRLW equation combines the explicit and implicit methods of solving the equations of motion of the model. This is due to the fact that the term in the equation which contains the highest time derivatives also possesses spatial derivatives. A similar problem was encountered by J. C. Eilbeck and G. R. McGuire when they numerically investigated the regularized long-wave equation [@Eilbeck]. Their paper provides a detailed discussion of how to use such methods. We followed their ideas, modifying them appropriately for our investigations, and we present a short discussion of our procedure in the following subsection.
Numerical approximation of mRLW equation {#Numerical_approx}
----------------------------------------
First, to simplify the mRLW equation, we introduce a new field $p$ as follows $$p = q_t \label{A.1}$$ so that the mRLW equation takes the form $$p_{xxt} + 2 q_{xx} p_t + 4 p_x^2 - p_x - p_t = 0 \,. \label{A.2}$$ We then take a finite set of points $x_0, x_1, \ldots, x_N$ and $t_0, t_1, \ldots, t_K$, and let $h$ denote the grid-spacing and $\tau$ the time-step. Furthermore, we denote the grid points as $(i h , m \tau) \equiv (i,m)$, where $i = 0,1,2,\ldots, N$ and $m = 0,1,2,\ldots, K$, and we employ the following notation: $p_i{}^m \equiv p(ih,m \tau)$ and $q_i{}^m \equiv q(ih,m \tau)$. Finally, we choose $v_i{}^m$ to denote our approximation to $p_i{}^m$, and $w_i{}^m$ to denote our approximation to $q_i{}^m$.
Next, we introduce the following central finite difference operators by their actions on $v_i{}^m$ as follows $$\begin{aligned}
\delta_x^2 v_i{}^m & = (v^m_{i+1} - 2 v_i{}^m + v^m_{i-1})/h^2 \,, \\
H_x v_i{}^m & = (v^m_{i+1}-v^m_{i-1}) / 2h \,, \\
H_t v_i{}^m & = (v^{m+1}_i-v^{m-1}_i) / 2 \tau \,,
\end{aligned}$$ and similarly on $w_i{}^m$. Applying these operators to equation (\[A.2\]) in a straightforward manner yields $$\delta_x^2 H_t v_i{}^m + 2 \delta_x^2 w_i{}^m H_t v_i{}^m + 4 \left( H_x v_i{}^m \right)^2 - H_x v_i{}^m - H_t v_i{}^m = 0 \,,$$ which can be rewritten as $$\begin{aligned}
\begin{split}
- \frac{v_i^{m+1}}{2 \tau} & + \frac{v_{i+1}^{m+1}-2v^{m+1}_i + v_{i-1}^{m+1}}{2h^2 \tau} + \frac{w_{i+1}^m v_i^{m+1} - 2 w_i{}^m v_i^{m+1} + w^m_{i-1}v_i^{m+1}}{h^2 \tau} \\& = - \frac{v_i^{m-1}}{2 \tau} + \frac{v_{i+1}^m- v_{i-1}^m}{2h} + \frac{v_{i+1}^{m-1}-2v_i^{m-1}+v_{i-1}^{m-1}}{2h^2 \tau} \\& \hspace{4.5 mm} + \frac{w_{i+1}^m v_i{}^{m-1} - 2w_i{}^m v_i^{m-1} + w_{i-1}^m v_i^{m-1}}{h^2 \tau} \\& \hspace{4.5 mm} - \frac{(v_{i+1}^m)^2 - 2 v_{i+1}^m v_{i-1}^m + (v_{i-1}^m)^2}{h^2} \,. \label{A.2.1}
\end{split}
\end{aligned}$$ Let us now introduce the following matrices $$A \equiv \begin{pmatrix} 1 & 0 & 0 & \cdots & & 0 \\
\frac{1}{2h^2 \tau} & a_1{}^m & \frac{1}{2h^2 \tau} & 0 & & \vdots \\
0 & \frac{1}{2h^2 \tau} & a_2{}^m & \frac{1}{2h^2 \tau} & \\
\vdots & & \ddots & \ddots & \ddots & 0 \\
& & & \frac{1}{2h^2 \tau} & a_{N-1}^m & \frac{1}{2h^2 \tau} \\
0 & \cdots & & 0 & 0 & 1
\end{pmatrix} \,,$$ $$B \equiv \begin{pmatrix} v_0^{m+1} \\ v_1^{m+1} \\ v_2^{m+1} \\ \vdots \\ v_{N-1}^{m+1} \\ v_N^{m+1} \end{pmatrix} \,,\quad C \equiv \begin{pmatrix} c_0{}^{m} \\ c_1{}^{m} \\ c_2{}^{m} \\ \vdots \\ c_{N-1}^{m} \\ c_N^{m} \end{pmatrix} \,,$$ where $$a_i{}^m \equiv \frac{-1 + w_{i+1}^m - 2 w_i{}^m + w_{i-1}^m}{h^2 \tau} - \frac{1}{2\tau} \,,$$ $$c_0{}^{m} \equiv v_0^{m+1} \,,\quad c_N^{m} \equiv v_N^{m+1}\,,$$ $$\begin{aligned}
\begin{split}
c_i{}^m \equiv & - \frac{v_i^{m-1}}{2 \tau} + \frac{v_{i+1}^m- v_{i-1}^m}{2h} + \frac{v_{i+1}^{m-1}-2v_i^{m-1}+v_{i-1}^{m-1}}{2h^2 \tau} \\& + \frac{w_{i+1}^m v_i{}^{m-1} - 2w_i{}^m v_i^{m-1} + w_{i-1}^m v_i^{m-1}}{h^2 \tau} \\& - \frac{(v_{i+1}^m)^2 - 2 v_{i+1}^m v_{i-1}^m + (v_{i-1}^m)^2}{h^2} \,,\quad i = 1, 2 , \hdots, N-1 \,,
\end{split}
\end{aligned}$$ so that equation (\[A.2.1\]) can be rewritten as the matrix equation $$AB = C \,.$$ Hence, we need to solve this equation for the vector $B$. This can done by using the well-known $LU$ decomposition method [@Schay].
Once we have solved for $B$, we have the values of $v_i^m$ at the next time level. One can then find the values of all $w_i^m$ by solving equation (\[A.1\]) using the central difference operator, that is, $$v_i{}^m = \frac{w^{m+1}_i - w_i^{m-1}}{2 \tau} \implies w_i^{m+1} = 2 \tau v_i{}^m + w_i^{m-1} \,. \label{A.3}$$ We then repeat this procedure for all time levels in order to calculate the time evolution of any initial configuration.
It is not too difficult to verify that this scheme is both second-order accurate in $\tau$ and in $h$. Furthermore, we have extensively tested this scheme against the analytical one- and two-soliton solutions. These tests have shown that the numerical simulations approximate the analytical solutions extremely well without any (visible) loss of radiation. This indicates that, for the values of $h$ and $\tau$ that we have used, the scheme is stable and we can trust its results.
Finally, let us add that had we substituted the finite difference operators directly into the mRLW equation without first making the substitution expressed by equation (\[A.1\]), then the $4q_{xt}^2$ term would have yielded a term $(-w_{i+1}^{m+1}w_{i-1}^{m+1})/(2h^2 \tau^2)$; and since this term contains two unknowns, ([*i.e.*]{}, $w_{i+1}^{m+1}$ and $w_{i-1}^{m+1}$), we would not have been able to solve for it. Moreover, for very similar reasons, we cannot use the well-known Crank-Nicolson method to numerically solve the mRLW equation.
Summary of parameters used to produce figures {#summary_of_variables}
---------------------------------------------
Table \[variables1\] shows all the parameters used to produce the figures shown in this paper.
[ l?[0.5mm]{} c | c | c | c | c]{} & Figure \[plot0\_2to0\_7\] & Figures \[plot1t200to1t675\] and \[1amplitude\_vs\_time\_and\_1location\_vs\_time\] & Figures \[plot3t200to1t775\] and \[3amplitude\_vs\_time\_and\_1location\_vs\_time\] & Figure \[plot8\_000to6\_250\] & Figure \[plot6\_000to6\_200\]\
$x_0$ & $-50$ & $-220$ & $-125$ & $-600$ & $-150$\
$x_N$ & $250$ & $250$ & $175$ & $1200$ & $350$\
$h$ & $0.1$ & $0.1$ & $0.1$ & $0.1$ & $0.1$\
$t_0$ & $0$ & $0$ & $0$ & $1$ & $1$\
$t_K$ & $40$ & $170$ & $42$ & $241$ & $81$\
$\tau$ & $0.001$ & $0.001$ & $0.001$ & $0.01$ & $0.001$\
$\omega_1$ & $5.00$ & $0.80$ & $0.80$ & N/A & N/A\
$\delta_1$ & $0.00$ & $66.51$ & $16.63$ & N/A & N/A\
$\omega_2$ & $3.00$ & $1.33$ & $3.07$ & N/A & N/A\
$\delta_2$ & $0.00$ & $110.90$ & $63.77$ & N/A & N/A\
$\omega_3$ & N/A & $1.84$ & $4.28$ & N/A & N/A\
$\delta_3$ & N/A & $152.88$ & $89.00$ & N/A & N/A \[variables1\]
[1]{}
J. C. Eilbeck and G. R. McGuire, *Numerical Study of the Regularized Longe-Wave Equation I: Numerical Methods*, J. Comp. Phys., **19** 1, 43-57 (1975)
L. A. Ferreira and W. J. Zakrzewski, *The concept of quasi-integrability: a concrete example*, JHEP, 1105 (2011)
L. A. Ferreira and W. J. Zakrzewski, *Numerical and analytical tests of quasi-integrability in modified sine-Gordon models*, JHEP, 58 (2014)
J. D. Gibbon, J. C. Eilbeck and R. K. Dodd, *A modified regularized long-wave equation with an exact two-soliton solution*, J. Phys. **A** 9, L127-L130 (1976)
J. Hietarinta, *Scattering of solitons and dromions* in *Scattering* edited by E. R. Pike and Pierre C. Sabatier, Academic Press, 1773-1791 (2002)
J. Hietarinta, *Introduction to the Hirota Bilinear Method*, Lecture Notes in Physics, Volume 638, Springer-Verlag, 101 (2004)
J. Hietarinta, *Hirota’s Bilinear Method and Its Connection with Integrability*, Lecture Notes in Physics, Volume 767, Springer Berlin Heidelberg, 279-314 (2009)
LMS symposium in Durham, *Geometric and Algebraic Aspects of Integrability*, Available at: http://www.maths.dur.ac.uk/events/Meetings/LMS/105/talks.html, (2016)
G. Schay, *A Concise Introduction to Linear Algebra*, Birkhäuser Boston, (2012)
T. Tao, *Why are solitons stable?*, \[online\] Arxiv.org. Available at: https://arxiv.org/abs/0802.2408v2 \[Accessed 12 May 2017\], (2008)
[^1]: We thank J. Hietarinta for drawing our attention to this model.
[^2]: We determine the phase-shift by finding the three highest points of each individual soliton at some $t$, and assume they fit a polynomial of degree $2$. Subsequently, we use each polynomial to estimate the position and height of its absolute maximum. We repeat this procedure for many values of time (see, for instance, figures \[1amplitude\_vs\_time\_and\_1location\_vs\_time\] and \[3amplitude\_vs\_time\_and\_1location\_vs\_time\] in the next section), and this has allowed us to determine the phase-shift experienced by the solitons.
[^3]: We thank A. Hone for drawing our attention to this conjecture.
[^4]: We thank L. A. Ferreira for this suggestion.
|
---
abstract: 'Fisher and Carpenter (*High-order entropy stable finite difference schemes for non-linear conservation laws: Finite domains, Journal of Computational Physics, 252:518–557, 2013*) found a remarkable equivalence of general diagonal norm high-order summation-by-parts operators to a subcell based high-order finite volume formulation. This equivalence enables the construction of provably entropy stable schemes'
address:
- 'Mathematisches Institut, Universität zu Köln, Weyertal 86-90, 50931 Köln, Germany'
- 'Department of Mathematics, The Florida State University, Tallahassee FL 32312, USA'
author:
- 'Gregor J. Gassner'
- 'Andrew R. Winters'
- David Kopriva
bibliography:
- 'References.bib'
title: 'Split Form Nodal Discontinuous Galerkin Schemes with Summation-By-Parts Property for the Compressible Euler Equations'
---
Ducros splitting ,Kennedy and Gruber splitting ,kinetic energy ,discontinuous Galerkin spectral element method ,3D compressible Euler equations ,Taylor-Green vortex ,split form
Introduction {#sec:intro}
============
This paper nodal discontinuous Galerkin (DG) approximations for the advective terms of the three dimensional compressible Navier-Stokes equations, namely the compressible Euler equations. In the discontinuous Galerkin community, stabilising an approximation is frequently done by . The typical DG implementation is under-integrated. It is well-known that when the numerical quadrature in the DG scheme is constructed so that flux functions that depend linearly on the solution (e.g. linear advection equation) are integrated exactly, the approximation retains the formal order of accuracy [@CS1; @CS2; @CS4; @CS5; @CS6]. The exact number of quadrature points depends on the polynomial ansatz space, on the element type and, of course, on the specific quadrature rule used. A similar concept is used for nodal DG schemes where the ansatz uses interpolation and (multi-)variate Lagrange-type basis functions. In many cases, DG discretisations for linear fluxes are directly applied to problems with non-linear flux functions by simply exchanging the linear flux function with the corresponding non-linear flux. The reasoning is clear, as the minimum number of quadrature (or interpolation) nodes necessary to obtain the expected order of convergence gives an implementation with the lowest number of arithmetic operations, and thus increased efficiency at first sight.
Stabilisation strategies for discontinuous Galerkin based discretisations {#sec:dg_dealiasing}
-------------------------------------------------------------------------
It turns out that the strategy of minimal effort has a drastic impact when the numerical solution is under-resolved. In such cases, e.g. under-resolved turbulence or shocks, aliasing errors due to variational errors corrupt the approximate solution and may even drive a non-linear instability. Whereas such a non-linear instability is often masked by excessive artificial viscosity when using low order approximations, high-order discretisations with their lower inherent numerical dissipation are prone to such instabilities. In fact, without additional counter-measures such high-order discretisations are unstable and crash [@Kirby2003]. It is of course arguable why one should even under-resolution at all, as the accuracy of results with low resolution is at least questionable. However, recent investigations show that it is possible to achieve quite accurate results with (very) high-order DG schemes, even with under-resolution as long as proper de-aliasing mechanisms are augmented [@tcfd2012].
A very popular strategy is the use of polynomial de-aliasing, as mentioned above. Motivated by spectral methods, the authors of [@Kirby2003] proposed to de-alias by increasing the number of quadrature nodes according to the non-linearity of the flux function so that the variational terms are evaluated exactly. As a prime example, the non-linear Burger’s equation has a quadratic flux function and hence roughly a factor of $1.5$ times the number of interpolation nodes is needed in each spatial direction to integrate the non-linear flux exactly. It is also remarkable that for DG discretisations with exact evaluations of the integral, Shu and Jiang [@cell_entropy_dg] proved a cell entropy inequality and, with this, non-linear $L_2$ stability. It is very important to note however, that Shu and Jiang consider *scalar* non-linear conservation laws and consequently non-linear stability of DG discretisations with exact integration is only valid for *scalar* conservation laws. Their stability estimate does not carry over to systems of non-linear conservation laws such as the compressible Euler equations, independent of the number of quadrature nodes used to evaluate the inner products.
In fact, recent results by [@rodrigo_iLES] show that even with up to four times the number of quadrature nodes in each spatial direction, DG discretisations with either local Lax-Friedrichs or Roe’s flux function crash for under-resolved turbulence simulations. The “up to four times the quadrature nodes" statements necessitates an additional remark: For the compressible Euler equations, the polynomial ansatz is typically done for the conserved quantities such as mass, momentum and energy. But, the flux functions are rational polynomials of the conserved quantities. This is problematic because the available quadrature rules are only exact for polynomial integrands. Thus, it is formally impossible to implement an exact integration based on standard quadrature rules for the compressible Euler equations.
To formally prove non-linear stability of DG discretisations for systems of conservation laws, it is necessary to reformulate the equations in terms of entropy variables. Here, the polynomial ansatz (and the test-function) space approximates the entropy variables and not the conserved variables. This is important, as it is now formally possible to show that the DG discretisation satisfies a cell entropy inequality for systems [@HM14_764]. However, it is important to note, that again, the stability proof relies on exact evaluation of the variational terms, similar to the proof of Shu and Jiang [@cell_entropy_dg]. As discussed above, in the case of the compressible Euler equations it is impossible (or impractical) to find a fixed number of quadrature nodes so that the variational errors due to the rational non-linearity of the flux functions with respect to the entropy variables are zero. At least an adaptive numerical quadrature approach would be necessary, which of course makes this strategy quite cumbersome with respect to implementation and efficiency.
Up to now, the only known stability proof without relying on exact evaluation of inner products for the compressible Euler equations is presented in Carpenter et al. [@carpenter_esdg]. The proof is possible because it uses a very specific form of the DG discretisation.
By choosing a nodal DG ansatz with Gauss-Lobatto (GL) nodes used for both interpolation and numerical integration, the so-called discontinuous Galerkin spectral element method with collocation (DGSEM) results, e.g. [@koprivabook]. In the present work, we always use interpolation and integration based on the GL nodes. This variant of the DG methodology is special because it possesses the summation-by-parts (SBP) property. The discrete mass matrix ${{\mathbf{{M}}}}$ and the discrete derivative matrix ${{\mathbf{D}}}$ of the DGSEM satisfy all formal definitions of a SBP operator [@Strand199447] $$\label{eq:sbp}
{{\mathbf{{Q}}}}:={{\mathbf{{M}}}}\,{{\mathbf{D}}}\quad\text{ with }\quad{{\mathbf{{Q}}}}+{{\mathbf{{Q}}}}^T={{\mathbf{{B}}}},$$ where ${{\mathbf{{B}}}}=\textrm{diag}([-1,0,\ldots,0,1])$ is the boundary evaluation operator , e.g. [@gassner_skew_burgers; @carpenter_esdg]. Furthermore, the mass matrix is diagonal and is used to define a discrete $L_2$-norm, e.g. to compute errors. To be more specific, the DGSEM operators form a so-called diagonal norm SBP operator. It was possible in [@carpenter_esdg; @fisher2013] to construct a specific form of the DGSEM that satisfies a cell entropy type inequality for all conservation laws while retaining the nodal nature of the discretisation, i.e. without relying on exact evaluation of the inner products. The only necessary ingredient to achieve a cell entropy type inequality is a two-point numerical flux function that gives exact entropy conservation in a standard finite volume type discretisation [@carpenter_esdg]. A remarkable advantage of these derivations are that they do not rely on any DG specifics, but only on the SBP property. Thus, the non-linear stability results directly carry over to all discretisations constructed with SBP operators, where the mass matrix ${{\mathbf{{M}}}}$ (often called norm matrix) is diagonal as in the case of DGSEM.
Alternative stabilisation strategies for the compressible Euler equations {#sec:fd_dealiasing}
-------------------------------------------------------------------------
We leave the DG community momentarily and investigate de-aliasing strategies used in other high-order communities such as finite differences, e.g. [@ducros2000; @Morinishi2010276; @kennedy2008; @FDaliasing; @larsson2007] and spectral methods, e.g. [@blaisdell1996effect; @Zang199127]. A very prominent example for de-aliasing is the use of alternative formulations of the non-linear advection terms, e.g. so-called split formulations. Due to the non-linear character of the advective terms of the Euler equations, there are many ways to re-write the equations. A good overview of the different split forms can be found in [@pirozzoli2011].
We consider here split forms of the three dimensional compressible Euler equations. In a general operator notation, we can write the Euler equations as $$\label{eq:operator_form}
U_t + {\mathcal{L}_x}(U)+ {\mathcal{L}_y}(U)+ {\mathcal{L}_z}(U) = 0,$$ with the conservative variables $$U=\begin{pmatrix}
u_1\\
u_2\\
u_3\\
u_4\\
u_5\\
\end{pmatrix}:=\begin{pmatrix}
\rho\\
\rho\,u\\
\rho\,v\\
\rho\,w\\
\rho\,e\\
\end{pmatrix},$$ where $\rho\,e=\rho\,\theta+\frac{1}{2}\rho\,(u^2+v^2+w^2)$ and $\rho$, $u,v,w$, $p$, $\theta$, $e$ are density, velocity, pressure, specific inner energy and specific total energy respectively, and ${\mathcal{L}}_{x,y,z}(U)$ are the non-linear spatial differential operators in the respective Cartesian direction $x,y,z$ acting on $U$. We close the system by considering a perfect gas equation which relates the internal energy and pressure as $p = (\gamma -1)\,\rho\theta$, where $\gamma$ denotes the adiabatic coefficient.
If we consider, for instance, the standard divergence form of the Euler equations, the non-linear operators are $$\label{eq:op_divergence}
\resizebox{0.925\textwidth}{!}{$
\begin{aligned}
{\mathcal{L}_x}^{div}(U):= F(U)_x= \begin{bmatrix}
\rho\,u\\
\rho\,u^2 +p\\
\rho\,u\,v\\
\rho\,u\,w\\
(\rho\,e+p)\,u
\end{bmatrix}_x,\quad
{\mathcal{L}_y}^{div}(U):= G(U)_y= \begin{bmatrix}
\rho\,v\\
\rho\,u\,v \\
\rho\,v^2+p\\
\rho\,v\,w\\
(\rho\,e+p)\,v
\end{bmatrix}_y,\quad
{\mathcal{L}_z}^{div}(U):= H(U)_z=\begin{bmatrix}
\rho\,w\\
\rho\,u\,w \\
\rho\,v\,w\\
\rho\,w^2+p\\
(\rho\,e+p)\,w
\end{bmatrix}_z.
\end{aligned}$}$$
The divergence form is typically used to construct a DG discretisation, as it directly gives discrete conservation of the resulting approximation. However, depending on how one interprets the non-linearities of the Euler flux (quadratic, cubic or rational) there are several ways to re-write the equations in an equivalent split form.
We use different split form approximations of derivatives of products to determine a particular splitting. The derivative of a product of two quantities is approximated by $$\label{eq:two_split}
\begin{split}
(a\,b)_x &= \frac{1}{2}\,(a\,b)_x + \frac{1}{2}\left(a_x\,b + a\,b_x\right),\end{split}$$ which originates from the general split form $$\label{eq:splitformquadratic}
\begin{split}
(a\,b)_x &= \alpha\,(a\,b)_x + (1-\alpha)\,\left(a_x\,b + a\,b_x\right),\quad \alpha\in\mathbb{R},
\end{split}$$ when choosing $\alpha=1/2$. We note that with the quadratic splitting it is possible to prove stability of linear variable coefficient problems [@kopriva2014].
For the derivative of cubic terms Kennedy and Gruber [@kennedy2008] proposed the following split form $$\resizebox{\textwidth}{!}{$
\begin{aligned}
{\frac{\partial }{\partial x}}(abc) = \alpha{\frac{\partial }{\partial x}}(abc) &+ \beta\left[a{\frac{\partial }{\partial x}}(bc) + bc{\frac{\partial }{\partial x}}(a)\right]+ \kappa\left[b{\frac{\partial }{\partial x}}(ac) + ac{\frac{\partial }{\partial x}}(b)\right]+ \delta\left[c{\frac{\partial }{\partial x}}(ab) + ab{\frac{\partial }{\partial x}}(c)\right] \\&+ \epsilon\left[bc{\frac{\partial }{\partial x}}(a) + ac{\frac{\partial }{\partial x}}(b) + ab{\frac{\partial }{\partial x}}(c)\right],
\end{aligned}$}$$ where $\epsilon = 1-\alpha-\beta-\kappa-\delta$ and $\alpha,\beta,\kappa,\delta\in\mathbb{R}$. From all the possible combinations, we choose the case $\alpha=\beta=\kappa=\delta=\frac{1}{4}$ and $\epsilon = 0$, which gives $$\label{eq:three_split}
\resizebox{\textwidth}{!}{$
\begin{aligned}
\begin{split}
(a\,b\,c)_x &= \frac{1}{4}\,(a\,b\,c)_x + \frac{1}{4}\left(a_x\,(b\,c) + b_x\,(a\,c)+c_x\,(a\,b)\right)+ \frac{1}{4}\left(a\,(b\,c)_x + b\,(a\,c)_x+c\,(a\,b)_x\right).\end{split}
\end{aligned}$}$$
With the two split forms and it is now possible to compose new equivalent forms of the non-linear operator ${\mathcal{L}}(U)$ to enhance the stability of discretisations. For instance, Morinishi [@Morinishi2010276] introduced a skew-symmetric form for the momentum equations, which was rewritten in Gassner [@gassner_kepdg] assuming time continuity as $$\label{eq:op_morinishi}
\resizebox{0.9\hsize}{!}{$
{\mathcal{L}_x}^{MO}(U) := \begin{bmatrix}
(\rho\,u)_x\\[0.1cm]
{\frac{1}{2}}\left((\rho u^2)_x+\rho u\,(u)_x + u\,(\rho u)_x\right) + p_x\\[0.1cm]
{\frac{1}{2}}\left((\rho u v)_x+\rho u\,(v)_x + v\,(\rho u)_x\right)\\[0.1cm]
{\frac{1}{2}}\left((\rho u w)_x+\rho u\,(w)_x + w\,(\rho u)_x\right)\\[0.1cm]
\left((\rho\theta+p)\,u\right)_x + {\frac{1}{2}}\left(\rho\,u^2\,(u)_x + u\,(\rho\,u^2)_x + \rho\,u\,v\,(v)_x + v\,(\rho\,u\,v)_x + \rho\,u\,w\,(w)_x + w\,(\rho\,u\,w)_x \right)
\end{bmatrix},$}$$ with the operators in the $y$ and $z$ direction defined similarly. It is important to note that the form of Morinishi does not use the split form of the quadratic terms ($\alpha=1/2$), but the pure advective form ($\alpha=0$ in ) in the energy equation. The form allows the construction of a formally kinetic energy preserving discontinuous Galerkin method [@gassner_kepdg], but we will show in the numerical results section that the positive stabilisation effect of this alternative form of Morinishi is the lowest in comparison to the other forms presented below. A possible reason could be that only the advective form is used in the total energy equation.
In contrast to Morinishi’s flux, Ducros et al. [@ducros2000] proposed the following form, where only the split form ($\alpha=1/2$) of the quadratic product is used $$\label{eq:op_ducros}
{\mathcal{L}_x}^{DU}(U) := \begin{bmatrix}
{\frac{1}{2}}\left((\rho u)_x+\rho(u)_x + u(\rho)_x\right)\\[0.1cm]
{\frac{1}{2}}\left((\rho u^2)_x+\rho u(u)_x + u(\rho u)_x\right) + p_x\\[0.1cm]
{\frac{1}{2}}\left((\rho u v)_x+\rho v(u)_x + u(\rho v)_x\right)\\[0.1cm]
{\frac{1}{2}}\left((\rho u w)_x+\rho w(u)_x + u(\rho w)_x\right)\\[0.1cm]
{\frac{1}{2}}\left(((\rho e +p)u)_x+(\rho e + p)(u)_x + u(\rho e + p)_x\right)
\end{bmatrix}.$$ However, this alternative formulation does not lead to a discretisation that is formally kinetic energy preserving.
The cubic form was first introduced and applied in [@kennedy2008]. The operator proposed by Kennedy and Gruber in the $x-$direction reads $$\label{eq:op_KG}
{\mathcal{L}_x}^{KG}(U) := \begin{bmatrix}
{\frac{1}{2}}\left((\rho u)_x+\rho(u)_x + u(\rho)_x\right)\\[0.1cm]
\frac{1}{4}\left[(\rho u^2)_x+\rho (u^2)_x + 2u(\rho u)_x + u^2(\rho)_x + 2\rho u(u)_x \right]+ p_x\\[0.1cm]
\frac{1}{4}\left[(\rho u v)_x+\rho (uv)_x + u(\rho v)_x + v(\rho u)_x + uv(\rho)_x + \rho v(u)_x + \rho u(v_x)\right]\\[0.1cm]
\frac{1}{4}\left[(\rho u w)_x+\rho (uw)_x + u(\rho w)_x + w(\rho u)_x + uw(\rho)_x + \rho w(u)_x + \rho u(w_x)\right]\\[0.1cm]
{\frac{1}{2}}\left((pu)_x+p(u)_x+u(p_x)\right)+\frac{1}{4}\left[(\rho e u)_x + \rho(e u)_x + e(\rho u)_x + u(\rho e)_x\right. \\
\left.\qquad\qquad\qquad\qquad\qquad\qquad+ eu(\rho)_x + \rho u (e)_x + \rho e (u)_x\right]
\end{bmatrix}.$$ The standard quadratic form is used for the continuity equation, whereas the cubic form is used for the advection terms in the momentum equation. In the energy equation, both the quadratic form for $p\,u$ and the cubic form for $\rho\,e\,u$ are applied. Again, this form allows one to construct formally kinetic energy preserving discretisations [@jameson2008].
In his overview, Pirozzoli [@pirozzoli2011] re-intrepreted the work of Kennedy and Gruber and used a similar but slightly different form of the flux splitting. Whereas the continuity and momentum equations don’t change, Pirozzoli re-wrote the energy term $(\rho\,e+p)u$ as $\rho\,h\,p$, where the specific enthalpy is given by $h=e+p/\rho$. With this, only the cubic form can be used for the energy equation, which results in $$\label{eq:op_pirozzoli}
\resizebox{0.9\textwidth}{!}{$
{\mathcal{L}_x}^{PI}(U) := \begin{bmatrix}
{\frac{1}{2}}\left((\rho u)_x+\rho(u)_x + u(\rho)_x\right)\\[0.1cm]
\frac{1}{4}\left[(\rho u^2)_x+\rho (u^2)_x + 2u(\rho u)_x + u^2(\rho)_x + 2\rho u(u)_x \right]+ p_x\\[0.1cm]
\frac{1}{4}\left[(\rho u v)_x+\rho (uv)_x + u(\rho v)_x + v(\rho u)_x + uv(\rho)_x + \rho v(u)_x + \rho u(v_x)\right]\\[0.1cm]
\frac{1}{4}\left[(\rho u w)_x+\rho (uw)_x + u(\rho w)_x + w(\rho u)_x + uw(\rho)_x + \rho w(u)_x + \rho u(w_x)\right]\\[0.1cm]
\frac{1}{4}\left[(\rho u h)_x + \rho(u h)_x + h(\rho u)_x + u(\rho h)_x+ u h(\rho)_x + \rho u(h)_x + \rho h (u)_x\right]
\end{bmatrix}.$}$$
There are other split forms available in literature, e.g. given by Kravchenko and Moin [@FDaliasing]. However, we will restrict this discussion to the forms presented above. The goal of the paper is to show how to discretise such forms for discontinuous Galerkin methods. More specifically, we focus on DGSEM (with GL nodes), as this specific variant satisfies the diagonal norm SBP which is key to achieve a conservative approximation for the split forms, e.g. [@kopriva2014].
The remainder of the paper is organised as follows: In the next section the nodal collocation spectral element framework is introduced. The main Sec. \[sec:DG-Disc2\] introduces the volume flux difference form and introduces specific numerical volume fluxes that generate known split forms of the compressible Euler equations. In Sec. \[sec:numerical results\] numerical experiments that support the theoretical findings are presented with our conclusions drawn in the last section.
The Nodal Discontinuous Galerkin Spectral Element Method {#sec:DG-Disc}
========================================================
In this section, we provide the basic construction of a nodal discontinuous Galerkin spectral element method (DGSEM) on tensor product elements. The implementation of the DGSEM for the compressible Euler equations is based on [@Hindenlang2012], which includes a detailed description of the standard weak form discretisation of the compressible Euler equations in divergence form. It is important to note we consider the strong form of the DGSEM in this work due to its close relation to the so-called simultaneous-approximation term (SAT) SBP finite difference methods [@gassner_skew_burgers]. In [@KoprivaGassner_GaussLob] it was shown that the weak and strong forms are algebraically equivalent. This equivalence of strong form and weak form is itself equivalent to the SBP property of an operator [@gassner_skew_burgers]. We specifically focus on the volume discretisation of the Euler terms. All other parts of the implementation remain unchanged. Thus, it is straightforward to extend an existing DGSEM code to the split form approximations presented in this paper. We restrict the following discussion and the numerical results section to Cartesian meshes as our focus is on the effect of the non-linear terms of the compressible Euler equations. For completeness, we provide the algebraic extensions to curvilinear meshes in \[sec:curvilinear\].
The first step in the discretisation is to subdivide the computational domain into non-overlapping hexahedral elements $C$. Each element is transformed to the reference element $C_0=[-1,1]^3$ with a polynomial mapping. The degree of the mapping is chosen to be at most the degree of the element-wise polynomial approximation $N$ to ensure free-stream preservation [@Kopriva:2006er]. For non-curved hexahedral elements, the mapping we use is trilinear and reads as $$\label{eq:StandardHexMap}
\begin{aligned}
{\underline{X}}({\underline{\xi}}) =(X({\underline{\xi}}),Y({\underline{\xi}}),Z({\underline{\xi}}))^T= &\frac{1}{8}\left\{{\underline{x}}_1(1-\xi)(1-\eta)(1-\zeta)+{\underline{x}}_2(1+\xi)(1-\eta)(1-\zeta)\right.\\
& + {\underline{x}}_3(1+\xi)(1+\eta)(1-\zeta)+{\underline{x}}_4(1-\xi)(1+\eta)(1-\zeta) \\
& + {\underline{x}}_5(1-\xi)(1-\eta)(1+\zeta)+{\underline{x}}_6(1+\xi)(1-\eta)(1+\zeta) \\
& + \left.{\underline{x}}_7(1+\xi)(1+\eta)(1+\zeta)+{\underline{x}}_8(1-\xi)(1+\eta)(1+\zeta)\right\},
\end{aligned}$$ where ${\underline{x}}_i$, $i=1,\ldots,8$ are the physical coordinates of the corners of the hexahedral element and ${\underline{\xi}}=(\xi,\eta,\zeta)^T$ are the reference coordinates. As we restrict ourselves here to Cartesian meshes, the Jacobian and metric terms simplify to $$\label{eq:cartesian_metric}
J = \frac{1}{8}\Delta x\Delta y\Delta z,\quad X_{\xi} = {\frac{1}{2}}\Delta x,\quad Y_{\eta} = {\frac{1}{2}}\Delta y,\quad Z_{\zeta} = {\frac{1}{2}}\Delta z,$$ with element side lengths $\Delta x$, $\Delta y$, and $\Delta z$.
For each element, each component of the conservative variable vector is approximated by a polynomial of degree $N$ in reference space, e.g. for the density $$\label{eq:polynom}
u_1(x,y,z,t)\big|_{C} = \rho(x,y,z,t)\big|_{C}\approx \rho(\xi,\eta,\zeta,t) := \sum\limits_{i,j,k=0}^N \rho^{i,j,k}(t)\,\ell_i(\xi)\,\ell_j(\eta)\ell_k(\zeta),$$ where $\{u_1^{i,j,k}(t)\}_{i,j,k=0}^{N}$ are the time dependent nodal degrees of freedom of the element $C$ at the nodes $(i,j,k)$. Each nodal interpolant is defined with $(N+1)^3$ GL nodes $\{\xi_i\}_{i=0}^N$, $\{\eta_j\}_{j=0}^N$, and $\{\zeta_j\}_{j=0}^N$ in the reference cube $C_0$. The associated Lagrange basis functions are given by $$\label{eq:lagrange_basis}
\ell_j(\xi)=\prod\limits_{i=0,i\neq j}^N\frac{\xi - \xi_i}{\xi_j-\xi_i},\qquad j=0,...,N,$$ and satisfy the cardinal property $$\label{cardinal}
\ell_j(\xi_i)=\delta_{ij},$$ where $\delta_{ij}$ denotes Kronecker’s symbol with $\delta_{ij}=1$ for $i=j$ and $\delta_{ij}=0$ for $i\neq j$. Integrating the polynomial Lagrange basis functions over the unit interval $[-1,1]$ gives the GL quadrature weights $\{\omega_j\}_{j=0}^N$. The GL nodes and weights form a quadrature rule with integration precision $2N-1$ that is used in the nodal DGSEM for the approximation of the weak form . The Lagrange basis functions, $\{\ell_j\}_{j=0}^N$, are discretely orthogonal to each other, and the mass matrix is diagonal $${{\mathbf{{M}}}}:=\textrm{diag}([\omega_0,...,\omega_N]).$$ In addition to the discrete integration, the polynomial basis functions and the interpolation nodes form a discrete differentiation. We introduce the polynomial derivative matrix $$\label{eq:DmatDef}
D_{ij}=\frac{\partial\ell_j}{\partial\xi}\Bigg|_{\xi=\xi_i},\quad i,j=0,\ldots,N.$$ As mentioned in the introduction, the derivative operator is special, as it satisfies the SBP-property for all polynomial degrees $N$, e.g. [@gassner_skew_burgers; @carpenter_esdg; @gassner_kepdg]. We define the matrix ${{\mathbf{{Q}}}}:={{\mathbf{{M}}}}{{\mathbf{D}}}$, which has the SBP-property ${{\mathbf{{Q}}}}+ {{\mathbf{{Q}}}}^T={\mathbf{B}}:=\textrm{diag}(-1,0,\ldots,0,1)$. The SBP-property is used to mimic integration-by-parts, by manipulating the derivative matrix to become $${{\mathbf{D}}}={{\mathbf{{M}}}}^{-1}{{\mathbf{{Q}}}}= {{\mathbf{{M}}}}^{-1}{\mathbf{B}}-{{\mathbf{{M}}}}^{-1}{{\mathbf{{Q}}}}^T.$$
With all these ingredients, we are able to formulate the standard DGSEM approximation of the divergence form of the flux equations . We use the elemental mapping and the metric terms to transform the system into the reference element $$\label{eq:_transformed_operator_form}
J\,U_t + \widetilde{{\mathcal{L}}}_\xi(U) + \widetilde{{\mathcal{L}}}_\eta(U) + \widetilde{{\mathcal{L}}}_\zeta(U) = 0,$$ with $$\widetilde{{\mathcal{L}}}_\xi(U)= Y_\eta Z_\zeta {\mathcal{L}}_\xi(U),\,\, \widetilde{{\mathcal{L}}}_\eta(U)= X_\xi Z_\zeta {\mathcal{L}}_\eta(U),\,\, \widetilde{{\mathcal{L}}}_\zeta(U)=X_\xi Y_\eta {\mathcal{L}}_\zeta(U),$$ where we used the restriction to Cartesian meshes to simplify the expressions.
We collect the three dimensional DGSEM approximation of the divergence form of the equations written in indicial notation. Consider a component $l$ of the system and a GL node $(i,j,k)$. The DGSEM approximation in strong form is $$\label{eq:RHSStrong}
\begin{aligned}
({\mathcal{L}}_\xi(U))_{ijk}^l&\approx \left[F^{*,l}(1,\eta_j,\zeta_k;{\underline{n}}) - F^l_{Njk}\right] - \left[F^{*,l}(-1,\eta_j,\zeta_k;{\underline{n}}) - F^l_{0jk}\right]+\sum_{m=0}^N D_{im}F^l_{mjk},\\
({\mathcal{L}}_\eta(U))_{ijk}^l&\approx \left[G^{*,l}(\xi_i,1,\zeta_k;{\underline{n}}) - G^l_{iNk}\right] - \left[G^{*,l}(\xi_i,-1,\zeta_k;{\underline{n}}) - G^l_{i0k}\right]+\sum_{m=0}^N D_{jm}G^l_{imk},\\
({\mathcal{L}}_\zeta(U))_{ijk}^l&\approx \left[H^{*,l}(\xi_i,\eta_j,1;{\underline{n}}) - H^l_{ijN}\right] - \left[H^{*,l}(\xi_i,\eta_j,-1;{\underline{n}}) - H^l_{ij0}\right]+\sum_{m=0}^N D_{km}H^l_{ijm},
\end{aligned}$$ where $l=1,\ldots,5$. We use collocation for the non-linear flux functions, e.g. $F^l_{ijk}:=f^l(U_{ijk})$, and denote the outward pointing normal vector by ${\underline{n}}$. Due to the discontinuous nature of the approximation space, it is necessary to introduce numerical surface flux functions, which resolve the jumps at the interface. These specific surface flux functions depend on the left and right state values at an interface and are marked by a $*$. We will specify the numerical surface fluxes in Sec. \[sec:numflux\].\
The resulting semi-discrete form is a coupled system of ordinary differential equations in time, which we integrate with a low storage 5-stage 4th order accurate explicit Runge-Kutta method, e.g. [@Kennedy1994].
Split form stabilisation for DGSEM {#sec:DG-Disc2}
==================================
In this section we demonstrate how to implement the different split forms. To do so, we begin with a standard strong form DGSEM implementation , and modify the discrete volume integrals that lead to $\sum_{m=0}^N D_{im}F^l_{mjk}$, $\sum_{m=0}^N D_{jm}G^l_{imk}$ and $\sum_{m=0}^N D_{km}H^l_{ijm}$ to include the different split forms presented in the introduction, Sec. \[sec:fd\_dealiasing\].
DGSEM with numerical volume flux function {#sec:DGSEM_fluxform}
-----------------------------------------
An important result that we use in this work is presented in Fisher et al. [@fisher2013] and Carpenter et al. [@carpenter_esdg]. These authors showed that it is possible to rewrite the application of a differencing operator ${{\mathbf{D}}}$ with the diagonal SBP property into an equivalent subcell based finite volume type differencing formulation $$\label{eq:flux-differencing_vol}
\begin{split}
&\sum_{m=0}^N D_{im}F^l_{mjk} = \frac{\bar{F}^l_{(i+1)jk} - \bar{F}^l_{(i)jk}}{\omega_i},\quad i,j,k=0,...,N,\\
&\sum_{m=0}^N D_{jm}G^l_{imk}= \frac{\bar{G}^l_{i(j+1)k} - \bar{G}^l_{i(j)k}}{\omega_j},\quad i,j,k=0,...,N,\\
&\sum_{m=0}^N D_{km}H^l_{ijm}= \frac{\bar{H}^l_{ij(k+1)} - \bar{H}^l_{ij(k)}}{\omega_k},\quad i,j,k=0,...,N,
\end{split}$$ with consistent auxiliary fluxes $\{\bar{F}^l_{(ii)jk}\}_{ii,j,k=0}^{(N+1),N,N}$, $\{\bar{G}^l_{i(jj)k}\}_{i,jj,k=0}^{N,(N+1),N}$ and $\{\bar{H}^l_{ij(kk)}\}_{i,j,kk=0}^{N,N,(N+1)}$. This result is important, as it directly gives conservation of diagonal norm SBP discretisations in the sense of Lax-Wendroff due to the telescoping flux differencing [@fisher2013]. In addition to the desired conservation property, the flux differencing interpretation enables the construction of entropy conserving discretisations without relying on exact integration.
Fisher et al. [@fisher2013] and Carpenter et al. [@carpenter_esdg] used a diagonal norm SBP operator, the volume flux differencing relation , and the existence of a two-point entropy conserving flux function $F_{EC}^{\#} = F_{EC}^{\#}(U_{ijk},U_{mjk})$ to obtain a high-order accurate discretisation $$\label{eq:highorder_flux}
\frac{\bar{F}^l_{(i+1)jk} - \bar{F}^l_{(i)jk}}{\omega_i} \approx 2\,\sum\limits_{m=0}^N D_{im}\,F^{\#,l}_{EC}(U_{ijk},U_{mjk}).$$ The important theorems and proofs can be found e.g. in Fisher et al. [@fisher2013] (Thms. 3.1 and 3.2 on page 15 and 18). By choosing the two-point entropy flux $F_{EC}^{\#}$ also as the numerical surface flux in a DGSEM discretisation, with the volume terms computed with , it was shown that the resulting DG discretisation conserves the (discrete) integral of the entropy [@carpenter_esdg]. Note that the goal of the work of Fisher and Carpenter et al. [@carpenter_esdg; @fisher2013] is not to construct an entropy conserving discretisation, but a discretisation that is entropy stable. However, from a scheme that exactly preserves entropy, it is possible to introduce dissipation terms, e.g. at the element interfaces so that the (discrete) integral of entropy is guaranteed to decrease (typically named *entropy stability*). Thus, entropy conservation can be seen as an intermediate step to entropy stability.
In the present work, we rewrite the volume terms using the flux differencing form . However, we keep the expression general and use the yet to be specified numerical volume fluxes $F^{\#}$, $G^{\#}$, and $H^{\#}$ $$\label{eq:RHSfluxform}
\begin{aligned}
({\mathcal{L}}_\xi(U))_{ijk}^l&\approx \left[F^{*,l}(1,\eta_j,\zeta_k;{\underline{n}}) - F^l_{Njk}\right] - \left[F^{*,l}(-1,\eta_j,\zeta_k;{\underline{n}}) - F^l_{0jk}\right]+2\,\sum\limits_{m=0}^N D_{im}\,F^{\#,l}(U_{ijk},U_{mjk}),\\
({\mathcal{L}}_\eta(U))_{ijk}^l&\approx \left[G^{*,l}(\xi_i,1,\zeta_k;{\underline{n}}) - G^l_{iNk}\right] - \left[G^{*,l}(\xi_i,-1,\zeta_k;{\underline{n}}) - G^l_{i0k}\right]+2\,\sum\limits_{m=0}^N D_{jm}\,G^{\#,l}(U_{ijk},U_{imk}),\\
({\mathcal{L}}_\zeta(U))_{ijk}^l&\approx \left[H^{*,l}(\xi_i,\eta_j,1;{\underline{n}}) - H^l_{ijN}\right] - \left[H^{*,l}(\xi_i,\eta_j,-1;{\underline{n}}) - H^l_{ij0}\right]+2\,\sum\limits_{m=0}^N D_{km}\,H^{\#,l}(U_{ijk},U_{ijm}).
\end{aligned}$$ We show in the next section that the reformulation of the volume integrals in plays an important role and that it is possible to generate many variants of DGSEM for the compressible Euler equations. We note that, in principle, it is possible to choose the numerical surface and volume flux differently. However, to restrict the myriads of possible combinations in this work, we couple the choice of the numerical volume flux and the numerical surface flux, detailed in Sec. \[sec:numflux\] below.
Split form DSGEM {#sec:split form}
----------------
In this section, we discuss three simple identities used to introduce specific numerical volume flux functions that recover the alternative compressible Euler operator split forms discussed in the introduction. To formulate these identities we consider three specific choices of the numerical volume flux, namely an arithmetic mean, the product of two arithmetic means and the product of three arithmetic means. We use the typical DG notation $${\left\{\!\left\{ a\right\}\!\right\}}_{im}:= \frac{1}{2}(a_i + a_m),$$ for an arithmetic mean of a generic nodal quantity $a_i,\,\,i=0,...,N$.
\[Lem\] Using the arithmetic mean, the product of two arithmetic means, or the product of three arithmetic means in the alternative numerical volume flux form of the DGSEM , it is possible to recover the standard DGSEM derivative form , the discrete quadratic split form and the discrete cubic split form respectively. More precisely, for generic nodal vector fields ${\underline{a}}=(a_0,...,a_N)^T$, ${\underline{b}}=(b_0,...,b_N)^T$, and ${\underline{c}}=(c_0,...,c_N)^T$, $$\label{eq:skewsym_identities}
\begin{aligned}
2\,\sum\limits_{m=0}^N D_{im}\,{\left\{\!\left\{ a\right\}\!\right\}}_{im} &= ({{\mathbf{D}}}\,{\underline{a}})_i,\\
2\,\sum\limits_{m=0}^N D_{im}\,{\left\{\!\left\{ a\right\}\!\right\}}_{im}\,{\left\{\!\left\{ b\right\}\!\right\}}_{im} &=\frac{1}{2}\left({{\mathbf{D}}}\,\uuline{a}\,{\underline{b}} + \uuline{a}\,{{\mathbf{D}}}\,{\underline{b}} + \uuline{b}\,{{\mathbf{D}}}\,{\underline{a}}\right)_i,\\
2\,\sum\limits_{m=0}^N D_{im}\,{\left\{\!\left\{ a\right\}\!\right\}}_{im}\,{\left\{\!\left\{ b\right\}\!\right\}}_{im}\,{\left\{\!\left\{ c\right\}\!\right\}}_{im} &=\frac{1}{4}\left({{\mathbf{D}}}\,\uuline{a}\,\uuline{b}\,{\underline{c}} + \uuline{a}\,{{\mathbf{D}}}\,\uuline{b}\,{\underline{c}} + \uuline{b}\,{{\mathbf{D}}}\,\uuline{a}\,{\underline{c}} + \uuline{c}\,{{\mathbf{D}}}\,\uuline{a}\,{\underline{b}} + \uuline{b}\,\uuline{c}\,{{\mathbf{D}}}\,{\underline{a}} + \uuline{a}\,\uuline{c}\,{{\mathbf{D}}}\,{\underline{b}} + \uuline{a}\,\uuline{b}\,{{\mathbf{D}}}\,{\underline{c}}\right)_i,
\end{aligned}$$ where we introduce the additional notation that double underline denotes a matrix with a vector along the diagonal, e.g., $\uuline{a} = diag({\underline{a}})$.
The proof of each of the split form identities is straightforward. However, each proof requires some algebraic manipulation unrelated to the focus of this paper. So we have moved the proofs to \[LemProof\].
We note that the right hand sides of are straightforward discretisations of the quadratic and the cubic split forms and , respectively.
\[rem:advective\_form\] By combining the split form identities, it is easy to see that the pure advective form of the quadratic product can be generated with the numerical volume flux of the form ${\left\{\!\left\{ a\right\}\!\right\}}_{im}\,{\left\{\!\left\{ b\right\}\!\right\}}_{im} -\frac{1}{2}{\left\{\!\left\{ a\,b\right\}\!\right\}}_{im}$.
We further note that all the derivations and proofs hold for general diagonal norm SBP operators and thus, for instance, all results in this paper directly carry over to finite difference diagonal norm SBP operators.
Analogously, it is possible to find equivalent forms of quartic products or products with even more terms when necessary.
With the identities presented in Lemma \[Lem\], it is now possible to state the second main result of this work, summarised in the following theorem.
\[Thm2\] A high-order accurate and consistent DGSEM discretisation of the alternative Euler formulations of Morinishi , Ducros et al. , Kennedy and Gruber , and Priozzoli can be generated by translating the quadratic and cubic split forms into the corresponding numerical volume flux. To compress the notation for the volume fluxes we note that the averages are taken in each spatial direction. For example, the average density in the $x$-direction or $z$-direction are given by $$\begin{aligned}
F^{\#,1}(U_{ijk},U_{mjk}) &= {\left\{\!\left\{ \rho\right\}\!\right\}} = {\frac{1}{2}}\left(\rho_{ijk} + \rho_{mjk}\right),\\[0.1cm]
H^{\#,1}(U_{ijk},U_{ijm}) &= {\left\{\!\left\{ \rho\right\}\!\right\}} = {\frac{1}{2}}\left(\rho_{ijk} + \rho_{ijm}\right),
\end{aligned}$$ respectively. The volume fluxes for the standard DGSEM in the divergence form and for each alternative formulation of the Euler equations are:
- : $$\begin{aligned}
F_{standard}^{\#}(U_{ijk},U_{mjk}) = &\begin{bmatrix} {\left\{\!\left\{ \rho u\right\}\!\right\}} \\[0.1cm]
{\left\{\!\left\{ \rho u^2\right\}\!\right\}} + {\left\{\!\left\{ p\right\}\!\right\}} \\[0.1cm]
{\left\{\!\left\{ \rho u v\right\}\!\right\}} \\[0.1cm]
{\left\{\!\left\{ \rho u w\right\}\!\right\}} \\[0.1cm]
{\left\{\!\left\{ u(\rho e + p)\right\}\!\right\}}\end{bmatrix},
\quad
G_{standard}^{\#}(U_{ijk},U_{imk}) = \begin{bmatrix} {\left\{\!\left\{ \rho v\right\}\!\right\}} \\[0.1cm]
{\left\{\!\left\{ \rho u v\right\}\!\right\}} \\[0.1cm]
{\left\{\!\left\{ \rho v^2\right\}\!\right\}} + {\left\{\!\left\{ p\right\}\!\right\}} \\[0.1cm]
{\left\{\!\left\{ \rho v w\right\}\!\right\}} \\[0.1cm]
{\left\{\!\left\{ v(\rho e + p)\right\}\!\right\}}\end{bmatrix},
\\ \\
&\qquad\qquad H_{standard}^{\#}(U_{ijk},U_{ijm})= \begin{bmatrix} {\left\{\!\left\{ \rho w\right\}\!\right\}} \\[0.1cm]
{\left\{\!\left\{ \rho u w\right\}\!\right\}}\\[0.1cm]
{\left\{\!\left\{ \rho v w\right\}\!\right\}} \\[0.1cm]
{\left\{\!\left\{ \rho w^2\right\}\!\right\}} + {\left\{\!\left\{ p\right\}\!\right\}} \\[0.1cm]
{\left\{\!\left\{ w(\rho e + p)\right\}\!\right\}}\end{bmatrix}.
\end{aligned}$$
- : $$\resizebox{0.85\textwidth}{!}{$
\begin{aligned}
F_{MO}^{\#}(U_{ijk},U_{mjk}) = &\begin{bmatrix} {\left\{\!\left\{ \rho\,u\right\}\!\right\}} \\[0.1cm]
{\left\{\!\left\{ \rho u\right\}\!\right\}}{\left\{\!\left\{ u\right\}\!\right\}} + {\left\{\!\left\{ p\right\}\!\right\}} \\[0.1cm]
{\left\{\!\left\{ \rho u\right\}\!\right\}}{\left\{\!\left\{ v\right\}\!\right\}} \\[0.1cm]
{\left\{\!\left\{ \rho u\right\}\!\right\}}{\left\{\!\left\{ w\right\}\!\right\}} \\[0.1cm]
{\left\{\!\left\{ (\rho\theta+p)\,u\right\}\!\right\}} + {\left\{\!\left\{ \rho\,u^2\right\}\!\right\}}{\left\{\!\left\{ u\right\}\!\right\}}+{\left\{\!\left\{ \rho\,u\,v\right\}\!\right\}}{\left\{\!\left\{ v\right\}\!\right\}}+{\left\{\!\left\{ \rho\,u\,w\right\}\!\right\}}{\left\{\!\left\{ w\right\}\!\right\}}-\frac{1}{2}\left({\left\{\!\left\{ \rho\,u^3\right\}\!\right\}}+{\left\{\!\left\{ \rho\,u\,v^2\right\}\!\right\}}+{\left\{\!\left\{ \rho\,u\,w^2\right\}\!\right\}} \right)\end{bmatrix},\\ \\
G_{MO}^{\#}(U_{ijk},U_{imk}) = &\begin{bmatrix} {\left\{\!\left\{ \rho\,v\right\}\!\right\}} \\[0.1cm]
{\left\{\!\left\{ \rho v\right\}\!\right\}}{\left\{\!\left\{ u\right\}\!\right\}}\\[0.1cm]
{\left\{\!\left\{ \rho v\right\}\!\right\}}{\left\{\!\left\{ v\right\}\!\right\}} + {\left\{\!\left\{ p\right\}\!\right\}} \\[0.1cm]
{\left\{\!\left\{ \rho v\right\}\!\right\}}{\left\{\!\left\{ w\right\}\!\right\}} \\[0.1cm]
{\left\{\!\left\{ (\rho\theta+p)\,v\right\}\!\right\}} + {\left\{\!\left\{ \rho\,u\,v\right\}\!\right\}}{\left\{\!\left\{ u\right\}\!\right\}}+{\left\{\!\left\{ \rho\,v^2\right\}\!\right\}}{\left\{\!\left\{ v\right\}\!\right\}}+{\left\{\!\left\{ \rho\,v\,w\right\}\!\right\}}{\left\{\!\left\{ w\right\}\!\right\}}-\frac{1}{2}\left({\left\{\!\left\{ \rho\,u^2\,v\right\}\!\right\}}+{\left\{\!\left\{ \rho\,v^3\right\}\!\right\}}+{\left\{\!\left\{ \rho\,v\,w^2\right\}\!\right\}} \right)\end{bmatrix},\\ \\
H_{MO}^{\#}(U_{ijk},U_{ijm}) = &\begin{bmatrix} {\left\{\!\left\{ \rho\,w\right\}\!\right\}} \\[0.1cm]
{\left\{\!\left\{ \rho w\right\}\!\right\}}{\left\{\!\left\{ u\right\}\!\right\}}\\[0.1cm]
{\left\{\!\left\{ \rho w\right\}\!\right\}}{\left\{\!\left\{ v\right\}\!\right\}} + {\left\{\!\left\{ p\right\}\!\right\}} \\[0.1cm]
{\left\{\!\left\{ \rho w\right\}\!\right\}}{\left\{\!\left\{ w\right\}\!\right\}} \\[0.1cm]
{\left\{\!\left\{ (\rho\theta+p)\,w\right\}\!\right\}} + {\left\{\!\left\{ \rho\,u\,w\right\}\!\right\}}{\left\{\!\left\{ u\right\}\!\right\}}+{\left\{\!\left\{ \rho\,v\,w\right\}\!\right\}}{\left\{\!\left\{ v\right\}\!\right\}}+{\left\{\!\left\{ \rho\,w^2\right\}\!\right\}}{\left\{\!\left\{ w\right\}\!\right\}}-\frac{1}{2}\left({\left\{\!\left\{ \rho\,u^2\,w\right\}\!\right\}}+{\left\{\!\left\{ \rho\,v^2\,w\right\}\!\right\}}+{\left\{\!\left\{ \rho\,w^3\right\}\!\right\}} \right)\end{bmatrix}.
\end{aligned}$}$$
- : $$\begin{aligned}
F_{DU}^{\#}(U_{ijk},U_{mjk}) = &\begin{bmatrix} {\left\{\!\left\{ \rho\right\}\!\right\}}{\left\{\!\left\{ u\right\}\!\right\}} \\[0.1cm]
{\left\{\!\left\{ \rho u\right\}\!\right\}}{\left\{\!\left\{ u\right\}\!\right\}} + {\left\{\!\left\{ p\right\}\!\right\}} \\[0.1cm]
{\left\{\!\left\{ \rho v\right\}\!\right\}}{\left\{\!\left\{ u\right\}\!\right\}} \\[0.1cm]
{\left\{\!\left\{ \rho w\right\}\!\right\}}{\left\{\!\left\{ u\right\}\!\right\}} \\[0.1cm]
({\left\{\!\left\{ \rho e\right\}\!\right\}} + {\left\{\!\left\{ p\right\}\!\right\}}){\left\{\!\left\{ u\right\}\!\right\}}\end{bmatrix},
\quad
G_{DU}^{\#}(U_{ijk},U_{imk}) = \begin{bmatrix} {\left\{\!\left\{ \rho\right\}\!\right\}}{\left\{\!\left\{ v\right\}\!\right\}} \\[0.1cm]
{\left\{\!\left\{ \rho u\right\}\!\right\}}{\left\{\!\left\{ v\right\}\!\right\}} \\[0.1cm]
{\left\{\!\left\{ \rho v\right\}\!\right\}}{\left\{\!\left\{ v\right\}\!\right\}} + {\left\{\!\left\{ p\right\}\!\right\}}\\[0.1cm]
{\left\{\!\left\{ \rho w\right\}\!\right\}}{\left\{\!\left\{ v\right\}\!\right\}} \\[0.1cm]
({\left\{\!\left\{ \rho e\right\}\!\right\}} + {\left\{\!\left\{ p\right\}\!\right\}}){\left\{\!\left\{ v\right\}\!\right\}}\end{bmatrix},
\\ \\
&\qquad\qquad H_{DU}^{\#}(U_{ijk},U_{ijm}) = \begin{bmatrix} {\left\{\!\left\{ \rho\right\}\!\right\}}{\left\{\!\left\{ w\right\}\!\right\}} \\[0.1cm]
{\left\{\!\left\{ \rho u\right\}\!\right\}}{\left\{\!\left\{ w\right\}\!\right\}} \\[0.1cm]
{\left\{\!\left\{ \rho v\right\}\!\right\}}{\left\{\!\left\{ w\right\}\!\right\}} \\[0.1cm]
{\left\{\!\left\{ \rho w\right\}\!\right\}}{\left\{\!\left\{ w\right\}\!\right\}}+ {\left\{\!\left\{ p\right\}\!\right\}} \\[0.1cm]
({\left\{\!\left\{ \rho e\right\}\!\right\}} + {\left\{\!\left\{ p\right\}\!\right\}}){\left\{\!\left\{ w\right\}\!\right\}}\end{bmatrix}.
\end{aligned}$$
- : $$\resizebox{0.86\textwidth}{!}{$
\begin{aligned}
F_{KG}^{\#}(U_{ijk},U_{mjk}) = &\begin{bmatrix} {\left\{\!\left\{ \rho\right\}\!\right\}}{\left\{\!\left\{ u\right\}\!\right\}} \\[0.1cm]
{\left\{\!\left\{ \rho\right\}\!\right\}}{\left\{\!\left\{ u\right\}\!\right\}}^2 + {\left\{\!\left\{ p\right\}\!\right\}} \\[0.1cm]
{\left\{\!\left\{ \rho\right\}\!\right\}}{\left\{\!\left\{ u\right\}\!\right\}}{\left\{\!\left\{ v\right\}\!\right\}} \\[0.1cm]
{\left\{\!\left\{ \rho\right\}\!\right\}}{\left\{\!\left\{ u\right\}\!\right\}}{\left\{\!\left\{ w\right\}\!\right\}} \\[0.1cm]
{\left\{\!\left\{ \rho\right\}\!\right\}}{\left\{\!\left\{ u\right\}\!\right\}}{\left\{\!\left\{ e\right\}\!\right\}} + {\left\{\!\left\{ p\right\}\!\right\}}{\left\{\!\left\{ u\right\}\!\right\}}\end{bmatrix},
\quad
G_{KG}^{\#}(U_{ijk},U_{imk}) = \begin{bmatrix} {\left\{\!\left\{ \rho\right\}\!\right\}}{\left\{\!\left\{ v\right\}\!\right\}} \\[0.1cm]
{\left\{\!\left\{ \rho\right\}\!\right\}}{\left\{\!\left\{ u\right\}\!\right\}}{\left\{\!\left\{ v\right\}\!\right\}} \\[0.1cm]
{\left\{\!\left\{ \rho\right\}\!\right\}}{\left\{\!\left\{ v\right\}\!\right\}}^2 + {\left\{\!\left\{ p\right\}\!\right\}}\\[0.1cm]
{\left\{\!\left\{ \rho\right\}\!\right\}}{\left\{\!\left\{ v\right\}\!\right\}}{\left\{\!\left\{ w\right\}\!\right\}}\\[0.1cm]
{\left\{\!\left\{ \rho\right\}\!\right\}}{\left\{\!\left\{ v\right\}\!\right\}}{\left\{\!\left\{ e\right\}\!\right\}} + {\left\{\!\left\{ p\right\}\!\right\}}{\left\{\!\left\{ v\right\}\!\right\}}\end{bmatrix},
\\ \\
&\qquad\qquad H_{KG}^{\#}(U_{ijk},U_{ijm}) = \begin{bmatrix} {\left\{\!\left\{ \rho\right\}\!\right\}}{\left\{\!\left\{ w\right\}\!\right\}} \\[0.1cm]
{\left\{\!\left\{ \rho\right\}\!\right\}}{\left\{\!\left\{ u\right\}\!\right\}}{\left\{\!\left\{ w\right\}\!\right\}} \\[0.1cm]
{\left\{\!\left\{ \rho\right\}\!\right\}}{\left\{\!\left\{ v\right\}\!\right\}}{\left\{\!\left\{ w\right\}\!\right\}} \\[0.1cm]
{\left\{\!\left\{ \rho\right\}\!\right\}}{\left\{\!\left\{ w\right\}\!\right\}}^2+ {\left\{\!\left\{ p\right\}\!\right\}} \\[0.1cm]
{\left\{\!\left\{ \rho\right\}\!\right\}}{\left\{\!\left\{ w\right\}\!\right\}}{\left\{\!\left\{ e\right\}\!\right\}} + {\left\{\!\left\{ p\right\}\!\right\}}{\left\{\!\left\{ w\right\}\!\right\}}\end{bmatrix}.
\end{aligned}$}$$
- : $$\label{eq:numflux_priozzli}
\begin{aligned}
F_{PI}^{\#}(U_{ijk},U_{mjk}) = &\begin{bmatrix} {\left\{\!\left\{ \rho\right\}\!\right\}}{\left\{\!\left\{ u\right\}\!\right\}} \\[0.1cm]
{\left\{\!\left\{ \rho\right\}\!\right\}}{\left\{\!\left\{ u\right\}\!\right\}}^2 + {\left\{\!\left\{ p\right\}\!\right\}} \\[0.1cm]
{\left\{\!\left\{ \rho\right\}\!\right\}}{\left\{\!\left\{ u\right\}\!\right\}}{\left\{\!\left\{ v\right\}\!\right\}} \\[0.1cm]
{\left\{\!\left\{ \rho\right\}\!\right\}}{\left\{\!\left\{ u\right\}\!\right\}}{\left\{\!\left\{ w\right\}\!\right\}} \\[0.1cm]
{\left\{\!\left\{ \rho\right\}\!\right\}}{\left\{\!\left\{ u\right\}\!\right\}}{\left\{\!\left\{ h\right\}\!\right\}}\end{bmatrix},
\quad
G_{PI}^{\#}(U_{ijk},U_{imk})= \begin{bmatrix} {\left\{\!\left\{ \rho\right\}\!\right\}}{\left\{\!\left\{ v\right\}\!\right\}} \\[0.1cm]
{\left\{\!\left\{ \rho\right\}\!\right\}}{\left\{\!\left\{ u\right\}\!\right\}}{\left\{\!\left\{ v\right\}\!\right\}} \\[0.1cm]
{\left\{\!\left\{ \rho\right\}\!\right\}}{\left\{\!\left\{ v\right\}\!\right\}}^2 + {\left\{\!\left\{ p\right\}\!\right\}}\\[0.1cm]
{\left\{\!\left\{ \rho\right\}\!\right\}}{\left\{\!\left\{ v\right\}\!\right\}}{\left\{\!\left\{ w\right\}\!\right\}}\\[0.1cm]
{\left\{\!\left\{ \rho\right\}\!\right\}}{\left\{\!\left\{ v\right\}\!\right\}}{\left\{\!\left\{ h\right\}\!\right\}}\end{bmatrix},
\\ \\
&\qquad\qquad H_{PI}^{\#}(U_{ijk},U_{ijm}) = \begin{bmatrix} {\left\{\!\left\{ \rho\right\}\!\right\}}{\left\{\!\left\{ w\right\}\!\right\}} \\[0.1cm]
{\left\{\!\left\{ \rho\right\}\!\right\}}{\left\{\!\left\{ u\right\}\!\right\}}{\left\{\!\left\{ w\right\}\!\right\}} \\[0.1cm]
{\left\{\!\left\{ \rho\right\}\!\right\}}{\left\{\!\left\{ v\right\}\!\right\}}{\left\{\!\left\{ w\right\}\!\right\}} \\[0.1cm]
{\left\{\!\left\{ \rho\right\}\!\right\}}{\left\{\!\left\{ w\right\}\!\right\}}^2+ {\left\{\!\left\{ p\right\}\!\right\}} \\[0.1cm]
{\left\{\!\left\{ \rho\right\}\!\right\}}{\left\{\!\left\{ w\right\}\!\right\}}{\left\{\!\left\{ h\right\}\!\right\}}\end{bmatrix}.
\end{aligned}$$
The proof follows directly when applying the results from Lem. \[Lem\] to the respective alternative Euler formulations. All of the alternative formulations are built from either the divergence form, the advective form, the quadratic or cubic split forms. We note that we use the first part of Lem. \[Lem\] to generate the standard flux divergence form. Furthermore, we need the pure advective form in the energy equation of the Morinishi (MO) form. To obtain the pure advective form we use the quadratic split form and subtract the divergence form scaled by one half, see Rem. \[rem:advective\_form\]. The other quadratic and cubic split forms are directly translated into their flux form using the second and third identity from the Lem. \[Lem\]. Note that all fluxes are consistent, i.e. $$F^{\#}(U_{ijk},U_{ijk}) = F(U_{ijk}),$$ and symmetric in their arguments, e.g. $$F^{\#}(U_{ijk},U_{mjk}) = F^{\#}(U_{mjk},U_{ijk}).$$ In the second part of the proof by Fisher et al. [@fisher2013] it was shown via Taylor expansion (second part of the proof of Thm. 3.1 on pages 15 and 16) that for general two-point entropy conserving flux functions the volume flux differencing approximation gives a high-order accurate scheme. However, the only properties that are used to prove high-order accuracy are consistency of the numerical volume flux and its symmetry. Thus, we can directly extend the proof by Fisher et al. to the numerical volume fluxes presented in Thm. \[Thm2\], as all numerical volume fluxes are indeed symmetric and consistent in their arguments, as noted above. Another way of proving the high-order accuracy is to use again the identities of Lem. \[Lem\], as it is clear that the equivalent split form discretisations (right hand side of the identities in Lem. \[Lem\]) are high-order accurate discretisations of the continuous split forms with the high-order accurate derivative matrix ${{\mathbf{D}}}$.
For central finite difference discretisations, Pirozzoli [@pirozzoli2010] already found a way to rewrite his alternative formulation into an equivalent flux differencing form. In fact, ignoring boundary conditions, central finite difference schemes considered by Pirozzoli have the SBP property and thus the numerical volume flux described here coincides with the one found by Pirozzoli.
To compare the split form DGSEM to entropy conservative methods, we use the variants introduced by Carpenter et al. [@carpenter_esdg]. As stated above, the motivation of the work by Fisher et al. [@fisher2013] and Carpenter et al. [@carpenter_esdg] is the construction of an entropy stable discretisation. The volume terms of the DGSEM are entropy conserving when using a two-point entropy conserving flux function in the volume flux differencing formulation [@carpenter_esdg]. The numerical surface fluxes can then be used to control the dissipation of the discretisation and to guarantee that entropy is decreasing, i.e. entropy stability. To generate two DGSEM variants with entropy conserving volume terms, we choose the following two numerical volume fluxes:
- : Introduce the parameter vector $$\label{zinProof}
{\underline{z}} = \left[\sqrt{\frac{\rho}{p}},\sqrt{\frac{\rho}{p}}u,\sqrt{\frac{\rho}{p}}v,\sqrt{\frac{\rho}{p}}w,\sqrt{\rho p}\right]^T,$$ and the averaged quantities for the primitive variables $$\begin{aligned}
&\qquad\qquad\qquad\hat{\rho} = {\left\{\!\left\{ z_1\right\}\!\right\}}z_5^{\ln},\;\;\hat{u}=\frac{{\left\{\!\left\{ z_2\right\}\!\right\}}}{{\left\{\!\left\{ z_1\right\}\!\right\}}},\;\; \hat{v}=\frac{{\left\{\!\left\{ z_3\right\}\!\right\}}}{{\left\{\!\left\{ z_1\right\}\!\right\}}},\;\;\hat{w} = \frac{{\left\{\!\left\{ z_4\right\}\!\right\}}}{{\left\{\!\left\{ z_1\right\}\!\right\}}},\\[0.1cm]
&\hat{p}_1 = \frac{{\left\{\!\left\{ z_5\right\}\!\right\}}}{{\left\{\!\left\{ z_1\right\}\!\right\}}},\;\;\hat{p}_2 = \frac{\gamma+1}{2\gamma}\frac{z_5^{\ln}}{z_1^{\ln}} + \frac{\gamma-1}{2\gamma}\frac{{\left\{\!\left\{ z_5\right\}\!\right\}}}{{\left\{\!\left\{ z_1\right\}\!\right\}}},\;\; \hat{h} = \frac{\gamma \hat{p}_2}{\hat{\rho}(\gamma-1)} + {\frac{1}{2}}(\hat{u}^2 + \hat{v}^2 + \hat{w}^2),
\end{aligned}$$ where we introduce the logarithmic mean $$\label{logMean}
(\cdot)^{\ln} \coloneqq \frac{(\cdot)_L - (\cdot)_R}{\ln((\cdot)_L) - \ln((\cdot)_R)},$$ needed for the entropy conserving flux functions. We note that a numerically stable procedure to compute the logarithmic mean is described by Ismail and Roe [@ismail2009 Appendix B]. The entropy conserving fluxes in each physical direction read as $$\label{eq:IRflux}
\resizebox{0.86\textwidth}{!}{$
F_{IR}^{\#}(U_{ijk},U_{mjk}) = \begin{bmatrix} \hat{\rho}\hat{u} \\[0.1cm]
\hat{\rho}\hat{u}^2 + \hat{p}_1 \\[0.1cm]
\hat{\rho}\hat{u}\hat{v} \\[0.1cm]
\hat{\rho}\hat{u}\hat{w} \\[0.1cm]
\hat{\rho}\hat{u}\hat{h}\end{bmatrix},
\quad
G_{IR}^{\#}(U_{ijk},U_{imk})= \begin{bmatrix} \hat{\rho}\hat{v} \\[0.1cm]
\hat{\rho}\hat{u}\hat{v} \\[0.1cm]
\hat{\rho}\hat{v}^2 + \hat{p}_1\\[0.1cm]
\hat{\rho}\hat{v}\hat{w}\\[0.1cm]
\hat{\rho}\hat{v}\hat{h}\end{bmatrix},
\quad H_{IR}^{\#}(U_{ijk},U_{ijm}) = \begin{bmatrix} \hat{\rho}\hat{w} \\[0.1cm]
\hat{\rho}\hat{u}\hat{w} \\[0.1cm]
\hat{\rho}\hat{v}\hat{w} \\[0.1cm]
\hat{\rho}\hat{w}^2+ \hat{p}_1 \\[0.1cm]
\hat{\rho}\hat{w}\hat{h}\end{bmatrix}.$}$$
- : Introduce notation for the inverse of the temperature $$\label{eq:beta}
\beta = \frac{1}{RT} = \frac{\rho}{2 p},$$ and the average pressure and enthalpy $$\resizebox{0.85\textwidth}{!}{$
\hat{p} = \frac{{\left\{\!\left\{ \rho\right\}\!\right\}}}{2{\left\{\!\left\{ \beta\right\}\!\right\}}},\quad \hat{h} = \frac{1}{2\beta^{\ln}(\gamma-1)} -{\frac{1}{2}}\left({\left\{\!\left\{ u^2\right\}\!\right\}} + {\left\{\!\left\{ v^2\right\}\!\right\}} + {\left\{\!\left\{ w^2\right\}\!\right\}}\right) + \frac{\hat{p}}{\rho^{\ln}} + {\left\{\!\left\{ u\right\}\!\right\}}^2+ {\left\{\!\left\{ v\right\}\!\right\}}^2+ {\left\{\!\left\{ w\right\}\!\right\}}^2,$}$$ then we have the entropy conserving and kinetic energy preserving flux components $$\label{EKEP}
\resizebox{0.85\textwidth}{!}{$
F_{CH}^{\#}(U_{ijk},U_{mjk}) = \begin{bmatrix} \rho^{\ln}{\left\{\!\left\{ u\right\}\!\right\}} \\[0.1cm]
{\rho^{\ln}}{\left\{\!\left\{ u\right\}\!\right\}}^2 + \hat{p} \\[0.1cm]
{\rho^{\ln}}{\left\{\!\left\{ u\right\}\!\right\}}{\left\{\!\left\{ v\right\}\!\right\}} \\[0.1cm]
{\rho^{\ln}}{\left\{\!\left\{ u\right\}\!\right\}}{\left\{\!\left\{ w\right\}\!\right\}} \\[0.1cm]
{\rho^{\ln}}{\left\{\!\left\{ u\right\}\!\right\}}\hat{h}\end{bmatrix},
\quad
G_{CH}^{\#}(U_{ijk},U_{imk})= \begin{bmatrix} \rho^{\ln}{\left\{\!\left\{ v\right\}\!\right\}} \\[0.1cm]
{\rho^{\ln}}{\left\{\!\left\{ u\right\}\!\right\}}{\left\{\!\left\{ v\right\}\!\right\}} \\[0.1cm]
{\rho^{\ln}}{\left\{\!\left\{ v\right\}\!\right\}}^2 + \hat{p}\\[0.1cm]
{\rho^{\ln}}{\left\{\!\left\{ v\right\}\!\right\}}{\left\{\!\left\{ w\right\}\!\right\}} \\[0.1cm]
{\rho^{\ln}}{\left\{\!\left\{ v\right\}\!\right\}}\hat{h}\end{bmatrix},
\quad
H_{CH}^{\#}(U_{ijk},U_{ijm}) = \begin{bmatrix} \rho^{\ln}{\left\{\!\left\{ w\right\}\!\right\}} \\[0.1cm]
{\rho^{\ln}}{\left\{\!\left\{ u\right\}\!\right\}}{\left\{\!\left\{ w\right\}\!\right\}} \\[0.1cm]
{\rho^{\ln}}{\left\{\!\left\{ v\right\}\!\right\}}{\left\{\!\left\{ w\right\}\!\right\}} \\[0.1cm]
{\rho^{\ln}}{\left\{\!\left\{ w\right\}\!\right\}}^2 + \hat{p} \\[0.1cm]
{\rho^{\ln}}{\left\{\!\left\{ w\right\}\!\right\}}\hat{h}\end{bmatrix}.$}$$
Kinetic energy preservation of the split forms {#sec:kep}
----------------------------------------------
In [@jameson2008] Jameson derived a general condition on a numerical flux function for finite volume schemes to generate kinetic energy preserving discretisations. The kinetic energy is $$\kappa:=\frac{1}{2}\rho\,(u^2+v^2+w^2) = \frac{(\rho\,u)^2+(\rho\,v)^2+(\rho\,w)^2}{2\,\rho}.$$ The kinetic energy balance can be directly derived from the compressible Euler equations and reads $$\label{eq:kep_balance_paper}
\resizebox{0.9\textwidth}{!}{$
\frac{\partial \kappa}{\partial t} + \left(\frac{1}{2}\rho\,u\,(u^2+v^2+w^2)\right)_x + \left(\frac{1}{2}\rho\,v\,(u^2+v^2+w^2)\right)_y+ \left(\frac{1}{2}\rho\,w\,(u^2+v^2+w^2)\right)_z +u\,p_x +v\,p_y +w\,p_z=0.
$}$$
Note, that the influence of the advective terms of the momentum flux $\rho\,u^2,\,\rho\,u\,v,\,\rho\,u\,w,...$ can be re-cast into flux form, whereas the pressure gradient in the momentum flux yields a non-conservative term (pressure work) in the kinetic energy balance . It is exactly this structure of the kinetic energy balance that is reflected in the definition of kinetic energy preserving by Jameson: Ignoring boundary conditions, a discretisation is termed kinetic energy preserving when the (discrete) integral of the kinetic energy is not changed by the advective terms, but only by the pressure work.
For finite volume schemes, Jameson [@jameson2008] found the following general form for the components of the numerical surface fluxes of the momentum equations that give kinetic energy preservation: $$\begin{aligned}
{3}\label{eq:kep_property}
&F^{*,2} = F^{*,1}\,{\left\{\!\left\{ u\right\}\!\right\}} + \widetilde{p},\quad &&F^{*,3} = F^{*,1}\,{\left\{\!\left\{ v\right\}\!\right\}},\quad &&F^{*,4} = F^{*,1}\,{\left\{\!\left\{ w\right\}\!\right\}},\nonumber\\
&G^{*,2} = G^{*,1}\,{\left\{\!\left\{ u\right\}\!\right\}} ,\quad &&G^{*,3} = G^{*,1}\,{\left\{\!\left\{ v\right\}\!\right\}}+\widetilde{p},\quad &&G^{*,4} = G^{*,1}\,{\left\{\!\left\{ w\right\}\!\right\}},\\
&H^{*,2} = H^{*,1}\,{\left\{\!\left\{ u\right\}\!\right\}} ,\quad &&H^{*,3} = H^{*,1}\,{\left\{\!\left\{ v\right\}\!\right\}},\quad &&H^{*,4} = H^{*,1}\,{\left\{\!\left\{ w\right\}\!\right\}}+\widetilde{p},\nonumber\end{aligned}$$ where $\widetilde{p}$ can be any consistent numerical trace approximation of the pressure. This structure is enough to guarantee that the advective terms in the resulting discrete kinetic energy balance are consistent and in conservative form.\
\
If we inspect the numerical volume flux functions presented in Thm. \[Thm2\], the conditions hold for the variants MO, KG, PI and CH. We will demonstrate that the conditions are enough to guarantee discrete kinetic energy preservation for the flux difference formulation .
Gassner [@gassner_kepdg] showed that Morinishi’s alternative formulation allows one to construct a kinetic energy preserving DG discretisation. Using the same approach, it is possible to prove that KG and PI are kinetic energy preserving as well. In contrast, the method of cannot be applied to the CH formulation, as no explicit split form is known. However, it is possible to make use of the flux differencing form of Sec. \[sec:DGSEM\_fluxform\] to prove that numerical volume fluxes satisfying the conditions generate high-order accurate schemes that are kinetic energy preserving in the sense of Jameson, i.e. that the advective terms can be re-cast into a conservative form . We summarise the third main result of this work in the following theorem.
\[Thm3\] If the numerical volume flux functions $F^\#$, $G^\#$, and $H^\#$ satisfy the condition , the volume terms of the resulting flux differencing discretisation are kinetic energy preserving in the sense of Jameson [@jameson2008], . More precisely, the volume terms of the discrete kinetic energy balance are given by $$\label{eq:kin_pres}
\begin{split}
\left(\frac{\partial \kappa}{\partial t}\right)_{ijk}\approx &-2\,\sum\limits_{m=0}^N D_{im}\,f^{\#,\kappa}(U_{ijk},U_{mjk})+D_{jm}\,g^{\#,\kappa}(U_{ijk},U_{imk})
+D_{km}\,h^{\#,\kappa}(U_{ijk},U_{ijm})\\
&-2\,\sum\limits_{m=0}^N u_{ijk}\,D_{im}\,\widetilde{p}(U_{ijk},U_{mjk})+v_{ijk}\,D_{jm}\,\widetilde{p}(U_{ijk},U_{imk})+w_{ijk}\,D_{km}\,\widetilde{p}(U_{ijk},U_{ijm}),
\end{split}$$ with a consistent and symmetric pressure discretisation $\widetilde{p}$ and with the consistent and symmetric advective fluxes of the discrete kinetic energy $$\begin{split}
f^{\#,\kappa}(U_{ijk},U_{mjk}) &:=\frac{1}{2}\,F^{\#,1}(U_{ijk},U_{mjk})\left(u_{ijk}\,u_{mjk}+v_{ijk}\,v_{mjk}+w_{ijk}\,w_{mjk}\right),\\
g^{\#,\kappa}(U_{ijk},U_{imk}) &:=\frac{1}{2}\,G^{\#,1}(U_{ijk},U_{imk})\left(u_{ijk}\,u_{imk}+v_{ijk}\,v_{imk}+w_{ijk}\,w_{imk}\right),\\
h^{\#,\kappa}(U_{ijk},U_{ijm}) &:=\frac{1}{2}\,H^{\#,1}(U_{ijk},U_{ijm})\left(u_{ijk}\,u_{ijm}+v_{ijk}\,v_{ijm}+w_{ijk}\,w_{ijm}\right).
\end{split}$$
Again, we move the proof to the \[Thm3Proof\].
Note, that the terms of the discrete kinetic energy balance are consistent to the continuous kinetic energy balance
The kinetic energy preserving flux function proposed by Jameson is identical to the $PI$ flux .
In Gassner [@gassner_kepdg] it was shown that when the volume terms of the DGSEM are kinetic energy preserving and the numerical surface flux function at element interfaces satisfies the general conditions then the multi-element discretisation is kinetic energy preserving as well. Thus, if the numerical volume and surface flux are the same it follows that the DGSEM approximation satisfies the kinetic energy preservation condition . In particular, the variants MO, KG, PI and CH are kinetic energy preserving in the sense that the (discrete) integral of the kinetic energy is only changed by the pressure work (ignoring boundary conditions).
Numerical surface flux functions {#sec:numflux}
--------------------------------
As mentioned above, we connect the choice of the volume numerical flux to the choice of the surface numerical flux. As a blueprint, we use the local Lax-Friedrichs numerical flux function, e.g. $$F^*(U_-,U_+) := \frac{1}{2}\left(F(U_-) + F(U_+)\right) - \frac{1}{2}\,\lambda_{max}\,\left[U_+ - U_-\right],$$ where $\lambda_{max}$ is an estimate of the fastest wave speed at the interface. Note that the structure of the local Lax-Friedrichs flux (LLF) is a consistent symmetric part $\frac{1}{2}\left(F(U_-) + F(U_+)\right)$ and a stabilisation term $- \frac{1}{2}\,\lambda_{max}\,\left[U_+ - U_-\right]$, which introduces dissipation that depends on the jump of the approximation at the interface.
Analogously, we replace the consistent symmetric part by the consistent symmetric numerical volume flux function $F^\#$, which is also used to generate the volume terms of the DGSEM. We add a stabilisation term $- Stab(U_-,U_+)$, that we design so that it introduces dissipation at the interface depending on the jump of the approximate solution $$\label{eq:numericalflux}
F^*(U_-,U_+) := F^\#(U_-,U_+) - Stab(U_-,U_+).$$ For the variants $MO$, $DU$, $KG$, and $PI$ we choose the same stabilisation term as in the local Lax Friedrichs flux $$\label{eq:llf}
Stab_{MO}(U_-,U_+)=Stab_{DU}(U_-,U_+)=Stab_{KG}(U_-,U_+)=Stab_{PI}(U_-,U_+):=\frac{1}{2}\,\lambda_{max}\,\left[U_+ - U_-\right],$$ where $\lambda_{max}$ is the maximum of the maximum wave speeds from the left and right at the interface. The stabilisation term for $CH$ is a direct 3D extension of the term introduced in [@chandrashekar2013] and is identical to for the first four components, but differs in the fifth component (energy equation): $$\resizebox{0.9\hsize}{!}{$\begin{split}
Stab^5_{CH}&(U_-,U_+):=\frac{1}{2}\lambda_{max}\Bigg\{ \left[\frac{1}{2(\gamma -1)\beta^{ln}} + (u^+u^-+v^+v^-+w^+w^-) \right]\,[\rho^+-\rho^-]\\
&+ {\left\{\!\left\{ \rho\right\}\!\right\}}{\left\{\!\left\{ u\right\}\!\right\}}[u^+-u^-] + {\left\{\!\left\{ \rho\right\}\!\right\}}{\left\{\!\left\{ v\right\}\!\right\}}[v^+-v^-]+ {\left\{\!\left\{ \rho\right\}\!\right\}}{\left\{\!\left\{ w\right\}\!\right\}}[w^+-w^-]+\frac{{\left\{\!\left\{ \rho\right\}\!\right\}}}{2(\gamma-1)}\left[\frac{1}{\beta^+} - \frac{1}{\beta^-} \right]\Bigg\},
\end{split}$}$$ where $\beta$ is introduced in .
Note that the standard LLF type dissipation terms in the first four components ensures guaranteed kinetic energy dissipation [@chandrashekar2013], whereas the specific fifth component of $CH$ stabilisation term additionally ensures guaranteed entropy dissipation [@chandrashekar2013]. For the $IR$ stabilisation, we use the term introduced in [@carpenter_esdg] $$Stab_{IR}(U_-,U_+) = \frac{1}{2}\lambda_{max}\,\mathcal{H}\,[V^+ - V^-],$$ where $V = [\frac{\gamma - s}{\gamma-1}-\frac{\rho\,\|u\|^2}{2\,p}, \frac{\rho\,u}{p}, \frac{\rho\,v}{p}, \frac{\rho\,w}{p}, -\frac{\rho}{p} ]^T$ are the entropy variables and $s=\ln(p) - \gamma\,\ln(\rho)$ is the specific thermodynamic entropy. The matrix $\mathcal{H}$ is the symmetric positive definite entropy Jacobian of the conservative variables $$\mathcal{H}:=\frac{\partial U}{\partial V}
= \begin{bmatrix}
\rho & \rho u & \rho v & \rho w & \rho e \\[0.1cm]
\rho u & \rho u^2 + p & \rho u v & \rho u w & \rho {h} u \\[0.1cm]
\rho v & \rho u v & \rho v^2 + p& \rho v w & \rho {h} v \\[0.1cm]
\rho w & \rho u w & \rho v w & \rho w^2 + p & \rho {h} w \\[0.1cm]
\rho e & \rho{h} u & \rho{h} v & \rho{h} w & \rho{h}^2 - \frac{a^2 p }{\gamma-1}\\[0.1cm]
\end{bmatrix},$$ where the sound speed is defined as $a^2 = \frac{\gamma\,p}{\rho}$. The maximum wave speed $\lambda_{max}$ and the entries of the entropy Jacobian $\mathcal{H}$ are evaluated at the average states used to compute the $IR$ numerical flux functions . Again, this stabilisation term is designed so that the discrete entropy is dissipated at element interfaces. Thus, the variants IR and CH with stabilisation terms are entropy stable.
Numerical Investigations {#sec:numerical results}
========================
The main purpose of this section is to investigate the of the new split form DGSEM with respect to different choices of grid resolution and/or polynomial degree. We compare the robustness and behaviour of the standard DGSEM, the variant DU, the new split form variants with kinetic energy preserving volume terms MO, KG and PI, the variant with entropy conserving volume terms IR (Carpenter et al. [@carpenter_esdg]) and the variant with entropy conserving and kinetic energy preserving volume terms CH.
We start however with an investigation of the experimental order of convergence, discrete entropy conservation and kinetic energy preservation. All simulation results are obtained with an explicit five stage fourth order accurate low storage Runge-Kutta scheme, where the time step is computed according to a CFL type condition with the local maximum wave speed and the relative grid size $\Delta x/(N+1)$, where $\Delta x$ is the element size. If not specified otherwise, we use $CFL=0.5$ for all computations. The adiabatic coefficient is chosen as $\gamma=1.4$.
$h$- and $p$-convergence
------------------------
We first investigate the $h$- and $p$-convergence properties of the schemes. We choose the following manufactured solution for the unsteady compressible Euler equations $$\begin{split}
\rho &= 2 + \frac{1}{10}\,\sin(\pi\,(x+y+z - 2\,t)),\\
u&=1,\\
v&=1,\\
w&=1,\\
\rho\,e &= \left(2 + \frac{1}{10}\,\sin(\pi\,(x+y+z - 2\,t))\right)^2,
\end{split}$$ with the corresponding source term $$\begin{split}
q_{\rho} &= c_1\,\cos(\pi\,(x+y+z - 2\,t))\\
q_{\rho\,u} &= c_2\,\cos(\pi\,(x+y+z - 2\,t))+c_3\,\cos(2\,\pi\,(x+y+z - 2\,t)),\\
q_{\rho\,v} &= c_2\,\cos(\pi\,(x+y+z - 2\,t))+c_3\,\cos(2\,\pi\,(x+y+z - 2\,t)),\\
q_{\rho\,w} &= c_2\,\cos(\pi\,(x+y+z - 2\,t))+c_3\,\cos(2\,\pi\,(x+y+z - 2\,t)),\\
q_{\rho\,e} &= c_4\,\cos(\pi\,(x+y+z - 2\,t))+c_5\,\cos(2\,\pi\,(x+y+z - 2\,t)),\\
\end{split}$$ where $c_1=\frac{\pi}{10} $, $c_2=-\frac{1}{5}\,\pi+\frac{1}{20}\,\pi\,(1+5\,\gamma)$, $c_3 =\frac{\pi}{100}\,(\gamma -1)$, $c_4 = \frac{1}{20}\left(-16\,\pi + \pi\,(9+15\,\gamma)\right)$, and $c_5 = \frac{1}{100}\,\left(3\,\pi\,\gamma-2\,\pi\right)$. The end time is set to $t_{end}=10$. The computational domain is a fully periodic box with the size $[-1,1]^3$.\
\
As stated above, derivations in this work are valid for all diagonal norm SBP operators, such as those from the finite difference community. A conceptional difference of a spectral element method to a finite difference approximation is the ability for $p$-convergence. In $p$-convergence studies the grid is fixed and the number of the nodes inside an element is increased. In a DGSEM, increasing the number of element nodes automatically increases the polynomial degree of the approximation and hence increases the approximation order in tandem with the resolution. The resulting *spectral* convergence can be observed in Fig. \[fig:p\_conv\], where all schemes are investigated with and without interface stabilisation on a regular $4^3$ grid. As an example, the $L_2$-errors in density are shown. All other quantities show a similar behaviour. The $L_2$-norm is computed discretely with the collocated GL quadrature used for the scheme.
For this test, the errors of the different schemes are remarkably close, especially when interface stabilisation is activated. Note that without interface stabilisation, the numerical surface flux is symmetric in its arguments. This causes an odd/even effect, which can be clearly observed in the left plot of Fig. \[fig:p\_conv\] and is in accordance to similar observations [@gassner_skew_burgers; @gassner_kepdg; @gassner2015]. The simulations for higher polynomial degree $N$ plateau out at about $10^{-8}$ in the right plot due to the accuracy of the time integration scheme. By decreasing the CFL from $0.5$ to $0.25$, we can see in the right plot of Fig. \[fig:p\_conv\] that the level of the plateau decreases exactly by a factor of $16$ as expected for the fourth order accurate RK method.
![\[fig:p\_conv\] $p$-convergence for all the schemes on a regular $4^3$ grid. Plot of the $L_2$-error in density is shown. Left: without interface stabilisation. Right: with interface stabilisation.](figures/eoc/p-conv_nostab "fig:"){width="45.00000%"}![\[fig:p\_conv\] $p$-convergence for all the schemes on a regular $4^3$ grid. Plot of the $L_2$-error in density is shown. Left: without interface stabilisation. Right: with interface stabilisation.](figures/eoc/p-conv_stab "fig:"){width="45.00000%"}
The next figure, Fig. \[fig:h\_conv\_nostab\], shows the $h$-convergence behaviour for the different split form DGSEM schemes without stabilisation for polynomial degree $N=3$ and $N=4$. Without interface stabilisation terms, we again obtain the odd/even behaviour: for odd polynomial degrees $N$ we do not get the optimal convergence rate $N+1$, but only a convergence rate of $N$.
![\[fig:h\_conv\_nostab\] $h$-convergence for all the schemes without interface stabilisation. Grid sequence goes from $2^3$ up to $16^3$. Plot of the $L_2$-error in density is shown. The odd/even effect can be seen. Left: $N=3$, experimental order of convergence $\approx 3$. Right: $N=4$, experimental order of convergence $\approx 5$.](figures/eoc/h-conv_nostabN3 "fig:"){width="45.00000%"}![\[fig:h\_conv\_nostab\] $h$-convergence for all the schemes without interface stabilisation. Grid sequence goes from $2^3$ up to $16^3$. Plot of the $L_2$-error in density is shown. The odd/even effect can be seen. Left: $N=3$, experimental order of convergence $\approx 3$. Right: $N=4$, experimental order of convergence $\approx 5$.](figures/eoc/h-conv_nostabN4 "fig:"){width="45.00000%"}
If we switch on the interface stabilisation terms, again, the errors produced by the different fluxes are almost the same. All the schemes show now the optimal order of convergence, i.e. for a polynomial of degree $N$, we get $N+1$st order convergence in $h$ as can be seen in Fig. \[fig:h\_conv\_stab\].
![\[fig:h\_conv\_stab\] $h$-convergence for the all schemes with interface stabilisation. Grid sequence goes from $2^3$ up to $16^3$. Plot of the $L_2$-error in density is shown. For polynomial degree $N$, we get $N+1$ convergence in $h$. Left: $N=3$. Right: $N=4$.](figures/eoc/h-conv_stabN3 "fig:"){width="45.00000%"}![\[fig:h\_conv\_stab\] $h$-convergence for the all schemes with interface stabilisation. Grid sequence goes from $2^3$ up to $16^3$. Plot of the $L_2$-error in density is shown. For polynomial degree $N$, we get $N+1$ convergence in $h$. Left: $N=3$. Right: $N=4$.](figures/eoc/h-conv_stabN4 "fig:"){width="45.00000%"}
In summary, all schemes show the expected $p$- and $h$-convergence behaviour.
Entropy conservation and kinetic energy preservation
----------------------------------------------------
All presented schemes conserve mass, momentum, and total energy by construction. In this section, we focus on the auxiliary conservation properties of the methods and investigate the entropy and kinetic energy of the schemes. To investigate the auxiliary conservation properties, we deactivate the numerical dissipation introduced by the surface stabilisation terms and only use the symmetric and consistent parts at element interfaces.
In contrast to the last section, it does not make sense to investigate the auxiliary conservation properties with well resolved smooth solutions, as in this case all methods would converge spectrally fast to them, if the solution supports it. To make this investigation challenging, we consider the inviscid Taylor-Green vortex. The initial condition in the periodic $[0,2\,\pi]^3$ box is $$\begin{split}
\rho &= 1,\\
u &= \sin(x)\,\cos(y)\,\cos(z),\\
v &= -\cos(x)\,\sin(y)\,\cos(z),\\
w &= 0,\\
p &= \frac{100}{\gamma} + \frac{1}{16}\,\left(\cos(2\,x)\,\cos(2\,z) + 2\,\cos(2\,y)+2\,\cos(2\,x)+\cos(2\,y)\,\cos(2\,z)\right).
\end{split}$$ The evolution of these simple initial conditions is quite challenging due to the non-linear interaction of scales as well as the transition to a turbulence like flow field that occurs for large enough times. Without physical viscosity, there is no small scale limit and thus approximations of the inviscid Taylor-Green vortex are always under-resolved for large enough times. This behaviour makes this test case a challenge for the robustness of high-order methods.
Before we investigate the robustness of the schemes, we use this test setup to investigate their auxiliary . The inviscid Taylor-Green vortex solution conserves both the total kinetic energy and the total entropy for all times.
As mentioned above, these investigations only make sense when the interface stabilisation is omitted. Otherwise, the numerical viscosity would dissipate kinetic energy and entropy when the approximation is under-resolved. However, without interface stabilisation the resulting discretisations are basically dissipation free and the robustness is fragile. A general observation is that the higher the polynomial degree and the higher the overall number of spatial degrees of freedom, the harder it is to successfully run the simulations until the final time. Without interface stabilisation, the highest polynomial degree we could choose and still get meaningful results is $N=3$. For $N=3$ it is interesting to note that all proposed split form schemes run for all tested resolutions (up to $256^3$), except for the standard and MO variants, which crash almost immediately. For higher polynomial degrees, the general trend is that low resolutions with $16^3$ or $32^3$ might run, but all schemes crash for resolutions greater than or equal to $64^3$.
Nevertheless, the choice of polynomial degree $N=3$ and $16^3$ grid cells ($64^3$ degrees of freedom) allows us to investigate the auxiliary conservation properties. The left part of Fig. \[fig:cons\] shows the time evolution of the total entropy density $$S = -\frac{\rho s}{\gamma-1},\quad s = \ln(p) - \gamma\ln(\rho),$$ for the different schemes. The right part shows the temporal evolution of the total kinetic energy.
![\[fig:cons\]Time evolution of the discrete total entropy and total kinetic energy for the case $N=3$ and $16^3$ grid cells. All results are obtained with $CFL=0.1$.](figures/kep_entropycons/EntropyCFLp1N3.pdf "fig:"){width="45.00000%"}![\[fig:cons\]Time evolution of the discrete total entropy and total kinetic energy for the case $N=3$ and $16^3$ grid cells. All results are obtained with $CFL=0.1$.](figures/kep_entropycons/KECFLp1N3.pdf "fig:"){width="45.00000%"}
Focusing first on the entropy conservation, as expected the IR and CH schemes conserve entropy basically to machine precision, whereas the other schemes show a small decay of the total entropy. However, the $y-$axis shows that the entropy dissipation of the other formulations is very small and arguably negligible.
The entropy conservation results lie in stark contrast to the kinetic energy preservation where substantial differences are observed. Note, that we have three schemes KG, PI and CH which, by construction, preserve the kinetic energy in the sense that the discrete pressure work changes the total kinetic energy, not the advective terms. We can see in the right part of Fig. \[fig:cons\] that KG and PI give similar results and indeed best preserve the total kinetic energy. Especially in comparison to the other discretisations, which show a loss of up to 10% of total kinetic energy. The IR scheme performs the worst.
There are two noteworthy remarks. First, the DU scheme conserves neither entropy nor preserves kinetic energy. However, this scheme has a lower loss of total kinetic energy compared to the entropy conserving schemes IR and CH. Second, although the CH scheme is formally entropy conserving *and* kinetic energy preserving, the results clearly show that the loss of kinetic energy is much larger than for the KG and PI schemes. This is an unexpected result and demands further investigation. It turns out that although the advective terms do not formally contribute to the evolution of the kinetic energy as desired, the discretisation of the pressure plays a crucial role for the kinetic energy. Both, KG and PI use a simple arithmetic mean for the pressure and it turns out that this seems to be important for the kinetic energy evolution.
The CH scheme chooses the discretisation of the pressure in such a way that entropy is discretely conserved. The pressure discretisation is much more complicated than the simple arithmetic mean and a possible explanation is that this introduces additional errors that affect the discrete total kinetic energy. To support this claim, we modify the CH scheme by changing the pressure discretisation to the simple arithmetic mean. The right part of Fig. \[fig:keep\] shows the evolution of the discrete total kinetic energy. The CH scheme with the pressure fix now behaves like the KG and PI scheme and significantly reduces the loss of kinetic energy in comparison to the original CH scheme. However, the CH scheme with pressure fix is, of course, not entropy conservative anymore as shown in the left part of Fig. \[fig:keep\].
![\[fig:keep\]Comparison of original CH scheme and the CH scheme where the pressure discretisation is changed to a simple arithmetic mean, as in the KG and PI variants. The plot shows the time evolution of the discrete total entropy (left) and the discrete total kinetic energy (right) for the case $N=3$ and $16^3$ grid cells. All results are obtained with $CFL=0.1$.](figures/kep_entropycons/EntropyPresFixN3.pdf "fig:"){width="45.00000%"}![\[fig:keep\]Comparison of original CH scheme and the CH scheme where the pressure discretisation is changed to a simple arithmetic mean, as in the KG and PI variants. The plot shows the time evolution of the discrete total entropy (left) and the discrete total kinetic energy (right) for the case $N=3$ and $16^3$ grid cells. All results are obtained with $CFL=0.1$.](figures/kep_entropycons/CHFixN3.pdf "fig:"){width="45.00000%"}
Summarising this section, the IR and CH schemes show machine precision entropy conservation as expected. The KG and PI schemes show near discrete kinetic energy . The CH scheme shows discrete kinetic energy conservation only with a modification of the pressure discretisation While the DU scheme does not possess any specific auxiliary conservation properties, it is still worth pointing out that it lost less of the total kinetic energy compared to the entropy conserving schemes.
Robustness
----------
In this main results section, the robustness of the schemes is investigated. In contrast to the former investigations, we now activate the interface stabilisation. The interface stabilisation terms depend on the jump of the solution at the interface and introduce dissipation. The main result of the numerical investigations is that with the stabilisation terms, all split schemes except the MO variant are robust in the sense that even for very high polynomial degrees the simulations do not crash even with severe under-resolution.
To make our point, we mimic a table presented in Moura et al. [@rodrigo_iLES], where the stability of an over-integrated DG method for the inviscid Taylor-Green vortex is reported. It is important to note that for high resolutions, the over-integrated DG discretisations crashed and it was concluded that the dissipation from the surface integrals is not enough to stabilise the discretisation. In contrast to the over-integration results, all simulations performed with the DGSEM split forms listed in Tbl. \[tab:TGVStability\] for the schemes KG, IR, PI, CH, and DU successfully run up to the final time of $t=14$. Note that the configurations marked with a box are the ones that run to the final time when over-integration is used [@rodrigo_iLES]. All other configurations could not be successfully completed, even when decreasing the time step and increasing the number of integration nodes up to a factor of four in each spatial direction \[personal communication with Rodrigo Moura\]. Thus, the entropy stable schemes and the new split form schemes offer a substantial advantage with respect to robustness compared to fully integrated DG.
------------------- --------------- -------------- -------------- -------------- --------------
DOFs/$N$ $3$ $4$ $5$ $6$ $7$
\[0.0cm\] $113^3$ [[$19$]{}]{} [[$16$]{}]{} [[$14$]{}]{}
\[0.0cm\]$159^3$ [[$28$]{}]{} [[$23$]{}]{} [[$19$]{}]{}
\[0.0cm\]$227^3$ [[$39$]{}]{} [[$32$]{}]{} [[$28$]{}]{}
\[0.0cm\]$320^3$ [[$80$]{}]{} [[$64$]{}]{} [[$53$]{}]{} [[$46$]{}]{} [[$40$]{}]{}
\[0.1cm\]$450^3$ [[$113$]{}]{} [[$90$]{}]{} [[$75$]{}]{} [[$64$]{}]{} [[$56$]{}]{}
------------------- --------------- -------------- -------------- -------------- --------------
: List of grid cell and polynomial degree configurations for KG, PI, IR, DU, CH schemes that were investigated. None of these schemes crashed. The number of DOFs was chosen to keep a factor of $\sqrt{2}$ between consecutive DOFs while allowing for (approximately) integer numbers of elements across the range of polynomial degrees considered. The configurations marked by a box are the ones that are stable with over-integration, whereas the other configuration could not be successfully finished with over-integration alone [@rodrigo_iLES].[]{data-label="tab:TGVStability"}
To further assess the behaviour of the schemes for the inviscid Taylor-Green vortex, we investigate the evolution of the total kinetic energy dissipation rate, $-d\,\kappa/dt$, and the enstrophy $$\sigma : = \frac{1}{|\Omega|}\int\limits_{\Omega}\frac{\rho}{2}\omega\cdot\omega\,d\Omega,$$ where $\omega$ is the vorticity vector. In the left part of Fig. \[fig:dkdt\_N3\], the resulting dissipation rate for the configuration $N=3$ with $16^3$ grid cells ($64^3$ degrees of freedom) for all schemes is plotted. Although there is a difference between the split form variants, it is fairly small. In the right part of Fig. \[fig:dkdt\_N3\] we focus on the KG variant only and increase the number of degrees of freedom to compare the $64^3$ results to the configuration with $32^3$ grid cells.
![\[fig:dkdt\_N3\] Plot of the total kinetic energy dissipation rate. Left: $N=3$ with $64^3$ degrees of freedom for all stable split forms. Right: KG scheme with $N=3$ and $16^3$ and $32^3$ grid cells respectively.](figures/dkdt_enstrophy/KEDecayN3.pdf "fig:"){width="45.00000%"}![\[fig:dkdt\_N3\] Plot of the total kinetic energy dissipation rate. Left: $N=3$ with $64^3$ degrees of freedom for all stable split forms. Right: KG scheme with $N=3$ and $16^3$ and $32^3$ grid cells respectively.](figures/dkdt_enstrophy/KEDecay_64VS128.pdf "fig:"){width="45.00000%"}
It is interesting to note that again, the dissipation rate does not change significantly.
It seems that the dissipation introduced by the numerical surface flux function has a certain limit . This is further supported by the results in Fig. \[fig:dkdt\_N7\_15\], where for configurations with fixed $64^3$ degrees of freedom the polynomial degrees are increased to $N=7$ and $N=15$. Note, that the dissipation rates do not significantly change.
![\[fig:dkdt\_N7\_15\] Plot of the total kinetic energy dissipation rate. Left: $N=7$ with $8^3$ grid cells for all stable split forms. Right: $N=15$ with $4^3$ grid cells for all stable split forms.](figures/dkdt_enstrophy/KEDecayN7.pdf "fig:"){width="45.00000%"}![\[fig:dkdt\_N7\_15\] Plot of the total kinetic energy dissipation rate. Left: $N=7$ with $8^3$ grid cells for all stable split forms. Right: $N=15$ with $4^3$ grid cells for all stable split forms.](figures/dkdt_enstrophy/KEDecayN15.pdf "fig:"){width="45.00000%"}
In contrast to the behaviour of the dissipation rates, the evolution of the enstrophy strongly depends on the spatial resolution and on the polynomial degree. Figure \[fig:enstrophy\] shows the enstrophy as a function of time for a fixed $64^3$ degrees of freedom in the left part. The difference between the split forms is again small, however the impact of the polynomial degree increases the magnitude of the total of enstrophy by an order of magnitude. The right part of Fig. \[fig:enstrophy\] considers again the KG scheme with $N=3$ and $16^3$ and $32^3$ grid cells, respectively. By comparing the left and right plot, it is interesting to note that the enstrophy of the configuration with $N=7$ and $8^3$ grid cells ($64^3$ degrees of freedom) has a higher maximum than the configuration with $N=3$ and $32^3$ grid cells ($128^3$ degrees of freedom).
![\[fig:enstrophy\] Temporal evolution of the enstrophy. Left: Fixed $64^3$ degrees of freedom for all stable split forms for configurations with polynomial degree $N=3,\,7,\,15$. Right: Enstrophy of the KG scheme with $N=3$ and $16^3$ and $32^3$ grid cells respectively.](figures/dkdt_enstrophy/EnstrophyCompare64.pdf "fig:"){width="45.00000%"}![\[fig:enstrophy\] Temporal evolution of the enstrophy. Left: Fixed $64^3$ degrees of freedom for all stable split forms for configurations with polynomial degree $N=3,\,7,\,15$. Right: Enstrophy of the KG scheme with $N=3$ and $16^3$ and $32^3$ grid cells respectively.](figures/dkdt_enstrophy/Enstrophy_64VS128.pdf "fig:"){width="45.00000%"}
The significant impact of the polynomial degree on the evolution of the enstrophy has a direct consequence for the estimated numerical viscosity of the discretisations. For solutions of the incompressible viscous Taylor-Green vortex, the dissipation rate is directly linked to the enstrophy via the physical viscosity $\mu$ [@tcfd2012] $$-\frac{d\,\kappa}{dt} = 2\,\mu\,\sigma.$$ We exploit this relationship to find an estimate for the viscosity introduced by the numerical discretisation. By relating the discrete dissipation rate and the enstrophy over time, we get an evolution of the numerical dissipation of the scheme $$\mu_{num} \approx \frac{-\frac{d\,\kappa}{dt}}{2\,\sigma}.$$ The numerical viscosity estimate is plotted in the left part of Fig. \[fig:numvisc\], where again the degrees of freedom are fixed to $64^3$. Due to the significantly higher enstrophy for the high polynomial degree discretisations, the estimate of the numerical viscosity is much lower. This fits to the apparent “high-order schemes have lower dissipation" paradigm. However, it is interesting to note that the actual dissipation rate of the kinetic energy does not vary much. In the right part of Fig. \[fig:numvisc\], we again compare the KG scheme for the $N=3$ and $16^3$ configuration.
![\[fig:numvisc\] Plot of the numerical viscosity over time. Left: Fixed $64^3$ degrees of freedom for all stable split forms for configurations with polynomial degree $N=3,\,7,\,15$. Right: Comparison of KG results with and without stabilisation for $N=3$ and $16^3$ grid cells.](figures/numerical_viscosity/NumViscCompare64_Stab.pdf "fig:"){width="45.00000%"}![\[fig:numvisc\] Plot of the numerical viscosity over time. Left: Fixed $64^3$ degrees of freedom for all stable split forms for configurations with polynomial degree $N=3,\,7,\,15$. Right: Comparison of KG results with and without stabilisation for $N=3$ and $16^3$ grid cells.](figures/numerical_viscosity/StabVSNoStab64.pdf "fig:"){width="45.00000%"}
We directly compare the impact of the interface stabilisation terms on the numerical viscosity and plot results from the simulation with interface stabilisation terms and without. It can be clearly observed that the no stabilisation run has almost zero numerical viscosity, while being stable for this severely under-resolved test case.
At the end of this section, we note that the variant KG (and PI) with cubic split forms was introduced by Kennedy and Gruber [@kennedy2008] to account for large density variations. Thus, it is interesting to investigate the difference of DU and KG/PI for a test configuration with higher compressibility. To do so we change the initial pressure for inviscid the Taylor-Green vortex so that the Mach number increases to $Ma=0.4$. This introduces more compressibility effects and as shown in Fig. \[fig:M04\_1\] and \[fig:M04\_2\], this causes problems for the DU variant. For the configuration $N=7$ with $16^3$ grid cells, we can observe that DU crashes at $t\approx 10.8$, even with very small CFL numbers. The variants KG, PI and IR, CH all run until the final time, demonstrating increased robustness of those variants for problems with higher density variations.
![\[fig:M04\_1\] Results of simulation with $16^3$ grid cells and $N=7$ ($128^3$ degrees of freedom) for the Taylor-Green vortex with a Mach number of $Ma=0.4$. Left: Plot of total enstrophy. Right: Plot of total kinetic energy. Note that DU crashes at $t\approx 10.8$, indicated by an extreme spike of the quantities.](figures/Machp4_128DOFs/EnstrophyN7Machp4.pdf "fig:"){width="45.00000%"}![\[fig:M04\_1\] Results of simulation with $16^3$ grid cells and $N=7$ ($128^3$ degrees of freedom) for the Taylor-Green vortex with a Mach number of $Ma=0.4$. Left: Plot of total enstrophy. Right: Plot of total kinetic energy. Note that DU crashes at $t\approx 10.8$, indicated by an extreme spike of the quantities.](figures/Machp4_128DOFs/KEN7Machp4.pdf "fig:"){width="45.00000%"}
![\[fig:M04\_2\] Results of simulation with $16^3$ grid cells and $N=7$ ($128^3$ degrees of freedom) for the Taylor-Green vortex with a Mach number of $Ma=0.4$. Left: Plot of dissipation rate of total kinetic energy. Right: Plot of Numerical viscosity estimation. Note that DU crashes at $t\approx 10.8$, indicated by an extreme spike of the quantities.](figures/Machp4_128DOFs/KEDecayN7Machp4.pdf "fig:"){width="45.00000%"}![\[fig:M04\_2\] Results of simulation with $16^3$ grid cells and $N=7$ ($128^3$ degrees of freedom) for the Taylor-Green vortex with a Mach number of $Ma=0.4$. Left: Plot of dissipation rate of total kinetic energy. Right: Plot of Numerical viscosity estimation. Note that DU crashes at $t\approx 10.8$, indicated by an extreme spike of the quantities.](figures/Machp4_128DOFs/NumViscN7Machp4.pdf "fig:"){width="45.00000%"}
Summarising the investigations of this section, the most important result is that all split form schemes, except the variant MO, significantly improve the robustness of high-order DG discretisations compared to polynomial de-aliasing with over-integration. We could demonstrate that the cubic split forms are more robust in comparison to the quadratic splitting when the flow is compressible. Furthermore, the influence of the numerical viscosity can be directly observed and tracked by comparison to nearly dissipation free variants of the scheme. This seems to be a beneficial and interesting feature for future investigations on turbulence modelling.
{#sec:conclusion}
{#section}
As a final discussion, the identities in Lem. \[Lem\] form the basis for a dictionary of a translation from split forms into flux forms and vice versa. We used these to translate known alternative formulations of the non-linear compressible Euler terms. However, this dictionary can be used to generate a multitude of new split forms by choosing symmetric and consistent flux approximations. To demonstrate this, we consider an example based on the Roe variables $$\begin{split}
q_1 &= \sqrt{\rho},\\
q_2 &= \sqrt{\rho}\,u,\\
q_3 &= \sqrt{\rho}\,v,\\
q_4 &= \sqrt{\rho}\,w,\\
q_5 &= \sqrt{\rho}\,h.
\end{split}$$ This specific set of variables is interesting, as it reformulates the flux into only quadratic products, e.g. the flux in the $x-$direction is $$F(U) =
\begin{pmatrix}
q_1\,q_2\\
q_2^2 + \frac{\gamma -1}{\gamma}\left(q_1\,q_5 - \frac{1}{2}\left(q_2^2+q_3^2+q_4^2\right) \right)\\
q_2\,q_3\\
q_2\,q_4\\
q_2\,q_5
\end{pmatrix}.$$ Assuming now a quadratic splitting of the products, we can directly translate this into an equivalent numerical volume flux $QU$, based on the quadratic identities of Lem. \[Lem\] $$F^{\#}_{QU}(U_{ijk},U_{mjk}) =
\begin{pmatrix}
{\left\{\!\left\{ q_1\right\}\!\right\}}\,{\left\{\!\left\{ q_2\right\}\!\right\}}\\
{\left\{\!\left\{ q_2\right\}\!\right\}}^2 + \frac{\gamma -1}{\gamma}\left({\left\{\!\left\{ q_1\right\}\!\right\}}\,{\left\{\!\left\{ q_5\right\}\!\right\}} - \frac{1}{2}\left({\left\{\!\left\{ q_2\right\}\!\right\}}^2+{\left\{\!\left\{ q_3\right\}\!\right\}}^2+{\left\{\!\left\{ q_4\right\}\!\right\}}^2\right) \right)\\
{\left\{\!\left\{ q_2\right\}\!\right\}}\,{\left\{\!\left\{ q_3\right\}\!\right\}}\\
{\left\{\!\left\{ q_2\right\}\!\right\}}\,{\left\{\!\left\{ q_4\right\}\!\right\}}\\
{\left\{\!\left\{ q_2\right\}\!\right\}}\,{\left\{\!\left\{ q_5\right\}\!\right\}}
\end{pmatrix},$$ and thus, generate a new scheme. In Fig. \[fig:finale\] we provide a comparison of the dissipation rates of the new scheme against the previous results.
![\[fig:finale\] Plot of dissipation rates for all variants including the Roe variable based splitting variant $QU$ for $N=3$ and $16^3$ grid cells.](figures/gw2/KEDecayN3withGW2.pdf){width="45.00000%"}
With this simple construction and translation guide, it is straightforward to introduce novel splittings, as demonstrated, and perhaps find forms that offer benefits over the ones currently available in the literature.
{#section-1}
In the first part of this paper, we exploit the volume flux difference form introduced by Carpenter et al. and Fisher et al. [@carpenter_esdg; @fisher2013] to construct a unified formulation for split forms of quadratic and cubic products. From a simple relationship between split forms for quadratic and cubic products, we showed that it is possible to directly translate known alternative compressible Euler formulations into the volume flux difference form by introducing a numerical volume flux $F^\#$, $G^\#$, $H^\#$. It is possible to simply change the volume flux and generate a new DGSEM. We prove that the kinetic energy preserving numerical volume fluxes generate high-order accurate discretisations that also preserve kinetic energy.
Several alternative formulations of the compressible Euler equations from the literature are investigated in the numerical results sections. All schemes show expected convergence behaviour and expected auxiliary conservation properties. An outcome of the numerical assessment is that the new DG schemes are more robust than DG with over-integration. Configurations that crash with over-integration can be successfully finished with the split form DGSEM as well as with the entropy stable DGSEM introduced by Carpenter et al. [@carpenter_esdg]. The numerical volume fluxes of the KG, PI and DU variants offer a computational advantage compared to the more complex entropy stable variants IR and CH. We note that the volume flux differencing form is computationally more intense than the standard DGSEM. On the other hand, we found that for the inviscid Taylor-Green vortex, all configurations for the standard DGSEM (without over-integration) immediately crash.
It is also worth noting that not all alternative formulations automatically improve the robustness, even if the split form is kinetic energy preserving, such as the MO variant. Similar to the standard DGSEM, this variant crashes for all configurations after a short simulation time. Whereas the DU variant is as stable as the KG, PI, IR and CH variants for low Mach numbers, for higher Mach numbers (more compressibility effects), DU is less robust. A conclusion so far is that a good balance between robustness and computational effort is offered by the variants KG and PI. However, it is clear that with the newly established framework for de-aliasing split forms of DG for the compressible Euler equations [@ducros2000; @kennedy2008; @FDaliasing; @larsson2007] more assessment of the many interesting aspects, such as robustness, accuracy and efficiency is necessary.
Additionally, the numerical viscosity of the different discretisations is investigated and it was possible to directly trace the influence of the stabilisation terms. The key for this is that even without interface stabilisation terms, some configurations of the schemes were stable. This offers interesting possibilities in the control and design of dissipation tailored for turbulence modelling.
Proofs
======
Proof of Lem. \[Lem\] \[Discrete split forms\] {#LemProof}
----------------------------------------------
First we consider and make use of the relation ${{\mathbf{D}}}={{\mathbf{{M}}}}^{-1}{{\mathbf{{Q}}}}$ to get $$\label{aAverage}
\begin{aligned}
2\,\sum\limits_{m=0}^N Q_{im}\,{\left\{\!\left\{ a\right\}\!\right\}}_{im} &= \sum_{m=0}^N Q_{im} (a_i + a_m),\\
&= a_i\sum_{m=0}^N Q_{im} + \sum_{m=0}^N Q_{im}a_m,\\
&= \sum_{m=0}^N Q_{im}a_m,\\
\end{aligned}$$ for $ i = 0,\ldots,N$ and we have used the fact that the rows of ${\mathbf{Q}}$ sum to zero [@Strand199447]. We compress the result into the matrix vector product form $$2\,\sum\limits_{m=0}^N D_{im}\,{\left\{\!\left\{ a\right\}\!\right\}}_{im} = ({\mathbf{D}}\,{\underline{a}})_i.$$
Next, we prove the identity for the product of two averages $$\label{substSimp}
\begin{aligned}
2\,\sum\limits_{m=0}^N Q_{im}\,{\left\{\!\left\{ a\right\}\!\right\}}_{im}\,{\left\{\!\left\{ b\right\}\!\right\}}_{im} &= {\frac{1}{2}}\sum_{m=0}^NQ_{im}(a_i+a_m)(b_i+b_m), \\
&=\frac{1}{2}\sum_{m=0}^NQ_{im}a_m\,b_m+\frac{a_i}{2}\sum_{m=0}^NQ_{im}\,b_m + \frac{b_i}{2}\sum_{m=0}^NQ_{im}\,a_m + \frac{a_i\,b_i}{2}\sum_{m=0}^NQ_{im},\\
&=\frac{1}{2}\left\{ \sum_{m=0}^NQ_{im}\,a_m\,b_m\, + a_i\sum_{m=0}^NQ_{im}\,b_m + b_i\sum_{m=0}^NQ_{im}\,a_m\right\},
\end{aligned}$$ for $i = 0,\ldots,N$. Again, we have used the property that the sum of each row of the SBP matrix ${\mathbf{Q}}$ vanishes. We see that after scaling by $i$-th term of ${{\mathbf{{M}}}}^{-1}$ the flux difference result is the $i^{\textrm{th}}$ row of the split form $$2\,\sum\limits_{m=0}^N D_{im}\,{\left\{\!\left\{ a\right\}\!\right\}}_{im}\,{\left\{\!\left\{ b\right\}\!\right\}}_{im} = \frac{1}{2}\left({{\mathbf{D}}}\,\uuline{a}\,{\underline{b}} +\uuline{a}\,{{\mathbf{D}}}\,{\underline{b}} + \uuline{b}\,{{\mathbf{D}}}\,{\underline{a}} \right)_i.$$
We prove the final identity of by considering a triple product of averages $$\label{tripleProdProof}
\resizebox{0.85\textwidth}{!}{$
\begin{aligned}
2\,\sum\limits_{m=0}^N Q_{im}{\left\{\!\left\{ a\right\}\!\right\}}_{im}{\left\{\!\left\{ b\right\}\!\right\}}_{im}{\left\{\!\left\{ c\right\}\!\right\}}_{im} &= \frac{1}{4}\sum_{m=0}^N Q_{im}(a_i+a_m)\,(b_i+b_m)\,(c_i+c_m),\\
&= \frac{1}{4}\sum_{m=0}^N Q_{im}\,\left(a_i b_i c_i + a_i b_i c_m + a_i b_m c_i + a_i b_m c_m \right.\\
&\qquad\qquad\quad\left.+ a_m\, b_i\, c_i+a_m\, b_i\, c_m + a_m\, b_m\, c_i + a_m\, b_m\, c_m\right),\\
&= \frac{a_i\,b_i\,c_i}{4}\sum_{m=0}^N Q_{im} + \frac{a_i\,b_i}{4}\sum_{m=0}^N Q_{im}\,c_m + \frac{a_i\,c_i}{4}\sum_{m=0}^N Q_{im}\,b_m + \frac{a_i}{4}\sum_{m=0}^N Q_{im}\,b_m\,c_m\\
&\qquad+\frac{b_i\,c_i}{4}\sum_{m=0}^N Q_{im}\, a_m+\frac{b_i}{4}\sum_{m=0}^N Q_{im}\,a_m\, c_m + \frac{c_i}{4}\sum_{m=0}^N Q_{im}\,a_m\,b_m + \sum_{m=0}^N Q_{im}\,a_m\,b_m\,c_m,\\
&= \frac{1}{4}\left\{\sum_{m=0}^N Q_{im}\,a_m\,b_m\,c_m+ a_i\sum_{m=0}^N Q_{im}\,b_m\,c_m + {b_i}\sum_{m=0}^N Q_{im}\,a_m\,c_m + {c_i}\sum_{m=0}^N Q_{im}\,a_m\,b_m \right.\\
&\qquad\qquad\left.+{b_i\,c_i}\sum_{m=0}^N Q_{im}\, a_m+ {a_i\,c_i}\sum_{m=0}^N Q_{im}\,b_m+{a_i\,b_i}\sum_{m=0}^N Q_{im}\,c_m\right\},\\
\end{aligned}$}$$ for $i = 0,\ldots,N$. The vanishing sum property of the rows of ${\mathbf{Q}}$ is, again, applied. We condense the final result of into matrix-vector form to obtain the final split form identity $$2\,\sum\limits_{m=0}^N D_{im}\,{\left\{\!\left\{ a\right\}\!\right\}}_{im}\,{\left\{\!\left\{ b\right\}\!\right\}}_{im}\,{\left\{\!\left\{ c\right\}\!\right\}}_{im} = \frac{1}{4}\left({{\mathbf{D}}}\,\uuline{a}\,\uuline{b}\,{\underline{c}} + \uuline{a}\,{{\mathbf{D}}}\,\uuline{b}\,{\underline{c}} + \uuline{b}\,{{\mathbf{D}}}\,\uuline{a}\,{\underline{c}} + \uuline{c}\,{{\mathbf{D}}}\,\uuline{a}\,{\underline{b}} + \uuline{b}\,\uuline{c}\,{{\mathbf{D}}}\,{\underline{a}} + \uuline{a}\,\uuline{c}\,{{\mathbf{D}}}\,{\underline{b}} + \uuline{a}\,\uuline{b}\,{{\mathbf{D}}}\,{\underline{c}}\right)_i.$$
Proof of Thm. \[Thm3\] \[Kinetic energy preservation\] {#Thm3Proof}
------------------------------------------------------
The kinetic energy is calculated as $$\kappa:=\frac{1}{2}\rho\,(u^2+v^2+w^2) = \frac{(\rho\,u)^2+(\rho\,v)^2+(\rho\,w)^2}{2\,\rho}.$$ The kinetic energy balance for the compressible Euler equations is $$\label{eq:kep_balance}
\resizebox{0.9\textwidth}{!}{$
\frac{\partial \kappa}{\partial t} + \left(\frac{1}{2}\rho\,u\,(u^2+v^2+w^2)\right)_x +u\,p_x + \left(\frac{1}{2}\rho\,v\,(u^2+v^2+w^2)\right)_y +v\,p_y+ \left(\frac{1}{2}\rho\,w\,(u^2+v^2+w^2)\right)_z +w\,p_z=0.
$}$$ We can also express the time derivative of the kinetic energy in terms of the time derivatives of the mass and momentum $$\label{eq:dkdt}
\begin{split}
\frac{\partial \kappa}{\partial t}:&= \frac{\partial k}{\partial\rho}\,(\rho)_t +\frac{\partial k}{\partial(\rho\,u)}\,(\rho\,u)_t +\frac{\partial k}{\partial(\rho\,v)}\,(\rho\,v)_t +\frac{\partial k}{\partial(\rho\,w)}\,(\rho\,w)_t,\\
&=-\left(\frac{u^2+v^2+w^2}{2}\right)(\rho)_t +(u)\,(\rho\,u)_t+(v)\,(\rho\,v)_t+(w)\,(\rho\,w)_t.
\end{split}$$
We focus on the volume parts of the continuity equation and consider a single node $ijk$ to get $$\label{eq:drdt}
\resizebox{0.9\textwidth}{!}{$
\begin{split}
-\left(\frac{u^2+v^2+w^2}{2}\right)(\rho)_t &\approx -\left(\frac{(u_{ijk})^2+(v_{ijk})^2+(w_{ijk})^2}{2}\right) 2\,\sum\limits_{m=0}^N D_{im}\,F^{\#,1}(U_{ijk},U_{mjk}) + D_{jm}\,G^{\#,1}(U_{ijk},U_{imk}) + D_{km}\,H^{\#,1}(U_{ijk},U_{ijm}) ,
\end{split}$}$$ and $$\resizebox{0.9\textwidth}{!}{$
\begin{split}
(u)\,(\rho\,u)_t &\approx u_{ijk}\,2\,\sum\limits_{m=0}^N D_{im}\, F^{\#,1}(U_{ijk},U_{mjk})\,{\left\{\!\left\{ u\right\}\!\right\}}_{(i,m)jk} + D_{jm}\, G^{\#,1}(U_{ijk},U_{imk})\,{\left\{\!\left\{ u\right\}\!\right\}}_{i(j,m)k}+D_{km}\, H^{\#,1}(U_{ijk},U_{ijm})\,{\left\{\!\left\{ u\right\}\!\right\}}_{ij(k,m)} \\
&+ u_{ijk}\, 2\,\sum\limits_{m=0}^N D_{im}\widetilde{p}(U_{ijk},U_{mjk}),\\
(v)\,(\rho\,v)_t &\approx v_{ijk}\,2\,\sum\limits_{m=0}^N D_{im}\, F^{\#,1}(U_{ijk},U_{mjk})\,{\left\{\!\left\{ v\right\}\!\right\}}_{(i,m)jk} + D_{jm}\, G^{\#,1}(U_{ijk},U_{imk})\,{\left\{\!\left\{ v\right\}\!\right\}}_{i(j,m)k}+D_{km}\, H^{\#,1}(U_{ijk},U_{ijm})\,{\left\{\!\left\{ v\right\}\!\right\}}_{ij(k,m)}\\
&+ v_{ijk}\, 2\,\sum\limits_{m=0}^N D_{jm}\widetilde{p}(U_{ijk},U_{imk}),\\
(w)\,(\rho\,w)_t &\approx w_{ijk}\,2\,\sum\limits_{m=0}^N D_{im}\, F^{\#,1}(U_{ijk},U_{mjk})\,{\left\{\!\left\{ w\right\}\!\right\}}_{(i,m)jk} + D_{jm}\, G^{\#,1}(U_{ijk},U_{imk})\,{\left\{\!\left\{ w\right\}\!\right\}}_{i(j,m)k}+D_{km}\, H^{\#,1}(U_{ijk},U_{ijm})\,{\left\{\!\left\{ w\right\}\!\right\}}_{ij(k,m)} \\
&+ w_{ijk}\, 2\,\sum\limits_{m=0}^N D_{km}\widetilde{p}(U_{ijk},U_{ijm}),
\end{split}$}$$ where we used a notation of the average of two states, e.g. ${\left\{\!\left\{ u\right\}\!\right\}}_{(i,m)jk} :=\frac{1}{2}(u_{ijk}+u_{mjk})$ and inserted the kinetic energy preserving structure of the numerical flux functions, see .
The next step is to sum all four expressions according to . We first note that as long as the numerical trace approximation of the pressure $\widetilde{p}$ is consistent and symmetric, it follows from the work of Fisher et al. and Carpenter et al. [@fisher2013; @carpenter_esdg] that the resulting discretisation of $u\,p_x + v\,p_y+w\,p_z$ of the pressure work is consistent and high-order accurate.
If we look at the advective terms, we note that the values $u_{ijk}$, $v_{ijk}$, and $w_{ijk}$ do not depend on the summation index $m$ and thus can be pulled into the sum. By doing this, we inspect the product of the velocities and the averages. Looking at the terms involving $F^{\#,1}(U_{ijk},U_{mjk})$, $$\resizebox{0.9\textwidth}{!}{$
\begin{split}
u_{ijk}{\left\{\!\left\{ u\right\}\!\right\}}_{(i,m)jk}\,F^{\#,1}(U_{ijk},U_{mjk}) &= u_{ijk}\,\frac{1}{2}(u_{ijk}+u_{mjk})\,F^{\#,1}(U_{ijk},U_{mjk}) = \left[\frac{(u_{ijk})^2}{2}+ \frac{u_{ijk}\,u_{mjk}}{2}\right]\,F^{\#,1}(U_{ijk},U_{mjk}),\\
v_{ijk}{\left\{\!\left\{ v\right\}\!\right\}}_{(i,m)jk}\,F^{\#,1}(U_{ijk},U_{mjk}) &=v_{ijk}\,\frac{1}{2}(v_{ijk}+v_{mjk})\,F^{\#,1}(U_{ijk},U_{mjk}) = \left[\frac{(v_{ijk})^2}{2}+ \frac{v_{ijk}\,v_{mjk}}{2}\right]\,F^{\#,1}(U_{ijk},U_{mjk}),\\
w_{ijk}{\left\{\!\left\{ w\right\}\!\right\}}_{(i,m)jk}\,F^{\#,1}(U_{ijk},U_{mjk}) &=w_{ijk}\,\frac{1}{2}(w_{ijk}+w_{mjk})\,F^{\#,1}(U_{ijk},U_{mjk}) = \left[\frac{(w_{ijk})^2}{2}+ \frac{w_{ijk}\,w_{mjk}}{2}\right]\,F^{\#,1}(U_{ijk},U_{mjk}).
\end{split}$}$$ Summing these three terms gives $$\resizebox{0.9\textwidth}{!}{$
\begin{split}
\left(u_{ijk}{\left\{\!\left\{ u\right\}\!\right\}}_{(i,m)jk} + v_{ijk}{\left\{\!\left\{ v\right\}\!\right\}}_{(i,m)jk} + w_{ijk}{\left\{\!\left\{ w\right\}\!\right\}}_{(i,m)jk}\right)\,F^{\#,1}(U_{ijk},U_{mjk})&=\frac{(u_{ijk})^2+(v_{ijk})^2+(w_{ijk})^2}{2}\,F^{\#,1}(U_{ijk},U_{mjk})\\
&+\frac{u_{ijk}\,u_{mjk}+v_{ijk}\,v_{mjk}+w_{ijk}\,w_{mjk}}{2}\,F^{\#,1}(U_{ijk},U_{mjk}).
\end{split}$}$$ Note, that the first term is identical to the first term on the right hand side of , except for the sign. Thus, when adding, this term cancels and we are left with the remainder term $$f^{\#,\kappa}(U_{ijk},U_{mjk}) :=\frac{1}{2}\,F^{\#,1}(U_{ijk},U_{mjk})\left(u_{ijk}\,u_{mjk}+v_{ijk}\,v_{mjk}+w_{ijk}\,w_{mjk}\right),$$ which is symmetric in the arguments and is consistent with the advective flux of the kinetic energy balance $f=\frac{1}{2}\rho\,u\,(u^2+v^2+w^2)$.\
\
Analogously, if we focus on the terms involving $G^{\#,1}(U_{ijk},U_{imk})$, we get $$g^{\#,\kappa}(U_{ijk},U_{imk}) :=\frac{1}{2}\,G^{\#,1}(U_{ijk},U_{imk})\left(u_{ijk}\,u_{imk}+v_{ijk}\,v_{imk}+w_{ijk}\,w_{imk}\right),$$ which is again symmetric in the arguments and consistent to $g=\frac{1}{2}\rho\,v\,(u^2+v^2+w^2)$. For the terms involving $H^{\#,1}(U_{ijk},U_{ijm})$ we get $$h^{\#,\kappa}(U_{ijk},U_{ijm}) :=\frac{1}{2}\,H^{\#,1}(U_{ijk},U_{ijm})\left(u_{ijk}\,u_{ijm}+v_{ijk}\,v_{ijm}+w_{ijk}\,w_{ijm}\right),$$ which is also symmetric and consistent to $h=\frac{1}{2}\rho\,w\,(u^2+v^2+w^2)$.
Thus, summarising, we get the following volume term in our discrete kinetic energy balance $$\begin{split}
\left(\frac{\partial \kappa}{\partial t}\right)_{ijk}\approx &-2\,\sum\limits_{m=0}^N D_{im}\,f^{\#,\kappa}(U_{ijk},U_{mjk})+D_{jm}\,g^{\#,\kappa}(U_{ijk},U_{imk})
+D_{km}\,h^{\#,\kappa}(U_{ijk},U_{ijm})\\
&-2\,\sum\limits_{m=0}^N u_{ijk}\,D_{im}\,\widetilde{p}(U_{ijk},U_{mjk})+v_{ijk}\,D_{jm}\,\widetilde{p}(U_{ijk},U_{imk})+w_{ijk}\,D_{km}\,\widetilde{p}(U_{ijk},U_{ijm}),
\end{split}$$ which is a consistent and high-order accurate approximation of the continuous kinetic energy balance , with the advective terms in conservative form.
Curvilinear flux differencing form {#sec:curvilinear}
==================================
Now the computational domain is divided into non-overlapping *curved* hexahedral elements $C$. We create a polynomial transformation ${\underline{X}}({\underline{\xi}})$ to map computational coordinates in the reference cube ${\underline{\xi}} = (\xi^1,\xi^2,\xi^3)= (\xi,\eta,\zeta)$ to physical coordinates ${\underline{x}} = (x,y,z)\in C$, for details see [@koprivabook].
The curvilinear coordinate system for ${\underline{\xi}}$ on the reference cube has three covariant basis vectors, ${\underline{a}}_i$, computed directly from the transformation $$\label{covariantVecs}
{\underline{a}}_i = {\frac{\partial {\underline{X}}}{\partial \xi^i}},\quad i=1,2,3.$$ From the covariant basis vectors we derive the contravariant basis vectors ${\underline{a}}^i$, scaled by the Jacobian of the transformation $J $ $$\label{contraVecs}
J {\underline{a}}^i = J \nabla{\xi^i} = {\underline{a}}_j\times{\underline{a}}_k,\quad (i,j,k)\textrm{ cyclic}.$$ Alternatively, there is an explicitly divergence-free form of the contravariant basis vectors derived in [@Kopriva:2006er] $$\label{divFreeContra}
J a^i_n = -\hat{x}_i\cdot\nabla_\xi\times(x_l\nabla_\xi x_m),\quad i = 1,2,3;\; n = 1,2,3;\; (n,m,l)\;\textrm{cyclic}.$$ The divergence-free form of the contravariant basis vectors is particularly important to prevent spurious oscillations in the solution on curved sided hexahedral elements [@Kopriva:2006er]. However, for two dimensional problems and straight-sided hexahedral meshes (e.g. Cartesian meshes) the cross product formulation is sufficient to prevent the generation of spurious waves by a mesh [@Kopriva:2006er].
The conservation law in the physical domain transforms to a conservation law equation in the reference domain of the form $$\label{newConsLaw}
J{U}_t + \widetilde{\mathcal{L}}^{div}_{\xi}(U) + \widetilde{\mathcal{L}}^{div}_{\eta}(U) + \widetilde{\mathcal{L}}^{div}_{\zeta}(U) = 0,$$ where the Jacobian values of the transformation are computed from the covariant basis vectors and the contravariant operators incorporate the metric terms $$\label{newVars}
\begin{aligned}
J &= {\underline{a}}_1\cdot({\underline{a}}_2\times {\underline{a}}_3),\\
\widetilde{\mathcal{L}}^{div}_{\xi}(U) &= \left(J {a}^1_1\,F(U)+J {a}^1_2\,G(U)+J {a}^1_3\,H(U)\right)_\xi,\\[0.1cm]
\widetilde{\mathcal{L}}^{div}_{\eta}(U) &= \left(J {a}^2_1\,F(U)+J {a}^2_2\,G(U)+J {a}^2_3\,H(U)\right)_\eta,\\[0.1cm]
\widetilde{\mathcal{L}}^{div}_{\zeta}(U) &= \left(J {a}^3_1\,F(U)+J {a}^3_2\,G(U)+J {a}^3_3\,H(U)\right)_\zeta.\\[0.1cm]
\end{aligned}$$ Next, consider the component $l$ of the mapped system and an GL node $(i,j,k)$, the DGSEM approximation in strong form is $$\label{eq:curve_operator}
\begin{aligned}
\left(\widetilde{\mathcal{L}}^{div}_{\xi}(U)\right)_{ijk}^l &\approx \left[\widetilde{F}^{*,l}(1,\eta_j,\zeta_k;{\underline{n}}) - \tilde{F}^l_{Njk}\right] - \left[\widetilde{F}^{*,l}(-1,\eta_j,\zeta_k;{\underline{n}}) - \widetilde{F}^l_{0jk}\right] + \sum_{m=0}^N D_{im}\widetilde{F}^l_{mjk},\\[0.1cm]
\widetilde{F}^l_{ijk} &\approx Ja^1_1\,F^l_{ijk} + Ja^1_2\,G^l_{ijk} + Ja_3^1\,H^l_{ijk},
\end{aligned}$$ where we use collocation for the non-linear flux functions, e.g. $F^l_{ijk} := F^l(U_{ijk})$ and denote the outward pointing normal vector by ${\underline{n}}$. Note that the terms $\left(\widetilde{\mathcal{L}}^{div}_{\eta}(U)\right)^l_{ijk}$ and $\left(\widetilde{\mathcal{L}}^{div}_{\zeta}(U)\right)^l_{ijk}$ have an analogous structure to .
Split form stabilisation for curvilinear DGSEM
----------------------------------------------
Just as in Sec. \[sec:DGSEM\_fluxform\], we consider alternative formulations for the volume part of the discretisation . Again, this means starting from a standard strong form DGSEM implementation and modifying the volume integrals. We note that for curvilinear meshes the metric terms are now polynomials. Thus, the non-linearity in the contravariant terms increases. The increased non-linearity due to the curvilinear grid is an additional cause for aliasing driven non-linear instabilities. For a detailed algorithmic description of the curvilinear flux difference formulation see [@wintermeyer2015].
In his PhD thesis, Fisher [@fisher2012] extended the de-aliasing flux differencing technique to curvilinear grids with corresponding element-wise mappings. For this extension, the contravariant operators in each direction are needed. We collect the main result below and proofs can be found in [@fisher2012]. In general curvilinear coordinates the high-order flux difference form has the structure $$\label{eq:highOrderFluxCurvilinear}
\begin{aligned}
\frac{\overline{\widetilde{F}^l}_{(i+1)jk}-\overline{\widetilde{F}^l}_{(i)jk}}{\omega_i} \approx 2\sum_{m=0}^N D_{im}&\left[F^{\#,l}(U_{ijk},U_{mjk}){\left\{\!\left\{ Ja^1_1\right\}\!\right\}}_{(i,m)jk}\right.\\[-0.2cm]
&\;+G^{\#,l}(U_{ijk},U_{mjk}){\left\{\!\left\{ Ja^1_2\right\}\!\right\}}_{(i,m)jk}\\
&\left.\;+H^{\#,l}(U_{ijk},U_{mjk}){\left\{\!\left\{ Ja^1_3\right\}\!\right\}}_{(i,m)jk}\right],
\end{aligned}$$ where $$\label{eq:metricTermAvg}
{\left\{\!\left\{ Ja^1_r\right\}\!\right\}}_{(i,m)jk} = {\frac{1}{2}}\left[\left(Ja_r^1\right)_{ijk} + \left(Ja_r^1\right)_{mjk}\right],\quad r = 1,2,3$$ are the average of the metric term components. For the approximation of the $\overline{\widetilde{G}^l}$ and $\overline{\widetilde{H}^l}$ contravariant flux differences the superscript on the metric term averages in change to $2$ and $3$ respectively. Each of the volume flux functions in are consistent and symmetric. Also, the resulting curvilinear approximation is high-order accurate and conservative in the Lax-Wendroff sense. We use the form of the volume discretisation rather than the standard discretisation $$\label{eq:curve_operatorFinished}
\resizebox{0.95\hsize}{!}{$
\begin{aligned}
\left(\widetilde{\mathcal{L}}^{div}_{\xi}(U)\right)_{ijk}^l \approx \left[\widetilde{F}^{*,l}(1,\eta_j,\zeta_k;{\underline{n}}) - \tilde{F}^l_{Njk}\right] - \left[\widetilde{F}^{*,l}(-1,\eta_j,\zeta_k;{\underline{n}}) - \widetilde{F}^l_{0jk}\right] +2\sum_{m=0}^N D_{im}&\left[F^{\#,l}(U_{ijk},U_{mjk}){\left\{\!\left\{ Ja^1_1\right\}\!\right\}}_{(i,m)jk}\right.\\[-0.2cm]
&\;+G^{\#,l}(U_{ijk},U_{mjk}){\left\{\!\left\{ Ja^1_2\right\}\!\right\}}_{(i,m)jk}\\
&\left.\;+H^{\#,l}(U_{ijk},U_{mjk}){\left\{\!\left\{ Ja^1_3\right\}\!\right\}}_{(i,m)jk}\right],
\\[0.1cm]
\left(\widetilde{\mathcal{L}}^{div}_{\eta}(U)\right)_{ijk}^l \approx \left[\widetilde{G}^{*,l}(\xi_i,1,\zeta_k;{\underline{n}}) - \tilde{G}^l_{iNk}\right] - \left[\widetilde{G}^{*,l}(\xi_i,-1,\zeta_k;{\underline{n}}) - \widetilde{G}^l_{i0k}\right] +2\sum_{m=0}^N D_{jm}&\left[F^{\#,l}(U_{ijk},U_{imk}){\left\{\!\left\{ Ja^2_1\right\}\!\right\}}_{i(j,m)k}\right.\\[-0.2cm]
&\;+G^{\#,l}(U_{ijk},U_{imk}){\left\{\!\left\{ Ja^2_2\right\}\!\right\}}_{i(j,m)k}\\
&\left.\;+H^{\#,l}(U_{ijk},U_{imk}){\left\{\!\left\{ Ja^2_3\right\}\!\right\}}_{i(j,m)k}\right],\\[0.1cm]
\left(\widetilde{\mathcal{L}}^{div}_{\zeta}(U)\right)_{ijk}^l \approx \left[\widetilde{H}^{*,l}(\xi_i,\eta_j,1;{\underline{n}}) - \tilde{H}^l_{ijN}\right] - \left[\widetilde{H}^{*,l}(\xi_i,\eta_j,-1;{\underline{n}}) - \widetilde{H}^l_{ij0}\right] +2\sum_{m=0}^N D_{km}&\left[F^{\#,l}(U_{ijk},U_{ijm}){\left\{\!\left\{ Ja^3_1\right\}\!\right\}}_{ij(k,m)}\right.\\[-0.2cm]
&\;+G^{\#,l}(U_{ijk},U_{ijm}){\left\{\!\left\{ Ja^3_2\right\}\!\right\}}_{ij(k,m)}\\
&\left.\;+H^{\#,l}(U_{ijk},U_{ijm}){\left\{\!\left\{ Ja^3_3\right\}\!\right\}}_{ij(k,m)}\right],
\end{aligned}$}$$ where the choice of the numerical surface fluxes in the normal direction, ${\underline{n}}$, for each of the $\#$ variants of DGSEM are identical to those discussed in Sec. \[sec:numflux\].
References {#references .unnumbered}
==========
|
---
abstract: 'In this paper we provide visual characterization of associative quasitrivial nondecreasing operations on finite chains. We also provide a characterization of bisymmetric quasitrivial nondecreasing binary operations on finite chains. Finally, we estimate the number of operations belonging to the previous classes.'
address: 'Mathematics Research Unit, University of Luxembourg, Maison du Nombre, 6, avenue de la Fonte, L-4364 Esch-sur-Alzette, Luxembourg'
author:
- Gergely Kiss
title: Visual characterization of associative quasitrivial nondecreasing operations on finite chains
---
Introduction {#s1}
============
The study of aggregation operations defined on finite ordinal scales (i.e, finite chains) have been in the center of interest in the last decades, e.g., [@C; @B; @Fo; @Ma1; @Ma2; @Ma3; @Ma4; @May2; @May; @Li; @RA; @Su1; @Su2]. Among these operations, discrete uninorms has an important role in fuzzy logic and decision making [@Be; @B1; @B2; @F].
In this paper we investigate associative quasitrivial nondecreasing operations on finite chains. In [@Beats2009; @Jimmy2017; @DKM] idempotent discrete uninorms (i.e. idempotent symmetric nondecreasing associative operations with neutral elements defined on finite chains) have been characterized. Since every idempotent uninorm is quasitrivial (see e.g. [@Czogala1984]), in some sense this paper is a continuation of these works where we eliminate the assumption of symmetry of the operations.
Now we recall the analogue results for the unit interval $[0,1]$ as follows. Czogała-Drewniak proved in [@Czogala1984] that an associative monotonic idempotent operation with neutral element is a combination of minimum and maximum, and thus these are quasitrivial. Martin, Mayor and Torrens in [@Martin2003] gave a complete characterization of associative quasitrivial nondecreasing operations on $[0,1]$. A refinement of their argument can be found in [@Beats2010]. (For the multivariable generalization of these results see [@GG2016].) We note that in [@Beats2009] the analogue of the result of Czogała-Drewniak for finite chains has been provided assuming of symmetry of such operations.
The study of $n$-ary operations $F\colon X^n\to X$ satisfying the associativity property (see Definition \[dbase\]) stemmed from the work of Dörnte [@Dor28] and Post [@Pos40]. In [@DudMuk96; @DM] the reducibility (see Definition \[defred\]) of associative $n$-ary operations have been studied by adjoining neutral elements. In [@A] a complete characterization of quasitrivial associative $n$-ary operations have been presented. In [@DKM] the quasitrivial symmetric nondecreasing associative $n$-ary operations defined on chains have been characterized. Recently, in [@KS] it was proved that associative idempotent nondecreasing $n$-ary operations defined on any chain are reducible. Using reducibility (see Theorem \[t2\]) a characterization of associative quasitrivial nondecreasing $n$-ary operations for any $2\le n\in \mathbb{N}$ can be obtained automatically by a characterization of associative quasitrivial nondecreasing binary operations.
The paper is organized as follows. In Section \[s2\] we present the most important definitions. In Section \[s3\], we recall ([@KS Theorem 4.8]) the reducibility of associative idempotent nondecreasing $n$-ary operations and, hence, in the sequel we mainly focus on the binary case. We introduce the basic concept of visualization for quasitrivial monotone binary operations and present some preliminary results due to this concept. Here we discuss an important visual test of non-associativity (Lemma \[lpic\]). Section \[s4\] is devoted to the visual characterization of associative quasitrivial nondecreasing operations with so-called ’downward-right paths’ (Theorems \[tfo1\] and \[thmchar\]). We also present an Algorithm which provides the contour plot of any associative quasitrivial nondecreasing operation. In Section \[s5\] we characterize the bisymmetric quasitrivial nondecreasing binary operations (Theorem \[tbfo1\]). In Section \[s6\] we calculate the number of associative quasitrivial nondecreasing operations defined on a finite chain of given size with and also without the assumption of the existence of neutral elements (Theorem \[tnumb\]). We get similar estimations for the number of bisymmetric quasitrivial nondecreasing binary operations defined on a finite chain of given size. (Proposition \[binumb\]). In Section \[s7\] we present some problems for further investigation. Finally, using a slight modification of the proof of [@KS Theorem 3.2], in the Appendix we show that every associative quasitrivial monotonic $n$-ary operations are nondecreasing.
Definition {#s2}
==========
Here we present the basic definitions and some preliminary results. First we introduce the following simplification. For any integer $l\geq 0$ and any $x\in X$, we set $l\cdot x= x,\ldots,x$ ($l$ times). For instance, we have $F(3\cdot x_1,2\cdot
x_2)=F(x_1,x_1,x_1,x_2,x_2)$.
\[dbase\] Let $X$ be an arbitrary nonempty set. A operation $F:X^n\to X$ is called
- *idempotent* if $F(n\cdot x)=x$ for all $x\in X$;
- *quasitrivial* (or *conservative*) if $$F(x_1,\ldots,x_n)\in\{x_1,\ldots,x_n\}$$ for all $x_1,\ldots,x_n\in X$;
- *($n$-ary) associative* if $$\begin{aligned}
\label{assoc}
\lefteqn{F(x_1,\ldots,x_{i-1},F(x_i,\ldots,x_{i+n-1}),x_{i+n},\ldots,x_{2n-1})}\\
&=&
F(x_1,\ldots,x_i,F(x_{i+1},\ldots,x_{i+n}),x_{i+n+1},\ldots,x_{2n-1})\end{aligned}$$ for all $x_1,\ldots,x_{2n-1}\in X$ and all $i\in \{1, \dots, n-1\}$;
- *($n$-ary) bisymmetric* if $$F(F(\mathbf{r}_1),\ldots,F(\mathbf{r}_n)) ~=~
F(F(\mathbf{c}_1),\ldots,F(\mathbf{c}_n))$$ for all $n\times n$ matrices $[\mathbf{r}_1 ~\cdots
~\mathbf{r}_n]=[\mathbf{c}_1 ~\cdots ~\mathbf{c}_n]^T\in X^{n\times
n}$.
We say that $F:X^n\to X$ has a [*neutral element*]{} $e\in X$ if for all $x\in X$ and all $i\in \{1,\dots, n\}$ $$F((i-1)\cdot e,x,(n-i)\cdot e) ~=~ x.$$
Hereinafter we simply write that an $n$-ary operation is associative or bisymmetric if the context clarifies the number of its variables. We also note that if $n=2$ we get the binary definition of associativity, quasitriviality, idempotency, and neutral element property.
Let $(X,\leq)$ be a nonempty chain (i.e, a totally ordered set). An operation $F\colon X^n\to X$ is said to be
- [*nondecreasing*]{} (resp. [*nonincreasing*]{}) if $$F(x_1,\ldots,x_n)\leq F(x'_1,\ldots,x'_n) \ \ \ (\textrm{resp.
} F(x_1,\ldots,x_n)\geq F(x'_1,\ldots,x'_n))$$ whenever $x_i\leq
x'_i$ for all $i\in \{1,\dots, n\}$,
- [*monotone in the $i$-th variable*]{} if for all fixed elements $a_1,\dots a_{i-1}, a_{i+1}, \dots, a_n$ of $X$, the $1$-ary function defined as $$f_i(x):=F(a_1,\dots,
a_{i-1},x,a_{i+1},\dots, a_n)$$ is nondecreasing or nonincreasing.
- [*monotone*]{} if it is monotone in each of its variables.
\[defred\]
We say that $F:X^n\to X$ [*is derived from*]{} a binary operation $G:X^2\to X$ if $F$ can be written of the form $$\label{eqjdef}
F(x_1, \dots, x_n)=x_1\circ \dots
\circ x_n,$$ where $x\circ y=G(x,y)$. It is easy to see that $G$ is associative (and $F$ is $n$-ary associative) if and only if is well-defined. If such a $G$ exists, then we say that $F$ is [*reducible*]{}.
We denote the [*diagonal*]{} of $X^2$ by $\Delta_X=\{(x,x):x\in
X\}$.
Let $L_k$ denote $\{1,\dots, k\}$ endowed with the natural ordering $(\le)$.
Then $L_k$ is a finite chain. Moreover, every finite chain with $k$ element can be identified with $L_k$ and the domain of an $n$-variable operation defined on a finite chain can be identified with $\underbrace{L_k\times\cdots\times L_k}_n=(L_k)^n$ for some $k\in \mathbb{N}$.
For an arbitrary poset $(X, \le)$ and $a\le b\in X$ we denote the elements between $a$ and $b$ by $[a,b]\subseteq X$. In particular, for $L_k$ $$[a,b]=\{m\in L_k: a \le m \le b\}.$$ We also introduce the lattice notion of the minimum ($\wedge$) and the maximum ($\vee$) as follows $$x_1\wedge\dots\wedge x_n=\wedge_{i=1}^n x_i=\min\{x_1, \dots, x_n\},$$ $$x_1\vee\dots\vee x_n=\vee_{i=1}^n x_i=\max\{x_1, \dots, x_n\}.$$
The binary operations Proj$_x$ and Proj$_y$ denote the projection to first and the second coordinate, respectively. Namely, Proj$_x(x, y)
= x$ and Proj$_y(x, y) = y$ for all $x, y \in X.$
Basic concept and preliminary results {#s3}
=====================================
The following general result was published as [@KS Theorem 4.8] recently.
\[t2\] Let $X$ be a nonempty chain and $F:X^n\to X$ $(n\ge 2)$ be an associative idempotent nondecreasing operation. Then there exists uniquely an associative idempotent nondecreasing binary operation $G:X^2\to X$ such that $F$ is derived from $G$. Moreover, $G$ can be defined by $$\label{eqgf}
G(a,b)=F(a, (n-1)\cdot b)=F((n-1)\cdot a, b)~~~~(a,b\in
X).$$
\[rem2\] By the definition of $G$, it is clear that if $F$ is quasitrivial, then $G$ is also.
According to Theorem \[t2\] and Remark \[rem2\], a characterization of associative quasitrivial nondecreasing binary operations automatically implies a characterization for the $n$-ary case. Therefore, from now on we deal with the binary case ($n=2$).
Visualization of binary operations {#s3.1}
----------------------------------
In this section we prove and reprove basic properties of quasitrivial associative nondecreasing binary operations in the spirit of visualization.
\[l1\] Let $X$ be a nonempty chain and let $F:X^2\to X$ be a quasitrivial monotone operation. If $F(x,t)=x$, then $F(x,s)=x$ for every $s\in
[x\wedge t, x\vee t]$ . Similarly, if $F(x,t)=t$, then $F(s,t)=t$ for every $s\in [x\wedge t, x\vee t]$.
A [*level-set*]{} of $F$ is a set of vertices of $L_k^2$ where $F$ has the same value. The [*contour plot*]{} of $F$ can be visualized by connecting the closest elements of the level-sets of $F$ by line segments. According to Observation \[l1\], this contour plot can be drawn using only horizontal and vertical line segments starting from the diagonal (as in Figure 1.). It is clear that these lines do not cross each other by the monotonicity of $F$.
(0,0)– (0,4)–(4,4)–(4,0) – cycle; (0,0)–(4,4); (1,1)–(3,1); (3,3)–(3,2); at (-0.3,1) [y]{}; at (-0.3,2) [z]{}; at (3,-0.3) [x]{}; (0,0) grid (4,4);
As a consequence we get the following.
\[cnem\] Let $X$ be a nonempty chain and $F:X^2 \to X$ be a quasitrivial operation. $$F \textrm{ is monotone } \Longleftrightarrow F \textrm{
is nondecreasing}.$$
We only need to prove that every monotone quasitrivial operation is nondecreasing.
As an easy consequence of Observation \[l1\] and the quasitriviality of $F$, we have $F(s,x)\le F(t,x)$ and $F(x,t)\le
F(s,t)$ for any $x,s,t\in X$ that satisfies $s\in[x,t]$. This implies that $F$ is nondecreasing in the first variable. Similar argument shows the statement for the second variable.
\[rnem\] The analogue of Corollary \[cnem\] holds whenever $n> 2$. The proof is essentially the same as the proof of [@KS Theorem 3.10]. Thus we present it in Appendix A.
In the sequel we are dealing with associative, quasitrivial and nondecreasing operations. There are several know forms of the following proposition. This type of results was first proved in [@Martin2003]. The form as stated here is [@Jimmy2017 Proposition 18].
\[prop:eqv\] Let $X$ be an arbitrary nonempty set and let $F:X^2 \to X$ be a quasitrivial operation. Then the following assertions are equivalent.
1. $F$ is [**not**]{} associative.
2. There exist pairwise distinct $x,y,z\in X$ such that $F(x,y), F(x,z), F(y,z)$ are pairwise distinct.
3. There exists a rectangle in $X^2$ such that one of the vertices is on $\Delta_X$ and the three remaining vertices are in $X^2\setminus \Delta_X$ and pairwise disconnected.
Now we present a visual version of the previous statement if $F$ is nondecreasing.
\[lpic\] Let $X$ be chain and $F:X^2\to X$ a quasitrivial, nondecreasing operation. Then $F$ is [**not**]{} associative if and only if there are pairwise distinct elements $x,y,z\in X$ that give one of the following pictures.
(0,0)– (0,2)–(2,2)–(2,0) – cycle; (0,0)–(2,2); (0.4,0.4)–(1,0.4); (1,1)–(1,1.7); (1.7,1.7)–(1.7,0.4); at (0.4,-0.3) [z]{}; at (-0.3,0.4) [z]{}; at (1,-0.3) [x]{}; at (-0.3,1) [x]{}; at (-0.3,1.7) [y]{}; at (1.7,-0.3) [y]{}; at (1,-1) [(a)]{};
(3,0)– (3,2)–(5,2)–(5,0) – cycle; (3,0)–(5,2); (3.4,0.4)–(3.4,1.7); (4,1)–(4,0.4); (4.7,1.7)–(4,1.7); at (3.4,-0.3) [y]{}; at (2.7,0.4) [y]{}; at (4,-0.3) [x]{}; at (2.7,1) [x]{}; at (2.7,1.7) [z]{}; at (4.7,-0.3) [z]{}; at (4,-1) [(b)]{};
(6,0)– (6,2)–(8,2)–(8,0) – cycle; (6,0)–(8,2); (6.4,0.4)–(6.4,1); (7,1)–(7.7,1); (7.7,1.7)–(6.4,1.7); at (6.4,-0.3) [x]{}; at (5.7,0.4) [x]{}; at (7,-0.3) [z]{}; at (5.7,1) [z]{}; at (5.7,1.7) [y]{}; at (7.7,-0.3) [y]{}; at (7,-1) [(c)]{};
(9,0)– (9,2)–(11,2)–(11,0) – cycle; (9,0)–(11,2); (9.4,0.4)–(10.7,0.4); (10,1)–(9.4,1); (10.7,1.7)–(10.7,1); at (9.4,-0.3) [y]{}; at (8.7,0.4) [y]{}; at (10,-0.3) [z]{}; at (8.7,1) [z]{}; at (8.7,1.7) [x]{}; at (10.7,-0.3) [x]{}; at (10,-1) [(d)]{};
By Proposition \[prop:eqv\], $F$ is not associative if and only if there exists distinct $x,y,z\in X$ satisfying one of the following cases: $$\label{eq1}
F(x,y)=x, F(x,z)=z, F(y,z)=y~~~~ \textrm{(Case 1),}$$ [or]{} $$\label{eq2}
F(x,y)=y, F(y,z)=z, F(x,z)=x~~~~\textrm{(Case 2)} .$$
Since $x,y,z\in X$ pairwise distinct elements, they can be ordered in 6 possible configuration of type $x<y<z$. For each case either or holds. Therefore we have 12 configurations as possible realizations of Case 1 or Case 2. Let us consider Case 1 (when equation holds) and assume $x<y<z$. This implies the situation of Figure \[f1\].
(0,0)– (0,3)–(3,3)–(3,0) – cycle; (0,0)–(3,3); (0.75,0.75)–(0.75,1.5); (1.5,1.5)–(1.5,2.25); (2.25,2.25)–(0.75,2.25); (1.5,2.25) circle (0.1cm); at (0.75,-0.3) [x]{}; at (-0.3,0.75) [x]{}; at (1.5,-0.3) [y]{}; at (-0.3,1.5) [y]{}; at (2.25,-0.3) [z]{}; at (-0.3,2.25) [z]{};
The red point signs the problem of this configuration, since two lines with different values cross each other. There is no such a quasitrivial monotone operation.
Thus this subcase provides ’fake’ example to study associativity. From the total, 8 cases are ’fake’ in this sense.
The remaining 4 cases are presented in the statement. Figure \[fig111\] (a) and (b) represent the cases when equation holds, and Figure \[fig111\] (c) and (d) represent the cases when holds.
Since for a 2-element set none of the cases of Figure \[fig111\] can be realized, as an immediate consequence of Lemma \[lpic\] we get the following.
\[cor2\] Every quasitrivial nondecreasing operation $F:L_2^2\to L_2$ is associative.
As a byproduct of this visualization we obtain a simple alternative proof for the following fact. This was proved first in [@Martin2003 Proposition 2].
Let $X$ be nonempty chain and $F:X^2\to X$ be a quasitrivial symmetric nondecreasing operation then $F$ is associative.
If we add the assumption of symmetry of $F$, each cases presented in Figure \[fig111\] have crossing lines (as in Figure \[figsym\]), which is not possible. Thus $F$ is automatically associative.
(0,0)– (0,4)–(4,4)–(4,0) – cycle; (0,0)–(4,4); (1,1)–(2,1); (2,2)–(2,3); (3,3)–(3,1); at (1,-0.3) [z]{}; at (-0.3,1) [z]{}; at (2,-0.3) [x]{}; at (-0.3,2) [x]{}; at (3,-0.3) [y]{}; at (-0.3,3) [y]{}; at (5.5, 2) [$\Longrightarrow$]{}; (7,0)– (7,4)–(11,4)–(11,0) – cycle; (7,0)–(11,4); (8,1)–(9,1); (9,2)–(9,3); (10,3)–(10,1); (8,1)–(8,2); (9,2)–(10,2); (10,3)–(8,3); (9,3) circle (0.1cm); (10,2) circle (0.1cm); at (8,-0.3) [z]{}; at (6.7,1) [z]{}; at (9,-0.3) [x]{}; at (6.7,2) [x]{}; at (10,-0.3) [y]{}; at (6.7,3) [y]{};
For finite chains more can be stated.
\[prop:ane\] If $F:L_k^2\to L_k$ is quasitrivial symmetric nondecreasing then it is associative and has a neutral element.
The conclusion that $F$ has a neutral element is not necessarily true when $X=[0,1]$ (see [@Martin2003]). This fact is one of the main difference between the cases $X=L_k$ and $X=[0,1]$.
If we assume that $F$ has a neutral element (as it follows by Proposition \[prop:ane\] for finite chains), then as a consequence of Observation \[l1\] we get the following pictures (Figure \[figsymne\]) for quasitrivial monotone operations having neutral elements. In Figure \[figsymne\] the neutral element is denoted by $e$.
(0,0)– (0,4)–(4,4)–(4,0) – cycle; (0,0)–(4,4); (0,0)–(1.5,0); (0,0)–(0,1.5); (1,1)–(1.5,1); (1,1)–(1,1.5); (2,2)–(1.5,2); (2,2)–(2,1.5); (3,3)–(1.5,3); (3,3)–(3,1.5); (4,4)–(1.5,4); (4,4)–(4,1.5); (0.5,0.5)–(0.5,1.5); (0.5,0.5)–(1.5,0.5); (2.5,2.5)–(2.5,1.5); (2.5,2.5)–(1.5,2.5); (3.5,3.5)–(3.5,1.5); (3.5,3.5)–(1.5,3.5); at (1.5,-0.3) [e]{}; at (-0.3,1.5) [e]{}; (1.5,1.5) circle\[radius=1.5pt\];
(5,1.5)– (5,4)–(6.5,4); (9,1.5)– (9,0)–(6.5,0); (6.5,1.5)–(9,1.5)–(9,4)–(6.5,4)–cycle; (6.5,1.5)–(5,1.5)–(5,0)–(6.5,0)–cycle; at (5.75,0.75) [$x\wedge y$]{}; at (7.75,2.75) [$x\vee y$]{}; at (6.5,-0.3) [e]{}; at (4.7,1.5) [e]{};
Visual characterization of associative quasitrivial nondecreasing operations defined on $L_k$ {#s4}
==============================================================================================
From now on we denote the upper and the lower ’triangle’ by $$T_1=\{(x,y): x,y\in L_k, x\le y\},\ \ \ \ T_2=\{(x,y): x,y\in L_k, x\ge y\},$$ respectively, as in Figure \[figtri\]. We note that $T_1\cap T_2$ is the diagonal $\Delta_{L_k}$.
(0,0)– (0,3)–(3,3)– cycle; (3,3)–(3,0)–(0,0); (7,3)–(7,0)–(4,0)– cycle; (4,0)– (4,3)–(7,3) ; at (1,2) [$\huge{T_1}$]{}; at (6,1) [$\huge{T_2}$]{};
For a operation $F:L_k^2\to L_k$ there can be defined the [*upper symmetrization $F_1$ and lower symmetrization $F_2$*]{} of $F$ as $$F_1(x,y)=\begin{cases}F(x,y) ~~~~&\textrm{ if } (x,y)\in T_1\\
F(y,x) ~~~~&\textrm{ if } (y,x)\in T_1
\end{cases}
~~~ \textrm{ and }~~~ F_2(x,y)=\begin{cases}F(x,y) ~~~~&\textrm{ if } (x,y)\in T_2\\
F(y,x) ~~~~&\textrm{ if } (y,x)\in T_2,
\end{cases}$$ Briefly, $F_1(x,y)=F(x\wedge y, x\vee y),\ F_2(x,y)=F(x\vee y,
x\wedge y)\ \ \forall x,y \in L_k$.
Fodor [@Fodor1996] (see also [@Sander Theorem 2.6]) shown the following statement.
\[prop:Fodor\] Let $X$ be a nonempty chain and $F:X^2\to X$ be an associative operation. Then $F_1$ and $F_2$, the upper and the lower symmetrization of $F$, are also associative.
This idea makes it possible to investigate the two ’parts’ of a non-symmetric associative operation as one-one half of two symmetric associative operations.
By Proposition \[prop:ane\], both symmetrization of a nondecreasing quasitrivial operation $F:L_k^2\to L_k$ has a neutral element.
We call an element [*upper (or lower) half-neutral element*]{} of $F$ if it is the neutral element of the upper (or the lower) symmetrization. For simplicity we always denote the upper and lower half-neutral element of $F$ by $e$ and $f$, respectively.
Summarizing the previous results we get following partial description.
\[cor\] Let $F:L_k^2\to L_k$ be an associative quasitrivial nondecreasing operation. Then it has an upper and an lower half-neutral element denoted by $e$ and $f$. Moreover, if $e\le f$ then $$F(x,y)=\begin{cases}
x\wedge y ~~~~&\textrm{ if } x\vee y\le e\\
y ~~~~&\textrm{ if } e \le x\le f\\
x \vee y ~~~~&\textrm{ if } f\le x\wedge y
\end{cases}$$ Analogously, if $f\le e$ then $$F(x,y)=\begin{cases}
x\wedge y ~~~~&\textrm{ if } x\vee y\le f\\
x ~~~~&\textrm{ if } f \le x\le e\\
x \vee y ~~~~&\textrm{ if } e\le x\wedge y
\end{cases}$$
(0,0)– (0,4)–(4,4)–(4,0) – cycle; (0,0)–(4,4); (0,0)–(3,0); (0,0)–(0,1.5); (1,1)–(3,1); (1,1)–(1,1.5); (2,2)–(3,2); (2,2)–(1.5,2); (3,3)–(1.5,3); (4,4)–(4,3); (4,4)–(1.5,4); (0.5,0.5)–(0.5,1.5); (0.5,0.5)–(3,0.5); (1.5,1.5)–(3,1.5); (2.5,2.5)–(1.5,2.5); (2.5,2.5)–(3,2.5); (3.5,3.5)–(1.5,3.5); (3.5,3.5)–(3.5,3); at (-0.3,1.5) [e]{}; at (3,-0.3) [f]{}; (1.5,1.5) circle\[radius=1.5pt\]; (3,3) circle\[radius=1.5pt\];
(5,1.5)– (5,4)–(6.5,4); (9,3)– (9,0)–(8,0); (6.5,1.5)–(6.5,4); (8,0)–(8,3); (6.5,0)–(8,0); (6.5,4)–(8,4); (9,4)–(9,3)–(8,3)–(8,4)–cycle; (6.5,1.5)–(5,1.5)–(5,0)–(6.5,0)–cycle; at (5.75,0.75) [$x\wedge y$]{}; at (8.5,3.5) [$x\vee y$ ]{};at (7.25,2) [Proj$_y$]{};
at (4.7,1.5) [e]{}; at (8,-0.3) [f]{};
We note that $e=f$ iff $F$ has a neutral element.
The following lemma is essential for the visual characterization.
\[limp\] Let $F:L_k^2\to L_k$ be an associative quasitrivial nondecreasing operation. Assume that there exists $a<b\in L_k$ such that $F(a,b)=a$ and $F(b,a)=b$. Then one of the following holds:
1. If $F(a+1, a)=a$, then $$F(x,b)=b \textrm{ and } F(y, a)=a$$ for every $x\in[a+1,b]$ and $y\in [a,b-1]$.
2. If $F(a+1,a)=a+1$, then $F(x,y)=x \ (=Proj_x)$ for all $x,y\in [a,b]$.
(2.5,4)– (2.5,7); (5.5,4)– (5.5,7); (2.5,7)–(5.5,7)–(5.5,4)–(2.5,4)–cycle; at (2.5,3.7) [a]{}; at (5.5,3.7) [b]{}; at (4,3.5) [$\huge{\Downarrow}$]{}; (0,0)– (0,3); (3,0)– (3,3); (2.6,0)–(0,0); (3,3)–(0.4, 3); at (0,-0.3) [a]{}; at (0.6,-0.3) [a+1]{}; at (3,-0.3) [b]{}; at (2.6,-0.3) [b-1]{}; at (4,1.5) [ or ]{};
(5,0)– (5,3); (5.4,0)– (5.4,3); (5.8,0)– (5.8,3); (6.2,0)– (6.2,3); (6.6,0)– (6.6,3); (7,0)– (7,3); (7.4,0)– (7.4,3); (7.8,0)– (7.8,3); at (5,-0.3) [a]{}; at (7.8,-0.3) [b]{};
Assume first that $F(a+1,a)=a$. Then it follows that $F(a+1,b)=b$, otherwise we get Figure \[fig111\] (a). Using Observation \[l1\] we have that $F(x,b)=b$ for every $x\in [a+1,b]$. The equation $F(b-1,b)=b$ implies that $F(b-1, a)=a$, otherwise we are in the situation of Figure \[fig111\] (b). Similarly, as above we get that $F(y,a)=a$ for every $y\in[a,b-1]$. Here we note that an analogue argument gives the same result if we assume originally that $F(b-1,b)=b$.
Now assume that $F(a+1,a)=a+1$. This immediately implies that $F(x,a)=x$ for every $x\in[a,b]$ by quasitriviality, since it cannot be $a$ by the nondecreasingness of $F$. Using Observation \[l1\] again, it follows that $F(x,y)=x$ for all $y\in [a,x]$. Since $F(b-1,b)=b$ also implies the previous case, the assumption $F(a+1,a)=a+1$ implies $F(b-1,b)=b-1$. Similarly as above, this condition implies that $F(x,b)=x$ for all $x\in[a,b]$ and, by Observation \[l1\], it follows that $F(x,y)=x$ for every $y\in
[x,b]$. Altogether we get that $F(x,y)=x=\textrm{Proj}_x(x,y)$ as we stated.
\[rimp\] Analogue of Lemma \[limp\] can be formalized as follows.
*Let $F:L_k^2\to L_k$ be an associative quasitrivial nondecreasing operation. Assume that there exists $a<b\in L_k$ such that $F(b,a)=a$ and $F(a,b)=b$. Then one of the following holds:*
1. If $F(a,a+1)=a$, then $$F(b,x)=b \textrm{ and } F(a,y)=a$$ for every $x\in[a+1,b]$ and $y\in [a,b-1]$.
2. If $F(a,a+1)=a+1$, then $F(x,y)=y(=Proj_y)$ for all $x,y\in [a,b]$.
(2.5,4)– (5.5,4); (2.5,7)– (5.5,7); (2.5,7)–(5.5,7)–(5.5,4)–(2.5,4)–cycle; at (2.2,4) [a]{}; at (2.2,7) [b]{}; at (4,3.5) [$\huge{\Downarrow}$]{}; (0,0)– (3,0); (0,3)– (3,3); (0,2.6)–(0,0); (3,3)–(3,0.4); at (-0.3,0) [a]{}; at (-0.4,0.4) [a+1]{}; at (-0.3,3) [b]{}; at (-0.4,2.6) [b-1]{}; at (4,1.5) [ or ]{};
(5,0)– (8,0); (5,0.4)– (8,0.4); (5,0.8)– (8,0.8); (5,1.2)– (8,1.2); (5,1.6)– (8,1.6); (5,2)– (8,2); (5,2.4)– (8,2.4); (5,2.8)– (8,2.8); at (4.7,0) [a]{}; at (4.7,3) [b]{};
The proof of this statement is analogue to Lemma \[limp\] using Figure \[fig111\](c) and (d) instead of Figure \[fig111\](a) and (b), respectively.
From the previous results we conclude the following.
\[labin\] Let $F:L_k^2\to L_k$ be an associative quasitrivial and nondecreasing operation and $e$ and $f$ the upper and the lower half-neutral elements, respectively, and let $a,b\in L_k$ ($a<b$) be given. If $F(x,y)=x$ for every $x,y\in [a,b]$ (i.e, Lemma \[limp\] (b) holds), then $f< e$ and $[a,b]\subseteq [f,e]$. Similarly, if $F(x,y)=y$ for every $x,y\in [a,b]$ (i.e, Remark \[rimp\] (b) holds), then $e< f$ and $[a,b]\subseteq [e,f]$.
This is a direct consequence of Proposition \[cor\]. If $a$ or $b$ is not in $[e\wedge f,e\vee f]$ then $\tilde{F}=F|_{[a,b]^2}$ contains a part where $\tilde{F}$ is a minimum or a maximum. Moreover, it is also easily follows that if $F(x,y)=x$ for every $x,y\in [a,b]$, then $f< e$ must hold. Similarly, $F(x,y)=y$ for every $x,y\in [a,b]$ implies $e< f$.
\[corcases\] Let $F, e, f$ be as in Lemma \[labin\] and assume that $a,b\in X$ such that $a<b$ and $F(a,b)\ne F(b,a)$. Then
1. Lemma \[limp\](b) holds iff $f< e$ and $a,b\in [f,e]$,
2. Remark \[rimp\](b) holds iff $e< f$ and $a,b\in [e,f]$.
3. Lemma \[limp\](a) or Remark \[rimp\](a) holds iff $a,b\not\in[e\wedge f,e\vee f]$.
With other words we have:
Let $F, e, f$ be as in Lemma \[labin\]. Then $F(a,b)=F(b,a)$, if $a\not\in [e\wedge f,e\vee f]$ and $b\in [e\wedge f,e\vee f]$, or $b\not\in [e\wedge f,e\vee f]$ and $a\in [e\wedge f,e\vee f]$.
This form makes it possible to extend the partial description. (See Figure \[figeprtde\] for the case $e<f$.)
(0,0)– (0,4)–(4,4)–(4,0) – cycle; (0,0)–(4,4); (0,0)–(3,0); (0,0)–(0,3); (1,1)–(3,1); (1,1)–(1,3); (2,2)–(3,2); (2,2)–(1.5,2); (3,3)–(1.5,3); (4,4)–(4,1.5); (4,4)–(1.5,4); (0.5,0.5)–(0.5,3); (0.5,0.5)–(3,0.5); (1.5,1.5)–(3,1.5); (2.5,2.5)–(1.5,2.5); (2.5,2.5)–(3,2.5); (3.5,3.5)–(1.5,3.5); (3.5,3.5)–(3.5,1.5); at (1.5,-0.3) [e]{}; at (-0.3,1.5) [e]{}; at (3,-0.3) [f]{}; at (-0.3,3) [f]{}; (1.5,1.5) circle\[radius=1.5pt\]; (3,3) circle\[radius=1.5pt\];
(5,3)– (5,4)–(6.5,4); (9,1.5)– (9,0)–(8,0); (6.5,3)– (8,3); (7.25,0.75)–(5.75, 2.1); (6.5,1.5)– (8,1.5); (7.25,3.5)–(8.5, 2.4);
(6.5,1.5)–(6.5,4); (8,0)–(8,3); (6.5,0)–(8,0); (6.5,4)–(8,4); (5,1.5)–(5,3)–(6.5,3)–(6.5,1.5)–cycle; (8,1.5)–(8,3)–(9,3)–(9,1.5)–cycle; (9,4)–(9,3)–(8,3)–(8,4)–cycle; (6.5,1.5)–(5,1.5)–(5,0)–(6.5,0)–cycle; at (5.75,0.75) [$x\wedge y$]{}; at (8.5,3.5) [$x\vee y$ ]{};at (7.25,2.25) [Proj$_y$]{}; at (5.75,2.25) [Proj$_x$]{}; at (8.5,2.25) [Proj$_x$]{};
at (4.7,1.5) [e]{}; at (6.5,-0.3) [e]{}; at (8,-0.3) [f]{}; at (4.7,3) [f]{};
Using Lemma \[limp\] and Remark \[rimp\] we can provide a visual characterization of associative quasitrivial nondecreasing operations. The characterization based on the following algorithm which outputs the contour plot of $F$. Before we present the algorithm we note that the letters indicated in the following figures represent the value of operation $F$ in the corresponding points or lines (not a coordinate of the points itself as usual).
[**Algorithm**]{}
1. Let $Q_1=L_k^2$ and $F:L_k^2\to L_k$ be an associative quasitrivial nondecreasing operation.
2. For $Q_i=[a,b]^2$ ($a\le b$) we distinguish cases according to the values of $F(a,b)$ and $F(b,a)$. Whenever $Q_i$ contains only 1 element ($a=b$) for some $i$, then we are done.
3. If $F(a,b)=F(b,a)=a$, then draw straight lines between the points $(b,a)$ and $(a,a)$ and between $(a,b)$ and $(a,a)$. Let $Q_{i+1}=[a+1,b]^2$. (See Figure \[figmove1\].)
at (3,-0.3) [a]{}; (3,0) circle\[radius=1.5pt\]; at (-0.3,3) [a]{}; (0,3) circle\[radius=1.5pt\];
(0,3)–(3,3)–(3,0)–(0,0)–cycle;
at (4,1.5) [ $\Longrightarrow$ ]{};
(5.4,0.4)–(5.4,3)–(8,3)–(8, 0.4)–cycle; at (6.5,-0.3) [a]{}; at (4.7,1.5) [a]{}; (5,3)–(5,0)–(8,0);
at (6.7,1.7) [$Q_{i+1}$]{};
4. If $F(a,b)=F(b,a)=b$, then draw straight lines between the points $(a,b)$ and $(b,b)$ and between $(b,a)$ and $(b,b)$. Let $Q_{i+1}=[a,b-1]^2$.
5. If $F(a,b)=a, F(b,a)=b$ and $F(a+1, a)=a+1$, then $F(x,y)=x$ for all $x,y\in [a,b]$ and we are done. (See Figure \[figmove2\])
at (3,-0.3) [b]{}; (3,0) circle\[radius=1.5pt\]; at (-0.3,3) [a]{}; (0,3) circle\[radius=1.5pt\]; at (0.4,-0.3) [a+1]{}; (0.4,0) circle\[radius=1.5pt\];
(0,3)–(3,3)–(3,0)–(0,0)–cycle;
at (4,1.5) [ $\Longrightarrow$ ]{};
(5,0)– (5,3); (5.4,0)– (5.4,3); (5.8,0)– (5.8,3); (6.2,0)– (6.2,3); (6.6,0)– (6.6,3); (7,0)– (7,3); (7.4,0)– (7.4,3); (7.8,0)– (7.8,3); at (6.5,-0.3) [Proj$_x$]{};
6. If $F(a,b)=b, F(b,a)=a$ and $F(a,a+1)=a+1$, then $F(x,y)=y$ for all $x,y\in [a,b]$ and we are also done.
7. If $F(a,b)=a, F(b,a)=b$ and $F(a+1, a)=a$, then Lemma \[limp\] (a) holds and we have Figure \[figmove3\]. Let $Q_{i+1}=[a+1,b-1]^2$.
at (3.3,0) [b]{}; (3,0) circle\[radius=1.5pt\]; at (-0.3,3) [a]{}; (0,3) circle\[radius=1.5pt\]; at (0.4,-0.3) [a]{}; (0.4,0) circle\[radius=1.5pt\];
(0,3)–(3,3)–(3,0)–(0,0)–cycle;
at (4,1.5) [ $\Longrightarrow$ ]{}; (5,0)– (5,3); (8,0)– (8,3); (7.6,0)–(5,0); (8,3)–(5.4, 3); at (6.5,-0.3) [a]{}; at (4.7,1.5) [a]{}; at (6.5,3.3) [b]{}; at (8.15,1.5) [b]{};
(5.4,0.4)–(5.4,2.6)–(7.6,2.6)–(7.6,0.4)–cycle; at (6.5,1.5) [$Q_{i+1}$]{};
8. If $F(a,b)=b, F(b,a)=a$ and $F(a,a+1)=a$, then Remark \[rimp\] (a) holds. Let $Q_{i+1}=[a+1,b-1]^2$.
It is clear that the algorithm is finished after finitely many steps. Let us denote this number of steps by $l\in \mathbb{N}$.
We also denote the top-left and the bottom-right corner of $Q_i$ by $p_i$ and $q_i$ ($i=1,\dots,l$), respectively.
Let $\mathcal{P}$ (and $\mathcal{Q}$) denote the path containing $p_i$ (and $q_i$) for $i\in \{1,\dots, l\}$ and line segments between consecutive $p_i$’s (and $q_i$’s). Let us denote the line segment between $p_i$ and $p_{i+1}$ by $\overline{p_i,p_{i+1}}$. We set the notation $\mathcal{P}=(p_j)_{j=1}^l$ and $\mathcal{Q}=(q_j)_{j=1}^l$.
Clearly, we get the path $\mathcal{P}$ if we start at the top-left corner of $L_k^2$ and in each step we move either one place to the right or one place downward or one place diagonally downward-right.
We say that a path is a [*downward-right path*]{} of $L_k$ if in each step it moves to the nearest point of $L_k^2$ either one place to the right or one place downward or one place diagonally downward-right.
(0,3) circle\[radius=1.5pt\];
(0.3,2.7) circle\[radius=1.5pt\]; (0.6,2.7) circle\[radius=1.5pt\]; at (0.6,2.4) [$\mathcal{P}$]{};
(0.9,2.7) circle\[radius=1.5pt\]; (1.2,2.4) circle\[radius=1.5pt\]; (0.9,2.7) – (0.9,2.4); (0.9,2.7) – (1.2,2.7); (0.9,2.7) – (1.15,2.45);
(1.2,2.1) circle\[radius=1.5pt\];
(0,3)–(3,3)–(3,0)–(0,0)–cycle; (0,3)–(0.3,2.7)–(0.6,2.7)–(0.9,2.7)–(1.2,2.4)–(1.2,2.1); (1.2,2.1)–(1.2,1.2)–(2.1,1.2)–(2.1,2.1)–cycle; (3,0) circle\[radius=1.5pt\]; (2.7,0.3) circle\[radius=1.5pt\]; (2.7,0.6) circle\[radius=1.5pt\]; at (2.4,0.6) [$\mathcal{Q}$]{}; (2.7,0.9) circle\[radius=1.5pt\]; (2.4,1.2) circle\[radius=1.5pt\]; (2.1,1.2) circle\[radius=1.5pt\]; (3,0)–(2.7,0.3) – (2.7,0.6)– (2.7,0.9)– (2.4,1.2) – (2.1,1.2);
If $\overline{p_i, p_{i+1}}$ is horizontal or vertical, then the reduction from $Q_i$ to $Q_{i+1}$ is uniquely determined. Moreover, if $\overline{p_i, p_{i+1}}$ is horizontal, then $F(x,y)=F(y,x)=x\wedge y$, where $p_i=(x,y)$ and $q_i=(y,x)$. Similarly, if $\overline{p_i, p_{i+1}}$ is vertical, then $F(x,y)=F(y,x)=x\vee y$, where $p_i=(x,y)$ and $q_i=(y,x)$. On the other hand if $\overline{p_i, p_{i+1}}$ is diagonal, then we have a free choice for the value of $F$ in $p_i$. This is determined by either Lemma \[limp\] (a) or Remark \[rimp\] (a). Since in this case the value of $F$ in $q_i$ is different from $p_i$, the value in $q_i$ is automatically defined. It is also clear from the algorithm that the path $\mathcal{Q}$ is the reflection of $\mathcal{P}$ to the diagonal $\Delta_{L_k}$.
Using the previous paragraph and Observation \[l1\] it is possible to reconstruct operations from a given downward-right path $\mathcal{P}$ which starts at $p_1=(1,k)$.
We illustrate the reconstruction on $L_6\times L_6$. The paths $\mathcal{P}=(p_j)_{j=1}^5$ and $\mathcal{Q}=(q_j)_{j=1}^5$ denoted by red and blue, respectively. According to the previous observations we get the following pictures (see Figure \[figrec\]). It can be clearly seen that $\mathcal{Q}$ is the reflection of $\mathcal{P}$ to the diagonal $\Delta_{L_6}$, and $4$ is the neutral element of the reconstructing operation, where $\mathcal{P}$ and $\mathcal{Q}$ touch each other and reach the diagonal $\Delta_{L_6}$. For the precise statement and proof see Theorem \[thmchar\].
in [0,1,...,5]{} [ in [0,1,...,5]{} [ at (.5\*,.5\*) ; ]{} ]{}
at (0,-0.3) [1]{}; at (0.5,-0.3) [2]{}; at (1,-0.3) [3]{}; at (1.5,-0.3) [4]{}; at (2,-0.3) [5]{}; at (2.5,-0.3) [6]{}; at (-0.3,0) [1]{}; at (-0.3,0.5) [2]{}; at (-0.3,1) [3]{}; at (-0.3,1.5) [4]{}; at (-0.3,2) [5]{}; at (-0.3,2.5) [6]{};
(0,2.5)–(0.5,2.5)–(0.5,2)–(1,1.5)–(1.5,1.5); (1.5,1.5)–(1.5,1)–(2,0.5)–(2.5,0.5)–(2.5,0); at (0,2.7) [$p_1$]{}; at (0.5,2.7) [$p_2$]{}; at (0.3,2) [$p_3$]{}; at (0.9,1.4) [$p_4$]{}; at (1.5,1.7) [$p_5(q_5)$]{}; at (2.7,0) [$q_1$]{}; at (2.7, 0.5) [$q_2$]{}; at (2, 0.7) [$q_3$]{}; at (1.7,1.1) [$q_4$]{}; in [0,1,...,5]{} [ in [0,1,...,5]{} [ at (.5\*+4.5,.5\*) ; ]{} ]{}
(4.5,2.5)–(5,2.5)–(5,2)–(5.5,1.5)–(6,1.5); (6,1.5)–(6,1)–(6.5,0.5)–(7,0.5)–(7,0);
(4.5,2.5)–(4.5,0)–(7,0); (5,2.5)–(7,2.5)–(7,0.5); (5,1.5)–(5,0.5)–(6,0.5); (5.5,2)–(6.5,2)–(6.5,1); (5.5,1.5)–(5.5,1)–(6,1);
(3,1.25)–(4,1.25); (7.5,2)–(8.5, 2.5); (7.5,0.5)–(8.5, 0); at (10.25,1.25) [ or ]{}; (5,2) circle\[radius= 0.5 em\]; (6.5,0.5) circle\[radius= 0.5 em\]; at (4.7,2) [?]{}; at (6.2, 0.5) [?]{};
in [0,1,...,5]{} [ in [0,1,...,5]{} [ at (.5\*+9,.5\*+2) ; ]{} ]{}
in [0,1,...,5]{} [ in [0,1,...,5]{} [ at (.5\*+9,.5\*-2) ; ]{} ]{}
(9,4.5)–(9.5,4.5)–(9.5,4)–(10,3.5)–(10.5,3.5); (10.5,3.5)–(10.5,3)–(11,2.5)–(11.5,2.5)–(11.5,2);
(9,4.5)–(9,2)–(11.5,2); (9.5,4.5)–(11.5,4.5)–(11.5,2.5); (9.5,4)–(9.5,2.5)–(10.5,2.5); (10,4)–(11,4)–(11,2.5); (10,3.5)–(10,3)–(10.5,3);
(9,0.5)–(9.5,0.5)–(9.5,0)–(10,-0.5)–(10.5,-0.5); (10.5,-0.5)–(10.5,-1)–(11,-1.5)–(11.5,-1.5)–(11.5,-2);
(9,0.5)–(9,-2)–(11.5,-2); (9.5,0.5)–(11.5,0.5)–(11.5,-1.5); (9.5,-0.5)–(9.5,-1.5)–(11,-1.5); (9.5,0)–(11,0)–(11,-1); (10,-0.5)–(10,-1)–(10.5,-1);
Let $\mathcal{P}\subset L_k^2$ be the downward-right path from $(1,k)$ to $(a,b)$ ($a<b$) and let $\mathcal{Q}$ be the reflection of $\mathcal{P}$ to the diagonal $\Delta_{L_k}$.
We say that $(x,y)\in L_k^2\setminus(\mathcal{P} \cup \mathcal{Q}
\cup [a,b]^2)$ is [*above*]{} $\mathcal{P} \cup \mathcal{Q}$ if there exists $p=(x,w)\in \mathcal{P}$ such that $y> w$ or $q=(w,y)\in \mathcal{Q}$ such that $x> w$.
Similarly, we say that $(x,y)\in L_k^2\setminus( \mathcal{P} \cup
\mathcal{Q} \cup [a,b]^2)$ is [*below*]{} $\mathcal{P} \cup
\mathcal{Q}$ if there exists a $p=(x,w)\in \mathcal{P}$ such that $y< w$ or a $q=(w,y)\in \mathcal{Q}$ such that $x< w$.
Using this terminology we can summarize the previous observations and we get the following characterization. The next statement can be seen as the analogue of theorem of Czogała-Drewiak [@Czogala1984 Theorem 3.] for finite chains.
\[tfo1\] For every associative quasitrivial nondecreasing operation $F:L_k^2\to L_k$ there exist half-neutral elements $a,b\in L_k$ ($a\le b$) and a downward-right path $\mathcal{P}=(p_j)_{j=1}^l$ (for some $l\in\mathbb{N},l<k$) from $(1,k)$ to $(a,b)$. We denote the reflection of $\mathcal{P}$ to the diagonal $\Delta_{L_k}$ by $\mathcal{Q}=(q_j)_{j=1}^l$. Then for every $(x,y)\not\in\mathcal{P}\cup\mathcal{Q}$ $$F(x,y)=\begin{cases}
x\vee y, &\textrm{ if } (x,y) \textrm{ is above } \mathcal{P} \cup \mathcal{Q}\\
x\wedge y, &\textrm{ if } (x,y) \textrm{ is below } \mathcal{P} \cup \mathcal{Q}\\
Proj_x \textrm{ or } Proj_y, &\textrm{ if } (x,y)\in [a,b]^2,
\end{cases}$$ and for every $(x,y)\in\mathcal{P}\cup\mathcal{Q}$ $$F(x,y)=\begin{cases}
x\wedge y &\textrm{ if } (x,y)=p_i \textrm{ or } q_i \textrm{ and } \overline{p_i,p_{i+1}} \textrm{ is horizontal}, \\
x\vee y, &\textrm{ if } (x,y)=p_i \textrm{ or } q_i \textrm{ and } \overline{p_i,p_{i+1}} \textrm{ is vertical,}\\
x \textrm{ or } y, &\textrm{ if } (x,y)=p_i \textrm{ and } \overline{p_i,p_{i+1}} \textrm{ is diagonal,}\\
x \textrm{ or } y, &\textrm{ if } (x,y)=q_i \textrm{ and } \overline{q_i,q_{i+1}} \textrm{ is diagonal.}\\
\end{cases}$$
If $a$ is the lower half-neutral element $f$ and $b$ is the upper half-neutral element $e$, then $F$ is $Proj_x$ on $[a,b]^2$, otherwise it is $Proj_y$.
Moreover $F$ is symmetric expect on $[a,b]^2$ and at the points $p_i\in \mathcal{P}$ and $q_i\in \mathcal{Q}$ where $\overline{p_i,
p_{i+1}}$ is diagonal ($i\in \{1, \dots,l-1\}$).
(0,3) circle\[radius=1.5pt\];
(0.3,2.7) circle\[radius=1.5pt\]; (0.6,2.7) circle\[radius=1.5pt\]; at (0.6,2.4) [$\mathcal{P}$]{};
(0.9,2.7) circle\[radius=1.5pt\]; (1.2,2.4) circle\[radius=1.5pt\]; (1.2,2.1) circle\[radius=1.5pt\];
(0,3)–(3,3)–(3,0)–(0,0)–cycle; (0,3)–(0.3,2.7)–(0.6,2.7)–(0.9,2.7)–(1.2,2.4)–(1.2,2.1); (1.2,2.1)–(1.2,1.2)–(2.1,1.2)–(2.1,2.1)–cycle; (3,0) circle\[radius=1.5pt\]; (2.7,0.3) circle\[radius=1.5pt\]; (2.7,0.6) circle\[radius=1.5pt\]; at (2.4,0.6) [$\mathcal{Q}$]{}; (2.7,0.9) circle\[radius=1.5pt\]; (2.4,1.2) circle\[radius=1.5pt\]; (2.1,1.2) circle\[radius=1.5pt\]; (3,0)–(2.7,0.3) – (2.7,0.6)– (2.7,0.9)– (2.4,1.2) – (2.1,1.2); at (0.8,0.8) [$x\wedge y$ ]{}; at (2.55,2.55) [$x\vee
y$]{}; at (1.65,1.65) [$Proj$]{};
The statement is clearly follows from the Algorithm and the definition of paths $\mathcal{P}$ and $\mathcal{Q}$.
The converse statement can be formalized as follows. This statement plays the role of theorem of Martin-Mayor-Torrens [@Martin2003 Theorem 4.] for finite chains.
\[thmchar\] Let $\mathcal{P}=(p_j)_{j=1}^l$ be a downward-right path in $T_1\subset L_k^2$ from $(1,k)$ to $(a,b)$ $(a\le b)$ and let $\mathcal{Q}=(q_j)_{j=1}^l$ be its reflection to the diagonal $\Delta_{L_k}$. Let $F:L_k^2\to L_k$ be defined for every $(x,y)\not\in\mathcal{P}\cup\mathcal{Q}$ as $$F(x,y)=\begin{cases}
x\vee y, &\textrm{ if } (x,y) \textrm{ is above } \mathcal{P} \cup \mathcal{Q},\\
x\wedge y, &\textrm{ if } (x,y) \textrm{ is below } \mathcal{P} \cup \mathcal{Q},\\
Proj_x \textrm{ or } Proj_y \textrm{ (uniformly)}, &\textrm{ for every } (x,y)\in [a,b]^2. \end{cases}$$ and for every $(x,y)\in\mathcal{P}\cup\mathcal{Q}$ $$F(x,y)=\begin{cases}
x\wedge y &\textrm{ if } (x,y)=p_i \textrm{ or } q_i \textrm{ and } \overline{p_i,p_{i+1}} \textrm{ is horizontal}, \\
x\vee y, &\textrm{ if } (x,y)=p_i \textrm{ or } q_i \textrm{ and } \overline{p_i,p_{i+1}} \textrm{ is vertical,}\\
x \textrm{ or } y \textrm{ (arbitrarily) }, &\textrm{ if } (x,y)=p_i
\textrm{ and } \overline{p_i,p_{i+1}} \textrm{ is diagonal.}
\end{cases}$$ If $(x,y)=q_i$ and $\overline{q_i,q_{i+1}} \textrm{ (or equivalently
}\overline{p_i,p_{i+1}}) \textrm{ is diagonal,}$ then $F(x,y)\in
\{x, y\}$ and $F(x,y)\ne F(y,x)$ uniquely define $F(x,y)$. Then $F$ is associative quasitrivial and nondecreasing.
It is clear that $F$ is defined for every $(x,y)\in L_k^2$ and $F$ is quasitrivial and nondecreasing. Now we show that $F$ is associative. If it is not the case, then by Lemma \[lpic\], one of the cases of Figure \[fig111\] is realized. Let $u,v,w \in L_k$ ($u<v<w$) denote the elements where its realized. Clearly $F(u,w)\ne
F(w,u)$ and $F$ is not a projection on $[u,w]^2$. Thus, by the definition of $F$, it follows that $(u,w)\in \mathcal{P}$ and $(w,u)\in \mathcal{Q}$. Hence $p_i=(u,w)$ for some $i=\{1, \dots,
l-1\}$ and $\overline{p_i, p_{i+1}}$ is diagonal. Thus we have one of the following situation (Figure \[figrem\]).
(0.9,2.7)– (0.9,0.9); (1.2,2.7)– (2.7,2.7); (2.7,0.9)–(2.7,2.7); (2.4,0.9)–(0.9, 0.9); at (0.9,-0.3) [u]{}; at (2.7,-0.3) [w]{}; at (-0.3,2.7) [w]{}; at (-0.3,0.9) [u]{}; (0,3) circle\[radius=1.5pt\];
(0.3,2.7) circle\[radius=1.5pt\]; (0.6,2.7) circle\[radius=1.5pt\]; at (0.45,2.4) [$\mathcal{P}$]{};
(0.9,2.7) circle\[radius=1.5pt\]; (1.2,2.4) circle\[radius=1.5pt\]; (1.2,2.1) circle\[radius=1.5pt\];
(0,3)–(3,3)–(3,0)–(0,0)–cycle; (0,3)–(0.3,2.7)–(0.6,2.7)–(0.9,2.7)–(1.2,2.4)–(1.2,2.1); (1.2,2.1)–(1.2,1.2)–(2.1,1.2)–(2.1,2.1)–cycle; (3,0) circle\[radius=1.5pt\]; (2.7,0.3) circle\[radius=1.5pt\]; (2.7,0.6) circle\[radius=1.5pt\]; at (2.4,0.45) [$\mathcal{Q}$]{}; (2.7,0.9) circle\[radius=1.5pt\]; (2.4,1.2) circle\[radius=1.5pt\]; (2.1,1.2) circle\[radius=1.5pt\]; (3,0)–(2.7,0.3) – (2.7,0.6)– (2.7,0.9)– (2.4,1.2) – (2.1,1.2);
(4.9,2.4)– (4.9,0.9); (4.9,2.7)– (6.7,2.7); (6.7,1.2)–(6.7,2.7); (6.7,0.9)–(4.9, 0.9); at (4.9,-0.3) [u]{}; at (6.7,-0.3) [w]{}; at (3.7,2.7) [w]{}; at (3.7,0.9) [u]{}; (4,3) circle\[radius=1.5pt\];
(4.3,2.7) circle\[radius=1.5pt\]; (4.6,2.7) circle\[radius=1.5pt\]; at (4.45,2.4) [$\mathcal{P}$]{};
(4.9,2.7) circle\[radius=1.5pt\]; (5.2,2.4) circle\[radius=1.5pt\]; (5.2,2.1) circle\[radius=1.5pt\];
(4,3)–(7,3)–(7,0)–(4,0)–cycle; (4,3)–(4.3,2.7)–(4.6,2.7)–(4.9,2.7)–(5.2,2.4)–(5.2,2.1); (5.2,2.1)–(5.2,1.2)–(6.1,1.2)–(6.1,2.1)–cycle; (7,0) circle\[radius=1.5pt\]; (6.7,0.3) circle\[radius=1.5pt\]; (6.7,0.6) circle\[radius=1.5pt\]; at (6.4,0.45) [$\mathcal{Q}$]{}; (6.7,0.9) circle\[radius=1.5pt\]; (6.4,1.2) circle\[radius=1.5pt\]; (6.1,1.2) circle\[radius=1.5pt\]; (7,0)–(6.7,0.3) – (6.7,0.6)– (6.7,0.9)– (6.4,1.2) – (6.1,1.2);
Therefore, since $u<v<w$, it follows that $F(u,v)\ne v, F(v,u)\ne v,
F(w,v)\ne v, F(v,w)\ne v$. Hence, none of the cases of Figure \[fig111\] can be realized. Thus $F$ is associative.
According to Theorems \[tfo1\] and \[thmchar\] it is clear that there is a surjection from the set of associative quasitrivial nondecreasing operations defined on $L_k$ to the downward-right paths defined on $T_1$ and started at $(1,k)$ (and ended somewhere in $T_1$). This surjection is a bijection if and only if the path $\mathcal{P}$ does not contain a diagonal move and $a=b$. This condition is equivalent that $F$ is symmetric (and has a neutral element).
\[cchar\] Let $F:L_k^2\to L_k$ be an associative quasitrivial nondecreasing operation. If $F$ is symmetric, then it is uniquely determined by a downward-right path $\mathcal{P}$ containing only horizontal and vertical line segments and it starts at $(1,k)$ and reaches the diagonal $\Delta_{L_k}$.
As a consequence of the previous corollary we obtain the result of [@Beats2009 Theorem 4.] (see also [@Jimmy2017 Theorem 14.]).
\[ccard\] The number of associative quasitrivial nondecreasing symmetric operation defined on $L_k$ is $2^{k-1}$.
Every path from $(1,k)$ to the diagonal $\Delta_{L_k}$ using right or downward moves contains $k$ points. According to Corollary \[cchar\], in each point of the path, except the last one, we have two options which direction we move further. This immediately implies that the number of associative quasitrivial nondecreasing symmetric operation defined on $L_k$ is $2^{k-1}$.
In Theorem \[tnumb\], as an application of the results of this section, we calculate the number of associative quasitrivial nondecreasing operations defined on $L_k$ and also the number of associative quasitrivial nondecreasing operations on $L_k$ that have neutral elements.
1. We note that from the proof of Lemma \[limp\] throughout this section we essentially use that $F$ is defined on a finite chain.
2. In the continuous case [@Czogala1984; @Martin2003] and also in the symmetric case [@Beats2009; @Jimmy2017] it is always possible to define a one variable function $g$, such that the extended graph of $g$ separates the points of the domain of the binary operation $F$ into two parts where $F$ is a minimum and a maximum, respectively. Now the paths $\mathcal{P}$ and $\mathcal{Q}$ play the role of the extended graph of $g$. Because of the diagonal moves of the path $\mathcal{P}$, it does not seems so clear how such a ’separating’ function can be defined in the non-symmetric discrete case.
Bisymmetric operations {#s5}
======================
In this section we show a characterization of bisymmetric quasitrivial nondecreasing binary operations based on the previous section.The following statement was proved as [@Jimmy2017 Lemma 22.].
\[lba\] Let $X$ be an arbitrary set and $F:X^2 \to X$ be an operation. Then the following assertions hold.
1. If $F$ is bisymmetric and has a neutral element, then it is associative and symmetric.
2. If $F$ is bisymmetric and quasitrivial, then $F$ is associative.
3. If $F$ is associative and symmetric, then it is bisymmetric.
Using also the results of Section \[s4\] we get the following statement.
\[thmbi\] Let $F:L_k^2\to L_k$ be a bisymmetric quasitrivial nondecreasing operation. Then there exists the upper half-neutral element $e$ and the lower half-neutral element $f$ and $F$ is symmetric on $(L_k\setminus[e\wedge f,e\vee f])^2$.
According to Lemma \[lba\](b), every quasitrivial bisymmetric operations are associative. Thus, by Proposition \[cor\] it has an upper and lower half-neutral element ($e$ and $f$, respectively).
Let us assume that $e\le f$ (the case when $f\le e$ can be handled similarly).
If there exists $u,v\in L_k$ such that $u<v$, $F(u,v)\ne F(v,u)$, then by Corollary \[corcases\], either $u,v \in [e,f]$ (then we do not need to prove anything) or $u,v\not\in [e,f]$. Moreover, if $u,v\not\in [e,f]$, then Lemma \[limp\](a) or Remark \[rimp\](a) holds. The existence of $e$ implies that $v-u\ge 2$.
If $$\label{equv} u=F(u,v) \ne F(v,u)=v$$ is satisfied, then Lemma \[limp\] (a) holds (i.e, $F(x,v)=v$ if $x\in [u+1,v]$ and $F(y,u)=u$ if $y\in[u,v-1]$). Since $v-u\ge 2$, $u+1\le v-1$, hence $F(u+1,u)=u$. On the other hand, $F$ is monotone and idempotent, thus by Observation \[l1\], $F(v,t)=v$ and $F(u,t)=u$ for all $t\in [u,v]$. Using bisymmetric equation we get the following $$u=F(u,v)=F(F(u+1,u),F(v,v-1))=F(F(u+1,v), F(u,v-1))=F(v,u)=v,$$ which is a contradiction.
Similarly, if $$\label{eqvu} v=F(u,v) \ne F(v,u)=u$$ is satisfied, then Remark \[rimp\] (a) holds (i.e, $F(v,x)=v$ if $x\in [u+1,v]$ and $F(u,y)=u$ if $y\in[u,v-1]$). Since $v-u\ge 2$, $u+1\le v-1$, hence $F(v-1,v)=v$. Applying Observation \[l1\] again, we have $F(t,v)=v$ and $F(t,u)=u$ for all $t\in
[u,v]$. Using bisymmetric equation we get a contradiction as $$u=F(v,u)=F(F(v-1,v),F(u,u+1))=F(F(v-1,u), F(v,u+1))=F(u,v)=v.$$
Applying Theorem \[thmbi\] we get the following characterization.
\[tbfo1\] Let $F:L_k^2\to L_k$ be a quasitrivial nondecreasing operation. Then $F$ is bisymmetric if and only if there exists $a,b\in L_k$ ($a\le
b$) and a downward-right path $\mathcal{P}=(p_j)_{j=1}^l$ (for some $l\in \mathbb{N}$) from $(1,k)$ to $(a,b)$ containing only horizontal and vertical line segments such that for every $(x,y)\not\in\mathcal{P}\cup\mathcal{Q}$ $$\label{eqbi1}F(x,y)=\begin{cases}
x\vee y, &\textrm{ if } (x,y) \textrm{ is above } \mathcal{P} \cup \mathcal{Q},\\
x\wedge y, &\textrm{ if } (x,y) \textrm{ is below } \mathcal{P} \cup \mathcal{Q},\\
Proj_x \textrm{ or } Proj_y \textrm{ (uniformly)}, &\textrm{ for every } (x,y)\in [a,b]^2. \end{cases}$$ and for every $(x,y)\in\mathcal{P}\cup\mathcal{Q}$ $$\label{eqbi2}F(x,y)=\begin{cases}
x\wedge y &\textrm{ if } (x,y)=p_i \textrm{ or } q_i \textrm{ and } \overline{p_i,p_{i+1}} \textrm{ is horizontal}, \\
x\vee y, &\textrm{ if } (x,y)=p_i \textrm{ or } q_i \textrm{ and }
\overline{p_i,p_{i+1}} \textrm{ is vertical,}
\end{cases}$$ where $\mathcal{Q}=(q_j)_{j=1}^l$ is the reflection of $\mathcal{P}$ to the diagonal $\Delta_{L_k}$.
In particular, $F$ is symmetric on $L_k^2\setminus[a,b]^2$ and one of the projections on $[a,b]^2$.
(0,3) circle\[radius=1.5pt\]; (0.3,3) circle\[radius=1.5pt\]; (0.3,2.7) circle\[radius=1.5pt\]; (0.6,2.7) circle\[radius=1.5pt\]; at (0.6,2.4) [$\mathcal{P}$]{}; (0.9,2.4) circle\[radius=1.5pt\]; (0.9,2.7) circle\[radius=1.5pt\]; (1.2,2.4) circle\[radius=1.5pt\]; (1.2,2.1) circle\[radius=1.5pt\];
(0,3)–(3,3)–(3,0)–(0,0)–cycle; (0,3)–(0.3,3)–(0.3,2.7)–(0.6,2.7)–(0.9,2.7)–(0.9,2.4)–(1.2,2.4)–(1.2,2.1); (1.2,2.1)–(1.2,1.2)–(2.1,1.2)–(2.1,2.1)–cycle; (3,0) circle\[radius=1.5pt\]; (3,0.3) circle\[radius=1.5pt\];
(2.7,0.3) circle\[radius=1.5pt\]; (2.7,0.6) circle\[radius=1.5pt\]; at (2.4,0.6) [$\mathcal{Q}$]{}; (2.7,0.9) circle\[radius=1.5pt\]; (2.4,0.9) circle\[radius=1.5pt\]; (2.4,1.2) circle\[radius=1.5pt\]; (2.1,1.2) circle\[radius=1.5pt\]; (3,0)–(3,0.3)–(2.7,0.3) – (2.7,0.6)– (2.7,0.9)–(2.4,0.9) –(2.4,1.2) – (2.1,1.2); at (0.8,0.8) [$x\wedge y$ ]{}; at (2.55,2.55) [$x\vee
y$]{}; at (1.65,1.65) [$Proj$]{};
(Necessity) Since $F$ is bisymmetric and quasitrivial, by Lemma \[lba\](b), $F$ is associative. By Theorem \[tfo1\], there exist half-neutral elements $a,b\in L_k$ ($a<b$) and a downward-right path $\mathcal{P}$ from $(1,k)$ to $(a,b)$. By Theorem \[tbfo1\], $F$ is symmetric on $L_k^2\setminus[a,b]^2$. Thus $\mathcal{P}$ does not contain a diagonal line segment. Hence, applying again Theorem \[tfo1\] we get that $F$ satisfies and .
(Sufficiency) The operation $F$ defined by and satisfies the conditions of Theorem \[thmchar\], thus $F$ is quasitrivial nondecreasing and associative. Now we show that $F$ is bisymmetric (i.e, $\forall u,v,w,z\in L_k$ $$\label{eqbia}
F(F(u,v),F(w,z))=F(F(u,w),F(v,z)).)$$ Let us assume that $F(x,y)=Proj_x$ on $[a,b]^2$ (for $F(x,y)=Proj_y$ on $[a,b]^2$ the proof is analogue). By Corollary \[corcases\], this implies that $a=f$ and $b=e$ ($f<e$) and, by Proposition \[cor\], it is clear that $$\label{eqproj}F(x,y)=x \ \forall x \in L_k,
\forall y \in [a,b].$$ Since $F$ is associative, we have $$F(F(u,v),F(w,z))=F(F(F(u,v),w),z)=F(F(u, F(v,w)),z)$$ and $$F(F(u,w),F(v,z))=F(F(F(u,w),v),z)=F(F(u, F(w,v)),z).$$
If $F(v,w)=F(w,v)$, then follows and we are done.
If $F(v,w)\ne F(w,v)$, then $v,w\in [a,b]^2$ and, since $F(x,y)=Proj_x$ on $[a,b]^2$, $F(v,w)=v$ and $F(w,v)=w$. Then, by , $$\begin{aligned}
&F(F(u, F(v,w)),z)=F(F(u,v),z)=F(u,z),\\
&F(F(u, F(v,w)),z)=F(F(u,w),z)=F(u,z).\end{aligned}$$ Thus $F$ is bisymmetric.
1. There is a one-to-one correspondence between downward-right paths containing only vertical and horizontal line segments and the quasitrivial nondecreasing bisymmetric operations if we fix that the operation is $Proj_x$ on $[a,b]^2$ ($a$ and $b$ are the half neutral-elements of the operation). The same is true, if the operation is $Proj_y$ on $[a,b]^2$.
2. The nondecreasing assumption can be substituted by monotonicity. Indeed, by Corollary \[cnem\], monotonicity is equivalent with nondecreasingness for quasitrivial operations.
The number of operations of given class {#s6}
=======================================
This section is devoted to calculate the number of associative quasitrivial nondecreasing operations. Byproduct of the following argument we also consider the number of associative quasitrivial nondecreasing operations having neutral elements. With the same technique one can easily deduce the number of bisymmetric quasitrivial nondecreasing binary operations (see Proposition \[binumb\]).
\[tnumb\] Let $A_k$ denote the number of associative quasitrivial nondecreasing operations defined on $L_k$ and $B_k$ denote the number of associative quasitrivial nondecreasing operations defined on $L_k$ and having neutral elements. Then $$A_k=\frac{1}{6}\big((2+\sqrt{3})(1+\sqrt{3})^k+(2-\sqrt{3})(1-\sqrt{3})^k-4\big),$$ $$B_k=\frac{1}{2\cdot\sqrt{3}}\big((1+\sqrt{3})^k-(1-\sqrt{3})^k\big).$$
The following observations show that these numbers are related to the downward-right path $\mathcal{P}=(p_j)_{j=1}^l$ (for some $l\le
k$) in $T_1$ starting from $(1,k)$. Let $m_{\mathcal{P}}$ be the number of diagonal line segments $\overline{p_i,p_{i+1}} \in
\mathcal{P}$ ($i\in\{1, \dots, l-1\}$). We say that the downward-right path $\mathcal{P}$ is [*weighted*]{} with weight $2^{m_{\mathcal{P}}}$.
\[lcount1\]
1. $B_k$ is the sum of the weights of weighted paths that starts at $(1,k)$ and reaches $\Delta_{L_k}$.
2. $A_k+B_k$ is twice the sum of the weights of weighted paths in $T_1$ that starts at $(1,k)$ and ends at any point of $T_1$.
<!-- -->
1. Applying Theorem \[tfo1\], it is clear that if an associative quasitrivial nondecreasing binary operation $F$ has a neutral element, then the downward-right path $\mathcal{P}$ defined for $F$ reaches the diagonal $\Delta_{L_k}$. By Theorem \[thmchar\], there can be defined $2^{m_{\mathcal{P}}}$ different operations for a given path $\mathcal{P}$ that reaches the diagonal, since we have a choice in each case when the path contains a diagonal line segment. This show the first part of the statement.
2. This statement follows from the fact that for any associative quasitrivial nondecreasing operation $F$ one can define a downward-right path which starts at $(1,k)$ and ends somewhere in $T_1$. If its end in $(a,b)$ where $a< b$ (not on $\Delta_{L_k}$), then $F$ is one of the projections in $[a,b]^2$, and $a$ and $b$ are the half-neutral elements of $F$. This makes the extra 2 factor in the statement.
Let $\Pi_1$ denote set of the weighted paths in $T_1$ that starts at $(1,k)$ and ends at $(a,b) $ where $a< b$. Similary, $\Pi_2$ denote the set of weighted paths that starts at $(1,k)$ and reaches $\Delta_{L_k}$. Hence, $$ A_k
=2\cdot\sum_{\mathcal{P}\in \Pi_1}2^{m_{\mathcal{P}}}+\sum_{\mathcal{P}\in \Pi_2}2^{m_{\mathcal{P}}} $$ According to the (a) part $$B_k=\sum_{\mathcal{P}\in \Pi_2}2^{m_{\mathcal{P}}}.$$ Adding these equations, we get the statement for $A_k+B_k$.
Now we present a recursive formula for $A_k$ and $B_k$.
\[lnumb\]
1. $B_1=1$, $B_2=2$ and $B_k=2\cdot B_{k-1}+2\cdot B_{k-2}$ for every $k\ge 3$.
2. $A_k=2\sum_{i=1}^k B_{i}-B_k$ for every $k\in \mathbb{N}$.
<!-- -->
1. $B_1=1$, $B_2=2$ are clear. The recursive formula follows from the Algorithm presented in Section \[s4\] and the definition of downward-right path $\mathcal{P}=(p_j)_{j=1}^l$. Now we assume that $k\ge 3$. If $\overline{p_1, p_2}$ is horizontal or vertical, then Case I. (a) or (b) of the Algorithm holds (see also Figure \[figmove1\]). Thus we reduce the square $Q_1$ of size $k$ to a square $Q_2$ of size $k-1$. If $\overline{p_1, p_2}$ is diagonal, then Case III (a) or (b) holds (see also Figure \[figmove3\]). Thus we reduce the square $Q_1$ of size $k$ to a square $Q_2$ of size $k-2$. By definition, the number of associative quasitrivial nondecreasing operations having neutral elements defined on a square of size $k$ is $B_k$. Thus we get that $B_k=2\cdot B_{k-1}+2\cdot B_{k-2}$.
2. This follows from Lemma \[lcount1\] (b) and the fact that ’sum of the weights of weighted paths from $(1, k)$ to any point of $T_1$’ is exactly $\sum_{i=1}^k B_i$. Indeed, let $s\in\{1, \dots, k\}$ be fixed. Then $B_s$ is equal to the sum of the weights of weighted paths $\mathcal{P}$ that starts at $(1,k)$ and ends at $(a,b)$ where $b-a=s$.
[*Proof of Theorem \[tnumb\].*]{} We use a standard method of second-order linear recurrence equations for the formula of Lemma \[lnumb\] (a). Therefore, $$B_k=c_1\cdot(\alpha_1)^k+c_2(\alpha_2)^k,$$ where $\alpha_1,
\alpha_2$ ($\alpha_1< \alpha_2$) are the solutions of the equation $x^2-2x-2=0$. Thus, $\alpha_1=1-\sqrt{3}, \alpha_2=1+\sqrt{3}$. By the initial condition $B_1=1$ and $B_2=2$, we get that $c_1=-c_2=\frac{1}{2 \sqrt{3}}$. Thus, $$B_k=\frac{1}{2\cdot\sqrt{3}}\big((1+\sqrt{3})^k-(1-\sqrt{3})^k\big).$$
According to Lemma \[lnumb\] (b), $A_k$ can be calculated as $2\cdot\sum_{i=1}^k B_i-B_k$.
This provides that $$A_k=\frac{1}{6}\big((2+\sqrt{3})(1+\sqrt{3})^k+(2-\sqrt{3})(1-\sqrt{3})^k-4\big)$$
Here we present a list of the first 10 value of $A_k$: $A_1=1$, $A_2=4$, $A_3=12, A_4=34, A_5=94, A_6=258, A_7=706, A_8=1930,
A_9=5274, A_{10}=14410$.
By Theorem \[t2\], we get the similar results for the $n$-ary case.
1. The number of associative quasitrivial nondecreasing operations $F:L_k^n\to L_k$ ($k\in \mathbb{N}$) having neutral elements is $$\frac{1}{2\cdot\sqrt{3}}\big((1+\sqrt{3})^k-(1-\sqrt{3})^k\big),$$
2. The number of associative quasitrivial nondecreasing operations $F:L_k^n\to L_k$ ($k\in \mathbb{N}$) is $$\frac{1}{6}\big((2+\sqrt{3})(1+\sqrt{3})^k+(2-\sqrt{3})(1-\sqrt{3})^k-4\big).$$
\[binumb\] Let $C_k$ denote number of bisymmetric quasitrivial nondecreasing binary operations defined in $L_k$ and $D_k$ denote the number of bisymmetric quasitrivial nondecreasing binary operations having neutral elements. Then $$D_k=2^{k-1},$$ $$C_k=3\cdot 2^{k-1}-2.$$
1. By Lemma \[lba\] and Proposition \[prop:ane\], bisymmetric quasitrivial nondecreasing binary operations having neutral elements defined on $L_k$ are exactly the associative quasitrivial symmetric nondecreasing binary operations. Thus by Corollary \[ccard\], we get that $D_k=2^{k-1}$.
2. Same argument as in Lemma \[lnumb\](b) shows that $C_k=2\sum_{i=1}^k{D_i}-D_k$. Using this we get that $C_k=2\cdot
(2^{k}-1)-2^{k-1}=3\cdot 2^{k-1}-2.$
During the finalization of this paper the author have been informed that Miguel Couceiro, Jimmy Devillet and Jean-Luc Marichal found an alternative and independent approach for similar estimations in their upcoming paper [@Jimmy].
Open problems and further perspectives {#s7}
======================================
First we summarize the most important results of our paper. In this article we introduced a geometric interpretation of quasitrivial nondecreasing associative binary operations. We gave a characterization of such operations on finite chains using downward-right paths. Combining this with a reducibility argument we provided characterization for the $n$-ary analogue of the problem. As a remarkable application of our visualization method we gave characterization of bisymmetric quasitrivial nondecreasing binary operation on finite chains. As a byproduct of our argument we estimated the number of operations belonging to these classes.
These results initiate the following open problems.
1. Characterize the $n$-ary bisymmetric quasitrivial nondecreasing operations. If these operations are also associative, then we can apply reducibility to deduce a characterization for them. On the other hand if $n\ge 3$, then not all of such operations are associative as the following example shows. Let $F\colon X^n\to X$ ($n\ge 3$) be the projection on the $i^{th}$ coordinate where $i$ is neither 1 or $n$. Then it is easy to show that it is bisymmetric quasitrivial nondecreasing but not associative.
2. Find a visual characterization of associative idempotent nondecreasing operations. Quasitrivial operations are automatically idempotent. Since idempotent operations are essentially important in fuzzy logic, this problem has its own interest.
Acknowledgements {#acknowledgements .unnumbered}
================
The author would like to thank Jimmy Devillet and the anonymous referee for the example given in the first open problem in Section 7. This research is supported by the internal research project R-AGR-0500 of the University of Luxembourg. The author was partially supported by the Hungarian Scientific Research Fund (OTKA) K104178.
Appendix {#appendix .unnumbered}
========
This section is devoted to prove the analogue of Corollary \[cnem\]. As it was already mentioned in Remark \[rnem\], the proof is just a slight modification of the proof of [@KS Theorem 3.2]. The difference is based on the following easy lemma.
\[lASP\] Let $X$ be a chain and $F: X^n\to X$ be an associative monotone operation. Then $F$ is non-decreasing in the first and the last variable.
The argument for the first and for the last variable is similar. We just consider it for the first variable. From the definition of associativity it is clear that an associative operation $F\colon X^n
\to X$ is satisfies $$\label{ASP}
\begin{split}
&F(F(x_1,\dots, x_n),x_{n+1}, \dots, x_{2n-1})= \\ & F(x_1, F(x_{2}, \dots, x_{n+1}), x_{n+2}, \dots, x_{2n-1}).
\end{split}$$ for every $x_1, \dots, x_{2n-1}\in X$. Now let us fix $x_2, \dots, x_{2n-1}\in X$ and define $$h(x)=F(F(x,
x_2,\dots ,x_n),x_{n+1}, \dots, x_{2n-1}).$$ The operation $F$ is monotonic in the first variable thus it is clear that $h(x)$ is nondecreasing, since we apply $F$ twice when $x$ is in the first variable. Then using we get that $F$ must be nondecreasing in the first variable.
As it was also mentioned in [@KS] the following condition is an easy application of [@A Theorem 1.4] using the statement therein for $A_2=\emptyset$.
\[thmAkk\] Let $X$ be an arbitrary set. Suppose $F:X^n\to X$ be a quasitrivial associative operation. If $F$ is not derived from a binary operation $G$, then $n$ is odd and there exist $b_1 , b_2$ $(b_1\ne b_2)$ such that for any $a_1, \dots, a_n \in \{b_1, b_2\}$ $$\label{eqbk}
F(a_1, \dots, a_n)=b_i~~(i=\{1,2\}),$$ where $b_i$ occurs odd number of times.
\[paqm\] Let $X$ be a totally ordered set and let $F\colon X^n\to X$ be an associative, quasitrivial, monotone operation. Then $F$ is reducible.
According to Theorem \[thmAkk\], if $F$ is not reducible, then $n$ is odd. Hence $n\ge 3$ and there exist $b_1, b_2$ satisfying equation . Since $b_1\ne b_2$, we may assume that $b_1<b_2$ (the case $b_2<b_1$ can be handled similarly). By the assumption for $b_1$ and $b_2$ we have $$\label{eq:b1b2}
F(n \cdot b_1)=b_1, ~ F(b_2, (n-1) \cdot b_1)=b_2, ~ F(b_2,
(n-2)\cdot b_1, b_2)=b_1.$$ By Lemma \[lASP\], $F$ is nondecreasing in the first and the last variable. Thus we have $$F(n \cdot b_1) \le F(b_2, (n-1) \cdot
b_1)\le F(b_2, (n-2)\cdot b_1, b_2).$$ This implies $b_1=b_2$, a contradiction.
The following was proved as [@KS Corollary 4.9].
\[cormain\] Let $X$ be a nonempty chain and $n\ge 2$ be an integer. An associative, idempotent, monotone operation $F:X^n\to X$ is reducible if and only if $F$ is nondecreasing.
Using Proposition \[paqm\] and Corollary \[cormain\] we get the statement.
\[cnemh\] Let $n\ge 2 \in \mathbb{N}$ be given, $X$ be a nonempty chain and $F:X^n \to X$ be an associative quasitrivial operation. $$F \textrm{
is monotone } \Longleftrightarrow F \textrm{ is nondecreasing}.$$
[11]{}
N. L. Ackerman. A characterization of quasitrivial $n$-semigroups, to appear in [*Algebra Universalis*]{}.
S. Berg and T. Perlinger. Single-peaked compatible preference profiles: some combinatorial results. [*Social Choice and Welfare*]{} 27(1), pp. 89-102, 2006. D. Black. On the rationale of group decision-making. [*J Polit Economy*]{}, 56(1), pp. 23-34, 1948. D. Black. [*The theory of committees and elections*]{}. Kluwer Academic Publishers, Dordrecht, 1987.
E. Czogała and J. Drewniak. Associative monotonic operations in fuzzy set theory. , 12(3), pp. 249-269, 1984.
M. Couceiro and J.-L. Marichal. Representations and characterizations of polynomial operations on chains. [*J. of Mult.-Valued Logic & Soft Computing*]{}, 16, pp. 65-86, 2010.
M. Couceiro, J. Devillet, and J.-L. Marichal. Characterizations of idempotent discrete uni- norms. (2018), 60-72.
M. Couceiro, J. Devillet and J-L. Marichal, Quasitrivial semigroups: characterizations and enumerations. Preprint. https://arxiv.org/pdf/1709.09162.pdf
B. De Baets and R. Mesiar. Discrete triangular norms. in [*Topological and Algebraic Structures in Fuzzy Sets, A Handbook of Recent Developments in the Mathematics of Fuzzy Sets*]{}, Trends in Logic, eds. S. Rodabaugh and E. P. Klement (Kluwer Academic Publishers), 20, pp. 389-400, 2003.
J. Devillet, G. Kiss and J.-L. Marichal. Characterizations of quasitrivial symmetric nondecreasing associative operations, submitted to [*Semigroup Forum*]{}.
W. Dörnte. Untersuchengen über einen verallgemeinerten Gruppenbegriff. 29, pp. 1-19, 1928.
W. A. Dudek and V. V. Mukhin. On topological $n$-ary semigroups. , 3, pp. 373-88, 1996.
W. A. Dudek and V. V. Mukhin. On $n$-ary semigroups with adjoint neutral element. , 14, pp. 163-168, 2006.
J. Fodor. Smooth associative operations on finite ordinal scales. [*IEEE Trans. Fuzzy Systems*]{}, 8, pp. 791-795, 2000.
J. Fodor An extension of Fung-Fu’s theorem. , 4(3), pp. 235-243, 1996. E. Foundas. Some results of Black’s permutations. [*Journal or Discrete Mathematical Sciences and Cryptography*]{}, 4(1), pp. 47-55, 2001.
G. Kiss and G. Somlai. A characterization of n-associative, monotone, idempotent functions on an interval that have neutral elements, , 96(3), pp. 438-451, 2018.
G. Kiss and G. Somlai. Associative idempotent nondecreasing functions are reducible, accepted at [*Semigroup Forum*]{}, https://arxiv.org/pdf/1707.04341.pdf.
G. Li, H.-W. Liu and J. Fodor. On weakly smooth uninorms on finite chain. [*Int. J. Intelligent Systems*]{}, 30, pp. 421-440, 2015.
J. Martín, G. Mayor and J. Torrens. On locally internal monotonic operations. , 137(1), pp. 27-42, 2003.
M. Mas, G. Mayor and J. Torrens. t-operators and uninorms on a finite totally ordered set. [*Int. J. Intelligent Systems*]{}, 14, pp. 909-922, 1999. M. Mas, M. Monserrat and J. Torrens. On bisymmetric operators on a finite chain. [*IEEE Trans. Fuzzy Systems*]{}, 11, pp. 647-651, 2003. M. Mas, M. Monserrat and J. Torrens. On left and right uninorms on a finite chain. [*Fuzzy Sets and Systems*]{}, 146, pp. 3-17, 2004. M. Mas, M. Monserrat and J. Torrens. Smooth t-subnorms on finite scales. [*Fuzzy Sets and Systems*]{}, 167, pp. 82-91, 2011. G. Mayor and J. Torrens. Triangular norms in discrete settings. in [*Logical, Algebraic, Analytic, and Probabilistic Aspects of Triangular Norms*]{}, eds. E. P. Klement and R. Mesiar (Elsevier, Amsterdam), pp. 189-230, 2005. G. Mayor, J. Su ner and J. Torrens. Copula-like operations on finite settings. [*IEEE Trans. Fuzzy Systems*]{}, 13, pp. 468-477, 2005.
E. L. Post. Polyadic groups, , [**48**]{}, pp. 208-350, 1940.
D. Ruiz-Aguilera and J. Torrens. A characterization of discrete uninorms having smooth underlying operators. [*Fuzzy Sets and Systems*]{}, 268, pp. 44-58, 2015.
D. Ruiz-Aguilera, J. Torrens, B. De Baets and J. Fodor. Idempotent uninorms on finite ordinal scales. [*Int. J. of Uncertainty, Fuzziness and Knowledge-Based Systems*]{} [**17**]{} (1), pp. 1-14, 2009.
D. Ruiz-Aguilera, J. Torrens, B. De Baets and J. Fodor. Some remarks on the characterization of idempotent uninorms. in: E. Hüllermeier, R. Kruse, F. Hoffmann (Eds.), [*Computational Intelligence for Knowledge-Based Systems Design, Proc. 13th IPMU 2010 Conference, LNAI, vol. 6178*]{}, Springer-Verlag, Berlin, Heidelberg, 2010, pp. 425-434. W. Sander Associative aggregation operators. In:[*Aggregation operators. New trends and applications*]{}, pp. 124-158. Stud. Fuzziness Soft Comput. Vol. 97. Physica-Verlag, Heidelberg, Germany, 2002. Y. Su and H.-W. Liu. Discrete aggregation operators with annihilator. [*Fuzzy Sets and Systems*]{}, 308, pp. 72-84, 2017.
Y. Su, H.-W. Liu, and W. Pedrycz. On the discrete bisymmetry. IEEE Trans. Fuzzy Systems. To appear. DOI:10.1109/TFUZZ.2016.2637376
|
---
abstract: 'We present resistivity and magnetization measurements on proton-irradiated crystals demonstrating that the superconducting state in the doped topological superconductor Nb$_x$Bi$_2$Se$_3$ (x = 0.25) is surprisingly robust against disorder-induced electron scattering. The superconducting transition temperature $T_c$ decreases without indication of saturation with increasing defect concentration, and the corresponding scattering rates far surpass expectations based on conventional theory. The low-temperature variation of the London penetration depth $\Delta\lambda(T)$ follows a power law ($\Delta\lambda(T)\sim T^2$) indicating the presence of symmetry-protected point nodes. Our results are consistent with the proposed robust nematic $E_u$ pairing state in this material.'
author:
- 'M. P. Smylie'
- 'K. Willa'
- 'H. Claus'
- 'A. Snezhko'
- 'I. Martin'
- 'W.-K. Kwok'
- 'Y. Qiu'
- 'Y. S. Hor'
- 'E. Bokari'
- 'P. Niraula'
- 'A. Kayani'
- 'V. Mishra'
- 'U. Welp'
title: 'Robust odd-parity superconductivity in the doped topological insulator Nb$_x$Bi$_2$Se$_3$'
---
Topological superconductors have attracted considerable interest [@Qi-Zhang-RevModPhys-TI-SC-review; @Ando-Fu-AnnualReview-TCI-and-TSC-review; @Sato-Ando-arXiv-TSC-review; @Sasaki-Mizushima-PhysicaC-SC-doped-TIs; @Fu-NatPhys-NewsAndViews-Bi2Se3-SC; @Matano-Ando-NatPhys-Knight-Shift-CBS] since they host gapless surface quasi-particle excitations in the form of Majorana fermions. The non-Abelian braiding properties of Majorana fermions constitute the basis for novel approaches to fault-tolerant quantum computing [@Wilczek-NatPhys-Majoranas; @Beenakker-AnnRevCMatt-Majoranas-in-TSCs]. The synthesis of topological superconductors is being pursued along two lines: proximity induced at the interface between conventional superconductors and certain semiconductors with large spin-orbit coupling [@Beenakker-AnnRevCMatt-Majoranas-in-TSCs], or as bulk material obtained by doping topological insulators, for instance Sn$_{1-x}$In$_x$Te [@Erickson-Geballe-Fisher-PRB-first-modern-TIT-study; @Novak-Ando-PRB-Unusual-gap-in-TIT; @Zhong-Gu-PRB-optimize-Tc-in-TIT] and M$_x$Bi$_2$Se$_3$ (M=Cu, Sr, Nb) [@Hor-Cava-CBS-discovery; @Liu-JACS-SBS-discovery; @Qiu-Hor-arXiv-NBS-discovery].
The emergence of topological superconductivity is determined by the symmetries and dimensionality of the material. In centro-symmetric and time-reversal invariant superconductors with complete gap [@Fu-Berg-PRL-TSC-and-CBS-model; @Chiu-Schnyder-RevModPhys-Classification-of-topological-materials] or with nodal gap [@Schnyder-Brydon-JPhysConMat-TopNodalSC; @Chiu-Schnyder-RevModPhys-Classification-of-topological-materials; @Sato-Fujimoto-PRL-Majoranas-in-nodal-TSCs], the superconducting state will have non-trivial topological characteristics if superconducting pairing has odd-parity, $\Delta$(-**k**) = -$\Delta$(**k**), and if the Fermi surface contains an odd number of time-reversal invariant momenta, **k** = -**k** + **G** with **G** a reciprocal lattice vector. For weak spin-orbit coupling (i.e., spin is a good quantum number), odd-parity pairing corresponds to spin-triplet pairing. Odd-parity pairing has been observed in the B-phase of superfluid $^3$He [@Vollhardt-He3-superfluidity] and is thought to be realized in several strongly correlated electron systems such as UPt$_3$ or UBe$_{13}$ [@Joynt-Taillefer-RevModPhys-UPt3; @Gross-Hirschfeld-ZPhysB-spin-triplet-and-lambda] as well as Sr$_2$RuO$_4$ [@Mackenzie-Maeno-RevModPhys-spin-triplet-in-SRO]. In contrast, conventional $s$-wave superconductors are not topological and do not support the Majorana surface mode.
An important unsettled question regarding the realization of topological superconductivity relates to its robustness against disorder in the material. The effect of electron scattering due to impurities and defects on the superconducting state crucially depends on the structure of the superconducting gap. Whereas an isotropic fully gapped $s$-wave state is robust against potential scattering due to nonmagnetic impurities [@Anderson-JPhysChemSol-Andersons-Theory; @Abrikosov-Gorkov-theory], unconventional superconductors are rather sensitive to disorder [@Balatsky-RevModPhys-Nodal-SC-weak-against-disorder]; therefore, one may have expected that topological superconductivity could only be achieved in extremely clean samples. However, recent theoretical considerations [@Nagai-Ota-PRB-TSC-may-be-robust; @Nagai-PRB-TSC-is-robust-in-CBS-theory; @Michaeli-Fu-PRL-Odd-parity-robustness] show odd-parity topological superconductivity with strong spin-orbit coupling may in fact be robust against disorder. Here, we present a study of the evolution of $T_c$, of the low-temperature London penetration depth $\lambda$, and of the resistivity of the candidate topological superconductor Nb$_x$Bi$_2$Se$_3$ with increasing disorder as introduced by proton irradiation. In the covered temperature range ($T/T_c \geq$ 0.12) the temperature variation of $\lambda(T)$ of the pristine samples as well as of all irradiated crystals is quadratic, $\Delta\lambda(T)\sim T^2$, indicative of symmetry-protected point nodes. $T_c$ is suppressed with increasing proton dose in all crystals, with no trend towards saturation at high doses. Concurrently, the residual resistivity, $\rho_0$, increases strongly. Within the conventional Abrikosov-Gor‘kov theory [@Abrikosov-Gorkov-theory], such increase of $\rho_0$ would induce two orders of magnitude stronger suppression of $T_c$, which suggests that the superconducting state is indeed robust against impurity scattering, contrary to more conventional nodal superconductors.
High-quality crystals of Nb$_x$Bi$_2$Se$_3$ with high superconducting volume fractions were grown by the same method used in Ref. , and show high superconducting volume fractions. Nb$_x$Bi$_2$Se$_3$ has the same trigonal space group $R\bar{3}m$ as the parent compound Bi$_2$Se$_3$, with slightly expanded $c$ axis to accommodate the Nb ion interstitially between adjacent Bi$_2$Se$_3$ quintuple layers (see Fig. 1). All samples were repeatedly irradiated along the $c$ axis with 5 MeV protons using the tandem Van de Graaff accelerator at Western Michigan University. The proton beam was passed through a gold foil to ensure homogeneous irradiation, and the sample was cooled to $\sim$-10C during irradiation. TRIM simulations [@SRIM-reference-book] for our irradiation geometry show that defect generation is uniform through the thickness of the samples. Irradiation with MeV-protons creates a distribution of defects including point defects in the form of interstitial-vacancy pairs as well as collision cascades and clusters [@Kirk-Yan-defect-generation; @LeiFang-Welp-Kwok-PRB-defect-generation-in-Ba122P; @Civale-PRL-proton-damage-in-YBCO].
![ a) The crystal structure of Nb$_x$Bi$_2$Se$_3$ derived from an ABC stacking of hexagonal sheets of Bi (green) and Se (red) atoms. The Nb ions (blue) sit in the van der Waals gap between quintuple layers of Bi$_2$Se$_3$ \[21\]. b) View down the $c$ axis, the $y$ axis is chosen to lie in the mirror plane. c) and d) Schematic presentations of the effect of defect scattering on an $s$-wave gap with symmetry-protected point nodes (d) and a gap with deep minima (c). Dark and light blue represent the gap amplitude before and after introduction of scattering, respectively. []{data-label="figStructure"}](Figure1-structure-fullsize.eps){width="1\columnwidth"}
We performed $ac$-susceptibility and London penetration depth measurements using the tunnel-diode oscillator (TDO) technique [@Prozorov-Giannetta-SST-TDO-reference] employing a custom-built TDO operating at 14.5 MHz. Here, the change in the resonator frequency $\Delta f(T)$ is proportional to the change of the London penetration depth $\Delta\lambda(T)$ such that $\Delta f(T)/\Delta f_0 = G\Delta\lambda(T) / \lambda_0$, where G is a calibration factor, $\Delta f_0$ is the total frequency change occuring between the lowest temperature and $T_c$, and $\lambda_0 = \lambda(T = 0)$. All crystals measured here showed $T_c \approx$ 3.4 K in the pristine state with minimal sample-to-sample variation.
![ Low-temperature resistivity of a single crystal of Nb$_x$Bi$_2$Se$_3$ showing suppression of $T_c$ and increase in the residual resistivity $\rho_0$ following multiple irradiations. The inset shows the resistivity up to room temperature with little change in curvature following repeated doses. []{data-label="figTransport"}](Figure2-CombinedTransportFig.eps){width="1\columnwidth"}
In Fig. 2, the temperature dependence of the resistivity for multiple irradiation levels measured up to room temperature is shown in the inset. The irradiation does not significantly affect the curvature of $\rho$ vs $T$, but instead offsets the curves, consistent with an increase in residual resistivity $\rho_0$. As the cumulative proton dose is increased, the transition temperature is clearly suppressed and the residual resistivity $\rho_0$, taken as the effectively temperature-independent value of the resistivity just above the transition onset, increases strongly. For all doses, the transitions remain reasonably sharp, indicating single-phase behavior throughout. The temperature dependent normalized magnetic susceptibility as determined from the TDO frequency shift of one sample is shown in Fig. 3 for multiple irradiation doses. The superconducting transition temperature $T_c$ is clearly suppressed with each dose. Nevertheless, the transitions remain sharp even at the highest cumulative irradiation dose. No secondary transitions from possible superconducting contaminants Nb or NbSe$_2$ were observed at higher temperatures.
![ Normalized magnetic susceptibility of a single crystal of Nb$_x$Bi$_2$Se$_3$ as a function of temperature for various values of cumulative p-irradiation dose. The transition temperature $T_c$ is clearly suppressed with each dose given in p/cm$^2$. []{data-label="figTDOdose"}](Figure3-renormalized-TDO-X-data.eps){width="1\columnwidth"}
Fig. 4 shows the low-temperature variation of the penetration depth $\Delta\lambda(T)$ of a Nb$_x$Bi$_2$Se$_3$ crystal irradiated to several cumulative doses versus reduced temperature squared, $(T/T_c)^2$. These data reveal that in the measured temperature range the penetration depth has a quadratic temperature dependence, $\Delta\lambda(T) \sim T^2$ for all doses of irradiation. The low-temperature variation of the London penetration depth is determined by the distribution and scattering of thermally activated quasi-particles on the Fermi surface. For an isotropic $s$-wave superconductor, $\Delta\lambda(T)$ at sufficiently low temperatures follows an exponential variation, $\Delta\lambda(T)/\lambda_0 \approx \sqrt{\pi\Delta_0 / 2T}$exp$(-\Delta_0 / T)$ where $\Delta_0$ is the zero temperature value of the energy gap. Nodes in the gap, however, will induce enhanced thermal excitation of low-lying quasi-particles, resulting in a power-law variation, $\Delta\lambda \sim T^n$, with the exponent depending on the type of node and on electron scattering. In particular, a quadratic temperature dependence is expected in a clean material with linear quasiparticle dispersion around point nodes in the superconducting gap [@Gross-Hirschfeld-ZPhysB-spin-triplet-and-lambda]. The observation [@Lawson-LuLi-dHvA-on-NBS-Fermi-surfaces] of quantum oscillations in Nb$_x$Bi$_2$Se$_3$ crystals similar to those used here shows that the unirradiated samples are fairly clean. Hence the quadratic temperature dependence of $\lambda$ is indicative of point nodes [@Smylie-TDO-on-NBS].
![ Low-temperature variation of the London penetration depth $\Delta\lambda(T)$ in a single crystal of Nb$_x$Bi$_2$Se$_3$ for multiple values of cumulative irradiation dose vs reduced temperature squared $(T/T_c)^2$. The linear fits (red, black lines) indicate quadratic behavior. As the dose increases, the temperature dependence remains quadratic, indicative of point nodes in the superconducting gap. Data are off-set vertically for clarity of presentation. []{data-label="figTDOlowT"}](Figure4-low-T-TDO-comparisons-Tsquared.eps){width="1\columnwidth"}
As shown in Fig. 4, the temperature dependence of $\lambda$ remains quadratic with increasing disorder. This finding is consistent with a theoretical analysis of the effect of impurity scattering on the gap structure of $p$-wave superconductors [@Gross-Hirschfeld-ZPhysB-spin-triplet-and-lambda]. For the axial $p$-wave gap (two point nodes) impurity scattering rates below a critical value do not affect the $T^2$-dependence. In contrast, the linear temperature dependence of $\lambda$ expected for the polar $p$-wave gap (equatorial line node) is expected to be strongly affected by impurity scattering.
A $T^2$-dependence of $\lambda$ could also arise in an anisotropic $s$-wave gap with deep gap minima such that the minimum gap is significantly smaller than the measurement temperature. However, potential scattering will make an anisotropic $s$-wave gap more isotropic implying an increase in the minimum gap value with increasing scattering [@Borkowski-Hirschfeld-PRB-Hirschfeld-scattering; @Fehrenbacher-Norman-PRB-scattering-in-anisotropic-materials] (see Fig. 1c) thereby altering the low-temperature variation of the penetration depth. In contrast, as symmetry-imposed nodes in the gap cannot be removed by electron scattering, the gap amplitude decreases rapidly with increasing scattering rate while the overall gap structure remains unchanged as indicated in Fig. 1d. Therefore, the persistent $T^2$-variation in the data in Fig. 4 rules out an anisotropic $s$-wave gap with deep minima, and is further support for an unconventional superconducting gap in Nb$_x$Bi$_2$Se$_3$.
Theoretical analysis of Bi$_2$Se$_3$-based superconductors [@Fu-Berg-PRL-TSC-and-CBS-model; @Fu-PRB-explaining-Knight-shift-in-CBS; @Venderbos-Kozii-Fu-PRB-Two-component-order-parameters-in-Bi2Se3s] shows that strong spin-orbit coupling can induce unconventional pairing symmetries in time-reversal symmetric systems, even if the pairing is mediated by conventional electron-phonon coupling. In particular, in a two-orbital model with short-range pairing interactions four pairing states that transform according to the four irreducible representations of the $D_{3d}$ crystal point group of Nb$_x$Bi$_2$Se$_3$ were identified. One is the fully symmetric conventional $s$-wave state, whereas the other three have odd-parity pairing. Among the latter, the state that corresponds to the two-dimensional representation $E_u$ has attracted considerable attention as it allows for a nematic state that would account for the surprising two-fold symmetry that emerges in several quantities below $T_c$ [@Fu-PRB-explaining-Knight-shift-in-CBS], i.e., the Knight shift and specific heat in Cu$_x$Bi$_2$Se$_3$ [@Matano-Ando-NatPhys-Knight-Shift-CBS; @Yonezawa-Ando-NatPhys-Rotational-breaking-via-calorimetry-in-CBS], magneto-transport [@Pan-deVisser-SciRep-Rotational-breaking-via-transport-in-SBS; @Du-Gu-HHWen-arXiv-Corbino-geometry-transport-rotational-breaking-in-SBS] in Sr$_x$Bi$_2$Se$_3$, and magnetic torque [@Asaba-Lawson-LuLi-PRX-Rotational-breaking-via-torque-magnetometry-in-NBS] in Nb$_x$Bi$_2$Se$_3$. The gap structure of the $E_u$-state depends on the orientation of the nematic director **n** (see Fig. 1); for **n** along an $x$-axis (perpendicular to the mirror plane) the $\Delta_{4x}$ state is realized with two symmetry-protected point nodes along **k**$_y$, whereas for **n** along a $y$-axis (parallel to the mirror plane) the $\Delta_{4y}$ state emerges with gap minima along **k**$_x$ [@Fu-PRB-explaining-Knight-shift-in-CBS]. A detailed study analogous to [@Borkowski-Hirschfeld-PRB-Hirschfeld-scattering; @Fehrenbacher-Norman-PRB-scattering-in-anisotropic-materials] of the response of the gap minima in the $\Delta_{4y}$ state to electron scattering has not been discussed yet in the literature to our knowledge. However, Fig. 4 shows that at reduced temperatures as low as 0.12 there is no indication of deviation from the $T^2$-dependence of $\lambda$ which would imply a very large ratio of maximum and minimum gap in a possible $\Delta_{4y}$ state of more than 10. Thus, while it is difficult to rule out $\Delta_{4y}$ completely, our results point towards the $\Delta_{4x}$ state as the superconducting ground state of Nb$_x$Bi$_2$Se$_3$.
Fig. 5 summarizes the evolution of $T_c$ with increasing proton irradiation dose as determined from resistivity and $ac$-susceptibility measurements. For the transport measurement samples, the increase of the residual resistivity, $\Delta\rho_0$, is directly obtained (see Fig. 2), whereas for the TDO samples, the $\Delta\rho_0$ values corresponding to a given p-dose are inferred from a fit of $\Delta\rho_0$ versus proton dose data obtained from the transport samples. Although there is some scatter in the data, the $T_c$ values of all Nb$_x$Bi$_2$Se$_3$ samples follow a smooth trend towards $T_c$ = 0 with increasing dose without any indication of saturation. The lack of saturation reinforces a model of odd-parity superconductivity in Nb$_x$Bi$_2$Se$_3$.
![ Evolution of $T_c$ with proton irradiation for several crystals of Nb$_x$Bi$_2$Se$_3$, as measured via transport (diamonds) and magnetic susceptibility (circles). The mustard diamonds and red circles are derived from the samples shown in Fig. 2 and 3, respectively. The inset shows $T_c/T_{c0}$ versus proton dose, whereas the main panel displays the same data as function of increase in residual resistivity, $\Delta\rho_0$ (lower x-axis) and normalized scattering rate, $g^\lambda$ (top x-axis). For comparison, the $T_c/T_{c0}$ data versus $g^\lambda$ of p-irradiated BaFe$_2$(As$_{1-x}$P$_x$)$_2$ (dashed line), He$^+$-irradiated YBCO (red line), and the Abrikosov-Gor‘kov prediction for $T_c/T_{c0}$ versus $g$ on a 100x expanded scale (green line) are included. []{data-label="figG"}](Figure5-TcTc0-vs-delta-rho0.eps){width="1\columnwidth"}
Further analysis of the data in Fig. 5 is based on the Abrikosov-Gor‘kov (AG) theory of pair-breaking scattering [@Abrikosov-Gorkov-theory; @Openov-PRB-Disorder-scattering-beyond-AG-theory]. For magnetic scattering in isotropic $s$-wave superconductors, or for potential scattering in superconductors with an anisotropic gap, the suppression of $T_c$ is given as ln $(T_c/T_{c0}) = \chi[\Psi(1/2) - \Psi(gT_{c0} / 2T_c)]$. Here, $\Psi$ is the digamma function, $\chi$ is a measure of the gap anisotropy, and $g = \hbar / 2\pi k_B T_{c0}\tau$ is the normalized scattering rate with $\tau$ the pairbreaking scattering time. For nonmagnetic defects, $\tau$ corresponds to the potential scattering time and for magnetic impurities to half the spin-flip scattering time. Since for odd-parity pairing, the Fermi surface average of $\Delta$(**k**) is zero, $\chi=1$, and $T_c$ is suppressed to zero at a critical value $g_c \approx$ 0.28. Linking the scattering rate to measurable quantities such as the increase in resistivity requires detailed information on the electronic band structure, transport and particle lifetimes, and the scattering potential. For instance, it has previously been observed that in multi-band superconductors the suppression of $T_c$ with disorder depends sensitively on the balance between inter and intra-band scattering rates [@Prozorov-Hirschfeld-PRX-Multiband-scattering]. As many of the microscopic parameters of Nb$_x$Bi$_2$Se$_3$ are currently unavailable we relate the measured increase in resistivity to the scattering rate using a simple single-band Drude model, $\Delta\rho_0 = m^* / (ne^2\tau_i)$ with $m^*$ and $n$ the effective mass and concentration of carriers, respectively, and 1/$\tau_i$ the scattering rate due to the irradiation-induced defects. Since the enhancement of the residual resistivity is large, we neglect the contribution from pre-existing defects in the total scattering rate. The parameter $m^* /ne^2$ can be estimated from values of the penetration depth, $\lambda^2 = m^* / \mu_0 ne^2$. We thus obtain the normalized scattering rate $g$ in terms of the London penetration depth as $g^\lambda = \hbar \Delta\rho_0 / 2 \pi k_B T_{c0} \mu_0 \lambda^2$, yielding $g^\lambda \approx 0.172~\Delta\rho_0/T_{c0}$, where $\Delta\rho_0$ is expressed in $\mu\Omega\cdot$cm and with a zero-temperature penetration depth of $\sim$ 240 nm [@Smylie-TDO-on-NBS].
The data in Fig. 5 show that the increase in resistivity required to induce a given reduction of $T_c$ is enhanced over predictions based on the AG theory by a very large margin. In AG-theory, each scattering event giving rise to enhanced resistivity is also pair-breaking. This implies that in Nb$_x$Bi$_2$Se$_3$ the majority of scattering events do not contribute to pair-breaking. Also included in Fig. 5 are the $T_c$/T$_{c0}$ vs $g^\lambda$ data on proton-irradiated BaFe$_2$(As$_{1-x}$P$_x$)$_2$ [@Smylie-PRB-TDO-on-Ba122P] and on He$^+$-irradiated YBCO [@Lang-YBCO-Tc-suppression]. These materials have sign-changing order parameters–$s_\pm$-gap symmetry with additional accidental line nodes and $d$-wave symmetry, respectively. Therefore, nonmagnetic potential scattering induces a rapid suppression of $T_c$. Similar behavior would be expected for Nb$_x$Bi$_2$Se$_3$ due to odd-parity pairing. Nevertheless, its $T_c$-suppression is in comparison remarkably weak. The reason for these surprising results lies in the particular electronic structure of Nb$_x$Bi$_2$Se$_3$, which has very strong spin-orbit coupling. In the relativistic limit of vanishing Dirac mass [@Nagai-Ota-PRB-TSC-may-be-robust; @Nagai-PRB-TSC-is-robust-in-CBS-theory; @Michaeli-Fu-PRL-Odd-parity-robustness], the emergent chiral symmetry effectively protects against impurity-induced scattering between two pseudo-chiral bands [@Nagai-PRB-TSC-is-robust-in-CBS-theory], if the scattering is non-magnetic and does not discriminate between the pseudo-chiral sectors. This effectively puts these odd-parity superconductors in the same category with respect to potential impurity scattering as $s$-wave superconductors, protected by the Anderson theorem [@Anderson-JPhysChemSol-Andersons-Theory]. Finite suppression of $T_c$ can result either from magnetic impurities, from disorder that couples differently to the bands, or from combined effect of a finite Dirac mass (that breaks chiral symmetry) and potential scattering. While irradiation is unlikely to introduce magnetic scattering, both latter mechanisms are most likely present. Further studies would be necessary to determine their relative importance.
In summary, 5-MeV proton irradiation has been shown to increase electron scattering in the candidate topological superconductor Nb$_x$Bi$_2$Se$_3$. A substantial increase in electron scattering is required to suppress $T_c$, far larger than anticipated via conventional theory, and the effect does not saturate even at large doses. The low-temperature variation of the London penetration depth $\Delta\lambda(T)$ remains quadratic in the pristine and disordered states. Together, these results suggest the presence of symmetry-protected point nodes in Nb$_x$Bi$_2$Se$_3$, further supporting the proposed nematic $E_u$ pairing state. Owing to the strong spin-orbit locking, these results are the first demonstration of an unconventional superconductor that is robust against nonmagnetic disorder suggesting that topological superconductivity can be realized in rather dirty materials.
TDO and magnetization measurements were supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division. MPS thanks ND Energy for supporting his research and professional development through the ND Energy Postdoctoral Fellowship Program. KW acknowledges support through an Early Postdoc Mobility Fellowship of the Swiss National Science Foundation. YSH acknowledges support from the National Science Foundation grant number DMR-1255607. VM was supported by the Laboratory Directed Research and Development Program of Oak Ridge National Laboratory, managed by UT-Battelle, LLC, for the U.S. Department of Energy.
[99]{} X.-L. Qi, S.-C. Zhang, Rev. Mod. Phys. **83**, 1057 (2011). Y. Ando, L. Fu, Annu. Rev. Condens. Matter Phys. **6**, 361 (2015). M. Sato, Y. Ando, arXiv:1608.03395v2. S. Sasaki, T. Mizushima, Physica C **514**, 206 (2015). L. Fu, Nat. Phys. **12**, 822 (2016). K. Matano, M. Kriener, K. Segawa, Y. Ando, and G. Zheng, Nat. Phys. **12**, 852 (2016). F. Wilczek, Nat. Phys. **5**, 614 (2009). C. W. J. Beenakker, Annu. Rev. Condens. Matter Phys. **4**, 113 (2013). A. S. Erickson, J. H. Chu, M. F. Toney, T. H. Geballe, I. R. Fisher, Phys. Rev. B **79**, 024520 (2009). M. Novak, S. Sasaki, M. Kriener, K. Segawa, Y. Ando, Phys. Rev. B **88**, 140502 (2013). R. D. Zhong, J. A. Schneeloch, X. Y. Shi, Z. J. Xu, C. Zhang, J. M. Tranquada, Q. Li, G. D. Gu, Phys. Rev. B **88**, 020505 (2013). Y. S. Hor, A. J. Williams, J. G. Checkelsky, P. Roushan, J. Seo, Q. Xu, H. W. Zandbergen, A. Yazdani, N. P. Ong, and R. J. Cava, Phys. Rev. Lett. **104**, 057001 (2010). Z. Liu, X. Yao, J. Shao, M. Zuo, L. Pi, S. Tan, C. Zhang, and Y. Zhang, J. Am. Chem. Soc. **137**, 10512 (2015). Y. Qiu, K. N. Sanders, J. Dai, J. E. Medvedeva, W. Wu, P. Ghaemi, T. Vojta, and Y. S. Hor, arXiv:1512.03519. L. Fu and E. Berg, Phys. Rev. Lett. **105**, 097001 (2010). C. K. Chiu, J. C. Y. Teo, A. P. Schnyder, S. Ryu, Rev. Mod. Phys. **88**, 035005 (2016). A. P. Schnyder and P. M. R. Brydon, J. Phys.: Condens. Matter **27**, 243201 (2015). M. Sato, S. Fujimoto, Phys. Rev. Lett. **105**, 217001 (2010). D. Vollhardt, P. W[ö]{}lfle, The Superfluid Phases of Helium 3, Dover Publications, 2013. R. Joynt and L. Taillefer, Rev. Mod. Phys. **74**, 235 (2002). F. Gross, B. S. Chandrasekhar, D. Einzel, K. Andres, P. J. Hirschfeld, H. R. Ott, J. Beuers, Z. Fisk, and J. L. Smith, Z. Phys. B: Condens. Matter **64**, 175 (1986). A. P. Mackenzie and Y. Maeno, Rev. Mod. Phys. **75**, 657 (2003). P. W. Anderson, J. Phys. Chem. Sol. **11**, 26 (1959). A. A. Abrikosov and L. P. Gor‘kov, Zh. Eksp. Teor. Fiz. **39**, 1781 (1960). \[Sov. Phys. JETP 12, 1243 (1961)\]. A. V. Balatsky, I. Vekther, and J.-X. Zhu, Rev. Mod. Phys. **78**, 373 (2006). Y. Nagai, Y. Ota, M. Machida, Phys. Rev. B **89**, 214506 (2014). Y. Nagai, Phys. Rev. B **91**, 060502(R) (2015). K. Michaeli, L. Fu, Phys. Rev. Lett. **109**, 187003 (2012). *SRIM, The Stopping and Range of Ions in Matter*, James F. Ziegler, Jochen P. Biersack, and Matthias D. Ziegler. M. A. Kirk, Y. Yan, Micron **30**, 507 (1999). L. Fang, Y. Jia, J. A. Schlueter, A. Kayani, Z. L. Xiao, H. Claus, U. Welp, A. E. Koshelev, G. W. Crabtree, and W.-K. Kwok, Phys. Rev. B **84**, 140504(R) (2011). L. Civale, A. D. Marwick, M. W. McElfresh, T. K. Worthington, A. P. Malozemoff, F. H. Holtzberg, J. R. Thompson, and M. A. Kirk, Phys. Rev. Lett. **65**, 1164 (1990). R. Prozorov and R. W. Giannetta, Supercond. Sci. Technol. **19**, R41 (2006). B. J. Lawson, P. Corbae, G. Li, F. Yu, T. Asaba, C. Tinsman, Y. Qiu, J. E. Medvedeva, Y. S. Hor, L. Li, Phys. Rev. B **94**, 041114 (2016). M. P. Smylie, H. Claus, U. Welp, W.-K. Kwok, Y. Qiu, Y. S. Hor, and A. Snezhko, Phys. Rev. B **94**, 180510(R) (2016). L. S. Borkowski, P. J. Hirschfeld, Phys. Rev. B **49**, 15404 (1994). R. Fehrenbacher, M.R. Norman, Phys. Rev. B **50**, 3495 (1994). L. Fu, Phys. Rev. B **90**, 100509(R) (2014). J. W. F. Venderbos, V. Kozii, and L. Fu, Phys. Rev. B **94**, 180504(R) (2016). S. Yonezawa, K. Tajiri, S. Nakata, Y. Nagai, Z. Wang, K. Segawa, Y. Ando, and Y. Maeno, Nat. Phys. **13**, 123 (2017). Y. Pan, A. M. Nikitin, G. K. Araizi, Y. K. Huang, Y. Matsushita, T. Naka, and A. de Visser, Sci. Rep. **6**, 28632 (2016). G. Du, Y. Li, J. A. Schneeloch, R. D. Zhong, G. D. Gu, H. Yang, H.-H. Wen, arxiv:1607.06357. T. Asaba, B. J. Lawson, C. Tinsman, L. Chen, P. Corbae, G. Li, Y. Qiu, Y. S. Hor, L. Fu, and L. Li, Phys. Rev. X **7**, 011009 (2017). L. A. Openov, Phys. Rev. B **58**, 9468 (1998). R. Prozorov, M. Konczykowski, M. A. Tanatar, A. Thaler, S. L. Bud‘ko, P. C. Canfield, V. Mishra, and P. J. Hirschfeld, Phys. Rev. X **4**, 041032 (2014). M. P. Smylie, M. Leroux, V. Mishra, L. Fang, K. M. Taddei, O. Chmaissem, H. Claus, A. Kayani, A. Snezhko, U. Welp, and W.-K. Kwok, Phys. Rev. B **93**, 115119 (2016). W. Lang, J. D. Pedarnik, in *Nanoscience and Engineering in Superconductivity*, edit. V. Moshchalkov, R. W[ö]{}rdenweber, W. Lang (Springer, Berlin, 2010) p. 81. There is a wide sample-to-sample variation in $T_c$-suppression data of YBCO; for an extensive list of references, see Ref.
|
---
abstract: 'We study the out-of-equilibrium dynamics of a BCS superconductor in the presence of a periodic drive with a frequency $\omega_d$ larger than the equilibrium superconducting gap and an external bath providing dissipation. Similar to the dissipationless case, a subset of quasiparticles, resonant with the drive, synchronizes and produces interesting non-linear phenomena. For small dissipation Rabi-Higgs oscillations \[Collado et al. Phys. Rev. B [**98**]{}, 214519 (2018)\] can be observed as a transient effect. At long-times in contrast, dissipation leads to a steady state with a population imbalance that increases quadratically for small drives and saturates for large drives as in non-linear quantum optics. We show also that second harmonic generation is allowed for a drive which acts on the BCS coupling constant. We also compute the intensity of time- and angle-resolved photo-emission spectroscopy (tr-ARPES) and time-resolved tunneling spectra. The excited quasiparticle population appears as a decrease (increase) in the photo-emission intensity at energy $-\omega_d/2$ ($+\omega_d/2$) measured from the chemical potential. The tunneling intensity shows a time-dependent depression at $\pm\omega_d/2$ due to the lacking of spectral density and population unbalance causing Pauli blockade. We predict that at short times, compared to the energy relaxation time, both experiments will show oscillations with the Rabi-Higgs frequency.'
author:
- 'H. P. [Ojeda Collado]{}'
- Gonzalo Usaj
- José Lorenzana
- 'C. A. Balseiro'
bibliography:
- 'library.bib'
title: 'Spectral fingerprints of the non-linear dynamics of driven superconductors with dissipation'
---
Introduction
============
The recent advances in laser technologies have opened new avenues for the study of collective behavior and emergent phenomena in condensed matter [@Fausti2011d; @Mansart2013; @Matsunaga2013; @Matsunaga2014; @Mankowsky2014; @Nicoletti2014; @Kaiser2014; @Mitrano2016; @Rajasekaran2018] and ultracold-atom systems [@Stoferle2004; @Haller2010; @Endres2012; @Chin2010; @Behrle2018; @Clark2015] far from the linear response paradigm. Theoretical examples of these strongly non-linear phenomena are dynamical phase transitions after a quantum quench [@Barankov2006a; @Eckstein2009] or under a periodic drive [@Collado2018].
While these studies are of interest per se, they can also, in principle, shine some light on the nature of the precursor equilibrium phases. In this context, superconducting condensates have drawn increasing attention, both in theoretical and experimental research, as challenging cases for the study of collective out of equilibrium states, especially in the case of superconductors with competing orders [@Fausti2011d; @Mansart2013; @Matsunaga2013; @Matsunaga2014; @Mankowsky2014; @Nicoletti2014; @Kaiser2014; @Mitrano2016; @Rajasekaran2018]. Much of the interest in the field was fueled by theoretical studies of quenched systems [@Volkov1974; @Barankov2006a; @Barankov2004] and experiments in which the superconductor is excited by a pump pulse without a complete suppression of the superconducting state. In the perturbed system, the superconducting order parameter $\Delta$ (Higgs mode) or the charge modes evolve with collective oscillations at frequency $2\Delta$ which are rapidly damped due to dephasing [@Mansart2013; @Matsunaga2013].
Another interesting way to manipulate a condensate is through periodic drives. In solid-state superconductors, different type of drives have been discussed [@Collado2018]. Among them are impulsive stimulated Raman scattering (ISRS) [@Lorenzana2013], phonon assisted modulation of the density of states (DOS-driving) as in the case of 2H-NbSe$_2$ [@Balseiro1980; @Littlewood1982] or of the coupling constant ($\lambda$-driving) as proposed for FeSe [@Collado2018], microwave drives and THz drives [@Cea2015] as already realized in Ref. \[\]. Periodic drives can also be achieved in ultra-cold atoms where there are well known techniques to modify Hamiltonian parameters at will: DOS-driving can be achieved modifying the depth of a periodic potential, as it has been done for bosons [@Stoferle2004; @Haller2010; @Endres2012], while $\lambda$-driven can be implemented in a variety of ways [@Chin2010; @Behrle2018; @Clark2015].
Previously, we have shown that within a BCS self-consistent dynamics, a periodic excitation at a frequency $\omega_d$ in resonance with quasiparticle excitations ($\omega_d>2\Delta$) can produce collective Rabi oscillations (“Rabi-Higgs” mode) of the quasiparticle population with a frequency proportional to the strength of the drive [@Collado2018]. This is due to a subset of quasiparticle excitations with energy $E_{\bm{k}}$ satisfying $\omega_d\approx 2 E_{\bm{k}}=2\sqrt{\xi_{\bm{k}}^2+\Delta^2}$ where $E_{\bm{k}}$ is the quasiparticle energy and $\xi_{\bm{k}}$ is the energy of the fermions measured from the chemical potential. A family of quasiparticles approximately satisfying the above condition synchronize among themselves and perform collective oscillations at the Rabi frequency.
The observation of a Rabi-Higgs mode requires entering into a highly nonlinear regime where damping and decoherence effects can be overcome. It is thus interesting to discuss under which conditions this non-linear regime can be achieved. It is well known from optical Bloch equations for a periodically driven two-level system [@Steck2019] that a finite energy relaxation time $\tau$ leads to a linear response regime at long times. The non-linear regime of Rabi oscillations can be accessed at short times when the Rabi frequency is larger than $1/\tau$. Similarly, the observation of Rabi-Higgs oscillations in a superconductor requires long relaxation times and/or strong drives (which result in Rabi-Higgs frequency $\Omega_R>1/\tau$). We mention that a similar phenomena has been proposed for driven graphene [@Mishchenko2009] where, however, the physics is much simpler as quasiparticle interactions have not been considered.
How long can $\tau$ be? Quasiparticle relaxation times in superconductors can be extremely long under favorable conditions. A simple estimate can be obtained through the Dynes parameter [@Dynes1978] in tunneling experiments which, as explicitly shown in Ref. \[\], is directly related to the damping of quasiparticles by the bath ultimately limiting the coherent dynamics. In aluminium samples, an inverse Dynes parameter $1/\gamma \equiv \tau\sim 10^{6}/\Delta\sim \mu$s has been measured [@Saira2012] which suggest that there is a large time window for coherent dynamics. Indeed, in Ref. \[\] we found that for a moderate drive strength the Rabi period $\tau_R\equiv 2\pi/\Omega_R$ is in the scale of tenths or hundreds of $1/\Delta\ll \tau$ (see also Fig. \[fig:fig1\] below). On the other hand, the above estimate for $\tau$ is probably too optimistic as the same out-of-equilibrium quasiparticles will open new relaxation channels [@Chang1978] and hence the whole out-of-equilibrium many-body problem can be considered.
Here we take a simple approach for the interplay of coherent non-linear phenomena and damping by considering a driven superconductor in the presence of a bath providing energy relaxation. We make the simplest possible assumption for the bath leaving a microscopic description for future work. Thus we treat the coupling with the bath as a phenomenological parameter and present the spectral signatures of the Rabi-Higgs modes varying the coupling and other experimental parameters. Differently from previous works, the dissipative dynamics is treated in a self-consistent way using the method introduced in Ref. \[\]. Both tr-ARPES and tunneling are analyzed in detail.
The time-dependent BCS model with dissipation
=============================================
In the absence of dissipation, $\lambda$-drive and DOS-drive produce qualitatively similar results [@Collado2018]. As we expect the same to be true in the presence of dissipation, we restrict hereon to study $\lambda$-driving.
In addition, in the dissipationless case, even for weak drives and for $\omega_d>2\Delta$ the Rabi-Higgs mode appears with frequency $\Omega_R$ proportional to the intensity of the drive. Apparently the Rabi-Higgs mode violates any linear response prescription but, as mention in the introduction, without dissipation the system becomes inherently non-linearly at long times no matter how weak the perturbation is. In other words, the order of limits is important. Taking first the Rabi frequency (proportional to the drive intensity) $\Omega_R\rightarrow 0$ and then energy relaxation times $\tau\rightarrow \infty$ a linear response regime is well defined. Inverting the order of limits it is not. In the following we study numerically the effect of dissipation on the Rabi-Higgs response.
Model and formalism
-------------------
We consider a single-band s-wave superconductor described by the Hamiltonian $$\label{eq:HBCS}
H_{\mathrm{BCS}}=\sum_{\bm{k},\sigma}\xi_{\bm{k}}c_{\bm{k}\sigma}^{\dagger}c_{\bm{k}\sigma}^{}-\lambda(t)\sum_{\bm{k},\bm{k^{\prime}}}c_{\bm{k}\uparrow}^{\dagger}c_{\bm{-k}\downarrow}^{\dagger}c_{\bm{-k^{\prime}}\downarrow}^{}c_{\bm{k^{\prime}}\uparrow}^{}$$ where $c_{\bm{k}\sigma}$ ($c_{\bm{k}\sigma}^{\dagger})$ destroys (creates) an electron with momentum $\bm{k}$, energy $\varepsilon_{\bm{k}}$ and spin $\sigma$. Here $\xi_{\bm{k}}=\varepsilon_{\bm{k}}-\mu$ measures the energy from the Fermi level $\mu$ and the pairing interaction $\lambda(t)$ is parameterized as $$\label{eq:ldt}
\lambda(t)=\lambda_0[1+\Theta\left(t\right)\alpha\sin\left(\omega_d t\right)]\,,$$ where $\Theta\left(t\right)$ is the Heaviside step function. In most of our calculations we take the parameter $\alpha\in\left[0,0.2\right]$, that corresponds to a modulation of up to $20$% of the equilibrium pairing interaction $\lambda_0$, to keep it within the range of experimental accessibility [@Collado2018; @Sentef2016]. In the thermodynamic limit, the Hamiltonian (\[eq:HBCS\]) is equivalent to the mean-field Hamiltonian, $$\begin{aligned}
\label{eq:hmf}
H_{\mathrm{MF}}=\sum_{\bm{k}}\psi_{\bm{k}}^{\dagger}\bm{H}_{\bm{k}}(t)\psi_{\bm{k}}, \end{aligned}$$ written in the Nambu spinor basis $\psi_{\bm{k}}=\left(c_{\bm{k}\uparrow},c_{-\bm{k}\downarrow}^{\dagger}\right)^{\mathrm{T}}$, where $$\label{eq:hkt}
\bm{H}_{\bm{k}}(t)=\left(\begin{array}{cc}
\xi_{\bm{k}} & -\Delta(t)\\
-\Delta(t)^{*} & -\xi_{\bm{k}}
\end{array}\right)\,,$$ and the instantaneous superconducting order parameter is given by $$\label{eq:ssd}
\Delta(t)=\lambda(t)\sum_{\bm{k}}\left\langle c_{\bm{k}\uparrow}^{\dagger}(t)c_{\bm{-k}\downarrow}^{\dagger}(t)\right\rangle .$$ Here $\left\langle \ldots\right\rangle $ denotes the expectation value on the initial state.
In order to consider dissipation we couple the superconductor to a reservoir. For simplicity we will take the bath to be at zero temperature, which means that if the superconductor is out of equilibrium at some time and it is allowed to evolve in the absence of the drive, it will eventually relax to the ground state with the bath absorbing all the excess energy. The method to treat the bath was explained in detail in Ref. \[\]. Here we summarize the main results.
To describe the reservoir effect, the self-consistent solution of the gap equation is written in terms of the Keldysh two-time contour Greens functions. In the Nambu spinor basis, the retarded and lesser Green functions are $2\times 2$ matrices with matrix elements given by $$\begin{aligned}
\label{eq:greendeff}
\notag
\bm{G}_{\bm{k}}^{R}\left(t,t^{\prime}\right)_{\alpha\beta}&=&-i\Theta\left(t-t^{\prime}\right)\left\langle \left\{ \psi_{\bm{k}{\alpha}}(t),\psi_{\bm{k}{\beta}}^{\dagger}\left(t^{\prime}\right)\right\} \right\rangle,\\
\bm{G}_{\bm{k}}^{<}\left(t,t^{\prime}\right)_{\alpha\beta}&=&i\left\langle \psi_{\bm{k}{\alpha}}^{\dagger}\left(t^{\prime}\right)\psi_{\bm{k}{\beta}}(t)\right\rangle, \end{aligned}$$ respectively. Thus, the superconducting gap equation \[Eq. (\[eq:ssd\])\] can be written as $$\label{eq:deltadef}
\Delta(t)=-i \lambda(t) \sum_{\bm{k}} {\bm{G}_{\bm{k}}^{<}\left(t,t\right)}_{12}\,.$$ When considering the coupling to a reservoir, the lesser Green function satisfies the Keldysh equation in time domain [@Antipekka1994; @Horacio1992; @Moore2019; @Collado2019] $$\label{eq:gl}
{\bm{G}}_{\bm{k}}^{<}(t,t^{\prime})=\int dt_{1}\int dt_{2}\,{\bm{G}}_{\bm{k}}^{R}(t,t_{1}){\bm{\Sigma}}_{\bm{k}}^{<}(t_{1},t_{2}){\bm{G}}_{\bm{k}}^{R}(t^{\prime},t_{2})^{\dagger}\,$$ where the dissipation effects are taken into account via the lesser self-energy ${\bm{\Sigma}}_{\bm{k}}^{<}(t_{1},t_{2})$ and the retarded Green function which is solution of the corresponding Dyson equation with a retarded self-energy ${\bm{\Sigma}}_{\bm{k}}^{R}(t_{1},t_{2})$. Following Refs. [@Moore2019; @Millis2017] we consider a mechanism for dissipation that couples each pair of states $\bm{k}\uparrow, -\bm{k}\downarrow$ with a reservoir described by a time-independent one body Hamiltonian $H_{b}=\sum_{\ell,\sigma}E_{\ell}d_{\ell\sigma}^{\dagger}d_{\ell\sigma}$ where $d_{\ell\sigma}^{\dagger}$ creates a state in a single particle bath state with energy $E_{\ell}$. In the limit of a wide-band reservoir with identical coupling $V_{\bm{k}\ell}=V_\ell$ for each $\bm{k}$ all details of the bath dropout and its effects can be described by a single frequency independent parameter $\gamma$ describing the effects of inelastic scattering and producing a level broadening $\sim\gamma$ and a finite lifetime $\tau=1/\gamma$. In this case, the retarded and lesser self-energy become momentum independent and diagonal in Nambu space, [*i.e.*]{} ${\bm{\Sigma}}_{\bm{k}}^{R}\equiv {\bf I}{{\Sigma}}^{R} $ and ${\bm{\Sigma}}_{\bm{k}}^{<}\equiv {\bf I}{{\Sigma}}^{<} $ with $$\begin{aligned}
\nonumber
{\Sigma}^{R}(t_{1},t_{2})&=&-i\gamma\delta(t_{1}-t_{2})/2\,,\\
\Sigma^{<}\left(t_{1},t_{2}\right)&=& i\gamma\int\frac{d\omega}{2\pi}f\left(\omega\right)e^{-i\omega\left(t_{1}-t_{2}\right)}\,.
\label{eq:selfl}\end{aligned}$$ Here $f(\omega)$ is the Fermi function evaluated at the bath temperature. Consequently, the Dyson equation for the retarded Green function can be easily integrated, being given by ${\bm{G}}_{\bm{k}}^{R}(t,t^{\prime})={\bm{G}}_{\bm{k}}^{R (0)}(t,t^{\prime})e^{-\gamma(t-t^{\prime})/2}$ where ${\bm{G}}_{\bm{k}}^{R (0)}(t,t^{\prime})$ is the retarded Green function in the absence of dissipation. The latter can be computed by solving the following differential equations (in matrix notation in the Nambu spinor basis and setting $\hbar=1$), $$\begin{aligned}
\label{eq:retev}
\notag
{\bm{G}}_{\bm{\bm{k}}}^{R(0)}\left(t,t\right)&=&-i\bm{I},\\
i\partial_{t}{\bm{G}}_{\bm{\bm{k}}}^{R(0)}\left(t,t^{\prime}\right)&=&\bm{H}_{\bm{k}}(t){\bm{G}}_{\bm{\bm{k}}}^{R(0)}\left(t,t^{\prime}\right),\;\;\;\;\;\; t>t^{\prime},\\
\notag
i\partial_{t^{\prime}}{\bm{G}}_{\bm{\bm{k}}}^{R(0)}\left(t,t^{\prime}\right)&=-&{\bm{G}}_{\bm{\bm{k}}}^{R(0)}\left(t,t^{\prime}\right)\bm{H}_{\bm{k}}\left(t^{\prime}\right),\;\;\;\;\;\; t>t^{\prime}.\end{aligned}$$ Replacing Eq. (\[eq:selfl\]) into Eq. (\[eq:gl\]) and assuming a reservoir at zero temperature we obtain $$\begin{aligned}
\label{eq:glf}
\nonumber
{\bm{G}}_{\bm{k}}^{<}(t,t^{\prime})=-\frac{\gamma}{2\pi}\int_{-\infty}^{t}\!dt_{1}\!\int_{-\infty}^{t^{\prime}}\!dt_{2}\,&&{\bm{ G}}_{ \bm{\bm{k}}}^{R(0)}\left(t,t_{1}\right){\bm{ G}}_{\bm{k}}^{R(0)}(t^{\prime},t_{2})^{\dagger}\\
&&\times\frac{e^{-\gamma(t-t_{1}+t^{\prime}-t_{2})/2}}{t_{1}-t_{2}+i0^{+}} \,.\end{aligned}$$ Hence, the time dependence of the order parameter \[Eq. (\[eq:deltadef\])\] can be obtained after computing the lesser Green function \[Eq. (\[eq:glf\])\] for $t^{\prime}=t$. This equal-time lesser Green function $\bm{G}_{\bm{k}}^{<}(t,t)\equiv\bm{G}_{\bm{k}}^{<}(t)$ satisfies the equation of motion $$\label{eq:gld}
\partial_{t}{\bm{G}}_{\bm{k}}^{<}(t)=-\gamma {\bm{G}}_{\bm{k}}^{<}(t)+\bm{\mathcal{I}}_{\bm{k}}(t)-i\left[\bm{H}_{\bm{k}}(t),{\bm{G}}_{\bm{k}}^{<}(t)\right]\,,$$ where $$\label{eq:ik}
\bm{\mathcal{I}}_{\bm{k}}(t)\!=\!\frac{i\gamma}{2\pi}\int_{-\infty}^{t}\!\!dt^{\prime}\left(\frac{{\bm{G}}_{ \bm{k}}^{R (0)}\left(t,t^{\prime}\right)}{t-t^{\prime}-i0^{+}}+\frac{{\bm{G}}_{ \bm{k}}^{R(0)}\left(t,t^{\prime}\right)^{\dagger}}{t-t^{\prime}+i0^{+}}\right)e^{-\frac{\gamma(t-t^{\prime})}{2}}.$$ The initial condition for the differential Eq. (\[eq:gld\]) is given by the equilibrium value of the lesser Green function $$\label{eq:gl0}
\bm{G}_{\bm{k}}^{<}(0)=\frac{i}{2}\bm{I}-\frac{i}{\pi E_{\bm{k}}}\arctan\left(\frac{2E_{\bm{k}}}{\gamma}\right)\left(\begin{array}{cc}
\xi_{\bm{k}} & -\Delta_{0}\\
-\Delta_{0} & -\xi_{\bm{k}}
\end{array}\right)\,,$$ where $E_{\bm{k}}=\sqrt{\xi_{\bm{k}}^2+\Delta_{0} ^2}$, which is time-independent and easily obtained after replacing the equilibrium retarded Green function in Eq. (\[eq:glf\]) (see Ref.\[\]). As a consequence, in the presence of dissipation, the equilibrium order parameter $\Delta_{0}$ is defined, via Eq. (\[eq:deltadef\]), by the gap equation $$\label{eq:gapee}
1=\frac{1}{\pi}\sum_{\bm{k}}\frac{\lambda_{0}}{E_{\bm{k}}}\arctan\left(\frac{2E_{\bm{k}}}{\gamma}\right)\,.$$ In the $\gamma\rightarrow 0$ limit Eq. (\[eq:gapee\]) becomes the standard BCS gap equation. In the presence of inelastic scattering ($\gamma$) the superconducting order parameter is reduced. As already mentioned, another important result is that at equilibrium the present formalism presents a rigorous justification for the Dynes formula for the density of states [@Collado2019]. As will be shows explicitly below, this provides a simple way to estimate the $\gamma$ parameter close to equilibrium directly from tunneling experiments [@Dynes1978; @Saira2012].
Rabi-Higgs modes in the superconducting response and dissipation effects
------------------------------------------------------------------------
We now present the numerical solution of our model for $\lambda-$driving. For concreteness we shall show simulations for $\omega_d=4\Delta_0$ but qualitative similar behavior is obtained for not too large frequencies above the gap ($\omega_d>2\Delta_0$).
At $t\leq0$ the system is in equilibrium and the order parameter $\Delta_{0}$ is given by Eq. (\[eq:gapee\]). At $t>0$ the drive switches on according to Eq. (\[eq:ldt\]) and we compute the equal-time lesser Green function via Eqs. (\[eq:gld\]), (\[eq:ik\]) and (\[eq:gl0\]) in order to self-consistently determine the superconducting order parameter evolution through Eq. (\[eq:deltadef\]).
### Non-linear effects in the undamped dynamics
We first briefly present the superconducting response in the absence of dissipation as a starting point to compare with those in which the reservoir effects play a role. In this case, the calculation can be made using either the Anderson pseudospin language [@Collado2018] or the present formalism by considering the evolution of the equal-time lesser Green function dictated only by the commutator with the Hamiltonian \[Eq. (\[eq:gld\]) without the two first terms in the r.h.s.\].
The dynamics of $\Delta\left(t\right)$ and the expectation value of the momentum distribution function, $$n_{\bm{k}}(t)=\sum_{\sigma}\left\langle c_{\bm{k}\sigma}^{\dagger}(t)c_{\bm{k}\sigma}(t)\right\rangle = 1-i \left[\bm{G}_{\bm{k}}^{<}(t)_{11}-\bm{G}_{\bm{k}}^{<}(t)_{22}\right]$$ are shown in Fig. \[fig:fig1\] for two different values of the perturbation amplitude $\alpha$. It is apparent from the figure that the order parameter oscillates with two fundamental frequencies and, after a short transient, averages to a smaller value respect to equilibrium. The drive frequency corresponds to a fast oscillation that can not be resolved on the scale of the figure and leads to the filled black regions of the gap dynamic. In addition, the amplitude of the gap shows the Rabi-Higgs oscillations with a frequency that increases approximately linearly with increasing $\alpha$. Indeed, $$\begin{aligned}
\label{eq:wr}
\frac{\Omega_R}{\Delta_0} \approx A(\omega_d/\Delta_0) \alpha\end{aligned}$$ with $A(\omega_d/\Delta_0)\sim 2.0$ for $\omega_d/\Delta_0=4$ \[see Eq. (29) in Ref. [@Collado2018] for an analytic approximation\].
The Rabi-Higgs mode is associated with a periodic inversion of the population of the quasiparticles resonant with the drive. For $\omega_d=4\Delta_0$, this is visible in the momentum distribution function $n_{\bm{k}}(t)$ as a narrow time dependent structure at quasiparticle energy $\xi_{\bm{k}}\approx\pm 2\Delta_{0}$ (Fig. \[fig:fig1\], middle panel) with a frequency that matches the Rabi-Higgs period of $\Delta(t)$. Notice that the inversion of colors along the anomaly represent a cyclic inversion of population of the resonant quasiparticles. Such time-dependent anomaly represent a clear hallmark of the Rabi-Higgs mode and opens the possibility to detect it through spectroscopies as we shall demonstrate in the next section.
![(Color online) Superconducting dynamics in the absence of dissipation for $\alpha=0.1$ (left column) and $\alpha=0.2$ (right column) and $\omega_d=4\Delta_0$. From top to bottom we show the superconducting order parameter as a function of time, the time-dependent momentum distribution function $n_{\bm{k}}(t)$ and the fast Fourier transform of $\Delta(t)$. We label the states by the normal state quasiparticle energy $\xi_{\bm{k}}$.[]{data-label="fig:fig1"}](SCgapf_light.pdf){width="0.97\columnwidth"}
Another non-linear effect is the generation of a second-harmonic. In the case of an electromagnetic drive, second harmonic generation is not allowed [@Cea2018]. This follows from the general fact that the current response to the vector potential $A$ is $J\sim \rho_s(A) A$ where $\rho_s(A)$ is the superfluid stiffness. In the absence of a steady state current, the free energy and $\rho_s(A)$ are even in $A$, so the lowest non-linear contribution to $J$ is order $A^3$, i.e. third harmonic generation [@Cea2018]. In contrast, not such symmetry exist in our case as for $\lambda>0$ terms odd in $\delta \lambda$ are allowed in the free energy and second harmonic generation is allowed. Indeed, as can be seen in the bottom panel of Fig. \[fig:fig1\] the superconducting response not only contains the driving frequency $\omega_d=4\Delta_{0}$ but also $2\omega_d=8\Delta_{0}$. This can also been seen in the middle panel of Fig. \[fig:fig1\], where a less intense Rabi-Higgs mode is developed associated with the second harmonic response, i.e. quasiparticles with $\xi_{\bm{k}}\approx\pm 4\Delta_{0}$ also show a narrow time dependent anomaly in the population but with a much smaller Rabi frequency.
![Time dependence of the superconducting order parameter in the presence of dissipation for $\alpha=0.1$ (left column) and $\alpha=0.2$ (right column). From top to bottom we use $\gamma=0$ (without dissipation effects), $\gamma=0.05\Delta_{0}$ and $\gamma=0.2\Delta_{0}$, respectively.[]{data-label="fig:fig2"}](SCgap.pdf){width="0.95\columnwidth"}
### Dissipative dynamics
The driven superconducting response ($\Delta(t)$) in the presence of a bath is shown in Fig. \[fig:fig2\] for different values of the bath parameter $\gamma$ and the perturbation amplitude $\alpha$. As in non-linear optics, in the presence of dissipation the Rabi oscillation is a transient effect. For sufficiently long times a steady state is achieved where only the drive frequency is present.
As $\gamma$ is increased (from top to bottom in Fig. \[fig:fig2\]) the system reaches a steady state more rapidly with a vanishing of Rabi-Higgs oscillations. By increasing $\alpha$, for a fixed value of $\gamma$, $\Omega_R$ increases \[cf. Eq. (\[eq:wr\])\] and several Rabi-Higgs oscillations are visible before they disappear as a consequence of relaxation (see for example Fig. \[fig:fig2\] (c,d)). Thus, in order to detect the Rabi-Higgs modes experimentally it is necessary to ensure that $\Omega_R \gtrsim\gamma$. In fact, for the $\gamma$ values used here, the slower Rabi-Higgs mode, associated with the second harmonic generation, is not visible.
Conversely, for small $\alpha$ and strong dissipation it is possible to get $\Omega_R\lesssim\gamma$ and only the oscillations synchronous with the drive are visible as expected from linear response theory (bottom panels of Fig. \[fig:fig2\] and Fig. \[fig:fig21\]). Indeed, in this regime, the amplitude of the oscillation in the order parameter increases linearly with $\alpha$ as shown in Fig. \[fig:fig21\]. However, it is important to note that by increasing $\alpha$, and after a very fast transient, the superconducting gap decreases in average which constitutes the first nonlinear effect arising in the dynamics. As a conclusion, in the presence of dissipation, we can distinguish two different regimes: one in which the linear response theory is applicable even at short times ($\Omega_R\lesssim\gamma$) and the other corresponding with a transient strong nonlinear behavior in which Rabi oscillations can be observed ($\Omega_R\gtrsim\gamma$).
![(Color online) (a) Time dependence of the order parameter for $\gamma=0.2\Delta_{0}$ and $\alpha=0.004, 0.01, 0.016, 0.022$. (b) Amplitude of the oscillation in the order parameter $\Delta_1$ as a function of the strength of the drive $\alpha$. The filled points correspond with the curves of panel (a).[]{data-label="fig:fig21"}](linearresponse.pdf){width="0.95\columnwidth"}
We now discuss in more detail the nonlinear regime, in particular the possibility of detection of the Rabi-Higgs mode in experiments. Previously, we have demonstrated that the Rabi-Higgs mode is associated with oscillations in the occupation values $n_{\bm{k}}$ (see Fig. \[fig:fig1\]). This charge fluctuations provide an efficient manner to detect the existence of this nonlinear mode with standard experimental techniques as we shall show in the next section.
Since for the simple electronic structure we are taking, $n_{\bm{k}}$ depends on ${\bm{k}}$ only through its distance from the Fermi surface ($\xi_{\bm{k}}$) it is useful to introduce the distribution function $ n(\xi_{\bm{k}})\equiv n_{\bm{k}}$. When convenient, in the following we will drop the momentum dependence and refer to $ n(\xi)$ loosely as the momentum distribution function keeping in mind the above equivalence.
In Fig. \[fig:fig3\] we show the time-dependent $n_{\bm{k}}$ distribution taking into account dissipative effects (middle and right column) in comparison with the dissipationless counterpart (left column) at different times within the first Rabi cycle (panels (d)-(f)) and in the steady state (panels (a)-(c)). We have used the same $\gamma$ and $\alpha$ parameters as in the right column of Fig. \[fig:fig2\].
For $\gamma=0$ the Rabi oscillations can be observed as oscillations in the occupation value $n_{\bm{k}}$ for $\xi_{\bm{k}}\approx\pm 2\Delta_{0}$, which we will refer to as $n({\pm}\xi_R)$. Small peaks also appear for $\xi_{\bm{k}}\approx\pm 4\Delta_{0}$, corresponding to the second-harmonic slower Rabi-Higgs mode that starts to develop in the temporal window used in Fig. \[fig:fig3\](d). At long-times, the two Rabi-Higgs modes can be seen as is shown in Fig. \[fig:fig3\](a).
In the presence of weak dissipation ($\gamma=0.05\Delta_{0}$), the transient dynamic only shows oscillations of $n(\pm\xi_R)$, corresponding to the fastest Rabi-Higgs mode. Finally, for strong dissipation ($\gamma=0.2\Delta_{0}$), a full Rabi cycle cannot be completed since the relaxation takes place in a very short time. As a consequence $n(\xi_R)$ ($n(-\xi_R)$) increases (decreases) during a short period of time and rapidly saturates without exhibiting Rabi oscillations.
![$n_{\bm{k}}$ distribution as a function of time for $\alpha=0.2$ and $\gamma=0$ (left column), $\gamma=0.05\Delta_{0}$ (middle column) and $\gamma=0.2\Delta_{0}$ (right column). In panels (d)-(f), the time increases from bottom to top as follows $t=0$ (equilibrium), $t=0.1\tau_R$, $t=0.25\tau_R$, $t=0.5\tau_R$, $t=0.75\tau_R$ and $t=\tau_R$ (Rabi-Higgs period $\tau_R \Delta_0=15$). For clarity, the zero of each curve was displaced vertically for a factor 2. The oscillating $n_{\bm{k}}$ distributions in the steady state are shown in panels (a)-(c) at different times within a drive period.[]{data-label="fig:fig3"}](nkt.pdf){width="0.95\columnwidth"}
The $n_{\bm{k}}$ distribution in the steady-state in the presence of dissipation are shown in Fig. \[fig:fig3\](b) and Fig. \[fig:fig3\](c) for different times within a period of the drive. In this case, $n_{\bm{k}}$ shows a peak at $\xi_{\bm{k}}\approx\pm 2\Delta_{0}$ such that $n(\pm\xi_R)\approx 1$ all the time. Taking into account spin this correspond to half-filled single particle states which, as shown below, corresponds to saturation of the population imbalance due to strong drive. Thus, ultimately the fate of the superconducting dynamics is a gap oscillation with the drive frequency (see Fig. \[fig:fig2\] (e,f)) and a stationary population unbalance at the quasiparticle energy $\xi_{R}$.
The population unbalance changes considerably as a function of the amplitude of the perturbation as can be appreciated in Fig. \[fig:fig31\]. Because the Rabi oscillation is only a transient effect, a full population inversion (e.g. a sign change of $n(\xi_R)-n(-\xi_R)$) is not possible in the steady state. As a consequence, the excited state population can not exceed the half-filled value (counting spin) $n(\xi_R)\approx1$ by considering a bath in the wide-band limit [@Ferron2012].
We have extracted the $n_{\bm{k}}$ distribution in the steady state for several values of $\alpha$ with $\gamma=0.2\Delta_{0}$. In accordance with the above statement, the value of $n(\xi_R)$ ($n(-\xi_R)$) in the steady state, increases (decreases) with $\alpha$ and saturates to $1$ in the limit of large $\alpha$ as it is shown in Fig. \[fig:fig31\]. Notice that the population imbalance is quadratic in $\alpha$ so it vanishes in the linear response regime which follows also from the fact that, by symmetry, it should be even in $\alpha$.
This saturation of the excited population is a common effect in nonlinear quantum optics. There, one usually considers a two-level system in the presence of a driving force and damping effects via a phenomenological parameter in the Bloch equation of motion. For a drive frequency in resonance with the two level system, the excited population in the steady state increases with the amplitude of the perturbation and in the limit of large intensity, the largest possible excited population is equal to the ground state population [@Steck2019]. Again, the difference in the present context is that the phenomena does not corresponds to a single two-level system but a collection of two-level systems interacting through the self-consistency that determines the superconducting order parameter.
![Steady-state $n_{\bm{k}}$ distributions for $\gamma=0.2\Delta_0$ and $\alpha=0.05$ (a), $\alpha=0.1$ (b), $\alpha=0.15$ (c) and $\alpha=0.2$ (d). In panel (e) the dashed and solid lines represent the (minimum, maximum) and average value of $n(\xi_R)$, respectively, as a function of $\alpha$. Notice that $n(\xi_R)$ is non zero even for $\alpha=0$ as an excess of occupation is inherent to the BCS ground state. The dot-dashed line is a quadratic fitting for the average of excited population for small $\alpha$ values. The population inversion starts when $n(\xi_R)$ exceeds the horizontal line $n_{\bm{k}}=1$ which is represented with arrows in the panel (d). As a consequence of the presence of dissipation, in the steady state regime this does not occur not matter how strong the perturbation is.[]{data-label="fig:fig31"}](nkt_ss_vs_alpha.pdf){width="0.95\columnwidth"}
Theoretical modelling of tr-ARPES and Tunneling experiments: Spectral fingerprints of Rabi-Higgs mode
=====================================================================================================
Our main aim in this section is to identify some spectral fingerprints of the Rabi-Higgs mode, and related non-linear phenomena, that could be experimentally detected in the presence of dissipation. Clearly, an emergent technique to detect time dependent phenomena is tr-ARPES and so we discuss such a case first. Yet, in addition, we shall demonstrate that also time resolved tunneling experiments could be useful to detect the non-linear mode discussed above.
![(Color online) tr-ARPES intensity for $\alpha=0.2$ at several photo-emission times inside the first period $\tau_R$ of Rabi-Higgs mode, from left to right, $t_0=0$ (equilibrium), $t_0=0.1\tau_R$, $t_0=0.25\tau_R$, $t_0=0.5\tau_R$, $t_0=0.75\tau_R$ and $t_0=\tau_R$. From top to bottom we show results for $\gamma=0$ (without dissipation), $\gamma=0.05\Delta_{0}$ and $\gamma=0.2\Delta_{0}$ as in Fig. \[fig:fig2\].[]{data-label="fig:fig4"}](trARPES_multiple_light.pdf){width="0.95\columnwidth"}
In all the calculations discussed below, we obtain the spectral signals in terms of the lesser Green function Eq. (\[eq:glf\]) as a function of $\alpha$ and $\gamma$.
tr-ARPES
--------
In our setting the $\lambda-$driving is turned on at time $t=0$ and the photoemission process is induced by a wave packet of photons centered at time $t_0$ and with central energy $\hbar \omega_{\bm q}$ larger than the work function of the solid $W$. For simplicity we use a Gaussian shape for this probe pulse, $s(t)=\exp(-(t-t_{0})^{2}/2\sigma^{2})/(\sigma\sqrt{2\pi})$ with standard deviation $\sigma$.
In a tr-ARPES experiment the momentum of the outgoing electrons ${\bm k}_e$ is measured. Energy conservation determines the excitation energy left in the system after the photoemission process, $\hbar \omega=\hbar\omega_{\bm q}-(\hbar k_e)^2/(2m_e)-W$ and momentum conservation yields information on the momentum of the excitations ${\bm{k}}$. The momentum resolved photocurrent in the detector at time $t$ is due to all electrons photoemitted before that time and is determined by [@Freericks2009], $$\label{eq:arpesint}
I_{\bm{k}}\left(\omega,t\right)=\mathrm{Im}\!\!\int_{-\infty}^{t}\!\!dt_{1}\!\!\int_{-\infty}^{t}\!\!dt_{2}s\left(t_{1}\right)s\left(t_{2}\right)e^{i\omega\left(t_{1}-t_{2}\right)}\bm{G}_{\bm{k}}^{<}\left(t_{1},t_{2}\right)_{11}\,.$$ In order to probe the Rabi-Higgs modes in real time in the following we use different probe times $t_{0}$ and take the integration limit from a lower cutoff time $t_l=t_0-5\sigma$ to $t=t_0+5\sigma$. Fig. \[fig:fig4\] shows the tr-ARPES intensity with and without dissipation for different $t_0$ during the first period $\tau_R$ associated with the Rabi-Higgs mode. For simplicity, we consider a parabolic band for electrons $\xi_{\bm{k}}\propto (k^{2}-k_F^{2})$ and use a probe pulse with a standard deviation $\sigma=0.1\tau_R$. From the photo-emission signal at equilibrium (left column in Fig. \[fig:fig4\]) there is an almost imperceptible broadening of the spectral line as $\gamma$ is increased, which leads to a decrease of superconducting order parameter according to Eq. (\[eq:gapee\]).
The presence of Rabi-Higgs oscillations are clearly visible in the top and middle panels of Fig. \[fig:fig4\] via a spectral weight around $\omega=\pm2\Delta_{0}$ that increases (decreases) above (below) the Fermi energy in the first half-period and has the opposite behavior in the second half-period of the Rabi-Higgs mode. One can visualize the process as an excitation of quasiparticles from the lower quasiparticle branch to the higher quasiparticle branch in the first half-cycle followed by a deexcitation in the second half-cycle or equivalently as a stimulated absorption phase followed by a stimulated emission phase.
If $\gamma$ is large enough, Rabi oscillations are not visible and only the spectral weight corresponding with the steady state is observed after a fast transient (bottom panel of Fig. \[fig:fig4\]).
Clearly the above dynamics is the photoemission image of the momentum distribution function imbalance discussed above. Indeed, integrating in frequency $$\begin{aligned}
\label{eq:sumrule} \int_{-\infty}^{\infty}\frac{d\omega}{2\pi}I_{\bm{k}}\left(\omega,t\right)&=&\mathrm{Im}\!\!\int_{-\infty}^{t}\!\!dt^{\prime}s\left(t^{\prime}\right)^2\bm{G}_{\bm{k}}^{<}\left(t^{\prime},t^{\prime}\right)_{11}\nonumber\\
&=&\frac12\int_{-\infty}^{t}\!\!dt^{\prime}s\left(t^{\prime}\right)^2 n_{\bm{k}}(t')\end{aligned}$$ which is clearly a moving average of $n_{\bm{k}}(t)$. For example, in the large $\gamma$ case the unbalance over time in the tr-ARPES intensity for $\omega=\pm2\Delta_{0}$ matches the $n_{\bm{k}}$ distribution in the steady state shown in Fig. \[fig:fig31\](d).
These results establish time-resolved photo-emission experiments as a tool to investigate steady state non-linearities, like population imbalance, and dynamic non-linearities like the Rabi-Higgs mode, in driven superconductors.
### Floquet Analysis
As already mentioned for $t\gtrsim\tau=1/\gamma$ one reaches a steady state and the superconducting gap oscillates only with the drive frequency. Therefore the mean-field Hamiltonian Eq. (\[eq:hmf\]) is time periodic at long times and we can use Floquet theorem to analyze the spectrum. This theorem guarantees the existence of a set of solutions of the time-dependent Schrodinger equation of the form $$\left|\psi_{\nu}\left(t\right)\right\rangle=\exp(-i\varepsilon_{\nu}t/\hbar)\left|\phi_{\nu}\left(t\right)\right\rangle$$ where $\left|\phi_{\nu}\left(t\right)\right\rangle$ has the same periodicity of the Hamiltonian Eq. (\[eq:hkt\]) [@Shirley65; @Sambe73]. The Floquet states $\left|\phi_{\nu}\left(t\right)\right\rangle$ are the solutions of $$\mathcal{H}_{F}\left|\phi_{\nu}\left(t\right)\right\rangle=\varepsilon_{\nu}\left|\phi_{\nu}\left(t\right)\right\rangle$$ where $\mathcal{H}_{F}=H_{MF}-i\hbar\partial/\partial t$ is the Floquet Hamiltonian and $\varepsilon_{\nu}$ is the quasienergy [@Kohler2005; @Grifoni98]. In the Floquet basis this Hamiltonian becomes a time-independent infinite matrix operator. Since we are interested in the low energy spectra we restrict the Floquet Hamiltonian to a large enough but finite subspace containing many multiphoton processes (finite number of replicas) [@Kohler2005; @Grifoni98].
For $\gamma=0.2\Delta_0$ and $\alpha=0.2$ the steady state condition is reached already for $t\sim 0.75 \tau_R= 11.25/\Delta_0$. The quasi-energy spectrum in the long time limit is well described by $\Delta(t)=\bar{\Delta}+\Delta_1 \cos(\omega_d t)$ with $\bar{\Delta}=0.86\Delta_0$ and $\Delta_1=0.42\Delta_0$ \[c.f. Fig. \[fig:fig2\](f)\].
In the upper panels of Fig \[fig:fig41\] we compare the tr-ARPES with the Floquet spectrum in the steady state. To analyze the details in the right panels we sacrifice temporal resolution to gain energy resolution by using a wider probe pulse. We see that the tr-ARPES signal nicely matches the Floquet spectrum, depicted by the dashed line. Thus the tr-ARPES signal essentially probes the occupied parts of the Floquet band structure.
Surprisingly, also in the transient dynamics the tr-ARPES intensity fits very well with a Floquet spectrum that is obtained from an effective $\Delta(t)$ with a monochromatic dependence (see lower panel of Fig. \[fig:fig41\]), even thought in this regime the superconducting gap shows several incommensurate frequencies (Rabi-Higgs and drive frequency) and Floquet theorem is not strictly applicable. For $\gamma=0$, we compute the Hamiltonian spectrum for a $\Delta(t)=0.63\Delta_0+0.35\Delta_0\cos(\omega_d t)$. Thus the transient response shown in Fig. \[fig:fig4\] and the lower panels in Fig. \[fig:fig41\] can be seen as a time-dependent change in the occupancy of the Floquet band structure. Clearly, the reason this analysis works in the Rabi-Higgs oscillation regime is the large separation of time scales between the slow Rabi-Higgs dynamics and the fast drive oscillations [@Lucila2018].
![(Color online) tr-ARPES intensity for drive strength $\alpha=0.2$, $t_0=0.75\tau_R$ and $\gamma=0.2\Delta_0$ (upper panel) and $\gamma=0$ (lower panel). We use a probe duration of $\sigma=0.1\tau_R$ (left column) and $\sigma=0.6\tau_R$ (right column). The dashed lines represent the quasi-energy spectrum assuming a periodic time dependence $\Delta(t)$ as defined in the main text.[]{data-label="fig:fig41"}](trARPESdeff.pdf){width="0.95\columnwidth"}
Time-resolved tunneling experiment
----------------------------------
We now discuss a possible setup to detect the Rabi-Higgs mode via tunneling measurements. In the last decades there have been efforts to add temporal resolution to scanning tunneling spectroscopy (STM) techniques and now sub-picosenconds resolution can be reached in experimental setups [@Morgenstern1609; @Nunes1029; @Mikio2004]. In principle one can radiate a sample while performing the measurement making STM a feasible time resolved technique at the same level that pump-probe measurements [@Mikio2004]. However, since the spatial resolution is not a requirement to reveal this nonlinear mode, a planar junction would probably be a more stable setting. An antenna could be used to couple the radiation with the superconductors as is done in detectors for photons with energy matching the superconducting gap [@Zmuidzinas1992; @Leridon1997]. In what follows, we will refer to STM signal but our formalism applies also for the case of a planar junction.
To obtain the STM signal, we now consider a tip located close above the out of equilibrium superconductor and tunnel-coupled locally to the $\bm{k}$ states in the system. The entire problem can be described by a generic Hamiltonian $H=H_{\mathrm{MF}}+H_{\mathrm{T}}+H_{\mathrm{TS}}$ where we have added the tip (or metallic contact) Hamiltonian $$H_{\mathrm{T}}=\sum_{\bm{p}}(\epsilon_{\bm{p}}+eV)a_{\bm{p}}^{\dagger}a_{\bm{p}}^{}\,,$$ and the coupling between both subsystem via the tunneling Hamiltonian $$H_{\mathrm{TS}}=\sum_{\bm{k},\bm{p}}\left(T_{\bm{k}\bm{p}}\,c_{\bm{k}}^{\dagger}a_{\bm{p}}^{}+h.c.\right).$$ The tip or the metallic contact is connected to an external voltage source with energy $eV$, $a_{\bm{p}}$ ($a_{\bm{p}}^{\dagger})$ destroys (creates) an electron with momentum $\bm{p}$ and energy $\epsilon_{\bm{p}}+eV$ in the tip and $T_{\bm{k}\bm{p}}$ is the strength of the tunneling. The electron current operator is given by $$\hat{I}=ie\left[\hat{N},\hat{H}\right]=ie\left(\hat{L}^{\dagger}-\hat{L} \right)\,,$$ where $e$ is the electron charge, $\hat{N}=\sum_{\bm{p}}a_{\bm{p}}^{\dagger}a_{\bm{p}}$ is the number operator and $\hat{L}=\sum_{\bm{k},\bm{p}}T_{\bm{k}\bm{p}}c_{\bm{k}}^{\dagger}a_{\bm{p}}^{}$. Assuming a weak coupling between tip and the superconductor we calculate the current through the tip in the linear response limit as $$\label{eq:itd}
I(t)=ie\int_{-\infty}^{\infty}\theta\left(t-t^{\prime}\right)\left\langle \left[\hat{I}(t),H_{TS}(t^{\prime})\right]\right\rangle_0$$ where $\left\langle ... \right\rangle_0 $ denotes the expectation value at zero order in the tunneling Hamiltonian. Notice that we have not consider the spin degree of freedom so far as the current is a spin conserved quantity (not spin-flip are allowed in the tunneling process). This will be included in the final expression of the time-dependent current as a factor $2$. From Eq. (\[eq:itd\]) we obtain
$$\label{eq:itf}
I(t)=2eT^{2}\int_{-\infty}^{t}dt^{\prime}\sum_{\bm{k,p}}\left(\left[e^{-i\epsilon_{\bm{p}}\left(t-t^{\prime}\right)}\left\langle c_{\bm{k}}^{\dagger}(t)c_{\bm{k}}^{}(t^{\prime})\right\rangle \Theta\left(\epsilon_{\bm{p}}-eV\right)-e^{i\epsilon_{\bm{p}}\left(t-t^{\prime}\right)}\left\langle c_{\bm{k}}^{}(t)c_{\bm{k}}^{\dagger}(t^{\prime})\right\rangle \Theta\left(-\epsilon_{\bm{p}}+eV\right)\right]+\left(t\leftrightarrow t^{\prime}\right)\right)\,,$$
where we have assumed a momentum-independent tunneling coupling $T_{\bm{k}\bm{p}}\equiv T$ and a zero-temperature Fermi distribution for the electrons on the tip via the Heaviside step function $\Theta(x)$. Taking the derivative of Eq. (\[eq:itf\]) with respect to the voltage $V$ and using Eq. (\[eq:greendeff\]), the time-dependent differential conductance can be written as $$\label{eq:conductance}
G(t)=\frac{dI}{dV}\propto \mathrm{Im}\sum_{\bm{k}}\int_{-\infty}^{t}dt^{\prime}\mathrm{Tr}\left[{\bm{G}}_{\bm{k}}^{<}(t,t^{\prime})\right]e^{-ieV\left(t-t^{\prime}\right)}$$ where $\mathrm{Tr}$ represents the trace in Nambu space. In the absence of the drive (that is, at equilibrium) the differential conductance becomes time-independent, being proportional to the well-know phenomenological Dynes density of states $$\label{eq:dos}
G\propto\rho(eV)=\rho_{0}\, \mathrm{Re} \left [ \frac{eV+i \Gamma}{\sqrt{(eV+i \Gamma)^2-\Delta_{0}^2}}\right ]\,,$$ where the Dynes parameter $\Gamma=\gamma/2$, $\Delta_{0}$ is the equilibrium order parameter and $\rho_{0}$ is the normal phase density of states. It is worth recalling that the Dynes formula was originally introduced phenomenologicaly [@Dynes1978]. There have been several theoretical proposals to provide a formal justification including Eliashberg physics [@Mikhailovsky1991], inelastic tunneling [@Pekola2010] and magnetic impurities [@Hlubina2016; @Hlubina2018]. In our previous work we have shown that the coupling with a bath [@Collado2019] provides a mechanism to justify Dynes formula and allows to link directly equilibrium tunneling with the $\gamma$ parameter. Of course since we are not providing a microscopic theory of the bath our justification is still semi-phenomenological.
Turning to the driven case, for small $\gamma$ values the out-of-equilibrium differential conductance is computationally very demanding since the whole dynamics has to be integrated in Eq. (\[eq:conductance\]), differently from the photoemission case where the integrals are cutoff by the probe pulse shape. To make the computations feasible we use $\gamma=0.2\Delta_0$ and $\alpha=0.1$. The nominal Rabi-Higgs period corresponds to $\tau_R\Delta_0\approx27$ but because of the large damping a steady state is reached before a full Rabi cycle is completed. Still as shown in Fig. \[fig:fig5\], the time-dependent differential conductance clearly shows a non trivial transient dynamics. Indeed, a clear depression in the time-resolved conductance around $eV=2\Delta_0$ can be observed as a function of time which can be identified with the beginning of the Rabi-Higgs oscillation.
As in the case of photoemission two effects determine the features in the spectrum. The upper and lower Bogoliubov bands hybridize with the first Floquet sideband of the lower and upper Bogoliubov bands respectively creating a pseudogap at $\pm\omega_d/2$ (see right column of Fig \[fig:fig41\]). Furthermore, the depression of the population at energy $-\omega_d/2$ depresses the probability to extract electrons and the excess of population at energy $\omega_d/2$ prevents injecting electrons due to Pauli blocking.
Another effect of the drive is that whole shape of the conductance gets modified. Due to the Dynes parameter, at equilibrium the differential conductance does not vanishes sharply for energies lower than $\Delta_0$. Once the time-dependent perturbation is turned on, the superconducting coherence peak decreases according to a decreasing in the average order parameter.
![Differential conductance for $\alpha=0.1$ at several values of times inside the first period $\tau_R$ of Rabi-Higgs mode, from bottom to top, $t=0$ (equilibrium), $t=0.1\tau_R$, $t=0.25\tau_R$, $t=0.5\tau_R$, $t=0.75\tau_R$ and $t=\tau_R$. We use $\gamma=0.2\Delta_{0}$ and the zero for each curve (horizontal dashed line) was displaced by factor 5 for clarity.[]{data-label="fig:fig5"}](stm.pdf){width="0.95\columnwidth"}
Summary and Outlooks {#sec:conc}
====================
We have studied the dynamic of a BCS superconductor subject to a periodic drive in the presence of dissipation focusing on interesting transient dynamics. For $\gamma\lesssim\omega_R$ the transient show Rabi-Higgs oscillations which persist forever when $\gamma=0$. In the opposite limit $\gamma\gtrsim\omega_R$ oscillations are not observed and the response is practically described by linear response at all times for a weak drive and show the phenomena of saturation of population imbalance for large drives. Another interesting non-linearity is second harmonic generation, which we have found it is allowed for $\lambda$-drive.
In all regimes the behavior resembles an isolated driven two level system. However one should keep in mind that the interactions are fundamental in synchronizing a finite fraction of the quasiparticles and make the mode collective. There are close analogies with phenomena in quantum optics where Rabi oscillations and saturation of population are common phenomena. This analogy paves the way to explore quantum optics protocols to control and manipulate the superconducting state of materials as already proposed in Ref. [@Collado2018].
The experimental detection of these highly nonlinear behaviors in driven superconductors could be a major step towards the quantum control and manipulation of quantum phases. We have proposed two techniques to detect Rabi-Higgs oscillations taking into account dissipation. A cyclic imbalance of quasiparticle population at specific locations in energy and momentum has been shown to appear in tr-ARPES. Due to technical reasons only the rise of the population imbalance could be explicitly computed for tunneling experiments but we expect that, in close analogy with photoemission, the anomalies oscillate with the Rabi frequency for small dissipation. Even detection of the steady state non-linearities would be a quite interesting experimental achievement.
The present formalism can be easily extended to take into account more interesting relaxation mechanisms via more sophisticated self-energies beyond the wide-band approximation used here for the bath. An interesting direction is to analyze how robust the Rabi-Higgs mode is in the presence of dephasing, decoherence and relaxation sources from a more microscopic point of view by considering residual Coulomb and electron-phonon interactions where heating effects could be relevant.
J.L. is in debt for enlightening discussions with L. Benfatto, C. Castellani, G. Seibold, B. Leridon, N. Bergeal and Jérôme Lesueur. We acknowledge financial support from Italian MAECI and Argentinian MINCYT through bilateral project AR17MO7 and from ANPCyT (grants PICTs 2013-1045 and 2016-0791), from CONICET (grant PIP 11220150100506) and from SeCyT-UNCuyo (grant 06/C603). J.L. acknowledges financial support from Italian MAECI thought collaborative project SUPERTOP-PGR04879, from Italian MIUR though Project No. PRIN 2017Z8TS5B, and from Regione Lazio (L. R. 13/08) under project SIMAP.
|
---
author:
- |
Christian Muise christian.muise@ibm.com\
IBM Research AI, Cambridge, USA\
Tathagata Chakraborti tchakra2@ibm.com\
IBM Research AI, Cambridge, USA\
Shubham Agarwal Shubham.Agarwal@ibm.com\
IBM Research AI, Cambridge, USA\
Ondrej Bajgar[^1] ondrej@bajgar.org\
Future of Humanity Institute, University of Oxford, UK\
Arunima Chaudhary arunima.chaudhary@ibm.com\
IBM Research AI, Cambridge, USA\
Luis A. Lastras-Montaño lastrasl@us.ibm.com\
IBM Research AI, Yorktown Heights, USA\
Josef Ondrej josef.ondrej@ibm.com\
IBM Watson, Praha, Czech Republic\
Miroslav Vodolán MVodolan@cz.ibm.com\
IBM Watson, Praha, Czech Republic\
Charlie Wiecha wiecha@us.ibm.com\
Watson Data and AI, Yorktown Heights, USA
bibliography:
- 'references.bib'
title: '[Planning for Goal-Oriented Dialogue Systems]{}'
---
[^1]: Work done when at IBM Watson
|
---
abstract: 'Given a point and an expanding map on the unit interval, we consider the set of points for which the forward orbit under this map is bounded away from the given point. For maps like multiplication by an integer modulo 1, such sets have full Hausdorff dimension. We prove that such sets have a large intersection property, that countable intersections of such sets also have full Hausdorff dimension. This result applies to maps like multiplication by integers modulo 1, but also to nonlinear maps like $x \mapsto 1/x$ modulo 1. We prove that the same thing holds for multiplication modulo 1 by a dense set of non-integer numbers between 1 and 2.'
author:
- |
David Färm\
\
\
title: |
Simultaneously Non-dense Orbits\
Under Different Expanding Maps
---
Introduction
============
Multiplication by integers modulo 1
-----------------------------------
It is well-know that for maps like $f_b \colon [0,1)\to [0,1)$ where $f\colon x \mapsto bx \mod 1$ and $b$ is an integer larger than one, the forward orbit $(f^n(x))_{n=0}^\infty$ is dense for almost all points with respect to the Lebesgue measure. It follows that sets like $$G_{f_b}(x) :=\Big\{\, y\in [0,1): x\notin \overline{\cup_{n=0}^\infty f_b^n(y)}\, \Big\},$$ where $x\in [0,1]$, have zero measure. On the other hand, it is not difficult to see that such sets have full Hausdorff dimension. In this paper we will consider what happens if we start intersecting such sets. For example we will prove a theorem that implies $$\dim_H(G_{f_2}(x) \cap G_{f_3}(x))=1$$ and even $$\dim_H \Big(\bigcap_{b=2}^\infty G_{f_b}(x_b) \Big)=1,$$ where $x_b\in [0,1]$ for all $b$. The key property of $f_b$ is that it generates a symbolic representation of $[0,1)$. Indeed, any number $x\in [0,1)$ can be represented as a sequence $(x_i)_{i=1}^\infty\in \{0,1,\dots b-1\}^{\mathbb N}$, where $x=\sum_{i=1}^\infty \frac{x_i}{b^i}$. This representation is unique except on a countable set. Since we are only interested in Hausdorff dimension this ambiguity can be disregarded. Now, we have a correspondence between $[0,1)$ and $\Sigma_b:=\{0,1,\dots b-1\}^{\mathbb N}$ where $f_b\colon [0,1)\to [0,1)$ corresponds to the left shift $\sigma \colon \Sigma_b \to \Sigma_b$, where $\sigma \colon (x_i)_{i=1}^\infty \mapsto (x_{i+1})_{i=1}^\infty$. Now, instead of considering the set $G_{f_b}(x)$ directly, we can consider the set $$\cup_{n=1}^\infty \Big\{(y_i)_{i=1}^\infty\in \Sigma_b: x_1\dots x_n \neq y_k \dots y_{k+n-1} \forall k\geq 1\Big\}.$$ We can handle much more general maps than these, but to state the main theorems we need to define the main tool of this paper, the $(\alpha,\beta)$-game.
The $\boldsymbol{(\alpha, \beta)}$-game
---------------------------------------
We will use a one dimensional version of a set theoretic game that was introduced by W. Schmidt in [@Schmidt1]. In our case, the game is played on the unit interval $[0,1]$ equipped with Euclidean metric. There are two players, Black and White, and two fixed numbers $\alpha,\beta \in (0,1)$. The rules are as follows.
- [*In the initial step*]{} Black chooses any closed interval $B_0$, and then White chooses a closed interval $W_0 \subset B_0$ such that $|W_0| = \alpha |B_0|$.
- [*Then the following step is repeated.*]{} At step $k$ Black choses a closed interval $B_{k} \subset W_{k-1}$ such that $|B_{k}| = \beta |W_{k-1}|$. Then White chooses a closed interval $W_{k} \subset B_k$ such that $|W_k| = \alpha |B_k|$.
It is clear that the set $$\bigcap_{k=0}^\infty W_k = \bigcap_{k=0}^\infty B_k$$ will always consist of exactly one point. A set $E$ is said to be $(\alpha, \beta)$-winning if White always can achieve that $$\bigcap_{k=0}^\infty W_k \subset E.$$ A set $E$ is said to be $\alpha$-winning if it is $(\alpha, \beta)$-winning for all $\beta$.
For us, the key property of $\alpha$-winning sets proved by Schmidt [@Schmidt1] can be summarised as follows.
\[winninggivesdim\] If the set $E\subset [0,1]$ is $\alpha$-winning for some $\alpha>0$, then $\dim_H(E)=1$.
\[intersectionproperty\] Let $\alpha>0$ and let $(E_i)_{i=1}^\infty$ be a sequence of $\alpha$-winning sets. Then the set $\cap_{i=1}^\infty E_i$ is also $\alpha$-winning.
Expanding maps generating full shifts {#givingfullshift}
-------------------------------------
Let $f\colon [0,1)\to [0,1)$ be such that there are finitely or countably many disjoint intervals $[a,b)\subset [0,1)$ such that $\sum |[a,b)|=1$ and $f|_{[a,b)}$ is monotone and onto for each of these intervals. Note that we do not assume that $f$ is well defined on $[0,1)$, only on each of the intervals $[a,b)$.
We take an enumeration of the intervals and associate each interval to the corresponding number so that we can refer to an interval as $[n]$ where $n$ is the appropriate number. Assume that for each of the intervals $[a,b)$ it holds that $|f(x)-f(y)|\geq |x-y|$ for all $x,y \in [a,b)$. Then we can define cylinders $$C_{\, x_1 \dots x_n}:=\Big\{x\in [0,1):\bigcap_{i=1}^n f^{-(i-1)}(x)\in [x_i] \, \Big\}.$$ If $\lim_{n\to \infty}|C_{x_1 \dots x_n}|= 0$ for all $(x_i)_{i=1}^\infty \in \Sigma:=\{0,1,\dots b-1\}^{\mathbb N}$ or $\{0,1,\dots \}^{\mathbb N}$ depending on if $[0,1)$ was split into finitely or infinitely many parts, we can represent $[0,1)$ by $\Sigma$. If the alphabet is infinite, some points in $[0,1)$ may not have a well-defined expansion. For example, with $f\colon x \mapsto \frac{1}{x} \mod 1$ we cannot represent the set $
\bigcup_{n=0}^\infty f^{-n}(\{0\})$ in $\Sigma$. It is clear that at least Lebesgue almost every point has a well defined expansion.
To study sets like $$G_{f}(x) :=\Big\{\, y\in [0,1): x\notin \overline{\cup_{n=0}^\infty f^n(y)}\, \Big\},$$ we will use their representation in $\Sigma$ which in this case is $$\bigcup_{n=1}^\infty \Big\{\, (y_i)_{i=1}^\infty\in \Sigma: x_1\dots x_n \neq y_k \dots y_{k+n-1} \ \forall k\geq 1\, \Big\}.$$
The key theorem of this paper is the following. We will discuss conditions $(i)$ and $(ii)$ in Section \[conditions\].
\[fullshiftinunitinterval\] Let $f$ be as described above and such that it satisfies the following conditions.
1. There exists an $\alpha_0>0$ such that for each $k \in \mathbb N$, each closed interval ${I\subset [0,1)}$ and each $\beta>0$, when playing the $(\alpha_0,\beta)$ game with ${B_0=I}$, after a finite number of turns White is able put his set $W_j$ in a generation $k$ cylinder for some $j$, thereby avoiding all endpoints of generation $k$ cylinders.
2. There is a positive function $g\colon \mathbb N \to [0, \infty)$ such that $g(m)\to 0$ as $m\to \infty$ and $$\frac{|C_{x_1\dots x_{n+m}}|}{|C_{x_1\dots x_n}|}\leq g(m)$$ for all $(x_i)_{i=1}^\infty\in \Sigma$ and all $n,m \in \mathbb N$.
Then for any $x\in [0,1)$ which has a well-defined expansion there is an $\alpha>0$ such that the set $$G_f(x)=\Big\{\, y\in [0,1): x\notin \overline{\cup_{n=0}^\infty f^n(y)}\, \Big\},$$ is $\alpha$-winning in $[0,1]$. In fact $\alpha=\min\{\alpha_0, \frac{1}{4}\}$ is small enough.
The main result of the paper is the following corollary which follows after using Proposition \[winninggivesdim\] and Proposition \[intersectionproperty\].
Let $(f_i)_{i=1}^\infty$ be a sequence of functions as in Theorem \[fullshiftinunitinterval\] and let $(x_i)_{i=1}^\infty$ be a sequence of points in $[0,1)$ with well-defined expansions. Then $$\dim_H \Big(\bigcap_{i=1}^\infty G_{f_i}(x_i)\Big)=1.$$
$\boldsymbol{\beta}$-shifts where the expansion of 1 terminates
---------------------------------------------------------------
The following method to expand real numbers in non-integer bases was introduced by Rényi [@Renyi] and Parry [@Parry]. For more details and proofs of the statements below, see their articles.
Let $[x]$ denote the integer part of the number $x$. Let $\beta \in (1,2)$. For any $x \in [0, 1]$ we associate the sequence $d(x,\beta) = \{d_n (x, \beta)\}_{n=0}^\infty \in \{0, 1\}^\mathbb N$ defined by $$d_n (x, \beta) :=
[\beta f_\beta^{n} (x)],$$ where $f_\beta (x) = \beta x \mod 1$. The closure of the set $$\{\, d (x,\beta) : x \in [0,1)\,\}$$ is denoted by $S_\beta$ and it is called the $\beta$-shift. It is invariant under the left-shift $\sigma \colon \{i_n\}_{n=0}^\infty \mapsto \{i_{n+1} \}_{n=1}^\infty$ and the map $d (\cdot, \beta) \colon x \mapsto d (x, \beta)$ satisfies $\sigma^n ( d (x, \beta) ) = d ( f_\beta^n (x), \beta)$. If we order $S_\beta$ with the lexicographical ordering then the map $d ( \cdot, \beta)$ is one-to-one and monotone increasing. The subshift $S_\beta$ satisfies $$\label{eq:Sbeta}
S_\beta = \{\, \{j_k\} : \sigma^n \{j_k\} < d (1, \beta) \ \forall n \,\}.$$
If $x \in [0,1]$ then $$x = \sum_{k=0}^\infty \frac{d_k (x, \beta) }{\beta^{k + 1}}.$$ We let $\pi_\beta$ be the map $\pi_\beta \colon S_\beta \to [0,1)$ defined by $$\pi_\beta \colon \{i_k\}_{k=0}^\infty \quad \mapsto \quad \sum_{k=0}^\infty \frac{i_k}{\beta^{k + 1}}.$$ Hence, $\pi_\beta ( d(x, \beta)) = x$ holds for any $x \in [0,1)$ and $\beta > 1$.
A cylinder $s$ is a subset of $[0,1)$ such that $$s :=\pi_\beta( \{\, \{j_k\}_{k=0}^\infty : i_k = j_k,\ 0\leq k < n \,\})$$ holds for some $n$ and some sequence $\{i_k\}_{k=0}^\infty$. We then say that $s$ is an $n$-cylinder or a cylinder of generation $n$ and write $$s = [i_0 \cdots i_{n-1}].$$ Consider $\beta$ such that the expansion of 1 terminates, such that $d(1, \beta)=j_0 \dots j_{k-1} 0^\infty$. The set of such $\beta$ is dense in $(1,2)$ and for such $\beta$ we can use (\[eq:Sbeta\]) to construct $S_\beta$ from the full shift $\Sigma_2=\{0,1\}^{\mathbb N}$ as follows. There are finitely many words $w$ of length $k$ such that $w<d(1, \beta)$. If we start with $\Sigma_2$ and remove all elements that contain any of these words, then by (\[eq:Sbeta\]) we get $S_\beta$. Thus $S_\beta$ is a subshift of finite type. Such shifts have have well-known properties that we can use to prove the following theorem.
\[SFTunitinterval\] Let $\beta \in (1,2)$ be such that the expansion of 1 terminates. Then for any $x\in [0,1]$ there is an $\alpha>0$ such that the set $$G_{f_\beta}(x)=\Big\{\, y\in [0,1): x\notin \overline{\cup_{n=0}^\infty f^n(y)}\, \Big\},$$ is $\alpha$-winning in $[0,1]$. In fact $\alpha=\frac{1}{4}$ is small enough.
Using Proposition \[winninggivesdim\] and Proposition \[intersectionproperty\] we get
Let $(\beta_i)_{i=1}^\infty$ be a sequence in $(1,2)$ such that that the expansion of 1 terminates for each $\beta_i$ and let $(x_i)_{i=1}^\infty$ be a sequence of points in $[0,1]$. Then $$\dim_H \Big(\bigcap_{i=1}^\infty G_{f_{\beta_i}}(x_i)\Big)=1.$$
Conditions on the maps {#conditions}
======================
Condition $\mathbf{(i)}$
------------------------
Assume that we did not have condition $(i)$. Depending on $f$, there might be points in $[0,1)$ which do not have well-defined representations as sequences. We will be playing the $(\alpha,\beta)$ game, trying to show that our sets are $\alpha$-winning. But if no further restrictions are put on $f$ this will not be possible, as the following example illustrates.
We are going to construct a function $f$ such that for each $\alpha>0$ there is a $\beta>0$ for which the set of points with well-defined representations as sequences is not $(\alpha, \beta)$-winning. First divide $(0,1)$ into the intervals $[\frac{1}{2^i},\frac{1}{2^{i-1}})$ where $i\in \mathbb N$. For each $i$ consider the corresponding interval. Split the interval into $4i$ subintervals of equal size. On every second of these let $f$ be linear onto $[0,1)$. Take all of the remaining subintervals and split them into $4i$ parts and continue this procedure indefinitely. After doing this for each $i$ we have defined a function $f$ except on a set of Lebesgue measure zero. Although this set is small with respect to Lebesgue measure we get into trouble.
For any $\alpha>0$, pick an $i \in \mathbb N$ such that $\frac{1}{i}<\alpha$. Let $\beta$ be such that $\alpha \beta = \frac{1}{4i}$. Let the player Black choose $B_0$ as the interval $[\frac{1}{2^i},\frac{1}{2^{i-1}})$. Then no matter how White chooses $W_0$, it is always possible for Black to choose $B_1$ as one of the $2i$ intervals on which $f$ was not defined until at smaller scale. The player Black can play so that this situation is repeated indefinitely. So, the points at which $f$ is well-defined is not $(\alpha,\beta)$-winning. Since $\alpha>0$ was arbitrary, this set is not $\alpha$-winning for any $\alpha>0$. So, with this $f$, we cannot use the $(\alpha,\beta)$-game.
It is clear that we avoid cases like this if we impose condition $(i)$ on $f$. For a given function $f$, condition $(i)$ may not be that easy to check so we give a sufficient condition for it to be satisfied.
\[endpointscondition\] Let $f$ be a function as described in Section \[givingfullshift\] and let $E(f)$ be the set of endpoints of generation $1$ cylinders. Let $Acc(E)$ denote the set of points of accumulation for a set $E$. If there is an $n\in \mathbb N$ such that $Acc^n(E(f))=\emptyset$, then condition $(i)$ is satisfied.
Assume that White is given an interval $I$ and wants to avoid all endpoints of generation $k$ cylinders. Since $Acc^n(E(f))$ is empty we know that $Acc^{n-1}(E(f))$ is finite. It is then easy for White to avoid this set in finitely many turns if $\alpha \leq \frac{1}{2}$. When this is done, White has placed a set $W_{j_1}$ such that it does not contain any points from $Acc^{n-1}(E(f))$. But then it can at most contain finitely many points from $Acc^{n-2}(E(f))$. Of course White can avoid these in the same way. By induction, White can avoid all points from $E(f)$ in a finite number of turns. This means that White can choose a set $W_{j_n}$ inside a generation $1$ cylinder $C_{x_1}$ after finitely many turns. Let $E(f^2)$ denote the set of endpoints of generation $2$ cylinders in $C_{x_1}$. If $n\geq 2$, White wants to avoid this set as well. But $E(f^2)$ is the inverse image of $E(f)$ under the homeomorphism $f|_{C_{x_1}}\colon C_{x_1} \mapsto [0,1)$. Thus, $E(f^2)$ has the same topological properties as $E(f)$. In particular, $Acc^n(E(f^2))=\emptyset$, so just as he avoided $E(f)$, White can avoid $E(f^2)$ in finitely many turns if $\alpha \leq \frac{1}{2}$. Repeating this argument, we get that White can avoid all endpoints of generation $k$ cylinders after a finite number of turns and place his set $W_j$ inside a generation $k$ cylinder for some finite $j$.
Note that while the condition in Lemma \[endpointscondition\] is sufficient to ensure condition $(i)$ it is by no means necessary. For example consider the middle third Cantor set. It is defined by repeatedly removing the middle third of each interval, starting with $[0,1]$. Let $f$ be the function obtained by letting $f$ be linear from $0$ to $1$ on each removed interval. Then $f$ is well-defined except on the middle third Cantor set which is a perfect set. Thus the conditions of Lemma \[endpointscondition\] are not fulfilled but it is obvious that in the $(\alpha,\beta)$-game, White only needs one turn to avoid the middle third Cantor set if $\alpha = \frac{1}{9}$. The set of endpoints of cylinders from higher generation will only be scalings of $E(f)$ since $f$ is linear on each cylinder. Thus, White can avoid the endpoints of the cylinders of any given generation in finitely many turns.
Condition $\mathbf{(ii)}$
-------------------------
To be able to prove Theorem \[fullshiftinunitinterval\] we need the cylinders to shrink in some uniform way. One way to get this is of course to require uniform expansion, that for some $\lambda>0$ it holds that $|f(x)-f(y)|\geq (1+\lambda)|x-y|$ for all $x,y$ in the same generation $1$ cylinder. We use the weaker Condition $\mbox{(ii)}$ to allow functions like $f\colon x\mapsto \frac{1}{x} \mod 1$.
The continued fraction expansion of numbers $x\in [0,1)$ which is given by the map $f\colon x\mapsto \frac{1}{x} \mod 1$, satisfies Condition $(ii)$.
Let $x\in (0,1)\setminus f^{-1}(0)$. Then $f^\prime (x)=-\frac{1}{x^2}$ and $|f^\prime (x)|\geq 1$. So, with $x\in (0,1)\setminus (f^{-1}(0)\cup f^{-2}(0))$ we have that if $|f^\prime (x)|\leq \frac{9}{4}$, then $$|f^\prime (x)|\leq \frac{9}{4} \quad \Rightarrow \quad x \geq \frac{2}{3} \quad \Rightarrow \quad f(x) \leq \frac{1}{2} \quad \Rightarrow \quad f^\prime (f(x))\geq 4.$$ So $|(f^2)^\prime (x)| \geq \frac{9}{4}> 2$ for all $x\in (0,1)\setminus (f^{-1}(0)\cup f^{-2}(0))$. This implies $$\frac{|C_{x_1\dots x_{n+m}}|}{|C_{x_1\dots x_n}|} \leq \sup_{x \in \tilde C_{x_1\dots x_{n+m}}} \frac{1}{(f^m)^\prime (x)} \leq 2^{-\lfloor \frac{m}{2}\rfloor} =g(m)$$ where $\tilde C_{x_1\dots x_{n+m}}$ means the interior of the cylinder $C_{x_1\dots x_{n+m}}$ and $\lfloor \frac{m}{2}\rfloor$ means the integer part of $\frac{m}{2}$.
Proofs
======
The idea we use to prove Theorem \[fullshiftinunitinterval\] and Theorem \[SFTunitinterval\] is to translate the $(\alpha,\beta)$-game into a game where the players are choosing symbols in a sequence rather than choosing intervals. By using a simple combinatorial argument we can then conclude that our sets are $\alpha$-winning.
A game of sequence building {#sequencebuildingsection}
---------------------------
Consider the following game for two players $\tilde B$ and $\tilde W$ with two parameters $c$ and $n$. The players are building a one sided infinite sequence $y=(y_i)_{i=1}^\infty$ in a finite or countable alphabet. First $\tilde B$ chooses $y=(y_i)_{i=1}^{b_0}$, where he can choose $b_0$ as large as he likes. Then, $(y_i)_{i=b_0+1}^\infty$ is divided into blocks of $n$ symbols.
(6,0.4)(0,0.4) (0,0.4)[(1,0)[3.6]{}]{} (0,0.8)[(1,0)[3.6]{}]{} (3.6,0.4)(0.2,0)[4]{}[(1,0)[0.1]{}]{} (3.6,0.8)(0.2,0)[4]{}[(1,0)[0.1]{}]{}
(0,0.4)[(0,1)[0.4]{}]{} (0.6,0.4)[(0,1)[0.4]{}]{} (1.6,0.4)[(0,1)[0.4]{}]{} (2.6,0.4)[(0,1)[0.4]{}]{} (3.6,0.4)[(0,1)[0.4]{}]{}
(-0.5,0.5)[$y:$]{} (0.2,0)[$b_0$]{} (1,0)[$n$]{} (2,0)[$n$]{} (3,0)[$n$]{}
The game is carried out in one block at a time, so we start in the first block. Consider a list of all possible words of length $n$. This might be infinite depending on whether or not the alphabet is finite. The player $\tilde B$ chooses two disjoint subsets of this list and lets $\tilde W$ pick any one of these two. After $\tilde W$ has made his choice, we have a new list of remaining words. Then $\tilde B$ chooses two disjoint subsets of this list and $\tilde W$ chooses one of these. The players continue like this and the game requires that $\tilde B$ plays so that regardless of how $\tilde W$ plays, this process ends after a finite number of turns, , that sooner or later only one word remains. This word is then put as $(y_i)_{i=b_0+1}^{b_0+n}$. The same procedure is carried out in each block and we get the sequence $y=(y_i)_{i=1}^\infty$. The game requires that $\tilde W$ gets to play at least $cn$ times in each block regardless of how he plays. This puts restrictions on how $\tilde B$ can construct his subsets. For example, at the first turn in a block, $\tilde B$ cannot choose one of his two subsets to consist of only one word.
\[digitgame\] Given any sequence $x=(x_i)_{i=1}^\infty$ and any $c>0$, there is a block size $n$ such that no matter how $\tilde B$ plays in the sequence building game, $\tilde W$ can make sure that there is a number $N$ such that $(x_i)_{i=1}^N \neq (y_i)_{i=k}^{k+N-1}$ for all $k \in \mathbb N$.
Assume that $\tilde B$ chooses the symbols $(y_i)_{i=1}^{b_0}$ and consider $(x_i)_{i=1}^{b_0+2n}$. If we want $y=(y_i)_{i=1}^\infty$ to be such that $(x_i)_{i=1}^{b_0+2n}$ does not occur anywhere in $y$, then it is enough to make sure that none of the $n$-blocks in $y$ occur in $(x_i)_{i=b_0+1}^{b_0+2n}$.
(6,1.8)(0,0)
(1.3,1.4)[(1,0)[2]{}]{} (1.3,1.8)[(1,0)[2]{}]{}
(1.3,1.4)[(0,1)[0.4]{}]{} (3.3,1.4)[(0,1)[0.4]{}]{}
(-0.5,1.5)[$(x_i)_{i=b_0+1}^{b_0+2n}$]{}
(1.6,0.8)(0,0.18)[6]{}[(0,1)[0.09]{}]{} (2.6,0.8)(0,0.18)[6]{}[(0,1)[0.09]{}]{}
(0,0.4)[(1,0)[3.6]{}]{} (0,0.8)[(1,0)[3.6]{}]{} (3.6,0.4)(0.2,0)[4]{}[(1,0)[0.1]{}]{} (3.6,0.8)(0.2,0)[4]{}[(1,0)[0.1]{}]{}
(0,0.4)[(0,1)[0.4]{}]{} (0.6,0.4)[(0,1)[0.4]{}]{} (1.6,0.4)[(0,1)[0.4]{}]{} (2.6,0.4)[(0,1)[0.4]{}]{} (3.6,0.4)[(0,1)[0.4]{}]{}
(-0.5,0.5)[$y$]{} (0.2,0)[$b_0$]{} (1,0)[$n$]{} (2,0)[$n$]{} (3,0)[$n$]{}
There are at most $n+1$ different words of length $n$ in $(x_i)_{i=b_0+1}^{b_0+2n}$ and it is sufficient for $\tilde W$ to avoid all these in each $n$-block. We will refer to the words that we want to avoid as dangerous words. In each $n$-block $\tilde W$ gets to make at least $cn$ choices between disjoint collections of words and thereby he can avoid many of the dangerous words. Indeed, the first time $\tilde W$ plays in a block he considers the two disjoint lists of words he is given by $\tilde B$. Since they are disjoint, at least one of the lists contains half or less of the dangerous words. By choosing this list, $\tilde W$ has avoided at least half of the dangerous words in just one play. The next time $\tilde W$ plays he is given two new disjoint lists of words to choose between. Remember that only at most half of the dangerous words are left among these, so $\tilde W$ can avoid at least half of the remaining dangerous words, leaving only at most $\frac{1}{4}$ of the original dangerous words after his second play. Continuing like this, if $2^{cn}>n+1$ he can avoid all the dangerous words in the $cn$ turns he has at each block. Since $c$ is fixed we can always find large enough $n$ such that this is true. It follows that if $\tilde W$ plays according to this strategy we have that $(x_i)_{i=b_0+1}^{b_0+2n} \neq (y_i)_{i=k}^{k+2n-1}$ for any $k\geq b_0+1$, so $(x_i)_{i=1}^{b_0+2n} \neq (y_i)_{i=k}^{k+b_0+2n-1}$ for any $k\geq 1$. Thus, $N=b_0+2n$ will do the job.
Proof of Theorem \[fullshiftinunitinterval\]
--------------------------------------------
The idea of this proof is to create a strategy for White in the $(\alpha,\beta)$-game so that White can play the role of $\tilde W$ in the sequence building game of Section \[sequencebuildingsection\]. We can then use Proposition \[digitgame\] to finish the proof. It might take several turns by White to be able to do what $\tilde W$ is supposed to do in one play. Each turn by $\tilde W$ will be divided into two phases consisting of turns by White. In the first phase, the task is to choose between disjoint collections of cylinders of some generation $k_i+k$. In the next phase, the task is to make sure that the game continues inside only one cylinder of generation $k_i+k$. This is to make sure that when we start over with phase one, the cylinders we are choosing between, all have the same coding up to the position $k_i+k$. Then choosing between disjoint collections of cylinders of generation $k_{i+1}+k$ is in fact the same thing as choosing between disjoint collections of codings of positions $k_i+k+1, \dots ,k_{i+1}+k$.
The $(\alpha,\beta)$-game starts when the player Black chooses his interval $B_0\subset [0,1]$.
$\mathbf{Phase 1:}$ Let $k_0$ be the largest generation for which there is a cylinder $C_{x_1 \dots x_{k_0}}$ intersecting $B_0$ such that $|C_{x_1 \dots x_{k_0}}|\geq |B_0|$. It might for example be that $k_0=0$, so that $C_{x_1 \dots x_{k_0}}=[0,1)$. By the maximality of $k_0$, all generation $k_0+1$ cylinders intersecting $B_0$ are smaller than $|B_0|$. By condition $(i)$ we know that all cylinders of generation $k_0+k$ intersecting $B_0$ are smaller than $g(k-1)|B_0|$. Let $k$ be a number such that $g(k-1)< \frac{1}{4}$. This is possible since $g(n)\to 0$ as $n\to \infty$, and it implies that the largest cylinder of generation $k_0+k$ intersecting $B_0$ is smaller than $\frac{|B_0|}{4}$.
Let $C'$ be the generation $k_0+k$ cylinder containing the center point of $B_0$. It follows that $B_0 \setminus C'$ consists of two intervals, each of length larger than $\frac{|B_0|}{4}$. Each of these intervals intersects a family of generation $k_0+k$ cylinders and these two families are disjoint. Each family of generation $k_0+k$ cylinders corresponds to a family of codings of positions $1,\dots,k_0+k$. Recall that in the $(\alpha,\beta)$-game, after Black chooses $B_0$, the other player, White, chooses a ball $W_0\subset B_0$ such that $|W_0|=\alpha|B_0|$. So, with $\alpha \leq \frac{1}{4}$, White can choose between two disjoint collections of codings at positions $1, \dots, k_0+k$ by placing $W_0$ to the left or right of $C'$.
(6,1)(0,0.2) (0,0.4)[(1,0)[6]{}]{} (0,0.8)[(1,0)[6]{}]{}
(0,0.4)[(0,1)[0.4]{}]{} (2.3,0.4)[(0,1)[0.4]{}]{} (3.5,0.4)[(0,1)[0.4]{}]{} (6,0.4)[(0,1)[0.4]{}]{}
(0,0.9)[$\overbrace{\rule{7.2cm}{0cm}}$]{} (2.9,1.2)[$B_0$]{} (2.8,0)[$C'$]{}
$\mathbf{Phase 2:}$ After this is done Black will choose an interval $B_1\subset W_0$ and it is up to White to place $W_1$ inside it. We want White to place $W_1$ inside a cylinder of generation $k_0+k$. It might happen that these generation $k_0+k$-cylinders are so small that White cannot do this right away. But by condition $(ii)$, we know that with $\alpha\leq \alpha_0$ then for every $\beta>0$ there is a strategy for the $(\alpha,\beta)$-game that White can use to place his set inside a generation $k_0+k$ cylinder after a finite number of turns, no matter how Black plays.
(6,1)(0,0.2) (0,0.4)[(1,0)[6]{}]{} (0,0.8)[(1,0)[6]{}]{}
(0,0.4)[(0,1)[0.4]{}]{} (0.4,0.4)[(0,1)[0.4]{}]{} (0.9,0.4)[(0,1)[0.4]{}]{} (1.2,0.4)[(0,1)[0.4]{}]{} (1.5,0.4)[(0,1)[0.4]{}]{} (2.3,0.4)[(0,1)[0.4]{}]{} (3.1,0.4)[(0,1)[0.4]{}]{} (3.3,0.4)[(0,1)[0.4]{}]{} (3.6,0.4)[(0,1)[0.4]{}]{} (4,0.4)[(0,1)[0.4]{}]{} (4.5,0.4)[(0,1)[0.4]{}]{} (5.1,0.4)[(0,1)[0.4]{}]{} (6,0.4)[(0,1)[0.4]{}]{}
(0,0.9)[$\overbrace{\rule{7.2cm}{0cm}}$]{} (2.9,1.2)[$B_1$]{} (1.3,0)[generation $k_0+k$-cylinders]{}
If White can place his set $W_1$ inside a cylinder $C_{x_1 \dots x_{k_0+k}}$, then he does. If he cannot, then he uses the following strategy.
First he places $W_1$ so that it only intersects cylinders $C_{x_1 \dots x_{k_0+k}}$ that are contained in $B_1$. This is possible since if there are subsets of cylinders $C_{x_1 \dots x_{k_0+k}}$ in $B_1$, then these parts together cannot constitute more than $2\alpha |B_1|$, otherwise White would have chosen $W_1$ inside one of them. At his next play, if he can place $W_2$ inside a cylinder $C_{x_1 \dots x_{k_0+k}}$ he does. Otherwise White chooses his set $W_2$ according to a $(\alpha,\alpha \beta^2)$-game strategy that allows him to avoid endpoints of generation $k_0+k$ cylinders after finitely many turns. At his next turn if White could not fit $W_3$ inside a cylinder $C_{x_1 \dots x_{k_0+k}}$ he plays $W_3$ so that he avoids generation $k_0+k$ cylinders that are not contained in $B_3$. As long as he cannot place his set inside a cylinder $C_{x_1 \dots x_{k_0+k}}$ White continues like this, every second turn playing to avoid endpoints of generation $k$ cylinders and the rest of the turns playing to avoid cylinders not contained in the set chosen but Black. Then sooner or later White will be able to place his set inside a cylinder $C_{x_1 \dots x_{k_0+k}}$ and he stops.
Let $j_0$ be the number of the turn at which White could play so that his set $W_j \subset C_{x_1 \dots x_{k_0+k}}$. If White needed more than one turn to accomplish this, it means that $C_{x_1 \dots x_{k_0+k}}\subset B_{j_0-2}$. Indeed, at every second play, White makes sure that all cylinders $C_{x_1 \dots x_{k_0+k}}$ that are not fully contained in the set chosen by Black are avoided. We conclude that in this case we have $|C_{x_1 \dots x_{k_0+k}}|\leq |B_{j_0-2}|$.
Now, White has used the turns $0,1,2,\dots ,j_0$ to make the first turn by $\tilde W$ in the sequence building game by choosing between disjoint collections of codings of positions $1, \dots, k_0+k$. He also uses these turns to make sure that the coding of positions $1, \dots, k_0+k$ is fixed after turn number $j_0$. Later on, this fact will allow White to to create the next turn by $\tilde W$.
After turn $j_0$ by White, Black will choose an interval $B_{j_0+1}\subset W_{j_0}$ and we start creating the next turn by $\tilde W$ in the sequence building game.
$\mathbf{Phase 1:}$ Let $k_1$ be the largest generation for which there is a cylinder $C_{x_1 \dots x_{k_1}}$ intersecting $B_{j_0+1}$ such that $|C_{x_1 \dots x_{k_1}}|\geq |B_{j_0+1}|$. Repeating what we did after finding $k_0$, we get that with $\alpha \leq \frac{1}{4}$, White can choose between two disjoint collections of generation $k_1+k$ cylinders. We know that all of these are in the same generation $k_0+k$ cylinder, so White can choose between two disjoint collections of codings at positions $k_0+k+1, \dots, k_1+k$.
$\mathbf{Phase 2:}$ We can then continue as before with $\alpha \leq \alpha_0$, finding a minimal $j_1$ such that $W_{j_1}$ can be placed in a cylinder $|C_{x_1 \dots x_{k_1+k}}|$. Again, if it took more than one turn by White to do this we have $|C_{x_1 \dots x_{k_1+k}}|\leq |B_{j_1-2}|$.
Now, White has used the turns $j_0+1,\dots, j_1$ to make the second turn by $\tilde W$ in the sequence building game by choosing between disjoint collections of codings of positions $k_0+k+1, \dots, k_1+k$ and prepared so that he will be able to make the next turn by $\tilde W$ later on.
We can continue repeating this procedure for each $i\geq 0$ constructing a turn by $\tilde W$ in which $\tilde W$ gets to choose between disjoint collections of codings at positions $k_i+k+1, \dots, k_{i+1}+k$.
(6,1)(1,0.2) (0,0.4)[(1,0)[6]{}]{} (0,0.8)[(1,0)[6]{}]{}
(0,0.4)[(0,1)[0.4]{}]{}
(6,0.4)[(0,1)[0.4]{}]{}
(0,0.4)(0.2,0)[30]{}[(0,1)[0.4]{}]{}
(-0.45,0.53)[$y:$]{} (0.2,-0.1)[$k_0$]{} (0.3,0.2)[(0,1)[0.25]{}]{} (1,-0.1)[$k_1$]{} (1.1,0.2)[(0,1)[0.25]{}]{} (1.9,-0.1)[$k_2$]{} (1.9,0.2)[(0,1)[0.25]{}]{} (4.3,-0.1)[$k_3$]{} (4.3,0.2)[(0,1)[0.25]{}]{} (5.5,-0.1)[$k_4$]{} (5.5,0.2)[(0,1)[0.25]{}]{}
(6,0.4)(0.2,0)[4]{}[(1,0)[0.1]{}]{} (6,0.8)(0.2,0)[4]{}[(1,0)[0.1]{}]{}
Next we will show that $k_{i+1}-k_i$ is bounded. We begin by recalling that we had a function $g$ that gave us a speed at which cylinders shrunk in size as the generation increased. We used this function to find a constant $k$ such that when we increased the generation by $k$ the size shrunk by at least a factor $4$. Since the number $k$ originates from potentially very crude estimates it tells us nothing about the size of $k_{i+1}-k_i$. In some cases, it might well happen that $k_{i+1}-k_i=1$ while for example $k=10$. When looking for a uniform bound on $k_{i+1}-k_i$ it will be convenient to consider only the case $k_{i+1}-k_i>k$. Since we are looking for an upper bound, the case $k_{i+1}-k_i\leq k$ is uninteresting.
We start at phase $1$ when constructing turn number $i$ for $\tilde W$ in the sequence building game. First Black plays by choosing a set $B_1$, then White chooses between two disjoint collections of generation $k_i+k$ cylinders. Then phase $2$ starts as Black plays again. Assume now that White is able to place his set inside a generation $k_i+k$ cylinder at his first turn in phase $2$. This ends phase $2$ and means the end of turn number $i$ for $\tilde W$ in the sequence building game.
After this, it is time to construct turn number $i+1$ by $\tilde W$. Black starts phase $1$ by choosing a set $B_2$. Then we find the maximal generation $k_{i+1}$ such that $B_2$ intersects a cylinder $C_{x_1 \dots x_{k_i+1}}$ such that $|C_{x_1 \dots x_{k_i+1}}|\geq |B_2|$. Since $|B_2|=(\alpha\beta)^2 |B_1|$ we get $$\begin{aligned}
|B_{2}| \leq &|C_{x_1\dots x_{k_{i+1}}}| \leq g(k_{i+1}-k_i-k)|C_{k_1 \dots x_{k_i+k}}|\\
< &g(k_{i+1}-k_i-k)|B_1|=\frac{g(k_{i+1}-k_i-k)|B_2|}{(\alpha \beta )^2}\end{aligned}$$ so $g(k_{i+1}-k_i-k) <(\alpha \beta)^2$. Since $g(n)\to 0$ as $n\to \infty$ this puts a bound on $k_{i+1}-k_i$.
Assume instead that when constructing turn number $i$ for $\tilde W$ in the sequence building game, White needed more than one turn in phase $2$, to place his set inside a generation $k_i+k$ cylinder. We recall that if $W_j$ is the last set chosen by White in this phase, then $|C_{x_1 \dots x_{k_i}}|< |B_{j-2}|$. After this, it is time to construct turn number $i+1$ by $\tilde W$. Black starts phase $1$ by choosing a set $B_{j+1}$. Then we find the maximal generation $k_{i+1}$ such that $B_{j+1}$ intersects a cylinder $C_{x_1 \dots x_{k_{i+1}}}$ such that $|C_{x_1 \dots x_{k_{i+1}}}|\geq |B_{j+1}|$. We get $$\begin{aligned}
|B_{j+1}| \leq &|C_{x_1\dots x_{k_{i+1}}}| \leq g(k_{i+1}-k_i-k) |C_{x_1 \dots x_{k_{i}+k}}| \\
< & g(k_{i+1}-k_i-k)|B_{j-2}|=\frac{g(k_{i+1}-k_i-k)|B_{j+1}|}{(\alpha \beta)^3},\end{aligned}$$ so $g(k_{i+1}-k_i-k) <(\alpha \beta)^3$. Since $g(n)\to 0$ as $n\to \infty$ this puts a bound on $k_{i+1}-k_i$.
What we have proven this far is that if we choose $\alpha\leq \min\{\alpha_0,\frac{1}{4}\}$, then the sequence $k_i$ has a maximal distance between its elements. This implies that if the block size $n$ is large enough, then in the following picture
(6,1)(1,0) (0,0.4)[(1,0)[3.6]{}]{} (0,0.8)[(1,0)[3.6]{}]{} (3.6,0.4)(0.2,0)[4]{}[(1,0)[0.1]{}]{} (3.6,0.8)(0.2,0)[4]{}[(1,0)[0.1]{}]{}
(0,0.4)[(0,1)[0.4]{}]{} (0.6,0.4)[(0,1)[0.4]{}]{} (1.6,0.4)[(0,1)[0.4]{}]{} (2.6,0.4)[(0,1)[0.4]{}]{} (3.6,0.4)[(0,1)[0.4]{}]{}
(-0.5,0.5)[$y:$]{} (0.2,0)[$b_0$]{} (1,0)[$n$]{} (2,0)[$n$]{} (3,0)[$n$]{}
there is at least one $k_i$ in each $n$-block. Increasing $n$ we can clearly make sure that there are at least $cn$ different $k_i$ in each $n$-block for some $c>0$. This implies that if we play the $(\alpha,\beta)$-game in $[0,1]$ with $\alpha\leq \min\{\alpha_0, \frac{1}{4}\}$, then White can use a strategy that transforms the game into the sequence building game. By Proposition \[digitgame\] the player $\tilde W$ can make sure that we get a number in $\Big\{\, z\in [0,1): x\notin \overline{\cup_{n=1}^\infty f^n(z)}\, \,\Big\}$ for any given $x\in [0,1)$ with well-defined expansion, by choosing the block size $n$ in the sequence building game. Since this can be done for any $\alpha\leq \min\{\alpha_0, \frac{1}{4}\}$ and any $\beta>0$ we conclude that $\Big\{\, z\in [0,1): x\notin \overline{\cup_{n=1}^\infty f^n(z)}\, \, \Big\}$ is $\alpha$-winning for all $x\in [0,1)$ with well-defined expansion, if $\alpha \leq \min\{\alpha_0, \frac{1}{4}\}$. This proves the theorem.
Proof of Theorem \[SFTunitinterval\]
------------------------------------
Since the symbol $\beta$ is already used in the $(\alpha,\beta)$-game we will use $b$ instead of $\beta$ to denote the base in the $\beta$-shift.
The method used to prove Theorem \[fullshiftinunitinterval\] works in this case as well, but now we do not have to worry about countable alphabets and points without well-defined expansions. Since $S_b$ is of finite type there is a constant $C_b$ such that $$C_b^{-1}<\frac{C_{x_0\dots x_{n-1}}}{b^{n}}<C_b$$ for all $n$ and all $(x_i)_{i=0}^\infty \in S_b$. This implies that $$\frac{|C_{x_0\dots x_{n+m}}|}{|C_{x_0\dots x_{n}}|} \leq \frac{C_b^2}{b^m}$$ for all $m,n$ and all $(x_i)_{i=0}^\infty \in S_b$. Thus we can let $\frac{C_b^2}{b^m}$ play the role of $g(m)$ from the proof of Theorem \[fullshiftinunitinterval\].
We will now briefly describe how White plays in the $(\alpha,\beta)$-game to construct turn number $i$ in the sequence building game. It all begins as usual with the player Black choosing a set $B_i$.
$\mathbf{Phase 1:}$ We do as in proof of Theorem \[fullshiftinunitinterval\]. We find a minimal $k_i$. Then we choose $k$ large enough so that by placing $W_i$ White can choose between two disjoint collections of generation $k_i+k$ cylinders. For example, $k\geq 1+\frac{4C_b^2}{\log b}$ will be enough.
$\mathbf{Phase 2:}$ We do as in the proof of Theorem \[fullshiftinunitinterval\]. We let White alternate between avoiding cylinders not contained in the sets chosen by Black and avoiding endpoints of generation $k_i+k$ cylinders until White can place his set in a generation $k_i+k$ cylinder.
Just as in the proof of Theorem \[fullshiftinunitinterval\] we conclude that with this sequence of turns, White is able to choose between disjoint collections of codings of positions $k_{i-1}+k+1, \dots k_i+k$. We then do as in proof of Theorem \[fullshiftinunitinterval\] to show that $k_{i+1}-k_i$ is bounded. We then apply Proposition \[digitgame\] to conclude that $\Big\{\, z\in [0,1): x\notin \overline{\cup_{n=1}^\infty f_b^n(z)}\, \, \Big\}$ is $\alpha$-winning for all $x\in [0,1]$ and all $\alpha \leq \frac{1}{4}$. This proves the theorem.
A note on the [$\boldsymbol{(\alpha,\beta)}$-game]{}
----------------------------------------------------
We note that in the proofs of Theorem \[fullshiftinunitinterval\] and Theorem \[SFTunitinterval\], the strategies we describe for White use the fact that Black can not zoom in more than a fixed factor $\gamma$ at each turn in each given game. It would not matter at all for the strategies if Black was allowed at each turn to choose $\gamma \in [\gamma_0,1] $ for some fixed $\gamma_0$. If we also allow White to choose $\alpha \in [\alpha_0,1] $, White can still use the same strategy. This leads us to consider the following modification of the $(\alpha,\beta)$-game.
Let $\alpha_0,\gamma_0 \in (0,1)$ be fixed.
- [*In the initial step*]{} Black chooses any closed interval $B_0$, and then White chooses an $\alpha \in [\alpha_0, 1]$ and a closed interval $W_0 \subset B_0$ such that $|W_0| = \alpha |B_0|$.
- [*Then the following step is repeated.*]{} At step $k$ Black choses $\gamma\in [\gamma_0,1]$ and a closed interval $B_{k} \subset W_{k-1}$ such that $|B_{k}| = \gamma |W_{k-1}|$. Then White chooses a new $\alpha \in [\alpha_0, 1]$ and a closed interval $W_{k} \subset B_k$ such that $|W_k| = \alpha |B_k|$.
The following observation now follows.
In Theorems \[fullshiftinunitinterval\] and \[SFTunitinterval\] with corollaries, the $(\alpha,\beta)$-game can be replaced by the modified $(\alpha,\beta)$-game described in this section.
[0]{}
W. Parry, [*On the $\beta$-expansion of real numbers*]{}, Acta Mathematica Academiae Scientiarum Hungaricae, 11 (1960), 401–416.
A. Rényi, [*Representations for real numbers and their ergodic properties*]{}, Acta Mathematica Academiae Scientiarum Hungaricae, 8 (1957), 477–493.
W. Schmidt, [*On badly approximable numbers and certain games*]{}, Trans. Am. Math. Soc. 123 (1966), 178–199.
|
---
abstract: |
Here we investigate some aspects of stochastic acceleration of ultrarelativistic electrons by magnetic turbulence. In particular, we discuss the steady-state energy spectra of particles undergoing momentum diffusion due to resonant interactions with turbulent MHD modes, taking rigorously into account direct energy losses connected with different radiative cooling processes. For the magnetic turbulence we assume a given power spectrum of the type $\mathcal{W}(k) \propto k^{-q}$. In contrast to the previous approaches, however, we assume a finite range of turbulent wavevectors $k$, consider a variety of turbulence spectral indexes $1 \leq q \leq 2$, and concentrate on the case of a very inefficient particle escape from the acceleration site. We find that for different cooling and injection conditions, stochastic acceleration processes tend to establish a modified ultrarelativistic Maxwellian distribution of radiating particles, with the high-energy exponential cut-off shaped by the interplay between cooling and acceleration rates. For example, if the timescale for the dominant radiative process scales with the electron momentum as $\propto p^r$, the resulting electron energy distribution is of the form $n_{\rm e}(p) \propto p^2 \, \exp\left[ - {1
\over a} \, \left(p / p_{\rm eq}\right)^a\right]$, where $a = 2-q-r$, and $p_{\rm eq}$ is the equilibrium momentum defined by the balance between stochastic acceleration and energy losses timescales. We also discuss in more detail the synchrotron and inverse-Compton emission spectra produced by such an electron energy distribution, taking into account Klein-Nishina effects. We point out that the curvature of the high frequency segments of these spectra, even though being produced by the same population of electrons, may be substantially different between the synchrotron and inverse-Compton components.
author:
- Łukasz Stawarz and Vahe Petrosian
title: On the Momentum Diffusion of Radiating Ultrarelativistic Electrons in a Turbulent Magnetic Field
---
Introduction
============
Stochastic acceleration of ultrarelativistic particles via scatterings by magnetic inhomogeneities was the first process discussed in the context of generation of a power-law energy distribution of cosmic rays [@fer49; @dav56]. Because the characteristic acceleration timescale for a given velocity of magnetic inhomogeneities, say Alfv[é]{}n velocity $v_A$, is $t_{\rm acc} \propto (v_{\rm A} / c)^{-2}$, the stochastic particle acceleration is often referred as a ‘2nd-order Fermi process’. For commonly occuring non-relativistic turbulence, $v_{\rm A} \ll c$, turbulent acceleration mechanism is often deemed less efficient when compared to acceleration by shocks where the rate of momentum change $\delta
p/p \sim v_{\rm sh} / c$ (hence the name 1st-order Fermi process). However, here one also needs repeated crossing of the shock front by the particles which can come about via scattering by turbulence upstream and downstream of the shock. Thus, again the acceleration rate or timescale is determined by the scattering time scale. For nonrelativistic turbulence $v_A\ll
c$, relativistic particles $p \gg m c^2$, and high-$\beta$ or weakly magnetized plasma, this time is shorter than the stochastic acceleration time, which may not be the case in many astrophisical plasmas. We note that in a relativistic regime, for example, 1st-order Fermi process encounters several difficulties in accelerating particles to high energies [e.g., @nie06a; @nie06b; @lem06], while at the same time stochastic particle energization may play a major role, since velocities of the turbulent modes may be high, $v_{\rm A} \lesssim c$. And indeed, 2nd-order Fermi processes were being discussed in the context of different astrophysical sources of high energy radiation and particles, such as accretion discs [e.g., @liu04; @liu06], clusters of galaxies [e.g., @pet01; @bru07], gamma-ray bursts [e.g., @ste04], solar flares [e.g., @pet99; @pet04], blazars [e.g. @kat06b; @gie07], or extragalactic large-scale jets [e.g., @sta02; @sta04]. We note, that although turbulent acceleration is often a process of choice in modeling high energy emission in different objects, and in fact there may be some other yet much less understood mechanisms responsible for generation of such (like magnetic reconnection), evidences for the distributed (or [*in situ*]{}) acceleration process taking place in several astrophysical systems are strong [see, e.g., @jes01; @ks06; @har07 in the context of extragalactic jets].
It was pointed out by @sch84 [@sch85], that continuous (stochastic) acceleration of high energy electrons undergoing radiative energy losses tends to establish their ultrarelativistic Maxwellian energy distribution, as long as particle escape from the acceleration site is inefficient. This analysis concerned a particular case of acceleration timescale independent on the electrons’ energy, and the dominant synchrotron-type energy losses. Interestingly, very flat (inverted) electron spectra of the ultrarelativistic Maxwellian-type — often approximated as a monoenergetic electron distribution — were discussed in the context of flat-spectrum radio emission observed from Sgr A$^{\star}$ and several active galactic nuclei [see, e.g., @bec97; @bir01 and references therein]. More recently, it was proposed that such ‘non-standard’ electron spectra can account for striking high-energy X-ray emission of large-scale jets observed by [*Chandra*]{} satellite [@sta02; @sta04], or correlated X-ray and $\gamma$-ray (TeV) emission from several BL Lac objects detected by the modern ground-base Cherenkov Telescopes [@kat06a; @gie07]. In addition, it was shown that narrow electron spectra, e.g. Maxwellian distribution, can explain properties of extragalactic high brightness temperature radio sources [@tsa07a; @tsa07b], alleviating the difficulties associated with the anticipated by not observed inverse-Compton catastrophe [@ost06].
Motivated by these most recent observational and theoretical results, in this paper we investigate further some aspects of stochastic acceleration of ultrarelativistic electrons by magnetic turbulence. In particular, we discuss steady-state energy spectra of particles undergoing momentum diffusion due to resonant interactions with turbulent MHD modes, taking rigorously into account direct [*in situ*]{} energy losses connected with different radiative cooling processes. As described in the next section §2, we use the quasilinear approximation for the wave-particle interactions, assuming a given power spectrum $\mathcal{W}(k) \propto k^{-q}$ for magnetic turbulence within some finite range of turbulent wavevector $k_1<k<k_2$, and consider turbulence spectral indexes in the range $1 \leq q \leq 2$. In section §3 we provide steady-state solutions to the momentum diffusion equation corresponding to the case of no particle escape but different cooling and injection conditions. In section §4 some particular solutions are given corresponding to the case of a finite particle escape from the acceleration site. In section §5 we discuss in more details synchrotron and inverse-Compton emission spectra of stochastically accelerated electrons, taking into account Klein-Nishina effects. Final discussion and conclusions are presented in the last section §6 of the paper.
General Description
===================
Let us denote the phase space density of ultrarelativistc particles by $f(\vec{x}, \vec{p}, t)$, such that the total number of particles is $\mathcal{N}(t) = \int d^3 x \int d^3 p \, f(\vec{x}, \vec{p}, t)$. Here the position coordinate $\vec{x}$ and the momentum coordinate $\vec{p}$ are not the position and the momentum of some particular particle, but are fixed to the chosen coordinate space, and therefore are independent variables. In the case of collisionless plasma, the function $f(\vec{x}, \vec{p}, t)$ satisfies the relativistic Vlasov equation with the acceleration term being determined by the Lorentz force due to the *average* plasma electromagnetic field acting on particles. This averaged field can be found, in principle, through the Maxwell equations, and such an approach would lead to the exact description of the considered system. However, due to strongly non-linear character of the resulting equations (and therefore substantial complexity of the problem), in most cases an approximate description is of interest. In the ‘test particle approach’, for example, one assumes configuration of electromagnetic field and solves the particle kinetic equation to determine particle spectrum. Further simplification can be achieved if one assumes presence of only a small-amplitude turbulence $(\delta \vec{E}, \delta \vec{B})$ in addition to the large-scale magnetic field[^1] $\vec{B}_0\gg \delta \vec{B}$, such that the total plasma fields are $\vec{B} =
\vec{B}_0 + \delta \vec{B}$ and $\vec{E} = \delta \vec{E}$.
In order to find the evolution of the particle distribution function in the phase space under the influence of such fluctuating electromagnetic field, it is convenient to consider an ensemble of the distribution functions (all equal at some initial time), such that the appropriate ensemble-averaging gives $\langle
\delta \vec{B} \rangle = \langle \delta \vec{E} \rangle = 0$ and $f(\vec{x},
\vec{p}, t) = \langle f(\vec{x}, \vec{p}, t) \rangle + \delta f(\vec{x},
\vec{p}, t)$. It can be then shown via the ‘quasilinear approximation’ of the Vlasov equation that the ensemble-average of the distribution function $\langle
f(\vec{x}, \vec{p}, t) \rangle$ satisfies the Fokker-Planck equation [@hal67; @mel68][^2]. If, in addition, the particle distribution function is only slowly varying in space (‘diffusion approximation’), and the scattering time (or mean free path) is shorter than all other relevant times (or mean free paths), the ensemble-averaged particle distribution function can be assumed to be spatially uniform and isotropic in $p$, namely $\langle
f(\vec{x}, \vec{p}, t) \rangle = \langle f(p, t) \rangle$, and the Fokker-Planck equation can be further reduced to the momentum diffusion equation [see @tsy77; @mel80; @sch02].
The resulting momentum diffusion equation describing evolution of the particle distribution can be written as $${\partial \over \partial t} \langle f(p, t) \rangle = {1 \over p^2} \, {\partial
\over \partial p} \, \left[ p^2 \, D(p) \, {\partial \over \partial p} \,
\langle f(p, t) \rangle \right] \, ,
\label{diff1}$$ where the momentum diffusion coefficients $D(p)$ approximates the rate of interaction with fluctuating electromagnetic field. Several other terms representing physical process that may influence evolution of the particle energy spectrum can be added to the diffusion equation (\[diff1\]). In particular, one can include continuous energy gains and losses due to direct acceleration (e.g., by shocks) and radiative cooling. Furthermore, if the diffusion of particles out of the turbulent region is approximated by a catastrophic escape rate (or time $t_{\rm esc}$), and if there is a source term $\widetilde{Q}(p, t)$ representing particle injection into the system, then the spatially integrated (over the turbulent region) one-dimensional particle momentum distribution, $n(p, t) \equiv 4 \pi \, p^2 \, \langle f(p, t) \rangle$, is obtained from [see, e.g., @pet04] $${\partial n(p, t) \over \partial t} = {\partial \over \partial p} \, \left[D(p)
\, {\partial n(p, t) \over \partial p} \right] - {\partial \over \partial p} \,
\left[ \left( {2 \, D(p) \over p} + \langle \dot{p} \rangle \right) \, n(p, t)
\right] - {n(p, t) \over t_{\rm esc}} + \widetilde{Q}(p, t).
\label{diff2}$$
Let us further assume presence of an isotropic Alfvénic turbulence described by the one-dimensional power spectrum $\mathcal{W}(k) \propto k^{-q}$ with $1
\leq q \leq 2$ in a *finite* wavevector range $k_1 \leq k \leq k_2$, such that the turbulence energy density $\int_{k_1}^{k_2} dk \, \mathcal{W}(k) =
(\delta B)^2 / 8 \pi$ is small compared with the ‘unperturbed’ magnetic field energy density, $\zeta \equiv (\delta B)^2 / B_0^2 < 1$. The momentum diffusion coefficient in equations (\[diff1\]-\[diff2\]) can be then evaluated [e.g., @mel68; @kul69; @sch89] as $$D(p) \approx {\zeta \, \beta_{\rm A}^2 \, p^2 \, c \over r_{\rm g}^{2-q} \,
\lambda_2^{q-1}} \, \propto p^q \, ,
\label{coef}$$ where $\lambda_2 = 2 \pi / k_1$ is the maximum wavelength of the Alfvén modes, $v_{\rm A} \equiv \beta_{\rm A} \, c$ is the Alfvén velocity, and $r_{\rm g} =
p c / e B_0$ is the gyroradius of *ultrarelativistic particles* of interest here. Similar formulae can be derived for the case of fast magnetosonic modes [e.g., @kul71; @ach81; @sch98]. This allows one to find the characteristic acceleration timescale due to stochastic particle-wave interactions, $t_{\rm acc} \equiv p^2 / D(p) \propto p^{2-q}/\beta_A^2$. Similarly, the escape timescale due to particle diffusion from the system of spatial scale $L$ can be evaluated as $t_{\rm esc} = L^2 / \kappa_{||} \propto
p^{q-2}$, where the spatial diffusion coefficient $\kappa_{||} = (1/3) \, c \,
\Lambda$ is given by the appropriate particle mean free path, $\Lambda \approx
(1/3) \, \zeta^{-1} \, r_{\rm g} \, (\lambda_2/r_{\rm g})^{q-1} \propto
p^{2-q}$, that can be found from the standard relation $D(p) = (1/3) \,
\beta_{\rm A}^2 \, p^2 \, c / \Lambda$ [for more details see, e.g., @sch02].
For convenience we define the dimensionless momentum variable $\chi \equiv p /
p_0$, where $p_0$ is some chosen (e.g., injection) particle momentum. With this, the (stochastic) acceleration and escape timescales can be written as $$\begin{aligned}
t_{\rm acc} & = & \tau_{\rm acc} \, \chi^{2-q} \, , \quad {\rm where} \quad
\tau_{\rm acc} \equiv {\lambda_2 \over \zeta \, \beta_{\rm A}^2 \, c} \,
\left({p_0 \, c \over e B_0 \, \lambda_2}\right)^{2-q} \, , \nonumber
\\
t_{\rm esc} & = & \tau_{\rm esc} \, \chi^{q-2} \, , \quad {\rm where} \quad
\tau_{\rm esc} \equiv {9 L^2 \, \zeta \over \lambda_2 \, c} \, \left({p_0 \, c
\over e B_0 \, \lambda_2}\right)^{q-2} \, . \label{timescales}\end{aligned}$$ Hereafter we also consider regular energy changes, strictly energy losses, being an arbitrary function of the particle energy as given by the appropriate timescale $t_{\rm loss}=t_{\rm loss}(p)$, namely $\langle \dot{p} \rangle = - (p
/ t_{\rm loss})$. We define further $\tau \equiv t / \tau_{\rm acc}$, $N(\chi,
\tau) \equiv p_0 \, n(p, t) \, V$, and $Q(\chi, \tau) \equiv \tau_{\rm acc} \,
p_0 \, \widetilde{Q}(p, t) \, V$, where $V = \int d^3 x$ is the volume of the system. With such, the momentum diffusion equation (\[diff2\]) reads as $${\partial N \over \partial \tau} = {\partial \over \partial \chi} \left[\chi^q
\, {\partial N \over \partial \chi} \right] - {\partial \over \partial \chi}
\left[\left(2 \, \chi^{q-1} - \chi\, \vartheta_{\chi}\right) N\right] -
\varepsilon \, \chi^{2-q} \, N + Q \, ,
\label{eqfinal}$$ or, in its steady-state ($\partial N / \partial \tau=0$) form, as $${\partial \over \partial \chi} \left[\chi^q \, {\partial N \over \partial \chi}
\right] - {\partial \over \partial \chi} \left[\left(2 \, \chi^{q-1} - \chi\,
\vartheta_{\chi}\right) N\right] - \varepsilon \, \chi^{2-q} \, N = - Q \, .
\label{steady}$$ In the above, we have introduced $$\vartheta_{\chi} \equiv {\tau_{\rm acc} \over t_{\rm loss}(\chi)} \quad {\rm
and} \quad \varepsilon \equiv {\tau_{\rm acc} \over \tau_{\rm esc}} \, .
\label{Gamma-definition}$$
Some specific solutions to the equation (\[eqfinal\]) were presented in the literature. Majority of investigations concentrated on the ‘hard-sphere approximation’ with $q=2$, i.e. with the mean free path for particle-wave interaction independent of particle energy ($\Lambda = \zeta \, \lambda_2 / 3$; ‘classical’ Fermi-II process). It was found, that in the absence of regular energy losses ($\vartheta_{\chi} = 0$), the steady-state solution of equation (\[steady\]) with the source term $Q(\chi) \propto \delta(\chi-\chi_{\rm
inj})$, where $\delta(\chi)$ is the Dirac delta, is of a power-law form $N(\chi>\chi_{\rm inj}) \propto \chi^{-\sigma}$ with $\sigma = -(1/2) + [(9/4) +
\varepsilon]^{1/2}$ [@dav56; @ach79; @par95]. Note, that for $\varepsilon \ll
1$ this can be approximated by $\sigma \approx 1 + \varepsilon/3$, which is the original result obtained by @fer49. In addition, with the increasing escape timescale, $\varepsilon \rightarrow 0$, the steady-state solution approaches $N(\chi>\chi_{\rm inj}) \propto \chi^{-1}$. This agrees with the general finding that for the range $1 \leq q < 2$ and the same injection conditions the steady-state particle energy distribution implied by the equation (\[steady\]) is $N(\chi>\chi_{\rm inj}) \propto \chi^{1-q}$, as long as the regular energy changes and particle escape can be neglected [$\vartheta_{\chi} = \varepsilon = 0$; @lac79; @bor86; @dro86; @bec06]. The whole energy range $0 \leq \chi \leq \infty$ with the appropriate (singular) boundary conditions is considered in @par95.
The analytic investigations of the momentum diffusion equation (\[eqfinal\]) in the $q=2$ limit including the radiative cooling have concentrated on the synchrotron-type losses $\vartheta_{\chi} \propto \chi$ [see, however, @sch87; @ste88]. The extended discussion on the time-dependent evolution for such a case (equation \[eqfinal\]) was presented by @kar62. As for the steady-state solution (equation \[steady\]), it was found that with $Q(\chi) \propto \delta(\chi-\chi_{\rm inj})$ and the range $0
\leq \chi \leq \infty$ $$N(\chi>\chi_{\rm inj}) \propto \chi^{\sigma+1} \, e^{- {\chi \over \chi_{\rm
eq}}} \, U\!\left[\sigma - 1, \, 2 \, \sigma +2, \, {\chi \over \chi_{\rm
eq}}\right] \, ,
\label{park}$$ where $\sigma$ is the energy spectral index introduced above, the equilibrium momentum $\chi_{\rm eq}$ is defined by the $t_{\rm acc} = t_{\rm loss}$ condition (yielding $\vartheta_{\chi} = \chi /
\chi_{\rm eq}$), and $U[a,b,z]$ is a Tricomi confluent hypergeometrical function [@jon70; @sch84; @par95][^3]. For $\chi \ll \chi_{\rm eq}$, i.e. for the particle momenta low enough to neglect radiative losses, the above distribution function has, as expected, a power-law form $N(\chi > \chi_{\rm inj}) \propto p^{ - \sigma}$. For $\chi \gtrsim
\chi_{\rm eq}$ and $\varepsilon \ll 1$, the particle energy spectrum approaches $N(\chi > \chi_{\rm inj}) \propto \chi^2 \, \exp \left( - \chi/\chi_{\rm
eq}\right)$. That is, as long as particle escape is inefficient, a two component stationary energy distribution is formed: a power-law $\propto \chi^{-1}$ at low ($\chi < \chi_{\rm eq}$) energies, and a pile-up bump (‘ultrarelativistic Maxwellian distribution’) around $\chi \sim \chi_{\rm eq}$. For shorter escape timescale no pile-up form appears, and the resulting particle spectral index depends on the ratio $\varepsilon$ of the escape and the acceleration time scales.
In the case of $q \neq 2$ and synchrotron-type energy losses $\vartheta_{\chi} \propto \chi$, the steady-state solution to the equation (\[steady\]) provided by @mel69 was questioned due to unclear boundary conditions applied [@tad71; @par95]. The special case of $q = 1$ with particle escape included (and the infinite energy range $0 \leq \chi \leq
\infty$) was considered further by @bog85. It was found, that with the injection of the $Q(\chi) \propto \delta(\chi-\chi_{\rm inj})$ type, the steady state solution of equation (\[steady\]) is[^4] $$N(\chi>\chi_{\rm inj}) \propto \chi^{2} \, e^{- {1\over 2} \, \left({\chi \over
\chi_{\rm eq}}\right)^2} \, U\!\left[{1 \over 2} \left({\chi_{\rm eq} \over
\chi_{\rm esc}}\right)^2 , \, 2 \, , \, {1 \over 2} \left({\chi \over \chi_{\rm
eq}}\right)^2\right] \, ,
\label{bogdan}$$ where the critical escape and equilibrium momenta $\chi_{\rm esc}$ and $\chi_{\rm eq}$ are defined by the conditions $t_{\rm esc} = t_{\rm acc}$ and $t_{\rm acc} = t_{\rm loss}$, respectively, yielding $\varepsilon = 1/ \chi_{\rm
esc}^{2}$ and $\vartheta_{\chi} = \chi / \chi_{\rm eq}^2$. This solution implies $N(\chi > \chi_{\rm inj}) \propto const$ at low particle momenta for which synchrotron energy losses are negligible ($\chi \ll \chi_{\rm eq}$), independent of the particular value of the escape timescale. At higher particle energies, an exponential dependence is expected, $N(\chi > \chi_{\rm
inj}) \propto \chi^{2 - (\chi_{\rm eq}/\chi_{\rm esc})^2} \, \exp \left[ - {1
\over 2} \, \left(\chi/\chi_{\rm eq}\right)^2 \right]$. Note, that with an increasing escape timescale this approaches $\sim \chi^2 \, \exp \left[ - {1 \over 2} \, \left(\chi/\chi_{\rm eq}\right)^2
\right]$.
Inefficient Particle Escape
===========================
In this section we are interested in steady-state solutions to the momentum diffusion equation (\[steady\]) in the case of a very inefficient particle escape and a general (i.e., not necessarily synchrotron-type) form of the regular energy changes $\vartheta_{\chi}$, which is however a continuous function of the particle energy. With $\varepsilon = 0$, the homogeneous form of this equation can be therefore transformed to the self-adjoint form $${d \over d \chi} \left[ P(\chi) \, {d \over d \chi} N(\chi)\right] - G(\chi) \,
N(\chi) = 0
\label{self}$$ with $$\begin{aligned}
P(\chi) & = & \chi^q \, S(\chi) \, , \nonumber \\
G(\chi) & = & \left[ 2 (q-1) \chi^{q-2} - {d \over d \chi}\left(\chi \,
\vartheta_{\chi}\right)\right] \, S(\chi) \, , \nonumber\\
S(\chi) & = & \chi^{-2} \, \exp\left[\int^\chi d \chi' \, \chi'^{1-q} \,
\vartheta_{\chi'}\right] \, . \label{s}\end{aligned}$$ We also restrict the analysis to the finite particle energy range $\chi \in
[\chi_1, \, \chi_2]$, where $0 < \chi_1, \, \chi_2 < \infty$. The justification for this is that for a finite range of the turbulent wavevectors, say $k
\in [k_1, \, k_2]$, the momentum diffusion coefficient as given in equation (\[coef\]) is well defined only for a finite range of particle energies (momenta). For example, gyroresonant interactions between the particles and the Alfénic turbulence require particles’ gyroradii comparable to the scale of the interacting modes, or $k\,r_{\rm g} \sim 2\pi$. Hence, the lower and upper limit of the particle energy range could be chosen as $\chi_1 =
2\pi e B_0 / k_1 c p_0$ and $\chi_2 = 2\pi e B_0 /k_2 c p_0$, respectively[^5]. Since all of the functions $P(\chi)$, $P'(\chi)$, $G(\chi)$, $S(\chi)$ are continuous, and $P(\chi)$, $S(\chi)$ are finite and strictly positive in the considered (closed) energy interval, the appropriate boundary value problem, $$\begin{aligned}
a_1 \, N(\chi_1) + a_2 \, \left. {d N(\chi) \over d \chi}\right|_{\chi_1} & = & 0 \,
, \nonumber \\
b_1 \, N(\chi_2) + b_2 \, \left. {d N(\chi) \over d \chi}\right|_{\chi_2} & = & 0 \,
, \label{bc}\end{aligned}$$ is regular. If one of these conditions is violated, which is the case for the infinite energy range $0 \leq \chi \leq \infty$, the problem becomes singular, and the extended analysis presented by @par95 has to be applied.
The two linearly-independent particular solutions to the homogeneous form of the equation (\[self\]) are $$\begin{aligned}
y_1(\chi) & = & S^{-1}(\chi) \, , \nonumber \\
y_2(\chi) & = & S^{-1}(\chi) \, \int^\chi d\chi' \, \chi'^{-q} \, S(\chi') \, ,
\label{y}\end{aligned}$$ or any linear combination of these, $$\begin{aligned}
u_1(\chi) & = & y_1(\chi) + \alpha \, y_2(\chi) \, , \nonumber \\
u_2(\chi) & = & y_1(\chi) + \beta \, y_2(\chi) \label{u}\end{aligned}$$ (each involving arbitrary multiplicative constants). By imposing the boundary conditions (\[bc\]) in a form $$\begin{aligned}
a_1 \, u_1(\chi_1) + a_2 \, \left .{d u_1(\chi) \over d \chi}\right|_{\chi_1} & = &
0 \, , \nonumber \\
b_1 \, u_2(\chi_2) + b_2 \, \left. {d u_2(\chi) \over d \chi}\right|_{\chi_2} & = &
0 \, , \label{bvp}\end{aligned}$$ parameters $\alpha$ and $\beta$ can be determined. With thus constructed particular solutions to the equation (\[self\]), one can define the Wronskian $w(\chi) \equiv u_1(\chi) \, u_2'(\chi) - u_1'(\chi) \, u_2(\chi)$, and next construct the Green’s function of the problem, $$\mathcal{G}(\chi, \chi_{\rm inj}) = {1 \over - \chi_{\rm inj}^q \, w(\chi_{\rm inj})} \times \left\{
\begin{array}{ccc}
u_1(\chi) \, u_2(\chi_{\rm inj}) & {\rm for} & \chi_1 \leq \chi < \chi_{\rm inj} \\
u_1(\chi_{\rm inj}) \, u_2(\chi) & {\rm for} & \chi_{\rm inj} < \chi \leq \chi_2
\end{array} \right. \, ,
\label{green1}$$ where $\chi_1 < \chi_{\rm inj} < \chi_2$. This gives the final solution to the equation (\[steady\]) $$N(\chi) = \int_{\chi_1}^{\chi_2} d\chi_{\rm inj} \, \mathcal{G}(\chi, \chi_{\rm inj}) \, Q(\chi_{\rm inj}) \, .
\label{solution}$$
Steady-state solutions exist, however, only for some particular forms of the injection function $Q(\chi,\tau)$. To investigate this issue, and to impose correct boundary conditions for the finite energy range $\chi_1 \leq \chi \leq
\chi_2$, let us integrate equation (\[eqfinal\]) over the energies and re-write it in a form of the continuity equation, $${\partial \mathcal{N} \over \partial \tau} + \left. \mathcal{F}\right|_{\chi_2}
- \left. \mathcal{F}\right|_{\chi_1} = \int_{\chi_1}^{\chi_2} d\chi \, Q(\chi,
\tau) \, .
\label{cont}$$ Here $\mathcal{N} \equiv \int_{\chi_1}^{\chi_2} d\chi \, N(\chi)$ is the total number of particles and the particle flux in the momentum space is defined as $$\mathcal{F}[N(\chi)] = \left(2 \chi^{q-1} - \chi \, \vartheta_{\chi} \right) N -
\chi^q \, {\partial N \over \partial \chi} \, .
\label{flux1}$$ Note, that with the particular solutions $u_1(\chi)$ and $u_2(\chi)$ given in (\[u\]), one has $$\begin{aligned}
\mathcal{F}[u_1(\chi)] & = & - \alpha \, , \nonumber \\
\mathcal{F}[u_2(\chi)] & = & - \beta \, , \label{flux2}\end{aligned}$$ independent of the momentum $\chi$ or of the particular form of the direct energy losses function $\vartheta_{\chi}$.
Let us comment in this context on the ‘zero-flux’ boundary conditions of the type (\[bc\]), namely $\left.\mathcal{F}\right|_{\chi_1} =
\left.\mathcal{F}\right|_{\chi_2} = 0$. These, with equations (\[bvp\]) and (\[flux2\]), imply $\alpha = \beta = 0$, i.e., $u_1(\chi) = u_2(\chi)$. In other words, one particular solution $y_1(\chi)$ satisfies the ‘no-flux’ boundary condition of the homogeneous form of the equation (\[self\]) for both $\chi_1$ and $\chi_2$. In such a case, the steady-state solution can be constructed using the function $y_1(\chi)$ only if it is orthogonal to the source function, $\int_{\chi_1}^{\chi_2} d\chi \, y_1(\chi) \, Q(\chi) = 0$. This condition, for any non-zero particle injection and $y_1(\chi) =
S^{-1}(\chi)$ as given in the equation (\[s\]), cannot be fulfilled [cf. @mel69; @tad71]. ‘Zero-flux’ boundary conditions for non-vanishing $Q(\chi)$ can be instead imposed if the particle injection is balanced by the particle escape from the system (see §3 below).
In the case of no particle escape, with the stationary injection such that $\int_{\chi_1}^{\chi_2} d\chi \, Q(\chi) \equiv A$ and with the direct (radiative) energy losses $\vartheta_{\chi} \neq 0$, the boundary conditions can be chosen as $$- \left.\mathcal{F}\right|_{\chi_1} = A \, , \quad {\rm and} \quad
\left.\mathcal{F}\right|_{\chi_2} = 0 \, ,
\label{conditions}$$ which give $\alpha = A$ and $\beta = 0$, and correspond to the conservation of the total number of particles within the energy range $[\chi_1, \chi_2]$. Let us justify this choice by noting that the radiative losses processes, unlike momentum diffusion strictly related to the turbulence spectrum, is well defined for particle momenta $\chi<\chi_1$ and $\chi> \chi_2$. Hence, with non-vanishing radiative losses (as expected for ultrarelativistic particles considered here), no flux of particles in the momentum space through the maximum value $\chi_2$ toward higher energies is possible (radiative losses in the absence of stochastic acceleration will always prevent from presence of particle flux above $\chi_2$). For the same reason, there is always a possibility for a non-zero particle flux toward lower energies through the $\chi_1$ point, since the stochastic acceleration timescale, even if being an increasing function of the particle energy, is always finite at $\chi_1 > 0$. Note in this context, that the particle flux at $\chi_1$ implied by the chosen boundary conditions (\[conditions\]) must be negative, $\left.\mathcal{F}\right|_{\chi_1} < 0$. That is, there is a continuous flux of particles through the $\chi_1$ point from high to low energies, which — in the absence of particle catastrophic escape from the system — balances particle injection. With these, one can find the Green’s function as $$\left. \mathcal{G}(\chi, \chi_{\rm inj})\right|_{\rm loss} = \left\{ \begin{array}{ccc}
S^{-1}(\chi) \, \left(A^{-1} + \int^\chi_{\chi_1} d\chi' \, \chi'^{-q} \,
S(\chi') \right) & {\rm for} & \chi_1 \leq \chi < \chi_{\rm inj} \\
S^{-1}(\chi) \, \left(A^{-1} + \int^{\chi_{\rm inj}}_{\chi_1} d\chi' \, \chi'^{-q} \,
S(\chi') \right) & {\rm for} & \chi_{\rm inj} < \chi \leq \chi_2
\end{array} \right. \, ,
\label{green-loss}$$ where $S(\chi)$, introduced in the equation (\[s\]), can be re-written as $$S(\chi) = \chi^{-2} \, \exp\left[\int^\chi {d \chi' \over \chi'} \, {t_{\rm
acc}(\chi') \over t_{\rm loss}(\chi')}\right] \, .
\label{s-loss}$$
Synchrotron Energy Losses
-------------------------
As an example let us consider synchrotron energy losses of ultrarelativistic electrons, which are characterized by the timescale $$t_{\rm syn} = \tau_{\rm syn} \, \chi^{-1} \, , \quad {\rm with} \quad \tau_{\rm
syn} \equiv {6 \pi \, m_{\rm e}^2 c^2 \over \sigma_{\rm T} \, p_0 \, B_0^2}
\label{syn}$$ [e.g., @blu70], and which define the equilibrium momentum $\chi_{\rm eq}
= (\tau_{\rm syn} / \tau_{\rm acc})^{1/(3-q)}$ through the condition $t_{\rm
acc} = t_{\rm syn}$, yelding $\vartheta = \chi / \chi_{\rm eq}^{3-q}$. The Green’s function (\[green-loss\]) adopts then the form $$\begin{aligned}
& & \left. \mathcal{G}(\chi, \chi_{\rm inj})\right|_{\rm syn} = \chi^2 \, e^{- \, {1 \over 3-q}
\, \left({\chi \over \chi_{\rm eq}}\right)^{3-q}} \, \left({1 \over A} +
\int^{\min[\chi_{\rm inj},\, \chi]}_{\chi_1} d\chi' \, \chi'^{-(2+q)} \, e^{{1 \over 3-q}
\, \left({\chi' \over \chi_{\rm eq}}\right)^{3-q}} \right) = \label{green-syn}
\\
& & = \chi^2 \, e^{- \, {1 \over 3-q} \, \left({\chi \over \chi_{\rm
eq}}\right)^{3-q}} \, \left( {1 \over A} + {\chi_{\rm eq}^{-1-q} \, (-1)^{4 /
(3-q)} \over (3-q)^{4 / (3-q)}} \, \Gamma\left[ - {1+q \over 3-q} \, , \, - \,
{\left(\min[\chi_{\rm inj},\, \chi] / \chi_{\rm eq}\right)^{3-q} \over 3-q} , \, - \,
{\left(\chi_1 / \chi_{\rm eq}\right)^{3-q} \over 3-q} \right] \right) \, ,
\nonumber\end{aligned}$$ where $\Gamma[a, z_1, z_2]$ is generalized incomplete Gamma function. By expressing the above solution in terms of Kummer confluent hypergeometrical functions $M[a,b,z]$ using the identity $\Gamma[a,z_1,z_2] = a^{-1} \, z_2^a \,
M[a, 1+a, -z_2] - a^{-1} \, z_1^a \, M[a, 1+a, -z_1]$ [@abr64], assuming $\chi_1 \ll \chi_{\rm eq}$, and noting that $M[a,b,z] \sim 1$ for $z \rightarrow
0$, one can rewrite it further as $$\begin{aligned}
& & \left. \mathcal{G}(\chi, \chi_{\rm inj})\right|_{\rm syn} \approx \chi^2 \, e^{- \, {1
\over 3-q} \, \left({\chi \over \chi_{\rm eq}}\right)^{3-q}} \times
\label{G-syn-app1} \\
& & \times \left({1 \over A} + {\chi_1^{-1-q} \over 1+q} - {\min(\chi_{\rm inj},\,\chi)^{-1-q}
\over 1+q} \, M\left[-{1+q \over 3-q},\, {2-2q \over 3-q},\, {1 \over 3-q}
\left({\min[\chi_{\rm inj},\, \chi] \over \chi_{\rm eq}}\right)^{3-q}\right]\right) \, .
\nonumber\end{aligned}$$ Finally, noting that $M[a,b,z] \sim \Gamma(b) \, e^z \, z^{a-b} / \Gamma(a)$ for $z \rightarrow \infty$, and neglecting the $A^{-1}$ term, one finds a rough approximation $$\left. \mathcal{G}(\chi, \chi_{\rm inj})\right|_{\rm syn} \sim
\left\{ \begin{array}{ccc}
{1 \over 1+q} \, \chi_1^{-1-q} \, \chi^2 \, e^{- \, {1 \over 3-q} \, \left(\chi
/ \chi_{\rm eq}\right)^{3-q}} \quad & {\rm for} & \min(\chi_{\rm inj}, \chi) \lesssim
\chi_{\rm eq} \\
\chi_{\rm eq}^{3-q} \, \chi^{-2} \quad & {\rm for} & \chi_{\rm eq} \ll \chi <
\chi_{\rm inj} \\
\chi_{\rm eq}^{3-q} \, \chi_{\rm inj}^{-4} \, e^{{1 \over 3-q} \, \left(\chi_{\rm inj} / \chi_{\rm
eq}\right)^{3-q}} \, \chi^2 \, e^{- \, {1 \over 3-q} \, \left(\chi / \chi_{\rm
eq}\right)^{3-q}} \quad & {\rm for} & \chi_{\rm eq} \ll \chi_{\rm inj} < \chi \end{array}
\right. \, .
\label{G-syn-app2}$$ Hence, as long as $\min[\chi_{\rm inj}, \, \chi] < \chi_{\rm eq}$, one has $\left.
\mathcal{G}(\chi, \chi_{\rm inj})\right|_{\rm syn} \propto \chi^2 \, \exp\left[- {1\over
3-q} \left(\chi / \chi_{\rm eq}\right)^{3-q}\right]$. If, however, $\min[\chi_{\rm inj}, \,
\chi] > \chi_{\rm eq}$, the Green’s function retains the spectral shape $\propto
\chi^2 \, \exp\left[- {1\over 3-q} \left(\chi / \chi_{\rm
eq}\right)^{3-q}\right]$ for $\chi_{\rm inj} < \chi$, while is of a power-law form $\left.
\mathcal{G}(\chi, \chi_{\rm inj})\right|_{\rm syn} \propto \chi^{-2}$ for $\chi < \chi_{\rm inj}$.
![[*Upper panel:*]{} Stochastic acceleration timescales for fixed plasma parameters ($B_0$, $\zeta$, $\beta_{\rm A}$, $\chi_1$, $\chi_2$) but different turbulence energy index: $q=2$ (thick solid lines), $q = 5/3$ (thick dashed lines), and $q=1$ (thick dotted lines). Thin solid line denotes radiative (synchrotron) energy losses timescale considered. [*Lower panel:*]{} Particle spectra resulting from joint stochastic acceleration and radiative (synchrotron) energy losses specified in the upper panel. The spectra correspond to the monoenergetic injection $Q(\chi) \propto \delta(\chi-1)$ with fixed $\int dp
\, \widetilde{Q}(p)$, and no particle escape. Thin solid line denotes particle spectrum expected for the same injection and cooling conditions, but with the momentum diffusion effects neglected, $\widetilde{N}(\chi)$.[]{data-label="delta"}](fig1.eps)
![The same as Figure (\[delta\]) except for $Q(\chi) \propto \delta(\chi-10^7)$.[]{data-label="powerlaw"}](fig2.eps)
In Figures (\[delta\]) and (\[powerlaw\]) we plot examples of particle spectra obtained from the above solution for the system with fixed plasma parameters ($B_0$, $\zeta$, $\beta_{\rm A}$, $\chi_1$, $\chi_2$) but different turbulence energy indices: $q=2$ (‘hard-sphere’ approximation; thick solid lines in the figures), $q = 5/3$ (Kolmogorov-type turbulence; thick dashed lines), and $q=1$ (Bohm limit; thick dotted lines). As for the source function, we consider two different forms, namely $Q(\chi) \propto \delta(\chi-1)$ in Figure \[delta\] and $Q(\chi) \propto \delta(\chi-10^7)$ in Figure \[powerlaw\], with the normalizations given in both cases by the same fixed $\int dp \,
\widetilde{Q}(p)$. The emerging particle spectra are compared with the ones expected for the same injection and cooling conditions, but with the momentum diffusion neglected, $\widetilde{N}(\chi)$. Such a steady-state electron distribution can be found from the appropriate equation $${\partial \over \partial \chi} \left[\chi\, \vartheta_{\chi} \,
\widetilde{N}(\chi)\right] + Q(\chi) = 0
\label{kinetic}$$ (see equation \[eqfinal\]), for which one has the straightforward solution [@kar62] $$\widetilde{N}(\chi) = {1 \over \chi \, \vartheta_{\chi}} \, \int_{\chi}^{\chi_2}
Q(\chi_{\rm inj}) \, d\chi_{\rm inj}
\label{kardashev}$$ (thin solid lines in the lower panels of Figures \[delta\]-\[powerlaw\]).
As shown in the figures and follows directly from the obtained solution \[green-syn\]–\[G-syn-app2\], joint stochastic acceleration and radiative (synchrotron-type) loss processes, in the absence of particle escape, tend to establish $N(\chi) \propto \chi^2 \, \exp\left[- {1 \over 3-q} \, \left(\chi /
\chi_{\rm eq}\right)^{3-q}\right]$ spectra independent of the energy of the injected particles and the form of the source function as long as it has a narrow distribution. Moreover, for $\chi \ll \chi_{\rm eq}$ the turbulence energy index $q$ does not influence the spectral shape of the electron energy distribution. Instead — with fixed normalization of the injection function $\widetilde{Q}(p)$ and fixed plasma parameters (including magnetic field intensities $B_0$ and $\zeta$) — turbulence power-law slope $q$ determines (i) the equilibrium momentum $\chi_{\rm eq}$, (ii) normalization of the electron energy distribution, and (iii) the spectral shape of the particle distribution for $\chi \geq \chi_{\rm eq}$. In particular, flatter turbulent spectrum leads to higher value of the equilibrium momentum $\chi_{\rm eq}$, lower normalization of $N(\chi)$, and steeper exponential cut-off at $\chi > \chi_{\rm eq}$. Note also, that if particles with momenta $\chi_{\rm inj} \gg \chi_{\rm eq}$ are being injected to the system, the resulting electron energy distribution may adopt the ‘standard’ form of the synchrotron-cooled source function (\[kardashev\]) at highest momenta $\chi_{\rm eq} \ll \chi < \chi_{\rm inj}$ (e.g., $\propto \chi^{-2}$ for the $Q(\chi) \propto \delta(\chi - 10^7)$ injection in Figure \[powerlaw\]).
Inverse-Compton Energy Losses and the Klein-Nishina Effects
-----------------------------------------------------------
Let us now investigate the effects of the inverse-Compton (IC) radiative energy losses in the presence of a turbulent particle acceleration. At low energies when the Klein-Nishina (KN) effects are negligible the IC case is identical to the synchrotron case with the magnetic energy density $B^2/8\pi$ replaced by the photon energy density $u_{\rm ph}$. The two cases differ when KN effects become important at high energies. To include these effects we approximate the radiative loss timescale as $$t_{\rm IC} = \tau_{\rm IC} \, \chi^{-1} \, \left(1 + {\chi \over \chi_{\rm
cr}}\right)^{1.5} \, , \quad {\rm where} \quad \tau_{\rm IC} \equiv {3 \, m_{\rm
e}^2 c^2 \over 4 \, \sigma_{\rm T} \, p_0 \, u_{\rm ph}} \quad {\rm and} \quad
\chi_{\rm cr} \equiv {m_{\rm e} c \over 4 \, p_0 \epsilon_0} \, .
\label{ic}$$ Here the radiation field involved in the IC scattering was assumed to be monoenergetic, with the total energy density $u_{\rm ph}$ and the dimensionless (i.e., expressed in the electron mass units) photon energy $\epsilon_0$. The above formula properly takes into account KN effect up to energies $\chi \leq 10^4 \,
\chi_{\rm cr}$ [@mod05]. Clearly, as long as $q < 1.5$, balance between acceleration and cooling timescales takes place at one particular momentum $\chi_{\rm eq}= \max(\chi_{\rm Th}, \, \chi_{\rm KN})$, depending on weather energy losses dominate over acceleration in the Thomson regime, $\chi_{\rm eq}=
\chi_{\rm Th} \equiv (\tau_{\rm IC}/\tau_{\rm acc})^{1/(3-q)}$, or in the KN regime, $\chi_{\rm eq}= \chi_{\rm KN} \equiv \chi_{\rm Th}^{(3-q)/(1.5-q)} \,
\chi_{\rm cr}^{-1.5 / (1.5 - q)}$. For $q > 1.5$, there may be instead two equilibrium momenta for a given one acceleration timescale, $\chi_{\rm eq, \, 1}
= \chi_{\rm Th}$ and $\chi_{\rm eq, \, 2} =\chi_{\rm KN}$, or no equilibrium momentum at all, if $t_{\rm acc} < t_{\rm ic}$ within the whole considered range $\chi < 10^4 \, \chi_{\rm cr}$. Finally, for $q = 1.5$ (that corresponds to the Kraichnan turbulence), the ratio between IC/KN and acceleration timescales is energy-independent, since both $t_{\rm acc} \propto \chi^{q-1}=\chi^{1/2}$ and, as given in (\[ic\]), $t_{\rm IC}(\chi > \chi_{\rm cr}) \propto \chi^{1/2}$.
Assuming hereafter $q \neq 3/2$, one can find from the equation (\[s-loss\]) that $$S(\chi) = \chi^{-2} \exp\left\{ {1 \over 3-q} \left({\chi \over \chi_{\rm
T}}\right)^{3-q} \, F\!\left[{3 \over 2}, \, 3 - q,\, 4-q,\, -{\chi \over
\chi_{\rm cr}}\right]\right\} \, ,
\label{Gauss}$$ where $F[a,b,c,z]$ is Gauss hypergeometric function. This gives the Green’s function $$\begin{aligned}
& & \left. \mathcal{G}(\chi, \chi_{\rm inj})\right|_{\rm ic}^{q \neq 1.5} = \chi^2 \,
\exp\left\{ - {1 \over 3-q} \left({\chi \over \chi_{\rm Th}}\right)^{3-q} \,
F\!\left[{3 \over 2}, \, 3 - q,\, 4-q,\, -{\chi \over \chi_{\rm
cr}}\right]\right\} \times \label{green-ic} \\
& & \times \left( {1 \over A} + \int_{\chi_1}^{\min(\chi_{\rm inj}, \chi)} d\chi' \,
\chi'^{-2-q} \, \exp\left\{ {1 \over 3-q} \left({\chi' \over \chi_{\rm
Th}}\right)^{3-q} \, F\!\left[{3 \over 2}, \, 3 - q,\, 4-q,\, -{\chi' \over
\chi_{\rm cr}}\right]\right\} \right) \, . \nonumber\end{aligned}$$ Below we discuss some properties of the obtained solution by expanding the Gauss hypergeometric functions as $F[a,b,c,z] \sim 1$ for $z \rightarrow 0$, and $F[a,b,c,z] \sim \left[\Gamma(c) \, \Gamma(b-a)/\Gamma(b) \, \Gamma(c-a)\right]
\, (-z)^{-a} + \left[\Gamma(c) \, \Gamma(a-b)/\Gamma(a) \, \Gamma(c-b)\right] \,
(-z)^{-b}$ for $z \rightarrow \infty$.
![[*Upper panel:*]{} Stochastic acceleration timescales for fixed $q=1$ and different plasma parameters (thick solid and dashed lines). Thin solid line denotes inverse-Compton energy losses timescale considered with the assumed $\chi_{\rm cr} = 10^4$. [*Lower panel:*]{} Particle spectra resulting from joint stochastic acceleration and inverse-Compton energy losses specified in the upper panel. The spectra correspond to the monoenergetic injection $Q(\chi) \propto
\delta(\chi-1)$ with fixed $\int dp \, \widetilde{Q}(p)$, and no particle escape. Thin solid line denotes particle spectrum expected for the same injection and cooling conditions, but with the momentum diffusion effects neglected, $\widetilde{N}(\chi)$.[]{data-label="KN-1"}](fig3.eps)
![The same as Figure (\[KN-1\]) except for $q=2$.[]{data-label="KN-2"}](fig4.eps)
Let us consider first the case of a low-energy injection, such that $\chi_{\rm inj} <
\min(\chi_{\rm Th}, \, \chi_{\rm cr})$. The Green’s function (\[green-ic\]) can be then approximated roughly by $$\left. \mathcal{G}(\chi, \chi_{\rm inj})\right|_{\rm ic; \, \chi_{\rm inj}<}^{q \neq 1.5} \sim
\left\{ \begin{array}{ccc}
{1 \over 1+q} \, \chi_1^{-1-q} \, \chi^2 \, e^{- \, {1 \over 3-q} \, \left(\chi
/ \chi_{\rm Th}\right)^{3-q}} \quad & {\rm for} & \chi < \chi_{\rm cr} \\
{1 \over 1+q} \, \chi_1^{-1-q} \,
e^{- \, {2 \over \sqrt{\pi}} \, \Gamma(3-q) \, \Gamma(q-1.5) \, \left(\chi_{\rm
cr} / \chi_{\rm Th}\right)^{3-q}} \, \chi^2 \, e^{- \, {1 \over 1.5-q} \,
\left(\chi / \chi_{\rm KN}\right)^{1.5-q}} \,\quad & {\rm for} & \chi > \chi_{\rm cr}
\end{array} \right. \, .
\label{G-ic-app1}$$ For $\chi < \chi_{\rm cr}$ the Green’s function has the same form as the synchrotron case, which is expected in the Thomson regime. However, $\chi >
\chi_{\rm cr}$, the KN effects modify the high-energy segment of the particle energy distribution. In particular, for $q < 1.5$ (e.g. $q=1$ in Fig. \[KN-1\]) the spectrum is always of a ‘single-bump’ form, possessing either a sharp or a smooth exponential cut-off depending on whether we are in the Thomson or KN cooling regime, respectively. On the other hand, with $q > 1.5$ (e.g. $q=2$ in Fig. \[KN-2\]) the acceleration and loss timescales can be equal at two different energies, in which case the particle spectra become concave, flattening smoothly from the exponential decrease $\propto
\chi^2 \, \exp\left[- {1\over 3-q} \left(\chi / \chi_{\rm
Th}\right)^{3-q}\right]$ at $\chi_{\rm Th} < \chi < \chi_{\rm cr}$ to the asymptoticaly approached $\propto \chi^2$ continuum at $\chi > \chi_{\rm KN}$. Such spectra are shown in Figure (\[KN-1\]-\[KN-2\]) for $q=1$ and $q=2$, respectively, assuming monoenergetic injection with $\chi_{\rm inj}=1$, $\chi_1 = 10^{-2}$, $\chi_{\rm KN} = 10^4$, and $\chi_2 = 10^{8}$. In each figure we use two different acceleration timescales (thick solid and dashed lines), but the radiative losses timescale, $t_{\rm loss}$, as well as the normalization of the injection function, $\int dp \,
\widetilde{Q}(p)$, are kept constant. The emerging spectra are compared with the electron energy distribution $\widetilde{N}(\chi)$ corresponding to the same injection and cooling conditions, but with the momentum diffusion neglected (equations \[kardashev\]; thin solid lines in the lower panels of the figures).
In the case of $q\neq 1.5$ and high-energy injection $\chi_{\rm inj} > \chi_{\rm cr}$, the appropriate Green’s fuction retains again familiar shape $\left.
\mathcal{G}(\chi, \chi_{\rm inj})\right|_{\rm ic; \, \chi_{\rm inj}>}^{q \neq 1.5} \sim {1 \over 1+q}
\, \chi_1^{-1-q} \, \chi^2 \, e^{- \, {1 \over 3-q} \, \left(\chi / \chi_{\rm
T}\right)^{3-q}}$ at low particle momenta $\chi < \chi_{\rm cr}$. And again, at $\chi > \chi_{\rm cr}$ significant deviations from such a form may be observed, as follows from the approximate form of the Green’s function $$\begin{aligned}
& & \left. \mathcal{G}(\chi, \chi_{\rm inj})\right|_{\rm ic; \, \chi_{\rm inj}>}^{q \neq 1.5} \approx
\chi^2 \, e^{ - {1 \over 1.5-q} \left(\chi / \chi_{\rm KN}\right)^{1.5-q}} \,
e^{- \, {2 \over \sqrt{\pi}} \, \Gamma(3-q) \, \Gamma(q-1.5) \, \left(\chi_{\rm
cr} / \chi_{\rm Th}\right)^{3-q}} \times \label{green-ic-high} \\
& & \times \int_{\chi_1}^{\min(\chi_{\rm inj}, \chi)} d\chi' \, \chi'^{-2-q} \, \exp\left\{ {1
\over 3-q} \left({\chi' \over \chi_{\rm Th}}\right)^{3-q} \, F\!\left[{3 \over
2}, \, 3 - q,\, 4-q,\, -{\chi' \over \chi_{\rm cr}}\right]\right\} \quad {\rm
for} \quad \chi > \chi_{\rm cr} \nonumber\end{aligned}$$ (see equation \[green-ic\] with the $A^{-1}$ term neglected). The resulting particle spectra are plotted in Figures (\[KN-3\]-\[KN-4\]), where we consider two limiting cases of $q=1$ and $q=2$, and assume monoenergetic injection $Q(\chi) \propto \delta(\chi-\chi_{\rm inj})$ with $\chi_{\rm inj} =
10^7$. All the other parameters are fixed as before. As shown, in addition to the spectral features discussed in the previous paragraph for the case of a low-energy injection (Figures \[KN-1\]-\[KN-2\]), the radiatively-cooled continuum may be observed at high particles energies $\chi < \chi_{\rm inj}$, depending on the efficiency of the acceleration process. The KN effects manifest thereby by means of a characteristic spectral flattening over the ‘standard’ power-law form $\propto \chi^{-2}$, obviously only within the momentum range $\chi_{\rm cr} <
\chi < \chi_{\rm inj}$, in agreement with the appropriate $\widetilde{N}(\chi)$ distribution (thin solid lines in the lower panels of Figures \[KN-3\]-\[KN-4\]). Such a feature, being a direct result of a dominant IC/KN-regime radiative cooling with the momentum diffusion effects negligible, was discussed previously by, e.g., @kus05 [@mod05; @man07].
![[*Upper panel:*]{} Stochastic acceleration timescales for fixed $q=1$ and different plasma parameters (thick solid and dashed lines). Thin solid line denotes inverse-Compton energy losses timescale considered with the assumed $\chi_{\rm cr} = 10^4$. [*Lower panel:*]{} Particle spectra resulting from joint stochastic acceleration and inverse-Compton energy losses specified in the upper panel. The spectra correspond to the monoenergetic injection $Q(\chi) \propto
\delta(\chi-10^7)$ with fixed $\int dp \, \widetilde{Q}(p)$, and no particle escape. Thin solid line denotes particle spectrum expected for the same injection and cooling conditions, but with the momentum diffusion effects neglected, $\widetilde{N}(\chi)$.[]{data-label="KN-3"}](fig5.eps)
![The same as Figure (\[KN-3\]) except for $q=2$.[]{data-label="KN-4"}](fig6.eps)
![[*Upper panel:*]{} Stochastic acceleration timescales for fixed $q=3/2$ and different plasma parameters (thick solid, dashed, and dotted lines). Thin solid line denotes inverse-Compton energy losses timescale considered with the assumed $\chi_{\rm cr} = 10^4$. [*Lower panel:*]{} Particle spectra resulting from joint stochastic acceleration and inverse-Compton energy losses specified in the upper panel. The spectra correspond to the monoenergetic injection $Q(\chi) \propto \delta(\chi-1)$ with fixed $\int dp \, \widetilde{Q}(p)$, and no particle escape. Thin solid line denotes particle spectrum expected for the same injection and cooling conditions, but with the momentum diffusion effects neglected, $\widetilde{N}(\chi)$.[]{data-label="KN-5"}](fig7.eps)
Finally, for completeness we note that with $q = 1.5$ one can solve equation (\[s-loss\]) to obtain $$S(\chi) = \chi^{-2} \exp\left\{ 2 \, \left({\chi_{\rm cr} \over \chi_{\rm
T}}\right)^{3/2} \, \left({\rm ArcSinh}\!\sqrt{\chi / \chi_{\rm cr}} -
{\sqrt{\chi / \chi_{\rm cr}} \over \sqrt{1 + (\chi / \chi_{\rm cr})}}
\right)\right\} \, .
\label{krei}$$ This reduces to $S(\chi) \sim \chi^{-2} \, \exp\left[{2\over 3} \left(\chi /
\chi_{\rm Th}\right)^{3/2}\right]$ for $\chi < \chi_{\rm cr}$, and can be approximated by $S(\chi) \sim 0.54 \, \chi^{-1} \, \chi_{\rm cr}^{-1}$ for $\chi
> \chi_{\rm cr}$. The resulting particle spectra, shown in Figure (\[KN-5\]) for the case of a low-energy injection $Q(\chi) \propto \delta(\chi - 1)$, are therefore $N(\chi<\chi_{\rm cr}) \propto \chi^{2} \, \exp\left[-{2\over 3}
\left(\chi / \chi_{\rm Th}\right)^{3/2}\right]$ at low momenta, or of the power-law form $N(\chi>\chi_{\rm cr}) \propto \chi^{-\sigma'}$ at higher momenta where the KN effects are important. Here $\sigma' \equiv {t_{\rm acc} \over
t_{\rm ic}(\chi > \chi_{\rm cr})} - 2 = {\tau_{\rm acc} \over \tau_{\rm ic}} \,
\chi_{\rm cr}^{1.5} -2 $.
Bremsstrahlung and Coulomb Energy Losses
----------------------------------------
At high densities or low magnetic field (in general low Alfv[é]{}n velocities) electron-electron and electron-ion interactions become important. These result in an elastic loss due to Coulomb collisions or radiative loss via bremsstrahlung. At low energies the bremsstrahlung loss rate is negligible when compared to the Coulomb loss rate, which is independent of energy for relativistic charge particles [see e.g. @pet73; @pet01]. However, since the bremsstrahlung rate increases nearly linearly with energy, above some critical energy bremsstarahlung becomes dominant. The time scales associated with these processes approximately are $$t_{\rm coul} = \tau_{\rm coul} \, \chi , \quad {\rm where} \quad \tau_{\rm coul}
\equiv {p_0 \over m_{\rm e} c} \, {2 \over 3 \, \sigma_{\rm Th} c \, n_{\rm g} \, \ln\Lambda}
\label{coul}$$ and $$t_{\rm brem} = \tau_{\rm brem} \, , \quad {\rm where} \quad \tau_{\rm brem}
\equiv {\pi \over 3 \, \alpha_{\rm fs} \sigma_{\rm Th} c \, n_{\rm g}} \, .
\label{brem}$$ Here $n_{\rm g}$ is the background plasma density, the Coulomb logarithm $\ln\Lambda$ varies from 10 to 40 for variety of astrophysical plasma, $\alpha_{\rm fs}=1/137$ is the fine structure constant, and the bremsstrahlung rate includes electron-ion and electron-electron bramsstrahlung, and assumes completely unscreened limit with approximately $10\%$ (fully ionized) helium abundance [@blu70]. The time scales are equal at energy $p_{\rm Coul} = \pi \, \ln \Lambda \, m_{\rm e} c/
(2 \, \alpha_{\rm fs})$. At higher energies the bremsstrahlung loss becomes unimportant compared to the synchrotron or IC losses. For example, the synchrotron loss becomes equal to and exceeds the bremsstrahlung loss at electron momenta $p\geq p_{\rm brem}
\equiv (m_e/m_p)(\alpha_{\rm fs}/\beta_{\rm A}^2) \, m_{\rm e} c$ so that for bremsstrahlung to be at all important we need $1000<p/(m_ec)< 10^{-5} \, \beta_{\rm A}^{-2}$, requiring $\beta_{\rm A}<0.003$. Below we investigate in some details stochastic acceleration for the conditions when the Coulomb and bremsstrahlung processes are the dominant loss processes.
At low energies, $p<p_{\rm Coul}$, Coulomb collision dominate. If $p_0\gg m_ec$ then in the range $m_ec\ll p\ll p_{\rm Coul}$ and for $q > 1$, the appropriate Green’s function becomes $$\begin{aligned}
& & \left. \mathcal{G}(\chi, \chi_{\rm inj})\right|_{\rm coul}^{q>1} = \chi^2 \, e^{{1 \over
1-q} \, \left[\left({\chi_1 \over \chi_{\rm eq}}\right)^{1-q} - \left({\chi
\over \chi_{\rm eq}}\right)^{1-q}\right]} \, \left({1 \over A} +
\int^{\min[\chi_{\rm inj},\, \chi]}_{\chi_1} d\chi' \, \chi'^{-(2+q)} \, e^{{1 \over 1-q}
\, \left[ \left({\chi' \over \chi_{\rm eq}}\right)^{1-q}- \left({\chi_1 \over
\chi_{\rm eq}}\right)^{1-q}\right]} \right) \approx \nonumber \\
& & \approx \chi^2 \, e^{- \, {1 \over 1-q} \, \left({\chi \over \chi_{\rm
eq}}\right)^{1-q}} \, {\chi_{\rm eq}^{-1-q} \, (-1)^{2 / (1-q)} \over (1-q)^{2 /
(1-q)}} \, \Gamma\left[ - {1+q \over 1-q} \, , \, - \, {\left(\min[\chi_{\rm inj},\, \chi]
/ \chi_{\rm eq}\right)^{1-q} \over 1-q} , \, - \, {\left(\chi_1 / \chi_{\rm
eq}\right)^{1-q} \over 1-q} \right] \label{green-coul}\end{aligned}$$ (see equations \[green-loss\]$-$\[s-loss\]), where the equilibrium momentum $\chi_{\rm eq}= (\tau_{\rm coul}/\tau_{\rm acc})^{1/(1-q)}$ is defined by the $t_{\rm acc} = t_{\rm coul}$ condition, yielding $\vartheta_{\chi} = \chi_{\rm
eq}^{q-1}/\chi$. Note that since $q>1$ are considered, the acceleration timescale is longer than the Coulomb interactions timescale for $\chi <
\chi_{\rm eq}$. Thus, in the case of a low-energy particle injection with $\chi_{\rm inj} < \chi_{\rm eq}$, the emerging particle spectra are of the ‘cooled’ form $N(\chi) = \widetilde{N}(\chi) \propto const$ (see equation \[kardashev\] with $\vartheta_{\chi} \propto \chi^{-1}$). If, however, higher-energy particles are injected to the system, an additional flat-spectrum component $N(\chi) \propto \chi^2$ is formed at $\chi > \chi_{\rm eq}$.
Let us finally note, that pure Coulomb energy losses and the Bohm limit $q=1$ correspond to the situation when $\vartheta_{\chi} = const$, and hence $S(\chi)
= \chi^{-2 + (\tau_{\rm acc}/\tau_{\rm coul})}$. The Green’s function (\[green-loss\]) adopts then the form $$\begin{aligned}
\left. \mathcal{G}(\chi, \chi_{\rm inj})\right|_{\rm coul}^{q=1} = \chi^{2-{\tau_{\rm acc}
\over \tau_{\rm coul}}} \, \left({1 \over A} + \int^{\min[\chi_{\rm inj},\, \chi]}_{\chi_1}
d\chi' \, \chi'^{-4+{\tau_{\rm acc} \over \tau_{\rm coul}}} \right) \sim
\nonumber \\
\sim {1 \over \sigma'} \, \chi^{-\sigma'} \, \left\{ \begin{array}{ccc}
\chi_1^{\sigma'} \quad & {\rm for} & \tau_{\rm acc}/\tau_{\rm coul} < 2 \\
\min^{\sigma'}(\chi_{\rm inj}, \,\chi) \quad & {\rm for} & \tau_{\rm acc}/\tau_{\rm coul} >
2 \end{array} \right. \, , \label{green-coul2}\end{aligned}$$ where $\sigma' \equiv {\tau_{\rm acc} \over \tau_{\rm coul}} - 2$. Hence, if only $\tau_{\rm acc} < 2 \, \tau_{\rm coul}$, a power-law particle energy distribution $N(\chi) \propto \chi^{-\sigma'}$ forms, with $-2 < \sigma' < 0$. For any longer acceleration timescale, $\tau_{\rm acc} > 2 \, \tau_{\rm coul}$, and for the source function $Q(\chi) \propto \delta (\chi - \chi_{\rm inj})$, the emerging electron spectra are $N(\chi) \propto const$ for $\chi < \chi_{\rm
inj}$, and $N(\chi) \propto \chi^{-\sigma'}$ with $\sigma' > 0$ for $\chi >
\chi_{\rm inj}$. This is consistent with the solution found by @bog85, who considered also synchrotron emission and finite escape timescale in addition to the Coulomb energy losses of ultrarelativistic electrons interacting with flat-spectrum turbulence $q=1$.
At higher energies and in the range $p_{\rm Coul}\ll p \ll p_{\rm brem}$ bremsstrahlung loss is the dominant process and the equilibrium momentum defined by the condition $t_{\rm acc} = t_{\rm brem}$ for $q<2$ becomes $\chi_{\rm eq}= (\tau_{\rm
brem}/\tau_{\rm acc})^{1/(2-q)}$, yielding $\vartheta_{\chi} = \chi_{\rm
eq}^{-(2-q)}$. Hence, the Green’s function (\[green-loss\]) is $$\begin{aligned}
& & \left. \mathcal{G}(\chi, \chi_{\rm inj})\right|_{\rm brem}^{q<2} = \chi^2 \, e^{- \, {1
\over 2-q} \, \left({\chi \over \chi_{\rm eq}}\right)^{2-q}} \, \left({1 \over
A} + \int^{\min[\chi_{\rm inj},\, \chi]}_{\chi_1} d\chi' \, \chi'^{-(2+q)} \, e^{{1 \over
2-q} \, \left({\chi' \over \chi_{\rm eq}}\right)^{2-q}} \right) \approx \label{green-brem} \\
& & \approx \chi^2 \, e^{- \, {1 \over 2-q} \, \left({\chi \over \chi_{\rm
eq}}\right)^{2-q}} \, {\chi_{\rm eq}^{-1-q} \, (-1)^{3 / (2-q)} \over (2-q)^{3 /
(2-q)}} \, \Gamma\left[ - {1+q \over 2-q} \, , \, - \, {\left(\min[\chi_{\rm inj},\, \chi]
/ \chi_{\rm eq}\right)^{2-q} \over 2-q} , \, - \, {\left(\chi_1 / \chi_{\rm
eq}\right)^{2-q} \over 2-q} \right] \nonumber\end{aligned}$$ (equations \[green-loss\]$-$\[s-loss\]). In other words, for any injection conditions the expected electron energy distribution is of the $N(\chi) \propto
\chi^2 \, \exp\left[- {1 \over 2-q} \left(\chi / \chi_{\rm
eq}\right)^{2-q}\right]$ form, except for the case when high energy particles with $\chi_{\rm inj} > \chi_{\rm eq}$ are injected to the system. Such high energy particles subjected to the bremsstrahlung energy losses form then an additional ‘cooled’ high-energy power-law tail $N(\chi) \propto \chi^{-1}$ in the momentum range between $\chi_{\rm eq}$ and $\chi_{\rm inj}$, in agreement with the appropriate form of $\widetilde{N}(\chi)$ with $\vartheta_{\chi} =
const$ (see equation \[kardashev\]).
The situation changes for $q=2$, since both the acceleration and cooling timescales are now independent of electrons’ energy. In this case $S(\chi) =
\chi^{-2 + (\tau_{\rm acc}/\tau_{\rm brem})}$, and the Green’s function (\[green-loss\]) adopts the form $$\begin{aligned}
\left. \mathcal{G}(\chi, \chi_{\rm inj})\right|_{\rm brem}^{q=2} = \chi^{2-{\tau_{\rm acc}
\over \tau_{\rm brem}}} \, \left({1 \over A} + \int^{\min[\chi_{\rm inj},\, \chi]}_{\chi_1}
d\chi' \, \chi'^{-4+{\tau_{\rm acc} \over \tau_{\rm brem}}} \right) \sim \nonumber \\
\sim {1 \over 1-\sigma'} \, \chi^{-\sigma'} \, \left\{ \begin{array}{ccc}
\chi_1^{-1+\sigma'} \quad & {\rm for} & \tau_{\rm acc}/\tau_{\rm brem} < 3 \\
\min^{-1+\sigma'}(\chi_{\rm inj}, \,\chi) \quad & {\rm for} & \tau_{\rm acc}/\tau_{\rm
brem} > 3 \end{array} \right. \, , \label{green-brem3}\end{aligned}$$ where $\sigma' \equiv {\tau_{\rm acc} \over \tau_{\rm brem}}-2$. This is consistent with the appropriate Green’s function found by @sch87 who, in a framework of the ‘hard-sphere’ approximation $q=2$, considered also synchrotron emission and particle escape in addition to the bremsstrahlung radiation. The solution (\[green-brem3\]) implies that within the whole energy range the expected electron energy distribution is of the power-law form $N(\chi) \propto \chi^{-\sigma'}$, with the power-law index $-2 < \sigma' < 1$. For any longer acceleration timescale, $\tau_{\rm acc} > 3 \, \tau_{\rm brem}$, and monoenergetic injection $Q(\chi) \propto \delta (\chi - \chi_{\rm inj})$, the emerging electron spectra are expected to be of the $N(\chi) \propto
\chi^{-1}$ form for $\chi < \chi_{\rm inj}$, while $N(\chi) \propto
\chi^{-\sigma'}$ with $\sigma' > 1$ for $\chi > \chi_{\rm inj}$.
Efficient Particle Escape
=========================
In this section we investigate steady-state solutions to the momentum diffusion equation of radiating ultrarelativistic particles with a finite escape timescale (equation \[steady\]). Our analytical approach force us to consider only the limiting cases of turbulent spectral indices $q=2$ or $q=1$, as well as to restrict the analysis of radiative losses to the synchrotron and/or IC-Thompson regime processes, ($\vartheta_{\chi} \propto \chi$). We note that the global approximation to the solution of the momentum diffusion equation not necessarily restricted to some particular values of the $q$ parameter, with the regular energy losses and particle escape terms included, were studied by @gal95 by using the WKBJ method. Just us before, we consider finite energy range of particles undergoing momentum diffusion, $0 < \chi_1, \, \chi_2 <
\infty$, strictly related to the finite wavelength range of interacting turbulent modes. We construct the Green’s function accordingly to the procedure outlined in the previous section §3, with addition of the escape term ($\varepsilon\neq 0$) and with a different boundary conditions. Specifically, we change equation (\[eqfinal\]) to $${\partial \mathcal{N} \over \partial \tau} + \left. \mathcal{F}\right|_{\chi_2}
- \left. \mathcal{F}\right|_{\chi_1} = \int_{\chi_1}^{\chi_2} d\chi \, Q(\chi,
\tau) - \varepsilon \, \int_{\chi_1}^{\chi_2} d\chi \, \chi^{2-q} \, N(\chi) \,
,
\label{cont-esc}$$ where the particle flux in the momentum space $\mathcal{F}[N(\chi)]$ is defined in the same way as previously (equation \[flux1\]). As evident, the no-flux boundary conditions, $\mathcal{F}[N(\chi_1)] = \mathcal{F}[N(\chi_2)] = 0$, and conservation of total number of particles, $\partial \mathcal{N} / \partial \tau
=0$, (within the energy range $[\chi_1, \chi_2]$) implies that the particle injection is completely balanced by the particle escape. We will assume this to be the case in this section. Physical realization of these would imply presence of an another efficient yet unspecified acceleration process operating at $\chi < \chi_1$, which prevent negative particle momentum flux through the $\chi_1$ boundary. As shown below, the solutions we obtain agree with the ones discussed in the literature for singular boundary conditions for the infinite momentum range [@jon70; @sch84; @bog85; @par95], as long as we are dealing with particle momenta $\chi \gg \chi_1$ and $\chi \ll \chi_2$.
‘Hard-Sphere’ Approximation
---------------------------
‘Hard-Sphere’ approximation for the momentum diffusion of ultrarelativistic electrons undergoing synchrotron energy losses corresponds to the fixed $q=2$ and $\vartheta_{\chi} = \chi/\chi_{\rm eq}$ (see equations \[timescales\] and \[syn\]). With these, the equation (\[steady\]) adopts the form $$\chi^2 \, N''(\chi) + \chi_{\rm eq}^{-1} \, \chi^2 \, N'(\chi) + \left(2 \,
\chi_{\rm eq}^{-1} \, \chi - 2 - \varepsilon\right) \, N(\chi) = -Q(\chi) \, .
\label{hss}$$ The two linearly-independent particular solutions to the homogeneous form of the above equation are $$\begin{aligned}
y_1(\chi) & = & \chi^{\sigma+1} \, e^{-{\chi \over \chi_{\rm eq}}} \,
U\!\left[\sigma-1, \, 2 \sigma +2, \, {\chi \over \chi_{\rm eq}}\right] \, ,
\nonumber \\
y_2(\chi) & = & \chi^{\sigma+1} \, e^{-{\chi \over \chi_{\rm eq}}} \,
M\!\left[\sigma-1, \, 2 \sigma +2, \, {\chi \over \chi_{\rm eq}}\right] \, ,
\label{hss-y}\end{aligned}$$ where $U[a,b,z]$ and $M[a,b,z]$ are Tricomi and Kummer confluent hypergeometrical functions, respectively, and $\sigma \equiv -(1/2) + [(9/4) +
\varepsilon]^{1/2}$. Introducing next their linear combinations, $u_1(\chi) =
y_1(\chi) + \alpha \, y_2(\chi)$ and $u_2(\chi) = y_1(\chi) + \beta \,
y_2(\chi)$, one may find that the no-flux boundary conditions $\mathcal{F}[u_1(\chi_1)]=\mathcal{F}[u_2(\chi_2)]=0$ are fulfilled for $$\alpha = (2+\sigma) \, {U\!\left[\sigma, \, 2 \sigma +2 , \, \chi_1 / \chi_{\rm
eq} \right] \over M\!\left[\sigma, \, 2 \sigma +2 , \, \chi_1 / \chi_{\rm eq}
\right]} \, , \quad {\rm and} \quad
\beta = (2+\sigma) \, {U\!\left[\sigma, \, 2 \sigma +2 , \, \chi_2 / \chi_{\rm
eq} \right] \over M\!\left[\sigma, \, 2 \sigma +2 , \, \chi_2 / \chi_{\rm eq}
\right]} \, .
\label{hss-bc}$$ This gives the Green’s function of the problem as $$\begin{aligned}
& & \left. \mathcal{G}(\chi, \chi_{\rm inj})\right|_{\rm esc}^{q=2} = {\Gamma(\sigma-1) \over
\Gamma(2\sigma+2)} \, \left(\alpha - \beta\right)^{-1} \, \chi_{\rm inj}^{-2} \, \chi_{\rm
eq}^{-2 \sigma - 1} \, e^{\chi_{\rm inj} / \chi_{\rm eq}} \times \nonumber \\
& & \times \quad \left\{ \begin{array}{ccc}
\left[y_1(\chi) + \alpha \, y_2(\chi)\right] \, \left[y_1(\chi_{\rm inj}) + \beta \,
y_2(\chi_{\rm inj})\right] & {\rm for} & \chi_1 \leq \chi < \chi_{\rm inj} \\
\left[y_1(\chi_{\rm inj}) + \alpha \, y_2(\chi_{\rm inj})\right] \, \left[y_1(\chi) + \beta \,
y_2(\chi)\right] & {\rm for} & \chi_{\rm inj} < \chi \leq \chi_2
\end{array} \right. \, . \label{hss-G} \end{aligned}$$
![‘Hard-sphere approximation’ ($q=2$): particle spectra resulting from joint stochastic acceleration, particle escape, and synchrotron energy losses. The spectra correspond to the monoenergetic injection $Q(\chi) \propto
\delta(\chi-\chi_{\rm inj})$ with fixed normalization, fixed acceleration and cooling rates, but different escape timescales (parameter $\varepsilon = 3$, $0.1$, $10^{-4}$; dotted, dashed, and solid lines, respectively). For illustration, $\chi_1 = 10^{-2}$, $\chi_{\rm inj} = 1$, $\chi_{\rm eq} = 10^6$, and $\chi_2 = 10^{8}$ have been selected. []{data-label="hss1"}](fig8.eps)
![The same as FIgure (\[hss1\]) except for $\chi_{\rm inj} = 10^6$.[]{data-label="hss2"}](fig9.eps)
In order to investigate the above solution, let us consider first the case $\chi_1 \ll \chi_{\rm inj} \ll \chi_{\rm eq} \ll \chi_2$, and use the standard expansion of the confluent hypergeometrical functions: $U[a,b,z] \sim z^{-a}$ and $M[a,b,z]
\sim \Gamma(b) \, e^z \, z^{a-b} / \Gamma(a)$ for $z \rightarrow \infty$, while $U[a,b,z] \sim \Gamma(b-1) \, z^{1-b} / \Gamma(a)$ and $M[a,b,z] \sim 1$ for $z
\rightarrow 0$ [@abr64]. In this limit one gets $$\left. \mathcal{G}(\chi, \chi_{\rm inj})\right|_{\rm esc, \, \chi_{\rm inj}<}^{q=2} \sim
\left\{ \begin{array}{ccc}
{1 \over 2 \sigma + 1} \, \chi_{\rm inj}^{-\sigma-2} \, \chi^{\sigma+1}
& {\rm for} & \chi_1 < \chi < \chi_{\rm inj} \\
{1 \over 2 \sigma + 1} \, \chi_{\rm inj}^{\sigma-1} \, \chi^{-\sigma}
& {\rm for} & \chi_{\rm inj} < \chi \ll \chi_{\rm eq} \\
{\Gamma(\sigma-1) \over \Gamma(2\sigma+2)} \, \chi_{\rm inj}^{\sigma-1} \, \chi_{\rm
eq}^{-\sigma-2} \, \chi^2 \, e^{-\chi / \chi_{\rm eq}} & {\rm for} & \chi_{\rm
eq} \lesssim \chi < \chi_2
\end{array} \right. \, .
\label{hss-G-approx}$$ Thus, by moving the critical momenta $\chi_1$ and $\chi_2$ toward $0$ and $\infty$, respectively, the resultant Green’s function approaches asymptotically — as expected — the corresponding Green’s function for singular boundary conditions obtained by @jon70 [@sch84] and @par95. In particular, one can find that with the monoenergetic injection $Q(\chi) \propto
\delta(\chi-\chi_{\rm inj})$, the resulting electron energy distribution is then of the form $N(\chi<\chi_{\rm inj}) \propto \chi^{\sigma+1}$ and $N(\chi>\chi_{\rm inj}) \propto \chi^{-\sigma}$ up to maximum momentum $\chi_{\rm eq}$. Moreover, for the increasing escape timescale $\varepsilon
\rightarrow 0$, one has $\sigma \approx 1$ and the pile-up bump $N(\chi) \propto
\chi^2 \, \exp\left[-\chi / \chi_{\rm eq}\right]$ emerging around $\chi \sim
\chi_{\rm eq}$ energies. This is shown in Figure (\[hss1\]), where we fixed normalization of the monoenergetic injection $\int dp \, \widetilde{Q}(p)$, acceleration and losses timescales, but varied the escape timescale ($\varepsilon = 3$, $0.1$, $10^{-4}$; dotted, dashed, and solid lines, respectively). For illustration we have selected $\chi_1 = 10^{-2}$, $\chi_{\rm
inj} = 1$, $\chi_{\rm eq} = 10^6$, and $\chi_2 = 10^{8}$.
When high energy particles are injected to the system, such that $\chi_1 \ll
\chi_{\rm eq} \ll \chi_{\rm inj} \ll \chi_2$, one may find useful asymptotic expansion of the Green’s function $$\left. \mathcal{G}(\chi, \chi_{\rm inj})\right|_{\rm esc, \, \chi_{\rm inj}>}^{q=2} \sim
\left\{ \begin{array}{ccc}
{\Gamma(\sigma-1) \over \Gamma(2 \sigma + 2)} \, \chi_{\rm eq}^{-\sigma-2} \,
\chi^{\sigma+1} \, e^{-\chi/\chi_{\rm eq}}
& {\rm for} & \chi_1 < \chi \lesssim \chi_{\rm eq} \\
\chi^{-2} \, \chi_{\rm eq} & {\rm for} & \chi_{\rm eq} \ll \chi < \chi_{\rm inj} \\
\chi_{\rm inj}^{-4} \, \chi_{\rm eq} \, e^{\chi_{\rm inj}/\chi_{\rm eq}} \, \chi^2 \,
e^{-\chi/\chi_{\rm eq}} & {\rm for} & \chi_{\rm inj} < \chi < \chi_2
\end{array} \right. \, .
\label{hss-G-approx2}$$ That is, for the monoenergetic injection $Q(\chi) \propto \delta(\chi-\chi_{\rm
inj})$ with $\chi_{\rm inj} > \chi_{\rm eq}$ the resulting electron energy distribution is of the form $N(\chi) \propto \chi^{\sigma+1} \,
\exp\left[-\chi/\chi_{\rm eq}\right]$ for $\chi \lesssim \chi_{\rm eq}$. However, within the energy range $\chi_{\rm eq} < \chi < \chi_{\rm inj}$ the power-law tail $N(\chi) \propto \chi^{-2}$ emerges, representing radiatively ($\vartheta_{\chi} \propto \chi$) cooled high-energy particles injected to the system, undergoing negligible (when compared to the energy loss rate) momentum diffusion. At even higher energies, $\chi > \chi_{\rm inj}$, the particle spectrum cuts-off rapidly. This is shown in Figure (\[hss2\]), where, as before, we fixed normalization of the monoenergetic injection $\int dp \,
\widetilde{Q}(p)$, acceleration and losses timescales, but varied the escape timescale ($\varepsilon = 3$, $0.1$, $10^{-4}$; dotted, dashed, and solid lines, respectively). For illustration we have selected $\chi_1 = 10^{-2}$, $\chi_{\rm
eq} = 10^2$, $\chi_{\rm inj} = 10^6$, and $\chi_2 = 10^{8}$. Note, that the esape timescale, and hence parameter $\varepsilon$, influences now the slope and normalization of particle energy distribution only in the ‘low-energy’ regime $\chi < \chi_{\rm eq}$, such that the spectrum approaches $\propto x^2$ for $\varepsilon \rightarrow 0$.
Bohm Limit
----------
Bohm limit for the momentum diffusion of ultrarelativistic electrons undergoing synchrotron energy losses corresponds to $q=1$ and $\vartheta_{\chi} =
\chi/\chi_{\rm eq}^2$ (see equations \[timescales\] and \[syn\]). The difference with the ‘hard-sphere’ approximation is that the balance between acceleration and escape timescales, $t_{\rm acc} = t_{\rm esc}$, define now yet another critical energy, $\chi_{\rm esc} = \varepsilon^{-1/2}$ and equation (\[steady\]) takes the form $$\chi \, N''(\chi) + \left(\chi_{\rm eq}^{-2} \, \chi^2 -1\right) \, N'(\chi) +
\left(2 \, \chi_{\rm eq}^{-2} \, \chi - \chi_{\rm esc}^{-2} \, \chi\right) \,
N(\chi) = -Q(\chi) \, .
\label{bohm-syn}$$ The two linearly-independent particular solutions to the homogeneous form of the above equation are $$\begin{aligned}
y_1(\chi) & = & \chi^{2} \, e^{-{1 \over 2} \, \left({\chi \over \chi_{\rm
eq}}\right)^2} \, U\!\left[\eta, \, 2, \, {1 \over 2} \left({\chi \over
\chi_{\rm eq}}\right)^2\right] \, , \nonumber \\
y_2(\chi) & = & \chi^{2} \, e^{-{1 \over 2} \, \left({\chi \over \chi_{\rm
eq}}\right)^2} \, M\!\left[\eta, \, 2, \, {1 \over 2} \left({\chi \over
\chi_{\rm eq}}\right)^2\right] \, , \label{bohm-syn-y}\end{aligned}$$ where $\eta \equiv {1 \over 2} (\chi_{\rm eq} / \chi_{\rm esc})^2$. Defining $u_1(\chi) = y_1(\chi) + \alpha \, y_2(\chi)$ and $u_2(\chi) = y_1(\chi) +
\beta \, y_2(\chi)$, one finds that the no-flux boundary conditions $\mathcal{F}[u_1(\chi_1)]=\mathcal{F}[u_2(\chi_2)]=0$ corresponds to $$\alpha = 2 \, {U\!\left[\eta+1, \, 3 , \, {1 \over 2} \left(\chi_1 / \chi_{\rm
eq}\right)^2 \right] \over M\!\left[\eta+1, \, 3 , \, {1 \over 2} \left(\chi_1 /
\chi_{\rm eq}\right)^2 \right]} \, , \quad {\rm and} \quad
\beta = 2 \, {U\!\left[\eta+1, \, 3 , \, {1 \over 2} \left(\chi_2 / \chi_{\rm
eq}\right)^2 \right] \over M\!\left[\eta+1, \, 3 , \, {1 \over 2} \left(\chi_2 /
\chi_{\rm eq}\right)^2 \right]} \, .
\label{bohm-syn-bc}$$ This gives the Green’s function of the problem as $$\begin{aligned}
& & \left. \mathcal{G}(\chi, \chi_{\rm inj})\right|_{\rm esc}^{q=1} =
{1 \over 4} \, \Gamma(\eta) \, \left(\alpha - \beta\right)^{-1} \, \chi_{\rm inj}^{-2} \,
\chi_{\rm eq}^{-2} \, e^{{1 \over 2} (\chi_{\rm inj} / \chi_{\rm eq})^2} \times
\nonumber \\
& & \times \quad \left\{ \begin{array}{ccc}
\left[y_1(\chi) + \alpha \, y_2(\chi)\right] \, \left[y_1(\chi_{\rm inj}) + \beta \,
y_2(\chi_{\rm inj})\right] & {\rm for} & \chi_1 \leq \chi < \chi_{\rm inj} \\
\left[y_1(\chi_{\rm inj}) + \alpha \, y_2(\chi_{\rm inj})\right] \, \left[y_1(\chi) + \beta \,
y_2(\chi)\right] & {\rm for} & \chi_{\rm inj} < \chi \leq \chi_2
\end{array} \right. \, . \label{bohm-syn-G}\end{aligned}$$
Let us consider first the case $\chi_1 \ll \chi_{\rm inj} \ll \chi_{\rm eq} \ll \chi_2$ for which he Green’s function of equation (\[bohm-syn-G\]) can be then approximated as $$\left. \mathcal{G}(\chi, \chi_{\rm inj})\right|_{\rm esc, \, \chi_{\rm inj}<}^{q=1} \sim
\left\{ \begin{array}{ccc}
{1 \over 2} \, \chi_{\rm inj}^{-2} \, \chi^{2} & {\rm for} & \chi_1 < \chi < \chi_{\rm inj} \\
{1 \over 2} & {\rm for} & \chi_{\rm inj} < \chi \ll \chi_{\rm eq} \\
2^{\eta -2} \, \Gamma(\eta) \, \chi_{\rm eq}^{-2+2 \, \eta} \, \chi^{2 - 2 \,
\eta} \, e^{-{1 \over 2} (\chi / \chi_{\rm eq})^2} & {\rm for} & \chi_{\rm eq}
\lesssim \chi < \chi_2
\end{array} \right. \, .
\label{bohm-G-approx}$$ Note that, as expected, in the limits $\chi_1 \rightarrow 0$ and $\chi_2 \rightarrow \infty$, the Green’s function (\[bohm-syn-G\]) approaches asymptotically the solution obtained for singular boundary conditions by @bog85. As shown in Figure (\[bohm-syn-1\]), for a monoenergetic injection $Q(\chi) \propto
\delta(\chi-\chi_{\rm inj})$, the resulting electron energy distribution is $N(\chi<\chi_{\rm inj}) \propto \chi^{2}$ and $N(\chi>\chi_{\rm inj}) \propto
const$ up to maximum momentum $\chi_{\rm eq}$, with the spectral indexes independent of the value of the escape timescale. However, for energies near and above $\chi_{\rm eq}$ the spectra depend on the value of $\eta$. For $\eta \rightarrow 0$, i.e. when the escape timescale is large, the familiar bump $N(\chi) \propto
\chi^2 \, \exp\left[-{1 \over 2} (\chi / \chi_{\rm eq})^2\right]$ emerges around $\chi \sim \chi_{\rm eq}$ energies (solid line). In the opposite case, when $\eta > 1$ (or $\chi_{\rm eq} > \chi_{\rm esc}$), no pile-up bump is present, and the electron spectrum cut-offs exponentially at $\chi_{\rm esc}$ momenta (dashed and dotted lines). Here, as before, we fixed the normalization of the monoenergetic injection $\int \widetilde{Q}(p) \, dp$, and the acceleration and loss timescales, but varied the escape timescale such that $\chi_{\rm esc} = 10^{5}$, $10^6$, and $10^7$ (dotted, dashed, and solid lines, respectively). We choose $\chi_1 = 10^{-2}$, $\chi_{\rm inj} = 1$, $\chi_{\rm eq} = 10^6$, and $\chi_2 =
10^{8}$.
![Bohm Limit ($q=1$): particle spectra resulting from joint stochastic acceleration, particle escape, and synchrotron energy losses. The spectra correspond to the monoenergetic injection $Q(\chi) \propto \delta(\chi-\chi_{\rm
inj})$ with fixed normalization, fixed acceleration and cooling rates, but different escape timescales (critical momenta $\chi_{\rm esc} = 10^{5}$, $10^6$, $10^7$; dotted, dashed, and solid lines, respectively). For illustration, $\chi_1 = 10^{-2}$, $\chi_{\rm inj} = 1$, $\chi_{\rm eq} = 10^6$, and $\chi_2 =
10^{8}$ have been selected.[]{data-label="bohm-syn-1"}](fig10.eps)
![The same as Figure (\[bohm-syn-1\]) except for $\chi_{\rm inj} = 10^6$.[]{data-label="bohm-syn-2"}](fig11.eps)
In the case when $\chi_1 \ll \chi_{\rm eq} \ll \chi_{\rm inj} \ll \chi_2$ the asymptotic expansion of the Green’s function (\[bohm-syn-G\]) yields $$\left. \mathcal{G}(\chi, \chi_{\rm inj})\right|_{\rm esc, \, \chi_{\rm inj}>}^{q=1} \sim
\left\{ \begin{array}{ccc}
2^{\eta - 2} \, \Gamma(\eta) \, \chi_{\rm inj}^{-2 \eta} \, \chi_{\rm eq}^{2 \eta -2} \,
\chi^2 \, e^{-{1 \over 2} \, (\chi / \chi_{\rm eq})^2} & {\rm for} & \chi_1 <
\chi \lesssim \chi_{\rm eq} \\
\chi_{\rm inj}^{-2 \eta} \, \chi_{\rm eq}^{2} \, \chi^{2 \eta -2} & {\rm for} & \chi_{\rm
eq} \ll \chi < \chi_{\rm inj} \\
\chi_{\rm inj}^{2 \eta - 4} \, \chi_{\rm eq}^{2} \, \chi^{2\eta -2} \, e^{{1 \over 2} \,
(\chi_{\rm inj} / \chi_{\rm eq})^2} \, e^{-{1 \over 2} \, (\chi / \chi_{\rm eq})^2} & {\rm
for} & \chi_{\rm inj} \lesssim \chi < \chi_2
\end{array} \right. \, .
\label{bohm-G-approx2}$$ Again as above, the spectrum is different in the case of high energy injection. For example, as shown in Figure (\[bohm-syn-2\]), for the monoenergetic injection $Q(\chi) \propto \delta(\chi-\chi_{\rm inj})$ with $\chi_{\rm inj} > \chi_{\rm eq}$ the resulting electron energy distribution is of the form $N(\chi) \propto \chi^{2} \,
\exp\left[-{1 \over 2} (\chi/\chi_{\rm eq})^2\right]$ for $\chi \lesssim
\chi_{\rm eq}$, while $N(\chi) \propto \chi^{2 \eta - 2}$ for $\chi_{\rm eq} \ll
\chi < \chi_{\rm inj}$. It is interesting to note that the Bohm limit case behaves differently from the $q=2$ case and analogous injection condition. The escape timescale affecs now (via the parameter $\chi_{\rm esc}$, or $\eta$) the normalization of the low-energy ($\chi <
\chi_{\rm eq}$) segment of the particle spectrum but not its power-law slope. It determines, on the other hand, the ‘radiatively-cooled’ part of the particle distribution in the range $\chi_{\rm eq} < \chi < \chi_{\rm inj}$, which is, however, very close to the standard $\propto \chi^{-2}$ for any $\chi_{\rm esc} \gg \chi_{\rm eq}$ (or $\eta \ll 1$). Here, as before, we fixed normalization of the monoenergetic injection $\int dp \,
\widetilde{Q}(p)$, and the acceleration and loss timescales, but varied the escape timescale such that $\chi_{\rm esc} = 10^{5}$, $10^6$, and $10^7$ (dotted, dashed, and solid lines, respectively). Also we set $\chi_1 = 10^{-2}$, $\chi_{\rm inj} = 1$, $\chi_{\rm eq} = 10^6$, and $\chi_2 = 10^{8}$.
Emission Spectra
================
In the previous sections §3 and §4, we showed that stochastic interactions of radiating ultrarelativistic electrons (Lorentz factors $\gamma \equiv p /
m_{\rm e} c \gg 1$) with turbulence characterized by a power-law spectrum $\mathcal{W}(k) \propto k^{-q}$ result in formation of a ‘universal’ high-energy electron energy distribution $$n_{\rm e}(\gamma) = n_0 \, \gamma^2 \, \exp\left[- {1\over a} \, \left({\gamma
\over \gamma_{\rm eq}}\right)^a\right] \, ,
\label{ele}$$ as long as particle escape from the system is inefficient and the radiative cooling rate scales with some power of electron energy. Here the equilibrium energy $\gamma_{\rm eq}$ is defined by the balance between acceleration and the energy losses timescales, while the parameter $a$ depends on the dominant radiative cooling process and the turbulence spectrum. In particular, for either synchrotron or IC/Thomson-regime cooling one has $a = 3 - q$. In the case of dominant IC/KN-regime energy losses (with $q < 1.5$) one has instead $a = 1.5 - q$. Below we investigate in more details emission spectra resulting from such an electron distribution.
Synchrotron Emission
--------------------
Assuming isotropic distribution of momenta of radiating electrons with energy spectrum $n_{\rm e}(\gamma)$, the synchrotron emissivity can be found as $$j_{\rm \nu, \, syn}(\nu) = {\sqrt{3} \, e^3 B \over 4 \pi \, m_{\rm e} c^2} \int
\, d\gamma \, \mathcal{R}\!\left({\nu \over \nu_c \, \gamma^2}\right) \, n_{\rm
e}(\gamma) \,
\label{syn-emissivity}$$ where $\nu_c = 3 e B / 4 \pi m_{\rm e} c$, $$\mathcal{R}(x) = {x^2 \over 2} \, K_{4/3}\left({x \over 2}\right) \,
K_{1/3}\left({x \over 2}\right) - 0.3 \, {x^3 \over 2} \, \left[
K_{4/3}^2\left({x \over 2}\right) - K_{1/3}^2\left({x \over 2}\right)\right] \,
,
\label{crusius}$$ and $K_{\mu}\left(z\right)$ is a modified Bessel function of the second order [@cru86]. Relatively complicated function (\[crusius\]) can be instead conveniently approximated by $\mathcal{R}(x) \approx 1.81 \times \left( 1.33 +
x^{-2/3} \right)^{-1/2} e^{-x}$ [@zir07], allowing for some analytical investigation of the integral (\[syn-emissivity\]). In particular, one may find that the synchrotron emissivity in a frequency range $\nu < \nu_{\rm syn}
\equiv \nu_c \, \gamma_{\rm eq}^2$ is of the form $j_{\rm \nu, \, syn}(\nu <
\nu_{\rm syn}) \propto \nu^{1/3}$, as expected in the case of a very hard (inverted) electron energy distribution at low energies, $n_{\rm e}(\gamma < \gamma_{\rm eq})
\propto \gamma^2$. At higher frequencies, however, the synchrotron spectrum steepens. In order to evaluate such a high-frequency spectral component, we use the introduced approximation for $\mathcal{R}(x)$, electron spectrum as given in (\[ele\]), and with these we re-write synchrotron emissivity (\[syn-emissivity\]) as $$j_{\rm \nu, \, syn}(\nu) \approx {1.81 \, \sqrt{3} \, e^3 B \, \gamma_{\rm eq}^3
n_0 \over 4 \pi \, m_{\rm e} c^2} \int \, dy \, g(\omega, y) \, \exp\left[-
\omega \, h(\omega, y)\right] \, ,
\label{integral}$$ where $\omega \equiv \nu / \nu_{\rm syn}$, $y \equiv \gamma / \gamma_{\rm eq}$, $g(\omega, y) \equiv y^2 \, (1.33 + \omega^{-2/3} y^{4/3})^{-1/2}$, and $h(\omega, y) \equiv y^{-2} + y^a / (\omega \, a)$.
![Synchrotron spectra resulting from the electron energy distribution (\[ele\]) for fixed parameters $B$, $n_0$, and $\gamma_{\rm eq}$. Solid lines correspond to integration of the exact form of $\mathcal{R}(x)$ as given in equation (\[crusius\]), while dashed lines to the approximate formulae following from (\[integral-approx\]). Different cases for the parameter $a$ are considered in the plot, namely (a) $a = 3-q$ with $q=1$, (b) $a = 3-q$ with $q=2$, and (c) $a = 1.5-q$ with $q=1$.[]{data-label="SYN-SED"}](fig12.eps)
With such a form it can be noted that for large $\omega$, i.e. for $\nu > \nu_{\rm syn}$, the integral of interest can be perform approximately using the steepest descent method [see @pet81]. This gives $$\begin{aligned}
& & j_{\rm \nu, \, syn}(\nu>\nu_{\rm syn}) \simeq {1.81 \, \sqrt{3} \, e^3 B \,
\gamma_{\rm eq}^3 n_0 \over 4 \pi \, m_{\rm e} c^2} \, \sqrt{{2 \pi \over \omega
\, h''(\omega, y_{\star})}} \,\, g(\omega, y_{\star}) \, \exp\left[- \omega \,
h(\omega, y_{\star})\right] \simeq
\nonumber \\
& & \simeq {0.54 \, e^3 B \, \gamma_{\rm eq}^3 n_0 \over m_{\rm e} c^2 \, \sqrt{2+a}}
\, \left({2 \nu \over \nu_{\rm syn}}\right)^{{6-a \over 4+2 a}} \, \left[1 +
\left({2 \nu \over \nu_{\rm syn}}\right)^{-{2 a \over 6 + 3 a}}\right]^{-1/2} \,
\exp\left[ - {2 + a \over 2 a} \, \left({2 \nu \over \nu_{\rm syn}}\right)^{{a
\over 2+a}}\right] \, , \label{integral-approx}\end{aligned}$$ where $y_{\star} = (2 \omega)^{1 / (2 +a)}$ is a global maximum of $h(\omega,
y)$, and $h''(\omega, y) = \partial^2 h(\omega, y) / \partial y^2$. Thus, the high-energy synchrotron component drops much less rapidly than suggested by the emissivity of a single electron, $\mathcal{R}(x) \propto e^{-x}$. For example, assuming synchrotron (and/or IC/Thompson-regime) dominance $a = 3-q$, the synchrotron emissivity reads very roughly as $$j_{\rm \nu, \, syn}\left(\nu > \nu_{\rm syn}\right) \propto
\nu^{1/2} \, \exp\left[-1.4 \, \left(\nu / \nu_{\rm syn}\right)^{1/2}\right] \,\,\,\,\,
{\rm for}\,\,\,\,\, q=1,$$ or[^6] $$j_{\rm \nu, \, syn}\left(\nu > \nu_{\rm syn}\right) \propto
\nu^{5/6} \, \exp\left[-1.9 \, \left(\nu / \nu_{\rm syn}\right)^{1/3}\right]
\,\,\,\,\, {\rm for}\,\,\,\,\, q=2.$$ In the case of the IC/KN-regime dominance, $a = 1.5 - q$, the emerging high-energy exponential cut-off in the synchrotron continuum can be even smoother than this, for example $j_{\rm \nu, \, syn}\left(\nu > \nu_{\rm
syn}\right) \propto \nu^{1.1} \, \exp\left[-2.9 \, \left(\nu / \nu_{\rm
syn}\right)^{0.2}\right]$ for $q=1$. These spectra are shown in Figure (\[SYN-SED\]) for fixed parameters $B$, $n_0$, and $\gamma_{\rm eq}$, where both integration of the exact form of $\mathcal{R}(x)$ as given in equation (\[crusius\]) was performed (solid lines), and also approximate formulae following from (\[integral-approx\]) were evaluated for comparison (dashed lines). Different cases for the parameter $a$ are considered in the plot, namely (a) $a = 3-q$ with $q=1$, (b) $a = 3-q$ with $q=2$, and (c) $a = 1.5-q$ with $q=1$. As shown, synchrotron spectra are curved and extend far beyond equilibrium frequency $\nu_{\rm syn}$. In the case of the dominant IC/KN-regime cooling with $q=1$, the $\nu j_{\nu}(\nu) -
\nu$ synchrotron spectrum peaks around $\sim 10^3 \nu_{\rm syn}$. We emphasize that the approximation (\[integral-approx\]), although obviously not accurate in a range $\nu \lesssim \nu_{\rm syn}$, works relatively well at higher frequencies, where the standard $\delta$-approximation for the synchrotron emissivity, $\nu
j_{\rm \nu, \, syn}(\nu) \propto [\gamma^3 \, n_{\rm e}(\gamma)]_{\gamma \propto
\nu^{1/2}}$, fails.
Inverse-Compton Emission
------------------------
Let us consider inverse-Compton emission of isotropic electrons up-scattering monoenergetic and isotropic photon field with energy density $u_{\rm ph}$ and dimensionless photon energy $\epsilon_0 \equiv h\nu_0 / m_{\rm e} c^2$. The appropriate emissivity can be then found from $$j_{\rm \nu, \, ic}(\nu) = {3 \, c h \sigma_{\rm T} \over 16 \pi \, m_{\rm e}
c^2} \, u_{\rm ph} \int_{{1 \over 2} \epsilon \, \left(1 + \sqrt{1 + (\epsilon \,
\epsilon_0)^{-1}}\right)} d\gamma \, {\epsilon \over \gamma^2 \epsilon_0^2} \,
\mathcal{J}(\epsilon, \epsilon_0, \gamma) \, n_{\rm e}(\gamma) \, ,
\label{ic-full1}$$ where $\epsilon \equiv h \nu / m_{\rm e} c^2$, and $\mathcal{J}(\epsilon,
\epsilon_0, \gamma)$ is the IC kernel $$\mathcal{J}(\epsilon, \epsilon_0, \gamma) = 2 \, \mathcal{I} \, \ln \mathcal{I}
+ \mathcal{I} + 1 - 2 \, \mathcal{I}^2 + {\mathcal{L}^2 \mathcal{I}^2 \, (1
-\mathcal{I}) \over 2 \, (1 - \mathcal{L} \, \mathcal{I})} \quad {\rm with}
\quad \mathcal{L} \equiv 4 \epsilon_0 \gamma \quad {\rm and} \quad \mathcal{I}
\equiv {\epsilon \over \mathcal{L} \, (\gamma - \epsilon)}
\label{ic-kernel}$$ [e.g., @blu70].
![Inverse-Compton spectra produced in the Thomson regime, resulting from the electron energy distribution (\[ele\]) for fixed parameters $B$, $n_0$, and $\gamma_{\rm eq}$. Solid lines correspond to the formulae (\[ic-full2\]), and dashed lines to the rough approximation (\[thom-high\]). Two different parameters $a = 3-q$ are considered in the plot, corresponding to the turbulence energy index $q=1$ and $q=2$ (cases (a) and (b), respectively).[]{data-label="IC-T-SED"}](fig13.eps)
![Inverse-Compton spectra produced in the KN regime, resulting from the electron energy distribution (\[ele\]) for fixed parameters $B$, $n_0$, and $\gamma_{\rm eq}$. Solid lines correspond to the exact evaluation of the integral (\[ic-full1\]), and dashed lines to the rough approximation (\[rough-KN\]). Different cases for the parameter $a$ are considered in the plot, namely (a) $a = 3-q$ with $q=1$, (b) $a = 3-q$ with $q=2$, and (c) $a =
1.5-q$ with $q=1$. For illustration $\gamma_{\rm cr}/\gamma_{\rm eq} = 0.01$ has been selected.[]{data-label="IC-KN-SED"}](fig14.eps)
Let us discuss first the case when the KN effects are negligible. The IC kernel can then be approximated by $\mathcal{J}(\epsilon, \epsilon_0, \gamma) \approx
{2 \over 3} (1-\omega / y^2)$, with $y \equiv \gamma / \gamma_{\rm eq}$, $\omega
\equiv \epsilon / \epsilon_{\rm ic/Th}$, and $\epsilon_{\rm ic/Th} \equiv 4
\epsilon_0 \, \gamma_{\rm eq}^2$ which is the characteristic energy of soft photon inverse-Compton up-scattered in a Thomson regime by electrons with Lorentz factor $\gamma_{\rm eq}$. Hence, with the electron energy distribution of the form (\[ele\]), one can find that $$\begin{aligned}
& & \epsilon j_{\rm \epsilon, \, ic/Th}(\epsilon) = {2 \over \pi} \, c \sigma_{\rm Th}
\, u_{\rm ph} \, n_0 \, \gamma_{\rm eq}^5 \, \int_{\sqrt{x}} \, dy \, \omega^2 \,
\left( 1 - {\omega \over y^2}\right) \, \exp\!\left[-{1 \over a} \, y^a\right] \approx
\nonumber \\
& & \approx {2 \over \pi} \, c \sigma_{\rm T} \, u_{\rm ph} \, n_0 \, \gamma_{\rm eq}^5 \,
a^{-1} \omega^2 \, \left\{ a^{1/a} \, \Gamma\!\left[a^{-1}, a^{-1} \,
\omega^{a/2}\right] - a^{-1/a} \, \omega \, \Gamma\!\left[- a^{-1}, a^{-1} \,
\omega^{a/2}\right]\right\} \, , \label{ic-full2}\end{aligned}$$ where $\Gamma[a,z]$ is incomplete Gamma function. With the expansion $\Gamma[a,z] \sim \Gamma[a]$ for $z \rightarrow 0$ [@abr64], one can approximate further $$\epsilon j_{\rm \epsilon, \, ic/Th}(\epsilon < \epsilon_{\rm ic/Th}) \sim {2 \over
\pi} \, c \sigma_{\rm Th} \, u_{\rm ph} \, n_0 \, \gamma_{\rm eq}^5 \, a^{{1-a \over a}}
\, \Gamma\!\left(a^{-1}\right) \, \left({\epsilon \over \epsilon_{\rm
ic/Th}}\right)^2 \, .
\label{thom-low}$$ In other words, the IC emissivity at low photon energies is of the form $j_{\rm
\epsilon, \, ic/Th}(\epsilon < \epsilon_{\rm ic/Th}) \propto \epsilon$. This is the flattest IC/Thomson-regime spectrum, being analogous to the flattest synchrotron one $j_{\rm \nu, \, syn}(\nu < \nu_{\rm syn}) \propto \nu^{1/3}$. At higher photon energies, noting that $\Gamma[a,z] \sim z^{a-1} \, e^{-z}$ for $z
\rightarrow \infty$, one may find instead $$\epsilon j_{\rm \epsilon, \, ic/Th}(\epsilon > \epsilon_{\rm ic/Th}) \sim {2 \over
\pi} \, c \sigma_{\rm Th} \, u_{\rm ph} \, n_0 \, \gamma_{\rm eq}^5 \, \left({\epsilon
\over \epsilon_{\rm ic/Th}}\right)^{{5-a \over 2}} \, \exp\left[- {1 \over a} \,
\left({\epsilon \over \epsilon_{\rm ic/Th}}\right)^{{a \over 2}} \right] \, .
\label{thom-high}$$ Therefore, the exponential cut-off of the IC/Thomson-regime component is now steeper than the exponential cut-off of the synchrotron component originating from the same particle distribution. In particular, with $a = 3-q$ one gets $j_{\rm \epsilon, \, ic/Th}(\epsilon > \epsilon_{\rm ic/Th}) \propto
\epsilon^{1/2} \, \exp[ - {1 \over 2} \, (\epsilon / \epsilon_{\rm ic/Th})]$ for $q=1$, while $j_{\rm \epsilon, \, ic/Th}(\epsilon > \epsilon_{\rm ic/Th}) \propto
\epsilon \, \exp[ - (\epsilon / \epsilon_{\rm ic/Th})^{1/2}]$ for $q=2$ (that can be compared with the corresponding synchrotron emissivities provided above). These spectra are shown in Figure (\[IC-T-SED\]) for fixed parameters $B$, $n_0$, and $\gamma_{\rm eq}$. Here the solid lines correspond to the formulae (\[ic-full2\]), and dashed lines to the rough approximation (\[thom-high\]). Two different parameters $a = 3-q$ are considered in the plot, corresponding to the turbulence energy index $q=1$ and $q=2$ (cases (a) and (b), respectively).
Finally, we comment on the emission spectra produced in a deep KN regime of the IC scattering, i.e. when $\gamma > \gamma_{\rm cr} \equiv 1/ 4 \epsilon_0$, by the highest-energy electrons $\gamma \gtrsim \gamma_{\rm eq}$. In such a case, the emissivity has to be evaluated by performing the integral (\[ic-full1\]) with the exact IC kernel as given in equation (\[ic-kernel\]). A rather crude approximation for such can be obtained by utilizing the $\delta$-approximation for the resulting IC/KN-regime photon energy, namely $\epsilon = \gamma$. In particular, with the electron energy distribution as given in (\[ele\]), and with all the previous assumptions regarding monoenergetic and isotropic soft photon field, one finds $$\begin{aligned}
& & \epsilon j_{\rm \epsilon, \, ic/KN}(\epsilon \gtrsim \gamma_{\rm eq}) \simeq
{m_{\rm e} c^2 \over 4 \pi}\left. {\gamma^2 \, n_{\rm e}(\gamma) \over t_{\rm
IC}(\gamma)}\right|_{\gamma = \epsilon} \simeq \nonumber \\
& & \simeq {1 \over 3 \pi} \, c \sigma_{\rm T} \, u_{\rm ph} \, n_0 \, \gamma_{\rm eq}^5 \,
\left({\epsilon \over \gamma_{\rm eq}}\right)^5 \, \left(1 + {\epsilon \over
\gamma_{\rm eq}} \, {\gamma_{\rm eq} \over \gamma_{\rm cr}}\right)^{-1.5} \,
\exp\left[- { 1 \over a} \, \left({\epsilon \over \gamma_{\rm
eq}}\right)^a\right] \, ,
\label{rough-KN}\end{aligned}$$ where $t_{\rm IC}(\gamma)$ is the inverse-Compton cooling timescale as introduced previously in equation (\[ic\]). As shown in Figure \[IC-KN-SED\], as a result the IC/KN-regime spectra cut-off sharply above $\epsilon = \gamma_{\rm eq}$ photon energies, imitating exponential cut-off in the energy distribution of radiating particles. Here the exact calculations are plotted as solid lines, and rough approximation (\[rough-KN\]) as dashed ones. We fix parameters $B$, $n_0$, $\gamma_{\rm eq}$, $\gamma_{\rm cr}/\gamma_{\rm eq} = 0.01$, and again, different cases for the parameter $a$ are considered; (a) $a = 3-q$ with $q=1$, (b) $a = 3-q$ with $q=2$, and (c) $a = 1.5-q$ with $q=1$. We also choose for illustration $\gamma_{\rm cr}/\gamma_{\rm eq} = 0.01$.
Discussion and Conclusions
==========================
In this paper we study steady-state spectra of ultrarelativistic electrons undergoing momentum diffusion due to resonant interactions with turbulent MHD waves. We assume a given power spectrum $\mathcal{W}(k) \propto k^{-q}$ for magnetic turbulence within some finite range of turbulent wavevectors $k$, and consider variety of turbulence spectral indices $1 \leq q \leq 2$. For example, $q=1$ corresponds to the ‘Bohm limit’ of the stochastic acceleration processes, $q = 2$ represents the ‘hard-sphere approximation’, while $q = 5/3$ and $q=3/2$ to the Kolmogorov or Kreichnan turbulence, respectively. Within the anticipated quasilinear approximation for particle-wave interactions, such a turbulent spectrum gives the momentum and pitch angle diffusion rates $\propto p^{q-2}$, or the acceleration and escape timescales $t_{\rm acc} \propto p^{2 - q}$ and $t_{\rm
esc} \propto p^{q-2}$. In the analysis, we also include radiative energy losses, being an arbitrary function of the electrons’ energy. In most of the cases, however, or at least in some particular energy ranges, the appropriate timescale for the radiative cooling scales simply with some power of the particle momentum, $t_{\rm loss} \propto p^r$. For example, $r = -1$ corresponds to synchrotron or inverse-Compton/Thomson-regime energy losses, $r=0$ (roughly) to the bremsstrahlung emission, $r = +1$ (roughly) to the Coulomb interactions of ultrarelativistic electrons, while $r = 1/2$ may conveniently approximate inverse-Compton cooling in the Klein-Nishina regime on monoenergetic background soft photon field.
We find that when the particles are confined to the turbulent acceleration region ($t_{\rm esc}\rightarrow\infty$), the resulting steady-state particle spectra (for a finite momentum range of interacting electrons) are in general of the modified ultrarelativistic Maxwellian type, $n_{\rm e}(p) \propto p^2 \,
\exp\left[ - {1 \over a} \, \left(p / p_{\rm eq}\right)^a\right]$, where $a =
2-q-r \neq 0 $. Here $p_{\rm eq}$ is the momentum at which the acceleration and radiative loss timescales are equal, $t_{\rm
acc}(p_{\rm eq}) = t_{\rm loss}(p_{\rm eq})$. This form is independent of the initial energy distribution of the electrons as long as this distribution is not very broad and the bulk of initial particles have $p<p_{\rm eq}$. However, if high energy particles with $p>p_{\rm eq}$ are injected to the system, there will be significant deviations from this simple form. For example, for a $\delta$-function initial distribution the spectrum will have a power-law tail $\propto
p^{r-1}$ in addition to the modified Maxwellian bump. Also, if the ratio of acceleration and energy losses timescales is independent of the electron energy, in other words, if $2-q = r$, then the resulting particle spectra are of the form $n_{\rm e}(p) \propto p^{-\sigma'}$, where $\sigma' \equiv
(t_{\rm acc}/t_{\rm loss}) - 2$. Finally, if the particle escape from the acceleration site is finite but still inefficient, a power-law tail $\propto
p^{1-q}$ may be present in the momentum range $p_{\rm inj} < p \ll p_{\rm eq}$, again in addition to the modified Maxwellian component. When the radiative losses timescale is not a simple power-law function of the electron energy, the emerging spectra may be of a more complex (e.g., concave) form.
We also analyze in more details synchrotron and inverse-Compton emission spectra of the electrons characterized by the modified ultrarelativistic Maxwellian energy distribution. In order to summarize briefly our findings, let us define the critical synchrotron frequency of the electrons with the equilibrium Lorentz factor $\gamma_{\rm eq} \equiv p_{\rm eq}/m_{\rm e} c$, namely $\nu_{\rm syn}
\equiv (3 e B / 4 \pi m_{\rm e} c) \, \gamma_{\rm eq}^2$, and the critical dimensionless energy of the monochromatic ($h\nu_0 \equiv \epsilon_0 \, m_{\rm
e}c^2$) soft photon field inverse-Compton up-scattered (in the Thomson regime) by the $\gamma_{\rm eq}$ electrons, $\epsilon_{\rm ic/Th} = 4 \,
\epsilon_0 \, \gamma_{\rm eq}^2$. With these, one can note that the low-frequency synchrotron emissivity is of the form $j_{\rm \nu, \, syn}(\nu <
\nu_{\rm syn}) \propto \nu^{1/3}$, as expected in the case of a very flat (or inverted) electron energy distribution $n_{\rm e}(\gamma < \gamma_{\rm eq})
\propto \gamma^2$. Such flat electron spectra seem to be required to explain several emission properties of relativistic jets in active galactic nuclei [@tsa07a; @tsa07b]. At higher frequencies, we find a rough approximation $j_{\rm \nu, \, syn}(\nu>\nu_{\rm syn}) \propto \nu^{(6-a) / (4+2 a)} \,
\exp\left[ - {2 + a \over 2 a} \, \left(2 \nu / \nu_{\rm syn}\right)^{a /
(2+a)}\right]$. Thus, the high-energy synchrotron component drops much less rapidly than suggested by the emissivity of a single electron, and the emerging high-frequency tail of the synchrotron spectrum is of a smoothly curved shape. It is therefore very interesting to note that almost exactly this kind of curvature is observed at synchrotron X-ray frequencies in several BL Lac objects [@mas04; @mas06; @per05; @tra07a; @tra07b; @gie07], in particular those detected also at TeV photon energies.
As for the inverse-Compton emission of ultrarelativistic electrons characterized by the modified Maxwellian energy distribution, we find that in the Thomson regime it is of the form $j_{\rm \epsilon, \, ic/Th}(\epsilon < \epsilon_{\rm
ic/Th}) \propto \epsilon$, and $j_{\rm \epsilon, \, ic/Th}(\epsilon >
\epsilon_{\rm ic/Th}) \propto \epsilon^{(3-a) / 2} \, \exp\left[ - {1 \over a} \,
\left(\epsilon / \epsilon_{\rm ic/Th}\right)^{a / 2}\right]$. Both very flat low-energy part of this component and also its curved high-energy segment may contribute to the observed $\gamma$-ray emission of some TeV blazars [@kat06a; @gie07][^7]. We also note, that the curvature of the high frequency segments of the synchrotron and inverse-Compton spectra, event though being produced by the same energy electrons and in the Thomson regime, are different. Such a difference is even more pronounce when the Klein-Nishina effects play a role, since in such a case an exponential decrease of the high-energy photon spectra is the strongest, $j_{\rm \epsilon, \, ic/KN}(\epsilon > \gamma_{\rm
eq}) \propto \epsilon^{7/2} \, \exp\left[ - {1 \over a} \, \left(\epsilon /
\gamma_{\rm eq}\right)^{a}\right]$, imitating exponential cut-off in the energy distribution of radiating particles.
Ł.S. was supported by MEiN through the research project 1-P03D-003-29 in years 2005-2008. Ł.S. acknowledges M. Ostrowski, R. Schlickeiser, and S. Fuerst for helpful comments and discussion.
Abramowitz, M., & Stegun, I.A. 1964, [*‘Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables’*]{}; Dover, New York (1964) Achterberg, A. 1979, A&A, 76, 276 Achterberg, A. 1981, A&A, 97, 259 Becker, P.A., Le, T., & Dermer, C.D. 2006, ApJ, 647, 539 Beckert, T., & Duschl, W.J. 1997, A&A, 328, 95 Birk, G.T., Crusius-Wätzel, A.R., & Lesch, H. 2001, ApJ, 559, 96 Blandford, R.D., & Eichler, D. 1987, Phys. Rep., 154, 1 Blumenthal, G.R., & Gould, R.J. 1970, Rev.Mod.Phys., 42, 237 Bogdan, T.G., & Schlickeiser, R. 1985, A&A, 143, 23 Borovsky, J.E., & Eilek, J.A. 1986, ApJ, 308, 929 Brunetti, G., & Lazarian, A. 2007, MNRAS, 378, 245 Crusius, A., & Schlickeiser, R. 1986, A&A, 164, L16 Davis, L. 1956, Phys. Rev., 101, 351 Dermer, C.D., Sturner, S.J., & Schlickeiser, R. 1997, ApJS, 109, 103 Dröge, W., & Schlickeiser, R. 1986, ApJ, 305, 909 Fermi, E. 1949, Phys. Rev., 75, 1169 Gallegos-Cruz, A., & Perez-Peraza, J. 1995, ApJ, 446, 400 Giebels, B., Dubus, G., & Khélifi, B. 2007, A&A, 462, 29 Hall, D.E., & Sturrock, P.A. 1967, Phys. Fluids, 10, 2620 Hardcastle, M.J., et al. 2007, ApJL, 670, L81 Jester, S., Röser, H.-J., Meisenheimer, K., Perley, R., & Conway, R. 2001, A&A, 373, 447 Jones, F.C. 1970, Phys. Rev. D, 2, 2787 Kardashev, N.S. 1962, Sov. Astron.-AJ, 6, 317 Kataoka, J., Stawarz, Ł., Aharonian, F., Takahara, F., Ostrowski, M., & Edwards, P.G. 2006, ApJ, 641, 158 Katarzyński, K., Ghisellini, G., Tavecchio, F., Gracia, J., & Maraschi, L. 2006a, MNRAS, 368, L52 Katarzyński, K., Ghisellini, G., Mastichiadis, A., Tavecchio, F., & Maraschi, L. 2006b, A&A, 453, 47 Kulsrud, R.M., & Pearce, W.P. 1969, ApJ, 156, 445 Kulsrud, R.M., & Ferrari, A. 1971, Ap&SS, 12, 302 Kusunose, M., & Takahara, F. 2005, ApJ, 621, 285 Lacombe, C. 1979, A&A, 71, 169 Lemoine, M., Pelletier, G., & Revenu, B. 2006, ApJ, 645, L129 Liu, S., Petrosian, V., & Melia, F. 2004, ApJ, 611, L101 Liu, S., Petrosian, V., Melia, F., & Fruer, C.L. 2006, ApJ, 648, L1020 Manolakou, K., Horns, D., & Kirk, J.G., 2007, A&A, 474, 689 Massaro, E., Perri, M., Giommi, P., & Nesci, R. 2004, A&A, 413, 489 Massaro, E., Tramacere, A., Perri, M., Giommi, P., & Tosti, G. 2006, A&A, 448, 861 Melrose, D.B. 1968, Ap&SS, 2, 171 Melrose, D.B. 1969, Ap&SS, 5, 131 Melrose, D.B. 1980, [*‘Plasma Astrophysics’*]{}, New York: Gordon & Breach Moderski, R., Sikora, M., Coppi, P.S., & Aharonian, F.A. 2005, MNRAS, 363, 954 Niemiec, J., & Ostrowski, M. 2006, ApJ, 641, 984 Niemiec, J., Ostrowski, M., & Pohl, M. 2006, ApJ, 650, 1020 Ostorero, L., et al. 2006, A&A, 451, 797 Park, B.T., & Petrosian, V. 1995, ApJ, 446, 699 Petrosian, V. 1973, ApJ, 186, 291 Petrosian, V. 1981, ApJ, 251, 727 Petrosian, V. 2001, ApJ, 557, 560 Petrosian, V., & Donaghy, T.Q. 1999, ApJ, 527, 945 Petrosian, V., & Liu, S. 2004, ApJ, 610, 550 Perlman, E.S., Madejski, G., Georganopoulos, M., Andersson, K., Daugherty, T., Krolik, J.H., Rector, T., Stocke, J.T., Koratkar, A., Wagner, S., Aller, M., Aller, H., & Allen, M.G. 2005, ApJ, 625, 727 Schlickeiser, R. 1984, A&A, 136, 227 Schlickeiser, R. 1985, A&A, 143, 431 Schlickeiser, R. 1989, ApJ, 336, 243 Schlickeiser, R. 2002, [*‘Cosmic Ray Astrophysics’*]{}, Springer-Verlag, Berlin-Heidelberg Schlickeiser, R., & Miller, J.A. 1998, ApJ, 492, 352 Schlickeiser, R., Sievers, A., & Thiemann, H. 1987, A&A, 182, 21 Stawarz, Ł., & Ostrowski, M. 2002, ApJ, 578, 763 Stawarz, Ł., Sikora, M., Ostrowski, M., & Begelman, M.C. 2004, ApJ, 608, 95 Steinacker, J., Schlickeiser, R., & Dröge, W. 1988, Solar Phys., 115, 313 Stern, B.E., & Poutanen, J. 2004, MNRAS, 352, L25 Tademaru, E., Newman, C.E., & Jones, F.C. 1971, Ap&SS, 10, 453 Tramacere, A., Giommi, P., Massaro, E., Perri, M., Nesci, R., Colafrancesco, S., Tagliaferri, G., Chincarini, G., Falcone, A., Burrows, D.N., Roming, P., McMath C.M., & Gehrels, N. 2007a, A&A, 467, 501 Tramacere, A., Massaro, F., & Cavaliere, A. 2007b, A&A, 466, 521 Tsang, O., & Kirk, J.G. 2007a, A&A, 463, 145 Tsang, O., & Kirk, J.G. 2007b, A&A, 476, 1151 Tsytovich, V.N. 1977, [*‘Theory of Turbulent Plasma’*]{}, Plenum Press, New York Zirakashvili, V.N., & Aharonian, F. 2007, A&A, 465, 695
[^1]: Due to the expected high conductivity of the plasma one can neglect the large-scale scale electric field, $\vec{E}_0 = 0$ .
[^2]: The Fokker-Planck equation can be also derived straight from the definition of the function $f(\vec{x}, \vec{p}, t)$, assuming that the interaction of the particles with turbulent waves is a Markov process in which every interaction (collision) changes the particle energy only by a small amount, and that the recoil of the turbulent modes during the collision can be neglected [@bla87].
[^3]: The effects of regular energy gains were omitted here for clarity. @jon70 [@sch84] and @par95 included in their investigations regular energy gains representing very idealized shock acceleration process. Within the anticipated ‘hard-sphere’ approximation, these gains were assumed to be characterized by the appropriate timescale independent on the particle energy, $\langle \dot{p}\rangle_{\rm gain} \propto p$ .
[^4]: The other (non-synchrotron) radiative losses terms included in the analysis presented by @bog85 were omitted here for clarity.
[^5]: In the case of the magnetosonic-type turbulence, interacting with particles via transit-time damping satisfying the Cherenkov condition $k\,r_{\rm g} \ll 1$, the low energy cut-off in the momentum diffusion coefficient could be chosen to be the energy of the particle whose velocity is comparable to the velocity of the fast magnetosonic mode, which is $\sim v_A$ for low-$\beta$, or magnetically dominated plasmas.
[^6]: As shown by @pet81, the following spectral form is also true for synchrotron emission by semirelativistic electrons.
[^7]: The caution here is that the computed in this paper high-energy spectra correspond to the situation of inverse-Comptonization of the monoenergetic seed photon field, which, in addition, is isotropically distributed in the emitting region rest frame. In the case of relativistic blazar jets, the external radiation (due to accretion disk, as well as circumnuclear gas and dust) is distributed anisotropically in the jet rest frame, while the isotropic synchrotron emission produced by the jet electrons is not strictly monochromatic [see, e.g., @der97 and references therein]. One the other hand, synchrotron radiation of ultrarelativistic electrons characterized by the Maxwellian-type energy distribution, as analyzed here, is not that far from the monoenergetic approximation, and the relativistic corrections regarding the anisotropic distribution of the soft photons in the emitting region rest frame are not supposed to influence substantially the spectral shape of the inverse-Compton emission. For these reasons, we believe that the main spectral features of the high-energy emission components computed in this paper are representative for the $\gamma$-ray emission of, e.g., TeV blazars.
|
---
abstract: 'Codes defined on graphs and their properties have been subjects of intense recent research. On the practical side, constructions for capacity-approaching codes are graphical. On the theoretical side, codes on graphs provide several intriguing problems in the intersection of coding theory and graph theory. In this paper, we study codes defined by planar Tanner graphs. We derive an upper bound on minimum distance $d$ of such codes as a function of the code rate $R$ for $R \ge 5/8$. The bound is given by $$d\le \left\lceil \frac{7-8R}{2(2R-1)} \right\rceil + 3\le 7.$$ Among the interesting conclusions of this result are the following: (1) planar graphs do not support asymptotically good codes, and (2) finite-length, high-rate codes on graphs with high minimum distance will necessarily be non-planar.'
author:
-
bibliography:
- 'docdb.bib'
title: Codes on Planar Graphs
---
Introduction
============
The spectacular success of codes on graphs has resulted in immense recent research activity on the practical and theoretical aspects of graphical codes. On the practical side, the powerful notion of representing parity constraints on Tanner graphs [@Gallager:1963xy][@Tanner:1981gd] has resulted in tremendous simplifications in the construction and implementation of capacity-approaching codes for various channels. On the theoretical side, the interplay of graph theory and coding theory has resulted in many intriguing problems.
In this paper, we are concerned with codes that are defined by planar Tanner graphs. Specifically, we study the minimum distance of codes that have a planar Tanner graph. Planarity of a graph, a classic notion in graph theory, allows for the embedding or rendering of a graph as a picture on a two-dimensional plane with no two edges intersecting. Specific examples of such graphs are trees and graphs with non-overlapping cycles. Interestingly, both these types of graphs have been shown to correspond to codes with poor minimum distance properties [@Etzion:1999jx][@Srimathy:2008qy]. In this paper, we show similar properties for high-rate codes that have planar Tanner graphs.
Specifically, the main result of this paper is that a code of rate $R\ge5/8$ with a planar Tanner graph has minimum distance bounded as $$d\le \left\lceil \frac{7-8R}{2(2R-1)} \right\rceil + 3\le 7,$$ where $\lceil x\rceil$ (for a real number $x$) is the smallest integer greater than or equal to $x$. Note that the result holds for any blocklength. Hence, non-planarity is vital for large minimum distance at high rates. This result provides justification for many known results on codes on highly non-planar graphs with large minimum distance [@Tanner:2001di], and suggests methods for other possible constructions.
The method of proof is novel and involves several steps. A given planar Tanner graph is modified through a series of construction steps to a planar Tanner graph with maximum bit node degree 3. For the modified graph, the existence of low-weight codewords is shown by an averaging argument. The existence is then extended to the original Tanner graph.
The rest of the paper is organized as follows. The construction of the modified Tanner graph is presented in Section \[sec:constructions\]. A simple version of the main result is proved in Section \[sec:distance-rate-bounds\] for the sake of clarity in exposition. A complete proof of the bound on minimum distance for codes on planar graphs is given in Section \[sec:proof-main-result\]. Finally, concluding remarks are made in Section \[sec:conclusion\].
Constructions {#sec:constructions}
=============
Consider a Tanner graph $G$ defining a linear code. The vertex set of $G$ is denoted $V(G)=V^b\cup V^c$, where $V^b$ and $V^c$ denote the set of bit and check nodes of the $G$. We will assume that the rate of the code defined by $G$ is $R=1-|V^c|/|V^b|$. Let $V^b_i$ denotes the set of degree-$i$ bit nodes. The edge set of $G$ is denoted $E(G)\subseteq V^b\times V^c$, and an edge of $G$ connecting bit node $v_b$ to check node $v_c$ is denoted $(v_b,v_c)$.
For a set of bit nodes $V^b_*\subseteq V^b$, $\mathcal{N}(V^b_*)$ denotes the set of check nodes connected to $V^b_*$ i.e. $\mathcal{N}(V^b_*)=\cup_{v_b\in V^b_*}\{v_c:(v_b,v_c)\in E(G)\}$. The degree of a bit node $v_b$ is $|\mathcal{N}(v_b)|$. For $V^c_*\subseteq V^c$, the set of induced bit nodes $\mathcal{I}(V^c_*)$ denotes the bit nodes whose neighbors are subsets of $V^c_*$ i.e. $\mathcal{I}(V^c_*)=\{v_b:\mathcal{N}(v_b)\subseteq V^c_*\}$. The following proposition (stated without proof) connects subsets of check nodes and their induced bit nodes to the minimum distance of the code.
\[prop:dmin\] Consider $V^c_*\subseteq V^c$ in a Tanner graph $G$ defining a code with minimum distance $d$. If $\mathcal{I}(V^c_*)> V^c_*$, then $d\le|V^{c}_*| +1$.
Following Proposition \[prop:dmin\], a subset of check nodes $V^c_*$ is said to be [*codeword-supporting*]{} whenever $\mathcal{I}(V^c_*)> V^c_*$.
In this paper, we provide bounds for the minimum distance of Tanner graphs that are planar i.e. Tanner graphs that can be embedded in a plane with no two edges intersecting [@Bondy:1976hl]. All Tanner graphs in the rest of the paper will be planar with a fixed embedding. For planar Tanner graphs, we use Proposition \[prop:dmin\] for bounding minimum distance by showing the existence of suitable codeword-supporting subsets of check nodes. For this purpose, we define a new planar graph involving the check nodes of the given planar Tanner graph.
Check graph of a planar Tanner graph
------------------------------------
Given a planar Tanner graph $G$, the *check graph* of $G$, denoted $\mathcal{C}(G)$, is a planar graph with vertex set $V^c$ (the set of check nodes of $G$). We use an embedding of $G$ in a plane, and place the nodes of $\mathcal{C}(G)$ in an isomorphic plane at the same locations as the check nodes of $G$ in the original plane. To aid in the construction, we identify the locations of the bit nodes of $G$ in the plane of $\mathcal{C}(G)$. The edges of $\mathcal{C}(G)$ are constructed as follows:
1. Consider a bit node $v_b\in V^b$ with degree $\lambda>1$ and $\mathcal{N}(v_b)=\{v_{c,0},v_{c,1},\cdots,v_{c,\lambda-1}\}$ labelled in a clockwise sequence in the planar embedding i.e. no edge out of $v_b$ lies in $v_{c,i}v_bv_{c,(i+1)_\lambda}$ ($(x)_{\lambda}$ denotes $x\mod \lambda$). Add edges $(v_{c,i},v_{c,(i+1)_\lambda})$ for $0\leq i\leq \lambda-1$ to form a simple cycle enclosing the location of $v_b$ in the plane of $\mathcal{C}(G)$.
2. In Step 1, edges causing a face enclosed by two edges should not be added.
3. Add more edges to make the graph maximal planar (a planar graph is maximal if one more edge will make the graph non-planar [@Bondy:1976hl]).
The construction of the check graph is illustrated in Fig. \[fig:cg\].
In Fig. \[fig:cg\], the nodes of the check graph $\mathcal{C}(G)$ are the check nodes of $G$ denoted $\{1,2,3,4\}$. In Step 1, for bit node ‘a’ of $G$, we connect the nodes $\{1,2,4,3\}$ of $\mathcal{C}(G)$ in a cycle. Step 1, for the nodes ‘b’, ‘c’ and ‘d’ of $G$ with degree larger than 1, results in faces with two edges. Hence, no other edges are added to $\mathcal{C}(G)$ as per Step 2. In Step 3, the edges (2,3) and (1,4) are added to make the check graph maximal planar. Note that there are four faces in $\mathcal{C}(G)$, labelled $f_1$, $f_2$, $f_3$ and $f_4$ in Fig. \[fig:cg\]. The face $f_4$ is the exterior or external face.
In general, maximal planarization in Step 3 is not unique. Hence, there can be many check graphs corresponding to a single Tanner graph. We fix one such check graph and call it the check graph of $G$.
The following correspondences between a planar Tanner graph $G$ and its check graph $\mathcal{C}(G)$ are vital for the minimum distance bounds.
- In Step 1, a degree-3 bit node $v_b$ of $G$ maps to a triangular face in $\mathcal{C}(G)$ connecting the three check nodes in $\mathcal{N}(v_b)$. Hence, we say that a degree-3 bit node is “identified" with a face in $\mathcal{C}(G)$. In some cases, this can be the external face.
- In Step 1, a degree-2 bit node $v_b$ of $G$ maps to an edge in $\mathcal{C}(G)$ connecting the two check nodes in $\mathcal{N}(v_b)$. We say that a degree-2 bit node is “identified" with an edge in $\mathcal{C}(G)$.
- A degree-1 bit node $v_b$ of $G$ does not result in any edges, but $v_b$ is represented by the one check node $\mathcal{N}(v_b)$ in $\mathcal{C}(G)$. We say that a degree-1 bit node $v_b$ is “identified" with the check node $\mathcal{N}(v_b)$.
- In Step 1, a bit node of degree $\lambda>3$ results in a face enclosed with $\lambda$ edges. In Step 3, maximum planarization converts such a face into $\lambda-2$ triangular faces.
In the example of Fig. \[fig:cg\], the degree-4 bit node of $G$ results in two triangular faces $f_1$ and $f_2$ in $\mathcal{C}(G)$. The check node 3 corresponds to the degree-1 bit node, while the edges $\{(1,2), (2,4), (3,4)\}$ correspond to the degree-2 bit nodes.
Two more examples to illustrate the construction of the check graph are shown in Fig. \[fig:ex2\].
For the graph $G$ in Fig. \[fig:ex2\], we get a triangle around the location of bit node ‘a’ in Step 1. For bit node ‘b’, the dotted lines show the possible edges in Step 1. However, no new edges are added as they result in faces enclosed by two edges. Notice that the circular dotted line would have resulted in the external face being “enclosed” by two edges. In $G$, the internal triangular face $f_1$ corresponds to the degree-3 bit node ‘a’, while the external triangular face $f_2$ corresponds to ‘b’.
For the graph $H$ in Fig. \[fig:ex2\], two triangles are added around the bit nodes ‘a’ and ‘c’ in Step 1. Note that the edge from check node 3 to check node 2 for bit node ‘b’ needs to be drawn in a circular fashion to enclose the location corresponding to ‘b’.
Since $\mathcal{C}(G)$ is maximal planar, by standard results in graph theory [@Bondy:1976hl], we know that there are $2|V^c|-4$ faces in $\mathcal{C}(G)$ and all faces are triangular (enclosed by three edges). Also, since faces in $\mathcal{C}(G)$ result from bit nodes of degree at least 3, we see that the maximum number of bit nodes of degree 3 in a planar Tanner graph is limited to $2|V^c|-4$. The faces of $\mathcal{C}(G)$ are denoted $F(\mathcal{C}(G))$. A face $f\in F(\mathcal{C}(G))$ is enclosed by three edges connecting three check nodes of $G$. The three check nodes of $G$ that form $f$ are denoted $V^c(f)$. In the example of Fig. \[fig:cg\], we have $V^c(f_1)=\{1,2,3\}$, $V^c(f_2)=\{2,3,4\}$, $V^c(f_3)=\{1,3,4\}$, and $V^c(f_4)=\{1,2,4\}$.
Prelude {#sec:prelude}
-------
To illustrate the usefulness of the check graph, we now present a bound on minimum distance of rate $\ge7/8$ codes with a planar Tanner graph whose maximum bit node degree is 3.
Let $G$ be a planar Tanner graph with $n$ bit nodes and $m$ check nodes defining a code with rate $R=1-m/n\ge7/8$ and minimum distance $d$. Let the maximum degree of a bit node in $G$ be 3. Then, $d\le3$. \[prop:prelude\]
Let $f_i$, $1\le i\le2m-4$, be the faces of the check graph $\mathcal{C}(G)$. Let $w_i=|\mathcal{I}(V^c(f_i))|$ be the number of bit nodes induced by the set of check nodes $V^c(f_i)$ forming the face $f_i$. Let $f_k$ be the face such that $w_k\ge w_i$, $1\le i\le2m-4$.
Since each bit node of $G$ has degree at most 3, it is induced at least once by some face $f_i$ in $\mathcal{C}(G)$. So, we have $$w_1 + w_2 + \cdots+ w_{2m-4} \ge n.$$ Since $w_k\ge w_i$ and $n\ge 8m$, we simplify as follows. $$\begin{aligned}
w_k(2m-4) & \ge n,\\
w_k \ge \frac{n}{2m-4} & > \frac{n}{2m} \ge 4.\end{aligned}$$ Hence, $w_k = |\mathcal{I}(V^c(f_k))| \ge 5$ and $|V^c(f_k)| = 3$. Thus, there exists a $(\ge 5, 3)$ subcode of the original code. This implies that $d\le3$.
The above proof uses the faces of $\mathcal{C}(G)$ to construct codeword-supporting set of check nodes in the original Tanner graph $G$. An averaging argument is used to show the existence of the codeword-supporting set. These themes will be used and extended in the remainder of this paper to prove more bounds on the minimum distance of codes with a planar Tanner graph.
Another crucial assumption in Proposition \[prop:prelude\] is on the maximum bit node degree. This assumption will be relaxed through another construction called the [*check inverse*]{}.
Check inverse of a planar Tanner graph
--------------------------------------
The same check graph can result from several planar Tanner graphs. Fig. \[fig:mcg\] illustrates one such example where two Tanner graphs $G$ and $H$ result in the same check graph $\mathcal{C}(G)$.
Given a check graph $\mathcal{C}(G)$ of a planar Tanner graph $G$, we can construct a special planar Tanner graph $G'$ with maximum bit node degree 3 such that $\mathcal{C}(G')=\mathcal{C}(G)$. The construction of this special Tanner graph, which we call the [*check inverse*]{} of $G$, is described next.
Given a planar Tanner graph $G$, the check inverse of $G$, denoted $G'$, is a planar Tanner graph with check node set $V'^{c} = V^{c}$ and bit node set $V'^{b}=V'^{b}_3\cup V'^{b}_2\cup V'^{b}_1$ constructed as follows:
1. $V'^{b}_3\cong F(\mathcal{C}(G))$. A bit node $v_b\in V'^{b}_3$ corresponds to a face $f\in F(\mathcal{C}(G))$, and is connected to the three check nodes in $V^c(f)$ that form the face $f$ in $\mathcal{C}(G)$. Bit nodes in $V'^{b}_3$ have degree 3, and $|V'^{b}_3|=2|V^c|-4$.
2. The set $V_2^{'b} \bigcup V_1^{'b}$ is an arbitrary subset of $V_2^b \bigcup V_1^b$ (the set of degree-2 and degree-1 bit nodes of $G$) of size $$|V'^{b}_2 \bigcup V_1^{'b}|=[|V^b|-(2|V^c|-4)]^+$$ where $[x]^+ = \max(x,0)$. A bit node $v_b \in V_2^{'b} \bigcup V_1^{'b} \subseteq V^b_2 \bigcup V_1^b$ is connected to the check node(s) in $\mathcal{N}(v_b)$.
\[ex:GtoG’\] The construction of the check inverse for a planar Tanner graph $G$ is illustrated in Fig. \[fig:chkinv\].
The Tanner graph $G$ has 9 bit nodes (1 of degree 4, 4 of degree 2, 4 of degree 1) and 4 check nodes. The check graph $\mathcal{C}(G)$ has 4 nodes, corresponding to the check nodes of $G$, and 4 faces (3 interior and 1 exterior) enclosed by three edges each. The check inverse $G'$ has 4 check nodes corresponding to the check nodes of $G$ or the nodes of $\mathcal{C}(G)$. The 4 bit nodes of degree 3 in $G'$ correspond to the 4 faces of $\mathcal{C}(G)$ from Step 1 of the construction of $G'$. These nodes are labelled with the labels of the faces in $\mathcal{C}(G)$. In Step 2, 5 bit nodes of degree-1 and 2 (the nodes ‘b’, ‘c’, ‘d’, ‘f’, ‘i’) are added to $G'$ with connections according to the corresponding connections in $G$. In Fig. \[fig:chkinv\], the dotted lines in $G'$ are the edges of $\mathcal{C}(G)=\mathcal{C}(G')$.
The following properties of the check inverse of a planar Tanner graph are important for future constructions:
- If $|V^b|>2|V^c|-4$, the check inverse $G'$ has the same number of check nodes and bit nodes as $G$. If the rate of the code defined by $G$ is greater than 1/2, we have $|V^b|>2|V^c|-4$.
- For rate greater than 1/2, we have $$\label{eq:deg12}
|V'^b_1|+|V'^b_2|=|V^b|-(2|V^c|-4).$$ From now on, we will restrict ourselves to planar Tanner graphs $G$ that define codes of rate greater than 1/2 so that (\[eq:deg12\]) always holds.
- The check graph of $G'$ is same as $\mathcal{C}(G)$ i.e, $\mathcal{C}(G')=\mathcal{C}(G)$. But each face in $\mathcal{C}(G')$ corresponds to a degree 3 bit node in $G'$ unlike in $\mathcal{C}(G)$ and $G$.
The check inverse plays a crucial role in the minimum distance bounds. Distance bounds will first be shown for the code represented by the Tanner graph $G'$. Then, the same bound will be seen to hold for the original graph $G$.
Dual of check graph and codeword-supporting subgraphs {#sec:dual}
-----------------------------------------------------
Since the check graph $\mathcal{C}(G)$ is planar, we can define its dual as defined for any planar graph [@Bondy:1976hl]. Since we are working on a particular embedding, the dual graph is unique.
The dual of $\mathcal{C}(G)$, denoted as $\mathcal{C}^\dagger(G)$, is a planar graph with vertex set $V^\dagger(G)$ that has a one-to-one correspondence with $F(\mathcal{C}(G))$. Two vertices of $\mathcal{C}^\dagger(G)$ are joined by an edge whenever the corresponding faces of $\mathcal{C}(G)$ share an edge. Since $\mathcal{C}(G)$ is maximal planar with $|V^c|$ vertices and $2|V^c|-4$ triangular faces, $\mathcal{C}^\dagger(G)$ has $2|V^c|-4$ vertices each of degree 3. Since there is a one-to-one correspondence between edges in a planar graph and its dual, let us denote the edge corresponding to $e$ in $\mathcal{C}(G)$ as $e^\dagger$ in $\mathcal{C}^\dagger(G)$.
If there are multiple edges between any two nodes of $\mathcal{C}^\dagger(G)$, we can show that the minimum distance of the code represented by $G$ is at most 2 (See Section \[sec:girth-condition\] for a proof). So, we consider planar Tanner graphs $G$ and check graphs $\mathcal{C}(G)$ such that there are no multiple edges in $\mathcal{C}^\dagger(G)$.
Subgraphs of the dual of the check graph play an important role in determining the existence of low-weight codewords in the code (or small codeword-supporting subsets of check nodes in the code’s Tanner graph). The basic idea is the following. Using a vertex-induced subgraph of the dual of the check graph $\mathcal{C}^\dagger(G)$, we define subsets of check nodes of $G$ and study when they are codeword-supporting. Let $U\subseteq V^\dagger(G)\cong F(\mathcal{C}(G))$ induce a subgraph $\mathcal{C}_U^\dagger(G)$ of $\mathcal{C}^\dagger(G)$. Because of the congruence, we will denote a vertex of $\mathcal{C}^\dagger(G)$ as a face $f\in F(\mathcal{C}(G))$. Hence, $U\subseteq F(\mathcal{C}(G))$. For each subset $U$, we associate a subset of check nodes $V^c_U$ of $G$ as given below: $$\label{eq:vcu}
V^c_U=\cup_{f\in U}V^c(f),$$ where $V^c(f)$ (as before) is the set of three nodes that form the face $f$ in $\mathcal{C}(G)$. Hence, every vertex subset $U$ in $\mathcal{C}^\dagger(G)$ corresponds to a subset of check nodes $V^c_U$ in $G$. We will prove existence of codeword-supporting subsets among the sets $V^c_U$ produced by different $U$.
The subgraph of $\mathcal{C}(G)$ induced by $V^c_U$ is denoted $\mathcal{C}_U(G)$. In spite of the notation, note that $\mathcal{C}_U^\dagger(G)$ is not necessarily the planar dual of $\mathcal{C}_U(G)$.
As before, the set of bit nodes induced by $V^c_U$ in the Tanner graph $G$ is denoted $\mathcal{I}(V^c_U)$. The [*weight*]{} of an induced subgraph $\mathcal{C}_U^\dagger(G)$, denoted $wt(\mathcal{C}_U^\dagger(G))$ or $wt(U)$, is defined as $wt(U) = |\mathcal{I}(V^c_U)|$, the number of induced bit nodes of $V^c_U$. An induced subgraph $\mathcal{C}_U^\dagger(G)$ of $\mathcal{C}^\dagger(G)$ is said to be [*codeword-supporting*]{} if $wt(U) > |V^c_U|$.
Consider a planar Tanner graph $G$ with $V^b = \{\text{a,b,c,d,e,f,g}\}$ and $V^c = \{1,2,3,4,5\}$ as shown in Fig. \[fig:induced\_bit\].
The check graph $\mathcal{C}(G)$ has 5 vertices corresponding to 5 check nodes of $G$ and 6 faces (including the external face). The 6 vertices in dual $\mathcal{C}^\dagger(G)$ are labelled according to the corresponding faces in $\mathcal{C}(G)$. Let $U=\{f_1,f_4\} \subset V^\dagger(G)$. Then, $V^c_U = \{1,2,3,4\}$ and $\mathcal{C}_U(G) = f_1 \bigcup f_3 \bigcup f_4$. The bit nodes induced by $V^c_U$, $\mathcal{I}(V^c_U) = \{\text{a,b,c,d}\}$ and hence $wt(U) = 4$.
In Fig. \[fig:induced\_bit\], the degree-3 bit nodes of $G$ $\{\text{c,d,e}\}$ are identified with the respective faces $\{f_1,f_4,f_5\}$ in the check graph $\mathcal{C}(G)$ and the corresponding vertices in the dual of the check graph $\mathcal{C}^\dagger(G)$. The degree-2 bit nodes of $G$ $\{\text{b,f,g}\}$ are identified with the respective edges $\{(1,4), (3,5), (1,5)\}$ in $\mathcal{C}(G)$ and the corresponding edges $\{(f_3,f_6), (f_2,f_5), (f_2,f_6)\}$ in $\mathcal{C}^\dagger(G)$. The degree-1 bit node $\{\text{a}\}$ of $G$ is identified with node 1 of $\mathcal{C}(G)$ and the external face of $\mathcal{C}^\dagger(G)$. \[ex:dual-check-graph\]
In the main result of this paper, we establish the existence of codeword-supporting subgraphs induced by a small subset $U$ in the dual of the check graph of a planar Tanner graph. The next two propositions show that the size of $V^c_U$ and the minimum distance of the code are bounded by the size of $U$ that induces a codeword-supporting subgraph in $\mathcal{C}^\dagger(G)$. Hence, small codeword-supporting subgraphs in the dual of the check graph result in low-weight codewords in the code.
Let $\mathcal{C}_U^\dagger(G)$ be a connected induced subgraph of $\mathcal{C}^\dagger(G)$ induced by a proper subset $U\subset F(\mathcal{C}(G))$. Then, for the subset of vertices $U$, $$|V^c_U| \le |U| + 2 - c(U),$$ where $c(U)$ is the number of simple cycles in $\mathcal{C}_U^\dagger(G)$. \[prop:vcu\]
As shown in Appendix \[sec:numbering\], the subgraph $\mathcal{C}_U^\dagger(G)$ can be constructed by adding nodes one at a time from the set $U$ in a suitable order. The order is such that, at each step of the construction, the most recently added node has degree 1 or 2 after inclusion. Hence, the resulting subgraph after each step is connected. With this particular ordering of nodes of $U$, we will prove the proposition by induction.
Let $|U| = 1$. Then, $|V^c_U| = 3 = |U| + 2 - 0$. Hence, the proposition is true for $|U| =1$. Assume that it is true for the subgraph induced by $P$ nodes where $P \subset U$ i.e. $$|V^c_P| \le |P| +2 - c(P),$$ where $c(P)$ is the number of simple cycles in $\mathcal{C}_P^\dagger(G)$. We will prove that the result holds when a new node $w$ is added. Consider the subgraph $\mathcal{C}_W^\dagger(G)$ induced by $W = P \bigcup w$. By the ordering of the nodes, $w$ has degree 1 or 2 in $\mathcal{C}_W^\dagger(G)$.
*Case* 1: $w$ has degree 1.
Since $w$ is connected to $\mathcal{C}_P^\dagger(G)$ by exactly one edge, the face in $\mathcal{C}_W(G)$ corresponding to $w$ results in the addition at most one new node to $\mathcal{C}_P(G)$. Also, the addition of $w$ does not create a new cycle in $\mathcal{C}_W^\dagger(G)$. Hence, $|V^c_W| \le |V^c_P| + 1 \le |P| +2 - c(P) + 1 = |W| +2 - c(W)$, where $c(W)$ is the number of simple cycles in $\mathcal{C}_W^\dagger(G)$ which is equal to $c(P)$.
*Case* 2: $w$ has degree 2
Since $w$ is connected to $\mathcal{C}_P^\dagger(G)$ by two edges, the nodes at the boundary of the face corresponding to $w$ in $\mathcal{C}_W(G)$ are already present in $\mathcal{C}_P(G)$. Thus number of nodes in $\mathcal{C}_P(G)$ is same as that in $\mathcal{C}_W(G)$. Also, the addition of $w$ increases the number of cycles in the resulting subgraph by 1. This is because $w$ connects two nodes in $P$ that are already connected in $\mathcal{C}_P^\dagger(G)$. Hence, $|V^c_W| = |V^c_P| \le |P| +2 - c(V) + 1 -1 = |W| +2 - c(W)$ where $c(W)$ is the number of simple cycles in $\mathcal{C}_W^\dagger(G)$.
The two cases are illustrated in Fig. \[fig:check\_count\].
The dotted lines in Fig. \[fig:check\_count\] are the edges induced in $\mathcal{C}^\dagger(G)$ by the addition of the node $w$.
Thus by induction, $$|V^c_U| \le |U| + 2 - c(U).$$
Let $G$ be a planar Tanner graph of a code with minimum distance $d$. If there is a codeword-supporting connected subgraph of $\mathcal{C}^\dagger(G)$ induced by a proper subset $U$, then $$d\le |U|+3.$$ \[prop:cu\]
Since $\mathcal{C}_U^\dagger(G)$ is codeword-supporting, $wt(U) = |\mathcal{I}(V^c_U)|> |V^c_U|$. By Proposition \[prop:dmin\], $d \le |V^c_U| +1$. Since $\mathcal{C}_U^\dagger(G)$ is connected, by Proposition \[prop:vcu\], $|V^c_U| = |U| + 2 -c\le |U| + 2$. Therefore, $d \le |U|+3$.
Therefore, to bound minimum distance, we search for subgraphs of $\mathcal{C}^\dagger(G)$ on minimum number of vertices that are codeword-supporting.
Distance-Rate Bounds {#sec:distance-rate-bounds}
====================
The main result of this paper is the following theorem.
Let $G$ be a planar Tanner graph representing a code with rate $R\ge5/8$ and minimum distance $d$. Then, $$d\le \left\lceil \frac{7-8R}{2(2R-1)} \right\rceil + 3.$$ \[thm:main\]
The proof involves multiple steps. In the first step, we construct the check inverse $G'$, check graph $\mathcal{C}(G')=\mathcal{C}(G)$ and its dual $\mathcal{C}^\dagger(G')$ as discussed in Section \[sec:constructions\]. In the second step, the distance bound of Theorem \[thm:main\] is shown for the code corresponding to the check inverse $G'$ by proving the existence of a suitable codeword-supporting subgraph in $\mathcal{C}^\dagger(G')$. In the third and final step, the same bound is shown to hold for $G$ by a series of graph manipulations.
For clarity of explanation, we first show the second and third steps in the proof for a weaker version of Theorem \[thm:main\]. For the weaker version, the codeword-supporting subgraph of $\mathcal{C}^\dagger(G')$ is simply an edge. However, the important ideas in the general proof are present in the weaker version as well. A general proof of Theorem \[thm:main\] is presented later in Section \[sec:proof-main-result\].
Illustrative proof
------------------
A weaker version of Theorem \[thm:main\] is the following.
Let $G$ be a planar Tanner graph representing a code with rate $R\ge11/16$ and minimum distance $d$. Then, $d\le5$. \[thm:weak\]
We first prove a few lemmas that are used in the final proof. Using the constructions in Section \[sec:constructions\], let $G'$ be the check inverse of $G$. Let $\mathcal{C}(G')$ be the check graph of $G'$ and let its dual be $\mathcal{C}^\dagger(G')$.
### Codeword-supporting subgraph for $G'$
Let $G'$ be a check inverse of $G$ supporting a code of rate $R \ge 11/16$. Then, there is an edge in $\mathcal{C}^\dagger(G')$ that is codeword-supporting. \[lemma:1\]
Consider an edge $e^\dagger=(f_1,f_2)\in E^\dagger(G')$, where $E^\dagger(G')$ denotes the edge set of $\mathcal{C}^\dagger(G')$. Since $\mathcal{C}(G')$ is maximal planar on $|V^c|$ vertices, we have $|E^\dagger(G')|=3|V^c|-6$. We set $U=\{f_1,f_2\}$ and get $\mathcal{C}_U^\dagger(G')=e^\dagger$ as shown in Fig \[fig:edge\_cs\]. For simplicity, the set $U$ is replaced with $e^\dagger$ in the notation. For instance, $V^c_U$ will be denoted $V^c_{e^\dagger}$ and so on.
Consider the following summation: $$\label{eq:y_p2}
\mathcal{Y}(G')= \sum_{e^\dagger \in E^\dagger(G')} \left(wt(e^\dagger) - |V^c_{e^\dagger}|\right).$$ We will show that $\mathcal{Y}(G')>0$, which implies that there exists an edge $e^\dagger$ such that $wt(e^\dagger) > |V^c_{e^\dagger}|$ proving the lemma.
From Fig. \[fig:edge\_cs\], $|V^c_{e^\dagger}|=4$ for all $e^\dagger$. So, to evaluate $\mathcal{Y}(G')$, we write $\sum_{e^\dagger \in E^\dagger(G')} wt(e^\dagger)$ as follows: $$\sum_{e^\dagger \in E^\dagger(G')} wt(e^\dagger)=\sum_{v_b\in V'^b}q(v_b),$$ where $q(v_b)$ is the number of times a node $v_b$ of $G'$ is induced by $V^c_{e^\dagger}$, $e^\dagger \in E^\dagger(G')$. Let $$\begin{aligned}
\eta_i = \sum_{{v_b \in V_i^{'b}}} q(v_b) \label{eta}\end{aligned}$$ where $V_i^{'b}$ is the set of degree-$i$ bit nodes in $G'$. Then, $$\sum_{e^\dagger \in E^\dagger(G')} wt(e^\dagger)=\eta_1 + \eta_2 + \eta_3$$
We can now evaluate $q(v_b)$ in the terms $\eta_i$ for $i=1,2,3$. A degree-3 bit node $v_b\in V'^b_3$ corresponds to a triangular face in $\mathcal{C}(G')$, which corresponds to a degree-3 node $f \in\mathcal{C}^\dagger(G')$. Whenever an edge $e^\dagger$ is incident on $f$, the node $v_b$ will be induced by $V^c_{e^\dagger}$. Since there are 3 edges incident on any node in $\mathcal{C}^\dagger(G')$, $q(v_b) = 3$ for $v_b\in V'^b_3$.
A degree-two bit node $v_b\in V'^b_2$ is identified with an edge in $\mathcal{C}(G')$. Note that this edge is common to two faces, say $f_1$ and $f_2$ in $\mathcal{C}(G')$. Let $f_1$ and $f_2$ be the corresponding nodes in $\mathcal{C}^\dagger(G')$. Then, $v_b$ is induced by $V^c_{e^\dagger}$, whenever $e^\dagger$ is incident to $f_1$ or $f_2$. Since there are 5 edges incident on two neighboring nodes $f_1$ and $f_2$ in $\mathcal{C}^\dagger(G')$, $q(v_b) = 5$ for $v_b\in V'^b_2$. This is illustrated in Fig \[fig:q\_deg2\_p2\].
A degree-1 bit node $v_b\in V'^b_2$ in $G'$ is identified by a check node to which it is connected to in $G'$. This check node corresponds to a face in $\mathcal{C}^\dagger(G')$. The node $v_b$ is induced by $V^c_{e^\dagger}$, whenever $e^\dagger$ is incident on one or more vertices of the face. Since there are at least 3 vertices in a face, note that $q(v_b)$ for a degree-1 bit node $v_b$ is greater than $q(v'_b)$ for a degree-2 bit node $v'_b$. Thus, $$\begin{aligned}
\sum_{e^\dagger \in E^\dagger(G')} wt(e^\dagger) &\ge& 3|V_3^{'b}| + 5 |V_2^{'b}| + 5|V_1^{'b}|,\\
&=& 3(2|V'^c|-4) + 5(|V'^b| - (2|V'^c|-4)),\\
&=&5|V'^b|-4|V'^c|+8,\end{aligned}$$ upon using (\[eq:deg12\]). Using in (\[eq:y\_p2\]), $$\begin{aligned}
\mathcal{Y}(G') &\ge& 5|V'^b|-4|V'^c|+8 - 4(3|V'^c|-6),\\
&=&5|V'^b|-16|V'^c|+32. \end{aligned}$$ We see that $\mathcal{Y}(G') >0$, whenever $R=1-\frac{|V'^c|}{|V'^b|} \ge \frac{11}{16}$.
By Lemma \[lemma:1\], we have shown that there is a codeword-supporting edge in $\mathcal{C}^\dagger(G')$.
### Codeword-supporting subgraph for $G$
To extend the proof to a general planar Tanner graph, we show that a series of simple modifications can transform the check inverse $G'$ to the original Tanner graph $G$. We begin by defining three basic operations on a generic planar Tanner graph $P$.
1. [*DS1:*]{} Remove a degree-3 bit node in $P$ and add a degree-1 bit node to some check node.
2. [*DS2:*]{} Remove a degree-3 bit node in $P$ and add a degree-2 bit node to a pair of check nodes keeping the resulting graph planar.
3. [*DE:*]{} Increase the degree of a degree-3 bit node by connecting it to one or more check nodes so that the resulting graph is still planar. The resulting increase in degree is called the [*expansion factor*]{} of DE.
The abbreviation DS stands for Degree Shrinking, and DE stands for Degree Expansion. These operations are illustrated in Fig \[fig:operations\].
In Fig. \[fig:operations\], the solid lines are the edges of the Tanner graph and the dotted lines are edges of the check graph. Note that the check graph is unaltered by the DS and DE operations.
Let $G$ be a planar Tanner graph with check inverse $G'$. Then, $G$ can be obtained from $G'$ by a series of DS1, DS2 and DE operations. \[prop:ds\]
In the process of constructing $G'$, the following observations can be made: (1) degree-3 nodes of $G$ are retained in $G'$, (2) some degree-2 and degree-1 nodes may be dropped, and (3) higher degree ($\ge4$) nodes of $G$ result in multiple degree-3 nodes in $G'$.
The operations DS1 and DS2 restore the dropped degree-2 and degree-1 nodes, while a following DE operation creates higher degree nodes. Note that DS1 and DS2 create “empty” faces in the check graph of $G'$ while DE makes a set of faces correspond to a single bit node of higher degree.
Also note that if we start with $G'$, DE by a factor of $x$ is always preceded by $x$ DS operations since $x$ empty faces should be created before DE in order to preserve planarity. These operations recursively position degree 2 and degree 1 bit nodes to the positions as in $G$ and also create bit nodes of higher degree matching to those in $G$.
Consider the Tanner graph $G$ and its check inverse $G'$ in Fig. \[fig:chkinv\] of Example \[ex:GtoG’\]. Starting with $G'$ we can obtain $G$ through a series of DS1, DS2 and DE operations. This is illustrated in Fig. \[fig:inv\_to\_G\]. Note that $\mathcal{C}(G)$ remains a valid check graph of the resulting graph after every operation.
We are now ready to prove the existence of a codeword-supporting edge in $\mathcal{C}^\dagger(G)$. The approach is to show that the operations DS1, DS2 and DE cannot decrease $\mathcal{Y}(G')$. Hence, at the end of the necessary series of DS and DE operations to get $G$ from $G'$, we have $\mathcal{Y}(G)>0$. The result is proved in the following lemma.
Let $G$ be a planar Tanner graph representing a code with rate $R\ge11/16$. Then there is an edge in $\mathcal{C}^\dagger(G)$ that is codeword-supporting. \[lemma:Gedge\]
We will prove by showing that the operations DS and DE on the bit nodes of $G'$ to get $G$ are such that $\mathcal{Y}(G)\ge \mathcal{Y}(G')$ (see (\[eq:y\_p2\]) for definition). Since both $\mathcal{C}^\dagger(G')$ and $\mathcal{C}^\dagger(G)$ have same structure when seen as graphs, the term $\sum_{e^\dagger \in E^\dagger(G')} |V^c_{e^\dagger}|=\sum_{e^\dagger \in E^\dagger(G)} |V^c_{e^\dagger}|$. The change will be in $\sum_{e^\dagger \in E^\dagger(G')} wt(e^\dagger)$. Let $$\Delta=\sum_{e^\dagger \in E^\dagger(G)} wt(e^\dagger)-\sum_{e^\dagger \in E^\dagger(G')} wt(e^\dagger).$$ We will show that $\Delta\ge0$ to claim the lemma.
Let $H$ be the planar graph obtained at some intermediate step in the transformation from $G'$ to $G$. Let us see how DS and DE operations affect $\sum_{e^\dagger \in E^\dagger(H)} wt(e^\dagger)$.
The operation DS2 reduces the number of degree-3 bit nodes by one, and increases the number of degree-2 bit nodes by one. Let $\eta_2^*$ and $\eta_3^*$ be the new values of the terms $\eta_2$ and $\eta_3$ in (\[eta\]) after the operation DS2. Let $\delta_{DS2}$ be the change in $\sum_{e^\dagger \in E^\dagger(H)} wt(e^\dagger)$ when a DS2 is performed. We see that $$\begin{aligned}
\eta_3^* &= \eta_3 - 3\\
\eta_2^* &= \eta_2 + 5\\
\delta_{DS2} &= (\eta_3^* + \eta_2^*) - (\eta_3 + \eta_2) = 2 >0\end{aligned}$$
Since DE by a factor of $x$ is preceded by $x$ DS operations, we will study the net effect. DE with expansion factor of $x$ preceded by $x$ DS2s results in the following:
1. reduces the number of degree-3 bit nodes by $x+1$
2. increases the number of degree-2 bit nodes by $x$
3. introduces a bit node of degree $x+3$.
Effect of (i) and (ii) can be readily derived as before. As (iii) involves introduction of a bit node, it increases $\mathcal{Y}(H)$ by a positive quantity, say $\alpha_x$. Let $\delta_{DE}$ be the change in $\sum_{e^\dagger \in E^\dagger(H)} wt(e^\dagger)$ when a DE is performed. Therefore, $$\delta_{DE} = -3(x+1) + 5(x) + \alpha_x.
\label{delta_p2}$$ Since $\delta_{DE}$ is non-negative for $x>1$, it is enough to compute $\alpha_1$ and show that $\delta_{DE}$ is non-negative for $x=1$. When the expansion factor is one, a degree-3 bit node $v_b$ becomes a degree-4 bit node, and $v_b$ is identified with two faces of the new check graph having a common edge. This is equivalent to saying that $v_b$ is identified with an edge in the dual of the check graph. Therefore, $\alpha_1 = 1$. Substituting in (\[delta\_p2\]), we get $\delta_{DE} =0$ for $x=1$.
As $q(v_b)$ of a degree-1 bit node $v_b$ is more than $q(v'_b)$ of a degree-2 bit node $v'_b$, $\delta_{DS1}$ is non-negative whenever DS1 is used in place of DS2 where $\delta_{DS1}$ is the change in $\sum_{e^\dagger \in E^\dagger(H)} wt(e^\dagger)$ when a DS1 is performed.
Since $G$ can be obtained from a series of DS1, DS2 and DE operations, and since $\delta_{DS1}$, $\delta_{DS2}$, and $\delta_{DE}$ are all non-negative, $\Delta \ge 0$. Hence, the lemma is proved.
### Proof of Theorem \[thm:weak\]
By Lemma \[lemma:Gedge\], $\mathcal{C}^\dagger(G)$ has an edge $e^\dagger$ such that $|\mathcal{I}(V^c_{e^\dagger}|)>|V^c_{e^\dagger}|$. We see that $|V^c_{e^\dagger}|=4$ by Proposition \[prop:vcu\]. Hence, by Proposition \[prop:dmin\], the code defined by $G$ has a codeword of weight at most $|V^c_{e^\dagger}| +1 =5$. This proves Theorem \[thm:weak\].
Proof of Main Result {#sec:proof-main-result}
====================
We now provide the proof of Theorem \[thm:main\]. The method and steps of proof are similar to that of the proof of the weaker Theorem \[thm:weak\]. A codeword-supporting subgraph will be shown to exist in the dual of check graph of check inverse by a similar counting argument. The result will then be extended by DS and DE operations to the original graph.
The main change is that an edge of the dual need not be codeword-supporting for lower rates. We will show that among the neighborhoods of vertices of the dual of check graph with $p$ nodes (for a suitably chosen $p$), there exists a codeword-supporting subgraph, which provides a bound on minimum distance.
We will impose a girth condition on the dual of check graph, so that the neighborhoods become trees for small $p$. Since the dual of check graph is regular with degree 3, the vertex neighborhoods will be rooted trees with mostly degree-3 nodes except near the leaves. The girth condition on the dual of the check graph will be later shown to translate into a condition on the rate of the code defined by the original Tanner graph.
The type of neighborhood structure in the dual check graph is captured by 3-trees defined below.
3-trees
-------
A [*3-tree*]{} rooted at a vertex $v_r$ is a rooted tree in which the root $v_r$ has at most 3 children and all other nodes have at most 2 children. The [*depth*]{} of a vertex $v$ is the length of the path from the root $v_r$ to the vertex $v$. The set of all nodes at a given depth is called a [*level*]{} of the tree. The root node is at depth zero. The [*height*]{} of a tree is the length of the path from the root to the deepest node in the tree. A [*complete*]{} 3-tree is one in which every level, except possibly the last, is completely filled.
The number of nodes at level $l$ (except possibly the last) of a complete 3-tree is $3.2^{l-1}$ and number of nodes up to and including level $l$ is $3.2^l - 2$. Fig. \[fig:3tree\] shows a picture of a complete 3-tree.
Here are a few results on complete 3-trees.
1. Let $p$ be the number of vertices of a complete 3-tree. Then $p=3.2^{l(p)-1} - 2 + z(p)$ for a suitable level $l(p)$ such that $0 \le z(p) < 3.2^{l(p)-1}$, where $z(p)$ denotes the number of nodes in the last level.
2. Height of the complete 3-tree with $p$ vertices is $$h(p) = \begin{cases}
l(p)\;\;\;\;\;\;, z(p) \neq 0\\
l(p)-1\;\;,z(p)=0
\end{cases}.$$
3. There are $t(p)= \binom{3.2^{l(p)-1}}{z(p)}$ complete 3-trees of height $h(p)$ rooted at a given vertex $v$. Each such tree is called a [*realization*]{} of the complete 3-tree rooted at $v$.
4. There are three branches from the root of a complete 3-tree on $p$ nodes. The number of nodes up to level $l<h(p)$ on one branch (excluding the root node) is $\frac{1}{3}(3.2^l - 3)=2^l-1$.
The parameters $l(p)$, $z(p)$, $h(p)$ and $t(p)$ are evaluated for some values of $p$ in Table \[Table:pvals\].
$p$ $l(p)$ $z(p)$ $h(p)$ $t(p)$
----- -------- -------- -------- --------
2 1 1 1 3
3 1 2 1 3
4 2 0 1 1
5 2 1 2 6
8 2 4 2 15
10 3 0 2 1
: Parameters of a complete 3-tree for some values of $p$[]{data-label="Table:pvals"}
Complete 3-graphs in $\mathcal{C}^\dagger(G)$
---------------------------------------------
Since $\mathcal{C}^\dagger(G)$ is planar with uniform degree 3, we can look for complete 3-trees on $p$ vertices in the vertex neighbourhoods of $\mathcal{C}^\dagger(G)$. Given $p$, each root has $t(p)$ complete 3-trees of height $h(p)$ only when the girth $g$ of $\mathcal{C}^\dagger(G)$ satisfies $$\label{eq:girth}
g \ge 2h(p)+1,$$ where $h(p)$ is the height defined as before.
Let $V^\dagger(G)$ be the vertex set of $\mathcal{C}^\dagger(G)$. Let $V_{i,j}^\dagger \subseteq V^\dagger(G)$ be the vertex set of the $j$-th realization of a complete 3-subtree rooted at node $v_i$ with $p$ vertices. Let the subgraph induced by $V_{i,j}^\dagger$ be denoted as $\mathcal{C}_{i,j}^\dagger(G)$ for simplicity. These induced subgraphs from now on will be referred to as [*complete 3-graphs*]{}.
Note that $\mathcal{C}_{i,j}^\dagger(G)$ need not be a tree due to the presence of one or more extra edges that can create cycles. However, under the girth condition, these additional edges can only be between leaves of the complete 3-tree creating cycles of length $2h(p)+1$. These additional edges between the leaves of a realization of a complete 3-tree in the neighborhood of a node are said to be [*cycle-creating*]{} with respect to that particular realization.
The number of realizations with root $v_i$ in which an edge $e^\dagger$ is cycle-creating is called the [*recurrence number*]{} of the edge $e^\dagger$ with respect to the vertex $v_i$, and is denoted as $r_{v_i}(e^\dagger)$. The total number of all realizations with respect to all roots in which $e^\dagger$ is cycle-creating is called the [*total recurrence number*]{} of the edge and denoted $r(e^\dagger)$. We readily see that $$\label{eq:rec}
r(e^\dagger) = \sum_{v_i \in V^\dagger(G)} r_{v_i}(e^\dagger).$$ An edge that is cycle-creating for at least one realization (or edge with positive total recurrence number) is called a [*cycle-edge*]{}. The set of all cycle-edges in $\mathcal{C}^\dagger(G)$ is denoted by $\mathfrak{B}(G)$.
Consider the dual graph $\mathcal{C}^\dagger(G)$ shown in Fig. \[fig:com3gph\_ex\].
We will show some complete 3-graphs in $\mathcal{C}^\dagger(G)$ of Fig. \[fig:com3gph\_ex\] for $p=4$. Fig. \[fig:com3gph\_ex\](a) shows a complete 3-graph rooted at $f_1$ in which $e_2^\dagger$ is a cycle-creating edge. Similarly, \[fig:com3gph\_ex\](b),(c) shows complete 3-graphs rooted at $f_4$ and $f_2$ respectively. The corresponding cycle creating edges are shown in dashed lines. Observe that $e_2^\dagger$ which is cycle-creating for the complete 3-graph rooted at $f_1$, but is not cycle-creating for the complete 3-graph rooted at $f_4$.
For the complete 3-graph rooted at $f_4$, $e_1^\dagger$ is cycle-creating. Since $p=4$, $t(p)=1$ i.e, there is only one realization per root node. Hence, $r_{f_4}(e_1^\dagger) = 1$. Also note that $e_1^\dagger$ is not cycle-creating for other complete 3-graphs rooted at other nodes. Therefore, $r(e_1^\dagger) = 1$. Using similar arguments, we can show that $r(e_i^\dagger) =1$ for $i=\{2,3,5,6,7\}$. These edges are cycle-edges in $\mathcal{C}^\dagger(G)$. All other edges are non-cycle-edges; hence, their recurrence number is zero.
Let $W_{v_i,j}=wt(\mathcal{C}_{i,j}^\dagger(G))$ be the weight of the $j$-th realization of a complete 3-graph rooted at $v_i$. Let $V^c_{i,j}$ be the set of check nodes forming the faces corresponding to the vertices $V_{i,j}^\dagger$ in $\mathcal{C}^\dagger(G)$ i.e. $V^c_{i,j}=V^c_U$ with $U=V^\dagger_{i,j}$ as in (\[eq:vcu\]). By Proposition \[prop:vcu\], $$\label{eq:vcij}
|V^c_{i,j}|\le p+2 - c_{v_i,j},$$ where $c_{v_i,j}$ is the number of cycles in the $j$-th realization of the complete 3-graph on $p$ nodes rooted at $v_i$. By definition, $\mathcal{C}_{i,j}^\dagger(G)$ is codeword-supporting if $W_{v_i,j} - |V^c_{i,j}| > 0$.
Consider the summation, $$\mathcal{Y}(G) = \sum_{v_i \in V^\dagger(G)} \sum_{j = 1}^{t(p)} \left(W_{v_i,j}-|V^c_{i,j}|\right) \label{eq:Ygen}$$ where $t(p)$ is the number of realizations of complete 3-trees on $p$ vertices rooted at a particular node assuming the girth condition (\[eq:girth\]). We will show that $\mathcal{Y}(G)>0$ to establish the existence of a codeword-supporting subgraph among the complete 3-graphs of $\mathcal{C}^\dagger(G)$. Let $m=|V^c|$ and $n=|V^b|$ be the number of check nodes and bit nodes of $G$. Hence, $|V^{\dagger}(G)|=2m-4$. Using (\[eq:vcij\]) in (\[eq:Ygen\]), we get $$\begin{aligned}
\mathcal{Y}(G) &\ge \sum_{v_i \in V^\dagger(G)} \sum_{j = 1}^{t(p)} W_{v_i,j} - [(2m-4)pt(p) + 2(2m-4)t(p)]\nonumber\\
&\phantom{= \sum_{v_i \in V^\dagger(G)} } + \sum_{v_i \in V^\dagger(G)} \sum_{j = 1}^{t(p)} c_{v_i,j},\label{eq:3} \end{aligned}$$ To show $\mathcal{Y}(G)>0$, we simplify the terms in the right hand side of (\[eq:3\]).
Simplifying $\sum_{v_i \in V^\dagger(G)} \sum_{j = 1}^{t(p)} c_{v_i,j}$
-----------------------------------------------------------------------
We begin by relating the number of cycles $c_{v_{i,j}}$ to recurrence number of edges.
The following equality holds: $$\sum_{v_i \in V^\dagger(G)} \sum_{j = 1}^{t(p)} c_{v_i,j} = \sum_{e_l^\dagger \in \mathfrak{B}(G)} r(e_l^\dagger)$$ \[prop:rec\]
We see that $$\sum_{v_i \in V^\dagger(G)} \sum_{j = 1}^{t(p)} c_{v_i,j}=\sum_{v_i \in V^\dagger(G)} \sum_{e_l^\dagger \in \mathfrak{B}(v_i)} r_{v_i}(e_l^\dagger),$$ where $\mathfrak{B}(v_i)$ is the set of edges that are cycle-creating in any of the complete 3-graphs rooted at $v_i$. Now, $$\begin{aligned}
\sum_{v_i \in V^\dagger(G)} \sum_{j = 1}^{t(p)} c_{v_i,j}&= \sum_{e_l^\dagger \in \mathfrak{B}(G)} \sum_{v_i \in V^\dagger(G)} r_{v_i}(e_l^\dagger)\nonumber\\
&= \sum_{e_l^\dagger \in \mathfrak{B}(G)} r(e_l^\dagger).\label{eq:1}\end{aligned}$$
### Occupied and unoccupied edges
In the computation of $\mathcal{Y}(G)$, the cycle edges identified with degree-2 bit nodes should be treated separately. Such cycle edges are classified next.
Each degree-2 bit node in $G$ is identified as an edge in $\mathcal{C}(G)$. An edge $e$ in $\mathcal{C}(G)$ is said to be [*occupied*]{} by a degree-2 bit node if the bit node is connected to the check nodes of $e$. Else it is said to be [*unoccupied*]{}. Since there is a one to one correspondence between edges of $\mathcal{C}(G)$ and edges of its dual $\mathcal{C}^\dagger(G)$, we use the terms occupied and unoccupied for edges of dual as well.
Similarly, we can talk of occupied and unoccupied check nodes. A check node of a planar Tanner graph is said to be occupied if there is a degree-1 bit node connected to it. The degree-1 bit node is said to occupy the check node. Otherwise, the check node is said to be unoccupied.
Let $\mathfrak{B}(G)$ be the set of cycle-edges, and let $\mathfrak{B}_o(G)$ and $\mathfrak{B}_u(G)$ be the set of occupied and unoccupied cycle-edges. We see that $\mathfrak{B}_o(G)$ and $\mathfrak{B}_u(G)$ partition $\mathfrak{B}(G)$ so that the following holds: $$|\mathfrak{B}(G)| = |\mathfrak{B}_o(G)| + |\mathfrak{B}_u(G)|. \label{eq:B}$$
In the Tanner graph of Figs. \[fig:induced\_bit\] and \[fig:com3gph\_ex\], the set of occupied edges are given by $\{e_4^\dagger,e_6^\dagger,e_7^\dagger\}$ as seen from Example \[ex:dual-check-graph\], and the set of cycle-edges is given by $\mathfrak{B}(G)=\{e_2^\dagger,e_3^\dagger,e_5^\dagger,e_6^\dagger,e_7^\dagger\}$. Hence, we see that $\mathfrak{B}_o(G)=\{e_6^\dagger,e_7^\dagger\}$ and $\mathfrak{B}_u(G)=\{e_2^\dagger,e_3^\dagger,e_5^\dagger\}$.
Using the partition of $\mathfrak{B}(G)$ in (\[eq:1\]), we see that $$\label{eq:oc}
\sum_{v_i \in V^\dagger(G)} \sum_{j = 1}^{t(p)} c_{v_i,j}=\sum_{e_l^\dagger \in \mathfrak{B}_o(G)} r(e_l^\dagger)+\sum_{e_l^\dagger \in \mathfrak{B}_u(G)} r(e_l^\dagger).$$
\[ex:occup-unocc-edges\] In the illustration of the DS and DE operations in Fig. \[fig:inv\_to\_G\], we stated that the check graph of $G$ and $G'$ are the same. However, the set of occupied edges and check nodes changes because of the changes in the number of degree-2 and degree-1 nodes. In the check graph $\mathcal{C}(G')$ in Fig. \[fig:inv\_to\_G\], the set of occupied edges is $\{(1,2),(2,4),(3,4)\}$, and the set of occupied nodes is $\{1,4\}$. However, in $\mathcal{C}(G)$ in Fig. \[fig:inv\_to\_G\], the set of occupied edges is $\{(1,2),(1,3),(2,4),(3,4)\}$, and the set of occupied nodes is $\{1,2,3,4\}$. Since new degree-2 and degree-1 nodes can possibly be added in $G$ through the DS operations, some unoccupied edges and check nodes in $\mathcal{C}(G')$ become occupied in $\mathcal{C}(G)$.
### Singular nodes {#sec:singular-nodes}
Another special situation arises with multiple edges in check graphs. Note that, by construction, multiple edges can arise in check graphs, if they do not result in a face enclosed by two edges. The effect of multiple edges in check graphs is characterized next.
*Definition:* Let $G$ be a planar Tanner graph, and let $\mathcal{C}(G)$ be its check graph. Let $\mathcal{C}_*(G)$ be a subgraph of $\mathcal{C}(G)$ consisting of a maximal planar graph on 4 vertices plus an edge that leads to an external face of length 2 in $\mathcal{C}_*(G)$ as shown in Fig. \[fig:sing\_defn\]. In addition, we impose the constraint that the interior faces of $\mathcal{C}_*(G)$ are faces in $\mathcal{C}(G)$.
A degree-1 bit node in $G$ is said to be *singular* if it occupies a check node in $G$ that corresponds to one of the interior nodes of $\mathcal{C}_*(G)$. In Fig. \[fig:sing\_defn\], a degree-1 bit node that occupies $v_c$ or $v'_c$ is singular.
The number of singular nodes in a planar Tanner graph $G$ is denoted $s_G$. The following proposition relates $s_G$ to the recurrence number.
Let $G$ be a planar Tanner graph with $s_G$ singular nodes. Let $\mathfrak{B}_u(G)$ be the set of unoccupied cycle-edges in $\mathcal{C}^\dagger(G)$ for the complete 3-graphs on 4 nodes. If $$\sum_{e_l^\dagger \in \mathfrak{B}_u(G)} r(e_l^\dagger) < s_{G},$$ there exists a codeword-supporting subgraph on 4 nodes in $\mathcal{C}^\dagger(G)$. \[prop:sing\]
Let $\sum_{e_l^\dagger \in \mathfrak{B}_u(G)} r(e_l^\dagger) < s_{G}$. Since $\sum_{e_l^\dagger \in \mathfrak{B_u}} r(e_l^\dagger)$ is non-negative $s_G$ is at least 1. Hence, there is at least one subgraph of the form $\mathcal{C}_*(G)$ as shown in Fig. \[fig:sing\_defn\] in $\mathcal{C}(G)$. Let $\mathfrak{B}$ be the set of edges in $\mathcal{C}^\dagger(G)$ corresponding to the set of interior edges $\{e_1,e_2,e_3,e_4,e_5\}$ in $\mathcal{C}_*(G)$. The edges of $\mathfrak{B}$ are shown as dashed lines in Fig. \[fig:cyc\_edges\_sing\]. Note these the edges are cycle-edges for $p=4$.
![Cycle edges in the dual for $p=4$.[]{data-label="fig:cyc_edges_sing"}](cyc_edges_sing.pdf)
Let $\mathcal{C}_*^i(G)$, $i=1,2...k$ be the subgraphs in $\mathcal{C}(G)$ of the form $\mathcal{C}_*(G)$. Let $s^i_G$ be the number of singular nodes in $\mathcal{C}_*^i(G)$ and $\mathfrak{B}^i$ be the set of 5 cycle-edges corresponding to the interior edges in $\mathcal{C}_*^i(G)$. Observe that $\mathfrak{B}^i$ are disjoint. Let $\mathfrak{B}^i_u$ be the subset of edges of $\mathfrak{B}^i$ that are unoccupied in $\mathcal{C}^\dagger(G)$. Let $\mathfrak{B}^*_u = \bigcup_{i=1}^k \mathfrak{B}^i_u$. Then, $\mathfrak{B}^*_u \subseteq \mathfrak{B}_u(G)$. Hence, $$\begin{aligned}
\sum_{e_l^\dagger \in \mathfrak{B}_u^*} r(e_l^\dagger) \le \sum_{e_l^\dagger \in \mathfrak{B}_u(G)} r(e_l^\dagger) < s_G,\\
\sum_{i=1}^k \sum_{e_l^\dagger \in \mathfrak{B}_u^i} r(e_l^\dagger) < \sum_{i=1}^k s^i_G, \\
\sum_{i=1}^k \left (\sum_{e_l^\dagger \in \mathfrak{B}_u^i} r(e_l^\dagger) - s^i_G \right) < 0.\end{aligned}$$ Thus, there exists $j$, $1 \le j\le k$, such that, $$\begin{aligned}
\sum_{e_l^\dagger \in \mathfrak{B}_u^j} r(e_l^\dagger) < s^j_G, \\
|\mathfrak{B}_u^j| \le \sum_{e_l^\dagger \in \mathfrak{B}_u^j} r(e_l^\dagger) < s^j_G, \\
|\mathfrak{B}_u^j| < s^j_G.\end{aligned}$$ Thus, there are at most $s^j_G -1$ edges in $\mathfrak{B}^j$ that are unoccupied. In other words, there are at least $6-s^j_G$ occupied edges in $\mathfrak{B}^j$. Therefore, the 4 nodes in $\mathcal{C}_*^j(G)$ induce at least $6-s^j_G +s^j_G = 6$ bit nodes in $G$. Since, there are 4 faces in $\mathcal{C}_*^j(G)$, there exists a codeword supporting subgraph on 4 nodes in $\mathcal{C}^\dagger(G)$.
Minimum distance bound for $G'$
-------------------------------
The next lemma is a generalization of Lemma \[lemma:1\] from edges to complete 3-graphs for the main result.
Let $G$ be a planar Tanner graph with check inverse $G'$. Let $G'$ define a code with rate $R$. Then, there exists a codeword-supporting subgraph on $p$ vertices in $\mathcal{C}^\dagger(G')$ where $$p= \left\lceil \frac{7-8R}{2(2R-1)} \right\rceil$$ is such that the girth condition (\[eq:girth\]) is satisfied. \[lemma:3g\]
As before, we will prove the result by showing that $\mathcal{Y}(G')>0$.
### Simplifying $\sum_{v_i \in V^\dagger(G')} \sum_{j = 1}^{t(p)} W_{v_i,j}$ {#simplifying-sum_v_i-in-vdaggerg-sum_j-1tp-w_v_ij .unnumbered}
Let us now consider the summation of weights term in (\[eq:3\]). As done in the illustrative proof of Theorem \[thm:weak\], we let $$\sum_{v_i \in V^\dagger(G')} \sum_{j = 1}^{t(p)} W_{v_i,j}=\sum_{v_b\in V'^b}q(v_b),$$ where $q(v_b)$ is the number of times a node $v_b$ of $G'$ is induced by $V^c_{i,j}$ for all $i$ and $j$. Let $$\label{eq:etadf}
\eta_i = \sum_{{v_b \in V_i^{'b}}} q(v_b),$$ where $V_i^{'b}$ is the set of degree-$i$ bit nodes in $G'$. Then, $$\begin{aligned}
\sum_{v_i \in V^\dagger(G')} \sum_{j = 1}^{t(p)} W_{v_i,j} = \eta_1 + \eta_2 + \eta_3. \label{eq:wexpn}\end{aligned}$$ We will begin with calculation of $\eta_3$. A degree-3 bit node $v_b$ in $G'$ is identified with a node $f$ in $\mathcal{C}^\dagger(G')$. So, $q(v_b)$ is same as the number of complete 3-graphs that contain $f$, which equals $pt(p)$. Hence, we see that $$q(v_b) = pt(p) \text{ for }v_b \in V^{'b}_3$$ Therefore, $$\eta_3 = (2m-4)pt(p). \label{q3}$$ The computation of $\eta_2$ and $\eta_1$, as shown in Appendices \[sec:eta\_2\] and \[sec:eta\_1\], results in the following. $$\begin{aligned}
\eta_2 & = |V'^b_2|(\frac {4} {3} p + \frac {2}{3})t(p) - \sum_{e_l^\dagger \in \mathfrak{B}_o(G')} r(e_l^\dagger) \label{q2}\\
\eta_1 & \ge |V'^b_1|(\frac {4} {3} p + \frac {2}{3})t(p) - s_{G'}(p), \label{q1}\end{aligned}$$ where $s_{G'}(p)$ is given by, $$s_{G'}(p) = \begin{cases}
s_{G'} \;\;\;\;\;\;\text{when~} p = 4\\
0 \;\;\;\;\;\;\text{else}
\end{cases}
\label{s(p)}$$ with $s_{G'}$ being the number of *singular nodes* in $G'$ as discussed in Section \[sec:singular-nodes\].
Using (\[q3\]), (\[q2\]) and (\[q1\]), we get $$\begin{gathered}
\sum_{v_i \in V^\dagger(G')} \sum_{j = 1}^{t(p)} W_{v_i,j}\ge(2m-4)pt(p)\\
+|V'^b_2|(\frac {4} {3} p + \frac {2}{3})t(p)+|V'^b_1|(\frac {4} {3} p + \frac {2}{3})t(p)\\
-\sum_{e_l^\dagger \in \mathfrak{B}_o(G')} r(e_l^\dagger)- s_{G'}(p).\end{gathered}$$ Using $|V'^b_1| + |V'^b_2| = n- (2m-4)$, we get $$\begin{gathered}
\label{eq:wsimp}
\sum_{v_i \in V^\dagger(G')} \sum_{j = 1}^{t(p)} W_{v_i,j}\ge[\frac{4}{3}pn+\frac{2}{3}n-\frac{1}{3}p(2m-4)-\frac{2}{3}(2m-4)]t(p)\\
-\sum_{e_l^\dagger \in \mathfrak{B}_o(G')} r(e_l^\dagger)- s_{G'}(p).\end{gathered}$$
Using (\[eq:oc\]) and (\[eq:wsimp\]) in the expression for $\mathcal{Y}(G')$ in (\[eq:3\]), we get $$\begin{aligned}
\mathcal{Y}(G') & \ge (\frac{4}{3}pn +\frac{2}{3}n - \frac{4}{3} p (2m-4) - \frac{8}{3} (2m-4))t(p) \nonumber \\
&\phantom{\frac{4}{3}pn +}+\sum_{e_l^\dagger \in \mathfrak{B}_u(G')} r(e_l^\dagger) - s_{G'}(p). \label{X}\end{aligned}$$ Let $X = (\frac{4}{3}pn +\frac{2}{3}n - \frac{4}{3} p (2m-4) - \frac{8}{3} (2m-4))t(p) $. Hence, $$\begin{aligned}
\mathcal{Y}(G') & \ge X + \sum_{e_l^\dagger \in \mathfrak{B}_u(G')} r(e_l^\dagger) - s_{G'}(p) \label{eq:yexpn_x}\end{aligned}$$
We fix $p$ to be the smallest integer that results in $X > 0$. This is readily seen to be $$p = \left \lceil \frac{7-8R}{2(2R-1)} \right \rceil. \label{pexpn}$$
*Case* 1: $p \ne 4$.\
When $p$ given by (\[pexpn\]) is not equal to 4, $s_{G'}(p)=0$ by (\[s(p)\]) and hence the term $\sum_{e_l^\dagger \in \mathfrak{B}_u(G')} r(e_l^\dagger) - s_{G'}(p)$ is non-negative. This implies that, for this $p$, $\mathcal{Y}(G') > 0$ and there exists a codeword-supporting complete 3-graph on $p = \left \lceil \frac{7-8R}{2(2R-1)} \right \rceil$ nodes.
*Case* 2: $p = 4$.\
If $\sum_{e_l^\dagger \in \mathfrak{B}_u(G')} r(e_l^\dagger) - s_{G'}(p) \ge 0$, $\mathcal{Y}(G') > 0$ for $p=4$ and there exists a codeword-supporting complete 3-graph on $p = \left \lceil \frac{7-8R}{2(2R-1)} \right \rceil = 4$ nodes. Even otherwise, by Proposition \[prop:sing\], there exists a codeword supporting subgraph on $4$ nodes.
Minimum distance bound for $G$
------------------------------
The bound for $G$ is obtained as in the proof of Theorem \[thm:weak\] by showing that the operations DS and DE do not reduce the value of $\mathcal{Y}(G')$. The graph $G$ is obtained from $G'$ through a series of recursive operations as shown in Proposition \[prop:ds\]. We will see how the DS and DE operations involved in transforming $G'$ to $G$ affect the summation $\mathcal{Y}(G')$. Since both $\mathcal{C}^\dagger(G')$ and $\mathcal{C}^\dagger(G)$ have same structure when seen as graphs, the term $\sum_{v_i \in V^\dagger(G)} \sum_{j = 1}^{t(p)} |V^c_{i,j}|=\sum_{v_i \in V^\dagger(G')} \sum_{j = 1}^{t(p)} |V^c_{i,j}|$. The only change will be in $\sum_{v_i \in V^\dagger(G')} \sum_{j = 1}^{t(p)} W_{v_i,j}$.
Let $H$ be a planar graph obtained at some intermediate step in the transformation from $G'$ to $G$. Since $\mathcal{C}(G')$ is a check graph for $H$, the girth condition is satisfied by $C^{\dagger}(H)$. We write $$\label{eq:wh}
W(H)=\sum_{v_i \in V^\dagger(H)} \sum_{j = 1}^{t(p)} W_{v_i,j} = \eta_1(H) + \eta_2(H) + \eta_3(H),$$ where $\eta_i(H) = \sum_{{v_b \in V_i^{b}}} q(v_b)$ with $V_i^{b}$ being the set of degree-$i$ bit nodes in $H$.
### Effect of DS2
The graph obtained after the DS1 operation on $H$ is denoted $\text{DS1}[H]$. Let $\Delta_{DS1}=W(\text{DS1}[H])-W(H)$ be the change in the weight summations because of the DS1 operation. Similar notation is used for the DS2 and DE operations.
The operation DS2 reduces the number of degree-three bit nodes by one and increases the number of degree-2 bit nodes by one. From (\[q3\]) and (\[q2\]), $$\begin{aligned}
\eta_3(\text{DS2}[H]) &= \eta_3(H) - pt(p),\\
\eta_2(\text{DS2}[H]) &= \eta_2(H) + \left(\frac{4}{3}p+\frac{2}{3} \right )t(p) - r(e^\dagger),\end{aligned}$$ where $e^\dagger$ is the edge in $\mathcal{C}^\dagger(\text{DS2}[H])$ identified with the new degree-2 bit node. Hence, $$\begin{aligned}
\Delta_{DS2} = \delta - r(e^\dagger), \label{eq:del_s2}\end{aligned}$$ where $\delta = \frac{1}{3}p.t(p) + \frac{2}{3} t(p)>0$.
### Effect of DS1
The operation DS1 reduces the number of degree-3 bit nodes by one and increases the number of degree-1 bit nodes by one. From Appendix \[sec:eta\_1\], $q(v_b)$ for a degree-1 bit node $v_b$ is at least $\left (\frac{4}{3}p+\frac{2}{3} \right )t(p)$ except for the case when $p=4$ and $v_b$ singular. Following calculations as before, we can show that $$\Delta_{DS1} = \delta - s'(p),
\label{eq:del_s1}$$ where $s'(p) = \begin{cases} 1, \; p = 4\text{ and }v_b\text{ singular},\\
0, \;\text{else}.
\end{cases}$
### Effect of DE
Let $\text{DE}(x)$ denote the DE operation by an expansion factor of $x$, which is necessarily preceded by $x$ DS operations. The operation DE$(x)$ results in the following:
1. reduces the number of degree-3 bit nodes by $x+1$.
2. increases the number of bit nodes of degree $\le 2$ by $x$.
3. introduces a bit node of degree $x+3$.
Let $\mathcal{E}_*^\dagger$ be the subset of edges of $\mathcal{C}^\dagger(H)$ identified with the new degree-2 bit nodes, and let $s^+$ be the number of new degree-1 bit nodes that are singular in $\text{DE}(x)[H]$. Effect of Steps (i) and (ii) can be readily derived as before. The introduction of a bit node in Step (iii) increases $W(H)$ by a positive quantity denoted $\alpha_x$. Hence, we see that $$\begin{gathered}
\Delta_{DE(x)} = x \left (\frac{1}{3}pt(p) + \frac{2}{3}t(p) \right) - p.t(p) \\
- \sum_{e^\dagger \in \mathcal{B}_*^{\dagger}}r(e^\dagger) - s^+(p) + \alpha_x, \end{gathered}$$ where $$s^+(p) = \begin{cases}
s^+, \; p = 4,\\
0, \;\text{else},\end{cases}$$ and $\mathcal{B}_*^\dagger$ is the set cycle edges in $\mathcal{E}_*^\dagger$. In the above equation, we use $\mathcal{B}_*^\dagger$ for the summation term instead of $\mathcal{E}_*^\dagger$ as recurrence number of a non cycle-edge is zero.
Let $\delta_{DE}(x) = x \left (\frac{1}{3}pt(p) + \frac{2}{3}t(p) \right) - pt(p) + \alpha_x $. Then, $$\Delta_{DE(x)} = \delta_{DE}(x) - \sum_{e^\dagger \in \mathcal{E}_*^\dagger}r(e^\dagger) - s^+(p). \label{eq:del_de2}$$ Since $\alpha_x$ is positive, $\delta_{DE}(x)$ is positive for $x\ge 3$. As shown in Appendix \[sec:gx\], $$\alpha_1 \ge \left (\frac{2}{3}p - \frac{2}{3}\right)t(p).$$ Substituting $\alpha_1$ in $\delta_{DE}(x)$ for $x=1$ gives $\delta_{DE}(1) \ge 0$. As shown in Appendix \[sec:gx\], $$\alpha_2 \ge \left (\frac{1}{2}p - \frac{1}{6}z(p) - 1 \right)t(p).$$ Substituting $\alpha_2$ in $\delta_{DE}(x)$ for $x=2$, we get $$\delta_{DE}(2) \ge \left ( \frac{1}{3} - \frac{1}{6}z(p) + \frac{1}{6}p \right )t(p) > 0,$$ since $p > z(p)$. Hence $\delta_{DE}(x) \ge 0$ for all $x$.
### Codeword-supporting complete 3-graph
We now present a generalization of Lemma \[lemma:Gedge\] for complete 3-graphs.
Let $G$ be a planar Tanner graph defining a code of rate $R$. Then, there exists a codeword-supporting complete 3-graph on $p = \left \lceil \frac{7-8R}{2(2R-1)} \right \rceil$ nodes in $\mathcal{C}^\dagger(G)$ provided the girth condition is satisfied. \[lem:G3tree\]
We will prove the lemma by showing that $\mathcal{Y}(G) > 0$ for $p = \left \lceil \frac{7-8R}{2(2R-1)} \right \rceil$.
Let $n_{DS1}$ and $n_{DS2}$ be the number of $DS1$ and $DS2$ operations (excluding those DS1 and DS2 operations that are performed to create empty faces before DEs) performed in obtaining $G$ from $G'$. Let $n_{DE}(x)$ be the number of DE operations with expansion factor $x$. Let $\mathcal{E}_*^\dagger$ be the subset of edges of $\mathcal{C}^\dagger(G')$ identified with the new degree-2 bit nodes resulting from these operations (see Example \[ex:occup-unocc-edges\]). Let $s^*$ be the number of singular nodes in $G$ that are not present in $G'$. Then, $$\begin{gathered}
\mathcal{Y}(G) = \mathcal{Y}(G') + n_{DS1}\delta + n_{DS2}\delta \\
+\sum_{x}(n_{DE(x)}\delta_{DE}(x)) - \sum_{e^\dagger \in \mathfrak{B}_*^\dagger}r(e^\dagger) - s^*(p),\end{gathered}$$ where $B_*^\dagger$ is the set of cycle edges in $\mathcal{E}_*^\dagger$ and $$s^*(p) = \begin{cases}
s^*,\; p=4, \\
0,\;\text{else.}
\end{cases}$$ Let $\Delta = n_{DS1}\delta + n_{DS2}\delta + \sum_{x}(n_{DE(x)}\delta_{DE}(x))$. From the previous section, we see that $\Delta \ge 0$. Substituting the expression for $\mathcal{Y}(G')$ from (\[eq:yexpn\_x\]), we get, $$\begin{gathered}
\mathcal{Y}(G) \ge X + \Delta + \sum_{e_l^\dagger \in \mathfrak{B}_u(G')} r(e_l^\dagger) - \sum_{e^\dagger \in \mathfrak{B}_*^\dagger}r(e^\dagger) \\
- (s_{G'}(p) + s^*(p)).\end{gathered}$$ We assume that the edges in $\mathcal{E}_*^\dagger$ are unoccupied in $\mathcal{C}^\dagger(G')$. Otherwise, there will be two or more degree-2 bit nodes in $G$ identified with the same edge and $d = 2$. Therefore, $$\sum_{e_l^\dagger \in \mathfrak{B}_u(G')} r(e_l^\dagger) - \sum_{e^\dagger \in \mathfrak{B}_*^\dagger}r(e^\dagger) = \sum_{e_l^\dagger \in \mathfrak{B}_u(G)} r(e_l^\dagger),$$ where $\mathfrak{B}_u(G)$ is the set of unoccupied cycle-edges in $C^\dagger(G)$. Similarly, we assume that the check nodes to which the new degree-1 singular nodes are connected to are unoccupied in $G'$ to avoid $d=2$. Hence, if $s_G$ is the number of singular nodes in $G$, we have $$s_{G'}(p) + s^*(p) = s_G(p),$$ where $$s_G(p) = \begin{cases}
s_G, ~~p = 4 \\
0, \;\text{~~else}.
\end{cases}$$ Substituting the above expressions in $\mathcal{Y}(G)$, we get, $$\mathcal{Y}(G) \ge X + \Delta + \sum_{e_l^\dagger \in \mathfrak{B}_u(G)} r(e_l^\dagger) - s_G(p).$$ From the proof of Lemma \[lemma:3g\] we see that, $X>0$ for $p = \left \lceil \frac{7-8R}{2(2R-1)} \right \rceil$.
*Case 1: $p \ne 4$*\
The term $\sum_{e_l^\dagger \in \mathfrak{B}_u(G)} r(e_l^\dagger) - s_G(p)$ is non-negative. Since $\Delta \ge 0$, $\mathcal{Y}(G) > 0$.
*Case 2: $p = 4$*\
If $\sum_{e_l^\dagger \in \mathfrak{B}_u(G)} r(e_l^\dagger) - s_{G}(p) \ge 0$, $\mathcal{Y}(G) > 0$ for $p=4$ and there exists a codeword-supporting complete 3-graph on $p = \left \lceil \frac{7-8R}{2(2R-1)} \right \rceil = 4$ nodes. Even otherwise, by Proposition \[prop:sing\], there exists a codeword supporting subgraph on $4$ nodes.
Girth condition and final proof {#sec:girth-condition}
-------------------------------
We now complete the proof of the main result by showing that the girth condition holds for suitable rates.
Consider a planar Tanner graph $G$ with minimum distance $d>2$. Let $g$ be the girth of $\mathcal{C}^\dagger (G)$. Then, $g \ge 3$. Hence, the girth condition is satisfied for $p\le 4$. \[prop:girth-condition-1\]
It is easy to see that there cannot be any loops in $\mathcal{C}^{\dagger}(G)$. We will show that multiple edges in $\mathcal{C}^{\dagger}(G)$ will result in $d=2$ to prove the proposition. Let $f_1$ and $f_2$ be two nodes in $C^{\dagger}(G)$ connected by two edges. An edge between $f_1$ and $f_2$ in the dual corresponds to a common edge between the two faces $f_1$ and $f_2$ in the $\mathcal{C}(G)$. Hence, the two edges connecting $f_1$ and $f_2$ correspond to two common edges between the faces $f_1$ and $f_2$. Since each face is of length 3 in $\mathcal{C}(G)$, both faces $f_1$ and $f_2$ have the same vertex set. Such a situation arises in the construction of $\mathcal{C}(G)$ only when two degree-3 bit nodes in $G$ have the same set of neighboring check nodes i.e. when $d=2$. Hence, when $d>2$ there are no multiple edges in $\mathcal{C}^{\dagger}(G)$.
Since $p \le 4$ results in $h(p) \le 1$, the girth condition for $p \le 4$ needs $g \ge 3$, which is satisfied for $\mathcal{C}^\dagger (G)$.
A situation with multiple edges is shown in Fig. \[fig:muledge\] where $U = \{f_1, f_2\}$.
In Fig. \[fig:muledge\], the subgraph $\mathcal{C}_U(G)$ of the check graph corresponding to multiple edges in $\mathcal{C}^\dagger(G)$ is shown. The dotted lines show the edges and nodes of $G$. We see that multiple edges in $\mathcal{C}^\dagger (G)$ result from two degree-3 bit nodes in $G$ having the same set of neighbouring check nodes, in which case $d=2$.
Observe that $p = \left\lceil \frac{7-8R}{2(2R-1)} \right\rceil \ge 0$. When $\frac{5}{8} \le R < \frac{7}{8}$, we have $1\le p \le 4$. By Proposition \[prop:girth-condition-1\] and Lemma \[lemma:3g\], there exists a codeword-supporting subgraph on $\left \lceil \frac{7-8R}{2(2R-1)} \right \rceil$ nodes. By Proposition \[prop:cu\], $$\begin{aligned}
d \le \left\lceil \frac{7-8R}{2(2R-1)} \right\rceil +3. \label{eq:dmin}\end{aligned}$$
When $R \ge \frac{7}{8}$, by Proposition \[prop:prelude\], $d' \le 3$ for the minimum distance $d'$ of the code defined by the check inverse $G'$. Since the DS and DE operations in the conversion from $G'$ to $G$ cannot decrease the sum $\sum_iw_i$ in Proposition \[prop:prelude\], the same bound holds for the minimum distance of $G$. So, we have $d\le 3= \left \lceil \frac{7-8R}{2(2R-1)} \right \rceil + 3$ for $R\ge\frac{7}{8}$. This concludes the proof of Theorem \[thm:main\], which is the main result of this paper.
A plot of the upper bound of Theorem \[thm:main\] on $d$ versus $R$ for planar codes is shown in Fig. \[fig:dvr\].
![Minimum distance versus rate for planar codes.[]{data-label="fig:dvr"}](dversusR.pdf){width="3.7in"}
The girth of $\mathcal{C}^\dagger (G)$ cannot be arbitrarily large. In fact, all $\mathcal{C}^\dagger (G)$ have girth $g \le 5$. This is because the girth of $\mathcal{C}^\dagger (G)$ is lesser than or equal to the minimum node degree in $\mathcal{C}(G)$, which is less than 6 by planarity; hence, $g$ can take a maximum value of 5.
Let $G$ be a planar Tanner graph which supports a code of rate $R \ge \frac{9}{16}$ with the girth of $\mathcal{C}^\dagger(G)$ greater than or equal to 5. Then, $$\begin{aligned}
d \le \left\lceil \frac{7-8R}{2(2R-1)} \right\rceil +3. \end{aligned}$$
Its easy to see that the girth condition is met for all $p \le 10$ (since $p=3.2^{l(p)-1} - 2 + z(p)$). With similar calculations as above, one can show that this places restriction on the rate as $R \ge \frac{9}{16}$.
Conclusion {#sec:conclusion}
==========
In this paper, we showed a bound on the minimum distance of high-rate ($\ge5/8$) codes that have planar Tanner graphs. The main result is the plot of the upper bound on minimum distance as a function of rate as shown in Fig. \[fig:dvr\]. In particular, we see that such codes have a maximum minimum distance of 7. Hence, non-planarity is essential for the construction of codes on graphs with high minimum distance.
The proof uses ideas from graph theory, coding theory and an averaging argument through a series of constructions that exploit the planarity of a Tanner graph. Ideas from the proof could be possibly employed in construction of codes on non-planar graphs in the future.
Extending the bound to codes of all rates with a planar Tanner graph is an interesting problem for future study. We conjecture that codes with planar Tanner graphs will not support codes with large minimum distance for any rate.
Ordering nodes in $\mathcal{C}_U^\dagger(G)$ {#sec:numbering}
============================================
In this appendix, we show that $\mathcal{C}_U^\dagger(G)$ for a proper subset of nodes $U$ can be recursively constructed by adding the nodes from $U$ one at a time in an order such that each newly added node has degree 1 or 2. For an ordered set $U=\{u_1,u_2\cdots,u_{|U|}\}$, let $U_i=\{u_1,u_2,\cdots,u_i\}$.
Let $\mathcal{C}_U^\dagger(G)$ be a connected subgraph of $\mathcal{C}^\dagger(G)$ induced by a proper node subset $U$. Then, there exists an ordering of the nodes in $U$, given by $U=\{u_1,u_2\cdots,u_{|U|}\}$, such that the degree of the node $u_i$ in $\mathcal{C}_{U_i}^\dagger(G)$ is either 1 or 2, and $\mathcal{C}_{U_i}^\dagger(G)$ is connected for $1\leq i\leq |U|$.
To prove the proposition, we first claim the following:
*Claim:* Let $\mathcal{C}_V^\dagger(G)$ be a subgraph of $\mathcal{C}^\dagger(G)$ induced by a proper subset $V$ of nodes. Then there exists either a degree-1 node in $\mathcal{C}_V^\dagger(G)$ or a degree-2 node that is not a cut-vertex of $\mathcal{C}_V^\dagger(G)$.
*Proof of claim:* We see that $\mathcal{C}_V^\dagger(G)$ has at least one node $v$ of degree $\le 2$, since $\mathcal{C}_V^\dagger(G)$ is a proper subgraph of $\mathcal{C}^\dagger(G)$. If $v$ is a degree-1 or a degree-2 non-cut vertex, we are done. Otherwise, if $v$ is a degree-2 cut-vertex node, let $V_1$ and $V_2$ be the vertex sets of the two components of $\mathcal{C}_V^\dagger(G) - v$.
If all the nodes in $V_1$ are of degree 3 in $\mathcal{C}_V^\dagger(G)$, then the edge joining $v$ and $V_1$ will be a cut edge in $\mathcal{C}^\dagger(G)$. A cut edge in $\mathcal{C}^\dagger(G)$ implies a loop in $\mathcal{C}(G)$ [@harary], which is not possible by construction. By similar arguments for $V_2$, we see that there is a node in $V_1$ and a node in $V_2$ with degree $\le 2$ in $\mathcal{C}_V^\dagger(G)$. Let $v_1$ be a node in $V_1$ of degree $\le 2$ in $\mathcal{C}_V^\dagger(G)$. If the node $v_1$ is of degree 1 or if it is a degree-2 non-cut vertex, we are done. Otherwise, proceed with $v_1$ in place of $v$ and $\mathcal{C}_{V_1}^\dagger(G)$ in place of $\mathcal{C}_V^\dagger(G)$. Since the vertex set is finite, we are guaranteed to find a degree-1 or a degree-2 non-cut vertex proceeding to smaller components.
We are now ready to prove the proposition.
In $\mathcal{C}_U^\dagger(G)$, let $u_{|U|}$ be a degree-1 or a degree-2 non-cut vertex. For $2\leq i\leq |U|-1$, let $u_i$ be a degree-1 or a degree-2 non-cut vertex in $\mathcal{C}_U^\dagger(G) - u_{|U|} - u_{|U|-1} \cdots - u_{i+1}$. Observe that $\mathcal{C}_U^\dagger(G) - u_{|U|} - u_{|U|-1} \cdots - u_{i+1}$ is connected. The required ordering is then given by $\{u_1,u_2,\cdots,u_{|U|}\}$.
Computation of $\eta_2$ {#sec:eta_2}
=======================
Let $G'$ be the check inverse of a planar Tanner graph $G$. Let $\eta_2$ be defined as in (\[eta\]). Then, $$\begin{aligned}
\eta_2 = |V'^b_2|(\frac {4} {3} p + \frac {2}{3})t(p) - \sum_{e_l^\dagger \in \mathfrak{B}_o(G')} r(e_l^\dagger)\end{aligned}$$ \[prop:q\_deg2\]
provided the girth condition is satisfied for $\mathcal{C}^\dagger(G)$.
A degree-two bit node $v_b$ is identified with an edge $e$ in $\mathcal{C}(G')$. Since $e$ is common to two faces, say $f_1$ and $f_2$, $v_b$ is counted whenever either or both the corresponding nodes (with same notation $f_1$ and $f_2$ as in Section \[sec:dual\] ) in $\mathcal{C}^\dagger(G')$ is in the vertex set of complete 3-graphs on $p$ vertices. Let $e^\dagger$ be the edge connecting $f_1$ and $f_2$ in $\mathcal{C}^\dagger(G')$ corresponding to $e$ i.e, the edge that is occupied by $v_b$. The situation is depicted in Fig. \[fig:eta2\].
Let $T_1$ and $T_2$ be the set of complete 3-graphs containing $f_1$ and $f_2$, respectively. Then, $$q(v^b) = |T_1 \bigcup T_2| = |T_1| + |T_2| - |T_1 \bigcap T_2|.$$ Since $|T_1|=|T_2|=pt(p)$ (assuming the girth condition), we are left with computing $|T_1 \bigcap T_2|$. Note that $T_1 \bigcap T_2$ is the set of complete 3-graphs that contain both $f_1$ and $f_2$ i.e, containing $e^\dagger$ as shown in Fig. \[fig:eta2\]. We will now count the number of 3-trees that will contain $e^{\dagger}$.
1. There are two branches rooted at $f_1$ and two at $f_2$ as seen in Fig. \[fig:eta2\]. The number of nodes up to level $l(p)-2$ from the respective roots in these four branches is $4(2^{l(p)-2}-1)$. Every complete 3-tree rooted at any of these nodes will contain $e^{\dagger}$.
2. The number of possible nodes in level $l(p)-1$ in the four branches is $4(2^{l(p)-2})$. Among the complete 3-trees rooted at these nodes, a fraction $\dfrac{z(p)}{3\cdot2^{l(p)-1}}$ will contain $e^{\dagger}$.
3. Finally, all complete 3-trees rooted at $f_1$ and $f_2$ will contain $e^{\dagger}$.
Hence, the total number of complete 3-trees that contain $e^{\dagger}$ is given by $$\left(4(2^{l(p)-2}-1)+4(2^{l(p)-2})\dfrac{z(p)}{3\cdot2^{l(p)-1}}+2\right)t(p).$$ In addition, there will be $r(e^{\dagger})$ complete 3-graphs that contain $e^{\dagger}$ as a cycle-creating edge. Therefore, $$\begin{aligned}
|T_1 \bigcap T_2| &= \left ( 2^{l(p)} + \frac{2}{3}z(p) - 2 \right )t(p) + r(e^\dagger), \nonumber\\
&=(\frac{2}{3}p - \frac{2}{3})t(p) + r(e^\dagger). \label{intersection}\end{aligned}$$ Hence, $q(v_b)$ for $v_b \in V_2^{'b}$ is $$q(v_b) = (\frac {4}{3} p + \frac {2}{3})t(p) - r(e^\dagger).$$ Letting $$\theta(p) = (\frac {4}{3} p + \frac {2}{3})t(p), \label{eq:theta}$$ we have $$\begin{aligned}
\eta_2&=&\sum_{v_b\in V_2^{'b}}\theta(p)-\sum_{e_l^\dagger\in\mathfrak{B}_o(G')}r(e^\dagger)\\
&=&|V'^b_2|\theta(p) - \sum_{e_l^\dagger \in \mathfrak{B}_o(G')} r(e_l^\dagger),\end{aligned}$$
Computation of $\eta_1$ {#sec:eta_1}
=======================
Let $G$ be a planar Tanner graph. Let $s_G$ be the number of singular nodes in $G$. Then, provided the girth condition on $\mathcal{C}^\dagger(G)$ is satisfied, $$\eta_1 \ge |V'^b_1|(\frac {4} {3} p + \frac {2}{3})t(p) - s_G(p),$$ where $\eta_1$ is defined in (\[eta\]) and $$s_G(p) = \begin{cases}
s_G, \; p = 4,\\
0, \;\text{else}.
\end{cases}$$ \[prop:eta2\]
A degree-1 bit node $v_b$ in $G$ is connected to one check node, and is identified with that node in $\mathcal{C}(G)$. Every node in $\mathcal{C}(G)$ maps to a face in $\mathcal{C}^\dagger(G)$. Let $V^\dagger (v_b)$ be the set of vertices of $\mathcal{C}^\dagger(G)$ forming the face corresponding to $\mathcal{N}(v_b)$. Then $v_b$ contributes to the summation of weights whenever one or more of the vertices in $V^\dagger (v_b)$ is in the vertex set of complete 3-graphs.
For $1\le p\le4$, the girth of $\mathcal{C}^\dagger(G)$ is at least 3. Two possible situations with girth 3 are shown in Fig. \[fig:eta1\].
Table \[tab:qvb\] enumerates $q(v_b)$ for $p=1,2,3,4$ for the two cases in Fig. \[fig:eta1\] and compares with $\theta(p)=(\frac {4} {3} p + \frac {2}{3})t(p)$.
$p$ Case from Fig. \[fig:eta1\] $q(v_b)$ $\theta(p)$
----- ----------------------------- ---------- -------------
1 (a),(b) 3 2
2 (a),(b) 12 10
3 (a) 15 14
3 (b) 14 14
4 (a) 6 6
4 (b) 5 6
: Enumeration of worst-case $q(v_b)$.[]{data-label="tab:qvb"}
From Table \[tab:qvb\], we see that the only case when $q(v_b)<\theta(p)$ is for $p=4$ in the situation of Case (b). Let $U=\{f_1,f_2,f_3,f_4\}$ in Fig. \[fig:eta1\]. Then for the subgraph $\mathcal{C}^\dagger_U(G)$, the corresponding check graph $\mathcal{C}_U(G)$ is shown in Fig. \[fig:sing\_check\] as dashed lines. Hence, by the definition in Section \[sec:singular-nodes\], $v_b$ is singular. For $p\ge5$, we have girth at least 5 according to the girth condition. Hence, the face of $\mathcal{C}^{\dagger}(G)$ corresponding to $v_b$ has at least five nodes connected to other neighbours. The values of $q(v_b)$ in these cases can be readily seen to be much larger than $\theta(p)$ in a similar fashion.
Therefore, we see that $$\eta_1 \ge |V'^b_1|(\frac {4} {3} p + \frac {2}{3})t(p)- s_G(p),$$ where $$s_G(p) = \begin{cases}
s_G, \;p = 4,\\
0, \;\text{else}.
\end{cases}$$
Computation of $\alpha_x$ {#sec:gx}
=========================
Let $G$ be a planar Tanner graph with a fixed embedding. Let $\mathcal{Y}(G)$ be defined as in (\[eq:Ygen\]). Let $\alpha_x$ be the increase in $\mathcal{Y}(G)$ when a new bit node of degree $x+3$ is embedded in $G$ such that the resulting graph is still planar. Then, $$\alpha_1 \ge \left (\frac{2}{3}p - \frac{2}{3}\right )t(p),$$ and $$\alpha_2 \ge \left (\frac{1}{2}p - \frac{1}{6}z(p) - 1 \right)t(p).$$ where $p$ is such that girth condition is satisfied for $\mathcal{C}^{\dagger}(G)$. \[prop:gx\]
When $x=1$, a degree-3 bit node $v_b$ becomes a degree-4 bit node and $v_b$ is identified with two faces of the check graph having a common edge. This is equivalent to saying that $v_b$ is identified with an edge in the dual. So, $\alpha_1$ is equal to the number of complete 3-graphs that contain the edge. This number is already derived in (\[intersection\]), and we get $$\alpha_1 \ge \left (\frac{2}{3}p - \frac{2}{3}\right )t(p).$$ When $x=2$, a degree-3 bit node $v_b$ becomes a degree-5 bit node as shown in Fig. \[fig:g2\]. The node $v_b$ is identified with three faces $f_1$, $f_2$ and $f_3$ of the check graph $\mathcal{C}(G)$. This corresponds to the connected subgraph $\mathcal{C}_U^\dagger(G)$ with $U=\{f_1,f_2,f_3\}$ in the dual as shown in Fig. \[fig:g2\].
So, $\alpha_2$ is equal to the number of complete 3-graphs which contain $\mathcal{C}_U^\dagger(G)$. Let us now compute $\alpha_2$. All the complete 3-graphs rooted at nodes at a depth of $l(p) -2$ or lesser from the node $f_2$ in the branches shown in Fig. \[fig:g2\] contain $\mathcal{C}_U^\dagger(G)$. Since girth condition is satisfied, there are $(3.2^{l(p)-2} -2)t(p)$ such complete 3-graphs. Now, consider only the child nodes of $f_1$ and $f_3$ in the branches shown in Fig. \[fig:g2\] that are at a depth of $l(p)-1$ from $f_2$. Since girth condition is satisfied, the number of such nodes is $2\cdot2^{l(p)-2}$ and there are $\frac{z(p)}{3\cdot2^{l(p)-1}}t(p)$ realizations of complete 3-graphs rooted at each such node containing $\mathcal{C}_U^\dagger(G)$. Thus, $$\begin{aligned}
\alpha_2 & \ge \left( 3.2^{l(p)-2} - 2 + 2.2^{l(p)-2} \frac{z(p)}{3.2^{l(p)-1}} \right )t(p), \\
%& ~~~+ \frac{1}{3} 3.2^{l(p)-2} \frac{z(p). (z(p)-1)}{(3.2^{l(p)-2})(3.2^{l(p)-2}) -1)}\\
& \ge \left (\frac{1}{2}p - \frac{1}{6}z(p) - 1 \right)t(p).\end{aligned}$$
|
---
abstract: 'Let $\psi$ be a conformal map of the unit disk $\mathbb{D}$ onto an unbounded domain and, for $\alpha >0$, let ${F_\alpha }=\left\{ {z \in \mathbb{D}:\left| {\psi \left( z \right)} \right| = \alpha } \right\}$. If ${H^p}\left( \mathbb{D} \right)$ denotes the classical Hardy space and $d_\mathbb{D} {\left( {0,{F_\alpha }} \right)}$ denotes the hyperbolic distance between $0$ and $F_\alpha$ in $\mathbb{D}$, we prove that $\psi$ belongs to ${H^p}\left( \mathbb{D} \right)$ if and only if $$\int_0^{ + \infty } {{\alpha ^{p - 1}}{e^{ - {d_{\mathbb{D}}}\left( {0,{F_\alpha }} \right)}}d\alpha } < + \infty .$$ This result answers a question posed by P. Poggi-Corradini.'
address: 'Department of Mathematics, Aristotle University of Thessaloniki, 54124, Thessaloniki, Greece'
author:
- Christina Karafyllia
title: Hyperbolic distance and membership of conformal maps in the Hardy space
---
[^1]
Introduction {#section1}
============
A classical problem in geometric function theory is to find the Hardy number of a region by looking at its geometric properties (see e.g. [@Ba], [@Ha] and [@Ra]). Answering a question of P. Poggi-Corradini ([@Co p. 36]), we give a necessary and sufficient integral condition for whether a conformal map of $\mathbb{D}$ belongs to ${H^p}\left( \mathbb{D} \right)$ by studying the hyperbolic metric in its image region. For a domain $D$, a point $z \in D$ and a Borel subset $E$ of $\overline D $, let ${\omega _D}\left( {z,E} \right)$ denote the harmonic measure at $z$ of $\overline E$ with respect to the component of $D \backslash \overline{E}$ containing $z$. The function ${\omega _D}\left( { \cdot ,E} \right)$ is exactly the solution of the generalized Dirichlet problem with boundary data $\varphi = {1_E}$ (see [@Ahl ch. 3] and [@Gar ch. 1]). The hyperbolic distance between two points $z,w$ in the unit disk $\mathbb{D}$ (see [@Ahl ch. 1], [@Bea p. 11-28]) is defined by $${d_\mathbb{D}}\left( {z,w} \right) = \log \frac{{1 + \left| {\frac{{z - w}}{{1 - z\bar w}}} \right|}}{{1 - \left| {\frac{{z - w}}{{1 - z\bar w}}} \right|}}$$ and the Green function for $\mathbb{D}$ (see [@Gar p. 41]) is directly related to the hyperbolic distance, as it is obvious by its definition $${g_\mathbb{D}}\left( {z,w } \right) = \log \left| {\frac{{1 - z\bar w }}{{z - w }}} \right|= \log \frac{{{e^{{d_\mathbb{D}}\left( {z,w } \right)}} + 1}}{{{e^{{d_\mathbb{D}}\left( {z,w} \right)}} - 1}}.$$ They are both conformally invariant and thus they can be also defined on any simply connected domain $D \ne \mathbb{C}$ by considering a conformal map $f$ of $\mathbb{D}$ onto $D$. Then ${d_D}\left( {z,w} \right) = {d_\mathbb{D}}\left( {{f^{ - 1}}\left( z \right),{f^{ - 1}}\left( w \right)} \right)$ and ${g_D}\left( {z,w} \right) = {g_\mathbb{D}}\left( {{f^{ - 1}}\left( z \right),{f^{ - 1}}\left( w \right)} \right)$ for every $z,w \in D$. Therefore, for $z,w \in D$,
$$\label{gree}
{g_D}\left( {z,w } \right) = \log \frac{{{e^{{d_D}\left( {z,w} \right)}} + 1}}{{{e^{{d_D}\left( {z,w} \right)}} - 1}}.$$
Also, for a set $E \subset D$, we define ${d_D}\left( {z,E} \right): = \inf \left\{ {{d_D}\left( {z,w} \right):w \in E} \right\}$.
The Hardy space with exponent $p$, $0<p<+\infty$, and norm ${\left\| \cdot \right\|_p}$ (see [@Du p. 1-2], [@Gar p. 435-441]) is defined to be $${H^p}\left( \mathbb{D} \right) = \left\{ {f \in H\left( \mathbb{D} \right):\left\| f \right\|_p^p = \mathop {\sup }\limits_{0 < r < 1} \int_0^{2\pi } {{{\left| {f\left( {r{e^{i\theta }}} \right)} \right|}^p}d\theta < + \infty } } \right\},$$ where $H\left( \mathbb{D} \right)$ denotes the family of all holomorphic functions on $\mathbb{D}$. S. Yamashita [@Ya] proved that a function $f \in H\left( \mathbb{D} \right)$ belongs to ${H^p}\left( \mathbb{D} \right)$, $0<p<+\infty$, if and only if $$\int_\mathbb{D} {{{\left| {f\left( z \right)} \right|}^{p - 2}}{{\left| {f'\left( z \right)} \right|}^2}\log \frac{1}{{\left| z \right|}}dA\left( z \right)} < + \infty,$$ where $dA$ is the Lebesgue measure on $\mathbb{D}$. The fact that a function $f$ belongs to ${H^p}\left( \mathbb{D} \right)$ imposes a restriction on the growth of $f$ and this restriction is stronger as $p$ increases. If $f$ is a conformal map on $\mathbb{D}$, then $f \in {H^p}\left( \mathbb{D} \right)$ for all $p<1/2$ ([@Du p. 50]).
Harmonic measure and hyperbolic distance are both conformally invariant and several Euclidean estimates are known for them. Thus, expressing the ${H^p}\left( \mathbb{D} \right)$-norms of a conformal map $\psi$ on $\mathbb{D}$ in terms of harmonic measure and hyperbolic distance, we are able to obtain information about the growth of the function by looking at the geometry of its image region $\psi \left( {\mathbb{D}} \right)$. Indeed, if $\psi $ is a conformal map on $\mathbb{D}$ and ${F_\alpha } = \left\{ {z \in \mathbb{D}:\left| {\psi \left( z \right)} \right| = \alpha } \right\}$ for $\alpha >0$, then P. Poggi-Corradini proved (see [@Co p. 33]) that $$\label{isod}
\psi \in {H^p}\left( \mathbb{D} \right) \Leftrightarrow \int_0^{ + \infty } {{\alpha ^{p - 1}}{\omega _{\mathbb{D}}}\left( {0,{F_\alpha }} \right)d\alpha } < + \infty.$$
He also proved that the Beurling-Nevanlinna projection theorem (see [@Ahl p. 43-44], [@Co p. 9-10, 35]) implies that
$$\label{1.1}
{e^{ - {d_\mathbb{D}}\left( {0,{F_\alpha }} \right)}}\le \frac{\pi}{2 }{\omega _\mathbb{D}}\left( {0,{F_\alpha }} \right) .$$
This observation led him to state two questions (see [@Co p. 36]):
\[quest\] Let $\psi $ be a conformal map of $\mathbb{D}$ onto an unbounded domain and, for $\alpha >0$, let ${F_\alpha } = \left\{ {z \in \mathbb{D}:\left| {\psi \left( z \right)} \right| = \alpha } \right\}$. Does there exist a positive constant $K$ such that for every $\alpha >0$, $${\omega _\mathbb{D}}\left( {0,{F_\alpha }} \right) \le K{e^{ - {d_\mathbb{D}}\left( {0,{F_\alpha }} \right)}}?$$
\[que\] More generally, is it true that $$\psi \in {H^p}\left( {\mathbb{D}} \right) \Leftrightarrow \int_0^{ + \infty } {{\alpha ^{p - 1}}{e^{ - {d_{\mathbb{D}}}\left( {0,{F_\alpha }} \right)}}d\alpha } < + \infty ?$$
We gave a negative answer to the first one in [@Ka] and the following theorem provides a positive answer to the second question.
\[theo\] Let $\psi $ be a conformal map of $\mathbb{D}$ onto an unbounded simply connected domain $D$ and ${F_\alpha } = \left\{ {z \in \mathbb{D}:\left| {\psi \left( z \right)} \right| = \alpha } \right\}$ for $\alpha >0$. If $0<p<+\infty$ then $$\psi \in {H^p}\left( {\mathbb{D}} \right) \Leftrightarrow \int_0^{ + \infty } {{\alpha ^{p - 1}}{e^{ - {d_{\mathbb{D}}}\left( {0,{F_\alpha }} \right)}}d\alpha } < + \infty .$$
Proof of Theorem \[theo\] {#section}
=========================
Suppose that $\psi \in {H^p}\left( {\mathbb{D}} \right)$ for some $0<p<+\infty$. This in conjunction with (\[isod\]) and (\[1.1\]) implies that $$\int_0^{ + \infty } {{\alpha ^{p - 1}}{e^{ - {d_{\mathbb{D}}}\left( {0,{F_\alpha }} \right)}}d\alpha } \le \frac{\pi }{2}\int_0^{ + \infty } {{\alpha ^{p - 1}}{\omega _{\mathbb{D}}}\left( {0,{F_\alpha }} \right)d\alpha } < + \infty.$$ Conversely, suppose that for some $0<p<+\infty$,
$$\label{hy}
\int_0^{ + \infty } {{\alpha ^{p - 1}}{e^{ - {d_{\mathbb{D}}}\left( {0,{F_\alpha }} \right)}}d\alpha } < + \infty.$$
Without loss of generality, we set $\psi \left( 0 \right) = 0$. Let $dA$ denote the Lebesgue measure on $\mathbb{D}$. For the Green function for $D$, set ${g_D}\left( {0,w} \right) = 0$ for $w \notin D$. By a change of variable and the conformal invariance of the Green function,
$$\begin{aligned}
\label{rel}
\int_\mathbb{D} {{{\left| {\psi \left( z \right)} \right|}^{p - 2}}{{\left| {\psi '\left( z \right)} \right|}^2}\log \frac{1}{{\left| z \right|}}dA\left( z \right)} &=&\int_\mathbb{D} {{{\left| {\psi \left( z \right)} \right|}^{p - 2}}{{\left| {\psi '\left( z \right)} \right|}^2}{g_\mathbb{D}}\left( {0,z} \right)dA\left( z \right)} \nonumber \\
&=&\int_D {{{\left| w \right|}^{p - 2}}{g_\mathbb{D}}\left( {0,{\psi ^{ - 1}}\left( w \right)} \right)dA\left( w \right)} \nonumber \\
&=& \int_D {{{\left| w \right|}^{p - 2}}{g_D}\left( {0,w } \right)dA\left( w \right)} \nonumber \\
&=& \int_0^{ + \infty } {\int_0^{2\pi } {{\alpha ^{p - 2}}{g_D}\left( {0,\alpha {e^{i\theta }}} \right)\alpha d\theta d\alpha } } \nonumber \\
&=&\int_0^{ + \infty } {{\alpha ^{p - 1}}\left( {\int_0^{2\pi } {{g_D}\left( {0,\alpha {e^{i\theta }}} \right)d\theta } } \right)d\alpha }.\end{aligned}$$
Applying elementary calculus, it is easily proved that there exist a positive constant $C$ and a point $x_0>0$ such that $$\label{cal}
\log \frac{{{e^x} + 1}}{{{e^x} - 1}} \le C {e^{-x}}$$ for every $x \ge {x_0}$. Note that for $D$ unbounded and simply connected, ${d_D}\left( {0,\psi \left( {{F_\alpha }} \right)} \right) \to + \infty $ as $\alpha \to + \infty$ which also follows from the hypothesis (\[hy\]). Therefore, by (\[cal\]) and (\[gree\]), we deduce that there exists an $\alpha_0>0$ such that for every $\alpha \ge \alpha_0$,
$${g_D}\left( {0,\alpha {e^{i\theta }}} \right) \le C{e^{ - {d_D}\left( {0,\alpha {e^{i\theta }}} \right)}} \le C{e^{ - {d_D}\left( {0,\psi \left( {{F_\alpha }} \right)} \right)}} = C{e^{ - {d_\mathbb{D}}\left( {0,{F_\alpha }} \right)}}.$$ Integrating with respect to $\theta$, we get
$$\label{in}
\int_0^{2\pi } {{g_D}\left( {0,\alpha {e^{i\theta }}} \right)d\theta } \le C\int_0^{2\pi } {{e^{ - {d_\mathbb{D}}\left( {0,{F_\alpha }} \right)}}d\theta } = 2\pi C{e^{ - {d_\mathbb{D}}\left( {0,{F_\alpha }} \right)}}$$
for every $\alpha \ge \alpha_0$. So, by (\[rel\]) and (\[in\]), we infer that
$$\label{fi}
\int_\mathbb{D} {{{\left| {\psi \left( z \right)} \right|}^{p - 2}}{{\left| {\psi '\left( z \right)} \right|}^2}\log \frac{1}{{\left| z \right|}}dA\left( z \right)} \le 2\pi C\int_{{\alpha _0}}^{ + \infty } {{\alpha ^{p - 1}}{e^{ - {d_\mathbb{D}}\left( {0,{F_\alpha }} \right)}}d\alpha }+ C',$$
where
$$C':=\int_0^{ \alpha_0 } {{\alpha ^{p - 1}}\left( {\int_0^{2\pi } {{g_D}\left( {0,\alpha {e^{i\theta }}} \right)d\theta } } \right)d\alpha }.$$ Finally, (\[hy\]) and (\[fi\]) give
$$\int_\mathbb{D} {{{\left| {\psi \left( z \right)} \right|}^{p - 2}}{{\left| {\psi '\left( z \right)} \right|}^2}\log \frac{1}{{\left| z \right|}}dA\left( z \right)}<+\infty$$ and thus by [@Ya] we conclude that $\psi \in {H^p\left( \mathbb{D} \right)}$.
[^1]: I thank Professor D. Betsakos, my thesis advisor, for his advice during the preparation of this work, and the Onassis Foundation for the scholarship I receive during my Ph.D. studies.
|
---
abstract: 'The Cu-doped topological insulator Bi$_2$Se$_3$ has recently been found to undergo a superconducting transition upon cooling, raising the possibilities that it is the first known “topological superconductor" or realizes a novel non-Abelian superconducting state. Its true nature depends critically on the bulk and surface state band topology. We present the first photoemission spectroscopy results where by examining the band topology at many different copper doping values we discover that the topologically protected spin-helical surface states remain well protected and separate from bulk Dirac bands at the Fermi level where Copper pairing occurs in the optimally doped topological insulator. The addition of copper is found to result in nonlinear electron doping and strong renormalization of the topological surface states. These highly unusual observations strongly suggest that superconductivity on the topological surface of Cu$_x$Bi$_2$Se$_3$ cannot be of any conventional type in account of the general topological theory. Characteristics of the three dimensional bulk Dirac band structure are reported for the first time with respect to the superconducting doping state and topological invariant properties which should help formulate a specific theory for this novel superconductor.'
author:
- 'L. Wray'
- 'S. Xu'
- 'J. Xiong'
- 'Y. Xia'
- 'D. Qian'
- 'H. Lin'
- 'A. Bansil'
- 'Y. Hor'
- 'R. Cava'
- 'M.Z. Hasan'
title: 'Observation of unconventional band topology in a superconducting doped topological insulator, Cu$_x$Bi$_2$Se$_3$: Topological or non-Abelian superconductor?'
---
Topological insulators embody a new state of matter characterized by topological invariants of the band structure rather than spontaneously broken symmetry, and feature massless Dirac-like conduction states on their surfaces [@moore1; @TIbasic; @MooreAndBal; @DavidNat1]. Bismuth selenide in particular has been found to be an ideal “hydrogen atom" topological insulator, realizing the simplest known case of topologically nontrivial band structure [@MatthewNatPhys; @DavidScience; @DavidTunable; @ZhangPred; @BiTeSbTe; @HorPtype; @ZhangFilm]. It has been proposed that inducing a superconducting gap in the surface states of topological insulators will lead to fault tolerant non-Abelian surface physics with potential application in spintronics and quantum computing [@FuSCproximity]. In this Letter, we use angle resolved photoemission spectroscopy (ARPES) to examine the effect of copper doping on the electron dynamics of Cu$_x$Bi$_2$Se$_3$, which has been shown to lead to a bulk superconducting transition at x$\geq$0.1 (max T$_C$=3.8$^o$ K) [@HorSC]. Band structure in the normal state superconductor is found to preserve a minimalist scenario for a topological metal, consisting of a single bulk band with nearly isotropic massive Dirac-like dispersion and a well defined, topologically protected surface Dirac cone. As a result of this band structure, we find that superconductivity at the sample surface *cannot be conventional*. Copper doping is systematically observed to have complex effects on the system, adding a small number of electrons and causing a strong renormalization of the surface bands in such a way as to preserve the topological character. The doped compound exhibits hexagonal dispersion anisotropy, with important implications for low energy interactions and potential device development [@FuHexagonal].
Undoped Bi$_2$Se$_3$ is a topological insulator with a large band gap ($>$300 meV) [@MatthewNatPhys], and belongs to a class of materials M$_2$X$_3$ (M=Bi,Sb; X=S,Se,Te) that includes at least two other topologically nontrivial materials, Bi$_2$Te$_3$ and Sb$_2$Te$_3$, with smaller band gaps and more complicated band structure [@ZhangPred; @BiTeSbTe]. These materials share a rhombohedral crystal structure, with a five atom unit cell arranged in quintuple layers, and have been investigated extensively in connection to thermoelectric applications [@DiSalvo]. Unlike band structure in topologically trivial materials, which is commonly more parabolic (“classical" E=$\frac{p^2}{2M}$), it is most natural for the bulk conduction bands of topological insulators to realize Dirac-like dispersion following an analogue of Einstein’s equation for the energy of a relativistic particle (E$^2$=M$^2$v$_C^4$+p$^2$v$_C^2$) with a critical velocity “v$_C$" analogous to the speed of light. The addition of electrons from copper doping allows us to quantitatively evaluate the bulk Dirac-like band character in the relativistic regime.
Angle resolved photoemission spectroscopy (ARPES) measurements were performed at the Advanced Light Source beamline 10.0.1 using 35.5-48 eV photons and Stanford Synchrotron Radiation Laboratory (7-22eV photons) with better than 15 meV energy resolution and overall angular resolution better than 1$\%$ of the Brillouin zone (BZ). Samples were cleaved and measured at 15$^o$K, in a vacuum maintained below 8$\times$10$^{-11}$ Torr. Momentum along the z-axis is determined using an inner potential of 9.5 eV, consistent with previous ARPES investigations of undoped Bi$_2$Se$_3$ [@MatthewNatPhys].
Surface and bulk state band calculations were performed for comparison with the experimental data, using the LAPW method implemented in the WIEN2K package [@wien2k]. Details of the calculation are identical to those described in Ref. [@MatthewNatPhys].
To better understand the highly nonlinear doping effect of copper, we present data on several copper-added Cu$_x$Bi$_2$Se$_3$ crystals (x=0, 0.01, 0.05, 0.12) and copper substituted Cu$_{0.1}$Bi$_{1.9}$Se$_3$, grown as described in Ref. [@HorSC]. Copper atoms intercalated between the van der Waals bonded selenium planes are thought to be single electron donors, while substitutional defects in which copper replaces bismuth in the lattice (Cu$_{Bi}$) are expected to each contribute two holes to the system [@CuAmphoteric]. Introducing x=0.12 copper doping for optimal superconductivity has been found to shift the z-axis lattice parameter by only 1.5$\%$ while leaving the in-plane lattice parameters and long-range crystalline order intact [@HorSC].
The qualitative effect of copper addition is an enlargement of the Fermi surface from electron doping and, surprisingly, a strong renormalization of the surface state. The surface state dispersion is flattened by 30$\%$ when copper doping of x=0.05 is added to the stoichiometric compound, and the Fermi surface becomes hexagonally anisotropic in a way consistent with numerical predictions in the local density approximation (LDA). Tunability of the surface state anisotropy is important for control of unconventional ordered states that may appear uniquely in topologically ordered materials [@FuHexagonal].
The bulk and surface-derived band structure of the normal state optimally doped superconductor (x=0.12) are explored in Fig-3. The bottom of the conduction band for electrons in the bulk is found at the three dimensional $\Gamma$-point (k$_z$=k$_x$=k$_y$=0), and can be seen inside the upper surface state (SS) cone. Due to the renormalization effect and less-than-linear electron doping, the occupied surface state band structure does not intersect with the bulk conduction band structure at any point in momentum space, making the surface state “well defined" and topologically protected. The bulk conduction band is only clearly visible when low photon energies (h$\nu$$<$20eV) are used to increase bulk penetration, probably due to a screening effect related to excess negative charge carriers in the surface state. (see analysis in Fig-2(d)) The gap between bulk valence and conduction bands appears to be unchanged upon copper doping.
Using 9.75eV photons to view the $\Gamma$-$\overline{M}$ and $\Gamma$-$\overline{K}$ directions shows Fermi momenta of 0.110$\pm$3 $\AA^{-1}$ and 0.106$\pm$3 $\AA^{-1}$ respectively. Varying incident energy to observe dispersion along the $\hat{z}$ axis ($\Gamma$-Z direction) reveals a Fermi momentum of 0.12 $\AA^{-1}$, suggesting that the bulk electron kinetics are three dimensionally isotropic. Carefully tracing the band (Fig-3(d-e)) yields a Fermi velocity of 3.5 eV$\times\AA$ along $\Gamma$-$\overline{M}$ and 4.1 eV$\times\AA$ along $\Gamma$-$\overline{K}$, estimated within 50 meV of the Fermi level. Assuming that the bulk conduction band forms a Fermi sea for superconductivity, the superconducting correlation length can be estimated based on the average Fermi velocity and superconducting critical temperature to be about $2000\AA$ ($\xi_0$$\sim0.2\times\hbar v_F/K_BT_C=0.2\times3.8eV\AA/(K_B\times3.8^oK)=2000\AA$), large enough for phase fluctuations to be neglected in the neighborhood of T$_C$. This is a typical coherence length for conventional superconducting materials, and much greater than that seen in other unconventional superconductors, such as strongly correlated cuprates ($\xi_0$$\sim100-200\AA$) or cobaltates ($\xi_0$$\sim200\AA$) [@SCproperties].
Along both the $\Gamma$-$\overline{M}$ and $\Gamma$-$\overline{K}$ directions, dispersion of the bulk conduction band appears to approach a limiting velocity at large momentum, rather than following a parabolic arc. This behavior, like the “massless" linear dispersion of the surface state bands, is mathematically analogous to the energy-momentum relationship of a relativistic particle approaching the speed of light. Classical (paraboloic, m=0.155 m$_e$) and relativistic Dirac-like (m=0.155 m$_e$, v$_c$=6 eV$\cdot\AA$) energy dispersions are plotted as fits for the bulk band in Fig-3(e). We find that a relativistic fit can more accurately reproduce the observed dispersion, however neither fit can account for a slight bend in the dispersion centered near 90 meV, which may suggest strong electron-phonon interactions in the system consistent with phonon-mediated superconductivity. The simplicity of band structure in the doped Bi$_2$Se$_3$ system clearly sets it apart from other known topological insulators (e.g. Bi$_2$Te$_3$) that have far greater deviation from pure Dirac-like kinetics [@DavidTunable; @BiTeSbTe; @FuHexagonal; @ChenBiTe].
The weakness of copper doping is critical to preserving topological order in the superconducting state, by keeping the Fermi level beneath the point of intersection between bulk and surface conduction bands. (near the top of the diagram in Fig-4(a,right)) The number of conducting charge carriers at the surface and in the bulk are estimated using the Luttinger count in Fig-2(d), by dividing the surface and bulk Fermi surface areas by the total area of the Brillouin zone. Although the doping increases monotonically as copper is added, the carrier density in the bulk is only $\sim$1/30th of what would be expected if all copper were intercalated between paired selenium layers. It is therefor likely that nearly one third of the added copper enters the sample through substitutional defects with bismuth, adding holes that counterbalance most of the electron doping from intercalation.
Attempting to force the creation of Cu$_{Bi}$ defects by adding less bismuth results in very weak hole doping for Cu$_{0.1}$Bi$_{1.9}$Se$_3$. In this case, the binding energy of the Dirac point decreases by more than 100 meV relative to the undoped compound, raising the bulk conduction band entirely above the Fermi level so that the material is a traditional topological insulator. An explanation for why hole doping is not stronger in this case may be that defects in which selenium fills bismuth vacancies will result in the addition of electrons.
The surface state renormalization that we have observed is also instrumental in separating the bulk and surface band structure. Renormalization is likely to result from a combination of increased carrier density on the surface relative to the bulk, evident from Fig-2(d), and relaxation of interlayer bond lengths near the cleaved surface. Changes to interatomic bonding lengths at the surface of strongly spin-orbit coupled systems can have a significant effect on surface state dispersion [@BiDistortion], and could be effected by the surface carrier density and presence of copper. We find that increasing the distance between the outermost bismuth and selenium layers by 0.2 $\AA$ can account for the difference in dispersion between undoped and superconducting doped crystals (Fig-2(c)).
A cartoon of band structure in a topological metal (insulator) is overlaid on ARPES data in Fig-4(a), illustrating that the gapped bulk bands are connected by a singly degenerate gapless surface state. When a bulk superconducting transition is introduced, there are two likely scenarios. If the parity eigenvalue of the bulk superconducting state is even (Fig-4(b,d)), electrons in the surface state will participate in bulk superconductivity through the proximity effect, resulting in the appearance of non-Abelian vortex states of great interest for quantum computing [@FuSCproximity]. If parity is odd, the system will be a topological superconductor [@FuNew], with new gapless states appearing beneath the T$_C$. ARPES measurements cannot resolve the bulk superconducting gap, which is expected to be only $\sim$0.6 meV from BCS theory (3.5$\times$K$_B$T$_C$/2=0.6 meV).
Although our results show that Bi$_2$Se$_3$ is a nearly ideal minimalist topological insulating system, many of the most interesting physical properties emerge due to its deviations from the simplest case of perfectly isotropic Dirac-like electron kinetics. Recent theoretical explorations have suggested that magnetic perturbations will not readily open a mass gap in the surface state of a topological insulator with perfectly linear dispersion [@ImpurityStates; @TopoFieldTheory], but can do so when the bands have some upward concavity as in Bi$_2$Se$_3$ [@FerroPosMassBiSe]. The hexagonal Fermi surface anisotropy demonstrated upon copper doping adds an out of plane component to spin polarization, and is a likely precursor to novel two dimensional ordered states unique to the surfaces of materials with topological order [@FuHexagonal]. Furthermore, our observation of large surface charge contained within the topologically protected surface state and a dramatic band renormalization effect have important implications for manipulation of topological insulator systems in future experiments and for technological applications.
In summary, we have investigated the surface and bulk band topology of the superconducting doped topological insulator Cu$_x$Bi$_2$Se$_3$. We discover that the copper-doped superconducting state unexpectedly preserves the band structure and topological order of undoped bismuth selenide, and that superconductivity at the sample surface cannot be conventional. The bulk conduction band conforms approximately to a massive Dirac dispersion, and has a slight bend near 90 meV that may be indicative of electron-boson coupling. We also observe strong doping dependent renormalization of surface states and hexagonal anisotropy that preserve the topological order via a $\pi$ Berry’s phase invariant and are of interest with respect to spin transport properties and potential device development.
We thank helpful conversation with L. Fu, B. A. Bernevig and P.W. Anderson.
\[
For an introduction to topological insulators, see, J.E. Moore, Nature Physics **5**, 378 (2009); S.-C. Zhang, Physics **1**, 6 (2008). L. Fu, C. L. Kane, and E. J. Mele, Phys. Rev. Lett. **98** 106803 (2007). J. E. Moore and L. Balents, Phys. Rev. B **75** 121306(R) (2007). D. Hsieh et al., Nature **452**, 970 (2008). Y. Xia *et al.*, Nature Physics **5**, 398 (2009). D. Hsieh *et al.*, Science **323**, 919 (2009). D. Hsieh *et al.* Nature (London) **460**, 1101 (2009). H. Zhang et al., Nature Phys. **5**, 438 (2009). D. Hsieh *et al.*, Phys. Rev. Lett. **103**, 146401 (2009). Y. S. Hor *et al.*, Phys. Rev. B **79**, 195208 (2009). Y. Zhang *et al.*, arXiv:0911.3706v2 (2009).
L. Fu and C. L. Kane, Phys. Rev. Lett. **100**, 096407 (2008). Liang Fu, arXiv:0908.1418 (2009). F.J. DiSalvo, Science **285**, 703 (1999).
P. Blaha *et al.*, computer code WIEN2K (Vienna University of Technology, Vienna, 2001). Y. S. Hor *et al.*, arXiv:0909.2890 (2009). A. Vasko, Appl. Phys. **5**, 217 (1974).
E.W. Carlson *et al.*, in *The Physics of Conventional and Unconventional Superconductors*, edited by K. H. Bennemann and J. B. Ketterson (Springer-Verlag, Berlin, 2002); A. Damascelli *et al.*, Rev. Mod. Phys. **75**, 473 (2003); A. Kaminski *et al.*, Phys. Rev. B **71**, 014517 (2005).
Y.L. Chen et al., Science **325**, 178 (2009). G. Bihlmayer, S. Blügel, and E. V. Chulkov, Phys. Rev. B **75**, 195414 (2007).
L. Fu and E. Berg, unpublished.
R. R. Biswas and A. V. Balatsky, arXiv:0910.460 (2009). X.-L. Qi, T. L. Hughes and S.-C. Zhang, Phys. Rev. B **78**, 195424 (2008). S.-Q. Shen, arXiv:0909.4125 (2009).
|
---
abstract: 'The two-dimensional topological insulating phase has been experimentally discovered in HgTe quantum wells (QWs). The low-energy physics of two-dimensional topological insulators (TIs) is described by the Bernevig-Hughes-Zhang (BHZ) model, where the realization of a topological or a normal insulating phase depends on the Dirac mass being negative or positive, respectively. We solve the BHZ model for a mass domain configuration, analyzing the effects on the edge modes of a finite Dirac mass in the normal insulating region (soft-wall boundary condition). We show that at a boundary between a TI and a normal insulator (NI), the Dirac point of the edge states appearing at the interface strongly depends on the ratio between the Dirac masses in the two regions. We also consider the case of multiple boundaries such as NI/TI/NI, TI/NI/TI and NI/TI/NI/TI.'
author:
- 'P. Michetti'
- 'P. H. Penteado'
- 'J. C. Egues'
- 'P. Recher'
title: Helical edge states in multiple topological mass domains
---
Introduction
============
Topological insulators (TIs) are time-reversal-symmetric materials featuring a topological phase characterized by a $\mathbb Z_2$ topological invariant [@kane2005a; @Moore2007]. In two-dimensions (2D), they exhibit the quantum spin Hall (QSH) phase [@kane2005a; @kane2005b]. The QSH phase has been theoretically predicted [@bernevig2006] and experimentally realized in HgTe/CdTe QWs [@konig2007]. The crucial ingredient of this narrow gap semiconductor material is the inverted band structure of HgTe. Similarly, 3D TIs supporting chiral fermions as surface states have been proposed and observed [@fuinversion2007; @hsieh2008; @zhang2009; @hsieh2009; @xia2009; @chen2009].
In HgTe/CdTe QWs, the topological phase is determined by the sign of the Dirac mass $M$. The gap between the E1 (s-like) and the H1 (p-like) subbands at the $\Gamma$ point is given by $2|M|$. The only experimentally accessible parameter tuning the Dirac mass from normal ($M>0$) to inverted ($M<0$) is the thickness of the HgTe QW. In particular, a topological transition from the normal to the topological insulating phase takes place when the QW thickness is increased above the critical thickness $t_C=6.3$ nm [@konig2007]. Recently, electrically driven topological insulating phase transitions have been proposed in heterostructures with gate tunable conduction-valence band energy separation. In particular, in Ref. a type-II InAs/GaSb/AlSb QW was proposed and recent experiments [@knez; @knez2] provided the first evidence pointing towards the presence of a topological insulating phase in these structures. In Ref. , double QW structures composed of narrow gap semiconductors are considered featuring a tunable topological transition with the application of a gate bias of the order of the gap of the individual trivial QWs. Both proposals pave the way to 2D systems where mass domains are designed with lithographic gates to create topological and normal regions. For example, ring-shaped TI regions are particularly interesting for the peculiar properties of their confined edge states, which can be controlled with a threading magnetic flux [@Maciejko2010; @Michetti2011]. On the other hand, the Dirac mass term in single HgTe QWs is related to the QW thickness and therefore thickness fluctuations of less than $1$ nm height can accidentally determine the formation of mass domains alternating TI regions, where $M<0$, to NI regions, where $M>0$.[@footnote1] These phenomena can be especially relevant for near zero-gap HgTe QWs [@Buttner2011].
![(Color online) Schemes of multiple topological mass domains analyzed in the present paper. According to the classification of the topological phase of each region into a TI or a NI we have $M<0$ or $M'>0$ and $M''>0$, respectively. []{data-label="fig:domains"}](fig/domains){width="7.5cm"}
The change in the topological invariant $\mathbb Z_2$ between two systems determines the presence of 1D helical edge states running along the boundary between the TI and the NI regions, a phenomenon referred to as the bulk-boundary correspondence. Such edge states are topologically protected against single particle elastic backscattering (as long as time reversal symmetry \[TRS\] is preserved) and are particularly interesting for their spin and charge transport properties. In the literature, these edge states are generally obtained by solving the Hamiltonian with *hard*-wall boundary conditions (BCs), i.e. by imposing that the wave function *vanishes* at the interface between a TI and a normal medium (an exception is the recent proposal of *natural* boundary conditions [@Medhi2012]). While this condition is appropriate to treat the interface between TIs and the $M\rightarrow+\infty$ vacuum, this is by no means a good approximation when dealing with electrically-induced mass domains [@liu2008; @michetti2012], where the normal regions have a finite positive Dirac mass. Finite mass-domains in Dirac systems have first been solved in the context of zero energy bound states in the 1+1 Dirac equation by Jackiw and Rebbi [@Jackiw76] and for interface states in band-inverting contacts based on HgCdTe and PbSnTe [@Volkov85; @Pankratov87]. More recently, finite mass-domains were proposed to induce valley-polarized metallic states in biased bilayer graphene [@Martin2008].
In the present paper we solve the edge states of a HgTe/CdTe QW-based TI for the case of *soft*-wall BCs, appropriate for describing systems with Dirac mass domains, where the wave function does *not* vanish at the interface, but its continuity and the continuity of its normal derivative are instead required. In Section II, we briefly review the BHZ model and describe the method used to solve the edge states in the topological mass domains. In Section III, we deal with a single NI/TI interface schematized in Fig. \[fig:domains\](a), with both hard-wall and soft-wall BCs. We show that soft-wall BCs quantitatively change the dispersion curves \[Fig.2(a)\] with respect to the case of hard-wall BCs. In particular, the hard-wall limit is only reproduced in the limit $M'\rightarrow\infty$. Even for $M'=10^4$ meV, which is of the order of magnitude of the electron extraction work-function of a crystal, an appreciable deviation from the hard-wall limit is still observed. The density profile of the edge state bound to the mass-domains (Fig. 3) is qualitatively affected by the soft-wall BCs, which allow it to extend into both the TI and the NI regions. In Section IV, we address the effect of a finite bulk inversion asymmetry (BIA) term, which introduces off-diagonal matrix elements to the otherwise block diagonal form of the BHZ model, and evaluate its effects on the bulk dispersion curves and on the edge state dispersion (Fig. \[fig:BIA\]). The effect of BIA on the edge states of a single NI/TI is shown to be tiny, so that BIA can be safely neglected in the more complex case of multiple NI/TI boundaries. In Section V, we analyze a system with two TI/NI interfaces. In particular, we investigate the case of a TI strip embedded in a normal system, sketched in Fig. \[fig:domains\](b), and vice versa: the case of a TI system where a strip region with $M>0$ is present \[Fig. \[fig:domains\](c)\] and analyze for the first time the edge coupling through the normal region. In both cases, the overlap of edge states bound to different boundaries leads to a fully gapped edge mode spectrum (Fig. \[fig:sw1\] and Fig. \[fig:lateral\]), with a minigap exponentially shrinking with the distance between the two interfaces.
Soft-wall BCs can also be used to couple three or more edge states, thus, in Section VI, we analyze the edge states for a system with three NI/TI boundaries \[Fig. \[fig:domains\](d)\]. We study how the properties of this system vary with the Dirac mass and geometrical parameters. We further argue that this system can describe a helical edge state at the sample boundary (vacuum/TI interface) in the presence of Dirac mass fluctuations in the TI composition giving rise to mass domains with bubbles having $M>0$ (i.e. normal character) in the bulk of the sample whose edge states could interact with the helical edge states at the sample boundary.
The BHZ model
=============
The spectrum of a HgTe QW near the $\Gamma$-point is effectively described in its low energy sector [@schmidt2009] by the 4-band model [@bernevig2006] $$\begin{aligned}
H_{\vec k}&=& \left(\begin{array}{cc}
h({\vec{k}}) &0\\
0 & h^*(-\vec{k})
\end{array}\right)\nonumber\\
h(\vec k)&=& \vec{d}\cdot\vec{\sigma}\label{eq:H0}\\
\vec{d}&=& \left( \varepsilon_k, A k_x , -A k_y , M_k \right)\nonumber\\
\varepsilon_k &=&C-D k^2 \hspace{1cm} M_k= M-B k^2, \nonumber
\end{aligned}$$ where $k=|\vec k|=\sqrt{k_x^2+k_y^2}$ and $\vec{\sigma}$ is the vector of Pauli matrices associated with the band-pseudospin degree of freedom (band $E_1$ or $H_1$) [@footnote2]. $H_{\vec{k}}$ is represented in the basis $\big\{|E_1+\rangle$, $|H_1+\rangle$, $|E_1-\rangle$, $|H_1-\rangle\big\}$, where the $E_1$ states ($J_z=\pm1/2$) are a mixture of the s-like $\Gamma_6$ band with the $\Gamma_8$ light-hole band, while $H_1$ ($J_z=\pm 3/2$) is basically the $\Gamma_8$ heavy-hole band. For later use in numerical simulations, we quote the following choice of parameters: $A=375$ meV nm, $B=-1120$ meV ${\rm nm}^2$ and $D=-730$ meV ${\rm nm}^2$. These parameters follow from the 8x8 Kane model [@novik2005]. Without loss of generality we also assume $C=0$. The Dirac mass $M$ depends on the QW thickness and $M<0$ corresponds to the inverted (QSH) regime whereas $M>0$ corresponds to the normal regime. In a first approximation, $H_{\vec k}$ is block diagonal in the spin degree of freedom [@bernevig2006], for which we define the corresponding vector of Pauli matrices $\vec{\tau}$. As we consider only systems with TRS, we can restrict ourselves to the block $h({\vec{k}})$. Results can be extended to the other Kramers block $h^{*}(-\vec{k})$ which is related to $h(\vec{k})$ by the time reversal operation $\hat{T}=i \tau_y \sigma_0 \hat{K}$, where $\hat{K}$ is the operator of complex conjugation.
The bulk dispersion curves obtained as the eigenvalues of Eq. (\[eq:H0\]) are described by $$E_\pm(\vec k) = \varepsilon_k \pm \sqrt{(M - B k^2)^2 + A^2k^2}.
\label{eq:bulk}$$ With a standard choice for the TI parameters (like the parameters stated above) the bulk dispersion relation displays a conduction band minimum (valence band maximum) at $k=0$ with energy $E=M$ (or $E=-M$). However, depending on the values of the parameters in Eq. (\[eq:H0\]), the bulk dispersion curves can also show a “Mexican hat” behavior. For a detailed analysis of the behavior of the 4-band model as a function of its parameters, see Appendix \[app1\] and Ref. .
Boundary conditions \[procedure\]
---------------------------------
We are interested in obtaining the eigenstates of Eq. (\[eq:H0\]) in real space for a semi-infinite geometry, invariant under translations along the $x$-axis. For such a system, $k_y$ is no longer a conserved quantity and should be replaced by the operator $-i\partial_y$. Compatible with a fixed energy $E$ and a real $k_x$, the secular equation $|h({\vec k})-E|=0$ provides four $k_y$-modes: $$k_y^2 \equiv k_\pm^2 = -k_x^2 -F \pm \sqrt{F^2-Q^2}
\label{modi}$$ with $$\begin{aligned}
F &=& \frac{A^2 -2(B M+DE)}{2(B^2-D^2)},\\
Q^2 &=& \frac{M^2-E^2}{B^2-D^2}.
\end{aligned}$$ With our choice of parameters, $k_\pm$ has one imaginary and one real solution for $E$ within the energy range of the bulk bands, whereas in the bandgap $\left(-|M|,|M| \right)$ both values for $k_y$ are imaginary.
For each $k_y$-mode, one can write the spinors satisfying the Schrödinger equation and corresponding to the Kramers blocks $\tau=\pm 1$ as $$\psi_{k_x, k_y,\tau}(x,y) = \frac{e^{i k_x x}}{\sqrt{L_x}}
\left(\begin{array}{l}
e^{i k_y y}\\
R_{\tau, k_y} e^{ i k_y y}
\end{array}
\right).
\label{eq:spinor0}$$ The ratio between the two components is $$R_{ \tau, k_y} = - \frac{A (\tau k_x - i k_y)}{-M - E - (D-B) (k_x^2+k_y^2)}.
\label{eq:erre}$$ The general solution of the Dirac equation with energy $E$ and wave vector $k_x$ is therefore given by a linear combination of the four solutions $k_y=\lambda k_\mu$ (with $\lambda,\mu=\pm$) obtained from Eq. (\[modi\]), $$\Psi_{k_x, \tau}^{(n)}(x,y) =
\sum_{\lambda, \mu=\pm}
c_{\lambda,\mu}^{(n)}
\psi_{k_x,\lambda k_\mu, \tau}(x,y),
\label{eq:spinor}$$ where we have introduced the index $n$ to refer, in what follows, to the $n$-th mass domain region.
Here we discuss the general procedure we use to solve the BHZ model in a system composed by $N$ mass domains with parallel boundaries at $y=y_n$ with $n=0, 1, \dots N$. Inside the $n$-th mass domain with the condition $y \in \left(y_{n-1},y_n\right)$, we consider the Dirac mass term (and all other parameters) as constant and a general expression for the spinor is given by Eq. (\[eq:spinor\]). At $y=y_n$ the value of the Dirac mass $M$ changes step-like [@adiabatic].
Hard-wall BCs at $y$ for the domain $n$ are expressed by $$\Psi_{k_x, \tau}^{(n)}(x,y)=0 \hspace{0.5cm} \forall x,\label{hard}$$ meaning that the edge state cannot extend beyond the boundary, being subject to a hard-wall confinement. Soft-wall BCs between two consecutive domains $n-1$ and $n$ are instead expressed by the continuity of the spinor and its normal derivative $$\begin{aligned}
\Psi_{k_x, \tau}^{(n-1)}(x,y_n)= \Psi_{k_x, \tau}^{(n)}(x,y_n)\hspace{0.5cm} \forall x,\nonumber\\
\partial_y \Psi_{k_x, \tau}^{(n-1)}(x,y_n)= \partial_y \Psi_{k_x, \tau}^{(n)}(x,y_n)\hspace{0.5cm} \forall x. \label{soft}
\end{aligned}$$
A system of BCs (either soft or hard) can be always expressed in a compact form as $$\mathbb{M}_{k_x}(E)~\vec{c}=0,
\label{boundary_c}$$ where $\vec{c}$ is a vector containing all the free coefficients $c_{\lambda,\mu}^{(n)}$ characterizing the wave function \[Eq. (\[eq:spinor\])\] in the $n=1,\dots N$ mass domain. $\mathbb{M}_{k_x}(E)$ can be constructed by appropriately using either Eq. (\[hard\]) or Eq. (\[soft\]) at each one of the boundaries of the system. As in standard quantum mechanics, the BCs determine the eigenenergies $E_{k_x}$ through the secular equation $$\det{\left[\mathbb{M}_{k_x}(E)\right]}=0.
\label{det}$$ The corresponding eigenspinors we obtain by solving Eq. (\[boundary\_c\]) for the coefficients $\vec{c}$.
An isolated boundary
====================
In this section, we consider a single interface between a TI ($y>0$) with $M<0$ and a NI ($y<0$) with $M'>0$ \[see Fig. \[fig:domains\](a)\] and calculate the resulting helical edge modes. In the limit $M'\rightarrow +\infty$, we recover the usual hard-wall BCs of Eq.(\[hard\]). For a finite positive $M'$, soft-wall BCs of Eqs. (\[soft\]) are employed.
Hard-wall boundary conditions
-----------------------------
We apply vanishing BCs at $y=0$ and search for modes with energy lying within the bandgap. Only two of the four $k_y$ solutions of Eq. (\[modi\]) with positive imaginary part, which we define as $\tilde{k}_\pm$, are normalizable in the region $y>0$ and contribute to the edge states. When $k_\pm$ are purely imaginary, we define $\tilde{k}_\pm = k_\pm$. When the parameters are such that $k_\pm$ are complex (see in Appendix \[app1\] Eq. \[complex\]), we instead have $\tilde{k}_\pm = \pm k_\pm$. Using Eqs. (\[hard\]) and (\[det\]), we obtain the relation $$\begin{aligned}
R_{\tau, \tilde{k}_+}&=&R_{\tau, \tilde{k}_-},
\label{eq:come}
\end{aligned}$$ that imposes a strict relation between the energy $E$ and momentum $k_x$. Isolating $k_x$ terms and squaring twice, we arrive after some algebra at the edge mode dispersion curves [@zhou2008; @wada2011] $$\begin{aligned}
E_{k_x} &=& -\frac{D}{B}M \pm k_x A\sqrt{\frac{B^2-D^2}{B^2}}.
\label{solution}
\end{aligned}$$ First of all, we observe that after the first squaring of Eq. (\[eq:come\]), we lose track of $\tau$, therefore only one of the $\pm$ signs in Eq. (\[solution\]) is actually a solution of Eq. (\[eq:come\]) for a given $\tau$. More important, because of the second squaring, the solutions in Eq. (\[solution\]) are not always allowed. For the usual parameter choice describing TIs, solutions are always admissible with $M<0$ [@zhou2008]. For $M>0$, the system admits no edge modes.
![(Color online) Probability density of the edge state for $k_x=0$ as a function of the distance from the NI/TI interface ($y=0$). The different curves represent the hard-wall (solid line) and soft-wall (dashed lines) boundary condition cases with $M'=10$, $100$ meV for a fixed TI mass $M=-10$ meV. The probability densities for the two blocks $\tau=\pm 1$ are degenerate. []{data-label="fig:onespinor"}](fig/SWone_edge_spinor){width="7cm"}
Soft-wall boundary conditions
-----------------------------
As mentioned previously, soft-wall BCs imply that at the interface ($y=0$) the spinor and its normal derivative are both continuous \[see Eq. (\[soft\])\]. For the $M'>0$ domain, i.e., $y<0$, only the $k_y$ modes of Eq. (\[modi\]) with negative imaginary part are allowed, while for $y>0$ possible solutions contain the modes with Im$\left(k_y\right)>0$. Using Eqs. (\[soft\]) and (\[boundary\_c\]) we obtain the $4\times4$ matrix $$\mathbb{M}_{E,k_x}=\left(
\begin{array}{cccc}
1 & 1 & - 1 & - 1 \\
R_{\tau, -\tilde{k}_+} & R_{\tau, -\tilde{k}_-} & -R'_{\tau,\tilde{k}_+} & -R'_{\tau, \tilde{k}_-} \\
\tilde{k}_+ & \tilde{k}_- & \tilde{k'}_+ & \tilde{k'}_- \\
\tilde{k}_+ R_{\tau, -\tilde{k}_+} & \tilde{k}_- R_{\tau, -\tilde{k}_-} & \tilde{k}_+ R'_{\tau,\tilde{k}_+} & \tilde{k}_- R'_{\tau, \tilde{k}_-} \\
\end{array}
\right),
\label{eq:2edges}$$ where the prime stands for both $\tilde{k}_{\pm}$ and $R_{\tau, \tilde{k}_{\pm}}$ calculated in the $M'$ domain. By numerically solving $\det{[\mathbb{M}_{E,k_x}]}=0$, we determine the energy dispersion relation of the helical edge states. In Fig. \[fig:energy\] we show the energy dispersions $E_{k_x}$ for a NI/TI interface keeping the TI Dirac mass $M=-10$ meV and varying the NI mass $M'$. Solid and dashed lines correspond to helical edge states of the spin-blocks $\tau$ and $-\tau$, respectively. The slope of the curves (velocity $v_x$) is not altered by varying $M'$. However, as shown in Fig. \[fig:Dirac\], the Dirac point rises with increasing $M'$ from $E_{k_x=0}=0$ (its limiting value for $M'\rightarrow0$) and eventually saturates at the value $E_{k_x=0}=-M\frac{D}{B}$ \[see Eq. (\[solution\])\] for $M'\rightarrow\infty$, reproducing the hard-wall case. In the inset of Fig. \[fig:Dirac\], we plot the decay length in the TI and NI regions as a function of $M'$.
Figure \[fig:onespinor\] displays the corresponding probability densities $|\Psi_{k_x,\tau}\left(x,y\right)|^2$ of edge states at the Dirac point for soft-wall BCs with $M'=10$, $100$ meV (dashed lines) and hard-wall BCs (solid line). In both cases, the wave functions are strongly peaked closely to the interface $y=0$ and exponentially decaying away from the interface. The characteristic decay length of the edge states are given by the inverse of the smaller $k_y$ mode calculated from Eq. (\[modi\]) compatible with their eigenenergies and $k_x$ values.
BIA effects \[BIA\]
===================
In the present section we address how bulk inversion asymmetry (BIA) affects the TI Hamiltonian in Eq. (\[eq:H0\]) and the edge mode solutions for hard-wall boundary conditions using the model of Ref. . We show that such BIA introduces a weak non-linearity in the edge dispersion, especially near the bulk band edges. The position of the Dirac point is not affected while the velocity of the helical particle is only slightly modified. We therefore will not consider the effect of BIA in the sections that follow.
![(Color online) Bulk-energy spectrum at ${\vec k}=0$ including the BIA-Hamiltonian Eq. (\[eq:BIA\]) as a function of Dirac mass parameter $M$. It shows an anticrossing with gap $2|\Delta|$.[]{data-label="fig:anticrossing"}](fig/BIA_anticrossing){width="6.5cm"}
The block diagonal form of the BHZ model is indeed an approximation for the low-energy physics of a HgTe QW. The presence of BIA introduces a coupling between $|E1,\pm\rangle$ and $|H1,\mp\rangle$ bands. The leading-order BIA perturbation term is expressed as [@konig2008] $$H_{BIA} = \left(
\begin{array}{cccc}
& & &-\Delta \\
& &\Delta & \\
&\Delta&& \\
-\Delta&&&
\end{array}
\right).
\label{eq:BIA}$$ It preserves TRS and therefore does not affect the topological properties of the BHZ model Eq. (\[eq:H0\]) [@konig2008], as long as the bulk gap is not closed. Note that Eq. (\[eq:BIA\]) introduces an anticrossing at ${\vec k}=0$ as a function of $M$, see Fig. \[fig:anticrossing\]. Such an anticrossing has been also found in Ref. as a function of the QW thickness and the inversion crossing of the $E_1$- and $H_1$-bands shifting to finite ${\vec k}$-values. Here we show (see Fig. \[fig:BIA\]) explicitly, that the helical edge states are still present in the model considered and only slightly modified, despite the anticrossing at ${\vec k}=0$.
Let us first rewrite the BHZ Hamiltonian in Eq. (\[eq:H0\]), including the BIA term in Eq. (\[eq:BIA\]), in the following form $$H = \varepsilon_k I_{4\times 4} + A \vec{k} \vec{\Sigma} + \Delta \Lambda_y - M_k \Lambda_z \Sigma_z,
\label{eq:BIABHZ}$$ where we have introduced two sets of unitary and Hermitian matrices $$\begin{aligned}
\Sigma_x = \tau_z \sigma_x ;\hspace{0.5cm} \Sigma_y = -\tau_0 \sigma_y;\hspace{0.5cm} \Sigma_z = -\tau_z \sigma_z \nonumber\\
\Lambda_x = \tau_x \sigma_y ;\hspace{0.5cm} \Lambda_y = \tau_y \sigma_y;\hspace{0.5cm} \Lambda_z = \tau_z \sigma_0
\label{eq:matrices}\end{aligned}$$ with the property that each set separately obeys Pauli commutation rules, while elements from the two sets commute. Note that the only matrices which are off-diagonal in the Kramers block pseudospin (i.e. containing $\tau_x$ or $\tau_y$) are $\Lambda_x$ and $\Lambda_y$. We now perform the following unitary transformation in the $\Lambda$-$\Sigma$ space which warrants Eq. (\[eq:BIABHZ\]) block-diagonal: $$U = \frac{1}{\sqrt{2}} \left[ -i(\zeta_y \Sigma_x -\zeta_x \Sigma_y ) \Lambda_y + \Sigma_z \Lambda_z \right]
\label{eq:U}$$ with $\vec{\zeta}=\vec{k}/|\vec{k}|$. After the transformation the Hamiltonian in Eq. (\[eq:BIABHZ\]) acquires the following form $$\begin{aligned}
H_{\vec k} &=& \varepsilon_k I_{4\times 4} + \left(-A |\vec{k}| I_{4\times 4} + \Delta \Lambda_z \right) \vec{\zeta} \vec{\Sigma} - M_k \Lambda_z \Sigma_z\nonumber\\
&=&\left(
\begin{array}{cc}
h_{+}(k) & 0\\
0 & h^*_{-}(-k)
\end{array}
\right),
\label{eq:BHZ}\end{aligned}$$ where we have introduced the helicity parameter $\eta=\pm 1$, and have defined $$\begin{aligned}
h_{\eta}(k) &=& \varepsilon_k I_{2\times 2} + \left(-A |\vec{k}| + \eta \Delta \right) \left(\zeta_x \sigma_x -\zeta_y \sigma_y \right) + M_k \sigma_z.\nonumber\\\end{aligned}$$ The helicity of the energy eigenstates is defined in the new basis by $\eta =\langle\tau_z\rangle$, which in the original basis of Eq. (\[eq:BIABHZ\]) is equivalent to $\eta =-\langle \left(\Sigma_x \zeta_x + \Sigma_y \zeta_y \right) \Lambda_y \rangle$.
![(Color online) Bulk bands dispersions including the effect of BIA terms, where full and dashed lines distinguish opposite helicities. Edge states for a single hard-wall boundary, numerically obtained by solving Eq. (\[eq:BIABHZ\]) are shown in dotted lines. The edge states dispersions for the corresponding system without BIA, obtained with Eq. (\[solution\]), are also displayed for comparison (full narrow line). We used the following parameters: $M=-2$ meV, $\Delta=1$ meV. []{data-label="fig:BIA"}](fig/bia){width="6.8cm"}
The bulk dispersion curves obtained from Eq. (\[eq:BHZ\]) are shown in Fig. \[fig:BIA\]. Similarly to an electronic system in the presence of the Rashba spin-orbit interaction, the dispersion curves can be classified through the helicity $\eta$. For a given wave vector, the effect of the BIA term is to lift the degeneracy of the two spin-blocks.
In order to solve for the edge states of the system with BIA, we need to treat Eq. (\[eq:BIABHZ\]) in real space \[we note that Eqs. (\[eq:U\]) and therefore Eq. (\[eq:BHZ\]) are well defined only in momentum space\]. We follow the procedure illustrated in Section \[procedure\], applying it to the $4\times4$ Hamiltonian in Eq. (\[eq:BIABHZ\]). In Fig. \[fig:BIA\], we show the edge states obtained for a system with $M=-2$ meV and $\Delta=1$ meV (dotted lines) and compare them with the edge states of the corresponding system with no BIA (full narrow lines), obtained analytically with Eq. (\[solution\]). These BIA terms do not change the position of the Dirac point, but slightly change the group velocity close to the Dirac point. Away from the Dirac point the edge dispersion in Fig. \[fig:BIA\] shows a weak non-linear distortion, accentuated near the bulk band edges. Projecting $H_{BIA}$ onto the unperturbed edge states, it is straightforward to show that the effects of $H_{BIA}$ are at least of order $\Delta^2$, see Ref. .
![(Color online) Energy dispersion of a TI strip (NI/TI/NI) calculated with hard-wall boundary conditions. The solid red lines correspond to $L=100$ nm, blue dashed curves to $L=200$ nm and black thin lines to $L=1000$ nm. []{data-label="fig:hw"}](fig/stripHW){width="6.3cm"}
Systems with two boundaries
===========================
A strip of TI
-------------
Here, we consider a NI/TI/NI mass domain shown in Fig. \[fig:domains\](b), where Dirac masses are $M'>0$, $M<0$ and $M''>0$, respectively. For very large $M'$ and $M''$ the use of hard-wall BCs is appropriate. For the entire subsection, the Dirac mass for the TI domain is $M=-10$ meV.
### Hard-wall boundary conditions
The case of a TI strip confined by hard-wall BCs has been first analyzed by Zhou et al. [@zhou2008]. We briefly comment in this section some of their results in order to set a benchmark for successive extension to soft-wall BCs. The TI strip has two pairs of helical edge states (it is not topologically protected) exponentially localized at the two boundaries which are separated by the width of the TI strip $L$. The decrease of the TI strip width leads to a finite overlap of edge modes belonging to different interfaces originating a minigap (a full gap in the edge mode dispersion curves) as shown in Fig. \[fig:hw\]. For $L=1000$ nm (thin full lines), the overlap is negligible, the minigap is exponentially suppressed. The dispersion curves are linear, just two copies of single-interface edge modes shown in Fig. \[fig:energy\] for $M'\rightarrow\infty$. For $L=200$ and $100$ nm, the overlap is instead substantial and the edge modes anticross at $k_x=0$ giving rise to a finite minigap.
![(Color online) Logarithmic plot of the minigap value as a function of the width $L$ of the TI strip for soft-wall boundary conditions (solid lines) with $M'=M''=10$, $50$, $100$ and $350$ meV. The dashed curve corresponds to the hard-wall boundary condition. []{data-label="fig:sw1"}](fig/gapxL){width="5.5cm"}
![(Color online) Electronic density of the edge mode of a TI strip of width $L=100$ nm at $k_x=0$. Considered NI Dirac masses are $M'=M''=10$, $100$ meV and $\infty$ (hard-wall confinement).[]{data-label="fig:2edgesspinor"}](fig/spinors2sw){width="6.5cm"}
### Symmetric soft-wall boundary conditions
We now consider $M'=M''$ to be finite, adopt soft-wall BCs and solve for the edge modes. The edge dispersion curves are qualitatively similar to the hard-wall case in Fig. \[fig:hw\], and an anticrossing behavior is found at $k_x=0$, with the opening of a minigap around the Dirac point of the corresponding one-boundary edge modes. We note that such an anticrossing point scales in energy with $M'$, similarly to the behavior of the Dirac point with soft-wall BCs shown in Fig. \[fig:Dirac\]. We plot in Fig. \[fig:sw1\] the minigap’s exponential decay as a function of the TI strip width $L$. The dashed line corresponds to the hard-wall case, and the soft-wall cases with finite Dirac masses of $350$, $100$ , $50$ and $10$ meV are also shown. With soft-wall BCs, the minigap is smaller and subject to have a faster decay, whose origin can be easily understood by looking at the edge states profile in Fig. \[fig:2edgesspinor\]. Here, we plot the profile of the electronic density of the $k_x=0$ edge modes in the direction perpendicular to the boundaries, for $L=100$ nm and for different NI masses ranging from $10$ meV to $\infty$ (hard-wall case). As one increases $M'$ the edge modes become more strongly confined into the TI strip, thus leading to an enhancement of the overlap between the edge modes at the two interfaces (signaled by the increase of the probability at $y=0$).
### Asymmetric confinement
We consider now NI/TI/NI mass domains which differ from the previous ones for the lack of the mirror symmetry at the center of the TI strip. This is the case whenever $M'\ne M''$. In particular we will focus on the prototypical situation of a hybrid confinement, when one interface is treated with hard-wall BCs and the other with soft-wall BCs. In Fig. \[fig:hybridmodes\], we present the dispersion curve for a TI strip with hybrid BCs, where $M''=\infty$ and for $M'=10$, $100$ and $1000$ meV. Due to the lack of the mirror symmetry the edge dispersion curves display minima at finite $k_x$. the effect of increasing $M'$ is to partially compensate the imbalance between the two boundaries ($M''=\infty$). As a consequence the minima shift towards $k_x=0$ and the center of the gap tends to the hard-wall Dirac point value in Fig. \[fig:Dirac\]. The gap is also increasing because of the stricter confinement from $M'$. In Fig. \[fig:hybridspinor\], we plot the edge modes profile for the case $M'=100$ meV of Fig. \[fig:hybridmodes\], corresponding to the points a, b, c and d.
![(Color online) Edge modes of a TI strip with hybrid confinement: $M''\rightarrow+\infty$ and $M'=10$, $100$, $1000$ meV for a TI region of width $L=100$ nm. The full and dashed lines distinguish $\tau=\pm 1$. []{data-label="fig:hybridmodes"}](fig/disp1sw1hw){width="6.8cm"}
![(Color online) Hybrid case – one hard-wall ($M''\rightarrow+\infty$) and one soft-wall ($M'=100$ meV) with $L=100$ nm. The panels (a)-(d) correspond to states labeled by a-d in Fig. \[fig:hybridmodes\].[]{data-label="fig:hybridspinor"}](fig/spinors1sw1hw){width="6.8cm"}
Laterally coupled TI edge states
--------------------------------
We consider also the possibility to laterally couple edge modes in a TI/NI/TI mass domain, where the overlap of the edge modes takes place in the central NI region \[see Fig. \[fig:domains\](c)\]. For concreteness, we assume TI regions with equal Dirac masses of $M=-10$ meV, while the NI mass is $M'>0$. This situation leads to qualitatively similar edge modes as in a TI strip with soft-wall confinement, with the opening of a minigap at $k_x=0$ as shown in Fig \[fig:lateral\](a). However, from a quantitative point of view, edge modes decay differently in the NI region, according to their characteristic penetration lengths \[see the inset in Fig. \[fig:Dirac\]\]. In Fig. \[fig:lateral\](b), we display the minigap value which exponentially shrinks as a function of the the NI mass $M'$.
![(Color online) In (a), edge modes for a TI/NI/TI mass domain with width of the central NI region of $L=100$ nm and NI mass of $M'=10$ and $20$ meV are shown. In (b), we present the value of the minigap as a function of $M'$. []{data-label="fig:lateral"}](fig/disp2lateral){width="6.8cm"}
Three mass domain system
========================
In this section we analyze the edge states for a mass domain NI/TI/NI/TI with three boundaries \[see Fig. \[fig:domains\](d)\] with Dirac masses $M''>0$, $M$, $M'>0$ and $M$, respectively. For simplicity we keep the same Dirac mass for the TI regions $M=-10$ meV. The three-boundary system can only be realized if $M'$ is finite, while $M''$ can be either finite or $\infty$ leading to soft-wall or hard-wall BCs at the first NI/TI interface. The edge dispersion curves are quite complex and are more easily understood by first considering the system assuming uncoupled edge states at each of the TI/NI interfaces. Without coupling, at each TI/NI interface we expect linear edge modes similar to that in Fig. \[fig:energy\], where the energy value of the Dirac point depends on the difference of the absolute value of the masses between NI and TI as described by Fig. \[fig:Dirac\]. When edge states belonging to different TI/NI interfaces overlap a minigap is formed due to anticrossing of the dispersions. The anticrossing takes place at $k_x=0$ if the two edge modes have equal Dirac point values (i.e. the two TI/NI interfaces share the same parameters), otherwise the anticrossing happens at finite a $k_x$.
\], calculated for a width of the first TI region $L=100$, $70$ and $50$ nm. The width of the second NI region is $d=70$ nm with Dirac mass $M'=10$ meV, while the first NI/TI interface is treated with hard-wall BCs. The second TI region is semi-infinite. Full and dashed lines distinguish $\tau=\pm 1$. []{data-label="fig:3edges3"}](fig/disp2sw1hw_L){width="7cm"}
We focus our analysis on a system where the first NI has a very large bandgap (e.g. the vacuum) and send $M''\rightarrow\infty$ (hard-wall BCs). We define $L$ as the width of the first TI domain (the second TI domain is considered semi-infinite) and $d$ the width of the second NI region (the one with mass $M'$). We note that this situation is qualitatively analogous to that of a HgTe QW in a TI phase. The first NI/TI interface is the physical edge at the interface vacuum/HgTe QW, correctly described with hard-wall BCs. The second NI region can be due to a large-scale (tens of nanometers in the 2D-plane) fluctuation in the QW thickness, leading to the appearance of a topologically trivial region. HgTe QWs are typically grown to have a thickness around the critical value of $6.3$ nm and it is generally sufficient to have a variation of the thickness of the order of fractions of a nanometer to induce a band inversion into a NI system.
\] with hard-wall BCs at the first NI/TI interface ($y=0$). The width of the first (from left to right) TI domain is $L=70$ nm, while that of the second NI domain is $d=70$ nm. The Dirac mass of the second NI domain is $M'=10$ meV. The density profiles in the panels (a)-(h) correspond to states labeled by letters a-h in Fig. \[fig:3edges3\]. []{data-label="fig:3edgesspinors"}](fig/spinors2sw1hwA "fig:"){width="7cm"}\
\] with hard-wall BCs at the first NI/TI interface ($y=0$). The width of the first (from left to right) TI domain is $L=70$ nm, while that of the second NI domain is $d=70$ nm. The Dirac mass of the second NI domain is $M'=10$ meV. The density profiles in the panels (a)-(h) correspond to states labeled by letters a-h in Fig. \[fig:3edges3\]. []{data-label="fig:3edgesspinors"}](fig/spinors2sw1hwB "fig:"){width="7cm"}
In Fig. \[fig:3edges3\], we plot the edge modes for a mass domain with $M'=10$ meV and $d=70$ nm and three different values of $L=100$, $70$ and $50$ nm. The edge mode of the first NI/TI interface would have by itself a Dirac point at around $6.5$ meV, however the overlap with the edge modes from the second TI/NI interface, which originates an anticrossing at $k_x\approx0.01$ nm$^{-1}$, pushes it up to around $8$ meV. This analysis is confirmed by the fact that the electronic density calculated at point $a$ and $f$ of Fig. \[fig:3edges3\] for the case with $L=70$ nm \[shown in Fig. \[fig:3edgesspinors\](a) and (f)\] is strongly peaked near $y=0$, corresponding to the first NI/TI boundary with hard-wall BCs. The anticrossing and the minigap opening at $k_x\approx0.01$ nm$^{-1}$ are due to the overlap of edge states between the first and the second boundaries as observed in the spinors in Fig. \[fig:3edgesspinors\](b) and (e). The gap opening at $k_x=0$ around $E=0$ is due to the overlap of the edge modes of the second and third boundary and the edge states resemble the case of the lateral coupling of edge states through a narrow NI region with mass $M'=10$ meV (analyzed in Fig. \[fig:lateral\] and related text), as can be seen in Fig. \[fig:3edgesspinors\](d) and (g). Spinors in Fig. \[fig:3edgesspinors\](c) and (h), which belong to the points $c$ and $h$ of Fig. \[fig:3edges3\] for the case with $L=70$ nm, resemble the edge modes of a symmetric TI strip with soft-wall BCs away from the minigap region of the dispersion curve.
The effect of the overlap of edge modes belonging to the second and third boundaries can be analyzed by varying $d$, which is done in Fig. \[fig:3edges2\]. Decreasing $d$ from $100$ nm to $50$ nm accentuates the anticrossing behavior at $k_x=0$ around $E=0$ meV due to the overlap of edge states belonging to the second and the third interfaces. Other features of the edge dispersion are only slightly affected. When the second NI region is thinner (see case with $d=20$ nm) the overlap of the edge states bound to it is so strong that they are energetically pushed into the bulk spectral range. As a result, the helical edge modes belonging to the first NI/TI interface (Dirac point at around 6.5 meV) are hardly affected by the presence of a second thin NI region.
![(Color online) Edge mode dispersion curves for a NI/TI/NI/TI mass domain with three boundaries, calculated for a width of the second NI region $d=100$, $50$ and $20$ nm for a fixed Dirac mass $M'=10$ meV. The width of the first TI region is $L=70$ nm, while the first NI/TI interface is treated with hard-wall BCs, and the second TI region is semi-infinite. Full and dashed lines distinguish $\tau=\pm 1$. []{data-label="fig:3edges2"}](fig/disp2sw1hw_d_2){width="7cm"}
The effect of varying the NI mass $M'$ is instead shown in Fig. \[fig:3edges4\]. The main effect of increasing $M'$ is the reduction of the edge state coupling through the NI strip with a corresponding decrease of the $k_x=0$ minigap at $E=0$. If we instead decrease $M'$ the minigap at $k_x=0$ increases and the 1D edge states are restricted to a smaller spectral region since $|E|<M'$.
As a final point, we note that the system with three interfaces, having an odd number of helical edge modes per spin is protected by TRS from opening a full gap in the edge state spectrum notwithstanding the finite overlap of individual edge states.
![(Color online) Edge mode dispersion curves for a NI/TI/NI/TI mass domain with three boundaries calculated for a Dirac mass of the second NI region $M'=10$, $5$ and $20$ meV. The width of the first TI region is $L=70$ nm, the width of the second NI region is $d=70$ nm, while the first NI/TI interface is treated with hard-wall BCs, and the second TI region is semi-infinite. Full and dashed lines distinguish $\tau=\pm 1$. []{data-label="fig:3edges4"}](fig/disp2sw1hw_M){width="7cm"}
At any given energy within the bulk-gap and for any given spin, there is an odd number (one or three) of propagating edge modes per spin. This is a direct consequence of the conservation of [*the parity*]{} of helical edge states linked to the $\mathbb Z_2$ topological invariant. To put these statements in relation to the configurations treated in this work, we note that the configuration shown in Fig. \[fig:domains\](a) with edge dispersions in Fig. \[fig:energy\] and the configuration shown in Fig. \[fig:domains\](d) with edge dispersions shown in Figs. \[fig:3edges3\], \[fig:3edges2\], \[fig:3edges4\] have both an [*odd*]{} number of Dirac mass domains (one and three, respectively) and correspondingly they are metallic. The configuration in Fig. \[fig:domains\](b) with dispersions in Figs. \[fig:hw\] and \[fig:hybridmodes\] and the configuration in Fig. \[fig:domains\](c) with dispersions in Fig. \[fig:lateral\](a) have an [*even*]{} number of Dirac mass domains and are gapped— correspondingly they are insulators.
Conclusion
==========
We have analyzed the edge states of a system described by the BHZ model where the Dirac mass varies spatially thus forming Dirac mass domains where topological insulating regions alternate with normal insulating regions. While for a TI/vacuum interface the use of hard-wall boundary conditions can be assumed, we show that at a TI/NI mass domain with a finite NI mass, soft-wall boundary conditions (characterized by the continuity of the spinor and its derivative) are required to correctly account for the edge state dispersion curves and for the shape of the corresponding wave functions. We solve the edge states for a system up to three TI/NI interfaces. For the case of two interfaces, we solve the problem of a TI strip with hard-wall, soft-wall and hybrid boundary conditions, extending the work in Ref. . We also have investigated the case of edge states that are laterally coupled via a narrow NI domain. While the edge mode spectrum is fully gapped in the two-boundary cases due to the edge mode overlap, we show that, as required by time-reversal symmetry, in the three boundary system, an odd number of edge modes (one or three) per spin is always present at any given energy within the bulk gap. The models solved in this work should be relevant to understand multiple Dirac mass domains induced by fractions of nanometer ranged thickness fluctuations in HgTe-based quantum wells or via tunable voltage-induced band-inversions, e.g. in double quantum well structures [@michetti2012]. Such a controlled creation of multiple helical edge states within a single structure could be used to create tunable spin- and charge-transport devices [@Liu2011; @Krueckl2011; @Dolcini2011; @Romeo2012].
We acknowledge useful discussions with Grigory Tkachov and financial support from the DFG grant RE 2978/1-1 (P.M. and P.R.) and from CNPq, CAPES and FAPESP (P.H.P and J.C.E). P.H.P. acknowledges the kind hospitality at the Institute of Theoretical Physics and Astrophysics, University of Würzburg, where part of this work was developed, and financial support for her visit.
The 4-band Model \[app1\]
=========================
We consider the usual TI Hamiltonian for one of the Kramers partners, given by the following $$h({\vec{k}})\hspace{-0.05cm} = \hspace{-0.03cm} \left( C \hspace{-0.07cm} -\hspace{-0.07cm} D k^2\right)\hspace{-0.05cm} \sigma_0 \hspace{-0.03cm}
+ \hspace{-0.03cm} ( M\hspace{-0.06cm} -\hspace{-0.06cm} B k^2)\sigma_z \hspace{-0.03cm}
+ \hspace{-0.03cm} A (k_x\sigma_x\hspace{-0.03cm}-\hspace{-0.03cm}k_y\sigma_y).$$ Let us define the following two conditions $$\begin{aligned}
\nu M < \nu \frac{A^2}{4B},
\label{condition1}\\
(k_{min}^{\pm})^2>0,
\label{condition2}
\end{aligned}$$ where $\nu={\rm sgn}\left[B/(D^2-B^2)\right]$. If both conditions in Eqs. (\[condition1\]) and (\[condition2\]) are satisfied, the bulk energy dispersion, given by Eq. (\[eq:bulk\]), has a “Mexican hat” form with a local maximum at $k=0$, and $$(k_{min}^{\pm})^2 = \frac{M}{B} - \frac{A^2}{2B^2} \pm \frac{|A| D}{2B^2} \sqrt{\frac{A^2 -4 M B}{D^2-B^2}},$$ are the valence band maxima ($-$) and conduction band minima ($+$), respectively, with energies $$\varepsilon_{\pm}^{C}\hspace{-0.03cm} =\hspace{-0.03cm} -\frac{D\hspace{-0.01cm} M}{B} \hspace{-0.01cm} + \hspace{-0.01cm} \frac{DA^2}{2\hspace{-0.01cm} B^2} \hspace{-0.01cm} \pm\hspace{-0.01cm}
\frac{|A|}{2\hspace{-0.01cm} B^2} \sqrt{\hspace{-0.05cm} (\hspace{-0.02cm} A^2 \hspace{-0.05cm} -\hspace{-0.05cm} 4\hspace{-0.01cm} M\hspace{-0.01cm} B\hspace{-0.01cm} )(\hspace{-0.01cm} D^2\hspace{-0.05cm} -\hspace{-0.05cm} B^2\hspace{-0.01cm} )}.
\label{minimi}$$
If Eq. (\[condition1\]) is satisfied but Eq. (\[condition2\]) is not for $k_{min}^{+}$($k_{min}^{-}$) then still the conduction (valence) band has a single minimum at $k=0$ with energy $|M|$ ($-|M|$).
With standard TI QW parameters the condition in Eq. (\[condition1\]) is generally not fulfilled; with our choice of parameters it would correspond to $$M < \frac{A^2}{4 B}\approx - 32~{\rm meV},$$ and therefore the valence (conduction) band has a maximum (minimum) at $k=0$ and energy $E=M$ ($E=-M$).
Compatible with a fixed energy $E$ and a real $k_x$, one generally obtains four complex values of $k_y$, given by Eq. (\[modi\]). Let us now analyze the domain of $k_y$ as a function of the TI parameters. Eq. (\[modi\]) leads to complex solutions if $$\begin{aligned}
F^2-Q^2 %= \frac{\left[A^2 -2(BE +DM)\right]^2 +4 A^2 (D-B)(M-E)}{4(B^2-D^2)^2}
<0,
\label{condition3}
\end{aligned}$$ otherwise the solutions are either purely real or purely imaginary. Such an analysis shows that complex $k_y$’s are found for $E \in \left( \varepsilon_-^{C},\varepsilon_+^{C}\right)$ if $|B|>|D|$ and in $E \notin\left( \varepsilon_-^{C},\varepsilon_+^{C}\right)$ for $|B|<|D|$ .
Note that for $|B|>|D|$, if both conduction and valence band have a Mexican hat form, then complex $k_y$’s are essentially found inside the gap region, bound by $\varepsilon^{C}_{\pm}$ in Eq. (\[minimi\]). In this case, $k_y=k_\pm$ \[defined in Eq. (\[modi\])\] are both complex in the gap spectral range with $|k_+|=|k_-|$. We define $$\begin{aligned}
\tilde{k}_\pm\hspace{-0.05cm} \doteq \hspace{-0.02cm} \pm \hspace{-0.01cm} k_\pm\hspace{-0.02cm}
= \hspace{-0.02cm} \pm \hspace{-0.01cm} \sqrt{\hspace{-0.03cm} -k_x^2 \hspace{-0.05cm} - \hspace{-0.03cm} F \hspace{-0.02cm}
\pm \hspace{-0.03cm} i \hspace{-0.01cm} \sqrt{\hspace{-0.02cm} |\hspace{-0.01cm}F^2\hspace{-0.05cm} -\hspace{-0.05cm} Q^2|}}\hspace{-0.05cm}
= \hspace{-0.01cm} \pm u \hspace{-0.01cm} +\hspace{-0.01cm} i v,
\label{complex}
\end{aligned}$$ where $u$ and $v$ are the real and imaginary parts of $k_{\pm}$ and $v>0$. We choose the present definition of $\tilde{k}_\pm$, so that they exponentially decay along the $y$-axis. Components corresponding to $k_y=\tilde{k}_\pm$ contribute to edge states (if existent) with a single decay length $1/v$ and an oscillatory behavior as $\sin{(uy)}$. If Eq. (\[condition1\]) is satisfied but Eq. (\[condition2\]) is not for $k_{min}^{+}$($k_{min}^{-}$) then in the interval between $\varepsilon_{+}^C$ ($\varepsilon_-^C$) and $-|M|$ ($|M|$) $k_y$ is purely imaginary. In this case we, instead, define $\tilde{k}_\pm \doteq k_\pm$ and, when existent, edge states will have two decaying lengths ($1/k_\pm$) and no oscillatory behavior [@Lu2012; @michetti2012].
[24]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{}
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
The conductivity of HgTe-based QWs in a randomly fluctuating Dirac mass potential has been considered in Refs. .
, ****, ().
B. Büttner [*et al.*]{}, Nature Phys. [**7**]{}, 418 (2011).
A. Medhi and V. B. Shenoy, J. Phys: Cond. Matter [**24**]{}, 355001 (2012).
, ****, ().
B.A. Volkov and O.A. Pankratov, Pis’ma v Zh. Eksp. Teor. Fiz. [**42**]{}, 145 (1985).
, ****, ().
I. Martin [*et al.*]{}, Phys. Rev. Lett. [**100**]{}, 036804 (2008).
, ****, ().
To make the notation more compact, we use four component Pauli matrices $\vec{\sigma}=\left(\mathbb{I}, \sigma_x, \sigma_y, \sigma_z \right)$.
, ****, ().
F. Lu [*et al.*]{}, Eur, Phys. Lett. [**98**]{} 17004 (2012).
If Dirac mass domains are induced by thickness variations of QWs, the width over which the mass domains develop should be smooth for the out-of-plane ($z$-axis) dynamics in order to avoid mixing with higher subbands, but still step-like for the in-plane motion which is governed by Eq. (1). Note that the typical in-plane extent of the edge states is of the order of $40$ nm, therefore we assume the Dirac mass domain to develop on a much shorter length scale.
, ****, ().
, ****, ().
, , ****, ().
R. Winkler [*et al.*]{}, arXiv:1011.4504.
P. Virtanen and P. Recher, Phys. Rev. B [**85**]{}, 035310 (2012).
C-. X-. Liu [*et al.*]{}, Phys. Rev. B [**83**]{}, 035407 (2011).
V. Krueckl and K. Richter, Phys. Rev. Lett. [**107**]{}, 086803 (2011).
F. Dolcini, Phys. Rev. B [**83**]{}, 165304 (2011).
F. Romeo [*et al.*]{}, Phys. Rev. B [**86**]{}, 165418 (2012).
|
---
abstract: 'In realistic situations, black hole spacetimes do not admit a global timlike Killing vector field. However, it is possible to describe the horizon in a quasi-local setting by introducing the notion of a quasi-local boundary with certain properties which mimic the properties of a black hole inner boundary. Isolated horzons and Killing horizons are examples of such kind. In this paper, we construct such a boundary of spacetime which is null and admits a conformal Killing vector field. Furthermore we construct the space of solutions (in general relativity) which admits such quasi-local conformal Killing boundaries. We also establish a form of first law for these quasi-local horizons.'
author:
- Ayan Chatterjee
- Avirup Ghosh
title: 'Quasi-local conformal Killing horizons: Classical phase space and the first law'
---
Introduction
============
A black hole is described to be a region of spacetime where the gravitational attraction is high enough to prevent even light from escaping to infinity. In asymptotically flat spacetimes, the impossibility of light escaping to future null infinity form the appropriate characterization of a black hole. In other words, this region lies outside the causal past of the future null infinity $\mathscr{I}^{+}$. The boundary of such a region is called the event horizon $\mathscr{H}$ [@Hawking:1973uf; @Wald:1984rg]. To be more precise, consider a strongly asymptotically predictable spacetime ($\mathcal{M}, g_{ab}$). The spacetime is said to contain a black hole if $\cal{M}$ is not contained in $J^{-}(\mathscr{I}^{+})$. The black hole region is denoted by $\mathscr{B}=\mathcal{M}-J^{-}(\mathscr{I}^{+})$ and the event horizon is the boundary of $\mathscr{B}$ (alternatively it may also be defined as the future boundary of past of future null infinity: $\mathscr{H}=\partial[\,J^{-}(\mathscr{I}^{+})\,]$). The definition of event horizon thus requires that we are able to construct the future null infinity $\mathscr{I}^{+}$. This implies that the entire future of the spacetime needs to be known beforehand to ensure the existence of an event horizon. Indeed, the condition of strong asymptotic predictibility of spacetime signifies that we have a complete knowledge of the future evolution. From the above consideration, it is clear that $\mathscr{H}$ is a global concept and it becomes difficult to proceed much further using this definition. However, the notions simplify for stationary spacetimes which are expected states of black holes in equilibrium. In equilibrium, these spacetimes admit Killing symmetries and thus exhibit a variety of interesting features. Indeed, the strong rigidity theorem implies that the event horizon of a stationary black hole is a Killing horizon [@Hawking:1971vc]. However not all Killing horizons are event horizons. Killing horizons only require a timelike Killing vector field in the neighbourhood of the horizon whereas construction of a stationary event horizon requires a global timelike Killing vector field.
The identification of the event horizon of a stationary black hole to a Killing horizon was useful to prove the laws of mechanics for event horizons [@Bardeen:1973gs]. It was shown that in general relativity, the surface gravity $\kappa_{H}$ of a stationary black hole must be a constant over the event horizon. The first law of black hole mechanics refers to stationary space-times admitting an event horizon and small perturbations about them. This law states that the differences in mass $M$, area $A$ and angular momentum $J$ to two nearby stationary black hole solutions are related through $\delta M=\kappa_{H} \delta A/8\pi + \Omega_{H}\delta J.$ One gets additional terms like charge if matter fields are present. Hawking’s proof that due to quantum particle creation, black holes radiate to infinity, particles of all species at a temperature $\kappa_{H}/2\pi$, implied that laws of black hole mechanics are the laws of thermodynamics of black holes [@Hawking:1974sw]. Moreover, the entropy of the black holes must be proportional to it’s area [@Bekenstein:1973ur; @Bekenstein:1974ax].
However, it was realised very soon that this identification of entropy to area leads to new difficulties. Classical general relativity gives rise to infinite number of degrees of freedom but it is not clear if the laws of thermodynamics can arise out of a statistical mechanical treatment of these classical information (see [@Wald:1995yp]). One must find ways to extract quantum degrees of freedom of general relativity. The framework of Killing Horizon was broadened to understand the origin of entropy and black hole thermodynamics [@Wald:1993nt; @Iyer:1994ys; @Jacobson:1993vj; @Youm:1997hw; @Carlip:1999cy; @Dreyer:2013noa; @Ghosh:2014pha]. It turned out that the framework of Isolated Horizons (IH) was more suited to address these questions from the perspective of loop quantum gravity [@Ashtekar:1998sp; @Ashtekar:2000sz; @Ashtekar:2000hw; @Ashtekar:2001is; @Ashtekar:2001jb; @Chatterjee:2006vy; @Chatterjee:2008if]. It is argued that the effective quantum degrees of freedom which capture the thermodynamic information of black holes are localised, more precisely, reside on the horizon. Isolated horizons are suited for this description since they capture only the local information; isolated horizons are local descriptions of horizons and unlike event horizons, do not require the global history of spacetime [@Smolin:1995vq; @Krasnov:1996tb; @Rovelli:1996dv; @Ashtekar:1997yu; @Ashtekar:1999wa; @Ghosh:2006ph; @Ghosh:2008jc; @Ghosh:2011fc; @Ghosh:2013iwa]. It arises that the effective field theory induced on a IH is a Chern- Simons theory whose quantisation and counting of states is consistent with the results of Bekenstein and Hawking. Moreover, since IH replaced the global notion of event horizons with a local description, the requirement of a knowledge of full space-time history as well as the asymptotics is avoided (see [@Corichi:2013zza; @Corichi:2014zoa] for a first order description of theories with topological terms). The underlying spacetime therefore might not admit a global Killing vector at all in the isolated framework. While this has been a significant development in the understanding of black hole mechanics, generalizations to dynamically evolving horizons has also been reported [@Ashtekar:2002ag; @Ashtekar:2003hk; @Ashtekar:2004cn]. These dynamical horizons are closely related to the notion of trapping horizons developed earlier [@Hayward:1993wb; @Hayward:1994yy]. Using the boundary conditions for dynamical horizon it was shown that a flux balance law, relating the change of area of the dynamical horizon to the flux of the matter energy, exists, reproducing an integrated version of a first law [@Ashtekar:2002ag; @Ashtekar:2003hk; @Ashtekar:2004cn]. Moreoever, it has also been shown that if the horizon is slowly evolving, a form of the first law arises [@Booth:2003ji; @Booth:2006bn; @Booth:2007wu]. The construction of a phase space for these horizons has also been carried out in the metric variables.
Another class of horizons that has been of interest are conformal Killing horizons (CKH). Though not a trapping horizon it essentially captures a dynamical situation. The notion of CKH and it’s properties were developed in [@DyerHonig; @Suldyer; @Sultana:2005tp; @Jacobson:1993pf; @Nielsen:2012xu]. These are null hypersurfaces whose null geodesics are orbits of a conformal Killing field. If $\xi^{a}$ is a vector field which satisfies $\lie_{\xi}g_{ab}=2fg_{ab}$, and is null, it generates a CKH for the metric $g_{ab}$. It has been shown that an analogue of the zeroth law holds for a conformal Killing horizon as well. More precisely, since $\xi^{a}$ generates a null surface, it is geodesic and one can define an accelration through $\xi^{b}\nabla_{b}\xi^{a}=\kappa_{\xi} \xi^{a}$. Then, the quantity $(\kappa_{\xi}-2f)$ which essentially is a combination of the acceleration of the conformal Killing vector and the conformal factor, can be shown to be Lie dragged along the horizon and can therefore be interpreted as a temperature. An analogue of the first law is therefore expected to hold in this case as well but has not been established in the literature. In this paper, we address the question if a form of the first law can be established at all for a CKH. As we discuss below, if such a law exists, it may lead to some important clues for a dynamically evolving horizon.
The plan of the paper is as follows. We start by developing the geometry of a quasi-local conformal Killing horizon. We assume that a spacetime time region $\mathcal{M}$ has a null boundary $\Delta$ which however may have non- zero expansion ($\theta =-2\rho\neq 0$). In other words we take the null generators of $\Delta$ to be only shear-free. We observe that these conditions are enough to ensure that the null generators $l^{a}$ are conformal Killing vectors on $\Delta$. Now, since these null surfaces are not expansion free, they may be growing; in fact $\lie_l ~{}^{2}\epsilon=\theta~{}^{2}\epsilon$ and hence, are good candidates for growing horizons. The situation in some sense mimics what one has at null infinity in an asymptotically flat space-time. However, we are more interested in an inner horizon. The physical situation for these horizons can be visualised as follows. Suppose matter falls in through a horizon as a result of which it grows (supposing that matter satisfies standard energy conditions) and hence has a non- zero positive expansion. When this matter flux stops to fall in through the horizon, by the Raychaudhuri equation, an initially positively expanding horizon will slow down it’s expansion and after some time reach the state of equilibrium. This equilibrium state has zero expansion and it’s geometrical set- up has been developed through the Isolated horizon formulation. We are interested to construct the space of solutions of only those dynamically evolving horizons which can be generated by a conformal Killing vector field. By construction, the CKH admit a limit to the IH formulation. We suppose that the matter flux across $\Delta$ be a real scalar field ($\varphi$) satisfying the condition $\lie_l\varphi =-2\rho\varphi$ on the horizon. The geometrical conditions ensures that a form of zeroth law exists. In the next section, we show that the action for general relativity admits a well defined variational principle in presence of the conformal Killing horizon boundary and proceed to construct the symplectic structure. An interesting outcome is the construction of the phase space, identification of a boundary symplectic structure and the existence of a first law. Further, it arises that gravity and matter together gives a well defined phase- space provided a balance condition holds. This balance condition turns out to be nothing but Einstein’s equation contracted with the null generators $l^a$ (say). We thus get a quasi-local analogue of a conformal Killing horizon.
Geometrical setting and boundary conditions
===========================================
In this section, we introduce the minimal set of boundary conditions which are suitable for a quasilocal conformal Killing horizon. We assume that all fields under consideration are smooth. Let $\mathcal{M}$ be a $4$- manifold equipped with a metric $g_{ab}$ of signature $(-,+,+,+)$. Consider a null hypersurface $\Delta$ of $\mathcal M$ with $l^a$ being it’s future directed null normal. Given this null normal $l^{a}$, one can introduce another future directed null vector field $n^{a}$ which is transverse to $\Delta$. Further, one has a set of complex null vector field $(m,\,\bar{m})$, which are tangential to $\Delta$. This null tetrad $(l,\, n\, m\, \bar{m})$ constitutes the Newman- Penrose basis. The vector fields satisfy the condition that $l.n=-1=-m.\bar{m}$, while all other scalar products vanish. Let $q_{ab}$ be the degenerate metric on the hypersurface. The expansion $\theta_{l}$ of the null normal is given by $q^{ab}\nabla_{a} l_b$. In terms of the Newman- Penrose co-effecients, $\theta_l=-2\rho$ (see appendix A and [@Chandrasekhar:1985kt] for details). The accelaration of $l^a$ follows from the expression $l^a\nabla_a~l_b=(\epsilon+\bar\epsilon)l_b$ and is given by $\kappa_l :=\epsilon+\bar\epsilon$. To avoid cumbersome notation, we will do away with the subscripts $(l)$ from now on if no confusion arises. It would be useful to define an equivalance class of null normals $[l^{a}]$ such that two null normals $l$ and $l'$ will be said to belong to the same equivalance class if $l'=cl$ where $c$ is a constant on $\Delta$.\
Quasi-local conformal horizon {#quasi-local-conformal-horizon .unnumbered}
-----------------------------
[*[Definition]{}*]{}: A null hypersurface $\Delta$ of $\mathcal{M}$ will be called quasi-local conformal horizon if the following conditions hold.:
1. $\Delta$ is topologically $S^2\times R$ and null.
2. The shear $\sigma$ of $l$ vanishes on $\Delta$ for any null normal $l$.
3. All equations of motion hold at $\Delta$ and the stress- energy tensor $T_{ab}$ on $\Delta$ is such that $-T^{a}{}_{b}\,l^b$ is future directed and causal.
4. If $\varphi$ is a matter field then it must satisfy $\lie_l\varphi=-2\rho\,\varphi$ on $\Delta$ for all null normals $l$.
5. The quantity $\left[2\rho+\epsilon + \bar{\epsilon}\right]$ is Lie dragged for any null-normal $l$.
Some comments on the boundary conditions are in order. The first condition imposes restrictions on the topology of the hypersurface. It is natural to motivate this condition from Hawking’s theorem on the topology of black holes in asymptotically flat stationary spacetimes or it’s extension [@Hawking:1971vc; @Galloway:2005mf]. But, we are also interested in spacetimes which are aymptotically non- flat or that they are non- stationary for which, these theorems may not hold true. However it is not unnatural to argue that since black hole horizons forming out of gravitational collapse have spherical topologies, such conditions might exist. This condition is also assumed in the Isolated Horizon formalism. For these isolated hypersurfaces, the expansion $\theta$ of the null normal $l^{a}$ vanishes (which is not true in our case). It is possible that cross- sections of such quasilocal horizons may admit other topologies. For the time being, we would not include such generalities and only retain the condition that the cross- sections of the hypersurfaces are spherical.
The second boundary condition on the shear is a simplification. Shear measures the amount of gravitational flux flowing across the surface, and we put the gravity flux to be vanishing. This boundary condition on the shear $\sigma$ of null normal $l^{a}$ has several consequences. First, since $l_a$ is hypersurface orthogonal, the Frobenius theorem implies that $\rho$ is real and $\kappa=0$. Secondly, the Ricci identity can be written as D-=(+|+3-| )-(-|+|+3)+\_0, where $D=l^{a}\nabla_{a}$, $\delta=m^{a}\nabla_{a}$, $\Psi_{0}$ is one of the Weyl scalars and the other quantities are the Newman- Penrose scalars (see [@Chandrasekhar:1985kt] for details). If $\sigma\stackrel{\Delta}{=}0$, it implies $\Psi_0\stackrel{\Delta}{=}0$. Next, since $\l^{a}$ is null normal to $\Delta$, it is twist- free and a geodetic vector field. The implications of $\l^{a}$ being twist- free has already been shown above. The accelaration of $l^a$ follows from the expression $l^a\nabla_a~l^b=(\epsilon+\bar\epsilon)l^b$ and is given by $\kappa_l :=\epsilon+\bar\epsilon$. The acceleration of the null normal varies over the equivalence class $[cl]$ where $c$ is a constant on $
\Delta$ . This is only natural that the acceleration varies in the class since in the absence of the knowledge of asymptotics, the acceleration cannot be fixed.
Further, it can be seen that the null normal $l^{a}$ is such that -2 m\_[(a]{}|[m]{}\_[b)]{} which implies that $l^a$ is a conformal Killing vector on $\Delta$. Moreover, the Raychaudhuri equation implies that $R_{ab}l^{a}l^{b}\neq 0$ and hence $-R^{a}{}_{b}l^{b}$ can have components which are tangential as well as transverse to $\Delta$.
The third boundary condition only implies that the field equations of gravity be satisfied and that the matter fields be such that their energy momentum tensor satisfies some mild energy conditions. The fourth and the fifth boundary conditions are somewhat adhoc but can be motivated. Let us first look at the fourth boundary condition. We have kept open the possibility that matter fields may cross the horizon and the horizon may grow. The matter field is taken to be a massless scalar field $\varphi$ which behaves in a certain way which mimics it’s conformal nature. The fifth condition is motivated by the fact that surface gravity remains invariant under conformal transformations [@Jacobson:1993pf; @Sultana:2005tp]. It can be shown that the quantity that is constant for these horizons is $\left(2\rho+\epsilon+\bar{\epsilon}\right)$. A conformal transformation of the metric amounts to a conformal transformation of the two-metric on $\Delta$. Under a conformal transformation $g_{ab}\rightarrow \Omega^2\,g_{ab}$ and one needs a new covariant derivative operator which annihilates the conformally transformed metric. Under such a conformal transformation $l^a\rightarrow l^a,l_a\rightarrow \Omega^2 l_a,n^a\rightarrow \Omega^{-2} n^a,n_a\rightarrow n_a,m^a\rightarrow \Omega^{-1}m^a,m_a\rightarrow \Omega m_a$. The new derivative operator is such that it transforms as $$\nabla_al_b\rightarrow\Omega^2\nabla_al_b+2\Omega\partial_a\Omega~l_b-\Omega^2\left[l_c\delta^c_a\partial_b\log{\Omega}+l_c
\delta^c_b\partial_a\log{\Omega}-g_{ab}g^{cd}l_c\partial_d\log{\Omega}\right]$$ If one defines a one- form $\omega_a\=-n^b\underleftarrow{\nabla_a}l_b$, it transforms under the conformal transformation as $$\tilde{\omega}_a\=\omega_a+2\partial_a\log{\Omega}-\partial_a\log{\Omega}-n_al^c\partial_c\log{\Omega}$$ It follows that the Newman- Penrose scalars transform in the following way $$\begin{aligned}
\widetilde{\left(\epsilon+\bar\epsilon\right)}&\=&(\epsilon+\bar\epsilon)
+2\lie_l\log{\Omega}\\
\tilde{\rho}&\=&\rho-\lie_l\log{\Omega}\\
\tilde{\sigma}&\=&\sigma\end{aligned}$$ where, $\rho=-m^a\bar{m}^b\nabla_al_b$ and $\sigma=-\bar{m}^a\bar{m}^b\nabla_al_b$. Thus it follows that $2\rho + \epsilon+\bar\epsilon$ remains invariant under a conformal transformation.
At this point, it would be useful to recall the boundary conditions of a weakly isolated horizon and note the important differences. A weakly isolated horizon is a null hypersurface which satisfies the first and the third boundary conditions given here and that the expansion of the null normal $l^{a}$ be zero. On such surface, there exists a one- form $\omega_{a}$ which is also assumed to be Lie dragged by the vector field $l^{a}$. Thus, instead of the condition on shear, for a WEH, the expansion of the null normal $l^{a}$ is taken to be vanishing, $\theta=0=2\rho$. By the Raychaudhuri equation, the boundary conditions imply that the shear is zero and that no matter field crosses the horizon (and hence the name isolated). However, here, we impose only the condition that the shear vanishes and keep the possibility that matter fields may fall through the surface (but no gravitational flux) and that the hypersurface may grow along the affine parameter. As we shall show, removing our last condition does not restrict one to define a well defined phase space, but is essential to get a first law. It is an analogue of the condition $\lie_l(\epsilon+\bar\epsilon)\stackrel{\Delta}{=}0$ assumed in the case of weakly isolated horizon. It may be useful to note that the fifth boundary condition as given above, can be recast is a form which is an analogue of that for a weakly isolated horizon by setting $\lie_l\tilde{\omega}=0$, where $\tilde{\omega}_a\=\omega_a+\partial_a\log{\Omega}-n_al^c\partial_c\log{\Omega}$ and the conformal factor is set such that $\lie_l\log{\Omega}=\rho$.\
Action principle and the classical phase space
==============================================
We are interested in constructing the space of solutions of general relativity, and we use the first order formalism in terms of tetrads and connections. This formalism is naturally adapted to the nature of the problem in the sense that the boundary conditions are easier to implement. Moreover it has the advantage that the construction of the covariant phase- space becomes simpler. For the first order theory, we take the fields on the manifold to be ($e_{a}{}^{I},\, A_{aI}{}^{J},\, \varphi$), where $e_{a}{}^{I}$ is the co- tetrad, $A_{aI}{}^{J}$ is the gravitational connection and $\varphi$ is the scalar field. The Palatini action in first order gravity with a scalar field is given by: $$\label{lagrangian1}
S_{G+M}=-\frac{1}{16\pi G}\int_{\mathcal{M}}\left(\Sigma^{I\!J}\wedge
F_{I\!J}\right)-\frac{1}{2}\int_{\mathcal{M}}d\varphi\wedge {}{\star}
d\varphi\;$$ where $\Sigma^{IJ}=\half\,\epsilon^{IJ}{}_{KL}e^K\wedge e^L$, $A_{IJ}$ is a Lorentz $SO(3,1)$ connection and $F_{IJ}$ is a curvature two-form corresponding to the connection given by $F_{IJ}=dA_{IJ}+A_{IK}\wedge A^{K}~_{J}$. The action might have to be supplemented with boundary terms to make the variation well defined.
Variation of the action {#variation-of-the-action .unnumbered}
-----------------------
For the variational principle, we consider the spacetime to be bounded by a null surface $\Delta$, two Cauchy surfaces $M_{+}$ and $M_{-}$ which extend to the asymptotic infinity. The boundary conditions on the fields are the following. At the asymptotic infinity, the fields satisfy appropriate boundary conditions. The fields on the hypersurfaces $M_{+}$ and $M_{-}$ are fixed so that their variations vanish. On the surface $\Delta$, we fix a set of internal null- tetrad $(l^{I}, n^{I}, m^{I}, \bar{m}^{I})$ such that the flat connection annihilates them. The fields on the manifold ($e_{a}{}^{I},\, A_{aI}{}^{J},\, \varphi$), must satisfy the following conditions. First, on $\Delta$, the configurations of the tetrads be such that $l^{a}=e_{I}^{a}l^{I}$ are the null vectors which satisfy the boundary conditions for quasi- local conformal horizon. Second, the possible connnections also satisfy the boundary conditions and be such that $\left(2\rho+\epsilon + \bar{\epsilon}\right)$ is constant. Thirdly, we consider all those configurations of scalar field which, on $\Delta$, satisfy $\lie_l\,\varphi=-2\rho\,\varphi$.
We now check that the variational principle is well- defined if the boundary conditions on the fields, as given above, hold. However, we need some expressions for tetrads and connections on $\Delta$, details of which are given in the appendix A. On the conformal horizon, the $\Sigma^{IJ}$ is given by $$\underleftarrow{\Sigma}^{IJ}\=2l^{[I}n^{J]}~^{2}\epsilon+2n\wedge(im~l^{[I}\bar{
m}^{J]}-i\bar{m}~l^{[I}m^{J]}),$$ and the connection is given by $$\begin{aligned}
\label{connection_delta}
\underleftarrow{A_{a}{}_{IJ}}&\stackrel{\Delta}{=}&
2\left[(\epsilon+\bar{\epsilon})n_a
-(\bar{\alpha}+\beta)\bar{m}_a-(\alpha+\bar{\beta})m_a\right]\,
l_{[I}n_{J]}+2(-\bar{\kappa}n_a
+\bar{\rho}\bar{m}_a)\, m_{[I}n_{J]}+2(-{\kappa}n_a
+{\rho}{m}_a)\,\bar{m}_{[I}n_{J]}\nn
&+& 2(\pi n_a+-\mu\bar{m}_a-\lambda m_a)\, m_{[I}l_{J]}+2(\bar{\pi} n_a
-\bar{\mu}{m}_a-\bar{\lambda}\bar{m}_a)\,\bar{m}_{[I}l_{J]}\nn
&+& 2\left[-(\epsilon-\bar{\epsilon})n_a +(\alpha-\bar{\beta}) m_a+(\beta-\bar{\alpha})\bar{m}_a
\right]\, m_{[I}\bar{m}_{J]}..\end{aligned}$$ The Lagrangian $4$- form for the fields ($e_{a}{}^{I},\, A_{aI}{}^{J},\,
\varphi$) is given in the following way. $$L_{G+M}=-\frac{1}{16\pi G}\left(\Sigma^{I\!J}\wedge
F_{I\!J}\right)-\frac{1}{2}d\varphi\wedge \star d\varphi .$$ The first variation of the action leads to equations of motion and boundary terms. The equations of motion consist of the following equations. First, variation of the action with respect to the connection implies that the curvature $F^{IJ}$ is related to the Riemann tensor $R^{cd}$, through the relation $F_{ab}{}^{IJ}=R_{ab}{}^{cd}\,e^{I}_{c}e^{J}_{d}$. Second, variation with respect to the tetrads lead to the Einstein equations and third, the first variation of the matter field gives the equation of motion of the matter field. On- shell, the first variation is given by the following boundary terms $$\delta L_{G+M} := d\Theta(\delta)=-\frac{1}{16\pi
G}d\left(\Sigma^{IJ}\wedge\delta A_{IJ}\right)-d(\delta\varphi\star d\varphi),$$ which are to be evaluated on the boundaries $M_{-}$, $M_{+}$, asymptotic infinity and $\Delta$. However, since fields are set fixed on the initial and the final hypersurfaces they vanish. The boundary conditions at infinity are assumed to be appropriately chosen and they can be suitably taken care of. The only terms which are of relevance for this case are the terms on the internal boundary. On the internal boundary $\Delta$, the boundary terms give (see appendix \[appb\] for details) 16G L\_[G+M]{}=-( n)\^2-(2 n\^2) + 8G( n) \^2Since Einstein’s equations give $R_{11}=8\pi G \, T_{11}$, the first and the third term cancel and only $\left(2\rho\, n\wedge\,^2\epsilon\right)$ remains. Thus, if one adds the term $16\pi G\,S^{'}= \int_{\Delta}\left(2\rho\, n\wedge\,^2\epsilon\right)$ to the action, it is well defined for the set of boundary conditions on $\Delta$. As we shall see below, since this is a boundary term, it does not contribute to the symplectic structure.
Covariant phase- space and the symplectic Structure {#covariant-phase--space-and-the-symplectic-structure .unnumbered}
---------------------------------------------------
For a general Lagrangian, the on-shell variation gives $\delta
L=d\Theta(\delta)$ where $\Theta$ is called the symplectic potential. It is a $3$-form in space-time and a $0$-form in phase space. Given the symplectic potential, one can construct the symplectic structure $\Omega (\delta_{1}, \,\delta_{2})$ on the space of solutions. One first constructs the symplectic current $J(\delta_1,\delta_2)=
\delta_1\Theta(\delta_2)-\delta_2\Theta(\delta_1)$, which, by definition, is closed on-shell. The symplectic structure is then defined to be: $$\Omega(\delta_1,\delta_2)=\int_{M}J(\delta_1,\delta_2)
$$ where $M$ is a space-like hypersurface. It follows that $dJ=0$ provided the equations of motion and linearized equations of motion hold. This implies that when integrated over a closed region of spacetime bounded by $M_+\cup M_-\cup
\Delta$ (where $\Delta$ is the inner boundary considered), $$\int_{M_+}J-\int_{M_-}J~+~\int_{\Delta}J=0,$$ where $M_+,M_-$ are the initial and the final space-like slices, respectively. If the third term vanishes then the bulk symplectic structure is independent of choice of hypersurface. However, if it does not vanish but turns out to be exact, $\int_{\Delta}J=\int_{\Delta}dj $ then the hypersurface independent symplectic structure is given by: $$\Omega(\delta_{1}, \,\delta_{2})= \int_MJ-\int_{S_\Delta}j$$ where $S_\Delta$ is the 2-surface at the intersection of the hypersurface $M$ with the boundary $\Delta$. The quantity $j(\delta_1,\delta_2)$ is called the boundary symplectic current and symplectic structure is also independent of the choice of hypersurface.
Our strategy shall be to construct the symplectic structure for the action given in eqn. . Let us first look at the Lagrangian for gravity. The symplectic potential in this case is given by, $16\pi G\Theta(\delta)=-\Sigma^{I\!J}\wedge \delta A_{I\!J}$. The symplectic current is therefore given by, $$\label{symplectic_current1}
J_G(\delta_1,\delta_2)=-\frac{1}{8\pi
G}\,\delta_{[1}\Sigma^{IJ}\wedge~\delta_{2]}A_{IJ}$$ The above expression eqn. , when pulled back and rescticted to the surface $\Delta$ gives \[symplec\_pulled\_back\] &&-2\_[\[1]{} \^2[****]{} \_[2\]]{}{(+|)n-(+| )m-(|+)|[m]{}}&&+2\_[\[1]{}(nim)\_[2\]]{}(||[m]{}) -2\_[\[1]{}(ni|[m]{})\_[2\]]{}(m) It can be shown that the symplectic current pulled back on to $\Delta$ for the gravity sector is given by (see the appendix for details)[^1]\
(\_1,\_2)&&- The first term in the above expression is exact but not others. Therefore the phase is well defined for our boundary conditions $\sigma\stackrel{\Delta}{=}0$ provided, if either $\Phi_{00}=0$, there is no matter flux across the horizon or if $\Phi_{00}/\rho$ gets cancelled with a contribution from the matter degrees of freedom through Einstein’s equation. We deal with a more general case. We show that the contribution of the scalar field is such that the symplectic current on $\Delta$ is again exact.
The symplectic current for the real scalar field is given by, $J_M(\delta_1,\delta_2)=2\,\delta_{[1}\varphi~\delta_{2]}\,{}\star d\varphi$. The symplectic current on the hypersurface $\Delta$ can be obtained as $$\underleftarrow{J_M}(\delta_1,\delta_2)=2\delta_{[1}\varphi~\delta_{2]}
(D\varphi~ n\wedge im\wedge\bar{m}),$$ where $D=l^{a}\nabla_{a}$. The boundary condition on the scalar field implies $D\varphi=-2\rho \,\varphi$ and hence, we get that $$\begin{aligned}
\underleftarrow{J_M}(\delta_1,\delta_2)&=&4\delta_{[1}\varphi~\delta_{2]}
(-\varphi\,\rho~n\wedge im\wedge\bar{m})\\
&=&-d\left\{\delta_{[1}\varphi^2~\delta_{2]}~^2\epsilon\right\}+\delta_{[1}\frac
{D\varphi D\varphi}{\rho}n\wedge~\delta_{2]}
~^2\epsilon\nn
&=&-d\left\{\delta_{[1}\varphi^2~\delta_{2]}~^2\epsilon\right\}+\delta_{
[1}~^2\epsilon\wedge~\delta_{2]}
\left(\frac{{\bf T}_{11}}{\rho}n\right)\end{aligned}$$ The combined expression is then given by: $$\underleftarrow{J_{M+G}}(\delta_1,\delta_2) \stackrel{\Delta}{=}-\frac{1}{4\pi
G}\left\{d\left(\delta_{[1}~^2{\bf\epsilon}~
\delta_{2]}\log{\rho}\right)\right\}-d\,\left\{\delta_{[1}\varphi^2~\delta_{2]}
~^2\epsilon\right\}$$ It follows that the hypersurface independent symplectic structure is given by: $$\begin{aligned}
\Omega(\delta_{1},
\delta_{2})=\int_{\mathcal{M}}J_{M+G}(\delta_1,\delta_2)-\int_{S_\Delta}
j&=&-\frac{1}{8\pi G}
\int_{\mathcal{M}}\delta_{[1}\Sigma^{IJ}\wedge~\delta_{2]}A_{IJ}+2\int_{\mathcal
{M}}\delta_{[1}\varphi~\delta_{2]}
(\star d\varphi)\nn
&+&\frac{1}{4\pi
G}\int_{S_{\Delta}}\left\{\delta_{[1}~^2{\bf\epsilon}~\delta_{2]}\log{\rho}
\right\}
+\int_{S_{\Delta}}\delta_{[1}\varphi^2~\delta_{2]}~^2\epsilon\end{aligned}$$ In the next section, we shall use this expression to derive the first law of mechanics for the conformal Killing horizon.
Hamiltonian evolution and the first law {#hamiltonian-evolution-and-the-first-law .unnumbered}
---------------------------------------
Given the symplectic structure, we can proceed to study the evolution of the system. We assume that there exists a vector which gives the time evolution on the spacetime. Given this vector field, one can define a corresponding vector field on the phase- space which can be interpreted as the infinitesimal generator of time evolution in the covariant phase- space. The Hamiltonian $H_l$ generating the time evolution is obtained as $\delta\, \tilde{H}_{l}= \Omega(\delta, \delta_{l})$, for all vector fields $\delta$ on the phase- space. Using the Einstein equations, we get that $$\begin{aligned}
\label{firstlaw2}
\Omega(\delta,\delta_l)&=&-\frac{1}{16\pi
G}\int_{S_{\Delta}}\left[(l.A_{IJ})\delta\Sigma^{IJ}-(l.\Sigma^{IJ})\wedge
\delta A_{IJ}\right]
+\int_{S_{\Delta}}\delta\varphi~(l\cdotp{}{\star} d\varphi)\nn
\nn
&&\hspace{1cm}+\frac{1}{8\pi
G}\int_{S_{\Delta}}\left(\delta~^2{\bf\epsilon}~\delta_{l}\log{\rho}-\delta_l~^2
{\bf\epsilon}
~\delta\log{\rho}\right)+\int_{S_{\Delta}}\frac{1}{2}(\delta\varphi^2~\delta_{l}
~^2\epsilon-\delta_l\varphi^2\delta~^2\epsilon)\nn\end{aligned}$$ We now need to impose a few conditions on the fields to make a well defined Hamiltonian. These conditions are to be imposed since the action of $\delta_{l}$ on some phase- space fields is not like $\lie_{l}$. This is because of $\rho, \epsilon+\bar\epsilon$ and $\varphi$ all cannot be free data on $\Delta$. First, we note the following equalities $$\begin{aligned}
\lie_{l}\left(\frac{1}{4\pi G}\log{\rho}-\frac{1}{8\pi
G}\log{\varphi}-\varphi^2\right)&=& \frac{1}{4\pi
G}(2\rho+\epsilon+\bar{\epsilon})\\
\lie_{l}\left(\frac{~^2\epsilon}{\varphi}\right)&=&0\end{aligned}$$ We assume that $\delta_l$ acts on $(2\rho+\epsilon+\bar{\epsilon})$ and $\left(\frac{~^2\epsilon}{\varphi}\right)$ like $\lie_l$. This can also be argued in the following fashion. Since $\delta_{l}\lie_{l}(2\rho+\epsilon+\bar{\epsilon})=0$ it immediately implies that $\lie_{l}\delta_{l}(2\rho+\epsilon+\bar{\epsilon})=0$. Hence, choosing $\delta_{l}(2\rho+\epsilon+\bar{\epsilon})=0$ at the initial cross-section implies that it remains zero throughout $\Delta$. Furthermore if we set $\delta_l\left(\frac{1}{4\pi G}\log{\rho}-\frac{1}{8\pi
G}\log{\varphi}-\varphi^2\right)$=0 at the initial cross-section, it remains zero everywhere on $\Delta$ and so, $$\begin{aligned}
\label{eqn_no1}
\frac{\delta_l\rho}{\rho}-8\pi
G\varphi\delta_l\varphi-\frac{\delta_l\varphi}{2\varphi}=0\end{aligned}$$ Another condition can be derived from the equation above $$\begin{aligned}
\delta_l\left(\frac{~^2\epsilon}{\varphi}\right)=\frac{1}{\varphi}
\delta_l~^2\epsilon-~^2\epsilon\frac{1}{\varphi^2}\delta_l\varphi=0\end{aligned}$$ The variations $\delta_l$ satisfy the following differential equations, which can be checked to be consistent with each other: $$\begin{aligned}
\label{eqn_no2}
\lie_l\delta_l\varphi&=&-2\delta_l\rho\varphi-2\rho\delta_l\varphi\\
\lie_l\delta_l~^2\epsilon&=&-2\delta_l\rho~^2\epsilon-2\rho\delta_l~^2\epsilon\end{aligned}$$ Putting condition $\eqref{eqn_no1}$ in $\eqref{eqn_no2}$, we get $$\delta_l\varphi=C(\theta,\phi)\exp\left[-{\int\left(16\pi
G\varphi^2+3\right)\rho dv}\right],$$ where $C(\theta,\phi)$, is a constant of integration. If we choose this constant $C(\theta,\phi)=0$, it immediately implies that $\delta_{l}\varphi=0=\delta_{l}{}^{2}\epsilon.$ With the choice of $\delta_l$ only the bulk symplectic structure survives and one gets from eq. $\eqref{firstlaw2}$ [^2] $$\begin{aligned}
\delta H_{l}&=&-\frac{1}{8\pi
G}\int_{S_\Delta}(2\rho+\epsilon+\bar{\epsilon})\delta~^2\epsilon+
\frac{1}{8\pi G}\int_{S_\Delta}~^2\epsilon~(-\delta\rho-8\pi G\,\delta\varphi
D\varphi)+\delta E^\infty\end{aligned}$$ where we have redefined our Hamiltonian $H_l=\tilde{H}_l+\int_{S_\Delta}\rho~^2\epsilon$. This redefination is possible since the definition of the Hamiltonian is ambiguous upto a total variation. Further, as expected $\Omega(\delta_{l},\delta_l)=0$. Next we define, $E^l_{\Delta}=E^\infty-H_l,$ as the horizon energy. It is clear from above that for $\rho\rightarrow 0$ (i.e in the isolated horizon limit) it matches with the definition in [@Ashtekar:2002ag; @Ashtekar:2003hk] if asymptotics is flat and $E^\infty=E_{ADM}$. It therefore follows that: -E\^l\_=-\_[S\_]{}(2++ |) \^2-\_[S\_]{}.
To recover the the more familiar form of first law known for a dynamical situation, we assume there is a vector field $\tilde{\delta}$ on phase space which acts only on the fields on $\Delta$ (and not in the bulk) such that it’s action on the boundary variables is to evolve the boundary fields along the affine parameter $v$ (it may be interpreted to be a time evolution, like $\lie$). Now demanding that $\tilde{\delta}$ to be Hamiltonian would give an integrability condition which also ensures that $\delta_l$ is Hamiltonian. So one can calculate $\Omega(\tilde{\delta},\delta_l):=\tilde{\delta}H_{l}$ which can be written in the following form[^3] $$\label{firstlaw}
\dot{E^l_\Delta}=\,\frac{1}{8\pi G}\left(2\rho+\epsilon+\bar{\epsilon}\right)\dot{A}+\frac{1}{8\pi
G}\int_{S_\Delta}\left[~^2\epsilon~(\dot\rho+8\pi
G\,\dot\varphi D\varphi)\right]$$ where dots imply changes in the variables produced by the action of $\tilde{\delta}$. Note that if $\tilde{\delta} =\lie_{l}$, then, $\tilde{\delta}\varphi D\varphi$ gives the expression $T_{ab}l^{a}l^{b}$. Equation is the form of evolution for the conformal Killing horizons. The first term in the above expression is the usual $TdS$ term while the second term is a flux term which takes into account the non-zero matter flux across $\Delta$.
Discussions
===========
In this paper, we have developed the geometrical set-up for a quasi-local description of a conformal Killing horizon. Further, we have also shown that one can understand these horizons to have a zeroth law (as was also discussed in [@Suldyer]) and a first law. This development of a notion of quasi-local conformal horizon should be taken in the same spirit as the development of the notion of isolated horizon from Killing horizons. A conformal Killing horizon is one which has a conformal Killing vector in the neighbourhood of the horizon. In contrast, a quasi-local conformal horizon only requires the existence of a null hypersurface generating vector which is shear free on the null hypersurface. The number of solutions of Einsteins’s equation for gravity and matter that admits a conformal Killing horizon may be small (examples of such kind has been constructed by [@Sultana:2005tp]). However the solutions admitting a quasi-local conformal horizon may be large. We do not comment on the nature of solutions that admits a quasi-local conformal horizon, we think that significant amount of insights may be obtained by numerical simulations and therefore falls in the regime of numerical relativity. The most useful application of these geometrical structures are in the dynamical evolution of black holes. Indeed, as matter falls in through the horizon and the black hole horizon grows, the expansion is non- zero. In such cases, it is important to understand if in this dynamical situation one can prove the existence of laws for black hole mechanics in some form.
We have taken a real scalar field as the matter field in question. The flux balance law is seen to be successfully implemented if it satisfies a condition $\lie_l\varphi\stackrel{\Delta}{=}-2\rho\,\varphi$. This assumption is motivated through the fact that $l^a$ is a conformal Killing vector on $\Delta$. Taking other matter fields will therefore be an immediate extension of our work. Further, from the onset we have ignored any space-like axial conformal Killing vector on $S_{\Delta}$. So a generalization to the rotational case seems to be another plausible extension. Since the case of an isolated horizon appears as a special case $\rho\rightarrow0$, the consistency of our analysis can actually be checked by taking the isolated horizon limit. In fact we perform this consistency check and find that the final expressions and the first law does give back the results obtained for an isolated horizon.
We should mention at this point that our construction does not capture the most general dynamical situation, as constructed in [@Hayward:1993wb; @Ashtekar:2002ag]. The horizons discussed in these references are spacelike boundaries foliated by partially trapped two surfaces which may not be shear-free. Further, an integrated version of the first law has been demonstrated to exist, which captures the dynamics of growing black hole horizons in full generality. However in these constructions, which use metric variables, the existence of a well defined phase- space has not been established and consequently the first law does not follow directly from the symplectic structure. In our case we have assumed that there is no gravitational flux (shear is zero) but only matter flows across the null boundary $\Delta$. In this simplified geometry, we have demonstrated that a space of solutions of Einstein’s equations exists which admit the boundary conditions of CKH and that a differential version of the first law of black hole mechanics can be obtained. Also, we have used the first order formalism for the construction of this symplectic structure. We do not know if one may get a well defined symplectic structure for boundary conditions discussed in [@Hayward:1993wb; @Ashtekar:2002ag]. Even if one is able to construct a phase- space, it is not possible to obtain a differential version the first law since there is no analogue of the zeroth law for such boundaries, but an integrated version of the first law is expected to hold.
Given a form of the first law, it is obvious to compare with the first law of thermodynamics. However, since the horizon is growing, it describes a non- equilibrium situation and hence may differ considerably from equilibrium thermodynamics where one studies the transition from one equilibrium state to a nearby equilibrium state. One should keep in view that thermodynamics arises out of microscopic dynamics of the underlying degrees of freedom and have universal validity (that are independent of the underlying dynamics of a particular system). For a general dynamical spacetime (when the gravitational degrees of freedom are excited), there is no time translation symmetry and hence no definition of entropy may be possible. Also in non- equilibrium cases, a system may not get enough time to relax back to the equilibrium state and hence no canonical definition of temperature exists. But, in the present scenario, though the horizon makes transition between two states which are far from equilibrium, because there exits a conformal Killing vector, this leads to a definite identification of temperature and a first law and possibly entropy. One may then enquire if dynamically growing horizons is attributed some entropy that can arise from some counting of microstates. The boundary symplectic structure has a natural interpretation of being the symplectic structure of a field theory residing on the boundary. In the case of an isolated horizon it turns out to be an $SU(2)$ or an $U(1)$ Chern-Simons theory. A quantization of the boundary theory therefore provides a microscopic description of the entropy of the isolated horizon. Since we explicitly construct the boundary symplectic structure it will be interesting to see if it does coincide with any known topological field theory. A complete answer to such questions shall have important implications for thermodynamics as well as black hole physics.
The Connection in terms of Newman-Penrose co-effecients
=======================================================
Fix a set a internal null vectors $(l_I,n_I,m_I,\bar m_I) $ on $\Delta$ such that $\partial_a (l_I,n_I,m_I,\bar m_I)\stackrel{\Delta}{=}0$. Given any tetrad $e^I_a$, the null tetrad $(l_a,n_a,m_a,\bar m_a)$ can be expanded as $l_a=e^I_a~l_I$. The expression for $\Sigma^{IJ}$ can now be readily calculated and is given as. \^[IJ]{}&=&2l\^[\[I]{}n\^[J\]]{} \^[2]{}+2n(im l\^[\[I]{}|[m]{}\^[J\]]{}-i|[m]{} l\^[\[I]{}m\^[J\]]{})&-&2i l n m\^[\[I]{}|[m]{}\^[J\]]{}-2l(im n\^[\[I]{}|[m]{}\^[J\]]{}-i|[m]{} n\^[\[I]{}m\^[J\]]{})
This is the full expression for $\Sigma^{IJ}$ where nothing has been been assumed regarding the nature of the boundary $\Delta$. If $\Delta$ is a null surface and $l_{a}$ is the null normal, we get that \^[IJ]{}&&2l\^[\[I]{}n\^[J\]]{} \^[2]{}+2n(im l\^[\[I]{}|[m]{}\^[J\]]{}-i|[m]{} l\^[\[I]{}m\^[J\]]{})
The covariant derivative is defined to be compatible with the tetrad *i.e.* $\nabla_b~e^I_a=0$. The covariant derivatives on the null tetrads can be written in terms of the Newman-Penrose coeffecients and are given by the following,
\_al\_b&=&-(+|)n\_al\_b+|n\_am\_b+n\_a|[m]{}\_b-(+|)l\_al\_b+|l\_am\_b+l\_a|[m]{}\_b&&+\[(|+)|[m]{}\_al\_b -||[m]{}\_am\_b-|[m]{}\_a|[m]{}\_b+(+| )m\_al\_b-m\_a|[m]{}\_b-|m\_a m\_b\]\
\_a n\_b&=&(+|)n\_an\_b- n\_am\_b-|n\_a|[m]{}\_b+(+|)l\_an\_b-l\_am\_b-| l\_a|[m]{}\_b&&-\[(|+)|[m]{}\_an\_b-|[m]{}\_am\_b-||[m]{}\_a|[ m]{}\_b+(+|)m\_an\_b-| m\_a|[m]{}\_b-m\_a m\_b\]\
\_am\_b&=&-|n\_al\_b+n\_an\_b-(-|)n\_am\_b-|l\_al\_b+ l\_an\_b-(-|)l\_am\_b&&+\[||[m]{}\_al\_b-|[m]{}\_an\_b+(-|)|[m]{} \_am\_b+| m\_al\_b-m\_an\_b+(-|)m\_a[m]{}\_b\]
Next, once we have fixed a set of null internal vectors on $\Delta$, the connection can be expanded in terms of these Newman- Penrose coefficients. Note that $\nabla_a~l_I=\partial_a~l_I+A_{aI}^J~l_J$. Therefore on $\Delta$, we have $e^b_I\nabla_al_b\stackrel{\Delta}{=}A_{aI}~^Jl_J$ and hence
A\^[(l)]{}\_[aI]{} \^Jl\_J&&-(+|)n\_al\_I+|n\_am\_I+n\_a|[m]{}\_I-(+|)l\_al\_I+|l\_am\_I+l\_a|[m]{}\_I&&+\[(|+)|[m]{}\_al\_I-||[m]{}\_am\_I-|[m]{}\_a|[ m]{}\_I+(+|)m\_al\_I-m\_a|[m]{}\_I-|m\_a m\_I\]\
A\^[(l)]{}\_[aIJ]{}&& 2l\_[\[I]{}n\_[J\]]{}&&+ 2m\_[\[I]{}n\_[J\]]{}+ 2|[m]{}\_[\[I]{}n\_[J\]]{}where the subscript $l$ in $A^{(l)}$ indicates that the only the vector field $l^{a}$ has been used to evaluate the connection. Similarly, we can proceed for other vector fields $n^{a}, m^{a}$ and $\bar{m}^{a}$. The resulting connections are given as follows A\^[(n)]{}\_[aIJ]{}&& 2n\_[\[I]{}l\_[J\]]{}&&+(n\_a+l\_a-|[m]{}\_a-m\_a) 2m\_[\[I]{}l\_[J\]]{}+(| n\_a+|l\_a-|[m]{}\_a-||[m]{}\_a) 2|[m]{}\_[\[I]{}l\_[J\]]{}\
A\^[(m)]{}\_[aIJ]{}&&(-|n\_a-|l\_a+||[m]{}\_a+| m\_a) 2l\_[\[I]{}|[m]{}\_[J\]]{}+(n\_a+l\_a-|[m]{}\_a-m\_a) 2n\_[\[I]{}|[m]{}\_[J\]]{}&&+ 2m\_[\[I]{}|[m]{}\_[J\]]{}
The full connection is then given by: A\_[aIJ]{}&&2 l\_[\[I]{}n\_[J\]]{}&&+2 m\_[\[I]{}n\_[J\]]{}+2 |[m]{}\_[\[I]{}n\_[J\]]{}&&+2 m\_[\[I]{}l\_[J\]]{}+2 |[m]{}\_[\[I]{}l\_[J\]]{}&&+2 m\_[\[I]{}|[m]{}\_[J\]]{}
Note that as in in the case of $\Sigma_{IJ}$ no boundary condition has been assumed in the above expression. In the main part of the paper this expression for the connection eqn shall be used but with the boundary conditions.
Further, we would be requiring the exterior derivatives on the null tetrads. We therefore give the expressions here. dn=\_[a]{}n\_[b]{} dx\^adx\^b&=&-n m-|n|[m]{}+(+|)ln-l m-| l|[m]{}&&-\[(|+)|[m]{}n-|[m]{} m+(+|)mn-| m|[m]{}\]\
dl=\_[a]{}l\_[b]{} dx\^adx\^b&=&-(+|)n l+|nm+n|[m]{}+|lm+ l|[m]{}&&+\[(|+)|[m]{}l-||[m]{} m+(+|)ml-m|[m]{}\]\
dm=\_[a]{}m\_[b]{} dx\^adx\^b&=&-|n l-(-|)nm+l n-(-|)lm&&+\[||[m]{}l-|[m]{} n+(-|)|[m]{}m+| ml-mn\]From the above expressions, it follows that for the area two- form which is given by ${}^{2}\epsilon=im\wedge\bar{m}$, we get that $d~^2\epsilon=2\rho \,n\wedge ~^2\epsilon \,\mbox{and} \, \lie_l
~^2\epsilon=-2\rho~~^2\epsilon$.
Variation of the action {#appb}
=======================
Since the boundary symplectic structure turned out to be exact, it is at once evident that the variation of the action should be well-defined with the the boundary conditions considered. However one may need to add an additional boundary term in order to it. As has been pointed out that such terms won’t affect the symplectic structure though. Therefore for completeness we consider the variation of the action and find out the necessary boundary term needed to make the variation well defined. We consider the action for gravity and a scalar field without any boundary terms a priori. The expression for $\Theta$ on $\Delta$ is calculated imposing the boundary conditions and the required boundary term can be obtained. We have L\_[M+G]{}=-(\^[IJ]{} F\_[IJ]{})-dd; It follows that d()=-d(\^[IJ]{} A\_[IJ]{})-d(d) Consider the gravity terms first[^4] &&-2 \^2+2(nim)(|[m]{})-2(n i|[m]{})()&=&-2 \^2+2(nim)(|[m]{})-2(n i|[m]{})()&=&d-4n\^2+2 \^2+4n\^2 +2 n \^2&=&d+2 \^2+(2 n \^2) The matter term gives
\(d) &=&-d(\^2 \^2)+( d) \^2&=&-d(\^2 \^2)-(n) \^2Adding everything up, one finds that, d()=-d(\^[IJ]{} A\_[IJ]{})-d(d)=-d( n \^2) So one needs to add $\frac{1}{8\pi G}\int_{\Delta}\left(\rho n\wedge~^2\epsilon\right)$ to the action to make the variation well-defined.
Boundary Symplectic Structure for Gravity
=========================================
The symplectic current in first order gravity is therefore given by, J\_G(\_1,\_2)&=&-\_[\[1]{}\^[IJ]{} \_[2\]]{}A\_[IJ]{}
We need to pull back the above expression on to the boundary and check if it is exact.
&&-2\_[\[1]{} \^2[ ****]{} \_[2\]]{}((+|)n-(+|)m-(|[ ]{}+)|[m]{})&&+2\_[\[1]{}(nim)\_[2\]]{}(||[m]{}) -2\_[\[1]{}(ni|[m]{})\_[2\]]{}(m)
We consider the first term in the above expression. By using the Ricci identity in terms of Newman-Penrose co-effecients D=\^2+(+|)+\_[00]{} we find that the first term can be written in the following form, -2\_[\[1]{} \^2[****]{}\_[2\]]{}((+|)n)&=&-2\_[\[1]{} \^2[****]{}\_[2\]]{}(( -- )n)&&=d(2\_[\[1]{} \^2[****]{} \_[2\]]{}log )-(2\_[\[1]{} d\^2[****]{}\_[2\]]{}log)&&+2\_[\[1]{} \^2[****]{}\_[2\]]{}((+)n)
Since the first term in the above expression is already exact, we leave it for the the moment and check if there is any simplication of the other terms when combined with the rest of the third and forth term in the symplectic current.
-2\_[\[1]{} d\^2[****]{}\_[2\]]{}log &&=-4\_[\[1]{} inm|[m]{} \_[2\]]{}log&&=-2\_[\[1]{} (nim)|[m]{} \_[2\]]{}-2(n im)\_[\[1]{}|[m]{} \_[2\]]{}&&+2\_[\[1]{} (ni|[m]{}) m \_[2\]]{}+2(ni|[m]{})\_[\[1]{} m \_[2\]]{}
The third and the fourth term in the symplectic current gives: &&2\_[\[1]{}(nim)\_[2\]]{}(|[m]{}) -2\_[\[1]{}(ni|[m]{})\_[2\]]{}(m)&&=2\_[\[1]{}(n im)|[m]{} \_[2\]]{}()+2\_[\[1]{}(n im)\_[2\]]{}|[m]{}&&-2\_[\[1]{}(ni|[m]{}) m \_[2\]]{}()-2\_[\[1]{}(ni|[m]{})\_[2\]]{}m
Adding the above two equations and then simplifying gives:
&&-2\_[\[1]{} d\^2[****]{}\_[2\]]{}log+2\_[\[1]{}(n im)\_[2\]]{}(|[m]{}) -2\_[\[1]{}(ni|[m]{})\_[2\]]{}(m)&&=-2n\_[\[1]{} \^2\_[2\]]{}+2\_[\[1]{} (n)\_[2\]]{} \^2&&=-2\_[\[1]{} \^2\_[2\]]{}(n)
So the boundary term becomes d(2\_[\[1]{} \^2[****]{} \_[2\]]{}log)+2\_[\[1]{} \^2[ ****]{}\_[2\]]{}(n)
Bulk Symplectic structure
=========================
For any vector field $\xi$ generating diffeomorphisms, the corresponding phase space variation $\delta_\xi$ acts in the bulk like $\lie_\xi$. It can then be shown that J\_G(,\_)&=&-+()e\^I Similarly for the matter fields, we get that J\_M(,\_) &=&d--.d (d)\
The second and the third term in the last expression enters Einstein’s equation. Therefore the full bulk symplectic structure is,
\_[M]{}J(,\_)&=&-\_[ M]{}+\_[M]{} (.d)
Acknowedgements {#acknowedgements .unnumbered}
---------------
The authors acknowledge the discussions with Amit Ghosh. The authors also thank the anonymous referee for suggestions that made the presentation better. AC is partially supported through the UGC- BSR start-up grant vide their letter no. F.20-1(30)/2013(BSR)/3082. AG is supported by Department of Atomic-Energy, Govt. Of India.
[99]{} S. W. Hawking and G. F. R. Ellis, “The Large scale structure of space-time,” Cambridge University Press, Cambridge, 1973 R. M. Wald, “General Relativity,” Chicago, USA: Chicago Univ. Pr. ( 1984) 491p S. W. Hawking, Commun. Math. Phys. [**25**]{}, 152 (1972). J. M. Bardeen, B. Carter and S. W. Hawking, “The Four laws of black hole mechanics,” Commun. Math. Phys. [**31**]{}, 161 (1973). S. W. Hawking, “Particle Creation by Black Holes,” Commun. Math. Phys. [**43**]{}, 199 (1975) \[Erratum-ibid. [**46**]{}, 206 (1976)\].
J. D. Bekenstein, “Black holes and entropy,” Phys. Rev. D [**7**]{}, 2333 (1973). J. D. Bekenstein, “Generalized second law of thermodynamics in black hole physics,” Phys. Rev. D [**9**]{}, 3292 (1974). R. M. Wald, “Quantum field theory in curved space-time and black hole thermodynamics,” Chicago, USA: Univ. Pr. (1994) 205 p
R. M. Wald, “Black hole entropy is the Noether charge,” Phys. Rev. D [**48**]{}, 3427 (1993) \[gr-qc/9307038\]. V. Iyer and R. M. Wald, “Some properties of Noether charge and a proposal for dynamical black hole entropy,” Phys. Rev. D [**50**]{}, 846 (1994) \[gr-qc/9403028\].
T. Jacobson, G. Kang and R. C. Myers, “On black hole entropy,” Phys. Rev. D [**49**]{}, 6587 (1994) \[gr-qc/9312023\]. D. Youm, “Black holes and solitons in string theory,” Phys. Rept. [**316**]{}, 1 (1999) \[hep-th/9710046\]. S. Carlip, “Entropy from conformal field theory at Killing horizons,” Class. Quant. Grav. [**16**]{}, 3327 (1999) \[gr-qc/9906126\].
O. Dreyer, A. Ghosh and A. Ghosh, “Entropy from near-horizon geometries of Killing horizons,” Phys. Rev. D [**89**]{}, 024035 (2014) \[arXiv:1306.5063 \[gr-qc\]\].
A. Ghosh, “Note on Kerr/CFT correspondence in a first order formalism,” Phys. Rev. D [**89**]{}, 124035 (2014) \[arXiv:1404.5260 \[gr-qc\]\].
A. Ashtekar, C. Beetle and S. Fairhurst, “Isolated horizons: A Generalization of black hole mechanics,” Class. Quant. Grav. [**16**]{}, L1 (1999) \[gr-qc/9812065\]. A. Ashtekar, C. Beetle, O. Dreyer, S. Fairhurst, B. Krishnan, J. Lewandowski and J. Wisniewski, “Isolated horizons and their applications,” Phys. Rev. Lett. [**85**]{} (2000) 3564 \[gr-qc/0006006\]. A. Ashtekar, S. Fairhurst and B. Krishnan, “Isolated horizons: Hamiltonian evolution and the first law,” Phys. Rev. D [**62**]{} (2000) 104025 \[gr-qc/0005083\].
A. Ashtekar, C. Beetle and J. Lewandowski, “Mechanics of rotating isolated horizons,” Phys. Rev. D [**64**]{}, 044016 (2001) \[gr-qc/0103026\]. A. Ashtekar, C. Beetle and J. Lewandowski, “Geometry of generic isolated horizons,” Class. Quant. Grav. [**19**]{}, 1195 (2002) \[gr-qc/0111067\]. A. Chatterjee and A. Ghosh, “Generic weak isolated horizons,” Class. Quant. Grav. [**23**]{}, 7521 (2006) \[gr-qc/0603023\]. A. Chatterjee and A. Ghosh, “Laws of Black Hole Mechanics from Holst Action,” Phys. Rev. D [**80**]{}, 064036 (2009) \[arXiv:0812.2121 \[gr-qc\]\].
L. Smolin, “Linking topological quantum field theory and nonperturbative quantum gravity,” J. Math. Phys. [**36**]{}, 6417 (1995) \[gr-qc/9505028\].
K. V. Krasnov, “Counting surface states in the loop quantum gravity,” Phys. Rev. D [**55**]{}, 3505 (1997) \[gr-qc/9603025\].
C. Rovelli, “Black hole entropy from loop quantum gravity,” Phys. Rev. Lett. [**77**]{}, 3288 (1996) \[gr-qc/9603063\].
A. Ashtekar, J. Baez, A. Corichi and K. Krasnov, “Quantum geometry and black hole entropy,” Phys. Rev. Lett. [**80**]{}, 904 (1998) \[gr-qc/9710007\]. A. Ashtekar, A. Corichi and K. Krasnov, “Isolated horizons: The Classical phase space,” Adv. Theor. Math. Phys. [**3**]{}, 419 (1999) \[gr-qc/9905089\]. A. Ghosh and P. Mitra, “Counting black hole microscopic states in loop quantum gravity,” Phys. Rev. D [**74**]{}, 064026 (2006) \[hep-th/0605125\].
A. Ghosh and P. Mitra, “Fine-grained state counting for black holes in loop quantum gravity,” Phys. Rev. Lett. [**102**]{}, 141302 (2009) \[arXiv:0809.4170 \[gr-qc\]\].
A. Ghosh and A. Perez, “Black hole entropy and isolated horizons thermodynamics,” Phys. Rev. Lett. [**107**]{}, 241301 (2011) \[Erratum-ibid. [**108**]{}, 169901 (2012)\] \[arXiv:1107.1320 \[gr-qc\]\]. A. Ghosh, K. Noui and A. Perez, “Statistics, holography, and black hole entropy in loop quantum gravity,” arXiv:1309.4563 \[gr-qc\]. A. Corichi, I. Rubalcava and T. Vukasinac, “First order gravity: Actions, topological terms and boundaries,” arXiv:1312.7828 \[gr-qc\]. A. Corichi, I. Rubalcava-García and T. Vukašinac, “Hamiltonian and Noether charges in first order gravity,” Gen. Rel. Grav. [**46**]{}, 1813 (2014).
S. A. Hayward, “General laws of black hole dynamics,” Phys. Rev. D [**49**]{}, 6467 (1994). S. A. Hayward, “Spin coefficient form of the new laws of black hole dynamics,” Class. Quant. Grav. [**11**]{}, 3025 (1994) \[gr-qc/9406033\].
A. Ashtekar and B. Krishnan, “Dynamical horizons: Energy, angular momentum, fluxes and balance laws,” Phys. Rev. Lett. [**89**]{}, 261101 (2002) \[gr-qc/0207080\].
A. Ashtekar and B. Krishnan, “Dynamical horizons and their properties,” Phys. Rev. D [**68**]{}, 104030 (2003) \[gr-qc/0308033\]. A. Ashtekar and B. Krishnan, “Isolated and dynamical horizons and their applications,” Living Rev. Rel. [**7**]{}, 10 (2004) \[gr-qc/0407042\].
I. Booth and S. Fairhurst, “The First law for slowly evolving horizons,” Phys. Rev. Lett. [**92**]{}, 011102 (2004) \[gr-qc/0307087\]. I. Booth and S. Fairhurst, “Isolated, slowly evolving, and dynamical trapping horizons: Geometry and mechanics from surface deformations,” Phys. Rev. D [**75**]{}, 084019 (2007) \[gr-qc/0610032\]. I. Booth and S. Fairhurst, “Extremality conditions for isolated and dynamical horizons,” Phys. Rev. D [**77**]{}, 084005 (2008) \[arXiv:0708.2209 \[gr-qc\]\]. C. C. Dyer and E. Honig, J. Math. Phys. 20, 409 (1979)
J. Sultana and C. C. Dyer, J. Math. Phys. 45, 4764 (2004)
J. Sultana and C. C. Dyer, “Cosmological black holes: A black hole in the Einstein-de Sitter universe,” Gen. Rel. Grav. [**37**]{}, 1347 (2005). T. Jacobson and G. Kang, “Conformal invariance of black hole temperature,” Class. Quant. Grav. [**10**]{}, L201 (1993) \[gr-qc/9307002\]. A. B. Nielsen and J. T. Firouzjaee, “Conformally rescaled spacetimes and Hawking radiation,” Gen. Rel. Grav. [**45**]{}, 1815 (2013) \[arXiv:1207.0064 \[gr-qc\]\].
S. Chandrasekhar, “The mathematical theory of black holes,” Oxford, UK: Clarendon (1985) 646 P.
G. J. Galloway and R. Schoen, “A Generalization of Hawking’s black hole topology theorem to higher dimensions,” Commun. Math. Phys. [**266**]{}, 571 (2006) \[gr-qc/0509107\].
[^1]: The entire construction and whatever follows goes through for negative $\rho$ with the replacement $\lvert\rho\rvert$ in place of $\rho$ in the argument of $\log$
[^2]: We assume that the contribution from the boundary at asymptotic infinity is a total variation $\delta E^\infty$.
[^3]: If the stress tensor satisfies the dominant enegy condition then $(2\rho+\epsilon+\bar\epsilon)$ is a constant on $\Delta$ [@Sultana:2005tp].
[^4]: In our case it might not be possible to define a unique covariant derivative on $\Delta$. However, since in the the calculations $l^a\nabla_a$ acts only on functions, the amibiguity do not play a role.
|
**Madness and regularity properties**
[Haim Horowitz and Saharon Shelah]{}
**Abstract**
Starting from an inaccessible cardinal, we construct a model of $ZF+DC$ where there exists a mad family and all sets of reals are $\mathbb Q$-measurable for $\omega^{\omega}$-bounding sufficiently absolute forcing notions $\mathbb Q$.[^1]
**Introduction**
Our study concerns the interactions between mad families and other types of pathological sets of reals. Given a forcing notion $\mathbb Q$ whose conditions are subtrees of $\omega^{<\omega}$ ordered by reverse inclusion, the notion of $\mathbb Q-$measurability is naturally defined. As the existence of mad families and non-$\mathbb Q-$measurable sets follows from the axiom of choice, one may consider the possible implications between the existence of mad families and the existence of non-$\mathbb Q-$measurable sets. The study of models of $ZF+DC$ where no mad families exist was initiated by Mathias in [\[]{}Ma[\]]{}, more results were obtained recently in [\[]{}HwSh1090[\]]{}, [\[]{}NN[\]]{} and [\[]{}To[\]]{}. Models of $ZF+DC$ where all sets of reals are $\mathbb Q$-measurable for various forcing notions $\mathbb Q$ were first studied by Solovay in [\[]{}So[\]]{}.
Our main goal is to show that $\mathbb Q-$measurability for $\omega^{\omega}$-boundning sufficiently absolute forcing notions does not imply the non-exsitence of mad families. In particular, as Random real forcing is $\omega^{\omega}$-bounding, it will follow that Lebesgue measurability for all sets of reals does not imply the non-existence of mad families.
We follow the strategy of [\[]{}Sh218[\]]{}, where a model of $ZF+DC+"$all sets of reals are Lebesgue measurable but there is a set without the Baire Property$"$ was constructed. Fixing an inaccessible cardinal $\kappa$, we define a partial order $AP$ consisting of pairs $(\mathbb P,\Gamma)$, where $\mathbb P$ is a forcing notion from $H(\kappa)$ and $\Gamma$ is an approximation of the desired mad family such that finite unions of members of $\Gamma$ are not dominated by reals from $V$. We shall obtain our model by forcing with this partial order and then with the partial order introduced generically by $AP$. The main point will be an amalgamation argument for $AP$ (over $\mathbb Q$-generic reals for an appropriate $\mathbb Q$), which will allow us to repeat Solovay’s argument from [\[]{}So[\]]{}.
Remark: It was brought to our attention by Paul Larson and Jindra Zapletal that a model of “every set of reals is Lebesgue measurable and there is a mad family” can also be constructed using the arguments from Section 5 of their paper [\[]{}LZ[\]]{}. However, they assume the existence of a proper class of Woodin cardinals, while in this paper we only assume the existence of an inaccessible cardinal.
$\\$
**The main result**
**Hypothesis 1:** Throughout the paper, $\bold f$ will be a fixed forcing frame (defined below) with $\kappa_{\bold f}=\kappa$ a fixed inaccessible cardinal.
**Definition 2:** Let $\bold f=(\kappa_{\bold f},\bold{P}_{\bold f},\bold{Q}_{\bold f})=(\kappa,\bold P,\bold Q)$ be a forcing frame when:
a\. $\kappa$ is the inaccessible cardinal from Hypothesis 1.
b\. $\bold P$ is the set of forcing notions from $H(\kappa)$.
c\. $\bold Q$ is a family of $\omega^{\omega}-$bounding forcing notions with sufficiently absolute definitions.
d\. If $\mathbb P \in \bold P$ and $V^{\mathbb P} \models "\mathbb Q \in \bold Q"$, then $\mathbb Q \in H(\kappa)^{(V^{\mathbb P})}$.
**Definition 3:** Let $AP=AP_{\bold f}$ be the partial order defined as follows:
a\. $a\in AP$ iff $a$ has the form $(\mathbb P,\Gamma)=(\mathbb{P}_a,\Gamma_a)$ where:
1\. $\mathbb P \in \bold P$ and $\Gamma$ is an infinite set of canonical $\mathbb P-$names of reals such that $\Vdash_{\mathbb P} "\Gamma$ is almost disjoint$"$.
2\. If $\underset{\sim}{\tau} \in \Gamma$, then $\Vdash_{\mathbb P} "\underset{\sim}{\tau}$ is an infinite subset of $\omega"$.
3\. For $a\in AP$, let $\Omega_a$ be the set of $\underset{\sim}{\tau} \in \Gamma_a$ which are objects and not just names.
4\. If $1\leq n$, $\underset{\sim}{a_0},...,\underset{\sim}{a_{n-1}} \in \Gamma_a \setminus \Omega_a$, $\underset{\sim}{a}=\underset{l<n}{\cup}\underset{\sim}{a_l}$ and $f_{\underset{\sim}{a}} : \omega \rightarrow \omega$ is the function enumerating $\underset{\sim}{a}$ in an increasing order, then $\Vdash_{\mathbb P} "f_{\underset{\sim}{a}}$ is not dominated by any $f\in (\omega^{\omega})^V"$.
b\. $a\leq_{AP} b$ iff
1\. $\mathbb{P}_a \lessdot \mathbb{P}_b$.
2\. $\Gamma_a \subseteq \Gamma_b$.
3\. If $\underset{\sim}{a_0},...,\underset{\sim}{a_{n-1}} \in \Gamma_b \setminus \Gamma_a$, $\underset{\sim}{a}=\underset{l<n}{\cup} \underset{\sim}{a_l}$ and $f_{\underset{\sim}{a}}$ enumerates $\underset{\sim}{a}$ in an increasing order, then $\Vdash_{\mathbb{P}_b} "f_{\underset{\sim}{a}}$ is not dominated by any member of $(\omega^{\omega})^{V[G\cap \mathbb{P}_a]}$.
**Observation 4:** $(AP,\leq)$ is indeed a partial order.
**Proof:** Suppose that $a\leq b$ and $b\leq c$. Let $\underset{\sim}{a_0},...,\underset{\sim}{a_{n-1}} \in \Gamma_c \setminus \Gamma_a$, and let $\underset{\sim}{a}$ and $f_{\underset{\sim}{a}}$ be as in Definition 3(b)(3). We may assume wlog that for some $0<m<n$, $\underset{\sim}{a_l} \in \Gamma_b$ iff $l<m$ (the cases $m=0$ and $m=n$ are trivial). Let $G_c \subseteq \mathbb{P}_c$ be $V$-generic and let $G_a=G_c \cap \mathbb{P}_a$ and $G_b=G_c \cap \mathbb{P}_b$. Let $g=(n_i : i<\omega) \in V[G_a]$, wlog $g$ is increasing. We shall prove that $f_{\underset{\sim}{a}}$ is not dominated by $g$.
Let $a_i=\underset{\sim}{a_i}[G_c]$, $a=\underset{\sim}{a}[G_c]$ and $b=\underset{l<m}{\cup}a_l$.
Subclaim 1: For infinitely many $i$, $[n_i,n_{i+1}) \cap (\underset{l<n}{\cup}a_l)=\emptyset$.
Subclaim 2: Subclaim 1 is equivalent to $"f_a$ is not dominated by $g"$.
Proof of Subclaim 1: Let $u=\{i : [n_i,n_{i+1}) \cap b=\emptyset\} \in V[G_b]$. By the fact that $a\leq b$ and by subclaim 2, $u$ is infinite. Let $(i(l) : l<\omega) \in V[G_b]$ be an increasing enumeration of $u$, so $(n_{i(l)} : l<\omega) \in V[G_b]$ is increasing. Let $c=\underset{m\leq l<n-1}{\cup}a_l$ and $v=\{l : [n_{i(l)},n_{i(l+1)}) \cap c=\emptyset\}$. As before, $v$ is infinite. If $l \in v$ then $c\cap [n_{i(l)},n_{i(l+1)})=\emptyset$ and therefore, $c\cap [n_{i(l)},n_{i(l)+1})=\emptyset$. Similarly, if $l\in v$ then $i(l) \in u$ and therefore $b\cap [n_{i(l)},n_{i(l)+1})$. It follows that $l\in v \rightarrow (b\cup c) \cap [n_{i(l)},n_{i(l)+1})=\emptyset$, and as $v$ is infinite, we’re done.
Proof of Subclaim 2: Suppose that $f_{\underset{\sim}{a}}$ is not dominated by any $g\in (\omega^{\omega})^{V^{\mathbb{P}_a}}$ and let $g=(n_i : i<\omega) \in V^{\mathbb{P}_a}$ be increasing. Choose $f\in V^{\mathbb{P}_c}$ such that $f$ is increasing, $l<f(l)$ for every $l$ and $|\{i : n_i \in [l,f(l))\}|$ is sufficiently large (e.g. $>2^l$). By our assumption, for infinitely many $l$, $f(l)\leq$ the $l$th member of $\underset{\sim}{a}$, and therefore $|\underset{\sim}{a} \cap f(l)| \leq l$. Let $u=\{ l : |\underset{\sim}{a} \cap f(l)| \leq l\}$, so $u$ is infinite. For $l\in u$, $l+1<|\{i : l\leq i, [n_i,n_{i+1}) \subseteq [l,f(l))\}|$, and as $u$ is infinite, for some $i$ such that $l\leq i$, $[n_i,n_{i+1}) \subseteq [l,f(l))$ and $[n_i,n_{i+1}) \cap \underset{\sim}{a}=\emptyset$. Therefore, for infinitely many $i$, $[n_i,n_{i+1}) \cap \underset{\sim}{a}=\emptyset$.
In the other direction, suppose that $f_{\underset{\sim}{a}}$ satisfies the condition of Subclaim 1. Let $g\in (\omega^{\omega})^{V^{\mathbb{P}_a}}$, we shall prove that $f_{\underset{\sim}{a}}$ is not dominated by $g$. We may assume wlog that $g$ is increasing. Choose the sequence $(n_i : i<\omega)$ by induction such that $n_0=0$ and $n_{i+1}>n_i+g(n_i)$, so $(n_i : i<\omega) \in V^{\mathbb{P}_a}$. By the assumption, the set $u=\{i : [n_i,n_{i+1}) \cap \underset{l<n}{\cup}a_l=\emptyset\}$ is infinite. For every $i\in u$, $|a\cap n_i| \leq n_i$, therefore $n_i < f_a(n_i)$. As $[n_i,n_{i+1}) \cap a=\emptyset$, it follows that $n_{i+1} \leq f_a(n_i)$, therefore $g(n_i)<n_{i+1} \leq f_a(n_{n_i})$, so $f_a$ is not dominated by $g$. $\square$
**Observation 4:** a. Every $\mathbb P \in \bold P$ is $\kappa-cc$, and $\bold P$ is closed under $\lessdot-$increasing unions of length $<\kappa$.
b\. If $\mathbb P \in \bold P$ and $\underset{\sim}{\mathbb Q}$ is a canonical $\mathbb{P}-$name of a case of $\bold Q$ which is in $H(\kappa)$, then $\mathbb P \star \underset{\sim}{\mathbb Q} \in \bold P$. $\square$
**Observation 5:** a. If $a\in AP$ then $(\{0\},\Omega_a) \in AP$ and $(\{0\},\Omega_a) \leq a$.
b\. $AP$ is $(<\kappa)-$complete. $\square$
**Claim 6:** $(AP,\leq)$ has the division property, namely, if $a\leq b$ and $\underset{\sim}{x}$ is a $\mathbb{P}_b$-name of a real such that $\Vdash_{\mathbb{P}_b} "(\omega^{\omega})^{V[\mathbb{P}_a]}$ is cofinal in $(\omega^{\omega})^{V[\mathbb{P}_a,\underset{\sim}{x}]}"$, then there is $a_1 \in AP$ such that:
a\. $a\leq a_1 \leq b$.
b\. $\Gamma_{a_1}=\Gamma_a$.
c\. $\mathbb{P}_{a_1}=\mathbb{P}_a \star \underset{\sim}{x}$ in the natural sense. $\square$
**Claim 7 ($(AP,\leq)$ has the amalgamation property):** Assume that $a_0 \leq a_l$ $(l=1,2)$, then there are $b_l$ $(l\leq 3)$ and $g_l$ $(l\leq 2)$ such that:
a\. $b_0 \leq b_l \leq b_3$ $(l=1,2)$.
b\. $g_l$ is an isomorphism from $b_l$ to $a_l$.
c\. $g_0 \subseteq g_l$ $(l=1,2)$.
**Proof:** We may assume wlog that $\mathbb{P}_{a_0}$ is trivial and that $\Omega_{a_1}=\Omega_{a_2}=\Gamma_{a_0}$ (as we can simply take the quotients).
We define $\mathbb{P}_{b_3}$ as follows:
a\. $p\in \mathbb{P}_{b_3}$ iff $p=(p_1,p_2) \in \mathbb{P}_{a_1} \times \mathbb{P}_{a_2}$ and for some $l(p), n_p, A_{p,1},A_{p,2},\underset{\sim}{a_{p,1}},\underset{\sim}{a_{p,2}}$ the following hold:
1\. $l(p) \in \{1,2\}$ and $n_p <\omega$.
2\. $A_{p,l}$ is a finite subset of $\Gamma_{a_l}$ with union $\underset{\sim}{a_{p,l}}$ $(l=1,2)$.
3\. For every $n>n_p$, there is $r_{n} \in \mathbb{P}_{a_{l(p)}}$ such that $\mathbb{P}_{a_{l(p)}} \models p_{l(p)} \leq r_{n}$ and $r_{n} \Vdash "\underset{\sim}{a_{p,l(p)}} \cap n \subseteq n_p"$.
b\. $\mathbb{P}_{b_3} \models p\leq q$ iff
1\. $p=(p_1,p_2), q=(q_1,q_2) \in \mathbb{P}_{a_1} \times \mathbb{P}_{a_2}$.
2\. $p_l \leq q_l$ $(l=1,2)$.
3\. $n_p \leq n_q$.
4\. $A_{p,l} \subseteq A_{q,l}$ $(l=1,2)$.
5\. There is no $n \in [n_p,n_q)$ such that $q_1 \nVdash "n\notin \underset{\sim}{a_{q,1}}"$ and $q_2 \nVdash "n\notin \underset{\sim}{a_{q,2}}"$.
We shall now define embeddings $f_l: \mathbb{P}_{a_l} \rightarrow \mathbb{P}_{b_3}$ $(l=1,2)$ as follows: For $p\in \mathbb{P}_{a_l}$, $f_l(p)=q \in \mathbb{P}_{b_3}$ will be the condition defined as follows:
a\. $q_l=p$ and $q_{3-l}=0_{\mathbb{P}_{a_{3-l}}} \in \mathbb{P}_{a_{3-l}}$.
b\. $l(q)=l$, $n_p=0$.
c\. $A_{q,1}=\emptyset=A_{q,2}$.
Subclaim 0: $\mathbb{P}_{b_3}$ is a partial order.
Subclaim 1: For every $p=(p_1,p_2)\in \mathbb{P}_{b_3}$ and open dense $I\subseteq \mathbb{P}_{3-l(p)}$, there is $q\in \mathbb{P}_{b_3}$ above $p$ such that $l(q)=3-l(p)$ and $q_{3-l(p)} \in I$.
Proof: Let $i=3-l(p)$ and let $p_i' \in I$ be above $p_i$. By the definition of $AP$, $f_{\underset{\sim}{a_{p,i}}}$ is not dominated by any function from $V$. We shall prove that there are $q_i \in \mathbb{P}_{a_i}$ above $p_i'$ and $n_*>n_p$ such that for every $n>n_*$, there is $q'$ above $q_i$ such that $q'\Vdash "\underset{\sim}{a_{p,i}} \cap [n_*,n)=\emptyset"$. Actually, $q_i=p_i'$ should work. Suppose not, then for every $n_*>n_p$ there is $n>n_*$ such that there is no $q'$ above $q_i$ forcing that $\underset{\sim}{a_{p,i}} \cap [n_*,n)=\emptyset$. Now choose $(n_j : j<\omega)$ by induction on $j$ as follows: $n_0=n_p+1$, and $n_{j+1}$ is the minimal $n>n_j$ such that there is no $q'$ above $q_i$ forcing that $\underset{\sim}{a_{p,i}} \cap [n_j,n)=\emptyset$. By the same argument as in the proof of observation 4, as $(n_j : j<\omega) \in V$, $p_i' \Vdash "\underset{\sim}{a_{p,i}} \cap [n_j,n_{j+1})=\emptyset$ for infinitely many $j"$. Therefore, there is $q'$ above $p_i'$ and $i_*$ such that $q' \Vdash "\underset{\sim}{a} \cap [n_{i_*},n_{i_*+1})=\emptyset"$, contradicting the choice of $n_{i_*+1}$.
Now define $q\in \mathbb{P}_{b_3}$ as follows:
1\. $q_i$ is as above.
2\. $q_{l(p)}$ is any member of $\mathbb{P}_{l(p)}$ which is above $p_{l(p)}$ and forces that $[n,n_*) \cap \underset{\sim}{a_{p,l(p)}}=\emptyset$ (such condition exists by clause (a)(3) in the definition of $\mathbb{P}_{b_3}$).
3\. $l(q)=i$.
4\. $n_q=n_*$.
5\. $A_{q,l}=A_{p,l}$ and $\underset{\sim}{a_{q,l}}=\underset{\sim}{a_{p,l}}$ for $l=1,2$.
It’s now easy to check that $q$ is as required.
Subclaim 2: a. $\{p \in \mathbb{P}_{b_3} : l(p)=i\}$ is dense in $\mathbb{P}_{b_3}$ for $i=1,2$.
b\. $I_n:=\{p \in \mathbb{P}_{b_3} : n_p>n\}$ is dense in $\mathbb{P}_{b_3}$.
Proof: (a) follows from Subclaim 1. (b) follows from the proof of Subclaim 1, as we note that $n_q=n_*>n_p$ in that proof.
Subclaim 3: $f_l: \mathbb{P}_{a_l} \rightarrow \mathbb{P}_{b_3}$ is a complete embedding for $l=1,2$.
Proof: It suffices to show that $f_l$ is a complete embedding into $\{p\in \mathbb{P}_{b_3} : l(p)=l\}$, which follows from the existence of a projection $\pi: \{p\in \mathbb{P}_{b_3} : l(p)=l\} \rightarrow \mathbb{P}_{a_l}$ defined in the natural way.
Subclaim 4: For every finite $A_1 \subseteq \Gamma_{a_1}$ and $A_2 \subseteq \Gamma_{a_2}$, the set $\{p \in \mathbb{P}_{b_3} : \underset{i=1,2}{\wedge}A_i \subseteq A_{p,i}\}$ is open dense.
Proof: In order to prove the claim by induction on $|A_1|+|A_2|$, it suffices to prove it when $A_i=\{\underset{\sim}{b}\}$ and $A_{3-i}=\emptyset$ for $i\in \{1,2\}$. Let $p\in \mathbb{P}_{b_3}$ and suppose that $l(p)=3-i$, it’s now easy to extend $p$ simply by adding $\underset{\sim}{b}$ to $A_{p,i}$. If $l(p)=i$, then by previous claims, there is $q$ above $p$ such that $l(q)=3-l(p)$, and now extend $q$ as in the previous case.
Subclaim 5: Let $\Gamma:=f_1(\Gamma_{a_1}) \cup f_2(\Gamma_{a_2})$, then $\Gamma$ is a set of canonical $\mathbb{P}_{b_3}$-names of infinite subsets of $\omega$ and $\Vdash_{\mathbb P} "\Gamma$ is almost disjoint$"$.
Proof: The first part follows by the fact that $f_1$ and $f_2$ are complete embeddings. In order to prove the second part, it suffices to show that if $\underset{\sim}{r} \in \Gamma_{a_1}$ and $\underset{\sim}{s} \in \Gamma_{a_2}$, then $\Vdash_{\mathbb P} "|\underset{\sim}{r} \cap \underset{\sim}{s}|< \aleph_0"$. Given $p \in \mathbb{P}_{b_3}$, by Subclaim 4, there is a stronger condition $q$ such that $\underset{\sim}{r} \in A_{q,1}$ and $\underset{\sim}{s} \in A_{q,2}$. We shall prove that $q\Vdash "|\underset{\sim}{r} \cap \underset{\sim}{s}|<\aleph_0"$.
Recall that for every $n$, the set $I_n=\{r \in \mathbb{P}_{b_3} : n\leq n_r\}$ is dense. Now let $G\subseteq \mathbb{P}_{b_3}$ be generic over $V$ such that $q\in G$, then for every $n_q <n$, there is $q_n \in G$ such that $n\leq n_{q_n}$. By the definition of the partial order $\leq_{\mathbb{P}_{b_3}}$ (clause (b)(5)), it follows that $q\Vdash_{\mathbb{P}_{b_3}} "|\underset{\sim}{r} \cap \underset{\sim}{s}|<\aleph_0"$.
Subclaim 6: Let $b_3=(\mathbb{P}_{b_3},\Gamma_{b_3})$ where $\Gamma_{b_3}$ is $\Gamma$ from the previous subclaim, then $b_3$ satisfies clauses $(1)+(2)$ from Definition (3)(a). As $\Omega_{a_1}=\Omega_{a_2}$, it follows that $\Omega_{b_3}=\Omega_{a_1}=\Omega_{a_2}$.
For $l=1,2$, let $b_l=f_l(a_l) \in AP$, then clauses $(1)+(2)$ from Definition (3)(b) hold for $b_l$ and $b_3$.
Subclaim 7: $b_3 \in AP$.
Proof: Let $A\subseteq \Gamma_{b_3} \setminus \Omega_{b_3}$ be finite, so there are finite sets $A_l \subseteq \Gamma_{a_l} \setminus \Omega_{a_l}$ $(l=1,2)$ such that $A=f_1(A_1) \cup f_2(A_2)$. Let $(n_i : i<\omega) \in (\omega^{\omega})^V$ be increasing and let $\underset{\sim}{u}=\{i : [n_i,n_{i+1}) \cap (\cup \{ \underset{\sim}{a} : \underset{\sim}{a} \in A \})=\emptyset\}$. Let $(p_1,p_2) \in \mathbb{P}_{b_3}$ and $n<\omega$, we shall find $(q_1,q_2)$ and $i>n$ such that $(p_1,p_2) \leq (q_1,q_2) \in \mathbb{P}_{b_3}$ and $(q_1,q_2) \Vdash_{\mathbb{P}_{b_3}} "i\in \underset{\sim}{u}"$. Without loss of generality, $l((p_1,p_2))=2$, and by Subclaim 4, wlog $A_i \subseteq A_{(p_1,p_2),i}$ $(i=1,2)$. For $l=1,2$, let $\underset{\sim}{a_l}=\cup \{ \underset{\sim}{a} : \underset{\sim}{a} \in A_l\}$, so $\underset{\sim}{a_l}$ is a $\mathbb{P}_{a_l}-$name and $\Vdash_{\mathbb{P}_{a_l}}"(\exists^{\infty}i)(\underset{\sim}{a_l} \cap [n_i,n_{i+1})=\emptyset)"$. Choose $(p_{1,l},j_{1,l} : l<\omega)$ by induction on $l<\omega$ such that:
1\. $p_{1,0}=p_1$.
2\. $\mathbb{P}_{a_1} \models p_{1,l} \leq p_{1,l+1}$.
3\. $j_{1,l}>l+\underset{k<l}{\Sigma}j_{1,k}$.
4\. $p_{1,l+1} \Vdash_{\mathbb{P}_{a_1}} "\underset{\sim}{a_1} \cap [n_{j_{1,l}},n_{j_{1,l}+1})=\emptyset"$.
For $l<\omega$, let $m_l=n_{j_{1,l}}$, so $(m_l : l<\omega) \in (\omega^{\omega})^V$ is increasing. Let $j$ be the minimal $j>n$ such that $n_{(p_1,p_2)} \leq m_j$. By the proof of Subclaim 1, there are $p_1'$ above $p_{1,j+1}$ and $k_*>n_{(p_1,p_2)}$ such that for every $k>k^*$ there is $p''$ above $p_1'$ forcing that $\underset{\sim}{a_{(p_1,p_2),1}} \cap [k^*,k)=\emptyset$. As $l((p_1,p_2))=2$, there is $p_2'$ above $p_2$ forcing that $\underset{\sim}{a_{(p_1,p_2),2}} \cap [n_{(p_1,p_2)},k^*+m_{j+1})=\emptyset$. Now let $(q_1,q_2)=(p_1',p_2')$, $n_{(q_1,q_2)}=k^*$, $l((q_1,q_2))=1$, $A_{(q_1,q_1),i}=A_{(p_1,p_2),i}$ $(i=1,2)$, it’s easy to see that $(q_1,q_2)$ and $j$ are as required.
Subclaim 8: $b_l \leq b_3$ where $b_l=f_l(a_l)$ $(l=1,2)$.
Proof: By symmetry, it suffices to prove the claim for $l=1$. Let $\underset{\sim}{a_0},...,\underset{\sim}{a_{n-1}} \in \Gamma_{b_3} \setminus \Gamma_{b_1}$, $\underset{\sim}{a}=\underset{l<n}{\cup} \underset{\sim}{a_l}$ and let $\underset{\sim}{g}$ be a $\mathbb{P}_{b_1}$-name of an increasing sequence from $\omega^{\omega}$, we shall prove that $\Vdash_{\mathbb{P}_{b_3}} "\underset{\sim}{u}:=\{i : \underset{\sim}{a} \cap [g(i),g(i+1))=\emptyset\}$ is infintie$"$. There are $\underset{\sim}{a_l'} \in \Gamma_{a_2} \setminus \Omega_{a_2}$ $(l<n)$ such that $\underset{l<n}{\wedge}f_2(\underset{\sim}{a_l'})=\underset{\sim}{a_l}$, let $\underset{\sim}{a'}=\underset{l<n}{\cup}\underset{\sim}{a_l'}$. Let $(\underset{\sim}{m_i} : i<\omega)$ be the $\mathbb{P}_{a_1}$-name for $f_1^{-1}((\underset{\sim}{g}(i) : i<\omega))$. Let $(p_1,p_2) \in \mathbb{P}_{b_3}$ and $n_*<\omega$, we shall find $(q_1,q_2) \in \mathbb{P}_{b_3}$ above $(p_1,p_2)$ and $n>n_*$ such that $(q_1,q_2) \Vdash_{\mathbb{P}_{b_3}} "n\in \underset{\sim}{u}"$. We can choose $(p_{1,i},m_{1,i} : i<\omega)$ by induction on $i<\omega$ such that $p_1 \leq p_{1,i} \in \mathbb{P}_{a_1}$, $p_{1,i} \leq p_{1,i+1}$ and $p_{1,i+1} \Vdash_{\mathbb{P}_{a_1}} "\underset{\sim}{m_i}=m_{1,i}"$. The rest of the proof is as in the previous subclaim. $\square$
**Claim 8:** For a dense set of $a\in AP$, $\Vdash_{\mathbb{P}_a} "\Gamma_a$ is mad$"$.
**Proof:** Let $\lambda_0=|\mathbb{P}_a|$ and $\lambda_1=2^{\lambda_0}$. Let $\mathbb{R}_1=Col(\aleph_0,\lambda_1)$ and $\mathbb P=\mathbb{P}_a \times \mathbb{R}_1 \in H(\kappa)$.
In $V^{\mathbb P}$, $\aleph_1^{V^{\mathbb P}}=\lambda_1^+$ and $\mathbb{P}_a \cup \mathcal{P}(\mathbb{P}_a)$ is countable, so $(\omega^{\omega})^{V^{\mathbb{P}_a}}$ is countable and $\Gamma:=\{ \underset{\sim}{\tau} : \underset{\sim}{\tau}$ is a canonical $\mathbb P-$name of a real such that the function listing $\underset{\sim}{\tau}$ dominates $(\omega^{\omega})^{V^{\mathbb{P}_a}} \}$ is dense in $[\omega]^{\omega}$. By the density of $\Gamma$, we can find $\Gamma' \subseteq \Gamma$ such that $\Vdash_{\mathbb P}"\Gamma' \cup \Gamma_a$ is mad$"$. Now let $b=(\mathbb P,\Gamma' \cup \Gamma_a)$, then (ignoring the obvious clauses) we need to prove that $b$ satisfies definition 3(a)(4) and that $a\leq b$ (for which we need to prove that the requirement from 3(b)(3) is satisfied). We shall prove that $a$ and $b$ satisfy requirement 3(b)(3), the proof that $b$ satisfies 3(a)(4) is similar. We shall work in $V^{\mathbb{P}_b}$. Let $\underset{\sim}{a_0},...,\underset{\sim}{a_n} \in \Gamma_b \setminus \Gamma_a$ and let $\underset{\sim}{a}=\underset{l\leq n}{\cup}\underset{\sim}{a_l}$. Suppose that $(m_i : i<\omega) \in V^{\mathbb{P}_a}$ is increasing, choose a sequence $(i(k) : k<\omega) \in V^{\mathbb{P}_a}$ such that $i(k+1)>m_{i(k)+1}+i(k)+(n+1)k$ and let $m_k'=m_{i(k)+1}$ $(k<\omega)$. For each $l\leq n$, the set $\underset{\sim}{u_l}=\{k<\omega : f_{\underset{\sim}{a_l}}(k)>m_{i(k+1)}\}$ is cofinite (by the definition of $\Gamma$). Therefore, for every $k$ large enough, $|\underset{\sim}{a_l} \cap m_{i(k+1)}|<k$ (for every $l\leq n$), hence $|\underset{\sim}{a} \cap m_{i(k+1)}|<(n+1)k$. For each such $k$, $|\{i : i\in [i(k),i(k+1)) \wedge \underset{\sim}{a} \cap [m_i,m_{i+1}) \neq \emptyset\}|<(n+1)k$. As $i(k+1)-i(k)>(n+1)k$, there is $i\in [i(k),i(k+1))$ such that $\underset{\sim}{a} \cap [m_i,m_{i+1})=\emptyset$. Therefore, $f_{\underset{\sim}{a}}$ is not dominated by a real from $V^{\mathbb{P}_a}$. $\square$
**Claim 9:** For every $a\in AP$ and a $\mathbb{P}_a$-name $\underset{\sim}{r}$ of a member of $[\omega]^{\omega}$, there is $b\in AP$ above $a$ such that $\Vdash_{\mathbb{P}_b} "$there is $\underset{\sim}{s} \in \Gamma_b$ such that $|\underset{\sim}{r} \cap \underset{\sim}{s}|=\aleph_0"$.
**Proof:** Follows directly from Claim 8. $\square$
**Observation 10:** Let $\mathbb{Q}$ be a forcing notion from $\bold Q$. **** Assume that $a_0 \leq a_l$, $\underset{\sim}{\eta_l}$ is a $\mathbb{P}_{a_l}$-name of a $\mathbb Q-$generic real over $V^{\mathbb{P}_{a_0}}$ $(l=1,2)$, and $\mathbb{P}_{a_0} \star \underset{\sim}{\eta_1}$ is isomorphic to $\mathbb{P}_{a_0} \star \underset{\sim}{\eta_2}$ over $\mathbb{P}_{a_0}$ (so wlog they’re equal to each other and we may denote the generic real by $\underset{\sim}{\eta}$). By Claim 6, there is $a_0' \in AP$ such that $a_0 \leq a_0' \leq a_l$ $(l=1,2)$, $\mathbb{P}_{a_0'}=\mathbb{P}_{a_0} \star \underset{\sim}{\eta}$ and $\Gamma_{a_0'}=\Gamma_{a_0}$. By Claim 7, there are $b_l$ $(l\leq 3)$ and $g_l$ $(l\leq 2)$ as there for $(a_0',a_1,a_2)$ here. $\square$
**Definition 11:** Let $H\subseteq AP$ be generic over $V$ and let $V_1=V[H]$. In $V_1$, let $\underset{\sim}{\mathbb P}[H]$ be $\underset{a\in H}{\cup}\mathbb{P}_a$.
**Claim 12:** $\Vdash_{AP} "\underset{\sim}{\mathbb P} \models \kappa-cc"$.
**Proof:** Suppose towards contradiction that $\Vdash_{AP} "\underset{\sim}{I} \subseteq \underset{\sim}{\mathbb P}$ is a maximal antichain of cardinality $\kappa"$. Choose by induction on $\alpha<\kappa$ a sequence $(a_{\alpha},p_{\alpha} : \alpha<\kappa)$ such that:
a\. $a_{\alpha} \in AP$.
b\. $(a_{\beta} : \beta<\alpha)$ is $\leq_{AP}$-increasing cotinuous.
c\. $a_{\beta+1} \Vdash_{AP} "p_{\beta} \in \underset{\sim}{I} \setminus \{p_{\gamma} : \gamma<\beta\}"$.
d\. $p_{\beta} \in \mathbb{P}_{\beta+1}$.
For every $\alpha<\kappa$, there is $q_{\alpha} \in \mathbb{P}_{a_{<\alpha}}:=\underset{\gamma<\alpha}{\cup}\mathbb{P}_{a_{\gamma}}$ such that $p_{\alpha}$ is compatible with every $r\in \mathbb{P}_{a_{<\alpha}}$ above $q_{\alpha}$. Let $\gamma(\alpha)<\alpha$ be the least $\gamma$ such that $q_{\alpha} \in \mathbb{P}_{a_{\gamma}}$. For some $\gamma(*)<\kappa$, $S:=\{ \alpha: \gamma(\alpha)=\gamma(*)\}$ is stationary. As $|\mathbb{P}_{a_{\gamma(*)}}|<\kappa$, there is $S'\subseteq S$ of cardinality $\kappa$ such that $\alpha_1<\alpha_2 \in S' \rightarrow q_{\alpha_1}=q_{\alpha_2}$, which leads to a contradiction. $\square$
**Definition 13:** Let $V_1$ be as in Definition 11 and let $G\subseteq \underset{\sim}{\mathbb{P}[H]}$ be generic over $V_1$, we shall denote $V[H,G]$ by $V_2$.
**Caim 14:** Every real in $V_2$ is from $V_1[G\cap \mathbb{P}_a]$ for some $a\in H$.
**Proof:** Let $\underset{\sim}{r}$ be a $AP \star \underset{\sim}{\mathbb P}$-name of a real. By Claim 12, $\underset{\sim}{\mathbb P}[H] \models \kappa-cc$ in $V_1$. Therefore, for every $n<\omega$ there are $AP-$names $\bar{p_n}=(\underset{\sim}{p_{n,\alpha}} : \alpha<\underset{\sim}{\alpha_n})$ and $\bar{t_n}=(\underset{\sim}{t_{n,\alpha}} : \alpha<\underset{\sim}{\alpha_n})$ such that:
a\. $\underset{\sim}{\alpha_n}<\kappa$.
b\. $\bar{p_n}$ is a maximal antichain in $\underset{\sim}{\mathbb{P}}[H]$.
c\. $\underset{\sim}{t_{n,\alpha}}$ is a $\underset{\sim}{\mathbb{P}}[H]-$name of an element of $\{0,1\}$.
d\. $\underset{\sim}{p_{n,\alpha}} \Vdash "n\in \underset{\sim}{r}$ iff $\underset{\sim}{t_{n,\alpha}}=1"$.
For every $n<\omega$ and $\alpha<\underset{\sim}{\alpha_n}$, there is $\underset{\sim}{a_{n,\alpha}} \in \underset{\sim}{H}$ such that $\underset{\sim}{p_{n,\alpha}} \in \mathbb{P}_{\underset{\sim}{a_{n,\alpha}}}$. Now let $a_0 \in AP$, we can find $\leq_{AP}$-increasing sequence $(a_n : n<\omega)$ such that $a_{n+1} \Vdash "\underset{\sim}{\alpha_n}=\alpha_n^*"$ for some $\alpha_n^*<\kappa$. Let $a_{\omega} \in AP$ be an upper bound, and now choose an increasing sequence $(a_{\omega+\alpha} : \alpha \leq \underset{n<\omega}{\Sigma} \alpha_n^*)$ by induction on $\alpha \leq \underset{n<\omega}{\Sigma} \alpha_n^*$ such that for every $n<\omega$ and $\beta<\alpha_n^*$, $a_{\omega+\underset{l<n}{\Sigma} \alpha_l^* +\beta+1} \Vdash "\underset{\sim}{a_{n,\beta}}=a_{n,\beta}^*$ and $\underset{\sim}{p_{n,\beta}}=p_{n,\beta}^*"$. We may assume wlog that $a_{n,\beta}^* \leq_{AP} a_{\omega+\underset{l<n}{\Sigma} \alpha_l^* +\beta+1}$, so $p_{n,\beta}^* \in \mathbb{P}_{a_{\omega++\underset{l<n}{\Sigma} \alpha_l^* +\beta+1}}$. It’s now easy to see that $\underset{\sim}{r}$ is a $\mathbb{P}_{a_{\omega+\underset{n<\omega}{\Sigma} \alpha_n^*}}$-name. $\square$
**Theorem 15:** a. In $V_2$, let $\mathcal A=\{\underset{\sim}{a}[G] : \underset{\sim}{a} \in \Gamma_b$ for some $b\in H\}$ and let $V_3=HOD(\mathbb R, \mathcal A)$, then $V_3 \models ZF+DC+"$there exists a mad family$"+"$all sets of reals are $\mathbb{Q}-$measurable for every $\mathbb Q \in \bold Q"$.
b\. $ZF+DC+"$every set of reals is Lebesgue measurable$"+"$there exists a mad family$"$ is consistent relative to an inaccessible cardinal.
**Proof:** a. The existence of a mad family follows by Claim 8. $\mathbb Q-$measurability for $\mathbb Q \in \bold Q$ follows from Claim 14 and Observation 10 as in Solovay’s proof.
b\. Apply the previous clause to $\mathbb Q=$Random real forcing. $\square$
As a corollary to the above theorem, we obtain an answer to a question of Henle, Mathias and Woodin from [\[]{}HMW[\]]{}:
**Corollary 16 $(ZF+DC)$:** The existence of a mad family does not imply that $\aleph_1 \leq \mathbb R$.
**Proof:** By Theorem 15 (applied to Random real forcing) and the fact that the existence of an $\omega_1$-sequence of distinct reals implies the existence of a non-Lebesgue measurable set of reals (see [\[]{}Sh176[\]]{}). $\square$
**Remark**: The above result was also obtained by Larson and Zapletal in [\[]{}LZ[\]]{} assuming the existence of a proper class of Woodin cardinals.
We conclude with a somewhat surprising observation, showing that the analog of Theorem 15 fails at the lower levels of the projective hierarchy:
**Observation 17:** If every $\Sigma^1_3$ set of reals is Lebesgue measurable, then there are no $\Sigma^1_2$-mad families.
**Proof:** By $[Sh176]$, $\Sigma^1_3$-Lebesgue measurability implies that $\omega_1^{L[x]}<\omega_1$ for every $x\in \omega^{\omega}$. By Theorem 1.3(2) in [\[]{}To[\]]{}, it follows that there are no $\Sigma^1_2$-mad families. $\square$
**On a question of Enayat**
We now address a question asked by Ali Enayat in [\[]{}En[\]]{}. The question is motivated by the problem of understanding the relationship between Freiling’s axiom of symmetry, the continuum hypothesis and the Lebesgue measurability of all sets of reals (see discussion in [\[]{}Ch[\]]{}).
As with the previous results, we were informed by Paul Larson that the following results can also be obtained under the assumption of a proper class of Woodin cardinals using the arguments from [\[]{}LZ[\]]{}.
**Definition 18:** a. Let $WCH$ (weak continuum hypothesis) be the statement that every uncountable set of reals can be put into 1-1 correspondence with $\mathbb R$.
b\. Let $AX$ (Freiling’s axiom of symmetry) be the following statement: Let $\mathcal F$ be the set of functions $f: [0,1] \rightarrow \mathcal{P}_{\omega_1}([0,1])$, then for every $f\in \mathcal F$ there exist $x,y \in [0,1]$ such that $x\notin f(y)$ and $y\notin f(x)$.
Remark: The term $WCH$ has a different meaning in several papers by other authors.
**Theorem 19:** $ZF+DC+\neg WCH+"$every set of reals is Lebesgue measurable$"$ is consistent relative to an inaccessible cardinal.
**Proof:** Let $V_3$ be the model from Theorem 15(b), we shall prove that $V_3 \models \neg WCH$ by showing that there is no injection from $\mathbb R$ to the mad family $\mathcal A$. Suppose toward contradiction that for some $(a,\underset{\sim}{p}) \in AP \star \underset{\sim}{\mathbb P}$ (where $\underset{\sim}{\mathbb P}$ is as in Definition 11), a canonical name for a real $\underset{\sim}{r}$ and a first order formula $\phi(x,y,z,\mathcal A)$, $(a,\underset{\sim}{p})\Vdash "\phi(x,y,\underset{\sim}{r},\mathcal A)$ defines an injection $F_{\underset{\sim}{r}}$ from $\mathbb R$ to $\mathcal A"$. We may assume wlog that $\underset{\sim}{r}$ is a canonical $\mathbb{P}_a$-name. We may also assume wlog that, for every $\underset{\sim}{s} \in \Gamma_a$, $(a,\underset{\sim}{p})\Vdash "$if $\underset{\sim}{s} \in Ran(F_{\underset{\sim}{r}})$, then $\underset{\sim}{s}=F_{\underset{\sim}{r}}(t)$ for some $t\in \mathbb{R}^{V^{\mathbb{P}_a}}"$. This is possible as $|\Gamma_a|<\kappa$, so we may construct an increasing sequence $(a_{\gamma} : \gamma<\beta)$ of length $<\kappa$, such that $a_0=a$ and such that the upper bound $(a_{\beta},\Gamma_{a_\beta})$ satisfies the above requirement. $((a_{\beta},\Gamma_a),\underset{\sim}{p})$ is then as required. By increasing $a$, we may assume wlog that $\underset{\sim}{p}$ is an object $p$ (and not just an $AP$-name) from $\mathbb{P}_a$. Now let $a_2 \in AP$ be defined as $a_2=(\mathbb{P}_a \star Cohen, \Gamma_a)$ and let $\underset{\sim}{\eta}$ be the $\mathbb{P}_{a_2}$-name for the Cohen real. There are $a_3 \in AP$ and a name $\underset{\sim}{\nu}$ such that $a_2 \leq a_3$ and $a_3 \Vdash "p \Vdash "\phi(\underset{\sim}{\eta},\underset{\sim}{\nu},\underset{\sim}{r},\mathcal A)""$, so $\underset{\sim}{\nu} \in \mathcal A$, and by the injectivity of $F_{\underset{\sim}{r}}$, $\underset{\sim}{\nu} \notin \Gamma_a$. We may assume wlog that $\underset{\sim}{\nu} \in \Gamma_{a_3}$.
Let $a_4$ be the amalgamation of two copies of $a_3$ over $a_2$ (i.e. as in the proof of Claim 7) and let $f_0: \mathbb{P}_{a_3} \rightarrow \mathbb{P}_{a_4}$ and $f_1: \mathbb{P}_{a_3} \rightarrow \mathbb{P}_{a_4}$ be the corresponding complete embeddings. As the amalgamation is over $a_2$, it follows that $f_0(\underset{\sim}{\eta})=f_1(\underset{\sim}{\eta})$ and $f_0(\underset{\sim}{r})=f_1(\underset{\sim}{r})$, and by the argument from the proof of Claim 7 (Subclaim 5), $f_0(\underset{\sim}{\nu}) \neq f_1(\underset{\sim}{\nu})$. As $f_l$ $(l=0,1)$ are isormorphisms between $a_3$ and $f_l(a_3) \leq a_4$ such that $f_l \restriction \mathbb{P}_{a_2}=Id$, they induce an automorphism of $(AP,\leq_{AP})$ mapping $a_3$ to $f_l(a_3)$ and $a_2$ to itself. Therefore, $a_4 \Vdash "" f_0(p) \Vdash"\phi(f_0(\underset{\sim}{\eta}),f_0(\underset{\sim}{\nu}),f_0(\underset{\sim}{r}),\mathcal A)""$,
$a_4 \Vdash ""f_1(p) \Vdash "\phi(f_1(\underset{\sim}{\eta}),f_1(\underset{\sim}{\nu}),f_1(\underset{\sim}{r}),\mathcal A)""$ and $f_0(p)=f_1(p)$, a contradiction. $\square$
**Theorem 20:** $WCH$ is independent of $ZF+DC+AX+"$all sets of reals are Lebesgue measurable$"$.
**Proof:** By [\[]{}We[\]]{}, $AX$ is implied by $ZF+DC+"$all sets of reals are Lebesgue measurable$"$. Therefore, $AX$ holds in the model $V_3$ from Theorem 15(b) and in Solovay’s model. By Corollary 19, $V_3 \models \neg WCH$. By the fact that all sets of reals in Solovay’s model have the perfect set property, it follows that $WCH$ holds in that model. $\square$
**References**
[\[]{}Ch[\]]{} Timothy Chow, Question about Freiling’s axiom of symmetry, cs.nyu.edu/pipermail/fom/2011-August/015676.html
[\[]{}En[\]]{} Ali Enayat, Lebesgue measurability and weak CH, mathoverflow.net/questions/72047/lebesgue-measurability-and-weak-ch
[\[]{}HwSh1090[\]]{} Haim Horowitz and Saharon Shelah, Can you take Toernquist inaccessible away?, arXiv:1605.02419
[\[]{}HMW[\]]{} James Henle, Adrian R. D. Mathias and W. Hugh Woodin, A barren extension. Methods in mathematical logic, Lecture Notes in Mathematics 1130, pages 195-207. Springer Verlag, New York, 1985
[\[]{}LZ[\]]{} Paul Larson and Jindrich Zapletal, Canonical models for fragments of the axiom of choice, users.miamioh.edu/larsonpb/lru5.pdf
[\[]{}Ma[\]]{} A. R. D Mathias, Happy families, Ann. Math. Logic **12** (1977), no. 1, 59-111
[\[]{}NN[\]]{} Itay Neeman and Zach Norwood, Happy and mad families in $L(\mathbb R)$, math.ucla.edu/\~ineeman/hmlr.pdf
[\[]{}Sh176[\]]{} Saharon Shelah, Can you take Solovay’s inaccessible away? Israel J. Math. 48 (1984), no. 1, 1-47
[\[]{}Sh218[\]]{} Saharon Shelah, On measure and category, Israel J. Math. 52 (1985) 110-114
[\[]{}So[\]]{} Robert M. Solovay, A model of set theory in which every set of reals is Lebesgue measurable, AM 92 (1970), 1-56
[\[]{}To[\]]{} Asger Toernquist, Definability and almost disjoint families, arXiv:1503.07577
[\[]{}We[\]]{} Galen Weitkamp, The $\Sigma^1_2$-theory of axioms of symmetry, J. Symbolic Logic 54 (1989), no. 3, 727-734
$\\$
(Haim Horowitz) Einstein Institute of Mathematics
Edmond J. Safra campus,
The Hebrew University of Jerusalem.
Givat Ram, Jerusalem, 91904, Israel.
E-mail address: haim.horowitz@mail.huji.ac.il
$\\$
(Saharon Shelah) Einstein Institute of Mathematics
Edmond J. Safra campus,
The Hebrew University of Jerusalem.
Givat Ram, Jerusalem, 91904, Israel.
Department of Mathematics
Hill Center - Busch Campus,
Rutgers, The State University of New Jersey.
110 Frelinghuysen road, Piscataway, NJ 08854-8019 USA
E-mail address: shelah@math.huji.ac.il
[^1]: Date: May 16, 2017
2000 Mathematics Subject Classification: 03E35, 03E15, 03E25
Keywords: mad families, Lebesgue measurability, amalgamation
Publication 1113 of the second author
Partially supported by European Research Council grant 338821 and NSF grant DMS-1362974.
|
---
abstract: 'In this work we have analyzed a novel concept of sequential binding based learning capable network based on the coupling of recurrent units with Bayesian prior definition. The coupling structure encodes to generate efficient tensor representations that can be decoded to generate efficient sentences and can describe certain events. These descriptions are derived from structural representations of visual features of images and media. An elaborated study of the different types of coupling recurrent structures are studied and some insights of their performance are provided. Supervised learning performance for natural language processing is judged based on statistical evaluations, however, the truth is perspective, and in this case the qualitative evaluations reveal the real capability of the different architectural strengths and variations. Bayesian prior definition of different embedding helps in better characterization of the sentences based on the natural language structure related to parts of speech and other semantic level categorization in a form which is machine interpret-able and inherits the characteristics of the Tensor Representation binding and unbinding based on the mutually orthogonality. Our approach has surpassed some of the existing basic works related to image captioning.'
author:
- |
Chiranjib Sur\
Computer & Information Science & Engineering Department, University of Florida.\
Email: chiranjibsur@gmail.com
title: 'CRUR: Coupled-Recurrent Unit for Unification, Conceptualization and Context Capture for Language Representation - A Generalization of Bi Directional LSTM'
---
[Shell : Bare Demo of IEEEtran.cls for IEEE Journals]{}
language modeling, dual context initialization, representation learning, tensor representation, memory networks
Introduction {#section:introduction}
============
short term (LSTM) memories are widely analyzed due to their high demand in industry to tackle huge volume of unlabeled data, and data analytic technologies greatly rely on them. Mere object detection and manual tagging failed to provide immense details of the activities and the events in the media data and to overcome the confusion created due to perception and language barriers between human interpretation capability and machine interpretation. Image captioning has progressed but slowed down to gain the optimum efficiency and in this work we have analyzed a new architecture that enhances the image captioning problem from visual features. In disguise, we introduced an effective way of coupling and decoupling tensors which can gather effective representations that can differentiate between different ways of writing and sentence constructions. The new architecture, named Coupled-Recurrent Unit Representation (CRUR) unit, is based on the entanglement of the representation of two recurrent units and passing the knowledge into a form of a structured Tensor Product Representations and decoupling it to the required sentences. The main idea behind this architecture is the fact that machine interpretation is based on the fact that machine can only understand orthogonal states and its interpretation can be easily processed and stored. Even, the whole segment of network and channel coding, signal detection and estimation theory is dependent on the orthogonal properties. After successful utilization of the Tensor Product Representation based on Hadamard matrix for question answering [@palangi2017deep] with high accuracy of prediction for answers, there was a need to informalize, regularize and generalize the concept to learnable representations that is scalable and can take forward the ultimate burden of deciphering huge amount of data and be helpful to mankind. [@huang2018tensor] provided some analysis on this architecture, but was limited to LSTM, while this work provided a much elaborated analysis and outperformed their work. Through experimentation, we were able to establish the fact that memory networks can learn the theoretical framework not only for better sentence generation applications, but also were able to gather the framework of sentences and thus will immensely help in revolutionizing the style and pattern of individuals and would be felt less like machines. Also the coupling effects of the units enhance the effectiveness of the memory and the generated sentences are quantitatively and qualitatively better than the individual units. The main reason of the enhancement is the functional ability of the network to generate representation to control the context of the situation and also the structure of the languages and thus effective in learning diversified representation from their product for the languages, a mimicry of human artifact, identity and difference.
In CRUR model, the information content of the overall structure is enhanced and each part can have a different narrative. The fusion of the narratives will help in better long term memory and better contextual learning. Coupling of the RNN helps in better retaining capability of the model and the sequential dependency creates better update for the variables and better mathematical model for the architecture. In traditional RNN, initialization was confined to limited sectors, while in CRUR, there are possibilities of dual initialization which can create dual narrative, while there are chances of diverse initialization which can itself be an advantage. However, there are structural differences between the LSTM to enforce non-uniformity and indifference in learning content. One can be regarded as a rule generator and the other is the rule enforcer. However, the basics of rules can never be generalized as they are part of narrative and differs for different samples. The rules narrative is structured internally and theoretically and must not be confused with rule based learning. In language generation, it is generally seen that the RNN structural learning is defined by the prediction of the next probable words than the categorical division. However, since languages are highly structured, researchers claim learning of structures of sentences are also important. But, if the prediction of next word is combined with the next probable rule, it can be more efficient. In fact, we can detect the part-of-speech of the word without knowing the word. The most important performance factor for CRUR unit is the effective tensor which represents complex relationships and is sensitive to compression and expansion of the embedding dimension. While expansion can be handled with proper dropout, compression can create limitation in variations of expression and can lead to error in part-of-speech and also shortening of description of events. While coupling is linear transformation of matrix, one part can be the driver for words occurrence while the other is for grammatical rules or part-of-speech. Initialization of the word predictor must come with context (like visual features) while the one with grammatical rules must be initialized with rule information (tagging distribution). While data driven training will not provide very stable and generalized part-of-speech representation, the perfection and variation in context depends on initialization and proper coupling of the two representations.
The rest of the document is arranged with problem description of language in Section \[section:problem\], theories and advent of the tensor product representation in Section \[section:theory\], discussion of the role of Bayesian Prior in Tensor Product Representation in Section \[section:Bayesian\], architectural details of the different CRUR models in Section \[section:CRUR\], methodology details of the application along with details of data in Section \[section:methodology\], experiment details, results and analysis details in Section \[section:results\], revisit of the existing works in the literature in Section \[section:literature\], conclusion and discussion in Section \[section:discussion\].
Our main contribution consists of the followings: 1) novel architecture for sentence representations where context features and language attribute feature cooperate for sentence generation 2) enhanced performance for architectures with just image features, achieved a BLEU\_4 value of 32.7% in comparison to the previous works 3) logical establishment of the mathematical modeling for tensor products, more than sequential establishments of bi-directional architecture 4) inclusion of language attribute influence and their representation for generation of sentences 5) enhanced predictive language attribute modeling from $\textbf{u}_t$, before decoding the words from $\textbf{f}_t$ with very high accuracy 6) language attributes based controlled analysis for generation of complex and compound sentences.
{width=".95\textwidth"}
{width="\textwidth"}
{width="\textwidth"}
Problem of Language Generation {#section:problem}
==============================
Machine’s ability to write based on events, contexts and facts, creates problem for many applications to directly convey their diverse messages to the users, the problem which was tackled through finite set of indications. Machines must come out of these finiteness with capability of generating texts from contexts and an infinite range of topics through different languages. While, large part of these non-deformed contexts comes from visual representations like images and videos, narrative contexts are associated with uncertainties and ambiguities for machines for inference. Ambiguities in visual contexts are relatively less, but the number of prospective increases. In this work, we have provided a new concept for generation of texts through the utilization of language attributes like parts-of-speech, semantics etc. In absence of concrete and generalized language rules, mathematically defining advanced models and data driven techniques are the best way to learn sentence construction. In this work, we have provided some instances of the approximation of the concepts through establishment of the topological relationships among different words. Generation of texts has many applications and each individual application is defined by different model approximation and is obsessed with their own problems.
Text Generation Problem
-----------------------
Text Generation is the ability of computer to generate texts from contexts like human beings sensing certain conditions and state space of the system. However, due to immense confusion in representations and structural differences in defining the functional approximation of the model, image captioning application found immense struggle in gaining high momentum for the considered statistical evaluations. However, if unique representation is produced and the models can be made sensitive to the minute variations in the representations, then caption generation can be made very effective. However, the irony is that deep learning deals with suppression of the variations and thus merges the samples to a distribution generated during the training sessions. Mathematically, we can define Text Generation model as $\{w_1,\ldots,w_n\} = f(T,\textbf{W})$ where $T = f(\textbf{v})$ with $w_i$ represent words, $T$ is the context representation, $\textbf{v}$ context features, $\textbf{W}$ are the estimated parameters. Text generation faces lots of problems including the biasness of memory networks to gather similar kinds of words after certain known words and lack of diversity in generation. This problem is generated from the suppression of variation of the representations, where the network conducts approximations through non-linear transformation of the contexts. However, the network should focus on suppression of variation of the individual features and that will help in determination of effective representation that can help in caption generations. The main cause of this problem is also the lack of definition of proper representation that can help in diversification of predictions and help in generation of words in sentences that were never used in training. Representation, mostly used, are something that is working, and not defined involving the existing embedding of word vectors.
Reply Generation
----------------
Reply Generation is the contextual generation of texts based on the textual query on a conversation. It is also a very important application of modern day world as the interaction between human and machine is not limited to one-way interception but two-way conversation. It requires both the agents interact with each other in a shared common language and understand what the other is saying. Mathematically, we can define Reply Generation problem as generating related sentences $\{w_1,\ldots,w_n\} = f(T,\textbf{W})$ where representation $T = f(\{p_1,\ldots,p_m\}) \subset f(\{q_1,\ldots,q_k\})$ is dependent on the query (context) series $\{q_1,\ldots,q_k\}$ being made in the interaction series $\{p_1,\ldots,p_m\}$.
Language & Style
----------------
Language differs in style and with change in style, the pattern of appearance of parts-of-speech (POS) changes. While we define an architecture, which can even provide a structural component for POS, it is eminent that the memory network is learning to generate sentences, which can be regarded as simple, complex and compound sentences based on the number of independent and dependent clause it contains. An independent clause can be regarded as a sentence, representing a complete thought, while dependent clause even though it has a subject and a verb, cannot be regarded as a sentence. Simple sentences would have one independent clause that is one subject and one predicate, while Complex sentences contain an independent clause and at least one dependent clause. Compound sentences has at least two joined independent clauses. The machine must learn these clauses and learn how to use them in the flow of conversation and sentence generation with logical construction of the action and activities.
Other aspects of the sentence construction are the language attributes that linguists have recognized through ages and classified the pattern, however failed to provide some concrete set of rules and algorithms for generation. This is where our work focused on to make machine learn the taste of aesthetic writing specific to disciplines. Knowledge of the grammar and part of speech is efficient but lack of concrete rules makes it very difficult for the machine to learn, while prediction memory model can predict easily based on the likelihood. While most of the language models are based on prediction of the next word, the likelihood of decision is dependent on one set of estimated parameters. In this work, we have utilized two models dedicated for context sequentiality and construction topology for word selection and parts-of-speech respectively. Also, while predicting the sentence, what we expect from the model to learn unique representations of the words and also the structural interpretation of the sentences and emphasis on machine comprehension through prediction models and later demonstrate to control the generation. Mathematically, we can define $\{w_1,\ldots,w_n\} = f(T,\textbf{W}, S)$ where $S$ is the style factor related to language attributes.
Theory of Tensor Product {#section:theory}
========================
A new tensor called Tensor Product $T$ is generated through the multiplication of two set of tensor series (filler $\textbf{f}$ and binder $\textbf{r}$) with prescribed interpretation and one set of tensor $\textbf{r}$ can help in perfect regeneration of the other $\textbf{f}$ from the new tensor $T$. Here $\textbf{f}$ is related to context interpretation and $\textbf{r}$ to language attribute interpretation and the combined give rise to the word level accuracy. Tensor Product and their generated Representation uses the concept of linearly independence with transformation and inverse transformation on the assumption that the inner product will help in localization. However, as large part of real world scenario and their associated problems are variational, the linearly independence criteria can be relaxed and represent the tensor product with other semi-independent vectors and rely on the assumption that tensors are far apart to interfere and the high dimension of $\textbf{f}$ (or if needed $\textbf{r}$) will provide adequate independence space for each of them. Generalized TPR can be represented as $\textbf{s}(\textbf{w})$ as, $$\textbf{s}(\textbf{w}) = \sum f_i \otimes r^T_i$$ where $\textbf{w}$ is the feature vector, and $\{\textbf{w} \rightarrow \textbf{f} : \textbf{w} \in \textbf{W}_{e} \}$ is the transformation, $\textbf{W}_{e}$ is the raw features or the embedding vectors for features which minimizes the context function such as Word2Vec for $k$ contexts as $W2V\_Fn = \min \sum\limits_{i} \sum\limits_{j = 1}^{k} ||\textbf{w}_i - \textbf{w}_j||^2 $, $\textbf{r}$ is the independence imposer for the TPR. This kind of tensor product creates a combined representations of the whole feature space and is unique, reduced in dimension and with the following decoupling equation, $$\textbf{f} = \textbf{s}(\textbf{w}) \otimes \textbf{r} = \textbf{s} \otimes \textbf{r}$$ where we can generate back the complete original vector $\textbf{w}$ from $\{\textbf{f} \rightarrow \textbf{w} : \min \limits_{\forall i \in N} \arg (\{\textbf{f}_i\} - \textbf{f}) \}$ without any error. We have $N$ sample instances and $\min \limits_{\forall i \in N} \arg (\{\textbf{f}_i\} - \textbf{f})$ points to the closest possible sample $i$. So what we can conclude that this $\textbf{s}(\textbf{w})$ instantiate a much better comprehensive and compressed state of the samples than the whole feature space and can be help many learning algorithms to create models that can understand and differentiate the representations without explicitly supervising it to learn that these are different and need to be differentiated. At the same time, the feature representation can be migrated to its original form in constant time. Previous approaches for transferring $\min \limits_{\forall i \in N} \arg (\{\textbf{f}_i\} - \textbf{f})$ point to the closest possible sample $i \in N$ were cosine distances or nearest neighbor with distance norm. However, the same task is possible in constant time as a transformation $\max\limits_{\forall i \in N} \arg \textbf{W}_f \textbf{f} = \textbf{f}_i$ through posing the problem as a probability distribution as we tune our model to gradient error rectification and learning schemes.
Tensor Product History
----------------------
Tensor Product has been widely used in signal processing and other applications related to mixing of signals and other network coding and channel coding applications where data packets are transmitted using the tensor product concept to retrieve at the other end. Wireless network coding also uses orthogonality coupling for transmission of data packets. Spectrum detection theory uses this orthogonality concept widely for many applications for mixing of the signals based on different basis and later uses the property of tensor product for detection and estimation. Here each basis consists of one representation and the combination of these representation create a higher level representation for others. The spectrum theory principle is based on the assumption that the basis $\textbf{b}_{p_i}$ tensors are orthogonal (orthonormal) or $p_i$ phase-shifted and when the signals $\textbf{s}_i$ are multiplied with these vector, they ($\textbf{D} = \sum \textbf{b}_{p_i} \textbf{s}_i$) are also phase-shifted and at the detection part, the multiplication of the same basis revert back the original signal $\textbf{s}'_i = \textbf{D} \textbf{b}_{p_i} \approx \textbf{s}_i $ without noise. The equations that explains this phenomenon are provided as follows. $$\textbf{d}_i = \textbf{b}_{p_i} \textbf{s}_i$$ $$\textbf{D} = \sum \textbf{d}_i = \sum \textbf{b}_{p_i} \textbf{s}_i$$ $$\textbf{s}'_i = \textbf{D} \textbf{b}_{p_i}$$ where we have $\textbf{s}_i$ as the original signal, $\textbf{b}_{p_i}$ is the basis transformer, $\textbf{d}_i$ is the transformed signal with phase shift and $\textbf{D}$ is the combined signal to be transmitted. $\textbf{s}'_i$ is the regenerated signal at the reception end. $\textbf{b}_{p_i}$ posses the property of orthogonal basis function like Cosine, Sine, etc.
Unification of Symbolism and Naturalism
---------------------------------------
Unification of Symbolism is marked by generation of global representation for symbols and these representations have the capability to learn the intricacies and rules of the operational procedures on the symbols. For example, in case of natural language processing, unification of symbolism is denoted by the capability to represent alphabets with orthogonal one-hot vectors and by continuous representations like GloVe, and tensor product representation can learn the grammatical rules of sentences through the use another vector that can be denoted as the dictator of the next probable parts-of-speech. These dictators will never be perfect for memory network based prediction due to the fact that the whole notion in memories is approximate representation and this is done to scale up the learning capabilities.
Naturalism is an important criteria for sentence generation and is a way to prevent language construction biasness. Construction biasness is defined as the appearance of similar kind sentence rules for sentence and the machine capability to learn only limited way of expressing themselves. This problem of construction biasness in machine generated sentences is known as the problem of Naturalism and need to be dealt with as we move towards more sophisticated systems and capability to generate meaningful sentences from contexts.
Hadamard Matrix
---------------
Generation of many orthogonal structures is difficult and hence Hadamard Matrix is used for initialization. Hadamard Matrix consists of series of orthogonal rows and columns and its generation ensures such functionality. While dealing with TPR generation and other prediction and detection frameworks, generation and maintenance of the mathematical constraints becomes important for the best performance. In general, Hadamard matrix is a $(2^n \times 2^n)$ square matrix, consisting of $\{-1, 1\}$ and each of the rows are orthogonal to all the others. The consequence is that, it can be used to generate mutually independent vectors for the TPRs. Hadamard Code TPR was build on top of Hadamard Coded matrix using the following equations. $$\label{eq:hadamard}
H_{2^n} = \frac{1}{c_{k-1}}
\left[{\begin{array}{cc}
H_{2^{n-1}} & H_{2^{n-1}} \\
H_{2^{n-1}} & -H_{2^{n-1}}
\end{array}}\right]
= \frac{1}{c_{k-1}} H_{2} \otimes H_{2^{n-1}}$$ $H_{2^n} \in \mathbb{R}^{nnnn \times nnn}_{0/1}$ mostly consists of zeros and ones. The rows and columns of $H_{2} \otimes H_{2^{n-1}}$ are symmetric and form bases of Hadamard matrix where we have $\otimes$ as the Kronecker product, $\frac{1}{c_{k-1}}$ is the normalization factor where $c_{k-1} = ({\sum |x_{i}|^2})^{\frac{1}{2}}$ with Frobenius norm or $L^2-$norm of any row as the normalizing coefficient. If we consider $(k-1) = 2$, then the most fundamental Hadamard matrix with $c_{2} = c_{(k-1) = 2} = ({\sum |x_{i}|^2})^{\frac{1}{2}}$ is denoted as the following, $$H_{2} = \{ \frac{1}{c_{2}} \}
\left[{\begin{array}{cc}
1 & 1 \\
1 & -1
\end{array}}\right]
=
\left[{\begin{array}{cc}
0.707 & 0.707 \\
0.707 & -0.707
\end{array}}\right]$$ This matix forms the starting matrix for all other high dimensional Hadamard matrix generation. In Hadamard Coding, the filler consists of multiplication of the Hadamard matrix row (Equation \[eq:hadamard\]) $r^T_i = (H_{2^n})_{i,}$ and the individual feature representations $f_i = (\textbf{W})_{i}$ from feature space $\textbf{W}$ like in case of natural languages, $f_i = (\textbf{W}_e)_{w_k}$ is the word embedding vector for corresponding word $w_k$ from $\textbf{W}_e$.
The Hadamard Code TPR individual is generated as an inner product of the rows $\{(H_{2^n})_{r_i}:i\in \{1,\ldots,p\}\}$ of $p-$level Hadamard matrix (with $\{1,p\}$ dimension) and $ \{F_j:j\in \{1,\ldots,\lceil \frac{d}{q} \rceil\}\} $, the corresponding segment vector (with $\{q,1\}$ dimension) of the $d-$dimensional features of the samples as denoted by $\{F_j\} \{(H_{2^n})_{r_i}\}^T$ to generate a $\{q,p\}$ matrix. Essentially we have $p=2^k$, $p \geq \lceil \frac{d}{q} \rceil$ and symbolically $F = f(\textbf{w})$. So the overall Hadamard Code TPR is denoted as, $$\textbf{s}_{H}(\textbf{w}) = \sum\limits_{i} \{F_i\} \otimes \{(H_{2^n})_{r_i}\}^T$$ where we can generate back the features $\textbf{w}$ as $\textbf{F}\rightarrow \textbf{w}$ and, $$\textbf{F} = \textbf{s}_{H} \otimes (H_{2^n})$$ This procedure helps in the easiest and efficient way of generating and dealing with tensor product representation through linear transformation of the weighted representations of the original features to the mutual orthogonal spaces. Also, TPRs ($\textbf{s}_{H}$), generated from this procedure, have very distinct, non-overlapping and unique feature space for the samples. This created discrete learning phenomenon, which, sometimes, goes against the variation tolerance and regularization compatible network based training models. In such models, connectedness and relatedness, how insignificant they may be, are inevitable part of the learning. This is why, directly dealing with Hadamard Code TPR may not help and there are some extra procedural requirement for the system to work. Next, we will describe in details the procedural flow for image captioning applications, mainly dealing with natural languages.
Let we have sentence with word $w_1,\ldots,w_n$ and word embedding $\textbf{W}_e \in \mathbb{R}^{v \times e}$, we can transfer one hot vector for each word $w_i$ as $(\textbf{W}_e)_i \in \mathbb{R}^{1 \times e}$, we have, $$\textbf{s}_{H} = \sum (\textbf{W}_e)_i * r_j$$ for $w_j = i$ and $\textbf{s}_{H}$ is the TPR. Conversely, to retrieve the information from the TPR, for each $j \in N$, we have, $$(w_p)_{j} = \textbf{s}_{H} * r_j^T$$ and if we consider the nearest neighbor for $(w_p)_{j} $ in $\textbf{W}_e$, we find that $$\begin{gathered}
(w_p)_{j} = \arg \min\limits_{k} \{ (\textbf{W}_e)_k \mid \min ||(\textbf{W}_e)_k - (w_p)_{j}|| \} = \\
(\textbf{W}_e)_{k=i} = (\textbf{W}_e)_{i} = w_{j}\end{gathered}$$ We have tested that the retrieval rate is 100% correct for word embedding like Word2Vec, GloVe for any dimension. The accuracy of the retrieval is not because of the dimension or the embedding, but due to the mutual orthogonal matrix which creates space for real $f_i$ to be segregated when $r_i$ is multiplied with $f_i r_i^T$ as $f_i r_i^T r_i$.
Experimental Framework for Hadamard Matrix
------------------------------------------
Normalized Hadamard Matrix as $\textbf{r}^T$ is used to couple the word embedding ($\textbf{f}$) of words to create TPR and then this TPR is used to generate the words using $\textbf{r}$ through 2-norm nearest neighbor estimation of the generated embedding with the embedding-to-word dictionary. The estimated sequence of words were generated with 100% accuracy for the training dataset. We tried to map the image embedding with TPR through deep MLP and this MLP can estimate the training data with high precision, but for testing data, the scheme failed because of the high sensitivity of the model for variations in estimation of the TPR. However, if we use a nearest neighbor model for TPR, where the generated TPR for the test data is taken replaced by an already established TPR, we can perform much better accuracy for the test data. If only the training data TPR are used, the accuracy can reach at around 70% for BLEU\_4 accuracy while if the testing data TPR is used then the accuracy can reach above 90% BLEU\_4 accuracy. However, nearest neighbor estimation is time and resource consuming and hence not a prefered solution for modern day applications. Also, nearest neighbor based systems will destroy the notion of generalization of representations and will prevent production of new representations that will create new set of word sequence. Overall, nearest neighbor based solutions do not provide scalability solutions for languages, where the scope of diversity of representations is practically infinite.
Approximation, Structuring & Mutual Orthogonal Problem
------------------------------------------------------
While pure Hadamard matrix row based encoding is sensitive to variations of image features and not scalable, we used LSTM based encoder and decoder. The memeory based models can be initialized and the end-to-end model can generate the perfect intermediate TPR which can be considered as a perfect approximation of representation, that is learned with time and back propagated feedback. The structuring of such representation is generated with the help of an approximately orthogonal vector $\textbf{r}$ and the language attribute $\textbf{f}$ which is associated with the language embedding. The whole system is based on approximation and the coupling and decoupling is deterministic approximation instead of deterministic overall. This assumption and phenomenon worked well for our experiments and had been found to produce better sentences.
Bayesian Prior {#section:Bayesian}
==============
Bayesian Prior estimation helps in better modeling and prediction, where the data is represented by a distribution or series of distribution, already estimated or known and the work of the model is to fit the distribution. Traditional statistical model already assume some kind of distributions for the independent variables and thus facilitates the effectiveness of prediction. But the problem becomes non-trivial when the optimization is multi-modal and best possible solution is not adding much to prediction due to inappropriate estimation of some distributions, data is non-linear though considered to be linear and so on. This problem of estimation of the distribution is caused due to the lack of transformation and processing of the data features, which required to be tamed for better estimation. This is where the deep network helps, where series of layers creates non-linear transformation and approximation of the feature sets to define a much finer and stable generator distribution. Our architecture estimates Bayesian Prior for both contexts $(\mathcal{P}(\textbf{f}))$ and language attributes $(\mathcal{P}(\textbf{r}))$. The TPR $(\mathcal{P}(\textbf{s}(\textbf{w})) = \mathcal{P}(\textbf{f})\mathcal{P}(\textbf{r}))$ is a joint Bayesian Prior generated from their product. Relative variations of $(\mathcal{P}(\textbf{f}))$ and $(\mathcal{P}(\textbf{f}))$ helps in better representation learning.
We define Bayesian Representation for TPR as an orthogonal set of variable representations that helps in better prediction. These representations can also be regarded as likelihood of contexts and events with sentence composition characteristics. What orthogonality adds to these likelihood is the main point of discussion. While tensor multiplication transforms the feature space to other subspace without significantly judging whether that is beneficial or not, involvement of a orthogonal space creates a discretization and prevents mixing of the different features and thus create enough opportunity to be segregated in the decoder, a phenomenon widely used in signal systems and spectrum detection theory. However, unfortunately there is no way to estimate the inter-working intricacies of the memory network except evaluation at the likelihood level. Moreover, we evaluate on the collective composition than individualism, unlike spectrum detection applications. However, we still work in that direction to establish the principle through estimation and analysis of the vector $\textbf{r} = \textbf{u}_t$ generated through Equation \[eq:ut\]. It has been proved that $\textbf{u}_t$ is helpful in full prediction of the language attributes and also helps in better estimation and diversification of $f_t$. In analysis we established that the relative orthogonality of $\textbf{u}_t$ vectors for instances of the sentence is beneficial and outperforms previous performances.
Representation Learning as Bayesian Prior
-----------------------------------------
Representation Learning as Bayesian Prior is an abstract concept to maintain the feature space and orthogonality will help in maintaining integrity and individuality of the feature space in the hope that it will help in segregate of the features during decoding and decoupling. So apparently, out of two set of vectors, $\textbf{S}_t$ will denote the likelihood for a contextual representation to be generated and $\textbf{u}_t$ will the likelihood that it belongs to a certain parts-of-speech or any other language attribute. Combined, they can help in establishing the likelihood of the word. This is the reason why, as a design constrain, $\textbf{S}_t$ is established as a large vector comparison to the smaller vector $\textbf{u}_t$ and to prevent $\textbf{u}_t$ to learn about the contextual representation. It must also be mentioned that representation, as Bayesian prior, aims at providing the best likelihood of the words to be generated and compose the sentence. This is perhaps way different from the end-to-end models where the intermediate representation is far more than a prior of individual likelihood as it needs to generate a series of interconnected sequence likelihood.
Tensor Binder as Bayesian Prior
-------------------------------
Deep learning has always been considered as a system which can approximate the prior estimation from the data based on the likelihood of the classes. This is the reason, in many cases, the amount of the data is important for establishment of the variation, whereas the prior estimation can be handled through repetition of similar data. Tensor binder or the orthogonal vector representation $\textbf{u}_t$ is such an approximation which works on the principle of gathering certain characteristics of sequential connection or topological dependency and can be regarded as Bayesian Prior. The main task of the tensor binder is to gather information related to the possibilities of a context and channelize the system with the best possibility. Like say a context of a ‘person’ can channelize it to ‘man’ or ‘woman’ or ‘boy’, but tensor binder will establish that the word should represent as a noun. The relationship for generation of a grammatically correct sentence is prediction of the next context representation and the parts-of-speech representation. Mathematical, we can define, $$\mathcal{P}(R) = \mathcal{P}(X)\mathcal{P}(Y)$$ where $\mathcal{P}(X)$ is context representation from image, $\mathcal{P}(Y)$ is the moderator or parts-of-speech representation, $\mathcal{P}(R)$ is the probability of the next event word representation as a noun and a word (say ’man’). This joint influence $\textbf{f}_t$ creates space for new interpretation and can be used to guide grammatically correct sentences and thus free from the short term memory of one LSTM. Coordination and collaboration of the two LSTMs improve performance. Here, $\mathcal{P}(X)$ helps in deciding the next context for the word, provided $\mathcal{P}(Y)$ helps in deciding the style of writing judged through language attributes like parts-of-speech etc.
CRUR vs Bi-Directional LSTM
---------------------------
Bi Directional LSTM (bi-LSTM) shares its architecture with CRUR, but limited to a specialized case which performed well for specific applications and does not hold good prospects for data, where the topological relationships hold immense information for inference. In other words, Bi Directional LSTM can be regarded as a special case of the CRUR architecture. Before we discuss CRUR, there is a need to discuss the bi-LSTM architecture and understand why CRUR cropped up as a generalized architecture and what kind of applications are more suitable for bi-LSTM. bi-LSTM was established for better prediction and inference ignoring certain aspects related to sequential relationship and topological significance. It was never defined for establishing unique data representation but to converge large part of the similarly classified entities to similar representation that will define some series of additive distributions. bi-LSTM equations can be denoted as the following, $$\textbf{h}_{1,t} = LSTM_1(\textbf{h}_{1,t-1},\textbf{x}_{1,t} \mid \textbf{h}_{1,0}, \textbf{x}_{1,t}=\{w_1,w_2,\ldots,w_n\})$$ $$\textbf{h}_{2,t} = LSTM_2(\textbf{h}_{2,t-1},\textbf{x}_{2,t} \mid \textbf{h}_{1,0}, \textbf{x}_{1,t}=\{w_n,\ldots,w_2,w_1\})$$ $$\textbf{y}_{p,t} = f(\textbf{W}_{f}\overrightarrow{\textbf{h}}, \textbf{W}_{b}\overleftarrow{\textbf{h}}) = \textbf{W}_{f}\textbf{h}_{1,t} + \textbf{W}_{b}\textbf{h}_{2,t}$$ where it is assumed that the sequences $\textbf{x}_{1,t}=\{w_1,w_2,\ldots,w_n\}$ and its reverse $\textbf{x}_{1,t}=\{w_n,\ldots,w_2,w_1\}$ will infer similar kind of prediction, whereas the real fact is that the two sequences depict completely different representations and can be trained to infer similar expectations. In fact, bi-LSTM tries to converge all combinations of the sequence $\textbf{x}_{1,t}=\{w_1,w_2,\ldots,w_n\}$ to similar kind of distributions, which will in parallel conceive both the ways equivalently.
But real world problems are much more complex and this kind of assumption can end up to two different inferences for the LSTMs, which can conflict with each other and end up with wrong inference. However, our defined CRUR model is aimed at defining unique representation for better reciprocity, regeneration of composition and global representation and in that respect, we generalize the representation instead of the generalization of the distribution. Mostly, deep learning is known to be efficient because of its capability to suppress of the variations for the representation to converge them to pertinent distribution, but we define our network to suppress the numerical at the feature level so that more stable representations are generated and can accommodate the infinite space of languages. Dual direction destroys the notion of uniqueness for TPR and will not be good option for natural language where ’I am’ and ’am I’ is different and must not be used as a converge for the notion of prediction and generation. Mathematically, for generative models we can define CRUR as, $$w_n = f(\textbf{f}_t,\textbf{W},\{w_1,\ldots,w_{n-1}\})$$ while we can define bi-LSTM generation as, $$w_n = f(\textbf{f}_t,\textbf{W},\{w_1,\ldots,w_{n-1}\},\{w_{n-1},\ldots,w_1\})$$ Other fundamental differences are defined in terms of capability of generation, where bi-LSTM is focused on inference while transforming the bi-LSTM to a generative one, we ended up with CRUR. Inialization and inter-cooperation or collaboration had been added advantage to CRUR closed model to avoid independent interpretation. In the next few sections, we will discuss more on the CRUR and the way to predict language attributes and control the generation. CRUR equations can be denoted as the followings, $$\begin{gathered}
\textbf{h}_{1,t} = LSTM_1(\textbf{h}_{1,t-1},\textbf{h}_{2,t-1},\textbf{x}_{1,t} \mid \textbf{h}_{1,0}, \\
\textbf{x}_{1,t}=\{w_1,w_2,\ldots,w_n\})\end{gathered}$$ $$\begin{gathered}
\textbf{h}_{2,t} = LSTM_2(\textbf{h}_{2,t-1},\textbf{h}_{1,t-1},\textbf{x}_{2,t} \mid \textbf{h}_{1,0}, \\
\textbf{x}_{1,t}=\{w_1,w_2,\ldots,w_n\})\end{gathered}$$ $$\textbf{y}_{p,t} = \textbf{W}_{f} \sigma(\textbf{W}_{f} \textbf{h}_{1,t}) \textbf{h}_{2,t}$$ where $\textbf{y}_{p,t}$ is interpreted at instances $t = \{1,2,\ldots,n\}$ as $\{w_1,w_2,\ldots,w_n\}$.
Coupled-Recurrent Unit Representatione {#section:CRUR}
======================================
The Coupled-Recurrent Unit Representation (CRUR) unit [@Sur2018Representation], [@palangi2017deep], [@huang2018tensor] is an entanglement or tensor product of different interpret-able tensors along with crafted initialization of the parameters. The overview of the generalized CRUR architecture has been pictured both in Figure \[fig:basicTPRa\] and Figure \[fig:basicTPRb\]. The success of CRUR depends on the hidden state sharing scheme and transfer of knowledge of one LSTM with the other and thus can coordinate and cooperate. That is the reason why, CRUR provided a much better effect than traditional individual LSTM. The main reason of better learning capability of CRUR is the wide range of dependencies and sharing of variables and intermediate states to complement each other and also due to generation of regularized and specialized tensor representations, which drive the architecture to generate better visual captions. Tensor product of different tensors diversifies the opportunity of different combinations of likelihood and prevents the model from learning biases. We will mostly deal with LSTM, but different recurrent units can be used to generate different architectures and we have done elaborated study of some of them to understand their learning capabilities.
The main differences of these architectures are based on the amount of interdependence of the state spaces, number of activation gates, which also define the diversity of substances and also the amount of knowledge it can learn. For example, GRU provided better learning due to the large range of simultaneous triggering of the non-linear functions and facilitating the incorporation of knowledge, while LSTM can produce much better sequences. In our new architecture, there are functionality related to learning generation of sentences and also the ability to interpret different definition of tensor representation. This representation is important as it helps in bringing together different aspects of the languages from visual features to combine and generate. Such representations are derived from individual recurrent units and are expected to evolve.
{width=".75\textwidth"}
{width=".75\textwidth"}
Basic Conceptual Model - Open Ended
-----------------------------------
Open Ended models are the most basic model and the bi-LSTM is a special case where the LSTMs are symmetrical in dimension, each is trained with either forward or backward sequence and the ensemble of the likelihood $\textbf{h}_{1,t}$ and $\textbf{h}_{2,t}$ can be generalized with addition $(\textbf{h}_{1,t}+\textbf{h}_{2,t})$, multiplication $(\textbf{h}_{1,t} \odot \textbf{h}_{2,t})$, concatenation $[\textbf{h}_{1,t},\textbf{h}_{2,t}]$ or even weighted combinations $(\textbf{W}_1\textbf{h}_{1,t}+\textbf{W}_2\textbf{h}_{2,t})$ of the two. However, open ended models are widely used for fusion of information and without any sharing of information open ended models tend to have different opinion of the same contextual relationship. This resulted in lower rate of learning, no mutual sharing of knowledge, lower number of variables and also low approximations. Mathematically, we can define Open Ended models as the followings, $$\textbf{h}_{1,t} = LSTM_1(\textbf{h}_{1,t-1},\textbf{x}_{1,t} \mid \textbf{h}_{1,0})$$ $$\textbf{h}_{2,t} = LSTM_2(\textbf{h}_{2,t-1},\textbf{x}_{2,t} \mid \textbf{h}_{2,0})$$ $$\textbf{y}_{p,t} = \textbf{W}_{1}\textbf{h}_{1,t} + \textbf{W}_{2}\textbf{h}_{2,t}$$ where $\textbf{h}_*$ is the generated hidden states of the memory network and $\textbf{x}_*$ are the inputs. The open ended models are described with detailed equations in the subsequent subsections. Nevertheless, some applications will find open ended model better due to the fact that traditional feature learning systems tend to perform well when ensemble of extracted feature (like boosting) are used for inference.
Tensor Representation Structure & Size Analysis
-----------------------------------------------
In our model, we have emphasized on asymmetrical LSTM structures for CRUR, which will help in reduction of variable estimations and at the same time will bound the interpretation of the representations and reduce sparsity. However, the dimension of the two LSTMs must be proportion so that the effect of one can be reflected in the other and can bring changes in the likelihood estimation of the sequence. Initially, we considered $\textbf{u}_t \in \mathbb{R}^{10}$ and $\textbf{S}_t \in \mathbb{R}^{10\times 10}$ while tensor product produced was $\textbf{f}_t \in \mathbb{R}^{10}$. Noticeably, $\textbf{f}_t \in \mathbb{R}^{10}$ was not enough for representation and converged the representation to non-variational tensors, although the theoretical framework supported that. We increased our model dimension to $\textbf{u}_t \in \mathbb{R}^{10}$ and $\textbf{S}_t \in \mathbb{R}^{10\times 10}$ while now tensor product provided was $\textbf{f}_t \in \mathbb{R}^{10\times 10}$. The first model, with $10 \times (10,10) = 10$, failed to work properly due to two reasons: no end-to-end framework (decoupling is possible there) and the disproportion fact that 10 was too low for the other to mingle around and act as a proper initial representation that can capitulate the image tensor as sentence. Later, we changed the dimension to $10 \times (10,10) = (10,10)$ or even $10 \times 10 = 100$ may work for some applications. This architecture performed better and we can structure the captions much more effectively than the previous model.
Open-End & Closed-End Schemes for CRUR
--------------------------------------
We have defined two architectures for the coupling decoder: one with a Multi-Layer Perceptron called shallow scheme and another with a recurrent unit called deep scheme. Recurrent units are widely accepted because of the end-to-end learning capability. While we compare the different schemes, it is important that we understand that the criteria of learning is both contextual and combination. While many caption strategies go for contexts, their low BLEU value indicate that they fail to generate the combination of visual content. We have provided some these incites when we do the qualitative analysis of the generated captions in Section \[subsec:Qualitative\]. The followings are the probable decoders and not generator and hence $\textbf{f}_t$ is used as attention and not as initialization and for each decouple, it is replaced by a new one. For MLP, the decouple equations are as follows, $$\textbf{y}_{t} = (\arg\max\sigma( \textbf{W}_{x}\textbf{f}_{t} ))$$ While LSTM, these are the decoupling equations for decoding, $$\textbf{y}_{t} = (\arg\max LSTM( \textbf{x}_{t}, \textbf{f}_{t}, \textbf{h}_{t-1} = \textbf{h}_{0} ))$$ where $\textbf{h}_{0}$ is initialized with constant. Even this equations may work for some applications. $$\textbf{y}_{t} = (\arg\max LSTM( \textbf{x}_{t}, \textbf{f}_{t}, \textbf{h}_{t-1} = f(\textbf{f}_{t}) ))$$ and is free from initialization. It must be mentioned that a LSTM decoder will be more sensitive and can differentiate between more among the visual features than the MLP layer and hence a default choice for many applications. Sensitive means that LSTM can facilitate more diverse caption generation and will help in segregation of more number of features to appear in the generated sentences.
![Open-End CRUR[]{data-label="fig:Open-End"}](Open-End.png){width=".5\textwidth"}
![Closed-End CRUR[]{data-label="fig:Closed-End"}](Closed-End.png){width=".5\textwidth"}
RNN Coupled CRUR
----------------
Coupled RNN is based on Recurrent Neural Network units and can be regarded as most traditional unit scheme. The RNN with equations $ h_{t} = x_{t} + h_{t-1} $ is much better for applications instead of equations $ h_{t} = x_{t} + y_{t-1} $ because the former has added advantage of dependency on previous latent state space than the dependency on the previous output space. Initialization of state variables like $\textbf{S}_t$ and $\textbf{p}_t$ is important for sequential learning. While $\textbf{S}_0$ and $\textbf{p}_0$ is initialized randomly with zero or very low range $[0.001,-0.001]$ tensors using constant seed. The initial $\textbf{S}_t$ and $\textbf{p}_t$ for the model may even get the touch of visual features to align itself with the contextual information like ($\textbf{S}_t = \textbf{W}_S \textbf{v}$ and $\textbf{p}_t = \textbf{W}_p \textbf{v}$). The model is initialization with the followings, $$\textbf{S}_{1} = \sigma(\mathds{1}(t=1)\textbf{v}\textbf{C}_{1,S})$$ $$\textbf{p}_{1} = \sigma(\mathds{1}(t=1)\textbf{w}\textbf{C}_{1,p})$$ where we have $\textbf{v}$, $\textbf{w} \in \mathbb{R}^{2048}$, $ \mathbb{R}^{1000}$ as the visual features and as the semantic information for the images components respectively.
#### Coupled Open-End RNN
Coupled-oRNN or Coupled Open-End RNN is characterized by no physical interaction between the parallel units and low interactions. This kind of phenomenon does not promote cooperation and with respect to distribution analysis, the units help in divergence of the different possibilities and thus helps in exploration, but unfortunately does not provide enough help in generation, but will provide better accuracy for supervised learning problems. However, if we do an analysis which has evaluations that measure the diversity and innovation of the generator, the Open-End version will be much better. Also, when in comes to distribution, it expands the working area, but whether it helps in generalization cannot be answered without experimentation. The iteration for generation starts with these equations. $$\textbf{S}_{t} = \sigma(\mathds{1}(t>1)\textbf{x}_{1,t-1}\textbf{D}_{1,S} + \textbf{S}_{t-1}\textbf{U}_{1,S})$$ $$\textbf{S}_{t} = RNN_S(\textbf{p}_{t-1},\textbf{p}_{t-1})$$ $$\textbf{p}_{t} = \sigma(\textbf{p}_{t-1}\textbf{W}_{1,p} + \mathds{1}(t>1)\textbf{x}_{2,t-1}\textbf{D}_{1,p})$$ $$\textbf{S}_{t} = RNN_S(\textbf{p}_{t-1},\textbf{p}_{t-1})$$ where we have the same nomenclature as the LSTM network detailed below.
#### Coupled Closed-End RNN
Coupled-cRNN or Coupled Closed-End RNN has the inter-connectivity among the hidden states and is marked by the exchange of the hidden states where the previous states of one unit help both the units to predict the next one. This model will be marked by the stability of the generation with high accuracy, but less exploration and dynamics of the model. However, when it comes to reoccurred of what has been learned by the model, this model have tendency of regeneration of those sequence and thus helps in better distribution modeling of the working space. When it comes to risk management of the working domain, this model will have much more stability in inference and what is being taught to learn. The generation iteration framework works on these following equations. $$\textbf{S}_{t} = \sigma(\textbf{p}_{t-1}\textbf{W}_{1,S} + \mathds{1}(t>1)\textbf{x}_{1,t-1}\textbf{D}_{1,S} + \textbf{S}_{t-1}\textbf{U}_{1,S})$$ $$\textbf{S}_{t} = RNN_S(\textbf{p}_{t-1},\textbf{p}_{t-1})$$ $$\textbf{p}_{t} = \sigma(\textbf{p}_{t-1}\textbf{W}_{1,p} + \mathds{1}(t>1)\textbf{x}_{2,t-1}\textbf{D}_{1,p} + \textbf{S}_{t-1}\textbf{U}_{1,p})$$ $$\textbf{p}_{t} = RNN_p(\textbf{p}_{t-1},\textbf{p}_{t-1})$$ The rest of the equation for estimation of occurrence $$\textbf{u}_{t} = \sigma(\textbf{W}_{u}\textbf{p}_{t})$$ $$\textbf{f}_{t} = \textbf{S}_{t} \textbf{u}_{t}$$ $$\textbf{x}_{t} = (\arg\max\sigma( \textbf{W}_{x}\textbf{f}_{t} )) \textbf{W}_e$$ where we have $\textbf{x}_{t}$ as the embedding of the predicted segment of the sentence.
LSTM Coupled CRUR
-----------------
Coupled LSTM consisted of LSTM units and had far reaching effects for different applications due to the domain intricacies, the model can incorporate to enhance learning capabilities. This kind of definition and interpretation of the variables of the model can help in better structuring and framing of sentences and can help in controlling the writing style and sentence complexity. When it comes to performance evaluation based on the reference sentences, this model has the highest accuracy based on different statistical models. Initialization and refined initialization for $\textbf{p}_0$ and $\textbf{S}_0$ as visual features $\textbf{v} \in \mathbb{R}^n$ is used as functional transformation or$f(\textbf{v})$ mainly involved in regularization and reduction of the dimension. Coupled LSTM can also defined as an open-ended and closed-ended models.
#### Coupled Open-End LSTM
Coupled-oLSTM or Coupled Open-End LSTM does not share any hidden state and processed the contexts independently and this is the reason why it is more sensitive to variations and when it comes to prediction, which is based on some reference sentence, it is not as fruitful as the closed architecture. In fact, bi directional is a sister of this framework. The equations for the Coupled-oLSTM are as follows, $$\label{eq:o_start1}
\textbf{i}_{1,t} = \sigma(\mathds{1}(t>1)\textbf{x}_{1,t-1}\textbf{D}_{1,i} + \textbf{S}_{t-1}\textbf{U}_{1,i})$$ $$\textbf{f}_{1,t} = \sigma(\mathds{1}(t>1)\textbf{x}_{1,t-1}\textbf{D}_{1,f} + \textbf{S}_{t-1}\textbf{U}_{1,f})$$ $$\textbf{o}_{1,t} = \sigma(\mathds{1}(t>1)\textbf{x}_{1,t-1}\textbf{D}_{1,o} + \textbf{S}_{t-1}\textbf{U}_{1,o})$$ $$\textbf{g}_{1,t} = \sigma(\mathds{1}(t>1)\textbf{x}_{1,t-1}\textbf{D}_{1,c} + \textbf{S}_{t-1}\textbf{U}_{1,c})$$ $$\textbf{c}_{1,t} = \textbf{f}_{1,t} \odot \textbf{c}_{1,t-1} + \textbf{i}_{1,t} \odot \textbf{g}_{1,t}$$ $$\label{eq:o_end1}
\textbf{S}_t = \textbf{o}_{1,t} \odot \sigma(\textbf{c}_{1,t})$$ where we replace Equation \[eq:o\_start1\] to Equation \[eq:o\_end1\] as the following equation. $$\textbf{S}_t = LSTM_S^o(\textbf{x}_{t-1},\textbf{S}_{t-1})$$ Similarly, the other unit, which sometimes are regarded as the converger of the context to the most effective likelihood for better prediction, is provided as the followings. $$\label{eq:o_start2}
\textbf{i}_{2,t} = \sigma(\textbf{p}_{t-1}\textbf{W}_{2,i} + \mathds{1}(t>1)\textbf{x}_{2,t-1}\textbf{D}_{2,i} )$$ $$\textbf{f}_{2,t} = \sigma(\textbf{p}_{t-1}\textbf{W}_{2,f} + \mathds{1}(t>1)\textbf{x}_{2,t-1}\textbf{D}_{2,f} )$$ $$\textbf{o}_{2,t} = \sigma(\textbf{p}_{t-1}\textbf{W}_{2,o} + \mathds{1}(t>1)\textbf{x}_{2,t-1}\textbf{D}_{2,o} )$$ $$\textbf{g}_{2,t} = \sigma(\textbf{p}_{t-1}\textbf{W}_{2,c} + \mathds{1}(t>1)\textbf{x}_{2,t-1}\textbf{D}_{2,c} )$$ $$\textbf{c}_{2,t} = \textbf{f}_{2,t} \odot \textbf{c}_{2,t-1} + \textbf{i}_{2,t} \odot \textbf{g}_{2,t}$$ $$\label{eq:o_end2}
\textbf{p}_t = \textbf{o}_{2,t} \odot \sigma(\textbf{c}_{2,t})$$ where we define Equation \[eq:o\_start2\] to Equation \[eq:o\_end2\] as the following equation. $$\textbf{p}_t = LSTM_p^o(\textbf{x}_{t-1},\textbf{p}_{t-1})$$
#### Coupled Closed-End LSTM
Coupled-cLSTM or Coupled Closed-End LSTM shares the intermediates and hence the effect of context initialzation is also doubled and the chances of a profound interpretation chance gets raised. A large number of applications literally depend on the initialzation of the network and this initializxatioon is interpreted as as weighted selection of some portion of the contexts that is selected heuristically and is learned with the training sequences. However, most of the time, the heuristic selection can be regarded as summation of the different segments of the contexts. The equations of the Coupled-cLSTM, which iterates through the sequence of the features, starts with the following equations. $$\label{eq:c_start1}
\textbf{i}_{1,t} = \sigma(\textbf{p}_{t-1}\textbf{W}_{1,i} + \mathds{1}(t>1)\textbf{x}_{1,t-1}\textbf{D}_{1,i} + \textbf{S}_{t-1}\textbf{U}_{1,i})$$ $$\textbf{f}_{1,t} = \sigma(\textbf{p}_{t-1}\textbf{W}_{1,f} + \mathds{1}(t>1)\textbf{x}_{1,t-1}\textbf{D}_{1,f} + \textbf{S}_{t-1}\textbf{U}_{1,f})$$ $$\textbf{o}_{1,t} = \sigma(\textbf{p}_{t-1}\textbf{W}_{1,o} + \mathds{1}(t>1)\textbf{x}_{1,t-1}\textbf{D}_{1,o} + \textbf{S}_{t-1}\textbf{U}_{1,o})$$ $$\textbf{g}_{1,t} = \sigma(\textbf{p}_{t-1}\textbf{W}_{1,c} + \mathds{1}(t>1)\textbf{x}_{1,t-1}\textbf{D}_{1,c} + \textbf{S}_{t-1}\textbf{U}_{1,c})$$ $$\textbf{c}_{1,t} = \textbf{f}_{1,t} \odot \textbf{c}_{1,t-1} + \textbf{i}_{1,t} \odot \textbf{g}_{1,t}$$ $$\label{eq:c_end1}
\textbf{S}_t = \textbf{o}_{1,t} \odot \sigma(\textbf{c}_{1,t})$$ where we represent Equation \[eq:c\_start1\] to Equation \[eq:c\_end1\] as the following equation. $$\textbf{S}_t = LSTM_S^c(\textbf{x}_{t-1},\textbf{S}_{t-1},\textbf{p}_{t-1})$$ The parallel unit, which contributes for the structuring of the sentences or the generated sequence is provided as the followings, $$\label{eq:c_start2}
\textbf{i}_{2,t} = \sigma(\textbf{p}_{t-1}\textbf{W}_{2,i} + \mathds{1}(t>1)\textbf{x}_{2,t-1}\textbf{D}_{2,i} + \textbf{S}_{t-1}\textbf{U}_{2,i})$$ $$\textbf{f}_{2,t} = \sigma(\textbf{p}_{t-1}\textbf{W}_{2,f} + \mathds{1}(t>1)\textbf{x}_{2,t-1}\textbf{D}_{2,f} + \textbf{S}_{t-1}\textbf{U}_{2,f})$$ $$\textbf{o}_{2,t} = \sigma(\textbf{p}_{t-1}\textbf{W}_{2,o} + \mathds{1}(t>1)\textbf{x}_{2,t-1}\textbf{D}_{2,o} + \textbf{S}_{t-1}\textbf{U}_{2,o})$$ $$\textbf{g}_{2,t} = \sigma(\textbf{p}_{t-1}\textbf{W}_{2,c} + \mathds{1}(t>1)\textbf{x}_{2,t-1}\textbf{D}_{2,c} + \textbf{S}_{t-1}\textbf{U}_{2,c})$$ $$\textbf{c}_{2,t} = \textbf{f}_{2,t} \odot \textbf{c}_{2,t-1} + \textbf{i}_{2,t} \odot \textbf{g}_{2,t}$$ $$\label{eq:c_end2}
\textbf{p}_t = \textbf{o}_{2,t} \odot \sigma(\textbf{c}_{2,t})$$ where we symbolize Equation \[eq:c\_start2\] to Equation \[eq:c\_end2\] as the following equation. $$\textbf{p}_t = LSTM_p^c(\textbf{x}_{t-1},\textbf{p}_{t-1},\textbf{S}_{t-1})$$ The final representation for estimation of the prediction likelihood of a word is provided as the joint tensor derived out the product of the individuals $\textbf{S}_{t}$ and $\textbf{u}_{t}$ which can be regarded as the context predictor and the structural component predictor. $$\label{eq:ut}
\textbf{u}_{t} = \sigma(\textbf{W}_{u}\textbf{p}_{t})$$ $$\textbf{f}_{t} = \textbf{S}_{t} \textbf{u}_{t}$$ $$\textbf{x}_{t} = (\arg\max \sigma( \textbf{W}_{x}\textbf{f}_{t} )) \textbf{W}_e$$ where $\{\textbf{x}_{1},\ldots,\textbf{x}_{n}\}$ is the generated sequence.
GRU Coupled CRUR
----------------
GRU Coupled CRUR is composed of Gated Recurrent Units. While LSTM is known to have the maximum effectiveness in long memory retention, GRU is known for its ability for better prediction and likelihood estimation and hence in many applications where the prediction required the final layer or the likelihood estimation layer to be sensitive, it performs better. While, most of the work is based on estimation of the sequence quality of the sentences, we have experimented GRU to determine the position of the GRU in the hierarchy of the memory unit processing and generation capability. Coupled architecture with GRU processes lesser number of weight estimations than Coupled LSTM and the analysis is more focused on whether we are gaining considerably with more weights or the convergence of some of the pipelines and activation units in GRU compensates for them.
#### Coupled Open-End GRU
Coupled-oGRU or or Coupled Open-End GRU follows the open structure principle of late fusion of the likelihood without any inter communication among the memory networks. The equations for GRU based CRUR is noted by the followings, $$\label{eq:o_start3}
\textbf{z}_{1,t} = \sigma(\mathds{1}(t>1)\textbf{x}_{1,t-1}\textbf{D}_{1,z} + \textbf{S}_{t-1}\textbf{U}_{1,z})$$ $$\textbf{r}_{1,t} = \sigma(\mathds{1}(t>1)\textbf{x}_{1,t-1}\textbf{D}_{1,r} + \textbf{S}_{t-1}\textbf{U}_{1,r})$$ $$\begin{gathered}
\label{eq:o_end3}
\textbf{S}_{t} = \textbf{z}_{1,t} \odot \textbf{S}_{t-1} + (1-\textbf{z}_{1,t}) \odot \\
\tanh( \mathds{1}(t>1)\textbf{x}_{1,t-1}\textbf{D}_{1,S} + (\textbf{r}_{1,t} \odot \textbf{S}_{t-1})\textbf{U}_{1,S})\end{gathered}$$ where we define Equation \[eq:o\_start3\] to Equation \[eq:o\_end3\] as the following equation, $$\textbf{S}_t = GRU_S^o(\textbf{x}_{t-1},\textbf{S}_{t-1},\textbf{p}_{t-1})$$ The other GRU unit, which learns the topological dependencies of the sequence is defined as the following equations, $$\label{eq:o_start4}
\textbf{z}_{2,t} = \sigma(\textbf{p}_{t-1}\textbf{W}_{2,z} + \mathds{1}(t>1)\textbf{x}_{2,t-1}\textbf{D}_{2,z} )$$ $$\textbf{r}_{2,t} = \sigma(\textbf{p}_{t-1}\textbf{W}_{2,r} + \mathds{1}(t>1)\textbf{x}_{2,t-1}\textbf{D}_{2,r} )$$ $$\begin{gathered}
\label{eq:o_end4}
\textbf{p}_{t} = \textbf{z}_{2,t} \odot \textbf{p}_{t-1} + (1-\textbf{z}_{2,t}) \odot \\
\tanh((\textbf{r}_{2,t} \odot \textbf{p}_{t-1})\textbf{W}_{2,S} + \mathds{1}(t>1)\textbf{x}_{2,t-1}\textbf{D}_{2,S} )\end{gathered}$$ where we replace Equation \[eq:o\_start4\] to Equation \[eq:o\_end4\] as the following equation, $$\textbf{S}_t = GRU_p^o(\textbf{x}_{t-1},\textbf{S}_{t-1},\textbf{p}_{t-1})$$
#### Coupled Closed-End GRU
Coupled-cGRU or Coupled Closed-End GRU iterates aound the following set of equations are likewise exchanges information though entanglement among the memory units. $$\label{eq:c_start3}
\textbf{z}_{1,t} = \sigma(\textbf{p}_{t-1}\textbf{W}_{1,z} + \mathds{1}(t>1)\textbf{x}_{1,t-1}\textbf{D}_{1,z} + \textbf{S}_{t-1}\textbf{U}_{1,z})$$ $$\textbf{r}_{1,t} = \sigma(\textbf{p}_{t-1}\textbf{W}_{1,r} + \mathds{1}(t>1)\textbf{x}_{1,t-1}\textbf{D}_{1,r} + \textbf{S}_{t-1}\textbf{U}_{1,r})$$ $$\begin{gathered}
\label{eq:c_end3}
\textbf{S}_{t} = \textbf{z}_{1,t} \odot \textbf{S}_{t-1} + (1-\textbf{z}_{1,t}) \odot \tanh((\textbf{z}_{1,t} \odot \textbf{S}_{t-1})\textbf{W}_{1,S} \\
+ \mathds{1}(t>1)\textbf{x}_{1,t-1}\textbf{D}_{1,S} + (\textbf{r}_{1,t} \odot \textbf{S}_{t-1})\textbf{U}_{1,S})\end{gathered}$$ where we represent Equation \[eq:c\_start3\] to Equation \[eq:c\_end3\] as the following equation, $$\textbf{S}_t = GRU_S^c(\textbf{x}_{t-1},\textbf{S}_{t-1},\textbf{p}_{t-1})$$ Similarly, for closed structure, the other GRU unit that governs the grammatical and part-of-speech integrity of the sentences is denoted as the followings, $$\label{eq:c_start4}
\textbf{z}_{2,t} = \sigma(\textbf{p}_{t-1}\textbf{W}_{2,z} + \mathds{1}(t>1)\textbf{x}_{2,t-1}\textbf{D}_{2,z} + \textbf{S}_{t-1}\textbf{U}_{2,z})$$ $$\textbf{r}_{2,t} = \sigma(\textbf{p}_{t-1}\textbf{W}_{2,r} + \mathds{1}(t>1)\textbf{x}_{2,t-1}\textbf{D}_{2,r} + \textbf{S}_{t-1}\textbf{U}_{2,r})$$ $$\begin{gathered}
\label{eq:c_end4}
\textbf{p}_{t} = \textbf{z}_{2,t} \odot \textbf{p}_{t-1} + (1-\textbf{z}_{2,t}) \odot \tanh((\textbf{z}_{2,t} \odot \textbf{p}_{t-1})\textbf{W}_{2,S} \\
+ \mathds{1}(t>1)\textbf{x}_{2,t-1}\textbf{D}_{2,S} + (\textbf{r}_{2,t} \odot \textbf{p}_{t-1})\textbf{U}_{2,S})\end{gathered}$$ where we define Equation \[eq:c\_start4\] to Equation \[eq:c\_end4\] as the following equation, $$\textbf{S}_t = GRU_p^c(\textbf{x}_{t-1},\textbf{S}_{t-1},\textbf{p}_{t-1})$$ The final representation $\textbf{f}_{t}$ is generated as a product of the tensors and is considered as a likelihood of the next word, as $\textbf{x}_{t}$, to be predicted and is directly relative to context $\textbf{S}_{t}$ and the parts of speech component $\textbf{u}_{t}$. $$\textbf{u}_{t} = \sigma(\textbf{W}_{u}\textbf{p}_{t})$$ $$\textbf{f}_{t} = \textbf{S}_{t} \textbf{u}_{t}$$ $$\textbf{x}_{t} = (\arg\max \sigma( \textbf{W}_{x}\textbf{f}_{t} )) \textbf{W}_e$$ generated at time $t$, $\{\textbf{x}_{1},\ldots,\textbf{x}_{n}\}$ is the sequence.
Lastly, it must be mentioned that overall, the main principle of the architecture is dependent on the fact that Open model promotes “Late Fusion" of the features which has been transformed non-linearly as likelihood and thus encourages pure processing of the features. These features are sensitive to variations and are itself are in pure form and this kind of late fusion of features and representation helps in better prediction of inference. They are more favorable to decision making and less influential to generative demonstrations. On the other hand, “Early Fusion" takes place in Closed models and hence, diverse opinions get framed at a very early part of the processing of the features. In early fusion, the chances of combination of a diverse sector of interpretation also gets enhanced and trained and thus these kinds of models are more sensitive to contexts and generation of sentences are not mere repetition of similar sequences.
Generalized and Customized Representation
-----------------------------------------
While describing the different dual models, we realized that looping around the same kind of feedback through the model can be detrimental as each of them has different roles for accomplishment. Hence we define two other ways of feedback and provided a comparative study. Mathematically, the most common notion is the followings, $$\textbf{x}_{1,t-1}, \textbf{x}_{2,t-1} = \textbf{W}_e\textbf{x}_{t-1}, \textbf{W}_e\textbf{x}_{t-1}$$ The other two training models can be regarded as generalized feedback as the feedback learns to adapt to the changes, the model goes through. These two feedback scheme can be regarded as a customized feedback as well as it provides better smoothness for the optimization for the generation of sequential dependencies. Mathematically, the other two schemes, with MLP and memory respectively, can be defined as follows, $$\textbf{x}_{1,t-1}, \textbf{x}_{2,t-1} = \textbf{W}_1\textbf{W}_e\textbf{x}_{t-1}, \textbf{W}_2\textbf{W}_e\textbf{x}_{t-1}$$ $$\textbf{x}_{1,t-1}, \textbf{x}_{2,t-1} = LSTM_1(\textbf{W}_e\textbf{x}_{t-1}), LSTM_2(\textbf{W}_e\textbf{x}_{t-1})$$ As future work, other feedback schemes like the POS structure embedding of the sentence can be used as feedback for the system. Figure \[fig:customizedCRUR\] provided a diagramtic overview of the two different feedback training schemes.
![Generalized (MLP) and Customized (RNN) Representation Feedback Based CRUR[]{data-label="fig:customizedCRUR"}](customizedCRUR.png){width=".5\textwidth"}
Methodology Specifics {#section:methodology}
=====================
Language attributes are different in characteristics and their prediction requires distinct modeling and interpretation while maintaining topological relationships among the different components. This section will mainly describe the details of the data, the experiments performed on them, numerical and qualitative results and interpretation of inference of the different statistical evaluations.
Application Description
-----------------------
Image captioning is an effective way of transforming visual and media images to meaningful sentences that describe certain actions and activities, detected in the media. Though there are thousands of possibilities of such captions, it is an effort to make the machine acquainted with what the possibilities it can see in the image and can reproduce in the captions. This has immense applications ranging from detection of object activities, answering questions about images and videos, story narration and commenting about the events in videos. In this work, we have focused on the prospect of language attribute control for the caption generator application.
Dataset Preparation
-------------------
MSCOCO dataset has been used for our experiments and has undergone much data engineering due to the involvement of immense interest and a large community from industry. MSCOCO consists of $123287$ training images and 566747 training sentence, where each image is associated with at least five sentences from a vocabulary of 8791 words. There are 5000 images (with 25010 sentences) for validation and 5000 images (with 25010 sentences) for testing. Each visual images corresponds to at least 5 different sentences, creating a pool of around (566+25+25)K sentences with vocabulary of $19K$. A large part of the words were under-represented and [@Gan2016] used a total of $8.7K$ words for the training. ResNet features description are used for visual images through transferring the learned knowledge from already trained Residual Network. Two set of features are being widely used for fusion, image features and the probability of the highly occurring objects in images, where ResNet features consists of $2048$ dimension feature vector while the other is Tag features with feature vector of $999$ dimension.
Different Tensor Regularization
-------------------------------
The dimension of Tensor must be considerable. Effective learning happens when the hidden layer dimension is optimum, which can be difficult to define. Hence proper regularization is necessary for many variables. We have used dropout value of $0.5$ for word embedding and the generated TPR tensor denoted as $\textbf{f}_t$. For effective decoding of the captions, an effective learning of the word embedding matrix $W_e$ is necessary along with a layer of Tensor Regularization through dropout. Magically, $0.5$ dropout rate happened to be better accuracy than $1$, $0.7$ and $0.4$ and it is difficult to figure out the reason.
Different features merge into the memory network to compose a perfect and favorable composition called representation that can be identified and decoupled into a sentence and a significant amount of dropout is required for each of them. A common dropout at the entry point of the network had worked but it is not enough as the combinational effect of each of them is selected and generalized. Different dropout combinations will not only help in proper selection and estimation of parameters but also help in generation of unique combination for different sequences. It must be noted that optimum amount of dropout rate is kept at $0.5$ because of the fact that $0.5$ helps in protecting at least more than $50\%$ of the feature participation for the tensors and thus scarcity of important contents is prevented while training the model. Also, $0.5$ helps in preventing over dominance of features and thus help in creation of diversity in the representation. It also makes the model sensitive to the variations of the features.
Normalization of Images Features
--------------------------------
In our experiments, we observed that the individual normalization of the ResNet visual features with mean of the features vector can enhance the BLEU\_4 accuracy of the model with an improvement of $(100\times 1/27 = 3.7 \%)$, which will ultimately enhance the overall context visualization situation for the model. This will bound the features to range of $\mathbb{R}^{2048} \in \mathcal{N}(\mu = 0,\,\sigma^{2})$ in comparison to the previous $\mathbb{R}^{2048} \in \mathcal{N}(\mu \neq 0,\,\sigma^{2})$. This is a significant event for the diverse representation of the image features, mainly when we are defining that suppression of the variations is not a good phenomenon in generative models as it reduces the sensitivity of the model to variations. However, it is fine as far as effectiveness is going high and the feature variations are suppressed at a very low level. Nevertheless, it must be mentioned that the most effectiveness of the handling features by a model is through normalization and had been well established fact in statistics. We have performed individual normalization and is noted mathematically as the following, $$\overline{\textbf{v}} = \textbf{v} - \frac{1}{n}\sum\limits_i^n v_i$$ where $v_i \in \{v_1,v_2,\ldots,v_n\} \in \textbf{v}$.
Normalization of Word Vector
----------------------------
Global vector for all machines, just like vocabulary can create generalization and cross-system interpretation possibility. However, if the global vocabulary representations are used like those provided by Word2Vec, GloVe etc, the effectiveness of the model gets enhanced if the whole set of representation is normalized. Normalization of the Word Vector helped in 1.5% improvement in BLEU\_4 accuracy which is an improvement of $(100\times 1.5/27 = 5.56 \%)$ improvement. Each word embedding vector $(\textbf{w}_e)_i$ is normalized as the following, $$(\overline{\textbf{w}}_e)_i = (\textbf{w}_e)_i - \frac{1}{(V*d)}\sum\limits_i^V \sum\limits_j^d (W_e)_{ij}$$ where, $W_e \in \mathbb{R}^{V\times d}$ is the word embedding matrix, $V$ is the vocabulary length and $d$ is the dimension of the continuous representation embedding for the words based on context. The final $\overline{\textbf{w}}_e \in \mathbb{R}^d \in \mathcal{N}(\mu=0,\,\sigma^{2})$ in comparison to ${\textbf{w}}_e \in \mathbb{R}^d \in \mathcal{N}(\mu\neq 0,\,\sigma^{2})$. Global vector representation will facilitate communication among different models and create feasible opportunities for machines in term of interpretation, storage and retrieval. However, locally trained embedding vector may be better for some cases of prediction and improve by 1% accuracy for BLEU\_4 metrics.
Beam Search
-----------
Beam Search provides the necessary search procedure considering variation of the prediction of the model and scope of error in sequential learning and prediction estimation and considered metrics. Beam Search works on the principle of expanding and trimming of the search space based on the evaluation criteria. Here the log of centered and scaled probability distribution [@Gan2016] of the softmax layer is used for evaluation. Beam Search helped in 8.93% improvement in BLEU\_4 accuracy which is an improvement of $(100\times 2.5/28.0 = 8.93 \%)$ improvement. Mathematically, beam search can be denoted as the following set of equations, $$\begin{split}
p & (\theta_1,\ldots,\theta_n|\textbf{v}) \\
\propto & \max \sum p(\theta_i|\theta_{i-1}\ldots\textbf{v}) \\
\propto & \max \sum p(\theta_i|p(\theta_{i-1}|\theta_{i-2}\ldots\textbf{v})) \\
\propto & \max \sum p(\theta_i|p(\theta_{i-1}|p(\theta_{i-2}|\ldots p_0(\theta_1|\textbf{v}))) \\
\end{split}$$ where $\textbf{v}$ is the extracted visual feature and $\{\theta_1,\ldots,\theta_n\}$ is the generated caption. Involving $\theta$ vocabulary set and $Sp$ as the special character set to indicate the start and end of sentence, maximum likelihood estimated as $\max p (\theta_1,\ldots,\theta_n|\textbf{v})$ $\forall$ $\theta_i \not\in Sp $ for the high value of $p(\theta_i \in Sp|\theta_{i-1}\ldots\textbf{v})$ will prune useful nodes in beam search tree to create the most probable sentence. Researchers has claimed ensemble to be efficient, only when the level of noise in the model is very high and variance is affecting the output. Our model didn’t gain much out of the ensemble of different model outputs. The main reason is that the probability of occurrence for $\{\theta_1,\ldots,\theta_n\}$ is varying and shifting, centering created different numerical range for different predictors with different weights. The training is mainly aimed at occurrence of truth at maximum.
Training Procedure
------------------
Supervised training for recurrent neural network is done mainly through feedback of the previous state(s) of the model like $\textbf{x}_{t-1} \in \mathbb{R}_{W_E}$ where $W_E$ is the dimension of the word embedding. $\textbf{x}_{t-1}$ is the real situation state output, but for better influence and better establishment of the sequential topology and low training error rate during learning phases, $\tilde{\textbf{x}}_{t-1}$ comes from the data and replaces $\textbf{x}_{t-1}$. In sequential learning, $\tilde{\textbf{x}}_{t-1}$ supervise the learning on the assumption that the learning is going well which ensures the long term learning structure is stable and concrete. During initial phases, when error rate is high, $\textbf{x}_{t-1}$ may be $\tilde{\textbf{x}}_{t-1}$ or may be $\textbf{x}_{error}$, but the feedback is return as $\tilde{\textbf{x}}_{t-1}$ where $\tilde{\textbf{x}}_{t-1} = \textbf{x}_{t-1}$ and $\tilde{\textbf{x}}_{t-1} \neq \textbf{x}_{t-1}$ respectively. Knowledge, in raw form from data, generates the subspace for learning while the expansion in variance of subspace is limited and thus creating bias. To increase the variance, error is added or regularized to increase the influence of the important variables. This phenomenon of supervised training creates a new environment, which is different from testing, sometimes termed as drifting and is an issue for sequential learning. Drifting shifts the learning experience biased away from the visual context. Constant injection of $\tilde{\textbf{x}}_{t-1}$ inhibits learning as the gradient diminishes and decrease in error stagnates. So we use $\textbf{x}_{t-1}$ instead of $\tilde{\textbf{x}}_{t-1}$ in some cases and this can create jerks or changes in the weights which will again try to reach a stable state. This problem can be acknowledge through the concept of jitter introduction. Jitter is some kind of noise which is expected during the normal progress of the operations. Introduction of jitter helps create a robust system while at the same time will prevent over-fitting. This concept is similar to the simulated annealing where diminished gradient is revived through phase transform which is one way to escape local optimum towards global one. The percentage of jitter must be very small compared to the normal training.
TPR Attention
-------------
TPR Attention is provided when we use the LSTM (or RNN) as decoder instead of the MLP, as shown in Figure \[fig:Closed-End\] and the performance evaluation is shown in Table \[table:table1\] as LSTM CRUR Attn. With just image features, this is perhaps the best possible performance (BLEU\_4 = .307) achieved so far, while these other similar performances used other kinds of features like semantic tag features etc.
Reinforcement Learning Through SCST
-----------------------------------
Self-critical Sequence Training (SCST) [@rennie2017self] works very well for image captioning applications, where the sequential dependencies help in providing an opportunity for enhancement in learning of the parameters. We achieved a performance of (BLEU\_4 = 0.327), whhen we used SCST with CIDEr-D as evaluation function for gradient feedback. We have denoted the result as LSTM CRUR Attn + RL in Table \[table:table1\]. Figure \[fig:ComparisonRL\] provided some improved instances of generated captions with attention model of CRUR and enhancement with RL. SCST based reinforcement learning can be represented as, $$\label{eq:CR1}
\frac{\delta L(\textbf{w})}{\delta \textbf{w}} = -\frac{1}{2b}\gamma \sum\limits_i \Phi(\textbf{y},\textbf{y}')$$ $$\label{eq:IR}
\frac{\delta L(\textbf{w})}{\delta \textbf{w}} = -\frac{1}{2b}\gamma \sum\limits_i \Phi(\{y_1,\ldots,y_c\}, \{y'_1,\ldots,y'_c\})$$ where $\Phi(.)$ is the evaluation function or the reward function that evaluates certain aspects of the generated captions $\{y_1,\ldots,y_c\} \in \textbf{y}$ and the baseline captions $\{y'_1,\ldots,y'_c\} \in \textbf{y}'$ and $b$ is the mini-batch size considered.
Results & Analysis {#section:results}
==================
This section is focused with the results on the architecture and its performance in comparison with other LSTM and bi-LSTM architectures. CRUR is a generative network and hence we concentrated our focus on generational criteria than mere prediction evaluations and its ability to guess the correct category. In fact, CRUR requires two types of compositional ingredients to be able to logically infer that both are participating in driving the generation and there are open scope to drive the sequential prediction with innovative sentences.
Assessment Procedures
---------------------
Assessment criteria is diversifies through a series of statistical criteria for natural languages as a single evaluation will never be able to judge the compositional ability of the model network in establishing the topological dependency of words and parts-of-speeches into a sentence. Bleu\_n calculates the statistical average of number of combined $n$ series of words that appear in the generated sentence compared to the original sentence. Other evaluation procedures include METEOR, ROUGE\_L, CIDEr-D and SPICE and mainly measure the overall sentence fluency.
Quantitative Analysis {#subsec:Quantitative}
---------------------
This part will mainly discuss the quantitative analysis for different architectures and based on different initialization with visual features. Table \[table:table1\] and Table \[table:table2\] provided a comparative study table for our architecture based on different initialization. From the results, we can clearly say that our model performed much better than the existing architectures and have promising prospects. In these experiments, we have used the training, validation and test set of [@Gan2016], which mainly follows Karpathy’s split and uses a 2048 dimension layer of ResNet101 as visual features.
[|c|c|c|c|c|c|c|c|c|]{} Algorithm & CIDEr-D & Bleu\_4 & Bleu\_3 & Bleu\_2 & Bleu\_1 & ROUGE\_L & METEOR & SPICE\
Human [@wu2017image] & 0.85 & 0.22 & 0.32 & 0.47 & 0.66 & 0.48 & 0.2 & –\
Neural Talk [@Karpathy2015Deep] & 0.66 & 0.23 & 0.32 & 0.45 & 0.63 & – & 0.20 & –\
Mind’sEye [@Chen2015Mind] & – & 0.19 & – & – & – & – & 0.20 & –\
Google [@vinyals2015show] & 0.94 & 0.31 & 0.41 & 0.54 & 0.71 & 0.53 & 0.25 & –\
LRCN [@Donahue2015Long-term] & 0.87 & 0.28 & 0.38 & 0.53 & 0.70 & 0.52 & 0.24 & –\
Montreal [@Xu2015Show] & 0.87 & 0.28 & 0.38 & 0.53 & 0.71 & 0.52 & 0.24 & –\
m-RNN [@Mao2014deep] & 0.79 & 0.27 & 0.37 & 0.51 & 0.68 & 0.50 & 0.23 & –\
[@Jia2015] & 0.81 & 0.26 & 0.36 & 0.49 & 0.67 & – & 0.23 & –\
MSR [@Fang2015captions] & 0.91 & 0.29 & 0.39 & 0.53 & 0.70 & 0.52 & 0.25 & –\
[@Jin2015Aligning] & 0.84 & 0.28 & 0.38 & 0.52 & 0.70 & – & 0.24 & –\
bi-LSTM [@wang2018image] & – & 0.244 & 0.352 & 0.492 & 0.672 & – & – & –\
MSR Captivator [@Devlin2015Language] & 0.93 & 0.31 & 0.41 & 0.54 & 0.72 & 0.53 & 0.25 & –\
Nearest Neighbor [@devlin2015exploring] & 0.89 & 0.28 & 0.38 & 0.52 & 0.70 & 0.51 & 0.24 & –\
MLBL [@kiros2014multimodal] & 0.74 & 0.26 & 0.36 & 0.50 & 0.67 & 0.50 & 0.22 & –\
ATT [@You2016Image] & 0.94 & 0.32 & 0.42 & 0.57 & 0.73 & 0.54 & 0.25 & –\
[@wu2017image] & 0.92 & 0.31 & 0.41 & 0.56 & 0.73 & 0.53 & 0.25 & –\
LSTM-R [@Gan2016] & 0.889 & 0.292 & 0.390 & 0.525 & 0.698 & – & 0.238 & –\
LSTM CRUR + $\textbf{p}_0$ Init & 0.845 & 0.290 & 0.392 & 0.527 & 0.690 & 0.512 & 0.228 & 0.156\
GRU CRUR + $\textbf{p}_0$ Init & 0.840 & 0.287 & 0.387 & 0.522 & 0.691 & 0.511 & 0.227 & 0.154\
LSTM CRUR & 0.860 & 0.294 & 0.391 & 0.523 & 0.690 & 0.510 & 0.229 & 0.155\
GRU CRUR & 0.808 & 0.273 & 0.370 & 0.499 & 0.660 & 0.499 & 0.220 & 0.153\
LSTM CRUR Attn$\dagger$ & 0.927 & 0.307 & 0.407 & 0.542 & 0.711 & 0.528 & 0.245 & 0.175\
LSTM CRUR Attn$\dagger$ + RL & 0.988 & 0.327 & 0.430 & 0.567 & 0.732 & 0.538 & 0.252 & 0.182\
\[table:table1\]
Algorithm CIDEr-D Bleu\_4 Bleu\_3 Bleu\_2 Bleu\_1 ROUGE\_L METEOR SPICE
-------------------------- --------- --------- --------- --------- --------- ---------- -------- -------
LSTM-R [@Gan2016] 0.889 0.292 0.390 0.525 0.698 – 0.238 –
bi-LSTM [@wang2018image] – 0.244 0.352 0.492 0.672 – – –
LSTM CRUR 0.800 0.254 0.357 0.494 0.663 0.462 0.217 0.157
GRU CRUR 0.647 0.220 0.311 0.435 .596 0.422 0.194 0.137
\[table:table2\]
Qualitative Analysis {#subsec:Qualitative}
--------------------
Quantitative never provides the best and aesthetic picture of languages and hence we borrowed qualitative analysis for evaluations. Here are some of the comparisons of the instances of different generated sentences from contexts in Figure \[fig:QualitativeAnalysis1\] and in Figure \[fig:QualitativeAnalysis2\]. Our new approach has produced much better and closely related captions for the images compared to the baseline captions. These generated captions are evidences that the architectures produce captions with novel compositions. This is also evident from the fact that the similarity is accounted with 35% with the original reference set sentences.
{width="\textwidth"}
{width="\textwidth"}
{width="\textwidth"}
Discussion {#section:discussion}
==========
In this work, we discussed the theoretical aspect of coupling different models, each representing different aspects likelihood and the joint fusion of the network will help in establishing the most effective likelihood of grammatically correct sequence of words as a sentence. This model is the generalization of the bi directional LSTM and provides much better insight of the architecture and their utility. Previous approach to deep learning architecture was limited to utilization and managing through evaluation of the end likelihood, an effort leading to no-understanding of the principles of why these architectures were made. However, in this work, we have discussed the different theoretical aspects that lead to an effective learning algorithm, mainly when handling fusion of different parallel architectures and sequence of topologically dependent data. While CRUR model is marked by its ability to learn different interpret-able aspects of the data, it gets rid of the pre-assumptions considered by bi-directional LSTMs, which are either misunderstood or not properly explained and documented and largely neglected. Some of the key take away points that can be said after these analysis:
1. We analyzed dual unit architecture and generalized the notion of product of tensors for exploratory generation.
2. The tensor products help the most for learning of the language attributes of the sentences simultaneously through representation that is different from the likelihood of prediction for a word.
3. We have done elaborated analysis of the language attributes and also came up with the controlling factor that help select sentence construction techniques, which was previously never tried before.
4. It is believed that detection will only be useful if we can use it for control. In that sense, we offered an approach to control different sentence constructions through likelihood of the next possible pasts-of-speech (and can be extended to simple, complex and compound).
Acknowledgments {#acknowledgments .unnumbered}
===============
The author has used University of Florida HiperGator, equipped with NVIDIA Tesla K80 GPU, extensively for the experiments. The author acknowledges University of Florida Research Computing for providing computational resources and support that have contributed to the research results reported in this publication. URL: http://researchcomputing.ufl.edu
[1]{}
Sur, C. (2018). Representation for Language Understanding. Gainesville: University of Florida, pp.1-90. Available at: <https://drive.google.com/file/d/15Fhmt5aM_b0J5jtE9mdWInQPfDS3TqVw> .
Yang, Z., Yuan, Y., Wu, Y., Salakhutdinov, R., Cohen, W. W. Encode, Review, and Decode: Reviewer Module for Caption Generation. arXiv 2016. arXiv preprint arXiv:1605.07912.
Kiros, R., Salakhutdinov, R., Zemel, R. (2014, January). Multimodal neural language models. In International Conference on Machine Learning (pp. 595-603).
Devlin, J., Gupta, S., Girshick, R., Mitchell, M., Zitnick, C. L. (2015). Exploring nearest neighbor approaches for image captioning. arXiv preprint arXiv:1505.04467.
Lu, D., Whitehead, S., Huang, L., Ji, H., & Chang, S. F. (2018). Entity-aware Image Caption Generation. arXiv preprint arXiv:1804.07889.
Lu, J., Yang, J., Batra, D., & Parikh, D. (2018, March). Neural Baby Talk. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (pp. 7219-7228).
You, Q., Jin, H., & Luo, J. (2018). Image Captioning at Will: A Versatile Scheme for Effectively Injecting Sentiments into Image Descriptions. arXiv preprint arXiv:1801.10121.
Melnyk, I., Sercu, T., Dognin, P. L., Ross, J., & Mroueh, Y. (2018). Improved Image Captioning with Adversarial Semantic Alignment. arXiv preprint arXiv:1805.00063.
Wu, J., Hu, Z., & Mooney, R. J. (2018). Joint Image Captioning and Question Answering. arXiv preprint arXiv:1805.08389.
Kilickaya, M., Akkus, B. K., Cakici, R., Erdem, A., Erdem, E., & Ikizler-Cinbis, N. (2017). Data-driven image captioning via salient region discovery. IET Computer Vision, 11(6), 398-406.
Chen, F., Ji, R., Su, J., Wu, Y., & Wu, Y. (2017, October). Structcap: Structured semantic embedding for image captioning. In Proceedings of the 2017 ACM on Multimedia Conference (pp. 46-54). ACM.
Jiang, W., Ma, L., Chen, X., Zhang, H., & Liu, W. (2018). Learning to guide decoding for image captioning. arXiv preprint arXiv:1804.00887.
Wu, C., Wei, Y., Chu, X., Su, F., & Wang, L. (2018). Modeling visual and word-conditional semantic attention for image captioning. Signal Processing: Image Communication.
Fu, K., Li, J., Jin, J., & Zhang, C. (2018). Image-Text Surgery: Efficient Concept Learning in Image Captioning by Generating Pseudopairs. IEEE Transactions on Neural Networks and Learning Systems, (99), 1-12.
Chen, H., Ding, G., Lin, Z., Zhao, S., & Han, J. (2018). Show, Observe and Tell: Attribute-driven Attention Model for Image Captioning. In IJCAI (pp. 606-612).
Cornia, M., Baraldi, L., Serra, G., & Cucchiara, R. (2018). Paying More Attention to Saliency: Image Captioning with Saliency and Context Attention. ACM Transactions on Multimedia Computing, Communications, and Applications (TOMM), 14(2), 48.
Zhao, W., Wang, B., Ye, J., Yang, M., Zhao, Z., Luo, R., & Qiao, Y. (2018). A Multi-task Learning Approach for Image Captioning. In IJCAI (pp. 1205-1211).
Li, X., Wang, X., Xu, C., Lan, W., Wei, Q., Yang, G., & Xu, J. (2018). COCO-CN for Cross-Lingual Image Tagging, Captioning and Retrieval. arXiv preprint arXiv:1805.08661.
Chen, M., Ding, G., Zhao, S., Chen, H., Liu, Q., & Han, J. (2017, February). Reference Based LSTM for Image Captioning. In AAAI (pp. 3981-3987).
Tavakoliy, H. R., Shetty, R., Borji, A., & Laaksonen, J. (2017, October). Paying attention to descriptions generated by image captioning models. In Computer Vision (ICCV), 2017 IEEE International Conference on (pp. 2506-2515). IEEE.
Chen, H., Zhang, H., Chen, P. Y., Yi, J., & Hsieh, C. J. (2017). Show-and-fool: Crafting adversarial examples for neural image captioning. arXiv preprint arXiv:1712.02051.
Ye, S., Liu, N., & Han, J. (2018). Attentive Linear Transformation for Image Captioning. IEEE Transactions on Image Processing.
Wang, Y., Lin, Z., Shen, X., Cohen, S., & Cottrell, G. W. (2017). Skeleton key: Image captioning by skeleton-attribute decomposition. arXiv preprint arXiv:1704.06972.
Chen, T., Zhang, Z., You, Q., Fang, C., Wang, Z., Jin, H., & Luo, J. (2018). “ Factual” or“ Emotional”: Stylized Image Captioning with Adaptive Learning and Attention. arXiv preprint arXiv:1807.03871.
Chen, F., Ji, R., Sun, X., Wu, Y., & Su, J. (2018). GroupCap: Group-Based Image Captioning With Structured Relevance and Diversity Constraints. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (pp. 1345-1353).
Liu, C., Sun, F., Wang, C., Wang, F., & Yuille, A. (2017). MAT: A multimodal attentive translator for image captioning. arXiv preprint arXiv:1702.05658.
Harzig, P., Brehm, S., Lienhart, R., Kaiser, C., & Schallner, R. (2018). Multimodal Image Captioning for Marketing Analysis. arXiv preprint arXiv:1802.01958.
Liu, X., Li, H., Shao, J., Chen, D., & Wang, X. (2018). Show, Tell and Discriminate: Image Captioning by Self-retrieval with Partially Labeled Data. arXiv preprint arXiv:1803.08314.
Chunseong Park, C., Kim, B., & Kim, G. (2017). Attend to you: Personalized image captioning with context sequence memory networks. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (pp. 895-903).
Sharma, P., Ding, N., Goodman, S., & Soricut, R. (2018). Conceptual Captions: A Cleaned, Hypernymed, Image Alt-text Dataset For Automatic Image Captioning. In Proceedings of the 56th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers) (Vol. 1, pp. 2556-2565).
Yao, T., Pan, Y., Li, Y., & Mei, T. (2017, July). Incorporating copying mechanism in image captioning for learning novel objects. In 2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (pp. 5263-5271). IEEE.
Zhang, L., Sung, F., Liu, F., Xiang, T., Gong, S., Yang, Y., & Hospedales, T. M. (2017). Actor-critic sequence training for image captioning. arXiv preprint arXiv:1706.09601.
Fu, K., Jin, J., Cui, R., Sha, F., & Zhang, C. (2017). Aligning where to see and what to tell: Image captioning with region-based attention and scene-specific contexts. IEEE transactions on pattern analysis and machine intelligence, 39(12), 2321-2334.
Ren, Z., Wang, X., Zhang, N., Lv, X., & Li, L. J. (2017). Deep reinforcement learning-based image captioning with embedding reward. arXiv preprint arXiv:1704.03899.
Liu, S., Zhu, Z., Ye, N., Guadarrama, S., & Murphy, K. (2017, October). Improved image captioning via policy gradient optimization of spider. In Proc. IEEE Int. Conf. Comp. Vis (Vol. 3, p. 3).
Cohn-Gordon, R., Goodman, N., & Potts, C. (2018). Pragmatically Informative Image Captioning with Character-Level Reference. arXiv preprint arXiv:1804.05417.
Liu, C., Mao, J., Sha, F., & Yuille, A. L. (2017, February). Attention Correctness in Neural Image Captioning. In AAAI (pp. 4176-4182).
Yao, T., Pan, Y., Li, Y., Qiu, Z., & Mei, T. (2017, October). Boosting image captioning with attributes. In IEEE International Conference on Computer Vision, ICCV (pp. 22-29).
Lu, J., Xiong, C., Parikh, D., & Socher, R. (2017, July). Knowing when to look: Adaptive attention via a visual sentinel for image captioning. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (Vol. 6, p. 2).
Vinyals, O., Toshev, A., Bengio, S., & Erhan, D. (2017). Show and tell: Lessons learned from the 2015 mscoco image captioning challenge. IEEE transactions on pattern analysis and machine intelligence, 39(4), 652-663.
Anderson, P., He, X., Buehler, C., Teney, D., Johnson, M., Gould, S., & Zhang, L. (2018). Bottom-up and top-down attention for image captioning and visual question answering. In CVPR (Vol. 3, No. 5, p. 6).
Zhang, M., Yang, Y., Zhang, H., Ji, Y., Shen, H. T., & Chua, T. S. (2018). More is Better: Precise and Detailed Image Captioning using Online Positive Recall and Missing Concepts Mining. IEEE Transactions on Image Processing.
Park, C. C., Kim, B., & Kim, G. (2018). Towards Personalized Image Captioning via Multimodal Memory Networks. IEEE Transactions on Pattern Analysis and Machine Intelligence.
Wang, Cheng, Haojin Yang, and Christoph Meinel. “Image Captioning with Deep Bidirectional LSTMs and Multi-Task Learning.” ACM Transactions on Multimedia Computing, Communications, and Applications (TOMM) 14.2s (2018): 40.
Rennie, S. J., Marcheret, E., Mroueh, Y., Ross, J., & Goel, V. (2017, July). Self-critical sequence training for image captioning. In CVPR (Vol. 1, No. 2, p. 3).
Wu, Q., Shen, C., Wang, P., Dick, A., & van den Hengel, A. (2017). Image captioning and visual question answering based on attributes and external knowledge. IEEE transactions on pattern analysis and machine intelligence.
Vinyals, Oriol, et al. “Show and tell: A neural image caption generator.” Proceedings of the IEEE conference on computer vision and pattern recognition. 2015.
Karpathy, Andrej, Armand Joulin, and Fei Fei F. Li. “Deep fragment embeddings for bidirectional image sentence mapping.” Advances in neural information processing systems. 2014.
Xu, Kelvin, et al. “Show, attend and tell: Neural image caption generation with visual attention.” International Conference on Machine Learning. 2015.
Fang, Hao, et al. “From captions to visual concepts and back.” Proceedings of the IEEE conference on computer vision and pattern recognition. 2015.
Karpathy, Andrej, and Li Fei-Fei. “Deep visual-semantic alignments for generating image descriptions.” Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. 2015.
Anne Hendricks, Lisa, et al. “Deep compositional captioning: Describing novel object categories without paired training data.” Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. 2016.
Chen, Xinlei, and C. Lawrence Zitnick. “Mind’s eye: A recurrent visual representation for image caption generation.” Proceedings of the IEEE conference on computer vision and pattern recognition. 2015.
Devlin, Jacob, et al. “Language models for image captioning: The quirks and what works.” arXiv preprint arXiv:1505.01809 (2015).
Donahue, Jeffrey, et al. “Long-term recurrent convolutional networks for visual recognition and description.” Proceedings of the IEEE conference on computer vision and pattern recognition. 2015.
Gan, Chuang, et al. “Stylenet: Generating attractive visual captions with styles.” CVPR, 2017.
Jin, Junqi, et al. “Aligning where to see and what to tell: image caption with region-based attention and scene factorization.” arXiv preprint arXiv:1506.06272 (2015).
Kiros, Ryan, Ruslan Salakhutdinov, and Richard S. Zemel. “Unifying visual-semantic embeddings with multimodal neural language models.” arXiv preprint arXiv:1411.2539 (2014).
Kiros, Ryan, Richard Zemel, and Ruslan R. Salakhutdinov. “A multiplicative model for learning distributed text-based attribute representations.” Advances in neural information processing systems. 2014.
Mao, Junhua, et al. “Deep captioning with multimodal recurrent neural networks (m-rnn).” arXiv preprint arXiv:1412.6632 (2014).
Memisevic, Roland, and Geoffrey Hinton. “Unsupervised learning of image transformations.” Computer Vision and Pattern Recognition, 2007. CVPR’07. IEEE Conference on. IEEE, 2007.
Pu, Yunchen, et al. “Variational autoencoder for deep learning of images, labels and captions.” Advances in Neural Information Processing Systems. 2016.
Socher, Richard, et al. “Grounded compositional semantics for finding and describing images with sentences.” Transactions of the Association for Computational Linguistics 2 (2014): 207-218.
Sutskever, Ilya, James Martens, and Geoffrey E. Hinton. “Generating text with recurrent neural networks.” Proceedings of the 28th International Conference on Machine Learning (ICML-11). 2011.
Sutskever, Ilya, Oriol Vinyals, and Quoc V. Le. “Sequence to sequence learning with neural networks.” Advances in neural information processing systems. 2014.
LTran, Du, et al. “Learning spatiotemporal features with 3d convolutional networks.” Proceedings of the IEEE international conference on computer vision. 2015.
Tran, Kenneth, et al. “Rich image captioning in the wild.” Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition Workshops. 2016.
Wu, Qi, et al. “What value do explicit high level concepts have in vision to language problems?.” Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. 2016.
Yang, Zhilin, et al. “Review networks for caption generation.” Advances in Neural Information Processing Systems. 2016.
You, Quanzeng, et al. “Image captioning with semantic attention.” Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. 2016.
Young, Peter, et al. “From image descriptions to visual denotations: New similarity metrics for semantic inference over event descriptions.” Transactions of the Association for Computational Linguistics 2 (2014): 67-78.
Farhadi, Ali, et al. “Every picture tells a story: Generating sentences from images.” European conference on computer vision. Springer, Berlin, Heidelberg, 2010.
Gan, Zhe, et al. “Semantic compositional networks for visual captioning.” arXiv preprint arXiv:1611.08002 (2016).
Girshick, Ross, et al. “Rich feature hierarchies for accurate object detection and semantic segmentation.” Proceedings of the IEEE conference on computer vision and pattern recognition. 2014.
Hodosh, Micah, Peter Young, and Julia Hockenmaier. “Framing image description as a ranking task: Data, models and evaluation metrics.” Journal of Artificial Intelligence Research47 (2013): 853-899.
Jia, Xu, et al. “Guiding the long-short term memory model for image caption generation.” Proceedings of the IEEE International Conference on Computer Vision. 2015.
Krishna, Ranjay, et al. “Visual genome: Connecting language and vision using crowdsourced dense image annotations.” International Journal of Computer Vision 123.1 (2017): 32-73.
Kulkarni, Girish, et al. “Babytalk: Understanding and generating simple image descriptions.” IEEE Transactions on Pattern Analysis and Machine Intelligence 35.12 (2013): 2891-2903.
Li, Siming, et al. “Composing simple image descriptions using web-scale n-grams.” Proceedings of the Fifteenth Conference on Computational Natural Language Learning. Association for Computational Linguistics, 2011.
Kuznetsova, Polina, et al. “TREETALK: Composition and Compression of Trees for Image Descriptions.” TACL 2.10 (2014): 351-362.
Mao, Junhua, et al. “Learning like a child: Fast novel visual concept learning from sentence descriptions of images.” Proceedings of the IEEE International Conference on Computer Vision. 2015.
Mathews, Alexander Patrick, Lexing Xie, and Xuming He. “SentiCap: Generating Image Descriptions with Sentiments.” AAAI. 2016.
Mitchell, Margaret, et al. “Midge: Generating image descriptions from computer vision detections.” Proceedings of the 13th Conference of the European Chapter of the Association for Computational Linguistics. Association for Computational Linguistics, 2012.
Ordonez, Vicente, Girish Kulkarni, and Tamara L. Berg. “Im2text: Describing images using 1 million captioned photographs.” Advances in Neural Information Processing Systems. 2011.
Yang, Yezhou, et al. “Corpus-guided sentence generation of natural images.” Proceedings of the Conference on Empirical Methods in Natural Language Processing. Association for Computational Linguistics, 2011.
Palangi, H., Smolensky, P., He, X., Deng, L., Redmond, W. A. (2017). Deep learning of grammatically-interpretable representations through question-answering. arXiv preprint arXiv:1705.08432.
Huang, Q., Smolensky, P., He, X., Deng, L., Wu, D. (2018). Tensor product generation networks for deep NLP modeling. In Proceedings of the 2018 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long Papers) (Vol. 1, pp. 1263-1273).
|
---
abstract: 'Here we examine the detectability of collisionless dark matter candidates that may constitute not all but only a subdominant component of galactic cold dark matter. We show that current axion searches are not suited for a subdominant component, while direct WIMP searches would not be severely affected by the reduced density. In fact, the direct detection rates of neutralinos stay almost constant even if neutralinos constitute 1% of the halo dark matter. Only for lower densities do the rates decrease with density. Even neutralinos accounting for only $10^{-4}$ of the local dark halo density are within proposed future discovery limits. We comment also on indirect WIMP searches.'
author:
- |
Gintaras Duda Graciela Gelmini Paolo Gondolo\
\
*${}^{1}$ Dept. of Physics and Astronomy, UCLA (University of California, Los Angeles),\
*405 Hilgard Ave., Los Angeles, CA 90095, USA\
*[gkduda,gelmini@physics.ucla.edu]{}\
\
*${}^{2}$ Dept. of Physics, Case Western Reserve University,\
*10900 Euclid Ave, Cleveland, OH 44106-7079, USA\
*[pxg26@po.cwru.edu]{}\
\
******
date:
title: |
-6pt\
-6pt\
-6pt\
-9pt \
Detectability of a subdominant density component of cold dark matter
---
Claims for the need of collisional cold dark matter [@NCCDM] as the main form of dark matter in the Universe have led us to consider the observability of collisionless cold dark matter (CCDM) when it is merely a subdominant component of the cold dark matter (CDM). Namely, if the previously favored CDM candidates, such as axions or Weakly Interacting Massive Particles (WIMPs), constitute only a fraction, say 1% or less, of the local dark matter density, would these particles still be observable in the current and proposed direct and indirect dark matter searches? This is a valid question even if non-CCDM is proven not to be necessary. In fact there is always the possibility of the CDM consisting of several populations, the one we are searching for not being the dominant one. We could even reverse our question in the following manner. If we see a CDM signal in any of our searches, could we be observing a subdominant component of the total CDM?
Naively one may claim that if the local CDM density is 1%, say, of the local halo density, the expected rates in CDM detectors, being proportional to the local number density, should decrease by the same amount. However, we note that a reduction in the relic CDM density implies in general an increase in the probability of interaction of CDM with the detector, for example an increase in the WIMP–nucleus cross section or an increase in the axion–photon coupling constant. Since the detection rate depends on the product of the interaction probability and the local CDM density, the increase in interaction probability may compensate the decrease in CDM density, and the detection rate would remain unchanged.
For axions, this argument is new; for WIMPs, it is not. It has been mentioned implicitly or explicitly in many papers on WIMP detectability since the inception of the subject[@MANY]. It is timely, we believe, to pinpoint, emphasize and update this argument, because it clearly points to the value of continuing WIMP searches even if WIMPs constitute only a small fraction of the dark matter.
We now present arguments that the compensation between interaction probability and local density occurs for axions and WIMPs, and point out some exceptions.
Unless there is segregation for different types of dark matter, the ratio of CCDM to total DM should be the same locally in the Galaxy and globally in the whole Universe. Thus in the following we assume that the local fraction of CCDM $f_{CCDM}$ is related to the CCDM relic density $\Omega_{CCDM}$ through $$\label{no-segregation}
f_{CCDM} = \frac{\rho_{CCDM}}{\rho_{\rm local}} =
\frac{\Omega_{CCDM}}{\Omega_{DM}},$$ where $\rho_{CCDM}$ is the local density of a particular CCDM candidate, $\rho_{\rm local}\simeq$ 0.3 GeV/cm$^3$ is the local halo density (at the location of the Earth), $\Omega_{CCDM}$ is the relic density of our particular CCDM candidate, and $\Omega_{DM} \simeq$ 0.3 is the total contribution of DM to the total energy density of the Universe.
Because the relic density of axions is directly related to its mass, and axion searches are tuned to the axion mass, current searches are not suited to look for a subdominant axion component. The axion relic density is directly related to its mass $m_a$. The usual relation (which has its caveats, see for example [@Sikivie] and references therein) between the axion relic density and its mass $m_a$ is, for a QCD constant of 200 MeV, $$\label{omega_a}
m_a \simeq \frac{0.6 \times 10^{-5} {\rm eV}}{(\Omega_a h^2)^{\frac {6}{7}}},$$ where $h$ is the reduced Hubble constant, $h\simeq 0.7$. A dominant component of axions with $\Omega_a= 0.3$ corresponds, according to this relation, to $m_a
=3 \times 10^{-5}$eV. Thus, we could decrease the density at most to $\Omega_a=
0.003$, so that axions contribute 1% of the total DM density, before encountering the upper bound of $3\times 10^{-3}$eV on the axion mass derived from the observed duration of the Supernova 1987A neutrino signal (and other bounds which exclude all heavier axions, see for example [@Sikivie] and references therein).
The power $P$ from axion to photon conversion in an electromagnetic cavity used for axion dark matter searches is proportional to the product $\rho_a m_a$ of the local axion density and the axion mass [@Sikivie]. In absence of segregation, eq. (\[no-segregation\]) shows that the power is also proportional to $\Omega_a m_a$, which using eq. (\[omega\_a\]) for the axion relic density gives $$P \propto \Omega_a^{1/7} ,$$ that is the power is proportional only to the 1/7th power of the axion relic density. For a decrease in $\Omega_a$ by a factor of 100, the power decreases only by a factor of 2. Of course, because the axion mass has shifted to keep relation (\[omega\_a\]) valid, this power is now at a frequency which is 500 times larger and one would need resonant cavities consequently smaller. The limiting factor of axion dark matter searches with electromagnetic cavities is not the axion to photon conversion power, but the size of the necessary cavities.
The relic density of WIMPs $\Omega_\chi$ is determined by their annihilation cross section $\sigma_a$ by the relation $$\Omega_{\chi} h^2 \simeq \frac{1 \times 10^{-37} {\rm cm}^2}
{\langle \sigma_a v \rangle}~,$$ where $\langle \sigma_a v\rangle$ is the thermal average of the annihilation cross section times the relative velocity of the WIMPs at freeze-out. A reduction in the relic WIMP density requires an increase in their annihilation cross section in the early Universe. This increase is often associated with an increase in the scattering cross section $\sigma_s$ of WIMPs off atomic nuclei. Since the interaction rate in detectors depends on the product $\sigma_s \rho_{\chi}$, if the scattering cross section increases as much as the annihilation cross section, the rate would be unchanged even if $\rho_{\chi}$ has decreased. Concerning indirect detection, the flux of rare cosmic rays and of gamma-rays produced in halo annihilations depends on the product of the square of the density and the annihilation cross section into a particular channel, $\sigma_a {\rho_\chi}^2$. Thus, even if an increase in the cross section would compensate the decrease in one of the powers of the density, the fluxes would still decrease linearly with the halo WIMP density. However, the intensity of the high-energy neutrino emission from the Sun and the Earth would in many cases decrease only slightly, because, to the extent that capture and annihilation of WIMPs in the Sun and the Earth have the time to equilibrate, the neutrino intensity depends only on the capture rate which in turn depends on the product $\sigma_s \rho_\chi$.
We can understand the relation between the scattering and annihilation cross sections $\sigma_s$ and $\sigma_a$ as follows. The scattering cross section of a WIMP of mass $m_{\chi}$ with a nucleus of mass $m_N$ is of the form $$\sigma_s \simeq \frac{m_{\chi}^2 m_N^2}{(m_{\chi} + m_N)^2} |A_s|^2~,$$ where $A_s$ is a reduced amplitude which depends on the dynamics of the collision. The annihilation cross section of WIMPs into light particles is $$\sigma_a \simeq N_a m_{\chi}^2 |A_a|^2~,$$ where $A_a$ is the corresponding reduced amplitude and $N_a$ is the number of annihilation channels. In the case of interactions of weak order, the amplitudes are of the order, $$|A_a|^2 \simeq \frac{\alpha^2}{M^2}~,~~~~~~ |A_s|^2 \simeq A^2
\frac{\alpha^2}{M^2}~,$$ where $\alpha$ is a coupling constant of weak order $\alpha \simeq 10^{-2}$, $M$ is a mass of the particles mediating the interaction, typically $M \simeq
100$GeV and $A$ is the atomic number of the interacting nucleus. Our expression for the scattering amplitude includes the nuclear coherent enhancement factor $A^2$ valid for spin-independent scattering; for spin-dependent scattering the factor $A^2$ should be dropped. Also, our expression for the annihilation cross section is valid for $m_{\chi} < M$, while in the opposite range, $m_{\chi} > M$, we expect $\sigma_a \simeq N_a
m_{\chi}^{-2}$.
The simplest case to consider is that of WIMPs lighter than the nuclei they interact with. From the above equations it is obvious that for these WIMPs $$\frac{\sigma_s} {\sigma_a} \simeq \frac{|A_s|^2}{|A_a|^2} \simeq {\rm const}$$ the ratio of cross sections is approximately constant. In fact, provided the main annihilation channel is into fermions, quarks in particular, crossing arguments insure that the reduced amplitudes of annihilation and scattering with nucleons are similar.
Heavier WIMPs may have other annihilation channels, such as Higgs bosons or vector boson pairs. The crossing argument then does not apply and we don’t expect the scattering amplitude to grow as much as the annihilation amplitude. Moreover, for WIMPs heavier than the nuclei they scatter from, the scattering cross section becomes largely independent of the WIMP mass, while the annihilation cross section always depends on $m_\chi$. In this case, while the annihilation cross section could be made larger by considering lighter (if $m_{\chi} > M$) or heavier (if $m_{\chi} < M$) WIMPs, the scattering cross section would remain largely unchanged.
Therefore, for relatively light WIMPs, and to a lesser extent for heavy WIMPs, we expect the scattering cross section to grow by the same factor $\Omega_{DM}/\Omega_{\chi}$ the annihilation cross section needs to grow to reduce the local CDM density by $\Omega_{\chi}/\Omega_{DM}$. So the rate, which is proportional to the product of the local CDM density and the scattering cross section, remains unchanged.
This argument ceases to be applicable at some small enough WIMP densities, because the necessary increase in cross sections is due to larger couplings and/or smaller mediator masses, which, at some point, encounter accelerator limits which exclude the model. In fact Fig. 1 (described below) shows that for neutralinos constituting 10% of the halo or more the direct detection rates are largely maintained (as evidenced by the behavior of the envelope of the highest rates), and for densities as low as 1% of the halo density, the highest rates only decrease by a factor of about three, showing that there is compensation in the interaction rates while densities decrease by a factor of up to 100. As mentioned, the compensation ceases to work for smaller densities, and for these (as can be seen in Figs. 1 and 2) the envelope of highest rates decreases linearly with the density.
To substantiate the general arguments presented so far, we have analyzed the concrete case of the lightest neutralino in usual variations of the Minimal Supersymmetric Standard model. We used a table of models allowed by all accelerator limits, produced with the DarkSUSY code [@DarkSUSY] over the last few years for other purposes, i.e. having in mind other issues which were addressed in the papers of Ref [@DarkSusypapers] for which the models were originally computed. We have, therefore, not done any particular sampling of the models to favor lower densities and higher detection rates. We restricted our attention to models with $\Omega_\chi \leq \Omega_{DM}= 0.3$ ($\Omega_\chi
h^2 \leq 0.15$) for which we found about 45,000 points in parameter space. For these models, using the spin-dependent and spin-independent neutralino-nucleon cross sections provided in the table, we computed the integrated interaction rates on Ge, following L. Bergström and P. Gondolo in ref. [@DarkSusypapers]. We plot the resulting integrated rates (in units of events per kg-day) in the first two figures of this paper.
Figs. 1 and 2 show the expected integrated rates in Ge detectors as function of the lightest neutralino relic density. Fig. 1 shows only a part of Fig. 2 (the part with the highest rates and densities) displaying the original points in the table of models. Fig. 2 shows the whole range of densities (which reach up to $\Omega_\chi h^2 \simeq 10^{-6}$) using a regular grid of points covering the region with models.
In Fig. 1 the change of the slope of the envelope of the points with maximal rate as the density diminishes is clearly evident. There is approximately no change in maximal rates in the first decade of decrease of density, from $\Omega_\chi h^2=0.15$ (for which neutralinos constitute the whole halo, $f_{CCDM}=1$) to 0.015 (for which neutralinos constitute 10% of the halo, $f_{CCDM} = 0.1$). There is only about a factor of 3 decrease in the next decade, from $\Omega_\chi h^2=0.015$ to 0.0015 ($f_{CCDM}$ from 0.1 to 0.01). For smaller densities the slope of the envelope clearly changes, and as evidenced by Fig. 3, the maximal rates decrease linearly with $\Omega_\chi h^2$ up to the smallest densities. Some of the points shown in Figs. 1 and 2, mostly among with the smallest densities in Fig. 3, should correspond to resonances in the annihilation cross section.
The compensation in the rates can be largely understood just by looking at the spin-independent neutralino-proton cross section $\sigma_{\chi-p}$ as a function of the lightest neutralino relic density, shown in Fig. 3, again with a regular grid showing the allowed region where points were found. Also from this figure, looking at the envelope of the highest cross sections, it is evident that for $\Omega_\chi h^2$ decreasing from 0.15 to 0.0015, i.e. in the first two decades of decrease in neutralino density, $\sigma_{\chi-p}$ increases with decreasing densities; this leads to a compensation in the direct rates. On the other hand, for smaller densities, $\sigma_{\chi-p}$ is about constant or decreases slightly with decreasing densities; this effect is due to accelerator bounds.
Since experimental upper bounds and discovery regions are at present given in terms of $\sigma_{\chi-p}$, Fig. 3 shows the approximate level of the claimed signal and present bounds (by the DAMA, CDMS, COSME-IGEX, and Heidelberg-Moscow collaborations [@DAMAetc]) and conceivable future discovery level (by the GENIUS proposal [@GENIUS]) which are of order $10^{-5}$ pbarns and $10^{-9}$ pbarns, respectively, for neutralinos which account for the whole local halo density, i.e. with $f_{CCDM}$=1. (These values depend on the neutralino mass, but to simplify the presentation we only take the most conservative bounds in our range of masses.) In our case these values must be understood as levels of $f_{CCDM} \sigma_{\chi-p}$, which are shown in Fig. 3 (with short-dashed and long dashed lines respectively). The present level of discovery lightly touches the boundary of the highest rates for densities reduced by up to a factor of about 10. This suggests the possibility that the DAMA claimed signal may correspond to subdominant neutralinos. It is very interesting to see that many models of subdominant neutralinos even with $10^{-4}$ of the total dark matter density, enter in the discovery limit proposed by Genius.
In conclusion, the main point of this paper is that the direct detection rates of neutralinos remain about constant for neutralino densities between 100% and 1% of the halo dark matter and only decrease linearly with the density for lower densities. Thus if a signal is found in direct detection experiments the question of which component of dark matter was found, the primary or a sub-dominant one, may remain open. We also note that neutralinos with density as small as $10^{-4}$ of the local dark halo density are within the discovery limits of proposed future experiments.
Acknowledgments {#acknowledgments .unnumbered}
===============
G.D. and G.G. were supported in part by the U.S. Department of Energy Grant No. DE-FG03-91ER40662, Task C. P.G. acknowledges that one of the points made in this paper, namely that in the MSSM some extreme low density neutralinos with $\Omega h^2 \simeq 10^{-3}-10^{-5}$ might be detectable, was made in conversations with Joe Silk and Joakim Edsjö in March 1999.
[99]{}
D. N. Spergel and P. J. Steinhardt, Phys. Rev. Lett. [**84**]{}, 3760 (2000)
See, for example, T. K. Gaisser, G. Steigman and S. Tilav, Phys. Rev. D [**34**]{}, 2206 (1986); B. Sadoulet, in Proc. of the “13th Texas Symposium on Relativistic Astrophysics", Chicago, Illinois, Dec. 1986, p. 260; K. Griest and B. Sadoulet, in Proc. of the “Second Particle Astrophysics School on Dark Matter", Erice, Italy, 1988; G. Gelmini, E. Roulet and P. Gondolo, Nucl. Phys. Proc. Suppl. [**14B**]{}, 251 (1990) and Phys. B [**351**]{}, 623 (1991); A. Bottino et al., Astropart. Phys. [**2**]{}, 77 (1994); F. Halzen, in “Int. Symp. on Particle Theory and Phenomenology", Ames, Iowa, May 1995, astro-ph/9508020; P. Gondolo, in XXXI Rencontre de Moriond “Dark Matter in Cosmology, Quantum Measurements, Experimental Gravitation”, Les Arcs, France, January 1996, astro-ph/9605290; V. Berezinsky et al. Astropart. Phys. [**5**]{}, 1 (1996); L. Bergström and P. Gondolo, Astropart. Phys. [**5**]{}, 183 (1996); A. Bottino, F. Donato, N. Fornengo and S. Scopel, Astropart. Phys. [**13**]{}, 215 (2000) and Phys. Rev. D [**63**]{}, 125003 (2001).
P. Sikivie, hep-ph/0002154 and references therein.
P. Gondolo, J. Edsjö, L. Bergström, P. Ullio, and E.A. Baltz, in preparation; [*ditto*]{}, astro-ph/0012234, to appear in the proceedings of the 3rd International Workshop on the Identification of Dark Matter (IDM2000) in York.
L. Bergström and P. Gondolo, Astropart. Phys. [**5**]{}, 183 (1996); J. Edsjö and P. Gondolo, Phys. Rev. [**D56**]{}, 1879 (1997); L. Bergström, P. Ullio, and J. H. Buckley, Astropart. Phys. [**9**]{}, 137 (1998); L. Bergström, J. Edsjö, P. Gondolo, Phys. Rev. [**D58**]{}, 103519 (1998); E.A. Baltz and J. Edsjö, Phys. Rev. [**D59**]{}, 023511 (1999).
R. Bernabei et al. \[DAMA col.\], Phys. Lett. [**B 480**]{}, 23 (2000); R. Abusaidi et al. \[CDMS col.\], Phys. Rev. Lett. [**84**]{}, 5699 (2000); I.G. Irastorza et al. \[COSME-IGEX col.\] hep-ph/0011318; L. Baudis et al. \[Heidelberg-Moscow col.\] Phys. Rev. [**D59**]{}, 022001 (1999).
L. Baudis et al. Phys. Rep. [**307**]{}, 301 (1998).
Figure Captions {#figure-captions .unnumbered}
===============
Fig. 1
: Integrated interaction rates of neutralinos in Ge detectors (computed as in L. Bergström and P. Gondolo Ref. [@DarkSusypapers]) in units of events per kg-day, as function of the neutralino relic density, for $\Omega_\chi h^2 \leq 0.15$. Each point represents an actual model.
Fig. 2
: Integrated interaction rates of neutralinos on Ge extended to the whole range of densities. A regular grid of points shows the region covered with models.
Fig. 3
: Spin-independent neutralino-proton cross section $\sigma_{\chi-p}$ as function of the lightest neutralino relic density. As in Fig.2, a regular grid of points shows the region where models were found. The short-dashed and long dashed lines of $f_{CCDM} \sigma_{\chi-p}$=$10^{-5}$ pbarns and $10^{-9}$ pbarns show the approximate level of DAMA claimed signal and the current bounds, and the conceivable future discovery level, respectively ($f_{CCDM}$ is the fraction of the local halo density consisting of neutralinos).
{width="\textwidth"} Figure 1.
{width="\textwidth"} Figure 2.
{width="\textwidth"} Figure 3.
|
---
abstract: 'The probabilistic serial (PS) rule is a prominent randomized rule for assigning indivisible goods to agents. Although it is well known for its good fairness and welfare properties, it is not strategyproof. In view of this, we address several fundamental questions regarding equilibria under PS. Firstly, we show that Nash deviations under the PS rule can cycle. Despite the possibilities of cycles, we prove that a pure Nash equilibrium is guaranteed to exist under the PS rule. We then show that verifying whether a given profile is a pure Nash equilibrium is coNP-complete, and computing a pure Nash equilibrium is NP-hard. For two agents, we present a linear-time algorithm to compute a pure Nash equilibrium which yields the same assignment as the truthful profile. Finally, we conduct experiments to evaluate the quality of the equilibria that exist under the PS rule, finding that the vast majority of pure Nash equilibria yield social welfare that is at least that of the truthful profile.'
author:
- Haris Aziz
- Serge Gaspers
- Simon Mackenzie
- |
Nicholas Mattei\
NICTA and UNSW, Sydney, Australia\
{haris.aziz, serge.gaspers, simon.mackenzie, nicholas.mattei}@nicta.com.au\
Nina Narodytska\
Carnegie Mellon University\
ninan@gmail.com Toby Walsh\
NICTA and UNSW, Sydney, Australia\
toby.walsh@nicta.com.au
title: Equilibria Under the Probabilistic Serial Rule
---
Introduction
============
Resource allocation is a fundamental and widely applicable area within AI and computer science. When resource allocation rules are not strategyproof and agents do not have incentive to report their preferences truthfully, it is important to understand the possible manipulations; Nash dynamics; and the existence and computation of equilibria.
In this paper we consider the *probabilistic serial (PS)* rule for the *assignment problem*. In the *assignment problem* we have a possibly unequal number of agents and objects where the agents express preferences over objects and, based on these preferences, the objects are allocated to the agents [@AGMW14a; @BoMo01a; @Gard73b; @HyZe79a]. The model is applicable to many resource allocation and fair division settings where the objects may be public houses, school seats, course enrollments, kidneys for transplant, car park spaces, chores, joint assets, or time slots in schedules. The *probabilistic serial (PS)* rule is a randomized (or fractional) assignment rule. A randomized or fractional assignment rule takes the preferences of the agents into account in order to allocate each agent a fraction of the object. If the objects are indivisible but allocated in a randomized way, the fraction can also be interpreted as the probability of receiving the object. Randomization is widespread in resource allocation as it is a natural way to ensure procedural fairness [@BCKM12a].
A prominent randomized assignment rule is the PS rule [@BoHe12a; @BoMo01a; @BCKM12a; @KaSe06a; @Koji09a; @Yilm10a; @SaSe13b]. PS works as follows: each agent expresses a linear order over the set of houses.[^1] Each house is considered to have a divisible probability weight of one. Agents simultaneously and at the same speed eat the probability weight of their most preferred house that has not yet been completely eaten. Once a house has been completely eaten by a subset of the agents, each of these agents starts eating his next most preferred house that has not been completely eaten (i.e., they may “join” other agents already eating a different house or begin eating new houses). The procedure terminates after all the houses have been completely eaten. The random allocation of an agent by PS is the amount of each object he has eaten. Although PS was originally defined for the setting where the number of houses is equal to the number of agents, it can be used without any modification for any number of houses relative to the number agents [@BoMo01a; @Koji09a].
In order to compare random allocations, an agent needs to consider relations between them. We consider two well-known relations between random allocation [@ScVa12a; @SaSe13b; @Cho12a]: $(i)$ *expected utility (EU)*, and $(ii)$ *downward lexicographic (DL)*. For EU, an agent prefers an allocation that yields more expected utility. For DL, an agent prefers an allocation that gives a higher probability to the most preferred alternative that has different probabilities in the two allocations. Throughout the paper, we assume that agents express *strict* preferences over houses, i.e., they are not indifferent between any two houses.
The PS rule fares well in terms of fairness and welfare [@BoHe12a; @BoMo01a; @BCKM12a; @Koji09a; @Yilm10a]. It satisfies strong envy-freeness and efficiency with respect to the DL relation [@BoMo01a; @ScVa12a; @Koji09a]. Generalizations of the PS rule have been recommended and applied in many settings [@AzSt14a; @BCKM12a]. The PS rule also satisfies some desirable incentive properties: if the number of objects is at most the number of agents, then PS is DL-strategyproof [@BoMo01a; @ScVa12a]. Another well-established rule, *random serial dictator (RSD)*, is not envy-free, not as efficient as PS [@BoMo01a], and the fractional allocations under RSD are \#P-complete to compute [@ABB13b].
Although PS performs well in terms of fairness and welfare, unlike RSD, it is not strategyproof. @AGM+15c showed that, in the scenario where one agent is strategic, computing his best response (manipulation) under complete information of the other agents’ strategies is NP-hard for the EU relation, but polynomial-time computable for the DL relation. In this paper, we consider the situation where *all* agents are strategic. We especially focus on pure Nash equilibria (PNE) — reported preferences profiles for which no agent has an incentive to report a different preference. We examine the following natural questions for the first time: *(i) What is the nature of best response dynamics under the PS rule? (ii) Is a (pure) Nash equilibrium always guaranteed to exist? (iii) How efficiently can a (pure) Nash equilibrium be computed? (iv) What is the difference in quality of the various equlibria that are possible under the PS rule?* In related work, @EkKe12a showed that when agents are not truthful, the outcome of PS may not satisfy desirable properties related to efficiency and envy-freeness. @HeMa12a provided a necessary and sufficient condition for implementability of Nash equilibrium for the random assignment problem. **Contributions.** For the PS rule we show that expected utility best responses can cycle for *any* cardinal utilities consistent with the ordinal preferences. This is significant as Nash dynamics in matching theory has been an active area of research, especially for the stable matching problem [@AGM+11a], and the presence of a cycle means that following a sequence of best responses is not guaranteed to result in an equilibrium profile. We then prove that a pure Nash equilibrium (PNE) is guaranteed to exist for any number of agents and houses and any utilities. To the best of our knowledge, this is the first proof of the existence of a Nash equilibrium for the PS rule. For the case of two agents we present a linear-time algorithm to compute a preference profile that is in PNE with respect to the original preferences. We show that the general problem for computing a PNE is NP-hard. Finally, we run a set of experiments on real and synthetic preference data to evaluate the welfare achieved by PNE profiles compared to the welfare achieved under the truthful profile.
Preliminaries
=============
An *assignment problem* $(N, H, {\succ \xspace})$ consists of a set of agents $N=\{1,\ldots, n\}$, a set of houses $H=\{h_1, \ldots, h_m\}$ and a preference profile ${\succ \xspace}=({\succ \xspace}_1,\ldots, {\succ \xspace}_n)$ in which ${\succ \xspace}_i$ denotes a complete, transitive and strict ordering on $H$ representing the preferences of agent $i$ over the houses in $H$. A *fractional assignment* is an $(n\times m)$ matrix $[p(i)(h_j)]_{\substack{1\leq i\leq n, 1\leq j\leq m}}$ such that for all $i\in N$, and $h_j\in H$, $0\leq p(i)(h_j)\leq 1$; and for all $j\in \{1,\ldots, m\}$, $\sum_{i\in N}p(i)(h_j)= 1$. The value $p(i)(h_j)$ is the fraction of house $h_j$ that agent $i$ gets. Each row $p(i)=(p(i)(h_1),\ldots, p(i)(h_m))$ represents the *allocation* of agent $i$. A fractional assignment can also be interpreted as a random assignment where $p(i)(h_j)$ is the probability of agent $i$ getting house $h_j$.
Given two random assignments $p$ and $q$, $p(i) {\succ \xspace}_i^{DL} q(i)$ i.e., a player $i$ *DL (downward lexicographic) prefers* allocation $p(i)$ to $q(i)$ if $p(i)\neq q(i)$ and for the most preferred house $h$ such that $p(i)(h)\neq q(i)(h)$, we have that $p(i)(h)>q(i)(h)$. When agents are considered to have cardinal utilities for the objects, we denote by $u_i(h)$ the utility that agent $i$ gets from house $h$. We will assume that the total utility of an agent equals the sum of the utilities that he gets from each of the houses. Given two random assignments $p$ and $q$, $p(i) {\succ \xspace}_i^{EU} q(i)$, i.e., a player $i$ *EU (expected utility) prefers* allocation $p(i)$ to $q(i)$ if $\sum_{h\in H}u_i(h) \cdot p(i)(h)> \sum_{h\in H}u_i(h) \cdot q(i)(h).$ Since for all $i\in N$, agent $i$ compares assignment $p$ with assignment $q$ only with respect to his allocations $p(i)$ and $q(i)$, we will sometimes abuse the notation and use $p{\succ \xspace}_i^{EU} q$ for $p(i){\succ \xspace}_i^{EU} q(i)$.
A *random assignment rule* takes as input an assignment problem $(N,H,{\succ \xspace})$ and returns a random assignment which specifies what fraction or probability of each house is allocated to each agent. We will primarily focus on the expected utility setting but will comment on and use DL wherever needed.
#### The Probabilistic Serial Rule and Equilibria.
The *Probabilistic Serial (PS) rule* is a random assignment algorithm in which we consider each house as infinitely divisible. At each point in time, each agent is eating (consuming the probability mass of) his most preferred house that has not been completely eaten. Each agent eats at the same unit speed. Hence all the houses are eaten at time $m/n$ and each agent receives a total of $m/n$ units of houses. The probability of house $h_j$ being allocated to $i$ is the fraction of house $h_j$ that $i$ has eaten. The PS fractional assignment can be computed in time $O(mn)$. We refer the reader to @BoMo01a or @Koji09a for alternative definitions of PS. The following example from @BoMo01a [@AGM+15c] shows how PS works.
\[example:PS\] Consider an assignment problem with the following preference profile. $$\begin{aligned}
\centering
\succ_1:&\quad h_1,h_2,h_3 \\ \succ_2:&\quad h_2,h_1,h_3 \\ \succ_3:&\quad h_2,h_3,h_1
\end{aligned}$$ Agents $2$ and $3$ start eating $h_2$ simultaneously whereas agent $1$ eats $h_1$. When $2$ and $3$ finish $h_2$, agent $3$ has only eaten half of $h_1$. The timing of the eating can be seen below.
(0,0) – (0,6); (0,0) – (20,0);
(20,6) – (20,0);
(0,2) – (20,2); (0,4) – (20,4); (20,0) – (20,6);
(10,0) – (10,6);
(0,6) – (20,6);
(15,0) – (15,6);
(0,-.8) node(c)[$0$]{}; (20/2,-1.2) node(c)[$\frac{1}{2}$]{};
(20/2,-2.5) node(c)[Time]{};
(20,-1) node(c)[$1$]{};
(15,-1.2) node(c)[$\frac{3}{4}$]{};
(-3,6) node(z)[Agent $1$]{}; (-3,4) node(z)[Agent $2$]{}; (-3,2) node(z)[Agent $3$]{};
(5,6.8) node(z)[$h_1$]{};
(5,4.8) node(z)[$h_2$]{};
(5,2.8) node(z)[$h_2$]{};
(12.5,6.8) node(z)[$h_1$]{};
(12.5,4.8) node(z)[$h_1$]{};
(12.5,2.8) node(z)[$h_3$]{};
(17.5,6.8) node(z)[$h_3$]{};
(17.5,4.8) node(z)[$h_3$]{};
(17.5,2.8) node(z)[$h_3$]{};
The final allocation computed by PS is $$PS(\succ_1,\succ_2,\succ_3)=\begin{pmatrix}
3/4&0&1/4\\
1/4&1/2& 1/4 \\
0&1/2 & 1/2
\end{pmatrix}.$$
Consider the assignment problem in Example \[example:PS\]. If agent $1$ misreports his preferences as follows: $\succ_1':\quad h_2,h_1,h_3,$ then $$PS(\succ_1',\succ_2,\succ_3)=\begin{pmatrix}
1/2&1/3&1/6\\
1/2 & 1/3 & 1/6 \\
0 & 1/3 & 2/3
\end{pmatrix}.$$ If we suppose that $u_1(h_1)=7$, $u_1(h_2)=6$, and $u_1(h_3)=0$, then agent $1$ gets more expected utility when he reports $\succ_1'$. In the example, the truthful profile is in PNE with respect to DL preferences but not expected utility.
We study the existence and computation of Nash equilibria. For a preference profile ${\succ \xspace}$, we denote by $({\succ \xspace}_{-i},{\succ \xspace}_i')$ the preference profile obtained from ${\succ \xspace}$ by replacing agent $i$’s preference by ${\succ \xspace}_i'$.
Nash Dynamics
=============
When considering Nash equilibria of any setting, one of the most natural ways of proving that a PNE always exists is to show that better or best responses do not cycle which implies that eventually, Nash dynamics terminate at a Nash equilibrium profile. Our first result is that DL and EU best responses can cycle. For EU best responses, this is even the case when agents have Borda utilities.
\[th:cycle\] With 2 agents and 5 houses where agents have Borda utilities, EU best responses can lead to a cycle in the profile.
The following 5 step sequence of best responses leads to a cycle. We use $U$ to denote the matrix of utilities of the agents over the houses such that $U[1,1]$ is the utility of agent $1$ for house $h_1$. Note that $P$ starts as the truthful reporting in our example. The initial preferences and utilities of the agents are:
$$\begin{aligned}
\succ_1:\quad & h_2,h_3,h_5,h_4,h_1 \\
\succ_2:\quad & h_5,h_3,h_4,h_1,h_2
\end{aligned}$$
$$U_0 = \begin{pmatrix}
0 & 4 & 3 & 1 & 2 \\
1 & 0 & 3 & 2 & 4
\end{pmatrix}.$$
This yields the following allocation and utilities at the start:
$$PS(\succ_1,\succ_2) = \begin{pmatrix}
1/2 & 1 & 1/2 & 1/2 & 0 \\
1/2 & 0 & 1/2 & 1/2 & 1 \\
\end{pmatrix}, EU_0 = \begin{pmatrix}
6 \\
7
\end{pmatrix}.$$ In Step 1, agent 1 deviates to increase his utility. He reports the preference $\succ_1': h_3,h_4,h_2,h_1,h_5$; which results in $$PS(\succ_1',\succ_2) = \begin{pmatrix}
0 & 1 & 1 & 1/2 & 0 \\
1 & 0 & 0 & 1/2 & 1 \\
\end{pmatrix}, EU_1 = \begin{pmatrix}
7.5 \\
6
\end{pmatrix}.$$
In Step 2, agent 2 changes his report to $\succ_2': h_3,h_4,h_5,h_1,h_2.$ This increases his utility to 7 and decreases the utility of agent 1 to 6.
In Step 3, Agent 1 changes his report to $\succ_1'': h_3,h_5,h_2,h_1,h_4.$ This increases the utility of agent 1 to 7.5 and decreases the utility of agent 2 to 4.5.
In Step 4, Agent 2 changes his report to $\succ_2'': h_5,h_3,h_4,h_1,h_2.$ which increases his expected utility to 6.5 while decreasing the expected utility of agent 1 to 7.
In Step 5, Agent 1 changes his report to $\succ_1''': h_3, h_4, h_2, h_1, h_5.$ Notice that $\succ_1''' = \succ_1'$ and $\succ_2'' = \succ_2$. This is the same profile as the one of Step 1, so we have cycled.
It can be verified that every response in this example is both an EU best response (with respect to any cardinal utilities consistent with the ordinal preferences) and also a DL best response. Hence, DL best responses and EU best responses (with respect to any cardinal utilities consistent with the ordinal preferences) can cycle.
The fact that best responses can cycle means that simply following best responses need not result in a PNE. Hence the normal form game induced by the PS rule is not a potential game [@MoSh96a]. Checking whether an instance has a Nash equilibrium appears to be a challenging problem. The naive method requires going through $O({m!}^n)$ profiles, which is super-polynomial even when $n=O(1)$ or $m=O(1)$.
Existence of Pure Nash Equilibria
=================================
Although it seems that computing a Nash equilibrium is a challenging problem (we give hardness results in the next section), we show that at least one (pure) Nash equilibrium is guaranteed to exist for any number of houses, any number of agents, and any preference relation over fractional allocations.[^2] The proof relies on showing that the PS rule can be modelled as a perfect information extensive form game.
A PNE is guaranteed to exist under the PS rule for any number of agents and houses, and for any relation between allocations.
Consider running PS on all possible ${m!}^n$ preference profiles for $n$ agents and $m$ objects. In each profile $i$, let $t_i^1,\dots, t_i^{k_i}$ be the $k_i$ different time points in the PS algorithm run for the $i$-th profile when at least one house is finished. Let $g=\text{GCD}(\{t_i^{j+1}-t_i^j{\mathbin{:}}j\in \{1,\ldots, k_i-1\}, i\in \{1,\dots,m!^n\})$ where GCD denotes the greatest common divisor. Since in each profile $i$, $t_i^{j+1}-t_i^j>0$ for all $j\in \{0,\ldots, k_i-1\}$, we have that $g$ is finite and greater than zero.
The time interval length $g$ is small enough such that each run of the PS rule can be considered to have $m/g$ stages of duration $g$. Each stage can be viewed as having $n$ sub-stages so that in each stage, agent $i$ eats $g/n$ units of a house in sub-stage $i$ of a stage. In each sub-stage only one agent eats $g/n$ units of the most favoured house that is available. Hence we now view PS as consisting of a total of $mn/g$ sub-stages and the agents keep coming in order $1,2,\ldots, n$ to eat $g$ units of the most preferred house that is still available. If an agent eats $g$ units of a house in a stage then it will eat $g$ units of the same house in his sub-stage of the next stage as long as the house has not been fully eaten. Consider a perfect information extensive form game tree. For a fixed reported preference profile, the PS rule unravels accordingly along a path starting at the root and ending at a leaf. Each level of the tree represents a sub-stage in which a certain agent has his turn to eat $g$ units of his most preferred available house. Note that there is a one-to-one correspondence between the paths in the tree and the ways the PS algorithm can be implemented, depending on the reported preference. A subgame perfect Nash equilibrium (SPNE) is guaranteed to exist for such a game via backward induction: starting from the leaves and moving towards the root of the tree, the agent at the specific node chooses an action that maximizes his utility given the actions determined for the children of the node. The SNPE identifies at least one such path from a leaf to the root of the game. The path can be used to read out the most preferred house of each agent at each point. The information provided is sufficient to construct a preference profile that is in Nash equilibrium. Those houses that an agent did not eat at all can conveniently be placed at the end of the preference list. Such a preference profile is in Nash equilibrium.
Complexity of Pure Nash Equilibrium
===================================
Our argument for the existence of a Nash equilibrium is constructive. However, naively constructing the extensive form game and then computing a subgame perfect Nash equilibrium requires exponential space and time. It is unclear whether a sub-game perfect Nash equilibrium or any Nash equilibrium preference profile can be computed in polynomial time.
General Complexity Results
--------------------------
In this section, we show that computing a PNE is NP-hard and verifying whether a profile is a PNE is coNP-complete. Recently it was shown that computing an expected utility best response is NP-hard [@AGM+15b; @AGM+15c]. Since equilibria and best responses are somewhat similar, one would expect that problems related to equilibria under PS are also hard. However, there is no general reduction from best response to equilibria computation or verification. In view of this, we prove results regarding PNE by closely analyzing the reduction in [@AGM+15b]. First, we show that checking whether a given preference profile is in PNE under the PS rule is coNP-complete.
Given agents’ utilities, checking whether a given preference profile is in PNE under the PS rule is coNP-complete.
Consider the reduction from 3SAT to an assignment setting from [@AGM+15b; @AGM+15c]. We show that checking whether the truthful preference profile is in PNE is coNP-complete. The problem is in coNP, since a Nash deviation is a polynomial time checkable No-certificate. The original reduction considers one manipulator (agent 1) while the other agents $N\setminus \{1\}$ are ‘non-manipulators’. In the original reduction, the utility functions of agents in $N'=N\setminus \{1\}$ are not specified. We specify the utility function of agents in $N'$ as follows: the utility of an agent in $N'$ for his $j$-th most preferred house is ${(8n)}^{m-j+1}$, where $n=|N|$ and $m$ is the number of houses. These utility functions can be represented in space that is polynomial in $O(n+m)$. We rely on 2 main observations about the original reduction. First, in the truthful profile, whenever an agent finishes eating a house all houses have either been fully allocated or are only at most half eaten. Second, in the truthful profile every house except the prize house (the last house that is eaten) is eaten by at least 2 agents. We now show that due to the utility function constructed, each agent from $N'$ is compelled to report truthfully. Assume for contradiction that this is not the case, and let us consider the earliest house (when running the PS rule) that some agent $i\in N'$ starts to eat although he prefers another available house $h$. Let $k$ denote the number of agents who eat a fraction of $h$ under the truthful profile. By reporting truthfully, we show that agent $i$ can get $\frac{1/n-1/{2n}}{2}=1/{4n}$ more of $h$ than by delaying eating $h$. Let us consider how much additional fraction of $h$ agent $i$ can consume by reporting truthfully. If he reports truthfully, he can start eating $h$ earlier and, in the worst case, he can only start $1/2n$ time units earlier. This means that $h$ is consumed earlier by a time of $1/2n$ if $i$ reports truthfully. Consider the time interval of length $1/2n$ between the time when $h$ is finished when $i$ is truthful about $h$ and the time $h$ is finished when $i$ delays eating $h$. In this last stretch of time interval $1/2n$, $i$ gets $\frac{1}{k}\cdot \frac{1}{2n}$ of $h$ extra when he does not report truthfully. Hence by reporting truthfully, $i$ gets at least $\frac{1/n-1/kn}{2}$ more of $h$ which is at least $1/4n$ since $k\geq 2$. Due to the utilities constructed, even if $i$ gets all the less preferred houses, he cannot make up for the loss in utility for getting only $1/4n$ of $h$.
Now that we have established that the agents in $N'$ report truthfully in a PNE, it follows that the truthful preference profile is in PNE iff the manipulator’s truthful report is his best EU response. Assuming that the agents in $N\setminus \{1\}$ report truthfully, checking whether the truthful preference is agent $1$’s best response was shown to be NP-hard. We have shown that the agents $N'$ report truthfully in a PNE. Hence checking whether the truthful profile is in PNE is coNP-hard.
Next, we show that computing a PNE with respect to the underlying utilities of the agents is NP-hard.
\[th:verifyPNE-NPhard\] Given agent’s utilities, computing a preference profile that is in PNE under the PS rule is NP-hard.
The same argument as above shows that the agents in $N'$ play truthfully in a PNE. Hence, a preference profile is in PNE iff agent $1$ reports his EU best response. It has already been shown that computing this EU best response is NP-hard [@AGM+15b] when the other agents are $N\setminus \{1\}$ and report truthfully. Thus computing a PNE is NP-hard.
Case of Two Agents
------------------
In this section, we consider the case of two agents since many disputes involve two parties. Since an EU best response can be computed in linear time for the case of two agents [@AGM+15b; @AGM+15c], it follows that it can be verified whether a profile is a PNE in polynomial time as well.
We can prove the following theorem for the “threat profile” whose construction is shown in Algorithm \[algo:2agent-DL-Nash\].
\[th:threat\] Under PS and for two agents, there exists a preference profile that is in DL-Nash equilibrium and results in the same assignment as the assignment based on the truthful preferences. Moreover, it can be computed in linear time.
The proof is by induction over the length of the constructed preference lists. The main idea of the proof is that if both agents compete for the same house then they do not have an incentive to delay eating it. If the most preferred houses do not coincide, then both agents get them with probability one but will not get them completely if they delay eating them. The algorithm is described as Algorithm \[algo:2agent-DL-Nash\].
We now prove that $Q_1$ is a DL best response against $Q_2$ and $Q_2$ is a DL best response against $Q_1$. The proof is by induction over the length of the preference lists. For the first elements in the preference lists $Q_1$ and $Q_2$, if the elements coincide, then no agent has an incentive to put the element later in the list since the element is both agents’ most preferred house. If the maximal elements do not coincide i.e. $h\neq h'$, then $1$ and $2$ get $h$ and $h'$ respectively with probability one. However they still need to express these houses as their most preferred houses because if they don’t, they will not get the house with probability one. The reason is that $h$ is the next most preferred house after $h'$ for agent $2$ and $h'$ is the next most preferred house after $h$ for agent $1$. Agent $1$ has no incentive to change the position of $h'$ since $h'$ is taken by agent $2$ completely before agent $1$ can eat it. Similarly, agent $2$ has no incentive to change the position of $h$ since $h$ is taken by agent $1$ completely before agent $2$ can eat it. Now that the positions of $h$ and $h'$ have been completely fixed, we do not need to consider them and can use induction over $Q_1$ and $Q_2$ where $h$ and $h'$ are deleted.
The desirable aspect of the threat profile is that since it results in the same assignment as the assignment based on the truthful preferences, the resulting assignment satisfies all the desirable properties of the PS outcome with respect to the original preferences. Since a DL best response algorithm is also an EU best response algorithm for the case of two agents [@AGM+15c], we get the following corollary.
Under PS and for 2 agents, there exists a preference profile that is in Nash equilibrium for any utilities consistent with the ordinal preferences. Moreover it can be computed in linear time.
**Input:** $(\{1,2\},H,(\succ_1,\succ_2))$\
**Output:** The *“threat profile”* $(Q_1,Q_2)$ where $Q_i$ is the preference list of agent $i$ for $i\in \{1,2\}$.
Let $P_i$ be the preference list of agent $i\in \{1,2\}$ Initialise $Q_1$ and $Q_2$ to empty lists. Let $h= \text{first}(P_1)$ and $h'=\text{first}(P_2)$ Append $h$ to $Q_1$; Append $h'$ to $Q_2$ Delete $h$ and $h'$ from $P_1$ and $P_2$ Append $h'$ to $Q_1$; Append $h$ to $Q_2$;
$(Q_1,Q_2)$.
{width="90.00000%"}
{width=".9\textwidth"} (A)
{width=".9\textwidth"} (B)
Experiments
===========
We conducted a series of experiments to understand the number and quality of equilibria that are possible under the PS rule. For quality, we use the utilitarian social welfare (SW) function, i.e., the sum of the utilities of the agents. We are limited by the large search space needed to examine equilibria. For instance, for each set of cardinal preferences we generate, we consider all misreports ($m!$) for all agents ($n$) leaving us with a search space of size $m!^n$ for each of the samples for each combination of parameters. Thus, we only report results for small numbers of agents and houses in this section. We generated 300 samples for each combination of preference model, number of agents, and number of items; reporting the aggregate statistics for these experiments for 4 agents in Figures \[fig:percent-change\] and \[fig:increase-num\]. Each individual sample with 4 agents and 4 houses took about 15 minutes to complete using one core on an Intel Xeon E5405 CPU running at 2.0 GHz with 4 GB of RAM running Debian 2.6.32. The results for 2 agents and up to 5 houses as well as 3 agents and up to 4 houses are similar.
We used a variety of common statistical models to generate data (see, e.g., [@Matt11a; @Mall57a; @LuBo11a; @Berg85a]): the Impartial Culture (**IC**) model generates all preferences uniformly at random; the Single Peaked Impartial Culture (**SP-IC**) generates all preference profiles that are single peaked uniformly at random; Mallows Models (**Mallows**) is a correlated preference model where the population is distributed around a reference ranking proportional to the Kendall-Tau distance; Polya-Eggenberger Urn Models (**Urn**) creates correlations between the agents, once a preference order has been randomly selected, it is subsequently selected with higher probability. In our experiments we set the probability that the second order is equivalent to the first to 0.5. We also used real world data from <span style="font-variant:small-caps;">PrefLib</span> [@MaWa13a]: AGH Course Selection (ED-00009). This data consists of students bidding on courses to attend in the next semester. We sampled students from this data (with replacement) as the agents after we restricted the preference profiles to a random set of houses of a specified size.
To compare the different allocations achieved under PS we need to give each agent not only a preference order but also a utility for each house. Formally we have, for all $i \in N$ and all $h_j \in H$, a value $u_i(h_j) \in {\ensuremath{\mathbb{R}}}$. To generate these utilities we use what we call the *Random* model: we uniformly at random generate a real number between $0$ and $1$ for each house. We sort this list in strictly decreasing order, if we cannot, we generate a new list (we discarded 0 lists in our experiments). We normalize these utilities such that each agent’s utility sums to a constant value (here, the number of houses) that is the same for all agents. In prior experiments we found the Random utility model to be the most manipulable and admit the worst equilibria. Therefore, we only focus on this utility model here (over Borda or Exponential utilities) as it represents, empirically, a worst case. We separate equilibria into three categories: those where the SW is the same as in the truthful profile, those where we have a decrease in SW, and those where we have an increase in SW. Given the social welfare of two different profiles, $SW_1$ and $SW_2$, we use percentage change ($\frac{|SW_1 - SW_2|}{SW_1}\cdot 100$) to understand the magnitude of this difference.
For all models, for all combinations of 2 to 4 agents and 2 to 5 houses there are, generally, slightly more equilibria that increase social welfare compared to the truthful profile than those that decrease it, as illustrated in Figure \[fig:percent-change\]. However, the vast majority of equilibria have the same social welfare as the truthful profile, and the best equilibria are, in general, slightly better than the worst equilibria, as illustrated in Figure \[fig:increase-num\]. Hence, if any or all of the agents manipulate, there may be a loss of SW at equilibria, but there is also the potential for large gains; and the most common outcome of all these agents being strategic is that, dynamically, we will wind up in an equilibria which provides the same SW as the truthful one. Our main observations are:
The vast majority of equilibria have social welfare equal to the social welfare in the truthful profile.
In general, the number of PNE that have increased social welfare (with respect to the truthful profile) is slightly more than the number of PNE that have decreased social welfare.
The maximum increase and decrease in SW in equilibria compared to the truthful profile was observed to be under 23% and 18% respectively .
There are very few profiles that are in equilibria, overall. Profiles with relatively high degrees of correlation between the preferences (Urn and AGH 2004) have fewer equilibrium profiles than the less correlated models (IC and SP-IC).
These trends appear stable with small numbers of agents and houses. We observed similar results for all combinations.
Conclusions
===========
We conducted a detailed analysis of strategic aspects of the PS rule including the complexity of computing and verifying PNE. The fact that PNE are computationally hard to compute in general may act as a disincentive or barrier to strategic behavior. Our experimental results show PS is relatively robust, in terms of social welfare, even in the presence of strategic behaviour. Our study leads to a number of new research directions. It will be interesting to extend our algorithmic results to the extension of PS for indifferences [@KaSe06a]. Additionally, studying *strong* Nash equilibria and a deeper analysis of Nash dynamics are other interesting directions.
Acknowledgments
===============
NICTA is funded by the Australian Government through the Department of Communications and the Australian Research Council through the ICT Centre of Excellence Program. Serge Gaspers is also supported by the Australian Research Council grant DE120101761.
H. Ackermann, P. W. Goldberg, V. S. Mirrokni, H. R[ö]{}glin, and B. V[ö]{}cking. Uncoordinated two-sided matching markets. , 40(1):92–106, 2011.
H. Aziz and P. Stursberg. A generalization of probabilistic serial to randomized social choice. In [*Proc. of 28th AAAI Conference*]{}, pages 559–565. AAAI Press, 2014.
H. Aziz, F. Brandt, and M. Brill. The computational complexity of random serial dictatorship. , 121(3):341–345, 2013.
H. Aziz, S. Gaspers, S. Mackenzie, and T. Walsh. Fair assignment of indivisible objects under ordinal preferences. In [*Proc. of 13th AAMAS Conference*]{}, pages 1305–1312, 2014.
H. Aziz, S. Gaspers, S. Mackenzie, N. Mattei, N. Narodytska, and T. Walsh. Manipulating the probabilistic serial rule. In [*Proc. of 14th AAMAS Conference*]{}, 2015.
H. Aziz, S. Gaspers, S. Mackenzie, N. Mattei, N. Narodytska, and T. Walsh. Manipulating the probabilistic serial rule. Technical report, arXiv.org, 2015.
S. Berg. Paradox of voting under an urn model: The effect of homogeneity. , 47:377–387, 1985.
A. Bogomolnaia and E. J. Heo. Probabilistic assignment of objects: Characterizing the serial rule. , 147:2072–2082, 2012.
A. Bogomolnaia and H. Moulin. A new solution to the random assignment problem. , 100(2):295–328, 2001.
E. Budish, Y.-K. Che, F. Kojima, and P. Milgrom. Designing random allocation mechanisms: [T]{}heory and applications. , 103(2):585–623, 2013.
W. J. Cho. Probabilistic assignment: A two-fold axiomatic approach. imeo, 2012.
O. Ekici and O. Kesten. An equilibrium analysis of the probabilistic serial mechanism. Technical report, [Ö]{}zye[ğ]{}in University, Istanbul, May 2012.
P. G[ä]{}rdenfors. Assignment problem based on ordinal preferences. , 20:331–340, 1973.
E. J. Heo and V. Manjunath. Probabilistic assignment: Implementation in stochastic dominance nash equilibria. Technical Report 1809204, SSRN, 2012.
A. Hylland and R. Zeckhauser. The efficient allocation of individuals to positions. , 87(2):293–314, 1979.
A-K. Katta and J. Sethuraman. A solution to the random assignment problem on the full preference domain. , 131(1):231–250, 2006.
F. Kojima. Random assignment of multiple indivisible objects. , 57(1):134—142, 2009.
T. Lu and C. Boutilier. Learning [M]{}allows models with pairwise preferences. In [*Proc. of 28th ICML*]{}, pages 145–152, 2011.
C. L. Mallows. Non-null ranking models. , 44(1/2):114–130, 1957.
N. Mattei and T. Walsh. Pref[L]{}ib: A library for preference data. In [*Proc. of 3rd ADT*]{}, volume 8176 of [*Lecture Notes in Artificial Intelligence (LNAI)*]{}, pages 259–270. Springer, 2013.
N. Mattei. Empirical evaluation of voting rules with strictly ordered preference data. In [*Proc. of 2nd ADT*]{}, pages 165–177. 2011.
D. Monderer and L. S. Shapley. Potential games. , 14(1):124–143, 1996.
D. Saban and J. Sethuraman. A note on object allocation under lexicographic preferences. , 50:283–289, 2014.
L. J. Schulman and V. V. Vazirani. Allocation of divisible goods under lexicographic preferences. Technical Report arXiv:1206.4366, arXiv.org, 2012.
O. Yilmaz. The probabilistic serial mechanism with private endowments. , 69(2):475–491, 2010.
[^1]: We use the term house throughout the paper though we stress any object could be allocated with these mechanisms.
[^2]: We already know from Nash’s original result that a *mixed* Nash equilibrium exists for any game.
|
---
abstract: 'We discuss the transition paths in a coupled bistable system consisting of interacting multiple identical bistable motifs. We propose a simple model of coupled bistable gene circuits as an example, and show that its transition paths are bifurcating. We then derive a criterion to predict the bifurcation of transition paths in a generalized coupled bistable system. We confirm the validity of the theory for the example system by numerical simulation. We also demonstrate in the example system that, if the steady states of individual gene circuits are not changed by the coupling, the bifurcation pattern is not dependent on the number of gene circuits. We further show that the transition rate exponentially decreases with the number of gene circuits when the transition path does not bifurcate, while a bifurcation facilitates the transition by lowering the quasi-potential energy barrier.'
author:
- Chengzhe Tian
- Namiko Mitarai
title: Bifurcation of Transition Paths Induced by Coupled Bistable Systems
---
Introduction
============
Bistable systems are widely utilized to model the biological processes which exhibit distinct phenotypes under homogeneous conditions [@Ferrell2001; @Ferrell2002; @Veening2008]. Switching between phenotypes (stable states) is facilitated by the stochasticity arising from molecular noise [@Balazsi2011]. The paths of switching have been studied in various systems, such as the $\lambda$-phage lysis-lysogeny decision [@Aurell2002] and cellular development and differentiation [@Wang2011], to gain insights into the molecular processes of biological decision making.
In nature, we sometimes observe situations where multiple bistable systems are coupled. For example in bacterial quorum sensing, every cell produces small signaling molecules whose production is regulated by a positive feedback to synchronize the population. This positive feedback may induce bistability between the high and low concentrations of the signaling molecules. The cells are further coupled by secreting and sensing the signaling molecules in the medium[@Miller2001; @Muller2006]. Another example is the Toxin-Antitoxin (TA) loci in [*Escherichia coli*]{}, where there are 10 known mRNase toxins and every pair may act as a bistable system allowing the cells to switch between the normal growing state and the dormant state which exhibits antibiotic persistence [@Maisonneuve2014; @Cataudella2012; @Fasani2013]. The TA systems may interact each other by interfering the protein synthesis and the cellular growth.
Motivated by these systems, in this paper we analyze a coupled bistable system that consists of interacting identical bistable motifs. We consider the case where coupling is such that the coupled system itself is also bistable and for each stable state the individual motifs are in the same steady state. In other word, the coupling is positive to allow all the bistable motifs to jointly switch from one state to the other state. While the individual bistable systems without coupling show the same pattern of transition paths by definition, the switching properties of the coupled system remain unclear. The individual systems may transit synchronously at the same pace, resulting in one transition path. However, it is also possible that individual systems lead the switching process. As a result, the transition paths of the coupled system are split into multiple ones, a phenomenon called “bifurcation of transition path”. Since many properties of the coupled system, such as the transition rates between the steady states, are dependent on the transition paths, it is interesting to study whether the transition paths of a given coupled bistable system bifurcate and how this bifurcation relates to the individual bistable systems and their coupling.
Bifurcation of transition paths was first demonstrated in the Maier-Stein model [@Maier1996], but its relevance and implication to biochemical systems remain to be explored. In this work, we first construct a model of coupled bistable gene circuits in Section \[modeling\]. Then we demonstrate that the transition paths bifurcate with appropriate parameter sets. In Section \[theory\], we construct a general formulation of coupled bistable systems. We consider the transition of this general model between its steady states as a noise-induced exit process from a metastable state and propose a criterion for the bifurcation of transition paths by extending the previous works on the Maier-Stein model [@Maier1996]. Finally in Section \[application\], we apply our criterion to the model of coupled bistable gene circuits. We confirm the theory numerically and discuss the transition rates.
Model of Coupled Bistable Gene Circuits {#modeling}
=======================================
Consider a model of coupled bistable gene circuits (Fig. \[model\]a). First we restrict our attention to one gene. The promoter of the gene is weak and the proteins of this gene bind to the promoter in the form of tetramers to activate the gene expression. We may model the proteins of this gene using $$\label{example1d}
\frac{\mathrm{d}x}{\mathrm{d}t} = k_0 + k_1 \frac{x^4}{x^4 + S^4} - \gamma x$$ where $x$ is the concentration of the protein, $k_0$ refers to the basal synthesis rate of the protein, the Hill term describes the activation of gene expression by the tetramers and the last term models the linear degradation. With appropriate parameter values, the positive feedback on gene expression allows Eq. \[example1d\] to show bistability.
We now couple $n$ such genes in one cell and we assume that all these genes (and their promoters and proteins) have the same kinetic properties. The genes are coupled in a way that the proteins of the genes are well-mixed in the cells and the mixture activates a cell by binding to a promoter in a tetramer. Multiple coupling strategies may be used. For example, if the proteins of the $n$ genes are identical, any four monomers may bind a promoter and we may model the coupled bistable gene circuits using $$\label{alpha1}
\frac{\mathrm{d}x_i}{\mathrm{d}t} = k_0 + k_1\frac{(x_1+\cdots+x_n)^4}{(x_1+\cdots+x_n)^4 + S_1^4} - \gamma x_i$$ where $x_i$ is the concentration of the products of the $i$-th gene and $S_1$ is the Hill constant for the coupled system. Meanwhile, if the genes are equipped with identical promoters but encode different proteins, and the tetramer activating the gene expression consists of four monomers from the same gene, we model the coupled system using $$\label{alpha4}
\frac{\mathrm{d}x_i}{\mathrm{d}t} = k_0 + k_1\frac{x_1^4+\cdots+x_n^4}{x_1^4+\cdots+x_n^4 + S_2^4} - \gamma x_i$$
We then generalize these examples and we propose the following model of coupled bistable gene circuits $$\label{example}
\frac{\mathrm{d}x_i}{\mathrm{d}t} = k_0 + k_1 \frac{ \bar{x}^4 }{ \bar{x}^4 + S^4 } - \gamma x_i, \bar{x}=\left(\frac{1}{n}\sum_{i=1}^n x_i^\alpha \right)^{1/\alpha}$$ where the parameter $\alpha$, called the configuration parameter, governs the general coupling strategy. It is straightforward that $\alpha=1$ corresponds to Eq. \[alpha1\] and $\alpha=4$ corresponds to Eq. \[alpha4\]. Here we allow $\alpha$ to be arbitrary positive values, though not all coupling strategies are biologically plausible. We also choose the values of the Hill constants ($S_1$ in Eq. \[alpha1\] and $S_2$ in Eq. \[alpha4\]) such that the steady states of every individual gene are not affected by the coupling. If the model for one gene (Eq. \[example1d\]) is bistable, it is straightforward that the general model of coupled bistable gene circuits (Eq. \[example\]) is also bistable and contains three steady states: two stable ones and one saddle. Furthermore, one can show that $x_1=x_2=\cdots=x_n$ holds at every steady state. For convenience, throughout this work we call the stable state where the concentrations of all gene products are low the “lower stable steady state” $\mathbf{x}_l$ and the other stable steady state the “higher stable steady state” $\mathbf{x}_h$.
If we set the volume of the cell to be $V$, the concentrations of proteins $x_i$ can be converted to the absolute numbers of molecules ($=Vx_i$, should be an integer) and the coupled system is governed by the chemical master equation. One may then sample the transition paths between the two steady states using the Gillespie algorithm [@Gillespie1977]. Here we restrict our attention to $n=2$ and the transition from the lower stable steady state to the higher one. As illustrated in Fig. \[model\]b, when $\alpha=1$, the transition paths are narrowly distributed around the diagonal (the line satisfying $x_1=x_2$), suggesting that the switching of the two genes is synchronized. The distribution becomes wider as the value of $\alpha$ increases, and at $\alpha=7.5$, the transition paths exhibit a visible bifurcation. Therefore, coupled bistable systems are capable of exhibiting both bifurcating and non-bifurcating transition paths, and we can modulate the bifurcation pattern for the model of coupled bistable gene circuits with the parameter $\alpha$.
 Model of coupled bistable gene circuits. [**a**]{}. A schematic illustration of the model. A cell contains $n$ genes with weak promoters and their products mix in the cell and activate the gene expression by binding to the promoters in the form of tetramers. [**b**]{}. Distribution of transition paths from the lower stable steady state ($\mathbf{x}_l$) to the higher one ($\mathbf{x}_h$). We set $V=45$ and we carry out 100 simulations using Gillespie algorithm. The distribution of the last instanton trajectories, i.e. the trajectories associated with the successful escapes, is calculated. We present the frequencies in the logarithmic scale in the form of heat plots. Red indicates high frequency and blue indicates low frequency. The black lines represent the most probable escape paths computed in the zero-noise limit. The parameter values are $k_0=0.1$, $k_1=1$, $S=1$ and $\gamma=0.5$. In Appendix D, we verify that the Gillespie simulation is carried out in the low-noise limit. ](Fig1.eps)
Theory
======
In this section we develop a criterion for coupled bistable systems to predict whether the transition paths bifurcate or not. To formulate a general model for coupled bistable systems, we notice that in the model of coupled bistable gene circuits (Eq. \[example\]), every gene is governed by the concentration of its own protein ($x_i$, in the degradation term) and the average concentration of all proteins ($\bar{x}$, in the production term). Here for a general coupled bistable system, we may model the deterministic drifts of the individual bistable systems in the same fashion and describe the effect of noises using $$\label{model_defn}
\frac{\mathrm{d}x_i}{\mathrm{d}t} = f(x_i, h(\mathbf{x})) + \sqrt{\epsilon}\sqrt{g(x_i, h(\mathbf{x}))} \xi_i, i=1,2,\cdots,n$$ which is interpreted as an Ito-type stochastic differential equation. Here $x_i$ is the state of the $i$-th bistable system and $n$ is the number of systems to be coupled. The function $h$, defined as $h(\mathbf{x})=(\sum_{i=1}^n x_i^\alpha/n)^{1/\alpha}$, computes the average state. We choose this formulation because it allows modulating of $\alpha$ and $n$ without changing the steady states. Obviously, the state of every individual bistable system is governed by its own state and the average state of all bistable systems, as modeled by the deterministic drift $f$. The second term of Eq. \[model\_defn\] arises from expanding the chemical master equation in the continuous limit and serves as the noise term for the coupled bistable systems. Here $\xi_i$ are independent Gaussian white noise sources ($\langle \xi_i(t)\xi_j(t')\rangle = \delta_{i,j}\delta(t-t')$) and we assume the multiplicative noise $g(x_i, h(\mathbf{x}))$ for the $i$-th system is also a function of its own state and the average state. The overall noise is modulated by a small parameter $0<\epsilon\ll 1$ in order to keep the validity of the continuous limit. We assume that the deterministic drift with $n=1$ gives three steady states - two stable ones and an unstable one. We further assume the coupled system also allows three steady states - two stable ones and a saddle point. At each steady state, the states of all bistable systems are assumed to be equal. A wide range of coupled bistable systems may be modeled in the fashion of Eq. \[model\_defn\]. For example, the model of coupled bistable system, as mentioned in Section \[modeling\], corresponds to $$\label{fuv}
f(u,v) = k_0 + k_1 v^4/(v^4+S^4) - \gamma u$$ and a noise amplitude of the form $$\label{guv}
g(u,v) = k_0 + k_1 v^4/(v^4+S^4) + \gamma u$$ [@Note]. The parameter $\alpha$ in $h(\mathbf{x})$ corresponds to the coupling mode of the genes and the small $\epsilon$ corresponds to a cell with a large volume $V$ (in the large volume limit, $\epsilon \propto V^{-1}$).
We derive our criterion by performing linear perturbation analysis in the zero-noise limit. When the noise level of a system reaches zero ($\epsilon\rightarrow 0$), the transition between stable steady states is dominated by the path(s) associated with the lowest energy cost (or “action” in standard terminology). This path is called the Most Probable Escape Path (MPEP)[@Maier1993]. The MPEP can be quantified analytically since it is governed by the Freidlin-Wentzell Hamiltonian [@Freidlin2012], allowing ones to use the tools of analytical mechanics for derivation.
One property of MPEP in conventional cases is that the MPEP of transition passes through the saddle of the system (refer to [@Maier1997]) and after that the system follows the deterministic flow to the final stable steady state [@Maier1993]. In our formulation of coupled bistable systems Eq. \[model\_defn\], the individual bistable systems are always driven by the same deterministic drift after passing the saddle. We then conclude that no bifurcation of transition paths will occur between saddle and the final steady state. Therefore, the transition capable of bifurcation is the one from the initial stable steady state to the saddle, and we may restrict our attention to this exit process.
The exit process may be studied with the approach of linear perturbation. To be precise, we first assume that the MPEP does not bifurcate, i.e. all the bistable systems take the same state during the transition. We then perturb the shape of MPEP by allowing some bistable systems to be in different states from the others during transition and we examine how the associated actions for the perturbed MPEP change. If the actions always increase regardless of the perturbation, the non-bifurcating path is locally energetically favorable. For the model of coupled bistable gene circuits, we confirm by numerical simulation that this path is indeed the MPEP. Meanwhile, if any perturbation leads to a decreased action, the non-bifurcating path is energetically unfavorable. The MPEP should then be some other paths. Due to the symmetry of the coupled system, multiple paths must exist. Therefore, the coupled system exhibits a bifurcation of transition paths.
We now follow this approach and we begin our analysis by deriving the equation governing the MPEP, closely following the procedure found in previous work[@Maier1996; @Maier1997]. The model of the coupled bistable system Eq. \[model\_defn\], viewed as a stochastic differential equation, gives the Fokker-Planck equation [@Risken1996] $$\label{Fokker-Planck}
\frac{\partial}{\partial t} P(\mathbf{x},t) = -\nabla\cdot (\mathbf{F}P(\mathbf{x},t)) + \frac{\epsilon}{2} \nabla\cdot[\nabla\cdot (\mathbf{G}P(\mathbf{x},t))]$$ where $P(\mathbf{x},t)$ denotes the probability of the coupled system at state $\mathbf{x}$ and time $t$. The drift vector $\mathbf{F}=[F_1,F_2,\cdots,F_n]'$ satisfies $F_i=f(x_i,h(\mathbf{x}))$ and the covariance $\mathbf{G}$ is a diagonal matrix satisfying $G_{i,i} = g(x_i, h(\mathbf{x}))$. As the initial condition, we use a delta function at the initial stable steady state. The separatrix of the basins gives an absorbing boundary condition. In the zero-noise limit, $\partial P(\mathbf{x},t)/\partial t\approx 0$ and we replace $P(\mathbf{x},t)$ by the quasi-steady state distribution $P_{ss}(\mathbf{x})$ governed by [@Naeh1990] $$\label{FP_quasi}
-\nabla\cdot (\mathbf{F}P_{ss}(\mathbf{x})) + \frac{\epsilon}{2} \nabla\cdot[\nabla\cdot (\mathbf{G}P_{ss}(\mathbf{x}))] = 0$$ We further assume the quasi-steady state distribution to take an Arrhenius form $P_{ss}(\mathbf{x}) \propto \exp\{-W(\mathbf{x})/\epsilon\}$, where $W(\mathbf{x})$ is the quasi-potential[@Kupferman1992]. We perform a WKB expansion by plugging the form into Eq. \[FP\_quasi\] and keeping the terms of the lowest order of $\epsilon$. One can show that the MPEP of the transition is a classical zero-energy trajectory of the Freidlin-Wentzell Hamiltonian $$\label{FW_Hamiltonian}
\mathcal{H}(\mathbf{x},\mathbf{p})=\frac{1}{2}\mathbf{p}^T \mathbf{G}(\mathbf{x}) \mathbf{p} + \mathbf{F}(\mathbf{x})^T\mathbf{p},$$ where the momentum vector can be computed by $\mathbf{p} = \nabla W(\mathbf{x})$[@Maier1997; @Note3]. The equation $\mathcal{H}=0$ can be viewed as the equation governing the MPEP, but it is worth noting that not all trajectories satisfying $\mathcal{H}=0$ are the MPEPs, as one has to examine the associated actions.
We now determine whether the MPEP of the transition bifurcates or not by linearly perturbing the non-bifurcating path and examining the actions. We can show that the non-bifurcating path is a zero-energy trajectory of the Freidlin-Wentzell Hamiltonian (Appendix A). The action associated with the non-bifurcating path $\bar{\mathbf{x}}=[\bar x, \bar x, \cdots, \bar x]^T$ can then be computed by the quasi-potential $W(\mathbf{x})$ and one may quantify how the actions change with perturbations to this path by considering the expansion $W(\bar{\mathbf{x}}+\Delta\mathbf{x}) = W(\bar{\mathbf{x}}) + \nabla W(\bar{\mathbf{x}})\cdot\Delta\mathbf{x} + \frac{1}{2} \Delta\mathbf{x}^T\nabla\nabla W(\bar{\mathbf{x}})\Delta\mathbf{x}$. Due to symmetry, the first-order term vanishes if the system is perturbed in the directions perpendicular to the non-bifurcating path. Note that these directions are of our primary interest, and we then discuss the second-order term and examine the eigenvalues of the hessian matrix $\mathbf{Z(\bar{\mathbf{x}})}=\nabla\nabla W(\bar{\mathbf{x}})$. The equation governing the hessian matrix, computed by differentiating the $\mathcal{H}=0$ twice over $\mathbf{x}$ and making use of the Hamiltonian equation $\mathrm{d}x_i/\mathrm{d}t = \partial\mathcal{H}/\partial p_i$ (also refer to Ref. [@Maier1996]), is $$\begin{aligned}
\frac{\mathrm{d}\mathbf{Z}}{\mathrm{d}t} =& -\mathbf{Z}\mathbf{G}\mathbf{Z} - \mathbf{B}^T\mathbf{Z} - \mathbf{Z}\mathbf{B} - \sum_k p_k\nabla\nabla F_k \notag\\
&- \mathbf{C}^T\mathbf{Z} - \mathbf{Z}\mathbf{C} - \frac{1}{2}\sum_k p_k^2\nabla\nabla G_{k,k} \label{hessian_full}\end{aligned}$$ where $B_{i,j} = \partial F_i/\partial x_j$ is a linearization of the drift vector. The linearization of the variances gives $C_{i,j} = p_i\partial G_{ii}/\partial x_j$.
We may simplify Eq. \[hessian\_full\] significantly (details in Appendix B). By utilizing the fact that all the bistable systems share the same state along the non-bifurcating path, we may express the hessian matrix $\mathbf{Z} = z_1\mathbf{I}_{n\times n}+z_2\mathbf{1}_{n\times n}$, where $\mathbf{I}_{n\times n}$ is an $n\times n$ identity matrix and $\mathbf{1}_{n\times n}$ is an $n\times n$ one matrix. We can further express the remaining terms, namely $\mathbf{B}$, $\mathbf{C}$, $\nabla\nabla F_k$ and $\nabla\nabla G_{k,k}$, in a similar fashion. By plugging the new forms into Eq. \[hessian\_full\], we obtain the equations governing $z_1$ and $z_2$: $z_1$ is shown to follow $$\begin{aligned}
\frac{\mathrm{d}z_1}{\mathrm{d}t} =& -2z_1g_1'p - g z_1^2 - \frac{1}{2}p^2\left(g_{11}''+g_2'\frac{\alpha-1}{x}\right) \notag\\
& - p\left(f_{11}''+f_2'\frac{\alpha-1}{x}\right) - 2z_1f_1'\label{z1_dt}\end{aligned}$$ and $z_2$ is shown to follow Eq. \[z2\_dt\]. Here $f_1'(u,v) = \partial f(u,v)/\partial u$, $f_2'(u,v)=\partial f(u,v)/\partial v$ and $f_{11}''(u,v) = \partial^2 f(u,v)/\partial u^2$ and similar for $g$. In Eq. \[z1\_dt\], $x$ and $p$ refer to the state and the momentum of every bistable system. The two arguments to the function $f$ and $g$ are $x$.
Here we analyze the eigenvalues of the hessian matrix $\mathbf{Z}$, namely $z_1$ (repeat $n-1$ times) and $z_1+nz_2$, and we need to determine the eigenvalues corresponding to perturbations that break the non-bifurcating assumption. In the Appendix C we show that the eigenvalue $z_1+nz_2$ is only governed by one individual bistable system and it represents the change in action along the non-bifurcating path. So this eigenvalue is not relevant in our context and we focus on the directions perpendicular to the non-bifurcating path, which is given by the eigenvalue $z_1$. We should examine the sign of $z_1$ along the non-bifurcating path from the initial stable steady state to the saddle. If $z_1$ is always positive, the perturbation is energetically unfavorable and the non-perturbing path is locally stable. If $z_1$ is negative somewhere, the MPEP bifurcates into multiple paths.
Sometimes it is convenient to express $z_1$ as a function of $x$ rather than time. Based on the Freidlin-Wentzell hamiltonian for the coupled bistable system, we have that the momentum vector may be computed by $\mathbf{p}=\mathbf{G}^{-1}(\dot{\mathbf{x}} - \mathbf{F})$. In addition, the transition for the individual bistable system satisfies that $\mathrm{d}x/\mathrm{d}t=-f(x,x)$ since it is moving against the deterministic drift [@Maier1996]. By plugging these relations into Eq. \[z1\_dt\], we show that $$\begin{aligned}
\frac{\mathrm{d}z_1}{\mathrm{d}x} =& \frac{2f}{g^2}\left(g_{11}''+g_2'\frac{\alpha-1}{x}\right) - \frac{2}{g}\left(f_{11}''+f_2'\frac{\alpha-1}{x}\right) \notag\\
& -\frac{4z_1g_1'}{g} + \frac{g}{f}z_1^2 + 2\frac{z_1f_1'}{f} \label{z1}\end{aligned}$$ Usually we assume that the system is equipped with a constant, non-zero noise at the initial stable steady states [@Maier1996]. We may then solve the equation $\mathrm{d}z_1/\mathrm{d}x=0$ at the initial stable steady states and we set the nontrivial solution to be the initial condition, which is $z_1=-2f_1'/g$. We can integrate Eq. \[z1\] from the initial stable steady state to the saddle and determine whether the MPEP bifurcates or not by looking at the sign of $z_1$.
Eq. \[z1\] generates several insights. The first two terms of Eq. \[z1\] illustrate that the nonlinearity induces the bifurcation of MPEP: $f_{11}''$ and $g_{11}''$ gives information about the nonlinearity for a bistable system without coupling, while $\alpha\neq 1$ gives the nonlinearity created by the coupling of the bistable system. We also notice that Eq. \[z1\] contains no terms of $n$, indicating that the bifurcation of MPEP is not dependent on the number of bistable systems to be coupled.
Application
===========
In this section we apply our theoretical criterion to the model of coupled bistable gene circuits (Eq. \[fuv\]-\[guv\]), and we first consider the MPEP of the model with two coupled genes. For the forward transition from the lower stable steady state $\mathbf{x}_l$ to the higher one $\mathbf{x}_h$, if we choose $\alpha=1$, i.e. no nonlinearity is generated from the coupling of the genes, and integrate Eq. \[z1\] from $\mathbf{x}_l$ to the saddle, we find that the eigenvalue $z_1$ monotonously decreases along the path (Fig. \[2d\_result\]a, left panel). $z_1$ remains positive throughout the path, indicating that at $\alpha=1$ the MPEP does not bifurcate. For $\alpha=2$, the nonlinearity from the coupling of the genes drives $z_1$ to a lower value, but $z_1$ remains positive and the MPEP does not bifurcate as well. For $\alpha=3$ and 4, the coupling is highly nonlinear and $z_1$ reaches negative values and diverges, suggesting that the MPEP bifurcates. Meanwhile, for the backward transition from the higher stable steady state $\mathbf{x}_h$ to the lower one $\mathbf{x}_l$, we may integrate Eq. \[z1\] from $\mathbf{x}_h$ to the saddle and we find that the eigenvalue $z_1$ remains positive regardless of the values of $\alpha$ (Fig. \[2d\_result\]b, left panel), suggesting that the MPEP never bifurcates.
To verify our prediction, we use the geometric Minimum Action Method (gMAM) [@Heymann2008PRL; @Heymann2008CPAM] to compute the MPEP numerically (Fig. \[2d\_result\]ab, right panels)[@Note2]. As predicted, the MPEP for the forward transition with $\alpha=1$ and $2$ do not bifurcate, the one with $\alpha=3$ and $4$ bifurcate to two symmetric ones with equal actions and no bifurcation can be observed for the backward transition. Therefore, the predictive power of the theoretical criterion is confirmed and this bifurcation pattern can explain the distribution of transition paths illustrated in Fig. \[model\]b.
We then examine the transition rates of two coupled genes. The action associated with the MPEP ($\Delta W$) is defined as the difference in the quasi-potential between the initial and final stable steady states, and one can show that the transition rate is proportional to $\exp\{-\Delta W/\epsilon\}$ [@Maier1997]. We sample the configuration $\alpha$ in a wide range and compute the actions. In accordance with the Maier-Stein model [@Maier1996], the actions for the model of coupled bistable gene circuits are not modulated by the value of $\alpha$ if no bifurcation of MPEP exists (Fig. \[nd\_result\]a, $n=2$). Meanwhile, in the presence of bifurcation, the actions are driven to a lower value by this configuration parameter and the transition rates increase. In short, the bifurcation of transition paths facilitates transition.
 Analysis of the coupling of two genes. The eigenvalues induced by the coupling of genes were integrated along the path with no bifurcation from the initial stable steady state to the saddle and plotted in the left panels. The right panels illustrate the most probable escape paths which are numerically computed with gMAM method. The dashed lines represent the separatrix of the basins surrounding the two stable steady states. The arrows represent the directions of transition. [**a**]{}. Forward transition from the lower stable steady state ($\mathbf{x}_l$) to the higher one ($\mathbf{x}_h$). [**b**]{}. Backward transition from the higher stable steady state ($\mathbf{x}_h$) to the lower one ($\mathbf{x}_l$). ](Fig2.eps)
Finally we examine the coupling of more genes. As predicted by the theory, the bifurcation in the transition paths is not dependent on the number of genes and the actions $\Delta W$ exhibit a similar dependency on $\alpha$ (Fig. \[nd\_result\]a, upper panels). Numerical simulation reveals that bifurcated MPEP has an interesting shape. As illustrated in Fig. \[nd\_result\]b for $n=5$ and $\alpha=4$, we plot how the states of individual genes differ from the arithmetic average of all states, and it shows that only one gene is in the leading position (blue curve) and the remaining ones follow with the same states (other curves). We numerically verify that the pattern of one leading/the rest following is the optimal one and other patterns are associated with a higher action. This finding is consistent with the probabilistic view that transient alternation in the state of one gene against the drift is more probable than alternating multiple simultaneously. Numerical analysis also shows that the actions associated with non-bifurcated MPEP scale linearly with the number of genes $n$, suggesting an exponential scaling law in the transition rate. The actions for bifurcated MPEP scales sublinearly with $n$ (Fig. \[nd\_result\]c). This pinpoints that the bifurcation in MPEP leads to a softer dependency between the transition rate and the size of the system.
 Analysis of the coupling of multiple genes. [**a**]{}. The actions $\Delta W$ (defined as the difference in the quasi-potential between the initial stable steady state and the saddle) are computed numerically for both forward transitions (left panels) and backward transitions (right panels). We consider the coupling of two to five genes and sample the configuration $\alpha$ in a wide range (the sampled values are shown in dots). The dash line represents the predicted critical $\alpha$ for bifurcation. The theory predicts that the transition paths for systems with $\alpha\approx2.5$ are bifurcated and the action becomes smaller. With the present numerical accuracy, we are able to show that the transition paths are bifurcated, but not able to find the correct action. [**b**]{}. The most probable escape path of the forward transition for systems with $n=5$ and $\alpha=4$. The horizontal axis represents the arithmetic average of the genes’ states ($x = \sum_{i=1}^n x_i/n$) and the vertical axis represents the difference between the state of individual genes and the average state ($\Delta x_i=x_i-x$). [**c**]{}. The actions $\Delta W$ of forward transitions for multi-dimensional systems with $\alpha=1$ (no bifurcation) and 4 (bifurcation). ](Fig3.eps)
Discussions {#discussion}
===========
In this work, taking the model of coupled bistable gene circuits as an example, we consider a general model of coupled bistable systems and derive a criterion to determine whether the most probable escape paths of the transition bifurcate or not. We show that in the present setup where the steady states are not modulated by $\alpha$ and $n$, this criterion is independent on the number of individual bistable systems. Furthermore, we apply our criterion to the model of coupled bistable gene circuits and verify the theory’s predictive power numerically. Numerical analysis reveals that only one bistable system takes the leading position in the bifurcated MPEP. We also show that the transition rates associated with non-bifurcated MPEP scale exponentially with the number of bistable systems while the coupled systems with bifurcated MPEP exhibit a softer scaling behavior.
The theoretical criterion in this work is developed for a restricted class of systems satisfying the conditions that every individual system is bistable and its steady states are not affected by the coupling and the MPEP passes through the saddle of the coupled system. One may study bifurcation of a wide range of systems by following our procedures and making appropriate changes. For example, it is possible to study the coupling of identical systems where every individual system is not bistable but the coupled system is, as well as to study the coupled systems whose the steady states are dependent on the coupling. The saddle point could also be replaced by the “global maximum along the dominant path” defined by the point along the MPEP where the drift along the path changes its sign (refer to [@Feng2014]). Finally, one may extend the theory to study the coupling of multidimensional bistable systems, though one needs to numerically search for the optimal transition path for one bistable system and integrate $z_1$ (now a matrix rather than a scalar) along this path. In summary, our work provides with a convenient method to study the bifurcation of transition paths of coupled bistable systems and may lead to wide range of practical insights.
Acknowledgments {#acknowledgments .unnumbered}
===============
C.T. and N.M. acknowledge Kim Sneppen and Erik Aurell for helpful discussions. This study is funded by the Danish National Research Foundation through the Center for Models of Life (C.T. and N.M.) and Center for Bacterial Stress Response and Persistence (N.M.).
Appendix: Detailed Derivations in Theory {#appendix-detailed-derivations-in-theory .unnumbered}
========================================
Non-bifurcating Path Is A Zero-Energy Path of the Freidlin-Wentzell Hamiltonian {#app1}
-------------------------------------------------------------------------------
Here we show that the non-bifurcating path of a coupled bistable system is a zero-energy trajectory of the corresponding Freidlin-Wentzell Hamiltonian. First we restrict our attention to one bistable system. Since it is one-dimensional, there exists only one transition path in the zero-noise limit and it is obviously a zero-energy path of the corresponding Freidlin-Wentzell Hamiltonian $\mathcal{H}_1 = \frac{1}{2} p^2 g(x,x) + p f(x,x) = 0$ where we write the drift and variance explicitly. We then discuss the non-bifurcating path of a coupled bistable system. Note that at every point along the non-bifurcating path, $x_1=x_2=\cdots=x_n=h(\mathbf{x})$ and $p_1=p_2=\cdots=p_n$ hold, and we have that the drifts for every individual bistable systems are the same and the diagonal elements of the covariance matrix $\mathbf{G}$ are equal as well. One may then express the corresponding Freidlin-Wentzell Hamiltonian in the form that $\mathcal{H} = \frac{1}{2} \mathbf{p}^T\mathbf{G}(\mathbf{x})\mathbf{p} + \mathbf{F}^T\mathbf{p} = n\left(\frac{1}{2}p^2g(x,x) + pf(x,x)\right)$ where we use $x$ and $p$ to represent the state and momentum of every bistable system. In other words, the coupling vanishes effectively and the Hamiltonian is only dependent on the transition of every individual bistable system. Since the transition path of every individual system is zero-energy, we can claim that the non-bifurcating path of the coupled system should satisfy $\mathcal{H}=0$.
Computation of the Hessian Matrix $\mathbf{Z}$ {#app2}
----------------------------------------------
We compute the eigenvalues of the hessian matrix $\mathbf{Z}=\nabla\nabla W$ along the non-bifurcating path. The element $z_{i,j}$ in the matrix $\mathbf{Z}$ contains information about the interaction between the $i$-th and the $j$-th bistable systems. Recall that all the bistable systems take the same state along the non-bifurcating path, and we have that all the diagonal elements should take the same value (denoted as $z_1+z_2$) and same for the off-diagonal elements (denoted as $z_2$), though we should notice that the diagonal elements may not equal to the off-diagonal ones. The hessian matrix $\mathbf{Z}$ then takes the form of $z_1\mathbf{I}_{n\times n} + z_2\mathbf{1}_{n\times n}$ and it is straightforward to show that the eigenvalues are $z_1$ (repeat $n-1$ times) and $z_1+nz_2$.
The non-bifurcating path also allows us to simplify the computation of the drifts and the noise levels in Eq. \[hessian\_full\]. Take the drifts as an example. Since all the bistable systems take the same state, for any bistable systems $i$, $j$, $k$ and $l$ ($i\neq j$, $k\neq l$), we have $x_i=x_j=x_k=x_l$ and therefore $F_i = F_j = F_k = F_l$. The linearization of the drifts is then $B_{i,j} = \partial f(x_i, h(\mathbf{x}))/\partial x_j = f_2'(x_i, h(\mathbf{x})) \partial h(\mathbf{x})/\partial x_j$. Note that the function $h$ is symmetric for all individual bistable systems, and we have $\partial h(\mathbf{x})/\partial x_j = \partial h(\mathbf{x})/\partial x_l$. We then show that $B_{i,j} = f_2'(x_k, h(\mathbf{x})) \partial h(\mathbf{x})/\partial x_l = \partial f(x_k,h(\mathbf{x}))/\partial x_l = B_{k,l}$. In other words, the off-diagonal elements of the linearization matrix $\mathbf{B}$ take the same value. Similarly, one can prove that the diagonal elements also take the same value and we can express $\mathbf{B}$ in the form of $\mathbf{B} = f_1'\mathbf{I}_{n\times n} + f_2'h'\mathbf{1}_{n\times n}$ where the arguments to the functions are $x$ and $h'$ is defined as $\partial h(\mathbf{x})/\partial x_i$. Meanwhile, we may also simplify the hessian of the drift $\nabla\nabla F_k$. By noticing that $p_1=p_2=\cdots=p_n$, we have that $$\begin{aligned}
&\sum_k p_k\nabla\nabla F_k = p(f_{11}''+nf_2'(h''_{s}-h''_{a}))\mathbf{I}_{n\times n} \\
& + p(2f_{12}''h' + nf_{22}''h'^2 + nf_2'h''_{a})\mathbf{1}_{n\times n}\end{aligned}$$ where $p$ is the momentum of one bistable system, $h''_{s}$ is defined as $\partial^2 h(\mathbf{x})/\partial x_i^2$ and $h''_{a}$ is defined as $\partial^2 h(\mathbf{x})/\partial x_i\partial x_j$ for $i\neq j$. Following the same procedure, one can show that $\mathbf{C} = g_1'p\mathbf{I}_{n\times n} + g_2'h'p\mathbf{1}_{n\times n}$ and $$\begin{aligned}
&\sum_k p_k^2\nabla\nabla G_{k,k} = p^2(g_{11}''+ng_2'(h''_s - h''_a))\mathbf{I}_{n\times n} \\
&+ p^2(2g_{12}''h'+ng_{22}''h'^2 + ng_2'h''_a)\mathbf{1}_{n\times n}\end{aligned}$$
We are readily to obtain the eigenvalues of the hessian matrix $\mathbf{Z}$ by plugging the equations above into Eq. \[hessian\_full\]. By making use the facts that $h' = 1/n$, $h''_a = (1-\alpha)/(n^2 x)$ and $h''_s = (1-\alpha)/(n^2 x) + (\alpha-1)/nx$ along the non-bifurcating path, we have Eq. \[z1\_dt\] and
$$\begin{aligned}
&\frac{\mathrm{d}z_2}{\mathrm{d}t} = - g(2z_1z_2+nz_2^2) - \frac{2}{n}p(z_1g_2' + nz_2g_1' + nz_2g_2')\notag\\
& - \frac{1}{2n}p^2 (2g_{12}'' + g_{22}'') - \frac{1-\alpha}{2nx}p^2 g_2' - \frac{1-\alpha}{nx}pf_2' \notag\\
& - \frac{1}{n}p(2f_{12}''+f_{22}'') - \frac{2}{n}(z_1f_2'+nz_2f_1'+nz_2f_2') \label{z2_dt}\end{aligned}$$
Analysis of $z_1+nz_2$ {#app3}
----------------------
We analyze the eigenvalue $z_1+nz_2$ along the non-bifurcating path. By making use of Eq. \[z1\_dt\] and Eq. \[z2\_dt\], we show that this eigenvalue follows $$\begin{aligned}
&\frac{\mathrm{d}(z_1+nz_2)}{\mathrm{d}t} =- g(z_1+nz_2)^2 - 2(z_1+nz_2)(g_1'+g_2')p\notag\\
& - p(f_{11}''+2f_{12}''+f_{22}'') - \frac{1}{2}p^2(g_{11}''+2g_{12}''+g_{22}'') \notag\\
& - 2(z_1+nz_2)(f_1'+f_2') \label{z1nz2_dt}\end{aligned}$$ In Eq. \[z1nz2\_dt\], the term $g_1'+g_2'$ is equivalent to $\mathrm{d}g(x,x)/\mathrm{d}x$, $g_{11}''+2g_{12}''+g_{22}''$ is equivalent to $\mathrm{d}^2 g(x,x)/\mathrm{d}x^2$ and similar for the terms of $f$. Since $f(x,x)$ and $g(x,x)$ are the drift and noise induced by one bistable system, we claim that the eigenvalue $z_1+nz_2$ corresponds to the direction of perturbation where the coupling of multiple bistable systems effectively vanishes. The only possible direction is the one along the non-bifurcating path. Therefore, the eigenvalue $z_1+nz_2$ should not be considered in our context as it does not break the non-bifurcating assumption. We should then focus on the eigenvalue $z_1$, as discussed in Section \[theory\].
Validation of Low-Noise Limit in Gillespie Simulation {#verification_FW}
-----------------------------------------------------
We verify that the Gillespie simulation in Fig. \[model\]b is carried out in the low-noise limit. We perform simulation with three volumes ($V=30, 40, 45$) and estimate the actions by fitting the simulation data to $T=C\mathrm{e}^{\Delta W/\epsilon}$ where $T$ is the mean transition time and $\epsilon=1/V$. We show that the estimated actions are in good agreement with the values associated with the MPEP which are computed in the low-noise limit (Fig. \[nd\_result\], Table \[comparison\_dW\]). Furthermore, we plot the MPEP on top of the heat plots in Fig. \[model\]b and we show that the transition paths sampled from Gillespie simulation match the MPEP. Therefore, we claim that our Gillespie simulation is in the low-noise limit.
$\alpha=1$ $\alpha=2$ $\alpha=4$ $\alpha=7.5$
-- ------------ ------------- ------------- -------------- -------------
$V=30$ $1.37*10^6$ $6.02*10^5$ $1.15*10^5$ $1.99*10^4$
$V=40$ $5.44*10^7$ $2.54*10^7$ $2.51*10^6$ $2.55*10^5$
$V=45$ $2.97*10^8$ $1.25*10^8$ $1.31*10^7$ $8.02*10^5$
0.360 0.358 0.315 0.248
0.360 0.360 0.325 0.259
: \[comparison\_dW\] Estimation of the actions ($\Delta W$) from Gillespie simulation.
[5]{} J.E. Ferrell, W. Xiong, Chaos. [**11**]{}, 227 (2001). J.E. Ferrell, Curr. Opin. Cell. Biol. [**14**]{}, 140 (2002). J.W. Veening, W.K. Smits, O.P. Kuipers, Annu. Rev. Micro. [**62**]{}, 193 (2008). G. Balazsi, A. van Oudenaarden, J.J. Collins, Cell. [**144**]{}, 910 (2011) E. Aurell, K. Sneppen, Phys. Rev. Lett. [**88**]{}, 048101 (2002) J. Wang, K. Zhang, L. Xu, E. Wang, Proc. Natl. Acad. Sci. U.S.A. [**108**]{}, 8257 (2011) M.B. Miller, B.L. Bassler, Annu. Rev. Microbiol. [**55**]{}, 165 (2001) J. Muller, C. Kuttler, B.A. Hense, M. Rothballer, A. Hartmann, J. Math. Biol. [**53**]{}, 672 (2006) E. Maisonneuve, K. Gerdes, Cell [**157**]{}, 539 (2014) I. Cataudella, A. Trusina, K. Sneppen, K. Gerdes, N. Mitarai, Nuc. Acids Res. [**40**]{}, 6424 (2012). R.A. Fasani, M.A. Savageau, Proc. Natl. Acad. Sci. U.S.A. [**110**]{}, E2528 (2013). R.S. Maier, D.L. Stein, J. Stat. Phys. [**83**]{}, 291 (1996) D.T. Gillespie, J. Phys. Chem. [**81**]{}, 2340 (1977). If we expand the chemical master equation for the model of coupled bistable gene circuits using Kramers-Moyal expansion and preserve up to the second order, we obtain a Fokker-Planck equation (Eq. \[Fokker-Planck\]) with a drift vector equal to the production rates minus the degradation rates and a diagonal diffusion matrix where the diagonal elements equal to the sum of the production and degradation rates. By defining the Langevin equation using Eq. \[fuv\] and \[guv\], we will obtain the same Fokker-Planck equation and thus the definition is physical. Alternatively, one may derive chemical Langevin equation from the master equation and it contains the sum of two noise terms: one for production $\sqrt{\epsilon}\sqrt{k_0+k_1v^4/(v^4+S^4)}\xi_i$ and one for degradation $\sqrt{\epsilon}\sqrt{\gamma u}\tilde\xi_i$ [@Gillespie2000]. Within the scope of this manuscript, these two formulations are equivalent as they give the same Fokker-Planck equation. R.S. Maier, D.L. Stein, Phys. Rev. E. [**48**]{}, 931 (1993) M.I. Freidlin, A.D. Wentzell, Random Perturbations of Dynamical Systems, 3rd Ed. (Springer-Verlag, Berlin, 2012) R.S. Maier, D.L. Stein, SIAM J. Appl. Math. [**57**]{}, 752 (1997) H. Risken, The Fokker-Planck Equation: Methods of Solution and Applications, 2nd Ed. (Springer-Verlag, Berlin, 1996) T. Naeh, M.M. Klosek, B.J. Matkowsky, Z. Schuss, SIAM J. Appl. Math. [**50**]{}, 595 (1990) R. Kupferman, M. Kaiser, Z. Schuss, E. Ben-Jacob, Phys. Rev. A. [**45**]{}, 745 (1992) Eq. \[FW\_Hamiltonian\] corresponds to the system constructed by preserving the Kramers-Moyal expansion of the chemical master equation up to the second-order term (cf. [@Note]). M. Heymann, E. Vanden-Eijnden, Phys. Rev. Lett. [**100**]{}, 140601 (2008). M. Heymann, E. Vanden-Eijnden, Commun. Pure Appl. Math. [**61**]{}, 1052 (2008). The numerical simulation is carried out with Eq. \[FW\_Hamiltonian\] which is a quadratic approximation to $\mathcal{H}_{full}=\sum_i \left[\left(k_0+k_1\frac{\bar{x}^4}{\bar{x}^4+S^4}\right)(\mathrm{e}^{p_i}-1)+\gamma x_i(\mathrm{e}^{-p_i}-1)\right]$ where $p_i$ is the conjugate momentum for $x_i$ [@Freidlin2012]. To verify the validity of the quadratic approximation, we perform numerical simulation with $\mathcal{H}_{full}$. With the present numerical accuracy, we do not observe any difference in the MPEP and the associated actions $\Delta W$ differ by less than 2% for the two regimes. All the conclusions presented in Section \[application\] remain valid for the non-approximating regime. We therefore claim that our work based on Eq. \[FW\_Hamiltonian\] is a valid analysis for the chemical master equation. D.T. Gillespie, J. Chem. Phys. [**113**]{}, 297 (2000). H. Feng, K. Zhang, J. Wang, Chem. Sci. [**5**]{}, 3761 (2014).
|
‘=11 makefntext\#1[ to 3.2pt [-.9pt $^{{\ninerm\@thefnmark}}$]{}\#1]{} makefnmark[to 0pt[$^{\@thefnmark}$]{}]{} PS. @myheadings[mkbothgobbletwo oddhead[ ]{} oddfootevenhead[ ]{}evenfoot \#\#1\#\#1]{}
6.0in 8.6in -0.25truein 0.30truein 0.30truein =1.5pc
**THE LAKE BAIKAL EXPERIMENT: SELECTED RESULTS**
*$^a$ Institute for Nuclear Research, Moscow, Russia*
$^b$ Irkutsk State University, Irkutsk, Russia
$^c$ Institute of Nuclear Physics, MSU, Moscow, Russia
$^d$ Nizhni Novgorod State Technical University, Nizhni Novgorod, Russia
$^e$ St.Petersburg State Marine Technical University, St.Petersburg, Russia
$^f$ Kurchatov Institute, Moscow, Russia
$^g$ Joint Institute for Nuclear Research, Dubna, Russia
$^h$ DESY-Zeuthen, Berlin/Zeuthen, Germany
$^i$ KFKI, Budapest, Hungary
presented by Zh.DZHILKIBAEV
E-mail: djilkib@pcbai10.inr.ruhep.ru
Detector
=========
The deep underwater Cherenkov detector [*NT-200*]{}, the medium-term goal of the BAIKAL collaboration [@Project; @APP; @APP2], was put into operation at April 6th, 1998. [*NT-200*]{} is deployed in Lake Baikal, Siberia, from shore at a depth of . The detector comprises 192 optical modules (OM) at 8 vertical strings, see Fig.1.
The OMs are grouped in pairs along the strings. They contain 37-cm diameter [*QUASAR*]{} PMTs which have been developed specially for our project [@Project; @APP; @OM2]. The two PMTs of a pair are switched in coincidence in order to suppress background from bioluminescence and PMT noise. A pair defines a [*channel*]{}.
All OMs face downward, with the exception of the OMs of the second and eleventh layers, which look upward. The distance between downward oriented layers is 6.25m, the distance between layers facing to each other (layers 1/2 and 10/11) is 7.5m, the distance between back-to-back layers (2/3 and 11/12) is 5.0m.
A [*muon-trigger*]{} is formed by the requirement of (with [*hit*]{} referring to a channel) within . $N$ is typically set to For such events, amplitude and time of all fired channels are digitized and sent to shore. A separate [*monopole trigger*]{} system searches for clusters of sequential hits in individual channels which are characteristic for the passage of slowly moving, bright objects like GUT monopoles.
In April 1993, the first part of [*NT-200*]{}, the detector [*NT-36*]{} with 36 OMs at 3 strings, was put into operation and took data up to March 1995. A 72-OM array, [**]{}, run in . In 1996 it was replaced by the four-string array [*NT-96*]{}. [ *NT-144*]{}, a six-string array with 144 OMs, was taking data in .
Analysis of experimental data taken with intermediate arrays, especially with [*NT-36*]{} and [*NT-96*]{}, proves the capability of the Baikal neutrino telescope to investigate various problems of neutrino and muon physics. Below we present results which illustrate the capability of the Baikal experiment to search for atmospheric muons and neutrinos, neutrinos induced by neutralino annihilation in the center of the Earth, magnetic monopoles and showers produced by high energy neutrinos.
Atmospheric Muons
=================
Muon angular distributions as well as depth dependence of the vertical flux obtained from data taken with [*NT-36*]{} have been presented earlier [@APP].
Another example which confirms the efficiency of track reconstruction procedure relates to the investigation of the shore “shadow” in muons with [*NT-96*]{}. The Baikal Neutrino Telescope is placed at a distance of 3.6 km to the nearby shore of the lake. The opposite shore is about 30 km away. This asymmetry opens the possibility to investigate the influence of the close shore to the azimuth distribution under large zenith angles, where reconstruction for the comparatively “thin” [*NT-96*]{} is most critical. A sharp decrease of the muon intensity at zenith angles of 70$^0$-90$^0$ is expected. The comparison of the experimental muon angular distribution with MC calculations gives us an estimation of the accuracy of the reconstruction error close to the horizontal direction. Indeed, the [*NT-96*]{} data show a pronounced dip of the muon flux in the direction of the shore and for zenith angles larger than 70$^0$ – in very good agreement with calculations which take into the effect of the shore.
Atmospheric Neutrinos
=====================
The main results have been obtained with the first small detector [*NT-36*]{} - investigation of atmospheric muon flux, searching for nearly vertically upward moving muons and searching for slowly moving GUT monopoles have been presented elsewhere [@APP; @FRST_vert; @GUT_monop]. Below we present selected results obtained with [*NT-96*]{}.
Identification of nearly vertically upward moving muons
-------------------------------------------------------
Different to the standard analysis [@APP], the method presented in this section relies on the application of a series of cuts which are tailored to the response of the telescope to nearly vertically upward moving muons [@FRST_vert; @INR_vert]. The cuts remove muon events far away from the opposite zenith as well as background events which are mostly due to pair and bremsstrahlung showers below the array and to naked downward moving atmospheric muons with zenith angles close to the horizon ($\theta>60^{\circ}$). The candidates identified by the cuts are afterwards fitted in order to determine the zenith angle.
We included all events with $\ge$4 hits along at least one of all hit strings. To this sample, a series of 6 cuts is applied. Firstly, the time differences of hit channels along each individual string have to be compatible with a particle close to the opposite zenith (1). The event length should be large enough (2), the maximum recorded amplitude should not exceed a certain value (3) and the center of gravity of hit channels should not be close to the detector bottom (4). The latter two cuts reject efficiently brems showers from downward muons. Finally, also time differences of hits along [*different*]{} strings have to correspond to a nearly vertical muon (5) and the time difference between top and bottom hit in an event has to be larger than a minimum value (6).
The effective area for muons moving close to opposite zenith and fulfilling all cuts exceeds $1000$ m$^2$.
Within 70 days of effective data taking, $8.4 \cdot 10^7$ events with the muon trigger $N_{hit} \ge 4$ have been selected.
Table1 summarizes the number of events from all 3 event samples (MC signal and background, and experiment) which survive the subsequent cuts. After applying all cuts, four events were selected as neutrino candidates, compared to 3.5 expected from MC. One of the four events has 19 hit channels on four strings and was selected as neutrino candidate by the standard analysis too. The zenith angular distribution of these four neutrino candidates is shown in the inner box of Fig.3.
[||c|c|c|c|c|c|c||]{} after cut [N]{}$^o$ $\rightarrow$ & 1 & 2 & 3 & 4 & 5 & 6\
atm. $\nu$, MC & 11.2 & 5.5 & 4.9 & 4.1 & 3.8 & 3.5\
background, MC & 7106 & 56 & 41 & 16 & 1.1 & 0.2\
experiment & 8608 & 87 & 66 & 28 & 5 & 4\
Regarding the four detected events as being due to atmospheric neutrinos, one can derive an the upper limit on the flux of muons from the center of the Earth due to annihilation of neutralinos - the favored candidate for cold dark matter.
The limits on the excess muon flux obtained with underground experiments [@Bak; @MACRO; @Kam] and [*NT-96*]{} are shown in Table 2. The limits obtained with [*NT-96*]{} are 4–7 times worse then the best underground limits since the data collecting time of [*NT-96*]{} was only $\approx 70$ days.
\[limit\]
-------------------- ------------- -------------- ------------- --------------
Zenith [*NT-96*]{} [*Baksan*]{} [*MACRO*]{} [*Kam-de*]{}
angles $>10GeV$ $>1GeV$ $>1.5GeV$ $>3GeV$
$\geq 150^{\circ}$ $11.0$ $2.1$ $2.67$ $4.0$
$\geq 155^{\circ}$ $9.3 $ $3.2$ $2.14$ $4.8$
$\geq 160^{\circ}$ $ 5.9-7.7 $ $2.4$ $1.72$ $3.4$
$\geq 165^{\circ}$ $4.8$ $1.6$ $1.44$ $3.3$
-------------------- ------------- -------------- ------------- --------------
This result, however, illustrates the capability of underwater experiments with respect to the search for muons due to neutralino annihilation in the center of the Earth.
Selection of neutrino events over a large solid angle
-----------------------------------------------------
The signature of neutrino induced events is a muon crossing the detector from below. With the flux of downward muons exceeding that of upward muons from atmospheric neutrino interactions by about 6 orders of magnitude, a careful reconstruction is of prime importance.
In contrast to first stages of the detector ([*NT-36*]{} [@FRST_vert]), [*NT-96*]{} can be considered as a real neutrino telescope for a wide region in zenith angle $\theta$. After the reconstruction of all events with $\ge$ 9 hits at $\ge$ 3 strings (trigger[*9/3*]{}), quality cuts have been applied in order to reject fake events. Furthermore, in order to guarantee a minimum lever arm for track fitting, events with a projection of the most distant channels on the track ($Z_{dist}$) less than 35 meters have been rejected. Due to the small transversal dimensions of [*NT-96*]{}, this cut excludes zenith angles close to the horizon.
The efficiency of the procedure has been tested with a sample of $ 1.8 \cdot 10^6$ MC-generated atmospheric muons, and with MC-generated upward muons due to atmospheric neutrinos. It turns out that the signal to noise ratio is $ > 1$ for this sample.
The reconstructed angular distribution of events taken with [*NT-96*]{} in April/September 1996 – after all cuts – is shown in Fig.3.
From 70 days of [**]{} data, 12 neutrino candidates have been found. Nine of them have been fully reconstructed. Three nearly upward vertical tracks (see subsection 3.1) hit only 2 strings and give a clear zenith angle but ambiguities in the azimuth angle – similar to the two events from [*NT-36*]{} [@APP]. This is in good agreement with MC expectations.
Search for Fast Monopoles ($\beta > 0.75$)
==========================================
Fast bare monopoles with unit magnetic Dirac charge and velocities greater than the Cherenkov threshold in water ($\beta = v/c > 0.75$) are promising survey objects for underwater neutrino telescopes. For a given velocity $\beta$ the monopole Cherenkov radiation exceeds that of a relativistic muon by a factor $(gn/e)^2=8.3\cdot10^3$ ($n=1.33$ - index of refraction for water) [@Fr; @DA]. Therefore fast monopoles with $\beta \ge 0.8$ can be detected up to distances $55$ m $\div$ $85$ m which corresponds to effective areas of (1–3)$\cdot 10^4$ m$^2$.
The natural way for fast monopole detection is based on the selection of events with high multiplicity of hits. In order to reduce the background from downward atmospheric muons we restrict ourself to monopoles coming from the lower hemisphere.
Two independent approaches have been used for selection of upward monopole candidates from the 70 days of [*NT-96*]{} data. The first one is similar to the method which was applied to upward moving muons (see subsection 3.1), with an additional cut $N_{hit}>25$ on the hit multiplicity. The second one cuts on the value of space-time correlation, followed by a cut $N_{hit}>35$ on the hit multiplicity. The upper limits on the monopole flux obtained with the two different methods coincide within errors.
The same type of analysis was applied to the data taken during $0.42$ years lifetime with the neutrino telescope [*NT-36*]{} [@INR].
The combined $90\%$ C.L. upper limit obtained by the Baikal experiment for an isotropic flux of bare fast magnetic monopoles is shown in Fig.4, together with the best limits from underground experiments Soudan2, KGF, MACRO and Ohya [@Oh; @MA; @KGF; @Sou] in Fig.4.
Search for Very High Energy Electron Neutrinos
==============================================
In this section we present very preliminary results with the aim to illustrate the capability of the Baikal Neutrino Telescope to search for extraterrestrial high energy neutrinos from AGNs, GRBs and other sources.
The idea used here to search for high energy electron neutrinos ($E_{\nu} > 100$ TeV) is to detect the Cherenkov light emitted by the electromagnetic and (or) hadronic particle cascade produced at the neutrino interaction vertex in the sensitive volume of the neutrino telescope. Earlier this idea has been used by DUMAND [@DUMAND] and to obtain upper limits on the diffuse flux of high energy neutrinos.
In order to reduce the background from downward moving atmospheric muons we restrict ourself to cascades produced in a sensitive volume below the detector (see Fig.5) and cause high multiplicity events in detector. The trigger conditions for event selection are the same as those which were used for fast monopole detection (see sec.4).
The effective volumes of [*NT-96*]{} averaged over neutrino directions for detection of cascades ith energy $E_{sh}$ are presented in Fig.6. The curves marked as “DOWN”, “UP” and “TOTAL” correspond to effective volumes averaged over lower and upper hemisphere and over all directions. Also effective volumes of detectors SPS (DUMAND) and AMANDA-A are presented in Fig.6.
After analysis of 70 days of [*NT-96*]{} data no evidence for any neutrino-induced cascades is found.
The limit to the $\tilde{\nu_e}$ flux at the W resonance energy
----------------------------------------------------------------
Although the neutrino-electron interactions can generally be neglected with respect to neutrino-nucleon interactions due to the small electron mass, the resonance cross section of $\tilde{\nu_{e}}e$ interaction at 6.3 PeV is larger than the $\nu N$ cross section at any energy up to $10^{21}$ eV.
The resonant cross section at 6.3 PeV for $\tilde{\nu_e}e$ scattering with a hadronic cascade in the final state:
$$\tilde{\nu_e} + e \rightarrow W^- \rightarrow hadrons$$
is $3.41 \times 10^{-31}$cm$^2$ [@Gandi]. The cross section averaged over the energy range
$$\Delta E=(M_w+2\Gamma_w)^2/2m_e - (M_w-2\Gamma_w)^2/2m_e,$$
$$M_w=80.22 GeV, \, \, \, \Gamma_w=2.08 GeV$$ is $\bar{\sigma}=1.12 \times 10^{-31}$cm$^2$. Eq.3 is used to calculate the upper limit on the diffuse flux of $\tilde{\nu_e}$:
$$\frac{dF_{\tilde{\nu}}}{dE_{\tilde{\nu}}} \leq \frac{2.3}{\frac{10}{18}\rho N_A \bar{\sigma} T \Omega_{eff} V_{eff}\Delta E }.$$
Here T is the detector livetime (70 days), $\Omega_{eff}$ and $ V_{eff}$ are the average effective solid angle and volume of the detector respectively.
The 90% CL limit at the W resonance energy is:
$$\frac{dF_{\tilde{\nu}}}{dE_{\tilde{\nu}}} \leq 3.7 \times 10^{-18} cm^{-2}s^{-1}sr^{-1}GeV^{-1}.$$
This limit lies between limits obtained by SPS ($1.1 \times 10^{-18}$ cm$^{-2}$s$^{-1}$sr$^{-1}$GeV$^{-1}$) and EAS-TOP ($7.6 \times 10^{-18}$cm$^{-2}$s$^{-1}$sr$^{-1}$GeV$^{-1}$).
The limit to the $\nu_e + \tilde{\nu_e}$ flux
---------------------------------------------
For setting a limit to the $\nu_e + \tilde{\nu_e}$ flux we have used the cross sections for $\nu_e$($\tilde{\nu_e}$) CC-interactions with nucleons [@Gandi]
$$\nu_e(\tilde{\nu_e}) + N \stackrel{CC}{\rightarrow} e^-(e^+) + hadrons$$
when all neutrino energy is transferred to the cascade. The energy dependence of neutrino absorption in the Earth has been taken into account.
Assuming the $F(E_{\nu})dE=A \delta (E_{\nu}-E)dE$ behavior of the differential neutrino flux the 90% CL limit has been obtained within the $10^{13} \div 6 \times 10^{15}$eV, see Fig.7.
To compare [*NT-96*]{} limit with those obtained by SPS and EAS-TOP [@EAS] we assume that a possible signal of 2.3 events originate in the energy interval from $10^5$ to $10^6$ GeV with an $E^{-2}$ differential spectrum of neutrinos.
This limit as well as limits obtained by other groups are shown in Fig.8. Also, the resulting neutrino fluxes from a number of different models \[22\] as well as backgrounds from atmospheric neutrinos \[23\] are shown in Fig.8.
Conclusions and Outlook
=======================
The results obtained with intermediate detector stages show the capability of Baikal Neutrino Telescope to search for the wide variety of phenomena in neutrino astrophysics, cosmic ray physics and particle physics.
The first atmospheric neutrinos have been identified. Also muon spectra have been measured, and limits on the fluxes of magnetic monopoles as well as of neutrinos from WIMP annihilation in the center of the Earth have been derived.
In the following years, [*NT-200*]{} will be operated as a neutrino telescope with an effective area between 1000 and 5000 m$^2$, depending on the energy and will investigate atmospheric neutrino spectra above 10 GeV. [*NT-200*]{} can be used to search for neutrinos from WIMP annihilation and for magnetic monopoles. It will also be a unique environmental laboratory to study water processes in Lake Baikal.
Apart from its own goals, [*NT-200*]{} is regarded to be a prototype for the development a telescope of next generation with an effective area of 50,000 to 100,000 m$^2$. The basic design of such a detector is under discussion at present.
[*This work was supported by the Russian Ministry of Research,the German Ministry of Education and Research and the Russian Fund of Fundamental Research ( grants* ]{} , , , and ).
[99]{} I.A.Sokalski and Ch.Spiering (eds.) [*The Baikal Neutrino Telescope NT-200, BAIKAL 92-03*]{} (1992)
I.A.Belolaptikov [*et al.*]{}, [*Astroparticle Physics*]{} 7 (1997) 263.
I.A.Belolaptikov [*et al.*]{}, [*astro-ph/9903341*]{} (1999) (accepted for publ. in [*Astropart. Phys.*]{}).
R.I.Bagduev [*et al.*]{} [*Preprint DESY*]{} 98-091, (1998)
L.B.Bezrukov [*et al.*]{}, [*Proc. of the 2nd Workshop on the Dark Side of the Universe*]{}, 221 (Rome, 1995) ([astro-ph/9601161]{})
L.B.Bezrukov [*et al.*]{}, [*Proc. of the 2nd Workshop on the Dark Side of the Universe*]{}, 221 (Rome, 1995) ([astro-ph/9601160]{})
V.A.Balkanov [*et al.*]{} 1998 [*Preprint INR 0972/98*]{} (in russian)
M.M.Boliev [*et al.*]{} [*Nucl.Phys.*]{} (Proc. Suppl.) [**48**]{} (1996) 83
T.Montaruli [*et al.*]{} [*Proc. 25-th ICRC*]{} Durban–South Africa, vol.7, (1997) 185
M.Mori [*et al.*]{} [*Phys. Rev.*]{} [**D48**]{} (1993) 5505
I.M.Frank 1988 [*Vavilov-Cherenkov Radiation*]{} (Moscow: Nauka) 192 (in russian)
D.A.Kirzhnits and V.V.Losjakov [*Pis’ma Zh. Eksp.Theor.Fz.*]{} [**42**]{} (1985) 226
V.A.Balkanov [*et al.*]{} [Preprint INR]{} (Moscow: INR) (1998) (in russian)
S.Orito [*et al.*]{} [*Phys. Rev. Lett.*]{} [**66**]{} (1992) 1951.
M.Ambrosio [*et al.*]{} [*MACRO Preprint*]{} MACRO/PUB 98/3 (1998)
H.Adarkar [*et al.*]{} [*Proc. 21st ICRC. Adelaide.*]{} (1990) 95
J.L.Thorn [*et al.*]{} [*Phys. Rev.*]{} [**D46**]{} (1992) 4846
J.W.Bolesta [*et al.*]{} [*Proc. 25-th ICRC*]{} Durban–South Africa, vol.7, (1997) 29
R.Porrata [*et al.*]{} [*Proc. 25-th ICRC*]{} Durban–South Africa, vol.7, (1997) 9
R.Gandhi [*et al.*]{} [*Astroparticle Physics*]{} [**5**]{} (1996) 81
M.Aglietta [*et al.*]{} [*Phys. Lett.*]{} [**B333**]{} (1994) 555
R.J.Protheroe [*e-preprint astro-ph/9809144*]{} (1998)
P.Lipari [*Astropart. Phys.*]{} [**1**]{} (1993) 195
|
---
abstract: 'In this paper, we identify a new phenomenon called *activation-divergence* which occurs in Federated Learning (FL) due to data heterogeneity (*i.e.*, data being non-IID) across multiple users. Specifically, we argue that the activation vectors in FL can diverge, even if subsets of users share a few common classes with data residing on different devices. To address the activation-divergence issue, we introduce a prior based on the principle of maximum entropy; this prior assumes minimal information about the per-device activation vectors and aims at making the activation vectors of same classes as similar as possible across multiple devices. Our results show that, for both IID and non-IID settings, our proposed approach results in better accuracy (due to the significantly more similar activation vectors across multiple devices), and is more communication-efficient than state-of-the-art approaches in FL. Finally, we illustrate the effectiveness of our approach on a few common benchmarks and two large medical datasets.'
author:
- 'Wei Chen[^1]'
- 'Kartikeya Bhardwaj[ ^fnsymbol[1]{}^]{}'
- Radu Marculescu
title: 'FedMAX: Mitigating Activation Divergence for Accurate and Communication-Efficient Federated Learning'
---
Introduction
============
Large amounts of data are increasingly generated nowadays on edge devices, such as phones, tablets, and wearable devices. If properly used, machine learning (ML) models trained using this data can significantly improve the intelligence of such devices [@yang2019federated]. However, since data on such personal devices is highly sensitive, training ML models by sending the users’ local data to a centralized server clearly involves significant privacy risks. Other examples of private datasets include personal medical records which must not be shared with third parties. Hence, in order to enable intelligence for these privacy-critical applications, Federated Learning (FL) has become the de facto paradigm for training ML models on local devices without sending data to the cloud [@konevcny2016federated; @konevcny2015federated].
As the state-of-the-art approach for FL, Federated Averaging (FedAvg) [@mcmahan2016communication] simply runs several *local* training epochs on a randomly selected subset of devices; these training epochs utilize only local data available on any user’s device. After local training, the models (not the local data!) are sent over to a server via a *communication round*; the server then averages all the parameters of these local models to update a *global* model. Unfortunately, FedAvg is not designed to handle the statistical heterogeneity in federated settings, *i.e.*, when data is *not* independent and identically distributed (non-IID) across the different devices. Not surprisingly, it has been recently reported that FedAvg can incur significant loss of accuracy when data is non-IID [@zhao2018federated; @sattler2019robust].
To deal with such non-IID settings, one approach called “data-sharing strategy" distributes global data across the local devices, such that the test accuracy can increase by making data look more IID [@zhao2018federated; @huang2018loadaboost]. However, obtaining this common global data is usually problematic in practice. Another approach called FedProx [@sahu2018convergence] targets the *weight-divergence* problem, *i.e.*, the local-weights diverge from the global model due to non-IID data at local devices (hence, the updates can go in different directions at different local devices). In this paper, we first identify a new phenomenon called *activation-divergence* and argue that the activation vectors in FL can diverge even if a subset of users share a few common classes of data. Since the activation vectors directly contribute to the model’s accuracy, making them as similar as possible *across all devices* should become an important objective in FL. To this end, we propose *FedMAX*, a new FL approach that introduces a new prior for local training. Specifically, our prior maximizes the entropy of local activation vectors across all devices. We show that our new prior:
1. Makes activation vectors across multiple devices more similar (for the same classes); in turn, this improves the classification accuracy of our approach;
2. Significantly reduces the number of total communication rounds needed (as one can perform more local training without losing accuracy). This is particularly important to save energy when training on edge devices.
Extensive experiments on five non-IID FL datasets demonstrate that our approach significantly outperforms both FedAvg [@mcmahan2016communication] and FedProx [@sahu2018convergence] (e.g. $5.64\% \sim 5.84\%$ better accuracy on CIFAR-10 dataset). We also observe up to $5\times$ reduction in communication rounds compared to FedAvg and FedProx.
The remainder of this paper is organized as follows. In Section \[sec2\], we provide some background information on FL and discuss the maximum entropy principle. We then present our proposed approach FedMAX in Section \[sec3\]. In Section \[sec4\], we provide a thorough evaluation of FedMAX, under both IID and non-IID settings, using three digit/object recognition and two medical datasets. Our results demonstrate the applicability of our idea and the practical benefits of FedMAX over other approaches.
Related Work and New Contribution {#sec2}
=================================
In FedAvg, after training on device’s own data, the updated local models are averaged at a central server in order to get a new global model. For non-IID data, the performance of FedAvg reduces significantly as the weights of different models often diverge [@zhao2018federated; @sattler2019robust]. To address this non-IID issue, several approaches propose to use some globally shared data to improve the accuracy by making the local data look more IID [@zhao2018federated; @huang2018loadaboost]. However, in practice, collecting this global data may be problematic (or even infeasible) due to privacy concerns; additionally, dealing with this global data can use up critical resources like the local storage space or network bandwidth. Consequently, another approach called FedProx [@sahu2018convergence] has been proposed to solve the weight-divergence problem by introducing a new loss function which constrains the local models to stay close to the global model.
In contrast to these prior approaches, we aim at constraining the activation-divergence across multiple devices. More precisely, our approach is based on the principle of maximum entropy which states that when there is no *a priori* information about a problem, the prior distribution should be chosen to maximize entropy [@jaynes1957information]. The core idea behind maximizing entropy is to obtain a prior which assumes the least amount of information about a given problem[^2]. We note that, while this principle has been exploited to solve traditional natural language processing problems [@rosenfeld1996maximum; @nigam1999using], it has never been used in the context of FL. In the next section, we explain the intuition behind using of this principle when dealing with non-IID data in FL and describe our newly proposed approach in detail.
We note that other studies exploit ML models [@wang2017chestx] and aim at addressing differential privacy [@triastcyn2019federated] of medical datasets. In practice, such samples of medical datasets are usually unbalanced and non-IID. Therefore, evaluating FL with medical datasets is necessary, especially when privacy issues are at stake [@triastcyn2019federated]. To this end, we perform multiple experiments on such two different medical datasets. The Chest X-ray dataset [@wang2017chestx] is one of the accessible medical image datasets for developing automated methods to identify and classify pneumonia. The APTOS dataset [@aptos] is also a well-known dataset for detecting the blindness with retina images taken using fundus photography. Our results show the effectiveness of our approach on these non-IID datasets.
Proposed Approach: FedMAX {#sec3}
=========================
FL aims to solve the learning task without explicitly sharing local data. More precisely, a central server coordinates the global learning across a network where each node is a device collecting data and performing a local learning task (as shown in Fig. \[fig1\](a)). The objective of FL is to minimize: $$ \underset{w}{\mathrm{min}} \ \ \ g\left(w\right)=\sum_{k=1}^{m} p_k \cdot g_k(w_k)
\label{fedMAX}$$ where $g_k(w_k)$ is the local objective which is typically the loss function of the prediction made with model parameters $w$; $m = C \cdot M$ is the number of devices selected at any given communication round, where $C$ is the proportion of selected devices and $M$ is the number of total devices; $\sum_{k=1}^{M} p_k = 1$, $p_k=\frac{n_k}{n}$ and $n_k$ is the number of samples available at the device $k$, $n=\sum_{k=1}^{M} n_k$ is the total number of samples.
In FedAvg, any local model is updated with its own data as $w^{t+1}_k \xleftarrow{} w^t_k-\eta \nabla g_k(w_k)$, where $\eta$ is the learning rate, $\nabla g_k(w_k)$ represents the gradient of $g_k(w_k)$; the global model is then formed by the averaging the parameters of all these local models, *i.e.*, $w^{t+1} \xleftarrow{} \sum_{k=1}^{M} \frac{n_k}{n} w^{t+1}_k$. For non-IID datasets, different local models will have different data. Although optimized with the same learning rate and the same number of local training epochs, the weights of these local models will likely diverge. Consequently, the accuracy of the global model decreases when its parameters are weight-averaged across these different local models. One possible solution to this problem is to constrain the local updates within a reasonable range, as FedProx proposed [@sahu2018convergence].
 for local training. The final logits and the activation vectors at the input of the last fully-connected layer are used in the objective function. $KL$ denotes Kullback-Leibler divergence, $a$ refers to the activation vector, $U$ is uniform distribution over activation vectors, and $F_{k}\left(w\right)$ is the cross-entropy loss on local data. We use similar activation vectors for other models such as ResNets for medical datasets.[]{data-label="fig1"}](schedule.pdf){width="100.00000%"}
Since activation vectors directly contribute to model accuracy, our new idea is to reduce the activation-divergence for the same classes across multiple devices. To this end, we propose a new prior for the local training that can help us achieve the above goal. More precisely, we use a Convolutional Neural Network (CNN) model consisting of five convolutional layers and two fully-connected layers (see Fig. \[fig1\](b)) for 3 digit/object recognition datasets and ResNet50 [@he2016deep] for two medical datasets, *i.e.*, APTOS and Chest X-ray. We also refer to the inputs of the last fully-connected layer as the *activation-vector*; for the 5-layer CNN, this activation-vector is 512-dimensional tensor which passes through the final fully-connected layer to yield logits (the unnormalized class probabilities). Hence, we propose a prior distribution that achieves similar activation vectors across all different devices.
#### $L^2$ Norm Regularization:
We initially consider the $L^2$ norm to constrain the activation vectors and argue that by preventing the activation vectors from taking large values, the $L^2$ norm should reduce the activation-divergence across different devices. We formulate the $L^2$ norm regularization as follows: $$ \underset{w}{\mathrm{min}} \ \ \ g_k(w_k) = F_k\left(w_k\right)+\beta\left\lVert a_i^k\right\rVert_2
\label{l2}$$ where $F_{k}\left(w_k\right)$ is the cross-entropy loss on local data (same as the cost function of FedAvg [@mcmahan2016communication] which tries to distinguish the various labels from each other), $k$ denotes to any local device in Fig. \[fig1\](a), $\left\lVert \cdot\right\rVert_2$ is $L^2$ norm, and $a_i^k$ refers to the activation vectors at the input of the last fully-connected layer (as shown in Fig. \[fig1\](b)) for sample $i$ on device $k$. Further, $\beta>0$ is a hyper-parameter used to control the scale of the $L^2$ norm regularization.
Intuitively, this $L^2$ norm regularization constrains the activation vectors and indirectly affects the parameters of other layers except the last fully-connected layer. However, reducing the activation to zero can lead to model underfitting, which results in poor performance. Therefore, we further propose another form of regularization to ensure more similar activation vectors across different devices.
#### Maximum Entropy Regularization:
The activation-divergence problem is more complex in the non-IID settings where different users deal with data from different classes. As such, we do not have any prior information about which users have data from which classes. Hence, in non-IID settings, we do *not* have any prior information about how the activation vectors at different users (for the given classes) should be distributed. Consequently, we propose to use the principle of maximum entropy [@jaynes1957information] and select a distribution for activation vectors that maximizes their entropy. Using such a prior, the local loss function for our FL problem is given by: $$ \underset{w}{\mathrm{min}} \ \ \ g_k(w_k) = F_k\left(w_k\right)-\beta\frac{1}{N}\sum_{i=1}^N \mathbb{H}(a_i^k)
\label{hh1}$$ where $N$ is a mini-batch size of local training data, and $\mathbb{H}$ denotes the entropy of activation vectors. Also, $\beta$ is a hyper-parameter that is used to control the scale of the entropy loss. Compared with , equation maximizes the entropy (hence it minimizes the negative entropy) of activation vectors $\mathbb{H}(a_i^k)$ instead of minimizing the $L^2$ norm of activation vectors $\left\lVert a_i^k\right\rVert_2$; therefore, we call this approach FedMAX.
Further, can be written using the Kullback-Leibler (KL) divergence as: $$ \underset{w}{\mathrm{min}} \ \ \ g_k\left(w_k\right)=F_k\left(w_k\right)+\beta \frac{1}{N}\sum_{i=1}^N KL\left(a_i^k||U\right)
\label{kl}$$ where $KL(\cdot||\cdot)$ denotes the KL divergence, and $U$ is uniform distribution over the activation vectors. Since equation is equivalent to equation up to a constant term, the new formulation does *not* affect the optimization process, thus resulting a maximum entropy too. As we shall see shortly, FedMAX is more stable than the $L^2$ norm-based regularization.
**Input**: $M, T, \beta, w^{0}, \eta, B, C, E$\
**Server:**
$m \xleftarrow{} max(C\cdot M,1)$ $S_{t} \xleftarrow{} $Random set of $m$ clients $w^{t+1}_{k}\xleftarrow{} Client(k, w^{t})$ $w^{t+1}\xleftarrow{} \sum_{k \in S_{t}} \frac{n_{k}}{n} w^{t+1}_{k}$\
**Client:**
$g\left(w;b\right)=F\left(w;b\right)+\beta \frac{1}{N}\sum_{i=1}^N KL\left(a_i||U\right)$ $w \xleftarrow{} w - \eta \nabla g(w;b)$ **return** $w$
The training process of FedMAX is similar to FedAvg (see Algorithm \[alg:algorithm\]). The initial model and weights $w^{0}$ are generated on a remote server. After selecting a subset of devices ($C$ represents the proportion of selected devices, as shown in Fig. \[fig1\](a)), the server sends the model (and the corresponding weights) only to these devices. The devices train the model for $E$ local epochs using their local data and then send the trained model back to the server. After averaging the models on the server, sending back the updated model to the newly selected devices finishes one communication round ($t$) – see Algorithm \[alg:algorithm\], where $M$ represents the number of devices, $B$ is the local training batch size, and $T$ represents the total number of communication rounds[^3]. This completes the newly proposed FedMAX; we next show its effectiveness on multiple datasets.
Experimental Setup and Results {#sec4}
==============================
We perform multiple experiments on five different datasets: FEMNIST\* [@caldas2018leaf], CIFAR-10, CIFAR-100 [@cifar10], APTOS [@triastcyn2019federated] and Chest X-ray [@kermany2018identifying]. The first three datasets are trained with the five layer CNN in Fig. \[fig1\](b), while the last two medical datasets are fine-tuned with ResNet50 [@he2016deep]. We consider a FL setting where we have a central server and a total of 100 local devices (*i.e.*, $M = 100$), each device containing only a subset of the entire dataset. At each communication round, only $10\%$ (*i.e.*, $C = 0.1$) of these devices are randomly selected by the server for local training. With different ways to separate data at the local devices, we can get either IID or non-IID of each dataset. In what follows, we show results for both IID and non-IID datasets.
Similarity of Activations
-------------------------
We first use synthetic data generated as in [@sahu2018convergence] to verify that the maximum entropy regularization leads to similar activations at different local devices. Samples $x_k \in \mathbb{R}^{1024}$ for $k$th device are drawn from a normal distribution $\mathcal{N}(v_k, \Sigma)$, which has two parameters: the mean vector $v_k$ and the covariance matrix $\Sigma$. Each element in the mean vector $v_k$ is generated from $\mathcal{N}(B_k, 1)$, and here $B_k\sim \mathcal{N}(0, \gamma_1)$. A larger $\gamma_1$ will lead to more varied mean vectors $v_k$ of the data distribution at each device, thus more non-IID data; the covariance matrix $\Sigma$ is a diagonal matrix where $\Sigma_{j,j}=\frac{1}{j^{1.2}}$ (similar to that used in [@shamir2014communication]).
Following the data-generation strategy presented in [@sahu2018convergence], we use a two-layer perceptron $y = argmax(w_2\cdot ReLU(w_1 \cdot x + b_1)+b_2)$ to generate the labels w.r.t the input samples[^4], where $w_1 \in \mathbb{R}^{10\times512}$, $w_2 \in \mathbb{R}^{512\times1024}$, $b_1 \in \mathbb{R}^{10}$, and $b_2 \in \mathbb{R}^{512}$. Each element in $w_1$, $w_2$, $b_1$, and $b_2$ is drawn from the normal distribution $\mathcal{N}(u_k, 1)$, where $u_k \sim \mathcal{N}(0, \gamma_2)$. The $\gamma_2$ controls the differences among the local models, thus indirectly influences the generated labels.
We use three different sets $(\gamma_1,\gamma_2)=(0,0), (0.5,0.5), (1,1)$ to generate the non-IID synthetic data. We train both FedAvg and FedMAX on the synthetic data with a two-layer perceptron which has the same structure as the model used to generate the labels. The training process lasts 200 communication rounds (*i.e.*, $T=200$), with one local training epoch (*i.e.*, $E=1$). For each communication round, the average activation $a_k$ of each local model is collected and the similarity between the local activation $a_k$ and the global activation $\overline{a}$ is calculated with KL-divergence $\delta_k = KL(\overline{a}||a_k)$. The global activation is calculated from the averages of all local activations $\overline{a} = \frac{1}{M} \sum_k a_k$, where $M$ is the total number of devices. The *overall similarity* per communication round is represented by the mean of the local similarity $\overline{\delta} = \frac{1}{M} \sum_k \delta_k$.
![The similarity effects of maximum entropy regularization, with different distributions of synthetic data $(\gamma_1,\gamma_2)=(0,0), (0.5,0.5), (1,1)$. As shown, FedMAX has relatively lower KL-divergence than FedAvg; this means that the maximum entropy regularization can make activation vectors more similar.[]{data-label="sim"}](200_similarity.pdf){width="100.00000%"}
As we can see from Fig. \[sim\], the maximum entropy regularization (FedMAX) can result in relatively lower KL-divergence of global and local activations, which means the activations from the model with maximum entropy regularization are similar to each other. Moreover, the values $\gamma_1 = 1,\gamma_2 = 1$ for synthetic data lead to a higher KL-divergence for both FedAvg and FedMAX during the first few epochs, which means that the more heterogeneous data distributions can cause activations very dissimilar from each other. Thus, constraining the activation within a reasonable range, or making the activations more similar to each other, can be benefit FL, especially for the non-IID case.
Comparison of $L^2$-norm Against Maximum Entropy
------------------------------------------------
We first compare our proposed FedMAX against the $L^2$ norm-based regularization on a non-IID CIFAR-10 dataset. For each regularization, we train a CNN like in Fig. \[fig1\] consisting of about 0.6 million parameters. The hyper-parameter $\beta$ of $L^2$ norm regularization varies from $10^{-4}$ to $10^{-1}$, and the $\beta$ of maximum entropy regularization varies from $1$ to $10^{4}$. Since the maximum entropy regularization is averaged over the activation, it has larger hyper-parameters than the $L^2$ norm.
![Test accuracy for different hyper-parameter value ($\beta$) on non-IID CIFAR-10 dataset with different regularizations ($L^2$ norm and maximum entropy), for 600 and 3000 communication rounds.[]{data-label="gamma"}](gamma_adjust.pdf){width="100.00000%"}
The results are shown in Fig. \[gamma\]. As we can see, both $L^2$ norm and maximum entropy regularization outperform the FedAvg, which is because that both methods enable more similar activation vectors across the devices. However, when compared against the $L^2$ norm, the accuracy of the maximum entropy regularization is more robust to hyper-parameter variation. Specifically, we found that for certain $\beta$ values, the $L^2$ norm results in extremely low accuracies (see Fig. \[gamma\](b)); this, in turn, can result in a much more time consuming hyper-parameter search for different datasets. Since FedMAX results in a significantly more stable behavior (see Fig. \[gamma\](a)), in the remaining of this paper, the experimental results are reported only for FedMAX using the maximum entropy regularization.
Digit/Object Recognition Datasets
---------------------------------
We first verify our approach on three different datasets: FEMNIST\* [@caldas2018leaf], CIFAR-10 and CIFAR-100. For each dataset, we train a CNN like in Fig. \[fig1\](b) consisting of about 0.6 million parameters. The training process lasts 3000 communication rounds (*i.e.*, $T=3000$) with a single local training epoch (*i.e.*, $E=1$); the mini-batch size $N$ at each selected device is 100. The learning rate $\eta$ is initialized to 0.1 and decays by $\times$0.9992 at each round. For reference, the decay rate in [@zhao2018federated] is 0.992[^5]. We also test the communication efficiency by setting the global communication rounds $T=600$, learning rate decay of 0.996, five local training epochs, and keep all other parameters the same; this way, the experimental settings remain consistent with the 3000 communication rounds setup.
For FedProx, the results are reported for the hyper-parameter $\mu=1$ [@sahu2018convergence]. We did try other $\mu$ values like $\{1, 2, 10, 20, 100\}$, but found that the results are very similar. Also, for our approach, we set $\beta$ = 1500. To split the datasets into the non-IID parts, we randomly assign 2 out of 10 classes (20 out of 100 classes) for CIFAR-10 (CIFAR-100) to each device. For FEMNIST\*, we follow the same setting as in [@sahu2018convergence], where data from 20 out of 26 classes are given to each device. For the IID case of all three datasets, labels are distributed uniformly across all users. In what follows, we present two sets of results: (*i*) Accuracy improvements and (*ii*) Communication-efficiency of FedMAX.
![Test accuracy for different datasets (both non-IID and IID) with different approaches, FedAvg, FedProx and our proposed approach (FedMAX), for 3000 communication rounds. FedMAX has a higher accuracy than the other approaches for all three datasets.[]{data-label="fig3000"}](3000.pdf){width="100.00000%"}
![Test accuracy for different datasets (both non-IID and IID) with different approaches, FedAvg, FedProx and our proposed approach (FedMAX), for 600 communication rounds. FedMAX has a higher accuracy than the other approaches for all three datasets.[]{data-label="fig600"}](600.pdf){width="100.00000%"}
#### Accuracy Comparison: More communication rounds, less local training
The test accuracy of the 3000 communication round experiment is shown in Fig. \[fig3000\]. As evident, our approach outperforms the other approaches for all three datasets. The test accuracy decreases accordingly as the datasets change from FEMNIST\* to CIFAR-100, where our CNN models become relatively smaller for the dataset. Since each device for CIFAR-10 has only two out of ten labels, this is an extreme non-IID case; this is why the test accuracy on CIFAR-10 varies much more rapidly (for all three approaches) compared to the other datasets. For the CIFAR-10 dataset, our model also converges significantly faster than the other approaches. The final accuracies across five runs for all experiments are shown in Table \[sample-table\]. As shown, our approach outperforms existing techniques for both IID and non-IID cases; the best results are highlighted with bold.
#### Communication-Efficiency: Less communication rounds, more local training
The test accuracy of the 600 communication rounds experiment is shown in Fig. \[fig600\]. With more local training, the weights of the models on different devices are expected to diverge more from the global model, which explains the loss of accuracy. However, FedMAX significantly outperforms the test accuracy of FedAvg [@mcmahan2016communication] and FedProx [@sahu2018convergence] by up to $8\%$ (see Table \[sample-table\], the better results are highlighted with bold.).
Another observation worth noting from Table \[sample-table\] is that for all three datasets, FedMAX with 600 communication rounds achieves comparable or even better accuracy than FedAvg and FedProx with 3000 communication rounds. This shows that, by relying on more local training, FedMAX significantly reduces communication rounds (by up to $5\times$) compared to prior techniques, without losing accuracy. This is particularlly important for edge computing where communication costs reduction is crucial for energy savings.
Medical Datasets
----------------
The APTOS dataset includes 38,788 samples, five labels describing the severity of blindness, and each class contains different numbers of retina images taken using fundus photography. The Chest X-ray dataset has 5,856 samples and two image categories (Pneumonia/Normal) graded by expert physicians. Each dataset is randomly split into 85% training data and 15% test data. Since these are unbalanced datasets, we use F1 macro score to measure the performance of the model.
The experiment setting is the same, but instead of training a five-layer CNN, we fine-tune a ResNet50 which is pre-trained on the ImageNet dataset. The activation of ResNet50 is the output of final average-pool layer, where the activation-vector is a 2048-dimensional tensor. The training process lasts 300 communication rounds (*i.e.*, $T=300$) with a single local training epoch; the mini-batch size $N$ at each selected device is 32. The learning rate $\eta$ is initialized to 0.001 and decays by $\times$0.992 at each round. To split the datasets into non-IID parts, we randomly assign different proportions of 5 classes (2 classes) for APTOS (Chest X-ray) to each device. For our approach, we set $\beta$ = 10,000 for APTOS dataset and 1,000 for Chest X-ray dataset.
![Test accuracy for different medical datasets for both non-IID and IID cases, APTOS and Chest X-ray, with different approaches, FedAvg and our proposed approach (FedMAX), for 300 communication rounds. FedMAX has a higher F1 score than FedAvg in APTOS dataset for the IID case. Both have similar scores as FedAvg in APTOS dataset for the non-IID case and Chest X-ray dataset.[]{data-label="fig300iid"}](300_medical_1.pdf){width="100.00000%"}
#### Accuracy Comparison:
The test accuracy of the IID and non-IID cases for the 300 communication-round experiment is shown in Fig. \[fig300iid\]. As evident, our approach FedMAX outperforms FedAvg on the APTOS IID case. On the non-IID case, our method yields similar results as FedAvg. The F1 score of the non-IID case varies more rapidly than the IID case. This is because the medical datasets are highly imbalanced; the non-IID partition by randomly separating the samples can lead to devices with only one class, which exacerbates the impact of the training process.
Compared with other datasets, the results of FedMAX on the Chest X-ray dataset are close to FedAvg. One possible reason is that since the Chest X-ray dataset has only two classes, it cannot really make the activations more similar among different labels across different devices. Besides, with fewer samples in the Chest X-ray dataset, after partitioning, each device contains only a small amount of data; this leads to a short local training process and comparably high frequent communication. As a result, the activation divergence may already be constrained, so that the FedAvg has a similar performance when compared against FedMAX. Final accuracy comparisons across the five runs for all our experiments are shown in Table \[medical-table\]. The better results are highlighted with bold.
Mitigating Activation Divergence
--------------------------------
We now analyze the impact of our proposed FedMAX on the activation-divergence that can happen in non-IID FL. We show 2-dimensional (2D) t-SNE plots of our 512-dimensional (512D) activation vectors for different devices (each device has two random classes from the CIFAR-10 dataset). Specifically, the t-SNE plots embed each 512D activation vector with a 2D point in such a way that similar objects are modeled by nearby points and dissimilar objects are modeled by distant points. We expect the activation vectors of the same class (even from different devices) to share more similarities, thus, their corresponding 2D points should be closer to each other and form a cluster on the t-SNE plots. To keep it simple, we perform the experiment on a total of 10 local devices, with all the devices training at every communication round.
![Two-dimensional tSNE plot of activation vectors (512D vector projected into 2D) for two approaches on CIFAR-10 dataset: FedAvg (top) and our proposed FedMAX (bottom). Left panel shows epoch 1, middle panel epoch 2, and right panel epoch 3. The numbers at the bottom show how the test accuracy of the two techniques varies with the training epochs. We note that initially, all t-SNE plots look similar and the test accuracy for both models is close to random accuracy ($\sim10\%$).[]{data-label="t1"}](tsne_01.pdf){width="100.00000%"}
In Fig. \[t1\], Fig. \[t2\], and Fig. \[t3\], the plots on the left show the activation vectors for FedAvg, and the ones on the right show those for FedMAX. Various colors represent the activation vectors for different classes, while the letters denote the device IDs. As the number of local epochs increases, we observe that: (*i*) FedMAX starts to gain accuracy, (*ii*) Activation vectors for FedMAX start to cluster together - see highlighted portions in Fig. \[t2\] and Fig. \[t3\] where the activation vectors from same classes (*i.e.*, the same color) come closer to each other across different devices (*i.e.*, letters A-J). In contrast, for FedAvg, clustering happens much more slowly and, hence, its accuracy is significantly lower than FedMAX.
![Similar to Fig. \[t1\], but left panel shows epoch 4, middle panel epoch 5, and right panel epoch 6. We see that same colors start coming together (*i.e.*, the activation vectors of same classes across different devices start to become more and more similar) in FedMAX. Consequently, in the accuracy of FedMAX ($\sim22\%$ until epoch 6) improves much faster than FedAvg ($10\%$ until epoch 6). However, clustering in FedAvg looks exactly the same as before.[]{data-label="t2"}](tsne_02.pdf){width="100.00000%"}
![Similar to the Figs. \[t1\] and \[t2\], but left panel shows epoch 9, middle panel epoch 12, and the right panel epoch 15. More and more clusters from same classes start forming for FedMax, while the clusters barely show up for FedAvg. This also results in the accuracy of FedMAX ($32\%$ until epoch 15) improving much faster than FedAvg ($16\%$ until epoch 15). We also see a significantly higher number of clusters formed for FedMAX compared to FedAvg.[]{data-label="t3"}](tsne_03.pdf){width="100.00000%"}
Conclusion
==========
In this paper, we have identified the activation-divergence phenomenon in FL and proposed FedMAX, a new approach for accurate and communication-aware FL in non-IID and IID settings. By exploiting the $L^2$ norm regularization and the principle of maximum entropy, we have introduced a new prior which assumes minimal information about the activation vectors at different devices.
With extensive experiments, we have shown that FedMAX improves the test accuracy and is significantly more communication-efficient than the state-of-the-art approaches running on FEMNIST\*, CIFAR-10, and CIFAR-100 for both non-IID and IID settings. Besides, we have presented experiments on two medical datasets, APTOS and Chest X-ray, and have shown the improvement of FedMAX on the APTOS IID case. We attribute the better performance of FedMAX to improving the similarity across the devices while regularizing the activation vectors. Finally, we note that FedAvg and FedMAX perform similarly on the Chest X-ray dataset due to the smaller number of samples which may hardly lead to activation divergence.
In future work, we plan to evaluate the FedMAX approach using different datasets which contain more classes and samples. Moreover, with the increasing need of multitasks learning, we also plan to implement FedMAX for different learning tasks such as language modeling.
[8]{} McMahan, H Brendan and Moore, Eider and Ramage, Daniel and Hampson, Seth and et al.: Communication-efficient learning of deep networks from decentralized data. arXiv preprint arXiv:1602.05629 (2016)
Zhao, Yue and Li, Meng and Lai, Liangzhen and Suda, Naveen and Civin, Damon and Chandra, Vikas: Federated learning with non-iid data. arXiv preprint arXiv:1806.00582 (2018)
Sattler, Felix and et al.: Robust and communication-efficient federated learning from non-iid data. arXiv preprint arXiv:1903.02891 (2019)
Sahu, Anit Kumar and Li, Tian and Sanjabi, Maziar and Zaheer, Manzil and Talwalkar, Ameet and Smith, Virginia: On the convergence of federated optimization in heterogeneous networks. arXiv preprint arXiv:1812.06127 (2018)
Yang, Qiang and et al.: Federated machine learning: Concept and applications. ACM Transactions on Intelligent Systems and Technology (TIST), vol. 10, pp. 12. ACM (2019)
Caldas, Sebastian and Wu, Peter and Li, Tian and Kone[č]{}n[y]{}, Jakub and McMahan, H Brendan and Smith, Virginia and Talwalkar, Ameet: Leaf: A benchmark for federated settings. arXiv preprint arXiv:1812.01097 (2018)
Kone[č]{}n[y]{}, Jakub and McMahan, H Brendan and Yu, Felix X and Richt[á]{}rik, Peter and Suresh, Ananda Theertha and Bacon, Dave: Federated learning: Strategies for improving communication efficiency. arXiv preprint arXiv:1610.05492 (2016)
Kone[č]{}n[y]{}, Jakub and McMahan, Brendan and Ramage, Daniel: Federated optimization: Distributed optimization beyond the datacenter. arXiv preprint arXiv:1511.03575 (2015)
Huang, Li and Yin, Yifeng and Fu, Zeng and Zhang, Shifa and Deng, Hao and Liu, Dianbo: LoAdaBoost: Loss-Based AdaBoost Federated Machine Learning on medical Data. arXiv preprint arXiv:1811.12629 (2018)
Kullback, Solomon: Information theory and statistics. Courier Corporation (1997)
Jaynes, Edwin T: Information theory and statistical mechanics. Physical review, vol. 106, pp. 620. APS (1957)
Rosenfeld, Roni: A maximum entropy approach to adaptive statistical language modeling. (1996)
Nigam, Kamal and Lafferty, John and McCallum, Andrew: Using maximum entropy for text classification. IJCAI-99 workshop on machine learning for information filtering, vol. 1, pp. 61–67. (1999)
Kermany, Daniel S and Goldbaum, Michael and Cai, Wenjia and Valentim, Carolina CS and Liang, Huiying and Baxter, Sally L and McKeown, Alex and Yang, Ge and Wu, Xiaokang and Yan, Fangbing and et al.: Identifying medical diagnoses and treatable diseases by image-based deep learning. Cell, vol. 172, pp. 1122–1131. Elsevier (2018)
Triastcyn, Aleksei and Faltings, Boi: Federated Learning with Bayesian Differential Privacy. arXiv preprint arXiv:1911.10071 (2019)
Alex Krizhevsky and Vinod Nair and Geoffrey Hinton: CIFAR-10 (Canadian Institute for Advanced Research). <http://www.cs.toronto.edu/~kriz/cifar.html> (2010)
Wang, Xiaosong and Peng, Yifan and Lu, Le and Lu, Zhiyong and Bagheri, Mohammadhadi and Summers, Ronald M: Chestx-ray8: Hospital-scale chest x-ray database and benchmarks on weakly-supervised classification and localization of common thorax diseases. Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 2097–2106. (2017)
Kaggle: APTOS. <https://www.kaggle.com/c/aptos2019-blindness-detection> (2019)
He, Kaiming and Zhang, Xiangyu and Ren, Shaoqing and Sun, Jian: Deep residual learning for image recognition. Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 770–778. (2016)
Shamir, Ohad and Srebro, Nati and Zhang, Tong: Communication-efficient distributed optimization using an approximate newton-type method. International conference on machine learning, pp. 1000–1008. (2014)
[^1]: Equal Contribution
[^2]: Making needless or unfounded prior assumptions about a problem can reduce the accuracy of the model, hence it is better to make minimal assumptions. For more information on maximum entropy, please refer to [@jaynes1957information; @kullback1997information]
[^3]: We note that this approach reduces to FedAvg if $\beta=0$.
[^4]: Once initialized, these models remain fixed .
[^5]: For our setup, since this decay rate results in an extremely small learning rate after thousands of epochs, we increase our learning rate decay to 0.9992.
|
---
abstract: 'Nonparametric estimation of the gap time distribution in a simple renewal process may be considered a problem in survival analysis under particular sampling frames corresponding to how the renewal process is observed. This note describes several such situations where simple product limit estimators, though inefficient, may still be useful. [^1]'
author:
- |
Richard D. Gill\
Department of Mathematics\
University of Leiden\
\
Niels Keiding\
Department of Biostatistics\
University of Copenhagen
date: 1 February 2010
title: 'Product-limit estimators of the gap time distribution of a renewal process under different sampling patterns'
---
Introduction
============
This note is about two classical problems in nonparametric statistical analysis of recurrent event data, both formalised within the framework of a simple, stationary renewal process.
We first consider observation around a fixed time point, i.e., we observe a backward recurrence time $R$ and a forward recurrence time $S$. It is well known that the nonparametric maximum likelihood estimator of the gap-time distribution is the Cox-Vardi estimator (Cox 1969, Vardi 1985) derived from the length-biased distribution of the gap time $R+S$. However, Winter & Földes (1988) proposed to use a product-limit estimator based on $S$, with delayed entry given by $R$. Keiding & Gill (1988) clarified the relation of that estimator to the standard left truncation problem. Unfortunately this discussion was omitted from the published version (Keiding & Gill, 1990). Since these simple relationships do not seem to be on record elsewhere, we offer them here.
The second observation scheme considers a stationary renewal process observed in a finite interval where the left endpoint does not necessarily correspond to an event. The full likelihood function is complicated, and we briefly survey possibilities for restricting attention to various partial likelihoods, in the nonparametric case again allowing the use of simple product-limit estimators.
Observation of a stationary renewal\
process around a fixed point
====================================
Winter & Földes (1988) studied the following estimation problem. Consider $n$ independent renewal processes in equilibrium with underlying distribution function $F$, which we shall assume absolutely continuous with density $f$, minimal support interval $(0,\infty)$, and hazard $\beta(t)=f(t)/(1-F(t))$, $t>0$. The reason for our unconventional choice $\beta$ for the hazard rate belonging to $F$ will become apparent later. Corresponding to a fixed time, say $0$, the backward and forward recurrence times $R_i$ and $S_i$, $i=1,...,n$, are observed; their sums $Q_i=R_i+S_i$ are length-biased observations from $F$, i.e., their density is proportional to $tf(t)$. Let $(R,S,Q)$ denote a generic triple $(R_i,S_i,Q_i)$. We quote the following distribution results: let $\mu$ be the expectation value corresponding to the the distribution $F$, $$\mu~=~\int_0^\infty u f(u) \mathrm du~=~\int_0^\infty(1-F(u))\mathrm du,$$ then the joint distribution of $R$ and $S$ has density $f(r+s)/\mu$ , the marginal distributions of $R$ and $S$ are equal with *density* $(1-F(r))/\mu$, and the marginal distribution of $Q=R+S$ has density $qf(q)/\mu$, the length-biased density corresponding to $f$.
Winter and Földes considered the product-limit estimator $$1-\widetilde F(t)~=~\prod_{i:Q_i\le t}\biggl(1-\frac 1 {Y(Q_i)}\biggr)$$ where $$Y(t)~=~\sum_{i=1}^nI\{R_i<t\le R_i+S_i\}$$ is the *number at risk* at time $t$. This estimator is the same as the Kaplan-Meier estimator for iid survival data $Q_1,\dots,Q_n$ left-truncated at $R_1,\dots,R_n$ (Kaplan & Meier 1958, Andersen et al. 1993). Winter & Földes showed that $1-\widetilde F$ is strongly consistent for the *underlying* survival function $1-F$ .
We shall show how the derivation of this estimator follows from a simple Markov process model similar to the one used by Keiding & Gill (1990) to study the random truncation model.
First notice that the conditional distribution of $Q=R+S$ given that $R=r$ has density $$\frac{ f(q)/\mu } { (1-F(r))/\mu },\quad r < q < \infty$$ that is, intensity (hazard) $f(q)/(1-F(q))$, $q>r$, which is just the hazard $\beta(q)$ corresponding to the underlying distribution $F$ left-truncated at $r$. Now define corresponding to $(R,S,Q)$ a stochastic process $U$ on $[0,\infty]$ with state space $\{0,1,2\}$ by $$U(t)~=~\Biggl\{
\begin{aligned}
& 0, & &~~t<~R,\\
& 1, & R~\le&~~t<~R+S,\\
& 2, & \quad R+S~\le &~~t .
\end{aligned}$$ Note that it takes in successsion the values $0$, $1$ and $2$. For $U(t)=0$, $$\begin{aligned}
P\bigl\{U&(t+h)=1\bigm|U(u),0\le u\le t\bigr\}\\
~&=~P\bigl\{R \le t+h\bigm|R>t\bigr\}\\
~&=~\alpha(h)h +o(h),\end{aligned}$$ where $\alpha$ is the hazard rate of the marginal distribution of $R$. For $U(t)=1$ (and hence $R\le t$) $$\begin{aligned}
P\bigl\{U&(t+h)=2\bigm|U(u),0\le u\le t\bigr\}\\
~&=~P\bigl\{R+S\le t+h\bigm|R=r\le t,R+S>t\bigr\}\\
~&=~\frac {f(t)} { 1-F(t) } h + o(h)\end{aligned}$$ by the above result on the conditional hazard of $R+S$ given $R$. For $U(t)=0$, $$P\bigl\{U(t+h)=2\bigm|U(u),0\le u\le t\bigr\}~=~o(h).$$ Other transitions are impossible.
That these conditional probabilities depend on $U(t)$ and $t$ only, but not on $U(u)$, $u<t$, proves that $U$ is a Markov process with intensities $$\alpha(t)~=~ \frac { 1-F(t) } { \int_t^\infty (1-F(r) ) \mathrm d r }$$ (the marginal hazard of $R$, equal to the residual mean lifetime function of the underlying distribution $F$) and $$\beta(t)~=~\frac {f(t) } { 1-F(t) },$$ see Figure 1.
The Markov process framework of Keiding & Gill (1990) now indicates that (ignoring information about $F$ in $\alpha$, and just focussing on the transition with rate $\beta$) the product limit estimator $1-\widetilde F$ is a natural estimator of the survivor function $1-F$ of interest, and consistency and asymptotic normality may be obtained as shown by Keiding & Gill (1990, Sec. 5).
Note that the backwards intensity $$\begin{aligned}
{\overline\alpha}(t)~&=~\alpha(t)\frac{ P\bigl\{ U(t)=0 \bigr\} } { P\bigl\{ U(t)=1 \bigr\} }\\
~&=~\alpha(t)\frac{ P\bigl\{ R>t \bigr\} } { P\bigl\{ R\le t<R+S \bigr\} }\\
~&=~\alpha(t)\frac{ \mu^{-1} \int_t^\infty(1-F(r))\mathrm d r } { \mu^{-1} \int_0^t\int_{t-r}^\infty f(r+s)\mathrm d s \mathrm d r }
\\
~&=~\frac {1-F(t) } { \int_t^\infty (1-F(r)) \mathrm d r } \frac{ \int_t^\infty(1-F(r))\mathrm d r } { \int_0^t(1-F(t)) \mathrm d r } ~=~\frac1t ,\end{aligned}$$ the *backwards* hazard-rate of a uniform distribution on a bounded interval $(0,A)$, $A<\infty$. Since it has been assumed that $R$ has support interval $(0,\infty)$, this shows that the present model *may not* be interpreted strictly as a left truncation model, which would require that ${\overline\alpha}(t)$ was the backwards hazard rate of some probability distribution on $(0,\infty)$. However, this distinction is not important to our discussion.
The fact that ${\overline\alpha}(t)$ does not depend on $F$ corresponds to Winter and Földes’ statement that $(R,S)$ contains no more information than $R+S$ about $F$. This already follows from sufficiency since the joint density of $(R,S)$ is $f(r+s)/\mu$. The likelihood function based on observation of $(R_1,S_1),\dots(R_n,S_n)$ is $$\mu^{-n}\prod_{i=1}^n f(r_i+s_i)$$ from which the NPMLE of $F$ is readily derived as $$\widehat F(t)~=~\sum_{i=1}^n \frac { I\bigl\{ R_i+S_i\le t\bigr\} } { R_i+S_i} \biggm / \sum_{i=1}^n\frac 1 { R_i+S_i},$$ that is the Cox-Vardi estimator in the terminology of Winter and Földes (Cox 1969, Vardi 1985).
It follows that the estimator $1-\widetilde F$ is *not* NPMLE. The important difference between the situation here and that of the random truncation model studied by Keiding & Gill (1990, Sec. 3) is that not only the intensity $\beta(t)$, but also $\alpha(t)$ depends only on the estimand $F$.
As already mentioned, weak convergence of $1-\widetilde F$ is immediate from Keiding & Gill (1990, Sec. 5). In particular, in order to achieve the extension to convergence on $[0,M]$ it should be required that $$\int_0^\varepsilon \mathrm d \Phi(s) /\nu_2(s)~<~\infty$$ in the terminology of Keiding & Gill (1990, Sec. 5c), and using $\mathrm d \Phi(t)=\beta(t)\mathrm d t$ and $$\nu_2(t)~=~P\bigl\{ U(t)=1\bigr\} ~=~\int_0^t\frac {1-F(s) } {\mu} \frac {1-F(t)}{1-F(s)}\mathrm d s~=\frac t\mu (1-F(t)),$$ the integrability condition translates into $$\int_0^\varepsilon \frac { \beta(t)} {P\bigl\{ U(t)=1\bigr\} } \mathrm d t~ < \infty$$ or finiteness of $E(1/X)$ where $X$ has the underlying (“length-unbiased”) interarrival time distribution $F$. It may easily be seen from Gill et al. (1988) that the same condition is needed to ensure weak convergence of the Cox-Vardi estimator.
A variation of the observation scheme of this section would be to allow also right censoring of the $S_i$. This can be immediately included in the Markov-process/counting process approach leading to the inefficient product-limit type estimator $1-\widetilde F$; the delayed-entry observations $S_i$ are simultaneously right-censored. See Vardi (1985, 1989) and Asgharian et al. (2002) for treatment of the full non-parametric maximum likelihood estimator of $F$, extending the Cox-Vardi estimator to allow right censoring.
Other ad hoc estimators and the rich relationships with a number of other important non-parametric estimation problems are discussed by Denby and Vardi (1985) and Vardi (1989).
Observation of a stationary renewal\
process in a finite interval
====================================
We consider again a stationary renewal process on the whole line and assume that we observe it in some interval $[t_1,t_2]$ determined independently of the process. Nonparametric estimation of the gap time distribution $F$ was definitively discussed by Vardi (1982) in discrete time and by Soon & Woodroofe (1996) in continuous time. Cook & Lawless (2007, Chapter 4) surveyed the general area of analysis of gap times emphasizing that the assumption of independent gap times is often unrealistic.
We shall here nevertheless work under the assumption of the simplest possible model as indicated above. Because the nonparametric maximum likelihood estimator is computationally involved it may sometimes be useful to calculate less efficient alternatives, and there are indeed such possibilities.
Under the observation scheme indicated above we may have the following four types of elementary observations\
1. Times $x_i$ from one renewal to the next, contributing the density $f(x_i)$ to the likelihood.\
2. Times from one renewal $T$ to $t_2$, which are right-censored versions of 1., contributing factors of the form $(1-F(t_2-T))$ to the likelihood.\
3. Times from $t_1$ to the first renewal $T$ (forward recurrence times), contributing factors of the form $(1-F(T-t_1))/\mu$ to the likelihood.\
4. Knowledge that no renewal happened in $[t_1,t_2]$ , actually a right-censored version of 3., contributing factors of the form $\int_{t_2-t_1}^\infty (1-F(u))\mathrm d u/\mu$ to the likelihood.\
McClean & Devine (1995) studied nonparametric maximum likelihood estimation in the conditional distribution given that there is at least one renewal in the interval, i.e., that there are no observations of type 4.
Our interest is in basing the estimation only on complete or right-censored gap times, i.e., observations of type 1 or 2. When this is possible, we have simple product-limit estimators in the one-sample situation, and we may use well-established regression models (such as Cox regression) to account for covariates. Peña et al. (2001) assumed that observation started at a renewal (thereby defining away observations of type 3 and 4) and gave a comprehensive discussion of exact and asymptotic properties of product-limit estimators with comparisons to alternatives, building in particular on results of Gill (1980, 1981) and Sellke (1988). The crucial point here is that calendar time and time since last renewal both need to be taken into account, so the straightforward martingale approach displayed by Andersen et al. (1993) is not available. Peña et al. also studied robustness to deviations from the assumption of independent gap times.
As noted by Aalen & Husebye (1991) in their attractive non-technical discussion of observation patterns, observation does however often start between renewals. (In the example of Keiding et al. (1998), auto insurance claims were considered in a fixed calendar period). As long as observation starts at a stopping time, inference is still valid, so by starting observation at the first renewal in the interval we can essentially refer back to Peña et al. (2001). A more formal argument could be based on the concept of the *Aalen filter*, see Andersen et al. (1993, p. 164). The resulting product-limit estimators will not be fully efficient, since the information in the backward recurrence time (types 3 and 4) is ignored. It is important to realize that the validity of this way of reducing the data depends critically on the independence assumptions of the model. Keiding et al. (1998), cf. Keiding (2002) for details, used this fact to base a goodness-of-fit test on a comparison of the full nonparametric maximum likelihood estimator with the product-limit estimator.
Similar terms appear in another model, called the *Laslett line segment problem* (Laslett, 1982). Suppose one has a stationary Poisson process, with intensity $\mu$, of points on the real line. We think of the real line as a calendar time axis, and the points of the Poisson process will be called *pseudo renewal times* or *birth times* of some population of individuals. Suppose the individuals have independent and identically distributed lifetimes, each one starting at the corresponding birth time. The corresponding calendar time of the end of each lifetime can of course be called a *death time*. Now suppose that *all we can observe* are the intersections of individuals’ lifetimes (thought of as time segments on the time axis) with an observational window $[t_1,t_2]$. In particular, we do not know the current age of an individual who is observed alive at time $t_1$. Again we have exactly the same four kinds of observations:\
1. Complete *proper* lifetimes corresponding to births within $[t_1,t_2]$ for which death occurred before time $t_2$.\
2. Censored *proper* lifetimes corresponding to births within $[t_1,t_2]$ for which death occurred after time $t_2$.\
3. Complete *residual* lifetimes corresponding to births which occurred at an unknown moment before time $t_1$, and for which death occurred after $t_1$ and before time $t_2$.\
4. Censored *residual* lifetimes corresponding to births which occurred at an unknown moment before time $t_1$, for which death occurred after time $t_2$, and which are therefore censored at time $t_2$.\
The *number* of at least partially observed lifetimes (proper or residual) is random, and Poisson distributed with mean equal to the intensity $\mu$ of the underlying Poisson process of birth times, times the factor $$t_2-t_1+\int_0^\infty (1 - F(y)) \mathrm d y .$$ This provides a fifth, “Poisson”, factor in the nonparametric likelihood function for parameters $\mu$ and $F$, based on all the available data. Maximizing over $\mu$ and $F$, the mean of the Poisson distribution is estimated by the observed number of partially observed lifetimes. Thus we find that the *profile likelihood* for $F$, and the *marginal likelihood* for $F$ based only on contributions 1.–4., are proportional to one another.
Nonparametric maximum likelihood estimation of $F$ was studied by Wijers (1995) and van der Laan (1996), cf. van der Laan & Gill (1999). Some of their results, and the calculations leading to this likelihood, were surveyed by Gill (1994, pp. 190 ff.). The nonparametric maximum likelihood estimator is consistent; whether or not it converges in distribution as $\mu$ tends to infinity is unknown, the model has a singularity coming from the vanishing probability density of complete lifetimes just larger than the length of the observation window corresponding to births just before the start of the observation window and deaths just after its end. Van der Laan showed that a mild reduction of the data by grouping or binning leads to a much better behaved nonparametric maximum likelihood estimator. If the amount of binning decreases at an appropriate rate as $\mu$ increases, this leads to an asymptotically efficient estimator of $F$. This procedure can be thought of as *regularization*, a procedure often needed in nonparametric inverse statistical problems, where maximum likelihood can be too greedy.
Both “unregularized” and regularized estimators are easy to compute with the EM algorithm; and the speed of the algorithm is not so painfully slow as in other inverse problems, since this is still a problem where “root $n$” rate estimation is possible.
The problem allows, just as we have seen in earlier sections, all the same inefficient but rapidly computable product-limit type estimators based on various marginal likelihoods. Moreover since the direction of time is basically irrelevant to the model, one can also look at the process “backwards”, leading to another plethora of inefficient but easy estimators. One can even combine in a formal way the censored survival data from a forward and a backward time point of view, which comes down to counting all uncensored observations twice, all singly censored once, and discarding all doubly censored data. (This idea was essentially suggested much earlier by R.C. Palmer and D.R. Cox, cf. Palmer(1948)). The attractive feature of this estimator is again the ease of computation, the fact that it only discards the doubly censored data, and its symmetry under reversing time. The asymptotic distribution theory of this estimator is of course not standard, but using the nonparametric delta method one can fairly easily give formulas for asymptotic variances and covariances. In practice one could easily and correctly use the nonparametric bootstrap, resampling from the partially observed lifetimes, where again a resampled complete lifetime is entered twice into the estimate.
The Laslett line segment problem has rather important extensions to observation of line segments (e.g., cracks in a rock surface) observed through an observational window in the plane. Under the assumption of a homogenous Poisson line segment process one can write down nonparametric likelihoods, maximize them with the EM algorithm; it seems that regularization may well be necessary to get optimal “root $n$” behaviour but in principle it is clear how this might be done. Again, we have the same plethora of inefficient but easy product-limit type estimators. Van Zwet (2004) studied the behaviour of such estimators when the line segment process is not Poisson, but merely stationary. The idea is to use the Poisson process likelihood as a quasi likelihood, i.e., as a basis for generating estimating equations, which will be unbiased but not efficient, just as in parametric quasi-likelihood. Van Zwet shows that this procedure works fine. Coming full circle, one can apply these ideas to the renewal process we first described in this section, and the other models described in earlier sections. All of them generate stationary line segment processes observed through a finite time window on the line. Thus the nonparametric quasi-likelihood approach can be used there too. Since in the renewal process case we are ignoring the fact that the intensity of the point process of births equals the inverse mean life-time, we do not get full efficiency. So it is disputable whether it is worth using an inefficient ad-hoc estimator which is difficult to compute when we have the options of Soon and Woodroofe’s fully efficient (but hard to compute) full nonparametric maximum likelihood estimator, and the many inefficient but easy and robust product-limit type estimators of this paper.
Acknowledgements {#acknowledgements .unnumbered}
================
This research was partially supported by a grant (RO1CA54706-12) from the National Cancer Institute and by the Danish Natural Sciences Council grant 272-06-0442 “Point process modelling and statistical inference”.
References {#references .unnumbered}
==========
Aalen, O.O. & Husebye, E. (1991). Statistical analysis of repeated events forming renewal processes. *Statistics in Medicine* **10**, 1227–1240.
Andersen, P.K., Borgan, Ø., Gill, R.D. & Keiding, N. (1993). *Statistical Models Based on Counting Processes*. Springer Verlag, New York.
Asgharian, M., Wolfson, D. B. & M’lan, C. E. (2002). Length-biased sampling with right censoring: an unconditional approach. *J. Amer. Statist. Assoc.* **97**, 201–209.
Cook, R.J. & Lawless, J.F. (2007). *The statistical analysis of recurrent events*. Springer Verlag, New York.
Cox, D.R. (1969). Some sampling problems in technology. In *New Developments in Survey Sampling* (N.L. Johnson and H. Smith, Jr., eds), 506–527. Wiley, New York.
Denby, L. & Vardi, Y. (1985). A short-cut method for estimation in renewal processes. *Technometrics* **27**, 361–373.
Gill, R.D. (1980). Nonparametric estimation based on censored observations of a Markov renewal process. *Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete* **53**, 97-116.
Gill, R.D. (1981). Testing with replacement and the product-limit estimator. *Annals of Statistics* **9**, 853–860.
Gill, R.D. (1994). Lectures on survival analysis. *Lecture notes in Mathematics* **1581**, 115–241. Springer, New York.
Gill, R.D., Vardi, Y. & Wellner, J.A. (1988). Large sample theory of empirical distributions in biased sampling models. *Annals of Statistics* **16**, 1069–1112.
Kaplan, E.L. & Meier, P. (1958). Non-parametric estimation from incomplete observations. *Journal of the American Statistical Association* **53**, 457–481.
Keiding, N. (2002). Two nonstandard examples of the classical stratification approach to graphically assessing proportionality of hazards. In: *Goodness-of-fit Tests and Model Validity* (eds. C. Huber-Carol, N. Balakrishnan, M.S. Nikulin and M. Mesbah). Boston, Birkhäuser, 301–308.
Keiding, N., Andersen, C. & Fledelius, P. (1998). The Cox regression model for claims data in non-life insurance. *ASTIN Bulletin* **28**, 95–118.
Keiding, N. & Gill, R.D. (1988). *Random truncation models and Markov processes*. Report MS- R8817, Centre for Mathematics and Computer Science, Amsterdam.
Keiding, N. & Gill, R.D. (1990). Random truncation models and Markov processes. *Annals of Statistics* **18**, 582–602.
Laslett, G.M. (1982). The survival curve under monotone density constraints with application to two-dimensional line segment processes. *Biometrika* **69**, 153–160.
McClean, S. & Devine, C. (1995). A nonparametric maximum likelihood estimator for incomplete renewal data. *Biometrika* **82**, 791–803.
Palmer, R.C. (1948). The dye sampling method of measuring fibre length distribution (with an appendix by R.C. Palmer and D.R. Cox). *Journal of the Textile Institute* **39**, T8–T22.
Peña, E. A., Strawderman, R. L. & Hollander, M. (2001). Nonparametric estimation with recurrent event data. *Journal of the American Statistical Association* **96**, 1299–1315.
Sellke, T. (1988). Weak convergence of the Aalen estimator for a censored renewal process, in *Statistical Decision Theory and Related Topics IV*, (eds. S. Gupta & J. Berger), Springer-Verlag, New York, Vol. 2, 183–194.
Soon, G. & Woodroofe, M. (1996). Nonparametric estimation and consistency for renewal processes. *Journal of Statistical Planning and Inference* **53**, 171–195.
van der Laan, M.J. (1996). Efficiency of the NPMLE in the line-segment problem. *Scandinavian Journal of Statistics* **23**, 527–550.
van der Laan, M.J. and Gill, R.D. (1999). Efficiency of NPMLE in nonparametric missing data models. *Mathematical Methods in Statistics* **8**, 251–276.
van Zwet, E.W. (2004). Laslett’s line segment problem. *Bernoulli* **10**, 377–396.
Vardi, Y. (1982). Nonparametric estimation in renewal processes. *Annals of Statistics* **10**, 772–785.
Vardi, Y. (1985). Empirical distributions in selection bias models. *Annals of Statistics* **13**, 178–203.
Vardi, Y. (1989). Multiplicative censoring, renewal processes, deconvolution and decreasing density: nonparametric estimation. *Biometrika* **76**, 751–761
Wijers, B.J. (1995). Consistent nonparametric estimation for a one-dimensional line-segment process observed in an interval. *Scandinavian Journal of Statistics* **22**, 335–360.
Winter, B.B. & Földes, A. (1988). A product-limit estimator for use with length-biased data. *Canadian Journal of Statistics* **16**, 337–355.
[^1]: Key words. Kaplan-Meier estimator, Cox-Vardi estimator, Laslett’s line segment problem, nonparametric maximum likelihood, Markov process
|
---
abstract: 'Membership identification is the first step to determine the properties of a star cluster. Low-mass members in particular could be used to trace the dynamical history, such as mass segregation, stellar evaporation, or tidal stripping, of a star cluster in its Galactic environment. We identified member candidates with stellar masses $\sim$0.11–2.4M$_\sun$ of the intermediate-age Praesepe cluster (M44), by using Pan-STARRS and 2MASS photometry, and PPMXL proper motions. Within a sky area of 3 deg radius, 1040 candidates are identified, of which 96 are new inclusions. Using the same set of selection criteria on field stars led to an estimate of a false positive rate 16%, suggesting 872 of the candidates being true members. This most complete and reliable membership list allows us to favor the BT-Settl model in comparison with other stellar models. The cluster shows a distinct binary track above the main sequence, with a binary frequency of 20–40%, and a high occurrence rate of similar mass pairs. The mass function is consistent with that of the disk population but shows a deficit of members below 0.3 solar masses. A clear mass segregation is evidenced, with the lowest-mass members in our sample being evaporated from this disintegrating cluster.'
author:
- 'P. F. Wang, W. P. Chen, C. C. Lin, A. K. Pandey, C. K. Huang, N. Panwar, C. H. Lee, M. F. Tsai, C.-H. Tang, B. Goldman, W. S. Burgett, K. C. Chambers, P. W. Draper, H. Flewelling, T. Grav, J. N. Heasley, K. W. Hodapp, M. E. Huber, R. Jedicke, N. Kaiser, R.-P. Kudritzki, G. A. Luppino, R. H. Lupton, E. A. Magnier, N. Metcalfe, D. G. Monet, J. S. Morgan, P. M. Onaka, P. A. Price, C. W. Stubbs, W. Sweeney, J. L. Tonry, R. J. Wainscoat, C. Waters'
title: 'Characterization of the Praesepe Star Cluster by Photometry and Proper Motions with 2MASS, PPMXL, and Pan-STARRS'
---
Introduction
============
A star cluster manifests itself as a density concentration of comoving stars in space. Born out of the same molecular cloud, the member stars have roughly the same age, similar chemical composition, and are at essentially the same distance from us. Star clusters, therefore, serve as good test beds to study stellar formation and evolution. In order to diagnose the properties of a star cluster, such as its age, distance, size, spatial distribution, mass function, etc., it is necessary to identify as completely as possible the member stars. In particular, with a sample of members including the lowest mass stars, or even substellar objects, one could trace the dynamical history of an open cluster, e.g., the effect of mass segregation, stellar evaporation, and tidal stripping in the Galactic environment.
Nearby open clusters are useful in study of their low-mass population. Praesepe (M44; NGC2632; the Beehive Cluster) is such a rich ($\sim1000$ members) and intermediate-age [757 Myr; @gas09] stellar aggregation in Cancer, as a member in the Hyades moving group [@egg60], also called the Hyades supercluster. Compared to Praesepe, the Hyades cluster itself has a scattered main sequence in the color-magnitude diagram (CMD) because of the significant depth with respect to its distance. The advantages of studying stars in Praesepe are numerous. First, with a distance determination ranging from 170 pc [@reg91] to 184 pc [@an07], the cluster is close enough to detect low-mass stars or even brown dwarfs. In this work, we adopted a distance $179\pm2$ pc [@gas09], and metallicity \[Fe/H\]=0.16 [@car11]. Second, the proper motion (PM) of the cluster is distinct from that of the field stars, so contamination is minimized when identifying member stars. Third, in contrast to a star cluster at birth, for which the spatial distribution of members is governed by the parental cloud structure, the stellar distribution in an evolved cluster depends mainly on the interaction between members, from which we could investigate the dynamical evolution of the cluster.
Early PM measurements of Praesepe included the pioneering work by @kle27 to identify bright members within a 1-deg radius of the cluster center, and by @jon83 who extended the detection limit to $V\sim17$ mag to include intermediate-mass members. @wan95 combined early data and presented a list of nearly 200 PM members. Using PMs and photometry, @jon91 identified a list of member candidates from $V\sim 9$ to 18 mag within 2 of the cluster center. Using optical and infrared photometry, @wil95 selected member candidates with mass $M > 0.08 M_\sun$ and concluded a mass function similar to the field, with no evidence of stellar evaporation. @wan11 summarized the photometric surveys of Praesepe members down to the hydrogen-burning limit. Notably, @ham95a, with a limiting magnitude of $ R \ga 20$ mag, thereby reaching the stellar mass of $\sim0.1$ M$_\sun$, derived a rising mass function toward the low-mass end, and presented evidence of mass segregation [@ham95b]. With the Two Micron All Sky Survey (2MASS) and Digital Sky Survey data covering a sky area of 100 deg$^2$, @ada02 extended the lower main sequence to 0.1 M$_\sun$, and determined the radial density profile of member stars. @kra07 surveyed a sky area of 300 deg$^2$ to identify members by optical to infrared spectral energy distribution, and by PM measurements taken from UCAC2 for bright stars or calculated from USNO-B1 and SDSS positions, reaching almost into the brown-dwarf regime. Their sample of early-type stars is incomplete because of the bright limit of UCAC2, whereas for later-type members the incompleteness is caused by the detection limits of USNO-B1 and 2MASS. Recently @kha13 used SDSS and PPMXL data to characterize the stellar members, including the mass segregation effect and binarity.
There have been efforts to identify brown dwarfs in Praesepe. @pin97 covered one deg$^2$ down to $I\sim21$ mag and identified 19 brown-dwarf candidates without spectral confirmation. @chap05 presented deep optical and near-infrared observations covering 2.6 deg$^2$ to a mass limit of 0.06 M$_\sun$. @gon06 explored the central 0.6 deg radius region, reaching a limit of $i_{\rm SDSS}\sim24.5$ mag corresponding to $\sim$0.05–0.13 M$_\sun$, and identified one substellar candidate. @bou10 performed an optical $I_c$ band and near-infrared $J$ and $K_s$ band photometric survey covering 3.1 deg$^2$ with detection limits of $I_c\sim23.4$ mag and $J\sim20.0$ mag, and found a handful of substellar candidates. The substellar census was augmented by @wan11 who, using very deep optical ($riz$ and $Y$-band) photometry of the central 0.59 deg$^{2}$ region of the cluster, identified a few dozen substellar member candidates. The first spectroscopically confirmed L dwarf member in Praesepe was secured by @bou13.
The stellar mass function of Praesepe was found to rise until 0.1 M$_\sun$ [@ham95b; @chap05; @bak10; @bou10], in contrast to the Hyades, which have about the same age but are deficient of very low-mass stars and brown dwarfs. Possible explanations include different initial mass functions for the two clusters, or that Praesepe somehow did not experience as much dynamical perturbation in its environments [@bou08]. A recent study with the UKIRT Infrared Deep Sky Survey (UKIDSS) Galactic Clusters Survey derived a declining mass function toward lower masses [@bou12]. One of the aims of this work is to secure a sample of highly probable members to address this issue.
The spatial distribution of star in a cluster is initially governed by the structure in the parental molecular cloud. As a star cluster ages, gravitational scattering by stellar encounters results in mass segregation [@spi75]; that is, massive stars tend to concentrate toward the center of the cluster, whereas lower mass stars, with a greater velocity dispersion, are distributed out to greater radii. For Praesepe, @ham95a combined their observations, complete to $R\sim20.0$ mag and $I\sim18.2$ mag, with those of @mer90 with $I\la12$ mag, to show a clear mass segregation effect. While brown dwarfs may have a preferred spatial distribution within a young star cluster [@cab08], they tend to be distributed uniformly as the cluster evolves @del00.
Observational attempts to find and characterize members in a star cluster often are sufficiently deep but limited in sky coverage, or cover wide areas but are restricted to only brighter (more massive) members. Studies with large sky coverages usually secure membership on the basis of photometry, lacking PM measurements for faint members. In this paper, we present photometric (2MASS and Pan-STARRS) and astrometric (PPMXL) diagnostics to select the member candidates in Praesepe. Our sample allows us to characterize the cluster including the binarity, its size, the mass function and the segregation effect. We describe the photometric and PM data in Section \[sec:data\], and how we identified probable members in Section \[sec:selection\]. The discussion is in Section \[sec:dis\], for which we compare our results with those in the literature. The binarity is discussed, and evidence of mass segregation and tidal stripping is presented. The paper ends with a short summary as Section \[sec:summary\].
Data Sources \[sec:data\]
=========================
Data used in this study include photometry and PM measurements within a 5-deg radius around the Praesepe center (R.A.=$08^{\rm h}40^{\rm m}$, Decl.$=+19\degr 42\arcmin$, J2000). Archival data were taken from the 2MASS Point Sources Catalog, PPMXL, and Pan-STARRS. The 2MASS Point Source Catalog [@skr06] has the 10$\sigma$ detection limits of $J\sim15.8$ mag, $H\sim15.1$ mag, and $K_s\sim14.3$ mag, and saturates around 4–5 mag. The typical astrometric accuracy for the brightest unsaturated sources is about 70–80 mas. PPMXL is an all-sky merged catalog based on the USNO-B1 and 2MASS positions of 900 million stars and galaxies, reaching a limiting $V\sim20$ mag [@roe10]. The typical error is less than 2 milliarcseconds (mas) per year for the brightest stars with Tycho-2 [@hog00] observations, and is more than 10 mas yr$^{-1}$ at the faint limit.
Pan-STARRS (the Panoramic Survey Telescope And Rapid Response System) is a wide field (7 deg$^2$) imaging system, with a 1.8 m, f/4.4 telescope [@hod04], equipped with a 1.4 giga-pixel camera [@ton08]. The prototype (PS1), located atop Haleakala, Maui, USA [@kai10], has been patrolling the entire sky north of $-30\degr$ declination since mid-2010. Repeated observations of the same patch of sky with a combination of $g_{\rm P1}$, $r_{\rm P1}$, $i_{\rm P1}$, $z_{\rm P1}$, and $y_{\rm P1}$ bands several times a month produce a huge inventory of celestial objects that vary in brightness or in position. Deep static sky images and catalog of stars and galaxies are also obtained. The PS1 filters differ slightly from those of the SDSS [@aba09]. The $g_{\rm P1}$ filter extends 20 nm redward of $g_{\rm SDSS}$ for greater sensitivity and lower systematics for photometric redshift estimates. SDSS has no corresponding $y$ filter [@ton12a]. The limiting magnitudes are $g_{\rm P1}\sim22.5$ mag, $r_{\rm P1}\sim22$ mag, $i_{\rm P1}\sim21.5$ mag, $z_{\rm P1}\sim21$ mag, and $y_{\rm P1}\sim19.5$ mag, with the saturation limit of $\sim14$ mag. Upon completion of its 3.5 year mission by early 2014, PS1 will provide reliable photometry and astrometry. While incremental photometry of PS1 is available at the moment, the calibration of astrometry, hence the PM measurements, will need yet to tie down the entire sky, so no PS1 PM data were used here. The photometric analysis and calibration is described in @mag13. PS1 photometry for each detected object has measurements at multiple epochs, but for the work reported here only the average magnitude is used. In our study, we therefore made use of the 2MASS photometry for stars too bright for PS1, plus the PS1 photometry for faint stars, and the PPMXL PMs to select and characterize stellar member candidates. In matching counterparts in different star catalogues, one arcsecond was used as the coincidence radius among PPMXL, PS1, and 2MASS sources.
Candidate Selection \[sec:selection\]
======================================
Our membership diagnosis relies on grouping in sky position, in PMs, and along the isochrones appropriate for the cluster in the infrared and optical CMDs. The sources with 2MASS photometric uncertainties greater than 0.05 mag, roughly reaching $J\sim15.2$ mag, $H \sim14.6$ mag, and $K_s \sim14.5$ mag, were removed from the sample. Candidacy was then further winnowed in the $J$ versus $J-K_s$ CMD by including only objects with $J-K_s$ colors within 0.3 mag from the Padova isochrones [@mar08]. This initial, wide range of colors allowed us not to adopt an *a priori* stellar evolutionary model, but in turn to put different models to test, as demonstrated below.
With the initial photometric sample, we then identified stars with PMs close to that of the cluster. Obviously the choice of the range is a compromise between the quality and the quantity of the candidate list. The optimal range was decided by how the cluster grouping is blended with the field. The PPMXL data toward Praesepe are shown in Figure \[fig:ppmxl2deg\]. The PM distribution has two peaks, one for the cluster ($\mu_\alpha\, \cos\delta \approx -36.5$masyr$^{-1}$, $\mu_\delta \approx -13.5$masyr$^{-1}$) and the other for field stars ($\mu_\alpha\, \cos\delta \approx -4$masyr$^{-1}$, $\mu_\delta \approx -3$masyr$^{-1}$). The latter is the reflex Galactic motion of the Sun toward this particular line of sight. The average PM we adopted for the cluster is close to those listed by SIMBAD, $\mu_\alpha\, \cos\delta \approx -35.99 \pm 0.14$masyr$^{-1}$, and $\mu_\delta \approx -12.92 \pm 0.14$masyr$^{-1}$ [@lok03]. Naturally, around the peak of the cluster, the distribution is dominated by members, and away from the peak the contamination by field stars becomes prominent. In fact, Praesepe is among a few cases where the cluster’s motion is clearly separated from that of the field, so the PM distribution exhibits a distinct secondary peak due to the cluster.
![The PPMXL proper motion vector point diagram of stars toward Praesepe. Stars within an angular distance of $5\degr$ of the cluster center are analyzed. Only stars spatially within the central $2\degr$ are displayed here for clarity. []{data-label="fig:ppmxl2deg"}](fig01){width="80.00000%"}
We exercised two levels of PM selection. First, a Gaussian function was fitted to the secondary (cluster) peak. Even through the distribution is known to be non-Gaussian [@gir89], the top part of the peak can be reasonably approximated by a Gaussian with a standard deviation of 9masyr$^{-1}$. This is the PM range, namely within $\Delta\mu=9$masyr$^{-1}$ of the cluster’s average PM, that we adopted to select PM membership. This range is similar to that used by @kra07 (8masyr$^{-1}$) or by @bou12 (8masyr$^{-1}$ in $\Delta\mu_\alpha\, \cos\delta$ and 12masyr$^{-1}$ in $\Delta \mu_\delta$). We note that @bou12 derived, using relative PMs on the basis of the UKIDDS data, a different mean motion ($\mu_\alpha\, \cos\delta = -34.17 \pm 2.74$masyr$^{-1}$, $\mu_\delta = -7.36 \pm 4.17$masyr$^{-1}$). The discrepancy may arise because these authors used the median value to choose the center of the PM range, yet the distribution is skewed because of the contribution from the field. The next level of PM selection is $\Delta\mu= 4$masyr$^{-1}$, at which there is about the same contribution from the cluster and from the field, i.e., a 50% contamination of the sample. Figure \[fig:pmtrans\] compares the cases of 4 versus 9masyr$^{-1}$. While bright candidates, including giant stars, are not much affected by the choice, the cluster sequence clearly stands out with the narrower PM range even without restrictions on position, color, or magnitude. The adoption of $ \Delta\mu < 9$masyr$^{-1}$ facilitates comparison between our results and previous works. But the $ \Delta\mu < 4$masyr$^{-1}$ sample was still kept for a more reliable selection of candidates. Figure \[fig:pmtrans\] also shows the PM distribution projected on the line connecting the peak of the field and the peak of the cluster. Even with this projection showing the maximum distinction between the two peaks, the distribution near the cluster is overwhelmed by that of the field.
![The 2MASS/PPMXL stars toward Praesepe. Top: The proper motion distribution. The two circles illustrate the cases of proper motion range of $ \Delta \mu =4$ mas yr$^{-1}$ and of $ \Delta \mu =9$ mas yr$^{-1}$, respectively. Stars within $ \Delta\mu =9$ mas yr$^{-1}$ but otherwise outside the cluster region (beyond $3\degr$) and photometrically not following the cluster isochrone, i.e., field stars, are marked with crosses. Bottom: The projected PM distribution along the line connecting the field centroid and the cluster centroid. The bump near $-35$mas yr$^{-1}$ is due to the cluster, which has a standard deviation of 9mas yr$^{-1}$ when fitted with a Gaussian function. []{data-label="fig:pmtrans"}](fig02){height="0.8\textheight"}
Figure \[fig:radden5d\] shows the radial density profile of stars following roughly the cluster’s isochrone and PM within the entire $5\degr$ field. The surface density decreases monotonically until around $3\degr$, then levels off. Our analysis therefore was conducted within a spatial radius of 3. At 179 pc, this corresponds to a linear dimension of $\sim18$ pc across. This size is relatively large among the 1657 entries with both angular diameter and distance determinations in the open cluster catalog compiled by @dia02[^1], with the majority having diameters of 2–4 pc.
![The radial density distribution of all stars within the entire 5 field satisfying the isochrone and PM criteria. The vertical line at the 3 radius marks where we consider the cluster region in our analysis. The region between radius 4 (shown by another vertical line) and 5 is used as the field region. []{data-label="fig:radden5d"}](fig03){width="70.00000%"}
Figure \[fig:compjjkggy\] shows the $J$ versus $J-K_s$ and the $g_{\rm P1}$ versus $g_{\rm P1} - y_{\rm P1}$ CMDs when the spatial (within or beyond 3 angular distance from the cluster center) and PM criteria (within 9 or 4masyr$^{-1}$) are applied. Even without a preselection by photometry or color, the cluster sequence is already evident. A subsample was chosen with a much restrictive set of parameters, namely with the angular distance within the central 30, and with $\Delta\mu = 4$ mas yr$^{-1}$. This subsample is incomplete, but consists of highly secured members, which validates our initial rough selection ranges of magnitude and colors, and can be used to compare various stellar atmospheric models.
For the 2MASS/PPMXL sample, photometric candidacy is selected in the $J$ versus $J-K_s$ CMD: (i) for stars brighter than $J\sim12$ mag, from 0.06 mag below to 0.18 mag above and perpendicular to the Padova track; for giants there is no photometric restriction, i.e., only the spatial and kinematic criteria were applied; (ii) for fainter stars, from 0.1 mag below to 0.1 mag above and perpendicular to the Siess isochrone.
For stars fainter than the 2MASS sensitivity, we resorted to the PS1 data collected up to January 2012. The luminosity function toward Praesepe reaches beyond $g_{\rm P1}\sim 21.5$ mag, but our data are limited by the sensitivity of the PPMXL dataset at around 21 mag. To avoid spurious detections, only sources that have been measured more than twice in both $g_{\rm P1}$ and $y_{\rm P1}$ bands were included in our analysis. The $g_{\rm P1}$ magnitudes were derived from the SDSS magnitudes [taken from @kra07] transformed to the PS1 photometric system [@ton12a], namely, by $g_{\rm p1} = g_{\rm SDSS} -0.012 -0.139\, x $, where $x = (g - r)_{\rm SDSS}$. For the $y_{\rm P1}$ magnitudes, because SDSS has no corresponding $y$, the transformation from $z_{\rm SDSS}$ was used, $y_{\rm P1} = z_{\rm SDSS} + 0.031 -0.095\, x $, where $x$ is again $(g - r)_{\rm SDSS}$. Because of this, plus the Paschen absorption, the transformation to $y_{\rm P1}$ (and to $z_{\rm P1}$) has a larger uncertainty than in other bands [@ton12a]. In the transformation to either $g_{\rm P1}$ or $y_{\rm P1}$, using the quadratic instead of the linear fit makes little difference. The bottom panel of Figure \[fig:compjjkggy\] plots $g_{\rm P1}$ versus $g_{\rm P1} - y_{\rm P1}$ together with the PS1 main sequence transformed from @kra07. For the PS1/PPMXL sample, the selection range is from 0.15 mag below to 0.4 mag above and perpendicular to the @kra07 main sequence transformed to the PS1 system [@ton12a].
![Top: The $J$ versus $J-K_s$ CMD for all the stars (gray dots), those with angular distances greater than 3 from the cluster center but with $ \Delta\mu < 9$ mas yr$^{-1}$ (small black crosses), those within 3 from the cluster center and with $ \Delta\mu < 9 $ mas yr$^{-1}$ (blue open circles), and those within 3 and with $ \Delta\mu < 4$ mas yr$^{-1}$ (blue filled circles). The stars at the very center of the cluster, namely within 30, and with $ \Delta\mu < 4$ mas yr$^{-1}$ are highly probable members and are marked as orange crosses. Note the group of blue stragglers beyond the main sequence turn-off point [@and98]. Bottom: The $g_{\rm P1}$ versus $g_{\rm P1} - y_{\rm P1}$ CMD, with the same symbols as in the top panel. The group of stars near $g_{\rm P1} = 18$mag, and $g_{\rm P1} - y_{\rm P1} = -1$mag include white dwarfs known in the cluster [@dob04; @dob06]. []{data-label="fig:compjjkggy"}](fig04){height="0.7\textheight"}
The combination of the 2MASS/PPMXL and the PS1/PPMXL samples contains a total of 1040 stars that satisfy all the criteria of photometry (along the isochrone), kinematics (consistent PMs), and spatial (within a 3 radius) grouping. In comparison, there are 168 stars satisfying the identical set of criteria except being with radii between 4 and $5\degr$ (which happens to have the same sky area as the 3 cluster radius, i.e., $9\pi$deg$^2$) — these are considered field stars and this number of stars should be subtracted from the cluster region. So our final list contains 1040 member candidates, among which about 872 ($\sim84\%$) should be true cluster members. Statistically a brighter candidate is more likely to be a true member than a fainter candidate because of the field contamination. If the stringent criterion of $ \Delta\mu =4$ mas yr$^{-1}$ had been used instead, the number of candidates would have become 547 within $3\degr$, and 33 between $4\degr$ and $5\degr$, yielding a net of 514 members within $3\degr$, yielding a 6% false positive rate.
The Updated Member List \[sec:dis\]
====================================
Table \[tab:members\] lists the properties of the 1040 candidates. The first two columns, (1) and (2), are the identification number and coordinates. Columns (3) and (4) give the PM measurements and errors in right ascension and in declination taken from the PPMXL catalog. Subsequent columns, from (5) to (12), list the photometric magnitudes and corresponding errors of PS1 $g_{\rm P1}$, $r_{\rm P1}$, $i_{\rm P1}$, $z_{\rm P1}$, and $y_{\rm P1}$, and 2MASS $J$, $H$, and $K_s$. The (13) column flags if the candidate is possibly binary. The last (14) column lists the common star name, if any. The 2MASS and PS1 CMDs of the members listed in Table \[tab:members\] are displayed in Figure \[fig:memjjkggy\], along with a selected stellar models: BT-Settl [@all13; @all14][^2], @sie00, Padova [@mar08], and @kra07. To convert the effective temperature in the @sie00 models to $J$, $H$, and $K_s$ magnitudes, we made use of the table presented in @ken95. While all isochrones follow roughly each other for $J \la 12$ mag, they differ noticeably toward faint magnitudes. The Padova isochrone is too blue to fit the data. This cannot be caused by reddening because Praesepe is very nearby, so is hardly reddened $E(B-V)=0.027$ mag [@tay06]. The rest four stellar models, though diverging toward the lowest mass end of our data, fit the data equally well. The highly secured list of candidates indicates a better fit with the BT-Settl model.
[rrrr r rrrr rrrr r]{} 413 & 129.7619871 19.7248670 & $-34.8 \pm 1.1$ & $-13.6 \pm 1.1$ & $12.120 \pm 0.001$ & $\ldots$ & $\ldots$ & $\ldots$ & $\ldots$ & $8.366 \pm 0.026$ & $8.126 \pm 0.021$ & $8.125 \pm 0.021$ & 0 & BD$+$202140\
414 & 129.7620587 19.5325438 & $-37.5 \pm 4.1$ & $-16.9 \pm 4.1$ & $17.618 \pm 0.005$ & $16.347 \pm 0.002$ & $15.095 \pm 0.600 $ & $14.364 \pm 0.001 $ & $ 14.060 \pm 0.003 $ & $ 12.829 \pm 0.022$ & $12.182 \pm 0.021$ & $11.962 \pm 0.019$ & 1 &\
415 & 129.7627808 19.4043081 & $-38.9 \pm 4.1$ & $-16.2 \pm 4.1$ & $19.373 \pm 0.018$ & $18.124 \pm 0.009$ & $16.549 \pm 0.003 $ & $15.841 \pm 0.002 $ & $ 15.494 \pm 0.003 $ & $ 14.261 \pm 0.027$ & $13.643 \pm 0.027$ & $13.407 \pm 0.035$ & 1 &\
416 & 129.7633143 20.0437781 & $-44.3 \pm 4.1$ & $-13.7 \pm 4.1$ & $14.975 \pm 0.001$ & $13.827 \pm 0.001$ & $13.489 \pm 0.600 $ & $13.074 \pm 0.001 $ & $ 12.927 \pm 0.001 $ & $ 11.867 \pm 0.023$ & $11.209 \pm 0.021$ & $11.051 \pm 0.020$ & 0 &\
417 & 129.7651196 19.9997784 & $-31.5 \pm 1.1$ & $-12.6 \pm 1.3$ & $ \ldots$ & $\ldots$ & $\ldots$ & $12.844 \pm 0.002 $ & $ 12.690 \pm 0.002 $ & $ 7.860 \pm 0.023$ & $ 7.819 \pm 0.016$ & $7.769 \pm 0.018 $ & 0 & HD73430\
418 & 129.7663424 20.5672773 & $-38.0 \pm 4.1$ & $-11.3 \pm 4.1$ & $17.691 \pm 0.006$ & $16.496 \pm 0.003$ & $15.336 \pm 0.600 $ & $14.615 \pm 0.001 $ & $ 14.342 \pm 0.002 $ & $ 13.108 \pm 0.025$ & $12.464 \pm 0.024$ & $12.276 \pm 0.021$ & 1 &\
419 & 129.7670607 19.5226714 & $-37.4 \pm 4.1$ & $-12.3 \pm 4.1$ & $14.274 \pm 0.001$ & $13.482 \pm 0.600$ & $13.076 \pm 0.600 $ & $12.831 \pm 0.600 $ & $ 12.601 \pm 0.001 $ & $ 11.562 \pm 0.022$ & $10.987 \pm 0.019$ & $10.857 \pm 0.016$ & 0 &\
420 & 129.7712342 19.7573463 & $-36.1 \pm 4.1$ & $-15.6 \pm 4.1$ & $19.064 \pm 0.016$ & $17.807 \pm 0.009$ & $16.395 \pm 0.600 $ & $15.616 \pm 0.001 $ & $15.289 \pm 0.003 $ & $ 14.010 \pm 0.024$ & $13.424 \pm 0.030$ & $13.164 \pm 0.028$ & 1 &\
421 & 129.7717692 20.1172023 & $-35.1 \pm 1.1$ & $-14.3 \pm 1.2$ & $9.489 \pm 0.600$ & $9.354 \pm 0.600$ & $9.347 \pm 0.600 $ & $9.375 \pm 0.600 $ & $9.383 \pm 0.600 $ & $ 8.603 \pm 0.030$ & $8.455 \pm 0.026$ & $ 8.413 \pm 0.027$ & 0 & HD73429\
422 & 129.7754141 19.6768137 & $-33.7 \pm 1.2$ & $-13.9 \pm 1.2$ & $7.539 \pm 0.600$ & $7.519 \pm 0.600$ & $7.559 \pm 0.600 $ & $7.573 \pm 0.600 $ & $7.586 \pm 0.600 $ & $ 6.857 \pm 0.026$ & $6.769 \pm 0.023$ & $ 6.708 \pm 0.018$ & 0 & HD73449\
\[tab:members\]
![Member candidates in Praesepe selected on the basis of position, proper motion, and magnitude/color. Top: The $J$ versus $J-K_s$ CMD, together with the stellar models of BT-Settl [@all13; @all14], Siess, Padova, and @kra07. Selected stellar mass values are labeled. Symbols are the same as in Figure \[fig:compjjkggy\]. Bottom: The $g_{\rm P1}$ versus $g_{\rm P1} - y_{\rm P1}$ CMD for candidates. The solid curve is the main sequence from @kra07 transformed to the PS1 system. Red symbols mark possible binaries. []{data-label="fig:memjjkggy"}](fig05){height="0.8\textheight"}
Our member candidates have been selected as grouping in five out of six-dimensional photometric and kinematic parameters, less only the radial velocity measurements. Our list hence is more reliable than using photometry alone, and is comprehensive in terms of stellar mass and sky area coverage than currently available. Among the 1040 candidates, 214 were selected by the 2MASS/PPMXL sample only, 82 by PS1/PPMXL only, and 742 by both. The reason that PS1/PPMXL does not find more candidates is, other than the limit at the bright end, because the faintest candidates are very red, $g_{\rm P1} - K_s \approx7$ mag — in favor of 2MASS detection — and because the PS1/PPMXL data are limited by the brightness limit of PPMXL. The situation will improve once PS1 produces its own PM measurements. A total of 890 of our candidates coincide with those by @kra07, 567 with those by @bou12, and 190 with neither. Of the latter, 96 candidates have not been identified in either @ham95b, @pin97, @ada02, or @bak10. Some of our candidates missed by @bou12 are located in the UKIDSS survey gap. Membership identification by photometry alone, e.g., by @gon06 and @bou10, is vulnerable to significant contamination by field stars, so reliable membership could be secured for bright stars only. To illustrate this, the entire PS1/PPMXL 5 sample contains 320,312 stars. There would have been 2445 candidates if only the photometric and positional criteria were set, but the number reduces drastically to 826 once the additional PM criterion ($\Delta\mu \leq 9$ mas yr$^{-1}$) is imposed.
Our member list includes the two stars recently reported by @qui12, BD$+$202184 (their Pr0201=NGC2632 KW418) and 2MASSJ08421149$+$1916373 (their Pr0211=NGC2632 KW448), to host exoplanets. A few candidates found in previous works did not pass our PM selection. For example, stars J083850.6$+$192317 and J084108.0$+$1914901, listed by @gon06 as members on the basis of optical and infrared photometry, have PMs ($\mu_\alpha\, \cos\delta =197.5$ mas yr$^{-1}$ and $\mu_\delta =79.6$ mas yr$^{-1}$ for J083850.6$+$192317, and $\mu_\alpha\, \cos\delta=-58.4$ mas yr$^{-1}$ and $\mu_\delta =24.9$ mas yr$^{-1}$ for J084108.0$+$1914901) inconsistent with being part of Praesepe. Another highly probable member suggested by @gon06, J084039.3$+$192840, already refuted by @bou10 because of its ($I_c - K_s$) color, is indeed not in our candidate list. Of the six brown dwarf candidates proposed by @bou10 [their Table 5], only three are found in our data, though the identification for either stars No. 099, or No. 909 is uncertain because of a nearby star in each case (see the finding charts in their Fig. 8). Only star No. 910 may have a PPMXL counterpart within 10, but it has a proper motion ($\mu_\alpha \cos\delta =-10.5 \pm 7.3$masyr$^{-1}$, $\mu_\delta = -10.7 \pm 7.3$masyr${-1}$) inconsistent with membership. The brown dwarf candidate found by @mag98, NGC2632 Roque Praesepe 1, was not in our list because of its faint magnitude ($J=21.0$ mag).
@van09 identified, but not tabulated, 24 [*Hipparcos*]{} members in Praesepe. With the identifications kindly provided by van Leeuwen, we confirm that they are all enlisted in our candidate sample. The blue stragglers in the cluster suggested by @and98, HD73666, HD73819, HD73618, HD73210, too bright for PS1, are all confirmed to be PM members. Our photometric selection precludes the white dwarfs known in the cluster [@dob04; @dob06]. They are too faint for 2MASS but have been recovered by PPMXL and PS1, illustrated in Figure \[fig:compjjkggy\]. One additional white dwarf candidate is identified in our data ($\alpha=127.166145\degr$, $\delta=+19.728674\degr$, J2000; $\mu_\alpha = -40.4 \pm 5.2$ masyr$^{-1}$, $\mu_\delta = -20.4 \pm 5.2$ masyr$^{-1}$) with $g_{\rm P1} = 18.15$ mag, and $y_{\rm P1} = 19.07$ mag. The white dwarf members follow the general cooling sequence from brighter/bluer to fainter/redder in the CMD. Scaled with white dwarfs in the field, studied by @ton12b also with PS1 data, the ones in Praesepe have a cooling time scale of 0.2–0.4 Gyr.
Binary Fraction
---------------
A binary system with identical component stars would have the brightness of either star overestimated by 0.75 mag. A binary sequence therefore is often seen as a swath up to 0.7–0.8 mag above the main sequence of a star cluster in a CMD. Multiple systems may have even larger magnitude differences. @ste95 and @hod99 estimated a multiplicity of $\sim0.5$ for low-mass members in Praesepe. In both the 2MASS and PS1 CMDs (see Figure \[fig:memjjkggy\]), the binary sequence stands out clearly. Such a distinct binary sequence was already noticed by @kra07. Note that the $J$ versus $J-K_s$ main sequence is characterized by a slanted upper part and turns nearly vertically below the mass of $\sim0.6$ M$_\sun$. While the upper main sequence allows us to gauge the distance (shifting vertically), the vertical segment provides a convenient tool to estimate the reddening of a cluster (shifting horizontally). This fact, however, also means the $J$ versus $J-K_s$ CMD cannot be used to evaluate the binarity at the lower main sequence. Instead, the PS1 CMD shows a monotonic track, so is useful for this purpose.
There is no clear dividing line above the main sequence to separate binaries from single stars. The bottom panel of Figure \[fig:memjjkggy\] demonstrates a magnitude difference of 0.5 mag above the main sequence as the dividing line. In this case, there are 242 stars above the line, or a binary fraction of about 23% of the total 1040 member candidates. No attempt was made to estimate separately the binarity of the 872 true member versus the 168 interloper samples. If the difference is lower to 0.4 mag or 0.3 mag, the number increases, respectively, to 302 (29%) or 389 (37%). The relatively small increase in the binary fraction is the consequence of a distinct binary sequence of this cluster; that is, the binaries in Praesepe tend to be of similar-mass systems, as noted, for example, by @pin03. Praesepe also seems to teem with multiple systems, as concluded by @kha13. @bou12 conducted an elaborative analysis on the binarity. Adopting a brightness range from 0.376 to 1.5 mag above the (single star) main sequence, these authors derived a binary frequency of $23.3\pm7.3\%$ for the mass range of 0.45 to 0.2 M$_\sun$, $19.6\pm3.8\%$ for 0.2 to 0.1 M$_\sun$, and $25.8\pm3.7\%$ for 0.1 to 0.07 M$_\sun$. Given the uncertainties in membership and binarity assignments, our data do not justify division of the sample into different mass bins, and we infer an overall binary frequency (or multiplicity) of at least 20–40%.
Cluster Mass Function \[sec:mass\]
----------------------------------
The stellar mass was interpolated via a least-square polynomial fitting to the $J$ (if too bright in PS1) or $g_{\rm P1}$ magnitude using the compilation of @kra07 (their Table 5), and adopting a distance modulus of 6.26 mag. The $g_{\rm P1}$ band observations saturate around $g_{\rm P1}\sim14$ mag, corresponding to $J\sim11.5$ mag in our sample, or about 0.6 M$_\sun$. The masses of our candidates range from $\sim0.11$ M$_\sun$ to $\sim2.39$ M$_\sun$.
The luminosity function of the cluster was derived by subtraction of the field contamination. For field stars, we selected the stars satisfying the same PM and isochrone criteria, but with angular distance between $4\degr$ and $5\degr$ from the cluster center. In Figure \[fig:glf\], the $g_{\rm P1}$ luminosity function of the member candidates listed in Table \[tab:members\] is subtracted by that of the field. The field distribution is flat, as expected, and contributes only as a small correction to the observed luminosity function. The corrected luminosity function rises spuriously near the PS1 saturation limit of $g_{\rm P1}\sim$11–15 mag, and then turns around near $g_{\rm P1}\sim18$ mag, or mass $\sim0.3$ M$_\sun$.
The mass function of Praesepe members is shown in Figure \[fig:mf\]. We note that this is the mass function for the stellar systems, i.e., with no binary correction. Using optical $I_c$ band and near-infrared $J$ and $K_s$ photometric data, @bou10 reported a rising mass function in the range from 0.6 M$_\sun$ to 0.1 M$_\sun$ then turning over, in agreement with previous works, e.g., by @ham95b. This increase in number with decreasing mass was shown by @wan11 to continue into the brown dwarf regime, peaking around $70$ M$_{\rm Jup}$ then decrease until about $50$ M$_{\rm Jup}$. @kra07 and @bak10 also derived a rising, but flatter, mass function. On the other hand, @bou12, using also the UKIDSS photometry, but adding additional proper motion information, obtained an opposite result, namely, a declining mass function between 0.6 M$_\sun$ and 0.1 M$_\sun$, different from those by @ham95b, @chab05, @kra07, @bak10, and @bou10. Our sample is more complete at the higher mass end than that by @bou12, but otherwise the mass function is consistent with theirs for stellar masses greater than around 0.3 M$_\sun$. Overall, the mass function we obtained resembles that of the disk population [@chab05] for the massive part, but shows a deficit of the lowest mass population ($\la 0.3$ M$_\sun$).
![ The observed $g_{\rm P1}$ luminosity function of member candidates (the red dash line) is subtracted by the field population with the same photometric and PM selection criteria (blue dotted line) to derive the corrected cluster luminosity function (solid blue line). The corresponding stellar mass is labeled at the top in unit of solar mass. []{data-label="fig:glf"}](fig06){width="60.00000%"}
![The mass function of Praesepe (solid line). Also shown are that by @chab05 for the disk population (long-dashed line), and those by @ham95b (a representative rising mass function) and @bou12 (representing a falling mass function) for Praesepe (dashed lines), each shifted vertically for display clarity. []{data-label="fig:mf"}](fig07){width="80.00000%"}
Spatial Distribution of Members
-------------------------------
Even the youngest star clusters may have elongated shape [@che04], likely a consequence of filamentary structure in the parental clouds. Subsequent encounters among member stars then circularize the core of a cluster. Mass segregation occurs as energy losing massive stars sink to the center, whereas lower-mass members gain energies and occupy a larger volume in space. Some stars may gain sufficient speed so as to escape the system. The lowest mass members are particularly vulnerable to such stellar “evaporation”. As the cluster evolves, the internal gravitational pull becomes weaker and external disturbances, such as differential rotation, or tidal force from passing molecular clouds and from the Galactic disk, act together to distort the shape of a cluster and eventually tear it apart. The deformation and tidal stripping are effective even for globular clusters [@che10].
Figure \[fig:segr\] shows how the stellar mass correlates with the spatial distribution. The radial density profiles have been computed for four different mass groups: M/M$_\sun \leq 0.2$ (129 stars), M/M$_\sun =$0.2–0.35 (256 stars), M/M$_\sun =$0.35–0.7 (332 stars), and M/M$_\sun \geq 0.7$ (323 stars). The top panel shows the observed density profiles, while the bottom panel compares the normalized profiles. Because of the normalization, no correction of the field contamination is necessary. Relatively massive members appear to be centrally concentrated, whereas lower mass members are more scattered spatially, a result of mass segregation.
Mass segregation in Praesepe was well demonstrated already by @ham95b, @kra07, and @kha13. Our result is consistent with that by @ham95b from 0.85 M$_\sun$ to 0.15 M$_\sun$. When the radial density distribution shown in Figure \[fig:segr\] is parameterized with an exponential form, $\sigma(r)~\propto~e^{-\alpha r}$, the least-squared fitting yields $\alpha=2.21$ (for members $>0.7~M_\odot$), 0.96 (0.35–0.7 M$_\sun$), and 0.42 (0.2–0.35 M$_\sun$). @cab08 suggested that a power-law function may be more appropriate. In any case, for the faintest sample, the density distribution is certainly not exponential. Instead, it exhibits a sharp truncation beyond 1. We interpret this as a consequence of stellar evaporation. This further supports the notion of a relative lack of low-mass stars in Praesepe, as already demonstrated in Figure \[fig:mf\].
Mass segregation is further manifested by the positional (Figure \[fig:pos\]) and PM distributions (Figure. \[fig:vel\]) of the members; namely, relatively massive members are concentrated in a smaller volume in space, and have a smaller velocity dispersion than lower-mass stars. The average stellar mass in our sample is $ \bar{m} \approx 0.59$ M$_\sun$, close to that for a Miller-Scalo initial mass function. With the total number of members $N=872$, the total stellar mass in the cluster then amounts to at least $\sim520~{\rm M}_\sun$. The lowest mass stars, with a declining mass function, do not contribute significantly to the total mass. With a radius $R=9$ pc, the velocity dispersion of the cluster then would be $v \approx (GN \bar{m} /R)^{1/2} =
0.5$ km s$^{-1}$, which is noticeably less than the typical value of 1–2 km s$^{-1}$ for Galactic open clusters. At the assumed distance of 179 pc to Praesepe, an intracluster PM dispersion of 1 mas yr$^{-1}$ corresponds to a velocity dispersion of 0.8 km s$^{-1}$. Our data thus are not precise enough to measure any PM gradient among members.
The evidence is mounting that Praesepe is dissolving. It is spatially extended with a sparse stellar density. @hol00 suggested that Praesepe might consist of two merging clusters. The relatively high fraction of equal mass pairs (and of multiples) may be the consequence of occasional stellar ejection during three-body encounters [@bin87], or during the merging process. Relevant time scales for a dissolving star cluster include: ($i$) the dynamical (crossing) time scale, $\tau_{\rm dyn} \approx 2R/v$, ($ii$) the relaxation time, $\tau_{\rm relax} \approx \tau_{\rm dyn} \, 0.1\, N/\ln N$, and (iii) the evaporation time, $\tau_{\rm evap} \approx 100\, \tau_{\rm relax}$ [@bin87]. For Praesepe, these time scales are $\tau_\mathrm{dyn} = 3.6 \times 10^{7}$ yr, $\tau_\mathrm{relax} = 4.6 \times 10^{8}$ yr, and $\tau_\mathrm{evap}=4.6\times 10^{10}$ yr, respectively. The lowest-mass members, having an average escape probability [@spi87] several times of that for the most massive stars, are particularly susceptible to ejection. The Praesepe cluster therefore is almost fully relaxed, and tidal stripping has occurred, starting with the lowest mass members being witnessed to escape from the cluster.
![The radial density distribution of the members. The lines with different colors show different magnitude ranges. The top panel shows each derived distribution and the bottom panel shows the same but normalized from unity at the center to zero at the edge of the cluster.[]{data-label="fig:segr"}](fig08){height="0.8\textheight"}
![ Positional distributions of stars more massive (open circles) and less massive (solid circles) than 0.35 M$_\sun$. []{data-label="fig:pos"}](fig09){height="0.8\textheight"}
![ Proper motion distributions for the same two mass groups of members as shown in Fig. \[fig:pos\]. []{data-label="fig:vel"}](fig10){height="0.8\textheight"}
Summary {#sec:summary}
=======
We have conducted a photometric and proper motion selection of member stars of the Galactic open cluster Praesepe, using 2MASS, PPMXL and Pan-STARRS data. Our sample is comprehensive in terms of sky area (3 radius), limiting magnitude ($g_{\rm P1} \sim21$ mag), and reliability ($\sim16\%$ false positive rate). A total of 1040 member candidates are identified, 872 of which are highly probable members, down to about 0.1 solar masses. While for members more massive than 0.6 M$_\sun$, the Padova isochrone works well, the BT-Settl atmospheric model fits better toward fainter magnitudes. The binary frequency of Praesepe members is about 20–40%, with a relatively high occurrence of similar mass pairs. The mass function is consistent with that of the disk population, but with a deficit of stars less massive than 0.3 M$_\sun$. Members show a clear evidence of mass segregation, with the lowest mass population being evaporated from the system. At the faint magnitude end, the bottleneck of membership selection for very faint objects remains the sensitivity of the PM measurements. Once the PS1 completes its survey in early 2014, increasing the photometric depth and the stellar PM baseline to more than 3.5 years, we expect to secure member lists for nearby star clusters well into the substellar regime.
We thank the referee, José A. Caballero, who provided very constructive comments on an earlier version to greatly improve the quality of the paper. We are grateful to Steve Boudreault for providing published data to produce Figure 7. The Pan-STARRS1 Surveys (PS1) have been made possible through contributions of the Institute for Astronomy, the University of Hawaii, the Pan-STARRS Project Office, the Max-Planck Society and its participating institutes, the Max Planck Institute for Astronomy, Heidelberg and the Max Planck Institute for Extraterrestrial Physics, Garching, The Johns Hopkins University, Durham University, the University of Edinburgh, Queen’s University Belfast, the Harvard-Smithsonian Center for Astrophysics, the Las Cumbres Observatory Global Telescope Network Incorporated, the National Central University of Taiwan, the Space Telescope Science Institute, the National Aeronautics and Space Administration under Grant No. NNX08AR22G issued through the Planetary Science Division of the NASA Science Mission Directorate, the National Science Foundation under Grant No. AST-1238877, and the University of Maryland. The NCU group is financially supported partially by the grant NSC101-2628-M-008-002.
Abazajian, K. N., Adelman-McCarthy, J. K., Ag[ü]{}eros, M. A., et al. 2009, , 182, 543 Adams, J. D., Stauffer, J. R., Skrutskie, M. F., et al. 2002, , 124, 1570 Ahumada, J., & Lapasset, E. 1995, , 109, 375 An, D., Terndrup, D. M., Pinsonneault, M. H., Paulson, D. B., Hanson, R. B.,& Stauffer, J. R. 2007, , 655, 233 Allard, F., Homeier, D., & Freytag, B. 2013, , 84, 1053 Allard, F. 2014, IAU Symposium, 299, 271 Andrievsky, S. M. 1998, , 334, 139 Asplund, M., Grevesse, N., Sauval, A. J., & Scott, P. 2009, , 47, 481 Baker, D. E. A., Jameson, R. F., Casewell, S. L., et al. 2010, , 408, 2457 Baraffe, I., Chabrier, G., Allard, F., & Hauschildt, P. H. 1998, , 337, 403 Binney, J., & Tremaine, S. 1987, Galactic Dynamics, Princeton, NJ, Princeton University Press, 1987 Boudreault, S., Bailer-Jones, C. A. L., Goldman, B., Henning, T., & Caballero, J. A. 2010, , 510, A27 Boudreault, S., Lodieu, N., Deacon, N. R., & Hambly, N. C. 2012, , 426, 3419 Boudreault, S., & Lodieu, N. 2013, , 434, 142 Bouvier, J., Kendall, T., Meeus, G., et al. 2008, , 481, 661 Caballero, J. A. 2008, , 383, 375 Carrera, R., & Pancino, E. 2011, , 535, A30 Chabrier, G. 2005, in The Initial Mass Function 50 Years Later, Eds. E. Corbelli and F. Palle (Springer, Dordrecht) Ap&SSL, 327, 41 Chappelle, R. J., Pinfield, D. J., Steele, I. A., Dobbie, P. D., & Magazz[ù]{}, A. 2005, , 361, 1323 Chen, W. P., Chen, C. W., & Shu, C. G. 2004, , 128, 2306 Chen, C. W., & Chen, W. P. 2010, , 721, 1790 de la Fuente Marcos, R., & de la Fuente Marcos, C. 2000, , 271, 127 Dias, W. S., et al. 2002, , 389, 871 Dobbie, P. D., Pinfield, D. J., Napiwotzki, R., et al. 2004, , 355, L39 Dobbie, P. D., Napiwotzki, R., Burleigh, M. R., et al. 2006, , 369, 383 Eggen, O. J. 1960, , 120, 540 Fossati, L., et al. 2008, , 483, 891 G[á]{}sp[á]{}r, A., Rieke, G. H., Su, K. Y. L., et al. 2009, , 697, 1578 Girard, T. M., Grundy, W. M., López, C. E., & van Altena, W. F. 1989, , 98, 227 Gonz[á]{}lez-Garc[í]{}a, B. M., Zapatero Osorio, M. R., B[é]{}jar, V. J. S., et al. 2006, , 460, 799 Hambly, N. C., Steele, I. A., Hawkins, M. R. S., & Jameson, R. F. 1995a, , 109, 29 Hambly, N. C., Steele, I. A., Hawkins, M. R. S., & Jameson, R. F. 1995b, , 273, 505 Hauschildt, P. H., Allard, F., & Baron, E. 1999, , 512, 377 Hodapp, K.W., Siegmund,W. A., Kaiser, N., et al. 2004, Proc. SPIE, 5489, 667 Hodgkin, S. T., Pinfield, D. J., Jameson, R. F., et al. 1999, , 310, 87 H[ø]{}g, E., Fabricius, C., Makarov, V. V., et al. 2000, , 355, L27 Holland, K., Jameson, R. F., Hodgkin, S., Davies, M. B., & Pinfield, D. 2000, , 319, 956 Jones, B. F., & Cudworth, K. 1983, , 88, 215 Jones, B. F., & Stauffer, J. R. 1991, , 102, 1080 Kaiser, N., et al. 2010, , 7733, 77330E Khalaj, P., & Baumgardt, H. 2013, , 434, 3236 Klein Wassink, W. J. 1927, Pub. of the Kapteyn Astron. Laboratory Groningen, 41, 1 Kenyon, S. J., & Hartmann, L. 1995, , 101, 117 Kraus, A. L., & Hillenbrand, L. A. 2007, , 134, 2340 Loktin, A. V., & Beshenov, G. V. 2003, Astron. Rep., 47, 6 Magazù, A., Rebolo, R., Zapatero Osorio, M. R., Martín, E. L., & Hodgkin, S. T. 1998, , 497, L47 Magnier, E. A., Schlafly, E., Finkbeiner, D., et al. 2013, , 205, 20 Marigo, P., Girardi, L., Bressan, A., et al. 2008, , 482, 883 Mermilliod, J.-C., Weis, E. W., Duquennoy, A., & Mayor, M. 1990, , 235, 114 Muench, A. A., Lada, E. A., Lada, C. J., & Alves, J. 2002, , 573, 366 Pinfield, D. J., Hodgkin, S. T., Jameson, R. F., Cossburn, M. R., & von Hippel, T. 1997, , 287, 180 Pinfield, D. J., Dobbie, P. D., Jameson, R. F., et al. 2003, , 342, 1241 Quinn, S. N., White, R. J., Latham, D. W., et al. 2012, , 756, L33 Reglero, V., & Fabregat, J. 1991, , 90, 25 Roeser, S., Demleitner, M., & Schilbach, E. 2010, , 139, 2440 Siess, L., Forestini, M., & Dougados, C. 1997, , 324, 556 Siess, L., Dufour, E., & Forestini, M. 2000, , 358, 593 Skrutskie, M. F., Cutri, R. M., Stiening, R., et al. 2006, , 131, 1163 Spitzer, L., Jr., & Shull, J. M. 1975, , 201, 773 Spitzer, L. 1987, “Dynamical Evolution of Globular Clusters”, Princeton, NJ, Princeton University Press, 1987 Steele, I. A., & Jameson, R. F. 1995, , 272, 630 Taylor, B. J. 2006, , 132, 2453 Tonry, J. L., Burke, B. E., Isani, S., Onaka, P. M., & Cooper, M. J. 2008, , 7021, 702105 Tonry, J. L., Stubbs, C. W., Lykke, K. R., et al. 2012, , 750, 99 Tonry, J. L., Stubbs, C. W., Kilic, M., et al. 2012, , 745, 42 van Leeuwen, F. 2007, , 474, 653 van Leeuwen, F. 2009, , 497, 209 Wang, J. J., Chen, L., Zhao, J. H., & Jiang, P. F. 1995, , 113, 419 Wang, W., Boudreault, S., Goldman, B., et al. 2011, , 531, A164 Williams, D. M., Rieke, G. H., & Stauffer, J. R. 1995, , 445, 359
[^1]: Updated to January 2013, available at [http://www.astro.iag.usp.br/\$\\sim\$wilton/](http://www.astro.iag.usp.br/$\sim$wilton/).
[^2]: <http://perso.ens-lyon.fr/france.allard/>, the latest of NextGen models by @hau99 using the solar abundance of @asp09
|
---
abstract: 'Transport events in turbulent tokamak plasmas often exhibit non-local or non-diffusive action at a distance features that so far have eluded a conclusive theoretical description. In this paper a theory of non-local transport is investigated through a Fokker-Planck equation with fractional velocity derivatives. A dispersion relation for density gradient driven linear drift modes is derived including the effects of the fractional velocity derivative in the Fokker-Planck equation. It is found that a small deviation (a few percent) from the Maxwellian distribution function alters the dispersion relation such that the growth rates are substantially increased and thereby may cause enhanced levels of transport.'
author:
- |
S. Moradi$^{1}$, J. Anderson$^{1}$ and B. Weyssow$^{2}$[*$^1$\
Department of Applied Physics, Nuclear Engineering, Chalmers University of Technology and Euratom-VR Association, Göteborg, Sweden\
$^{2}$EFDA-CSU, D-85748 Garching, München, Germany*]{}
title: 'A theory of non-local linear drift wave transport'
---
Introduction
============
Understanding anomalous transport in magnetically confined plasmas is an outstanding issue in controlled fusion research. A satisfactorily understanding of the non-local features as well as the non-Gaussian probability distribution functions (PDFs) found in experimental measurements of particle and heat fluxes is still lacking. In particular, experimental observations of the edge turbulence in the fusion devices [@Zweben] show that in the Scrape of Layer (SOL) the plasma fluctuations are characterized by non-Gaussian PDFs. It has been recognized that the nature of the cross-field transport through the SOL is dominated by turbulence with a significant ballistic or non-local component where a diffusive description is improper [@Naulin]. Moreover, the scaling of the confinement time $\tau \propto L^{\alpha}$ with $\alpha < 2$ [@kaye] is typical in low-confinement mode discharges, instead of the diffusion induced result $\tau \propto L^2$, where $L$ is the system size. There is a considerable amount of experimental evidence [@cardozo1995; @gentle1995; @callen1997; @mantica1999; @vanmilligen2002; @BalescuBook] and recent numerical gyrokinetic [@pradalier; @sanchez2008] and fluid [@negrete2005] simulations that plasma turbulence in tokamaks is highly non-local.
In addition, intermittent turbulence is characterized by patchy spatial structure that is bursty in time. The PDFs of these intermittent events shows unimodal structure with “elevated” tails that deviates from a Gaussian prediction. The understanding of these events are at best limited [@Zweben; @bramwell2009; @carreras1996; @carreras1999; @anderson1; @anderson2]. Moreover, the high possibility of confinement degradation by intermittency strongly calls for a predictive theory.
A prominent candidate for explaining the suggestive non-local features of plasma turbulence is the inclusion of a fractional velocity derivative in the Fokker-Planck (FP) equation leading to an inherently non-local description as well as giving rise to non-Gaussian PDFs of e.g. densities and heat flux. The non-locality is introduced through the integral description of the fractional derivative [@zaslavsky; @sanchez; @negrete] and the non-Maxwellian distribution function drives the observed PDFs of densities and heat flux far from Gaussian.
The aim of this study is to elucidate the effects of a non-Maxwellian distribution function induced by the fractional velocity derivative in the Fokker-Planck equation. Some previous papers on plasma transport have used models including a fractional derivative where the fractional derivative is introduced on phenomenological premises [@sanchez; @negrete]. In the present work we introduce the Levy statistics into the Langevin equation thus yielding a fractional FP description. This approach is similar to that of Ref. [@sanchez2006] resulting in a phenomenological description of the non-local effects in plasma turbulence. Using fractional generalizations of the Liouville equation, kinetic descriptions have been developed previously [@zaslovsky; @tarasov]. It has been shown that the chaotic dynamics can be described by using the FP equation with coordinate fractional derivatives as a possible tool for the description of anomalous diffusion [@zaslovsky2]. Much work has been devoted on investigation of the Langevin equation with Levy white noise, see References [@West; @Jeperson; @Fogedby; @Vlad], or related fractional FP equation [@West]. Furthermore, fractional derivatives have been introduced into the FP framework in a similar manner as the present work [@chechkin2000; @chechkin2002] but a study including drift waves is still called for. To this end we quantify the effects of the fractional derivative in the FP equation in terms of a modified dispersion relation for density gradient driven linear plasma drift waves where we have considered a case with constant external magnetic field and a shear-less slab geometry. In order to calculate an equilibrium PDF we use a model based on the motion of a charged Levy particle in a constant external magnetic field obeying non-Gaussian, Levy statistics. This assumption is the natural generalization of the classical example of the motion of a charged Brownian particle with the usual Gaussian statistics [@chandrasekhar]. The fractional derivative is represented with the Fourier transform containing a fractional exponent. We find a relation for the deviation from Maxwellian distribution described by $\epsilon$ through the quasi-neutrality condition and the characteristics of the plasma drift wave are fundamentally changed, i.e. the values of the growth-rate $\gamma$ and real frequency $\omega$ are significantly altered. A deviation from the Maxwellian distribution function alters the dispersion relation for the density gradient drift waves such that the growth rates are substantially increased and thereby may cause enhanced levels of transport.
The paper is organized as follows. In Sec. 2 the mathematical framework of the fractional FP equation (FFPE) is introduced. In Sec. 3 a dispersion relation for the density gradient driven drift modes using the FFPE are derived. In Sec 4 the deviations from a Maxwellian distribution function are investigated and the dispersion relation is solved in Sec. 5. We conclude the paper with a results and discussion in Sec. 6.
Fractional Fokker-Planck Equation
=================================
Following the theory of Brownian motion we write an equation of motion for a colloidal particle in a background medium as a Langevin equation of the following form [@chandrasekhar] $$\begin{aligned}
\label{eq:1.1}
\frac{d\mathbf{v}}{dt}=-\nu\mathbf{v}+A(t)\end{aligned}$$ Here, we assumed that the influence of the background medium can be split into a dynamical friction, $-\nu\mathbf{v}$, and a fluctuating part, $A(t)$ which is a Gaussian white noise. The Gaussian white noise assumption is usually imposed in order to obtain a Maxwellian velocity distribution describing the equilibrium of the Brownian particle. This connection is due to the relation between the Gaussian central limit theorem and classical Boltzmann-Gibbs statistics [@Khintchine]. However, the Gaussian central limit theorem is not unique and a generalization of the Gaussian central limit theorem to the case of summation of independent identically distributed random variables described by long tailed distributions is performed by Lévy [@levy], and Khintchine [@Khintchine]. In this case the Lévy distributions replace the Gaussian in a generalized central limit theorem.
The simplest case of generalized Brownian motion considered by West and Seshadri [@seshadri] is to assume for fluctuation part, $A(t)$, in Equation (\[eq:1.1\]) to be a white Lévy noise. Following the approach used by Barkai [@barkai] we find the Fractional Fokker-Planck Equation (FFPE) with fractional velocity derivatives for shear-less slab geometry in the presence of a constant external force as $$\begin{aligned}
\label{eq:1.2}
\frac{\partial F_{s}}{\partial t}+\mathbf{v}\frac{\partial F_{s}}{\partial \mathbf{r}}+\frac{\mathbf{F}}{m_{s}}\frac{\partial F_{s}}{\partial \mathbf{v}}=\nu\frac{\partial }{\partial \mathbf{v}}(\mathbf{v}F_{s})+D\frac{\partial^{\alpha} F_{s}}{\partial |\mathbf{v}|^{\alpha}},\end{aligned}$$ where $s(=e,i)$ represents the particle species and $0\le\alpha\le 2$. Here, the term $\frac{\partial^{\alpha} F_{s}}{\partial |\mathbf{v}|^{\alpha}}$ is the fractional Riesz derivative. The fractional differentiation may be represented through singular integrals or by its Fourier transform as we will see later in Equation (\[eq:1.4\]). Note that the connection to the integral representation indicates that the model is inherently non-local in velocity space. The diffusion coefficient, $D$, is related to the damping term $\nu$, according to a generalized Einstein relation [@barkai] $$\begin{aligned}
\label{eq:1.3}
D=\frac{2^{\alpha-1}T_{\alpha}\nu}{\Gamma(1+\alpha)m_{s}^{\alpha-1}}.\end{aligned}$$ Here, $T_{\alpha}$ is a generalized temperature, and taking force $\mathbf{F}$ to represent the Lorentz force (due to a constant magnetic field and a zero-averaged electric field) acting on the particles of species $s$ with mass $m_{s}$ and $\Gamma(1+\alpha)$ is the Euler gamma function. We find the solution by using the Fourier representation of equation (\[eq:1.2\]) above as $$\begin{aligned}
\label{eq:1.4}
\frac{\partial \mathcal{F}_{s}}{\partial t}+(-\mathbf{k}+\Omega_{s}(\mathbf{k}^{v}\times \hat{b})+\nu\mathbf{k}^{v})\frac{\partial\mathcal{F}_{s}}{\partial \mathbf{k}^{v}}=-D|\mathbf{k}^{v}|^{\alpha} \mathcal{F}_{s},\end{aligned}$$ where $\Omega_{s}=e_{s}B/m_{s}c$ is the Larmor frequency of species $s$, $\hat{b}=\mathbf{B}/B$ is the unit vector in the direction of magnetic field and $\mathcal{F}_{s}$ is the characteristic function $$\begin{aligned}
\label{eq:1.5}
\mathcal{F}_{s}(\mathbf{k},\mathbf{k}^{v};t)=\int\int d\mathbf{r}\;d\mathbf{v}\exp(i\mathbf{k}\cdot\mathbf{r}+i\mathbf{k}^{v}\cdot\mathbf{v})F_{s}(\mathbf{r},\mathbf{v};t),\end{aligned}$$ where we have denoted the wave-vector by $\mathbf{k}$ and the corresponding wave vector for the velocity as $\mathbf{k}^v$. We can rewrite the kinetic equation by identification of time derivatives of the wave vectors as $$\begin{aligned}
\label{eq:1.6}
\frac{d\mathcal{F}_{s}}{dt}=\frac{\partial \mathcal{F}_{s}}{\partial t}+\frac{d\mathbf{k}^{v}}{dt}\frac{\partial\mathcal{F}_{s}}{\partial \mathbf{k}^{v}}+\frac{d\mathbf{k}}{dt}\frac{\partial\mathcal{F}_{s}}{\partial \mathbf{k}}=0.\end{aligned}$$ We use the method of characteristics on the Equation (\[eq:1.4\]) and (\[eq:1.6\]) whereby we find that the characteristics are $$\begin{aligned}
\label{eq:1.7}
\frac{\partial \mathcal{F}_{s}}{\partial t}=-D|\mathbf{k}^{v}|^{\alpha} \mathcal{F}_{s},\\
\frac{d\mathbf{k}^{v}}{dt}=-\mathbf{k}+\Omega_{s}(\mathbf{k}^{v}\times \hat{b})+\nu\mathbf{k}^{v},\\
\frac{d\mathbf{k}}{dt}=0.\end{aligned}$$ Following the method used in Ref. [@chechkin2002; @chechkin2000] the solution corresponding to the homogenous and steady state system in Fourier space is $$\begin{aligned}
\label{eq:2.14}
\mathcal{F}_{s}(\mathbf{k}^{v}, t)=e^{-\frac{D}{\alpha\nu}(|\mathbf{k}^{v}_{\bot}|^{\alpha}+|\mathbf{k}^{v}_{\parallel}|^{\alpha})}.\end{aligned}$$ In order to find the solution in real space we compute the inverse Fourier transform of Equation (\[eq:2.14\]) $$\begin{aligned}
\label{eq:2.15}
F_{s}(\mathbf{r},\mathbf{v})=C(\mathbf{r})\int \frac{d\mathbf{k}_{\bot}^{v}d\mathbf{k}_{\parallel}^{v}}{(2\pi)^{3/2}}e^{-i(\mathbf{k}_{\bot}^{v}\mathbf{v}_{\bot}+\mathbf{k}_{\parallel}^{v}\mathbf{v}_{\parallel})}e^{-\frac{D}{\alpha\nu}(|\mathbf{k}^{v}_{\bot}|^{\alpha}+|\mathbf{k}^{v}_{\parallel}|^{\alpha})}.\end{aligned}$$ We define a new variable $\mathcal{D}=\frac{D}{\nu}$ where coefficient $D$ is given by the expression in Equation (\[eq:1.3\]). $C(\mathbf{r})$ is a normalization factor which remains to be defined. Taking the inverse Fourier transform of the Equation (\[eq:2.15\]) for $\alpha=2$ we get $$\begin{aligned}
\label{eq:2.16}
F_{s}(\mathbf{r},\mathbf{v})=\frac{C(\mathbf{r})}{\mathcal{D}}e^{-(\frac{v_{\bot}^2+v_{\parallel}^2}{4\mathcal{D}})}.\end{aligned}$$ The unknown normalization factor $C$ can be determined by comparing the integrals of the Maxwellian distribution and our distribution. In comparison the Maxwellian distribution is defined as $$\begin{aligned}
\label{eq:2.17}
F^{M}_{s}(\mathbf{r},\mathbf{v})=\frac{n_{s}(\mathbf{r})}{(\sqrt{\pi}V_{T,s}(\mathbf{r}))^3}e^{-(v_{\bot}^2+v_{\parallel}^2)/V_{T,s}^2(\mathbf{r})},\end{aligned}$$ where $V_{T,s}(\mathbf{r})=\sqrt{2T_{s}(\mathbf{r})/m_{s}}$ is the thermal velocity of species $s$. By integrating the Maxwellian distribution over the velocity space we find the density as $$\begin{aligned}
\label{eq:2.18}
\int d\mathbf{v}F^{M}_{s}(\mathbf{r},\mathbf{v})=2\pi\int_{0}^{\infty}v_{\bot}dv_{\bot}\int_{-\infty}^{\infty}dv_{\parallel}\frac{n_{s}(\mathbf{r})}{(\sqrt{\pi}V_{T,s}(\mathbf{r}))^3}e^{-(v_{\bot}^2+v_{\parallel}^2)/V_{T,s}^2(\mathbf{r})}=n_{s}(\mathbf{r}),\end{aligned}$$ whereas performing the same integration of the expression in Equation(\[eq:2.16\]) we obtain $$\begin{aligned}
\label{eq:2.19}
\int d\mathbf{v}F_{s}(\mathbf{r},\mathbf{v})=2\pi\int_{0}^{\infty}v_{\bot}dv_{\bot}\int_{-\infty}^{\infty}dv_{\parallel}\frac{C(\mathbf{r})}{ \mathcal{D}}e^{-(\frac{v_{\bot}^2+v_{\parallel}^2}{4\mathcal{D}})}=2\pi^{3/2}\sqrt{2\mathcal{D}} C(\mathbf{r}). \end{aligned}$$ We can now compare the two results obtained in Equations (\[eq:2.18\]) - (\[eq:2.19\]) and we find the following relation $$\begin{aligned}
\label{eq:1.21}
C(\mathbf{r})=\frac{n_{s}(\mathbf{r})}{2\pi^{3/2}\sqrt{2\mathcal{D}}}. \end{aligned}$$ The distribution function can now be determined by replacing this expression into Equation (\[eq:2.16\]) for $C(\mathbf{r})$ yielding $$\begin{aligned}
\label{eq:1.22}
F_{s}(\mathbf{r},\mathbf{v})=\frac{n_{s}(\mathbf{r})}{2\pi^{3/2}\mathcal{D}\sqrt{2\mathcal{D}}} e^{-(\frac{v_{\bot}^2+v_{\parallel}^2}{4\mathcal{D}})}.\end{aligned}$$ We can easily recover the Maxwellian distribution in Equation (\[eq:2.17\]) by setting $\alpha=2$ in the definition for $D$ in Equation (\[eq:1.3\]) and using that $\Gamma(3)=2$. Note that for a general $\alpha$, the equilibrium distribution is as follows $$\begin{aligned}
\label{eq:2.24.1}
F_{s}(\mathbf{r},\mathbf{v})=\frac{n_{s}(\mathbf{r})}{2\pi^{3/2}\sqrt{2\mathcal{D}}} \int \frac{d\mathbf{k}_{\bot}^{v}d\mathbf{k}_{\parallel}^{v}}{(2\pi)^{3/2}}e^{-i(\mathbf{k}_{\bot}^{v}\mathbf{v}_{\bot}+\mathbf{k}_{\parallel}^{v}\mathbf{v}_{\parallel})}e^{-\frac{\mathcal{D}}{\alpha}(|\mathbf{k}^{v}_{\bot}|^{\alpha}+|\mathbf{k}^{v}_{\parallel}|^{\alpha})},\end{aligned}$$ where $$\begin{aligned}
\label{eq:2.25.1}
\mathcal{D}=\frac{V_{T,s}^{\alpha}}{\Gamma(1+\alpha)},\end{aligned}$$ and we have introduced a generalized thermal velocity as $$\begin{aligned}
\label{eq:2.25.2}
V_{T,s}^{\alpha}=\frac{2^{\alpha-1}T_{\alpha}}{m_{s}^{\alpha-1}}.\end{aligned}$$ The generalized equilibrium distribution including the effects of the fractional velocity derivative in Equation (\[eq:2.24.1\]) becomes $$\begin{aligned}
\label{eq:2.24}
F_{s}(\mathbf{r},\mathbf{v})=\frac{n_{s}(\mathbf{r})}{2\pi^{3/2}(\Gamma(1+\alpha))^{-1/2}\sqrt{2V_{T,s}^{\alpha}}}
\int \frac{d\mathbf{k}_{\bot}^{v}d\mathbf{k}_{\parallel}^{v}}{(2\pi)^{3/2}}e^{-i(\mathbf{k}_{\bot}^{v}\mathbf{v}_{\bot}+\mathbf{k}_{\parallel}^{v}\mathbf{v}_{\parallel})}e^{-\frac{V_{T,s}^{\alpha}}{\Gamma(1+\alpha)\alpha}(|\mathbf{k}^{v}_{\bot}|^{\alpha}+|\mathbf{k}^{v}_{\parallel}|^{\alpha})}.\end{aligned}$$ We will now determine the dispersion relation for density gradient driven drift waves including the effects of the fractional velocity differential operator.
The dispersion relation
=======================
In order to quantify the non-local effects on drift waves induced by the fractional differential operator we will determine the dispersion relation for density gradient driven drift modes. We start by formulating the linearized gyro-kinetic theory where the particle distribution function, averaged over gyro-phase is of the form (see Ref. [@Balescu1991]) $$\begin{aligned}
\label{eq:2.26}
f_{s}(\mathbf{r},\mathbf{v})=F_{s}(\mathbf{r},\mathbf{v})+(2\pi)^{-4}\times\int\int d\mathbf{k}\;d\omega\exp(i\mathbf{k}\cdot\mathbf{r}-i\omega t)\delta f^{s}_{\mathbf{k},\omega}(\mathbf{v}).\end{aligned}$$ We assume that the turbulence is purely electrostatic and neglect magnetic field fluctuations $(\delta \mathbf{B}=0)$. For small deviations from the local equilibrium we find the linearized gyro-kinetic equation of the form $$\begin{aligned}
\label{eq:2.27}
(-\omega+k_{\parallel}v_{\parallel})\delta f^{s}_{\mathbf{k},\omega}(v_{\parallel},v_{\bot})+(\omega-\omega_{*s})\frac{e_{s}}{T_{s}}J_{0}(|\Omega_{s}|^{-1}k_{\bot}v_{\bot})F_{s}(x,\mathbf{v})\delta \phi_{\mathbf{k},\omega}=0,\end{aligned}$$ where $\omega_{*s}=\frac{cT_{s}}{e_{s}B}k_{y}\cdot\frac{d\; ln\;n(x)}{d x}$ is the drift wave frequency of species $s$, and we assumed that the space dependence of $F_{s}$ is only in the $x$ direction perpendicular to the magnetic field as well as for the density gradient. In the equation above, $J_{0}$ is the Bessel function of order zero, $v_{\parallel}$ is the parallel velocity, $v_{\bot}\equiv (v_{x}^{2}+v_{y}^{2})^{1/2}$ is the perpendicular velocity and hence we write the total speed as $v=(v_{\bot}^{2}+v_{\parallel}^{2})^{1/2}$. Inserting the expression for $F_{s}$ from the Equation (\[eq:2.24\]) in Equation (\[eq:2.27\]) and rearranging the terms we find the perturbed distribution $\delta f_{\mathbf{k},\omega}$ as $$\begin{aligned}
\label{eq:2.28}
\delta f^{s}_{\mathbf{k},\omega}(v_{\parallel},v_{\bot})=
-\frac{e_{s}}{T_{s}}[\frac{\omega-\omega_{*s}}{k_{\parallel}v_{\parallel}-\omega}]J_{0}(|\Omega_{s}|^{-1}k_{\bot}v_{\bot})\delta \phi_{\mathbf{k},\omega}
\frac{n_{s}(\mathbf{r})}{2\pi^{3/2}(\Gamma(1+\alpha))^{-1/2}\sqrt{2V_{T,s}^{\alpha}}} \times\nonumber\\
\int \frac{d\mathbf{k}_{\bot}^{v}d\mathbf{k}_{\parallel}^{v}}{(2\pi)^{3/2}}e^{-i(\mathbf{k}_{\bot}^{v}\mathbf{v}_{\bot}+\mathbf{k}_{\parallel}^{v}\mathbf{v}_{\parallel})}e^{-\frac{V_{T,s}^{\alpha}}{\Gamma(1+\alpha)\alpha}(|\mathbf{k}^{v}_{\bot}|^{\alpha}+|\mathbf{k}^{v}_{\parallel}|^{\alpha})}.\end{aligned}$$ Here, the wave vector perpendicular to magnetic field is $k_{\bot}=(k^2_{x}+k^2_{y})^{1/2}$. The gyro-kinetic Equation (\[eq:2.28\]) is complemented with Poisson equation for the electric potential. For fluctuations with wave vectors much smaller than the Debye wave vector, the Poisson equation becomes the quasi-neutrality condition $$\begin{aligned}
\label{eq:2.29}
\sum_{s} e_{s}\delta n^{s}_{\mathbf{k},\omega}=0,\end{aligned}$$ where the density fluctuation is related to the distribution function through $$\begin{aligned}
\label{eq:2.31}
\delta n^{s}_{\mathbf{k},\omega}=-\frac{e_{s}}{T_{s}}n_{s}\delta\phi_{\mathbf{k},\omega} + \int d\mathbf{v}
J_{0}(|\Omega_{s}|^{-1}k_{\bot}v_{\bot})\delta f^{s}_{\mathbf{k},\omega}(v_{\parallel},v_{\bot}).\end{aligned}$$ In the above equation we have separated the adiabatic response (first term on the right hand side) from the non-adiabatic response (second term on the right hand side). We have to keep in mind that the density $n_{s}$ coming from the $F_{s}(x,\mathbf{v})$ in the adiabatic response is also given by Equation (\[eq:2.24\]) and for a general $0\le\alpha\le2$ the adiabatic response can be different than that calculated by Maxwellian distribution of Equation (\[eq:2.17\]). Using the quasi-neutrality condition (\[eq:2.29\]) we find the dispersion equation which determines the eigenfrequencies as a function of the wave vector, $\omega=\omega(\mathbf{k})=\omega_{r}(\mathbf{k})+i\gamma(\mathbf{k})$. In the simplest case we consider a plasma consisting of electrons and a single species of singly charged ions with the equal temperatures. For the density fluctuation therefore we have $$\begin{aligned}
\label{eq:2.32}
\delta n^{s}_{\mathbf{k},\omega}=-n_{s}(\mathbf{r})\frac{e_{s}}{T_{s}}\delta\phi_{\mathbf{k},\omega}[M^{ad,s}+M^{s}_{\mathbf{k},\omega}].\end{aligned}$$ Therefore, the dispersion equation as in the Ref. [@Balescu1991] is $$\begin{aligned}
\label{eq:2.33}
M^{ad,e}+M^{e}_{\mathbf{k},\omega}=-M^{ad,i}-M^{i}_{\mathbf{k},\omega},\end{aligned}$$ where $$\begin{aligned}
\label{eq:2.34}
M^{ad,s}=\int d\mathbf{v}
\frac{1}{2\pi^{3/2}(\Gamma(1+\alpha))^{-1/2}\sqrt{2V_{T,s}^{\alpha}}}
\int \frac{d\mathbf{k}_{\bot}^{v}d\mathbf{k}_{\parallel}^{v}}{(2\pi)^{3/2}}e^{-i(\mathbf{k}_{\bot}^{v}\mathbf{v}_{\bot}+\mathbf{k}_{\parallel}^{v}\mathbf{v}_{\parallel})}e^{-\frac{V_{T,s}^{\alpha}}{\Gamma(1+\alpha)\alpha}(|\mathbf{k}^{v}_{\bot}|^{\alpha}+|\mathbf{k}^{v}_{\parallel}|^{\alpha})},\end{aligned}$$ gives the adiabatic contribution, and $$\begin{aligned}
\label{eq:2.35}
M^{s}_{\mathbf{k},\omega}=\int d\mathbf{v}[\frac{\omega-\omega_{*s}}{k_{\parallel}v_{\parallel}-\omega}]J_{0}(b_{s}v_{\bot}/V_{Ts})\times\nonumber\\
\frac{1}{2\pi^{3/2}(\Gamma(1+\alpha))^{-1/2}\sqrt{2V_{T,s}^{\alpha}}}
\int \frac{d\mathbf{k}_{\bot}^{v}d\mathbf{k}_{\parallel}^{v}}{(2\pi)^{3/2}}e^{-i(\mathbf{k}_{\bot}^{v}\mathbf{v}_{\bot}+\mathbf{k}_{\parallel}^{v}\mathbf{v}_{\parallel})}e^{-\frac{V_{T,s}^{\alpha}}{\Gamma(1+\alpha)\alpha}(|\mathbf{k}^{v}_{\bot}|^{\alpha}+|\mathbf{k}^{v}_{\parallel}|^{\alpha})},\end{aligned}$$ gives the non-adiabatic contribution. Here, $b_{s}=k_{\bot}V_{T,s}/\Omega_{s}$. If we take $\alpha=2$ in the Equation(\[eq:2.33\]) we recover the dispersion equation for a Maxwellian distribution as in the Ref. [@Balescu1991].
Adiabatic response
------------------
First, we may analyze the contribution from the adiabatic parts of the dispersion relation only by ignoring all fluctuations, yielding $$\begin{aligned}
\label{eq:2.37}
|M^{ad,e}|=|M^{ad,i}|.\end{aligned}$$ In addition, utilizing the quasi-neutrality condition while neglecting the density gradient in the system we have $n_{i}=n_{e}$, therefore $\alpha_{e}$ and $\alpha_{i}$ becomes connected through Equation (\[eq:2.37\]). This indicates that the deviation from a Maxwellian distribution described by $\alpha$ for electrons and ions becomes dependent on each other. We will get back to this relation in later sections.
Deviations from a Maxwellian distribution function
==================================================
We will now turn our attention to the problem of solving the dispersion relation described by Equation (\[eq:2.33\]). In order to solve this dispersion equation we use the method proposed in Ref. [@Balescu1991] with the difference that here we have to perform additional integrations over $\mathbf{k}^{v}$. We have $$\begin{aligned}
\label{eq:3.1}
M^{s}_{\mathbf{k},\omega}=\frac{\omega-\omega_{*,s}}{|k_{\parallel}|V_{T,s}}Z(\xi_{s})\Gamma(b_{s}),\end{aligned}$$ where the plasma dispersion function is $$\begin{aligned}
\label{eq:3.2}
Z(\xi_{s})=\frac{V_{T,s}}{\sqrt{\pi}}Lim_{\sigma\rightarrow 0}\int_{-\infty}^{\infty}du[\frac{\Phi(v_{\parallel})}{u-\xi_{s}-i\sigma}],\end{aligned}$$ with $u=v_{\parallel}/V_{Ts}$, $\xi_{s}=\omega/(|k_{\parallel}|V_{Ts})$ and the function $\Phi(v_{\parallel})$ is $$\begin{aligned}
\label{eq:3.3}
\Phi(v_{\parallel})=\frac{1}{\sqrt{2(\Gamma(1+\alpha))^{-1/2}\sqrt{2V_{T,s}^{\alpha}}}}
\int \frac{d\mathbf{k}_{\parallel}^{v}}{(2\pi)^{1/2}}e^{-i\mathbf{k}_{\parallel}^{v}\mathbf{v}_{\parallel}}e^{-\frac{V_{T,s}^{\alpha}}{\Gamma(1+\alpha)\alpha}(|\mathbf{k}^{v}_{\parallel}|^{\alpha})}.\end{aligned}$$ The integral over $v_{\bot}$ can be written in a general way as $$\begin{aligned}
\label{eq:3.4}
\Gamma(b_{s})=2V_{T,s}^2\int_{0}^{\infty}dw w \Psi_{s}(b_{s}w)\Phi(v_{\bot}),\end{aligned}$$ where $w=v_{\bot}/V_{Ts}$, $\Psi_{s}=J_{0}^2(b_{s}v_{\bot}/V_{Ts})$ and, $$\begin{aligned}
\label{eq:3.5}
\Phi(v_{\bot})=\frac{1}{\sqrt{2(\Gamma(1+\alpha))^{-1/2}\sqrt{2V_{T,s}^{\alpha}}}}
\int \frac{d\mathbf{k}_{\bot}^{v}}{(2\pi)}e^{-i\mathbf{k}_{\bot}^{v}\mathbf{v}_{\bot}}e^{-\frac{V_{T,s}^{\alpha}}{\Gamma(1+\alpha)\alpha}(|\mathbf{k}^{v}_{\bot}|^{\alpha})}.\end{aligned}$$ The analytical solutions for integrals over $\mathbf{k}^{v}$ with an arbitrary $\alpha$ in the Equations (\[eq:3.3\]) and (\[eq:3.5\]) requires rather tedious calculations. Instead we consider an infinitesimal deviation of the form $\alpha=2-\epsilon$, where $0\le\epsilon\ll 2$ and expand the terms depending on $\alpha$ in the Equations (\[eq:3.3\]) and (\[eq:3.5\]) around $\epsilon=0$ as follows $$\begin{aligned}
\label{eq:3.6}
\frac{1}{\sqrt{(\Gamma(1+\alpha))^{-1/2}\sqrt{V_{T,s}^{\alpha}}}} e^{-\frac{V_{T,s}^{\alpha}}{\Gamma(1+\alpha)\alpha}(|k^{v}|^{\alpha})}=
\frac{2^{1/4}e^{-\frac{1}{4}V_{T,s}^{2}|k^{v}|^{2}}}{\sqrt{V_{T,s}}}+\epsilon\Lambda(k^{v})+\mathcal{O}[\epsilon^2],\end{aligned}$$ where $$\begin{aligned}
\label{eq:3.7}
\Lambda(k^{v})=\frac{e^{-\frac{1}{4}V_{T,s}^{2}|k^{v}|^{2}}}{2^{11/4}\sqrt{V_{T,s}}}\{
-3+2{\gamma_E}-4V_{T,s}^2|k^{v}|^2+2 {\gamma_E} V_{T,s}^2|k^{v}|^2\nonumber\\
+2{\log}[V_{T,s}]+2V_{T,s}^2 {\log}[V_{T,s}]|k^{v}|^2+2V_{T,s}^2|k^{v}|^2 {\log}[|k^{v}|]\}.\end{aligned}$$ Here, we have used the Euler-Mascheroni constant $\gamma_E = 0.57721$. The first term in Equation (\[eq:3.6\]) will produce $$\begin{aligned}
\label{eq:3.3.1}
\Phi(u)=\frac{e^{-u^2}}{V_{T,s}^{3/2}},\;\;\;\;\;\;\; \mbox{and} \;\;\;\;\;\;\;\;\Phi(w)=\frac{e^{-w^2}}{V_{T,s}^{3/2}}\end{aligned}$$ which give the Maxwellian adiabatic response $$\begin{aligned}
\label{eq:3.1.1}
M^{ad,s}=1.\end{aligned}$$ By using the expansion defined by the expression (\[eq:3.6\]) in Equations (\[eq:2.34\]) and (\[eq:2.34\]), the adiabatic and non-adiabatic part of the dispersion relation $M^{ad,s}$ and $M^{s}_{\mathbf{k},\omega}$ are as follows $$\begin{aligned}
\label{eq:3.8}
M^{ad,s}=1+(2\pi\int_{-\infty}^{\infty}dv_{\parallel}\int_{0}^{\infty}dv_{\bot}v_{\bot}\times\nonumber\\
\frac{1}{2\sqrt{2}\pi^{3/2}} \int \frac{d\mathbf{k}_{\bot}^{v}d\mathbf{k}_{\parallel}^{v}}{(2\pi)^{3/2}}e^{-i(\mathbf{k}_{\bot}^{v}\mathbf{v}_{\bot}+\mathbf{k}_{\parallel}^{v}\mathbf{v}_{\parallel})}\Lambda(k_{\bot}^{v})\Lambda(k_{\parallel}^{v}))\epsilon+\mathcal{O}[\epsilon]^2=1+\epsilon W^{ad,s}.\nonumber\\\end{aligned}$$ and $$\begin{aligned}
\label{eq:3.9}
M^{s}_{\mathbf{k},\omega}=2\pi\int_{-\infty}^{\infty}dv_{\parallel}\int_{0}^{\infty}dv_{\bot}v_{\bot}[\frac{\omega-\omega_{*s}}{k_{\parallel}v_{\parallel}-\omega}]\Psi_{s}(b_{s}v_{\bot}/V_{Ts})\times\nonumber\\
\frac{1}{(\sqrt{\pi}V_{T,s}(\mathbf{r}))^3}e^{-(v_{\bot}^2+v_{\parallel}^2)/V_{T,s}^2(\mathbf{r})}+\nonumber\\
(2\pi\int_{-\infty}^{\infty}dv_{\parallel}\int_{0}^{\infty}dv_{\bot}v_{\bot}[\frac{\omega-\omega_{*s}}{k_{\parallel}v_{\parallel}-\omega}]\Psi_{s}(b_{s}v_{\bot}/V_{Ts})\times\nonumber\\
\frac{1}{2\sqrt{2}\pi^{3/2}} \int \frac{d\mathbf{k}_{\bot}^{v}d\mathbf{k}_{\parallel}^{v}}{(2\pi)^{3/2}}e^{-i(\mathbf{k}_{\bot}^{v}\mathbf{v}_{\bot}+\mathbf{k}_{\parallel}^{v}\mathbf{v}_{\parallel})}\Lambda(k_{\bot}^{v})\Lambda(k_{\parallel}^{v}))\epsilon+\mathcal{O}[\epsilon]^2=N^{s}_{\mathbf{k},\omega}+\epsilon W^{s}_{\mathbf{k},\omega}.\nonumber\\\end{aligned}$$ Inserting these relations we may rewrite the dispersion relation (\[eq:2.33\]) in the form $$\begin{aligned}
\label{eq:3.10}
(1+N^{e}_{\mathbf{k},\omega})+\epsilon (W^{ad,e}+W^{e}_{\mathbf{k},\omega})=-(1+N^{i}_{\mathbf{k},\omega})-\epsilon (W^{ad,i}+W^{i}_{\mathbf{k},\omega}).\end{aligned}$$ The first terms on the right and left hand sides generate the usual contributions to the dispersion equation as in Ref. [@Balescu1991] and the terms proportional to $\epsilon$ generate the non-Maxwellian contributions where we have $$\begin{aligned}
\label{eq:3.11}
N^{s}_{\mathbf{k},\omega}=\frac{\omega-\omega_{*,s}}{|k_{\parallel}|V_{T,s}}Z(\xi_{s})\Gamma(b_{s}),\end{aligned}$$ with the usual plasma dispersion function $Z(\xi_{s})$ written as $$\begin{aligned}
\label{eq:3.12}
Z(\xi_{s})=\frac{1}{\sqrt{\pi}}Lim_{\sigma\rightarrow 0}\int_{-\infty}^{\infty}du e^{-u^2}[\frac{1}{u-\xi_{s}-i\sigma}],\end{aligned}$$ and $$\begin{aligned}
\label{eq:3.13}
\Gamma(b_{s})=2\int_{0}^{\infty}dw w e^{-w^2}\Psi_{s}(b_{s}w).\end{aligned}$$ The effects of the fractional velocity derivative can be boiled down to a non-Maxwellian contribution of the form $$\begin{aligned}
\label{eq:3.14}
W^{s}_{\mathbf{k},\omega}=\frac{\omega-\omega_{*,s}}{|k_{\parallel}|V_{T,s}}Z_{\epsilon}(\xi_{s})\Gamma_{\epsilon}(b_{s}),\end{aligned}$$ where the non-Maxwellian plasma dispersion function is given by $$\begin{aligned}
\label{eq:3.15}
Z_{\epsilon}(\xi_{s})=\frac{V_{T,s}}{\sqrt{\pi}}Lim_{\sigma\rightarrow 0}\int_{-\infty}^{\infty}du[\frac{\Phi(v_{\parallel})}{u-\xi_{s}-i\sigma}],\end{aligned}$$ with the function $\Phi(v_{\parallel})$ being $$\begin{aligned}
\label{eq:3.16}
\Phi(v_{\parallel})=\frac{1}{2^{3/4}}\int \frac{dk^{v}_{\parallel}}{(2\pi)^{1/2}}\exp(-ik^{v}_{\parallel}v_{\parallel})\Lambda(k_{\parallel}^{v}).\end{aligned}$$ It is important to note that the deviation from Maxwellian is different for the different species (electrons and ions). In the rest of Sec. 4, we will quantify the deviations. The non-Maxwellian contribution to Equation (\[eq:3.4\]) is $$\begin{aligned}
\label{eq:3.17}
\Gamma_{\epsilon}(b_{s})=2V_{T,s}^2\int_{0}^{\infty}dw w \Psi_{s}(b_{s}w)\Phi(v_{\bot}),\end{aligned}$$ where $$\begin{aligned}
\label{eq:3.18}
\Phi(v_{\bot})=\frac{1}{2^{3/4}}\int \frac{dk^{v}_{\bot}}{(2\pi)}\exp(-ik^{v}_{\bot}v_{\bot})\Lambda(k_{\bot}^{v}).\end{aligned}$$ To extimate the non-Maxwellian contribution we need to determine the inverse Fourier transforms of the Equations (\[eq:3.16\]) and (\[eq:3.18\]) resulting in $$\begin{aligned}
\label{eq:3.18.1}
\Phi(z)=\frac{1}{8V_{T,s}^{3/2}}e^{-z^2} \nonumber \\
\left\{-4(-2+{\gamma_E})z^2+(-7+4\; {\gamma_E})+ 2 \log[V_{T,s}]+2e^{z^2}{_1 F_1}[\frac{3}{2},\frac{1}{2},-z^2]\right\}\end{aligned}$$ with $z=\{u,w\}$ and ${_1 F_1}[a;b;z]$ denoting Kummer’s confluent hypergeometric function. Therefore we can write $$\begin{aligned}
\label{eq:3.18.2}
W^{ad,s}= \frac{2V_{T,s}^3}{\sqrt{\pi}} \int_{-\infty}^{\infty}du\int_{0}^{\infty}wdw \Phi(u)\Phi(w).\end{aligned}$$ By inserting typical values for the plasma parameters from Ref. [@BalescuBook] we find the velocities as $V_{T,e}=5.93\times 10^{9} [cm/s]$ and $V_{T,i}=1.38\times 10^{8} [cm/s]$ and we obtain $$\begin{aligned}
\label{eq:3.18.3}
W^{ad,e}=33.724\;\;\;\;\;\;\;\;\;,W^{ad,i}=23.6591.\end{aligned}$$ Following the adiabatic condition in Equation (\[eq:2.37\]) and the expanded dispersion relation in Equation (\[eq:3.10\]) we obtain the following ratio between the non-Maxwellian contributions $$\begin{aligned}
\label{eq:3.18.4}
\frac{\epsilon_{i}}{\epsilon_{e}}=\frac{W^{ad,e}}{W^{ad,i}}=1.42541.\end{aligned}$$ This relation means that if there is a deviation of the distribution function from the Maxwellian for plasma electrons, the deviation from the Maxwellian for ions will be $\sim 1.4$ larger.
Solutions of the dispersion relation
====================================
We will solve the dispersion relation in terms of expansions of the plasma dispersion function by noting that the drift waves are defined in the frequency range $|k_{\parallel}|V_{Ti}\ll\omega \ll |k_{\parallel}|V_{Te}$ in evaluating Equations (\[eq:3.12\]) and (\[eq:3.15\]). We define the expansion parameter for electrons in powers of $\xi_{e}=\omega/(|k_{\parallel}|V_{Te})\ll 1$ and for ions we expand it in powers of $\xi_{i}^{-1}=(|k_{\parallel}|V_{Ti})/\omega\ll 1$, respectively. The Maxwellian dispersion function $Z(\xi_{s})$ has the same definition as in Ref. [@Balescu1991] $$\begin{aligned}
\label{eq:3.19}
Z(\xi_{e})=\frac{1}{\sqrt{\pi}}Lim_{\sigma\rightarrow 0}\int_{-\infty}^{\infty}du e^{-u^2}[\frac{1}{u-\xi_{e}-i\sigma}]
=-2\xi_{e}+\frac{4\xi_{e}^3}{3}+i\sqrt{\pi}(1-\xi_{e}^2)+\mathcal{O}[\xi_{e}^4],\end{aligned}$$ whereas the non-Maxwellian plasma dispersion function $Z_{\epsilon}(\xi_{e})$ becomes $$\begin{aligned}
\label{eq:3.20}
Z_{\epsilon}(\xi_{e})=\frac{V_{T,e}}{\sqrt{\pi}}Lim_{\sigma\rightarrow 0}\int_{-\infty}^{\infty}du[\frac{\Phi(u)}{u-\xi_{e}-i\sigma}]
=\frac{V_{T,e}}{\sqrt{\pi}}Lim_{\sigma\rightarrow 0}\nonumber\\
\int_{-\infty}^{\infty}du\Phi(u)[\frac{1}{u-i\sigma}+\frac{\xi_{e}}{(u-i\sigma)^2}+\frac{\xi_{e}^2}{(u-i\sigma)^3}+\frac{\xi_{e}^3}{(u-i\sigma)^4}+\mathcal{O}[\xi_{e}^4]].\end{aligned}$$ For ions, using the expansion in powers of $\xi_{i}^{-1}$ we can rewrite the above integrals as a function of the expansion parameter as $$\begin{aligned}
\label{eq:3.21}
Z(\xi_{i})=\frac{1}{\sqrt{\pi}}Lim_{\sigma\rightarrow 0}\int_{-\infty}^{\infty}du e^{-u^2}[\frac{1}{u-\xi_{i}-i\sigma}]
=-\xi_{i}^{-1}-\frac{1}{2}\xi_{i}^{-3}+\mathcal{O}[\xi_{i}^{-5}],\end{aligned}$$ and the non-Maxwellian $Z_{\epsilon}(\xi_{i})$ becomes $$\begin{aligned}
\label{eq:3.22}
Z_{\epsilon}(\xi_{i})=\frac{V_{T,i}}{\sqrt{\pi}}Lim_{\sigma\rightarrow 0}\int_{-\infty}^{\infty}du[\frac{\Phi(u)}{u-\xi_{i}-i\sigma}]
=\frac{V_{T,i}}{\sqrt{\pi}}Lim_{\sigma\rightarrow 0}\nonumber\\
\int_{-\infty}^{\infty}du\Phi(u)[\frac{1}{(-\xi_{i}-i\sigma)}-\frac{u}{(\xi_{i}+i\sigma)^2}+\frac{u^2}{(-\xi_{i}-i\sigma)^3}-\frac{u^3}{(\xi_{i}+i\sigma)^4}+\mathcal{O}[\xi_{i}^{-5}]].\end{aligned}$$ We can now evaluate he Maxwellian integrals of the forms $\Gamma(b_{e})$ and $\Gamma(b_{i})$ assuming $\Psi_{e}=1$, $\Psi_{i}=J_{0}^2(b_{i}v_{\bot}/V_{Ti})$ we get $$\begin{aligned}
\label{eq:3.23}
\Gamma(b_{e})=2\int_{0}^{\infty}dw w e^{-w^2}=1,\end{aligned}$$ and $$\begin{aligned}
\label{eq:3.24}
\Gamma(b_{i})=2\int_{0}^{\infty}dw w e^{-w^2}\Psi_{e}(b_{i}w)=e^{-b_{i}/2}\mathcal{I}_{0}(b_{i}),\end{aligned}$$ where $\mathcal{I}_{0}$ denotes modified Bessel function of the zeroth order. The final result will be found after evaluating the non-Maxwellian $\Gamma_{\epsilon}(b_{e})$ and $\Gamma_{\epsilon}(b_{i})$ are given as $$\begin{aligned}
\label{eq:3.25}
\Gamma_{\epsilon}(b_{e})=2V_{T,e}^2\int_{0}^{\infty}dw w \Phi(w)=4.8\times 10^{5},\end{aligned}$$ and $$\begin{aligned}
\label{eq:3.26}
\Gamma_{\epsilon}(b_{i})=2V_{T,i}^2\int_{0}^{\infty}dw w \Psi(b_{i}w)\Phi(w)=6.1\times 10^{4},\end{aligned}$$ where we have used $V_{T,e}=5.93\;10^{9} [cm/s]$, $V_{T,i}=1.38\;10^{8} [cm/s]$ and $b_{i}=0.1$. Finally we can summarize different terms in the dispersion relation (\[eq:3.10\]) as $$\begin{aligned}
\label{eq:3.27}
N^{e}_{\mathbf{k},\omega}=(\xi_{e}-\bar{\omega}_{*,e})(-2\xi_{e}+\frac{4\xi_{e}^3}{3}+i\sqrt{\pi}(1-\xi_{e}^2)),\nonumber\\
N^{i}_{\mathbf{k},\omega}=(\xi_{i}-\bar{\omega}_{*,i})(-\xi_{i}^{-1}-\frac{1}{2}\xi_{i}^{-3})e^{-b_{i}/2}\mathcal{I}_{0}(b_{i}),\nonumber\\
W^{e}_{\mathbf{k},\omega}=(\xi_{e}-\bar{\omega}_{*,e})Z_{\epsilon}(\xi_{e})\Gamma_{\epsilon}(b_{e}),\nonumber\\
W^{i}_{\mathbf{k},\omega}=(\xi_{i}-\bar{\omega}_{*,i})Z_{\epsilon}(\xi_{i})\Gamma_{\epsilon}(b_{i}),\end{aligned}$$ where $\bar{\omega}_{*,s}=\omega_{*,s}/|k_{\parallel}|V_{T,s}$. Note that the non-Maxwellian contributions in Equations (\[eq:3.20\]), (\[eq:3.22\]), (\[eq:3.25\]) and (\[eq:3.26\]) have been calculated numerically. By utilizing the found values of the integrals above we rewrite the dispersion relation (\[eq:3.10\]) as follows $$\begin{aligned}
\label{eq:3.28}
(1+\epsilon_{e} W^{ad,e})+(\xi_{e}-\bar{\omega}_{*,e})\{-2\xi_{e}+\frac{4\xi_{e}^3}{3}+i\sqrt{\pi}(1-\xi_{e}^2)+\epsilon_{e} Z_{\epsilon}(\xi_{e}) \Gamma_{\epsilon}(b_{e})\}=\nonumber\\
-(1+\epsilon_{i} W^{ad,i})-(\xi_{i}-\bar{\omega}_{*,i})\{(-\xi_{i}^{-1}-\frac{1}{2}\xi_{i}^{-3})e^{-b_{i}/2}\mathcal{I}_{0}(b_{i})+\epsilon_{i} Z_{\epsilon}(\xi_{i})\Gamma_{\epsilon}(b_{i})\}\end{aligned}$$ where $W^{ad,s}$ are given in Equation (\[eq:3.18.3\]) and we will use the ratio between $\epsilon_{e}$ and $\epsilon_{i}$ from Equation (\[eq:3.18.4\]).
Results and discussion
======================
We have derived a dispersion relation for drift waves driven by a density gradient in a shear-less slab geometry with constant magnetic field where the small deviation from a Maxwellian distribution is described by $\epsilon$. Here we will determine the quantitative effects on the real frequency and growth rate as a function of this deviation. We start by assuming that we have adiabatic electrons for which the dispersion Equation (\[eq:3.28\]) is, $$\begin{aligned}
\label{eq:4.1}
2+\epsilon_{i} (2.35\;W^{ad,e}+ W^{ad,i})=\nonumber\\
-(\xi_{i}-\bar{\omega}_{*,i})\{(-\xi_{i}^{-1}-\frac{1}{2}\xi_{i}^{-3})e^{-b_{i}/2}\mathcal{I}_{0}(b_{i})+\epsilon_{i} Z_{\epsilon}(\xi_{i})\Gamma_{\epsilon}(b_{i})\}.\end{aligned}$$ After rearranging the terms in the above equation we finally get the following relation for $\epsilon_{i}$: $$\begin{aligned}
\label{eq:4.1.1}
\epsilon_{i} =\frac{-2\xi_{i}^3+[\xi_{i}^3+0.5\xi_{i}-\bar{\omega}_{*,i}\xi_{i}^2-0.5\bar{\omega}_{*,i}]e^{-b_{i}/2}\mathcal{I}_{0}(b_{i})}{W^{ad,tot}\xi_{i}^3+(\bar{\omega}_{*,i}\xi_{i}^3-\xi_{i}^4)Z_{\epsilon}(\xi_{i})\Gamma_{\epsilon}(b_{i})}\end{aligned}$$ where $W^{ad,tot}=2.35\;W^{ad,e}+ W^{ad,i}$. This relation gives the possible deviation of the equilibrium PDF from the Maxwellian PDF for a given plasma turbulence, i.e $\xi_{i}$. One has to remember that only positive values of $\mathbf{Re}[\epsilon]$ are physically meaningful.
Using the same plasma parameters as was used in Equations (\[eq:3.18.3\]) and (\[eq:3.24\],\[eq:3.26\]) we compute the term $Z_{\epsilon}(\xi_{i})$, and from Equation (\[eq:3.22\]) we get $$\begin{aligned}
\label{eq:4.2}
Z_{\epsilon}(\xi_{i})=\frac{V_{T,i}}{\sqrt{\pi}}Lim_{\sigma\rightarrow 0}\{\frac{1}{(-\xi_{i}-i\sigma)}\int_{-\infty}^{\infty}du\Phi(u)+\frac{1}{(-\xi_{i}-i\sigma)^3}\int_{-\infty}^{\infty}u^2du\Phi(u)\}\nonumber\\
=\frac{-6.5\times 10^{-9} - 3.8 \times 10^{-9} \xi_{i}^2}{\xi_{i}^3}.\end{aligned}$$ Here, those integrations omitted resulted in zero contributions and rewriting Equation (\[eq:4.1\]) by using these explicit values results in the expression for the deviation in Equation (\[eq:4.1.1\]) we obtain $$\begin{aligned}
\label{eq:4.4}
\epsilon_{i} =\frac{-2\xi_{i}^3+[\xi_{i}^3+0.5\xi_{i}-\bar{\omega}_{*,i}\xi_{i}^2-0.5\bar{\omega}_{*,i}]e^{-b_{i}/2}\mathcal{I}_{0}(b_{i})}{66.3\xi_{i}^3-39.2\xi_{i}+39.2\bar{\omega}_{*,i}+23.0 \bar{\omega}_{*,i}\xi_{i}^2}\end{aligned}$$ Figure \[fig1\] shows $\epsilon_{i}$ from Equation (\[eq:4.4\]) where $\xi_{i}=\omega+i \gamma$. Here, the values of $\omega, \gamma$ are normalized to $|k_{\parallel}|V_{T,i}$. We have assumed parameter values $b_{i}=0.1$, $k_{\parallel}=10^{-3}$ and $\bar{\omega}_{*,i}=-7.1\times 10^{2}$ with $d\;ln\;n/dx=1$. It is found that there is a threshold in the growth rate $\gamma$ close to $\gamma = 0.7$ and that increasing to $1.0$ only increases the deviation from a Maxwellian from 0 to 0.03. It should be noted that $\epsilon$ increases the excess kurtosis of the distribution function by a similar amount thus a quite small deviation from a Maxwellian can have a rather significant impact.
In figure \[fig2\], the mode growth rate as a function of $\epsilon_{i}$ is shown. Note that in this figure the values of growth rate are the solutions of the Equation (\[eq:4.4\]) for a given $\epsilon_{i}$ while in the figure \[fig1\] we solve Equation (\[eq:4.4\]) for $\epsilon$ at a given $\xi_{i}$. As the dispersion equation is of 3rd order in $\bar{\omega}$ three possible solutions exist, however we are only interested in the solutions with non-zero imaginary value, $\gamma>0$ corresponding to unstable situations. It is shown in figure \[fig2\] that a deviation of $\epsilon_{i}=0.01$ yield an increase of about $20\%$ in the growth rate. Furthermore, the growth rate increases almost linearly with increasing $\epsilon_{i}$ and such an increase in the growth rate will lead to a significant increase in the level of anomalous flux.
In summary, we have derived a dispersion relation for density gradient driven linear drift waves including the effects coming from the inclusion of a fractional velocity derivative in the Fokker-Planck equation in the case of constant magnetic field and a shear-less slab geometry. The solutions of this Fokker-Planck equation are the alpha-stable distributions. It has not yet been shown that in a direct way one can derive the alpha-stable distribution function [@BalescuBook; @montroll] from the classical form of collision operator [@Gatto]. One way may be to construct a new type of collisional operator by considering a fractal phase space and reformulate the collision operator on this new space. However, such a discussion is outside the scope of the present paper. Interestingly enough, we note that non-local effects are observed in non-linear collisionless fluid simulations of plasma turbulence where the non-local transport showing Levy features are induced by the interaction of the non-linear terms in the dynamical equations [@negrete2005]. The non-local features of non-linear fluid models are indicated by recent analytical theories using path-integral methods to derive probability density fucntions of fluxes [@anderson1].
The fractional derivative is represented with the Fourier transform containing a fractional exponent that we are able to connect to the deviation from a Maxwellian distribution described by $\epsilon$. The characteristics of the plasma drift wave are fundamentally changed, i.e. the values of the growth-rate $\gamma$ and real frequency $\omega$ are significantly altered. A deviation from the Maxwellian distribution function alters the dispersion relation for the density gradient drift waves such that the growth rates are substantially increased and thereby may cause enhanced levels of transport.
[*Acknowledgements*]{} The authors would like to thank professor T. Fülöp for her helpful comments. This work was funded by the European Communities under Association Contract between EURATOM and [*Vetenskapsr[å]{}det*]{}.
References {#references .unnumbered}
==========
[11]{} S. J. Zweben, J. A. Boedo, O. Grulke, C. Hidalgo, B. LaBombard, R. J. Maqueda, P. Scarin and J. L. Terry [*Plasma Phys. Control. Fusion*]{} [**49**]{} S1-S23 (2007) V. Naulin [*Journal of Nucl. Mater.*]{} [**363-365**]{} 24-31 (2007) S. M. Kaye, Cris W. Barnes, M. G. Bell, J. C. DeBoo, M. Greenwald, K. Riedel, D. Sigmar, N. Uckan, and R. Waltz [*Phys. fluids B*]{} [**2**]{} 2926 (1990) N. Lopez Cardozo [*Plasma Phys. contr. fusion*]{} [**37**]{} 799 (1995) K. W. Gentle, R. V. Bravenec, G. Cima, H. Gasquet, G. A. Hallock, P. E. Phillips , D. W. Ross, W. L. Rowan, and A. J. Wootton [*Phys. Plasmas*]{} [**2**]{} 2292 (1995) J. D. Callen and M. W. Kissick [*Plasma Phys. Control. Fusion*]{} [**39**]{} B173 (1997) P. Mantica, P. Galli, G. Gorini, GMD Hogeweij, J. de Kloe, and NJ Lopes Cardozo [*Phys. Rev. Lett.*]{} [**82**]{} 5048 (1999) B. Ph. van-Milligen, E. de la Luna, F.L. Tabars, E. Ascasbar, T. Estrada, F. Castejn, J. Castellano, I. Garcia-Cortes, J. Herranz, C. Hidalgo, J.A. Jimenez, F. Medina, M. Ochando, I. Pastor, M.A. Pedrosa, D. Tafalla, L. Garca, R. Sanchez, A. Petrov, K. Sarksian and N. Skvortsova [*Nucl. Fusion*]{} [**42**]{} 787 (2002) R. Balescu [*Aspects of Anomalous Transport in Plasmas*]{} (IoP publishing) (2005) G. Dif-Pradalier, P. H. Diamond, V. Grandgirard, Y. Sarazin, J. Abiteboul, X. Garbet, Ph. Ghendrih, A. Strugarek, S. Ku and C. S. Chang [*Phys. Rev. E*]{} [**82**]{} 025401 (2010) R. Sanchez, D. E. Newman, J.-N Leboeuf, V. K. Decyk and B. A. Carreras [*Phys. Rev. Lett.*]{} [**101**]{}, 205002 (2008) D. del-Castillo-Negrete, B. A. Carreras and V. E. Lynch [*Phys. Rev. Lett.*]{} [**94**]{}, 065003 (2005) S. T. Bramwell [*Nat. Phys.*]{} [5]{} 443 (2009) B. A. Carreras, C. Hidalgo, E. Sanchez, M. A. Pedrosa, R. Balbin, I. Garcia-Corts, B. van Milligen, D. E. Newman, and V. E. Lynch [*Phys. Plasmas*]{} [**3**]{} 2664 (1996) B. A. Carreras, B. Ph. van Milligen, C. Hidalgo, R. Balbin, E. Sanchez, I. Garcia-Cortes, M. A. Pedrosa, J. Bleuel and M. Endler [*Phys. Rev. Lett.*]{} [**83**]{} 3653 (1999) J. Anderson and E. Kim [*Plasma Phys. contr. Fusion*]{} [**52**]{} 012001 (2010) J. Anderson and P. Xanthopoulos [*Phys. Plasmas*]{} [**17**]{} 110702 (2010) G. M. Zaslavsky, [*Hamiltonian Chaos and Fractional Dynamics*]{} (Oxford University Press, Oxford) (2005) R. Sanchez, B. Ph. van-Milligen and B. A. Carreras [*Phys. Plasmas*]{} [**12**]{} 056105 (2005) D. del-Castillo-Negrete [*Phys. Plasmas*]{} [**13**]{} 082308 (2006) R. Sanchez, B. A. Carreras, D. E. Newman, V. E. Lynch and B. Ph. van Milligen [*Phys. Rev. E*]{} [**74**]{}, 016305 (2006) G. M. Zaslavsky [*Phys. Rep.*]{} [**371**]{} 461 (2002) V. E. Tarasov [*J. Phys. Conference series*]{} [**7**]{} 17 (2005) G. M. Zaslavsky [*Physica D*]{} [**76**]{} 110 (1994) B. J. West and V. Seshadri [*Physica A*]{} [**113**]{} 203 (1982) R. Metzler , E. Barkai and J. Klafter [*Phys. Rev. Lett.*]{} [**82**]{} 3563 (1999) H. C. Fogedby [*Phys. Rev. Lett.*]{} [**73**]{} 2517 (1994) M. O. Vlad, J. Ross and F. W. Schneider [*Phys. Rev. E*]{} [**62**]{} 1743 (2000) A. V. Chechkin and V. Yu Gonchar [*Open Sys. & Information Dyn.*]{} [**7**]{} 375-390 (2000) A. V. Chechkin and V. Yu Gonchar [*Phys. of Plasmas*]{} v9, [**1**]{} 78 (2002) S. Chandrasekhar [*Rev. Modern Phys.*]{} [**21**]{} 383 (1949) A. Y. Khintchine [*The Mathematical Foundation of Statistical Mechanics*]{} Dover (New York) (1948) P. Lévy [*Theorie del ’Addition des Variables*]{} (Gauthier-Villiers, Paris) (1937) B. J. West and V. Seshadri [*Linear Systems with Levy Fluctuations*]{}, Physica A, [**113**]{} 203 (1982) E. Barkai [*Stable Equilibrium Based on Lévy Statistics: Stochastic Collision Models Approach*]{}, Phys. Rev. E. Rapid Communication, [**68**]{} 055104(R) (2003) R. Balescu [*Phys. Fluids*]{} **B 4** 91 (1992) E. W. Montroll and J. T. Bendler [*J. Stat. Phys.*]{} **34**129 (1984) R. Gatto and H. E. Mynick [*Theory and application of the generalized Balescu-Lenard transport formalism, Nova Publishers*]{} (2008)
|
16.0cm 24.0cm -1.5cm +0.2cm -1.0cm
[CERN-TH.7237/94]{}\
[[ Dominance of the light-quark condensate\
in the heavy-to-light exclusive decays]{}]{} {#dominance-of-the-light-quark-condensate-in-the-heavy-to-light-exclusive-decays .unnumbered}
============================================
\
Theoretical Physics Division, CERN\
CH - 1211 Geneva 23\
and\
Laboratoire de Physique Mathématique\
Université de Montpellier II\
Place Eugène Bataillon\
F-34095 - Montpellier Cedex 05\
\
Using the QCD [*hybrid*]{} (moments-Laplace) sum rule, we show $semi$-$analytically$ that, in the limit $M_b \rar \infty$, the $q^2$ and $M_b$ behaviours of the heavy-to-light exclusive ($\bar B\rar \rho~(\pi)$ semileptonic as well as the $ B\rar \rho\gamma$ rare) decay–form factors are $universally$ dominated by the contribution of the soft light-quark condensate rather than that of the hard perturbative diagram. The QCD-analytic $q^2$ behaviour of the form factors is a polynomial in $q^2/M^2_b$, which mimics quite well the usual pole parametrization, except in the case of the $A_1^B$ form factor, where there is a significant deviation from this polar form. The $M_b$-dependence of the form factors expected from HQET and lattice results is recovered. We extract with a good accuracy the ratios: $V^B(0)/A^B_1(0) \simeq A^B_2(0)/A^B_1(0) \simeq 1.11\pm 0.01$, and $A^B_1(0)/F^B_1(0) \simeq 1.18 \pm 0.06$; combined with the “world average" value of $f^B_+(0)$ or/and $F^B_1(0)$, these ratios lead to the decay rates: $\Gamma_{\bar B\rar \pi e\bar \nu} \simeq (4.3 \pm 0.7)
\times|V_{ub}|^2 \times 10^{12
}$ s$^{-1}$, $\Gamma_{\bar B\rar \rho e\bar \nu}/
\Gamma_{\bar B\rar \pi e\bar \nu} \simeq .9 \pm 0.2$, and to the ratios of the $\rho-$polarised rates: $\Gamma_+/\Gamma_- \simeq 0.20 \pm 0.01,
{}~\alpha \equiv 2\Gamma_L/\Gamma_T-1 \simeq -(0.60 \pm 0.01)$.
CERN-TH.7237/94\
Introduction
============
In previous papers [@SN1; @SN2], we have introduced the [*hybrid*]{} (moments-Laplace) sum rule (HSR), which is more appropriate than the [*popular*]{} double exponential Laplace (Borel) sum rule (DLSR) for studying the form factors of a heavy-to-light quark transition; indeed, the [*hybrid*]{} sum rule has a well-defined behaviour when the heavy quark mass tends to infinity. In [@SN2], we studied analytically with the HSR the $M_b$-dependence of the $B\rar K^*
\gamma$ form factor and found that it is dominated by the light-quark condensate and behaves like $\sqrt{M_b}$ at $q^2=0$. We have also noticed in [@SN1] that the light-quark condensate effect is important in the numerical evaluation of the $\bar B \rar \rho~(\pi)~
$ semileptonic form factors, while it has been noticed numerically in [@DOSCH] using the DLSR that for the $\bar B \rar \rho$ semi-leptonic decays, the $q^2$ behaviour of the $A^B_1$ form factor in the time-like region is very different from the one expected from the $standard$ pole representation. In this paper, we shall study analytically the $M_b$-behaviour of the different form factors for a better understanding of the previous numerical observations. As a consequence, we shall re-examine with our analytic expression the validity of the $q^2$-dependence obtained numerically in [@DOSCH], although we shall mainly concentrate our analysis in the Euclidian region ($q^2 \le 0$). There, the QCD calculations of the three-point function are reliable; also the lattice results have more data points. For this purpose, we shall analyse the form factors of the $\bar B \rar \pi (\rho)~
$ semileptonic and $B\rar \rho\gamma$ rare processes defined in a standard way as: (p’)|u \_(1-\_5) b B(p)&=&(M\_B+M\_)A\_1 \^\*\_-\^\*p’(p+p’)\_\
&&+ \_p\^p’\^,\
(p’)|u\_bB(p)&=& f\_+(p+p’)\_+f\_-(p-p’)\_,\
<(p’)|s \_q\^bB(p)> &=& i\_\^[\*]{}p\^p’\^F\^[B]{}\_1\
&&+ \^\*\_(M\^2\_B-M\^2\_)-\^\*q(p+p’)\_ . In the QCD spectral sum rules (QSSR) evaluation of the form factors, we shall consider the generic three-point function: V(p,p’,q) = -d\^4x d\^4y (ip’x-ipy) 0TJ\_L(x)O(0)J\_b(y)0, whose Lorentz decompositions are analogous to the previous hadronic amplitudes. Here $J_L \equiv \bar u\gamma_\mu d
{}~~(J_L\equiv (m_u+m_d) \bar u i\gamma_5 d)$ is the bilinear quark current having the quantum numbers of the $\rho~(\pi) $ mesons; $J_b \equiv (M_b +m_d)
\bar d i\gamma_5 b$ is the quark current associated to the $B$-meson; $O\equiv \bar b\gamma_\mu
u$ is the charged weak current for the semileptonic transition, while $O\equiv \bar
b\frac{1}{2}\sigma^{\mu\nu}q_\nu$ is the penguin operator for the rare decay. The vertex function obeys the double dispersion relation: V(p\^2,p’\^2,q\^2)= \_[M\^2\_b]{}\^ \_[0]{}\^ V(s,s’,q\^2)+... As already emphasized in [@SN2], we shall work with the HSR: (n,) && \^n \_[p\^2=0]{}V(p\^2,p’\^2,q\^2)\
&=& \_[M\^2\_b]{}\^ \_[0]{}\^ ds’ (-’ s’) V(s,s’,q\^2), rather than with the DLSR ($\cal{L}$ is the Laplace transform operator). This sum rule guarantees that terms of the type: , which appear in the successive evaluation of the Wilson coefficients of high-dimension operators, will not spoil the OPE for $M_b \rar \infty$ unlike the case of the double Laplace transform sum rule, which blows up in this limit for some of its applications in the heavy-to-light transitions.
In order to come to observables, we insert intermediate states between the charged weak and hadronic currents in (2), while we smear the higher-states effects with the discontinuity of the QCD graphs from a threshold $t_c$ ($t'_c$) for the heavy (light) mesons. Therefore, we have the sum rule: \_[res]{}& & 2C\_L f\_B (-M\^2\_L)\
&& \_[M\^2\_b]{}\^[t\_c]{} \_[0]{}\^[t’\_c]{} ds’ (-s’) V\_[[PT]{}]{}(s,s’,q\^2) + [NPT]{}. $PT~ (NPT)$ refers to perturbative (non–perturbative) contributions; $C_L \equiv f_P M_P^2$ for light pseudoscalar mesons, while $C_L\equiv M_V^2/(2\gamma_V)$ for light vector mesons; $M_L$ is the light meson mass. The decay constants are normalized as: & & (m\_q+M\_Q)0|q (i\_5)QP= 2 M\^2\_Pf\_P\
& &0|q \_QV =\^\*\_2 . $F(q^2)$ is the form factor of interest. For our purpose, we shall consider the expression of the decay constant $f_B$ from moments sum rule at the same order (i.e. to leading order) [@SN3]: M\_b\^2 \_[M\^2\_b]{}\^[t\_c]{} -1- . For convenience, we shall work with the non–relativistic energy parameters $E$ and $\delta M_{(b)}$: s (M\_b+E)\^2 M\_[(b)]{} M\_B-M\_b, where, as we saw in the analysis of the two-point correlator, the continuum energy $E_c$ is [@SN3]: E\^D\_c && (1.08 0.26) \
E\^B\_c && (1.30 0.10) \
E\^\_c && (1.5 \~1.7) . In terms of these continuum energies, and at large values of $M_b $, the decay constant reads [@SN3]: f\^2\_B && \^[2n\_2-1]{} { 1-(n\_2+1)\
&&+(2n\_2+3)(n\_2+1)+ \^2 - 1- },\
The semileptonic decay
=======================
The corresponding form factors defined in (1) have been estimated with the HSR [@SN1] and the DLSR [@SN1], [@DOSCH]. Instead of taking the average values from the two methods as was done in [@SN1], we shall only consider the HSR estimates, because of the drawbacks previously found in the DLSR approach: A\^B\_1(0) 0.16 - 0.41, A\^B\_2(0) 0.26 - 0.58, V\^B(0) 0.28 - 0.61. The errors in these numbers are large, as the HSR has no $n$-stability. In the following, we derive semi-analytic formulae for the form factors. Using the leading order in $\alpha_s$ QCD results of the three-point function, and including the effect of the dimension-5 operators as given in [@OVI], one deduces the sum rule ($q^2 \le 0$): A\^B\_1(q\^2) - [|qq ]{} \^[2n]{} 1- +\^[(5)]{}+ , with: \_1 && (M\^2\_’)\
\^[(5)]{}&& { n- 1-n-n\^2\
&&-(n+1)n-1+2’ M\^2\_b(1+2n)+2(n+1)q\^2’}\
where $\cal{I}_1$ is the integral from the perturbative expression of the spectral function. It is constant for $M_b \rar \infty$. Its value and behaviour at finite values of $M_b$ and for $q^2=0$ is given in Fig. 1. At $M_b=4.6$ GeV, it reads: $\cal{I}_1 \simeq (3.6 \pm 1.2)~\mbox{GeV}^2$ and behaves to leading order in $1/M_b$ as $t'^2_cE_c/\la \bar qq \ra$, which is reassuring as it gives a clear meaning of the expansion in (13). The other values of the QCD parameters are [@SN4]: $\la \bar qq \ra =-(189~\mbox{MeV})^3 \ga\log {M_b/\Lambda}\dr^{12/23}$ and $M^2_0 = (0.80\pm 0.01)~ \mbox{GeV}^2$ from the analysis of the $B,B^*$ sum rules. The $\rho$-meson coupling is $\gamma_\rho \simeq 2.55$.
One can deduce from the previous expression that $A^B_1$ is dominated by the light-quark condensate in the $1/M_b$-expansion counting rule. Moreover, the perturbative contribution is also numerically small at the $b$-mass. The absence of the $n$-stability is explicit from our formula, due to the meson-quark mass difference entering the overall factor. This effect could be however minimized by using the expression of $f_B$ in (11) and by imposing that the effects due to the meson–quark mass differences from the three- and two-point functions compensate each other to leading order. This is realized by choosing: 2n=n\_2-, which, fixes $n$ to be about 2, in view of the fact that the two-point function stabilizes for $n_2 \simeq$4-5. In this way, one would obtain the leading-order result in $\alpha_s$: A\_1\^B 0.3 - 0.6, where we have used the leading-order value $f^{L.O}_B \simeq 1.24 f_\pi$. However, although this result is consistent with previous numerical fits in (12) and in [@DOSCH], we only consider it as an indication of a consistency rather than a safe estimate because of the previous drawbacks for the $n-$stability. One should also keep it in mind that the values given in (12) correspond to the value of $f_B \simeq 1.6 f_\pi$, which includes the radiative corrections of the two-point correlator and which corresponds to smaller values of $n$. Improvements of the result in (16) need (of course) an evaluation of the radiative corrections for the three-point function. The $q^2$–dependence of $A^B_1$ can be obtained with good accuracy, without imposing the previous constraint. We obtain the numerical result in Fig. 2, which is well approximated by the effect from the light–quark condensate alone: R\^B\_1(q\^2) 1-. Performing an analytic continuation of this result in the time-like region, we reproduce the numerical result from the DLSR [@DOSCH](see Fig. 2), which indicates that the result is independent of the form of the sum rule used, while in the time-like region the perturbative contribution still remains a small correction of the light-quark condensate one. This result is clearly in contradiction with the $standard$ pole-dominance parametrization, as, indeed, the form factor decreases for increasing $q^2$-values. A test of this result needs improved lattice measurements over the ones available in [@SACH]. From the previous expressions, and using the fact that $f_B$ behaves as $1/\sqrt{M_b}$, one can also predict the $M_b$-behaviour of the form factor at $q^2_{max} \simeq M^2_b+2M_\rho M_b$: A\^B\_1(q\^2\_[max]{}) \~, in accordance with the expectations from HQET [@HQET] and the lattice results [@SACH]. The analysis of the $V^B$ and $A^B_2$ form factors will be done in the same way. Here, one can realize that the inclusion of the higher dimension-5 and -6 condensates tends to destabilize the results, although these still remain small corrections to the leading-order results. Then, neglecting these destabilizing terms, one has: V\^B(q\^2) - |qq \^[2n]{} 1 + +...\
A\^B\_2(q\^2) - |qq \^[2n]{} 1 + +... with: \_V && (M\^2\_’)\
\_2 && (M\^2\_’). $\cal{I}_{V,2}$ are integrals from the perturbative spectral functions, which also behave like $\cal{I}_1$ to leading order in $1/M_b$. They are given in Fig. 1 for $q^2=0$ and for different values of $M_b$. As expected, they are constant when $M_b \rar \infty$, although, as in the previous case, the asymptotic limit is reached very slowly. Here, the $n$-stability of the analysis is also destroyed by the overall $(M_B/M_b)^{2n}$ factor, which hopefully disappears when we work with the ratios of form factors. We show in Fig. 2 the $q^2$-dependence of the normalized $V^B$ and $A^B_2$, which is very weak since the dominant light-quark condensate contribution has no $q^2$-dependence. The small increase with $q^2$ is due to the $q^2$-dependence of the small and non-leading contribution from the perturbative graph. Lattice points in the Euclidian $q^2$-region [@SACH] agree with our results. An analytic continuation of our results at time-like $q^2$ agrees qualitatively with the one in [@DOSCH]. The numerical difference in this region is due to the relative increase of the perturbative contribution in the time-like region due to the effect of the additional non-Landau-type singularities. However, this effect does not influence the $M_b$ behaviour of the form factors at $q^2_{max}$, which can be safely obtained from the leading-order expression given by the light-quark condensate. One can deduce: V\^B(q\^2\_[max]{}) \~, A\^B\_2(q\^2\_[max]{}) \~. This result is in agreement with HQET and lattice data points. Finally, we can also extract the ratios of form factors. At the $\tau'$-maxima and at the $n$-maxima or inflexion point, we obtain from Fig. 3: r\_2 r\_V 1.11 0.01, where the accuracy is obviously due to the cancellation of systematics in the ratios. This result is again consistent with the lattice results [@SACH], but more accurate.
The semileptonic decay
=======================
The relevant form factor defined in (1) has been numerically estimated within the HSR with the value [@SN1]: f\^B\_+(0) 0.20 0.05, where the contribution of the $\pi'$(1.3) meson has been included for improving the sum rule variable stability of the result. In this paper, we propose to explain the meaning of this numerical result from an analytic expression of the sum rule. Using the QCD expression given in [@OVI], we obtain for a pseudoscalar current describing the pion: f\^B\_+(q\^2) - \^[2n]{} 1+\^[(5)]{}+ , where $\cal{I}_\pi$ is the spectral integral coming from the perturbative graph. Its value at $q^2=0$ for different values of $M_b$ is shown in Fig. 1. It indicates that at $M_b=4.6$ GeV, the perturbative contribution, although large, still remains a correction compared with the light-quark condensate term; $\delta^{(5)}$ is the correction due to the dimension-5 condensate and reads: \^[(5)]{} - 2n+ (n+1) n-1. One can use the well-known PCAC relation (m\_u+m\_d) |qq = -m\^2\_f\^2\_, f\_=93.3 into the previous sum rule in order to express $f^B_+$ in terms of the meson couplings. Unlike the case of the $B\rar \rho$ form factors where the scale dependence is contained in $\la \bar qq \ra$, $f^B_+$ is manifestly renormalization-group-invariant. It should be noted, as in the case of the sum rule determination of the $\omega\rho\pi$ coupling [@SN4], that the $f_\pi$-dependence appears indirectly via (26) in a correlator evaluated in the deep Euclidian region, while the pion is off shell, which is quite different from soft-pion techniques with an on-shell Goldstone boson. One can also deduce from (24) that for large $M_b$, $f^B_+$ behaves like $\sqrt{M_b}$. In this limit the $q^2$-dependence is rather weak, as it comes only from the non-leading $1/M_b$ contributions; we therefore have, to a good accuracy: f\^B\_+(q\^2\_[max]{}) f\^B\_+(0) \~. As in the previous case, the slight difference between the $q^2$-behaviour in the time-like region and the one from that obtained in [@DOSCH], at a finite value of $M_b$(=4.6 GeV), is only due to a numerical enhancement caused by the non-Landau singularities of the perturbative contribution in this region, but does not disturb the $M_b$-behaviour of the form factor. Finally, we extract the ratio of the form factor: r\_. Unfortunately, we do not have stabilities, as the stability points are different for each form factor, which is mainly due to the huge mass-difference between the $\rho$ and $\pi$ mesons.
rare decay
===========
We can use the previous results into the HQET [@HQET] relation among the different form factors of the rare $B \rar \rho \gamma$ decay ($F_1^{B}\equiv
F_1^{B \rar \rho}$) and the semileptonic ones. This relation reads around $q^2_{max}$: F\_1\^[B ]{}(q\^2) = +A\^B\_1(q\^2), from which we deduce: F\_1\^[B]{}(q\^2\_[max]{}) \~. However, we can also study, directly from the sum rule, the $q^2$-dependence of $F_1^{B }$. Using the fact that the corresponding sum rule is also dominated by the light-quark condensate for $M_b \rar \infty$ [@SN2], an evaluation of this contribution, at $q^2 \not= 0$, shows that the light-quark condensate effect has no $q^2$-dependence to leading order. Then, we can deduce, to a good accuracy: F\_1\^B(q\^2\_[max]{}) F\_1\^B(0) \~. Let us now come back to the parametrization of the form factor at $q^2=0$. We have given in [@SN2] an expanded interpolating formula that involves $1/M_b$ and $1/M^2_b$ corrections due to the meson-quark mass difference, to $f_B$ and to higher-dimension condensates. Here, we present a slightly modified expression, which is: F\_1\^B(0) - |qq \^[2n]{} 1 + +... , with: \_&& (M\^2\_’),\
\_&& (204) \^2 M\_b 4.6 , where we have neglected the effects of higher-dimension condensates; $\cal{I}_\gamma$ is the perturbative spectral integral. One should notice that unlike the other spectral integrals in Fig. 1, $\cal{I}_\gamma$ reaches quickly the asymptotic limit when $M_b \rar \infty$. Using the estimated value of $F^B_1(0)$ in [@SN2], we can have, in units of GeV: F\^B\_1= 1+ , which leads of course to the same formula at large $M_b$ as in [@SN2]. However, due to the large coefficient of the perturbative contribution, it indicates that an extrapolation of the result obtained at low values of $M_c$ is quite dangerous, as it may lead to a wrong $M_b$-behaviour of the form factor at large mass. One should notice that (34) and the one in [@SN2] lead to the same numerical value of $F^D_1(0)$. Proceeding as for the former cases, we can also extract the ratio: r\_ 1.180.06, from the analysis of the $\tau'$- and $n$-stability shown in Fig. 3.
Values of the -form factors
===========================
The safest prediction of the absolute value of the form factors available at present, where different versions of the sum rules and lattice calculations have a consensus, is the one for $f^B_+(0)$: f\^B\_+(0) & &0.26 0.12 0.04 [@SACH]\
& &0.26 0.03 [@DOSCH] ( [@OVCH])\
& &0.23 0.02 [@SN1]\
& &0.27 0.03 [@RUCKL],\
from which one can deduce the “world average": f\^B\_+(0) 0.25 0.02 . For estimating $A^B_1(0)$, one can use the present most reliable estimate of $F^B_1$ [@SN2], [@ALI]: F\^B\_1(0) 0.27 0.03, where we have used the strength of the $SU(3)$-breakings obtained in [@SN2], in order to convert the result for $B\rar K^*\gamma$ of [@ALI] into the $B\rar \rho\gamma$ of interest here. Then,we deduce: A\_1\^B(0) 0.32 0.02, which is consistent with a direct estimate [@SN1; @ALI], but the result is again more accurate.
-semileptonic-decay rates
=========================
We are now in a good position to predict the different decay rates. In so doing, we shall use the pole parametrization, except for the $A^B_1$ form factor. For the $B\rar \pi$, we shall use the experimental value 5.32 GeV of the $B^*$ mass. For the $B\rar \rho$, we shall use the fitted value ($6.6\pm 0.6$) GeV [@DOSCH] for the pole mass associated to $A^B_2$ and $V^B$. For $A^B_1$, we use the linear form suggested by (13), with an effective mass of ($5.3\pm 0.7$) GeV, which we have adjusted from the numerical behaviour given in [@DOSCH] (we have not tried to reproduce the change of the behaviours for $t\simeq
(0.76-0.95)M^2_b$ obtained in [@DOSCH], which is a minor effect). Using the standard definitions and notations, we obtain: \_[|Be|]{} (3.6 0.6) |V\_[ub]{}|\^210\^[12]{} \^[-1]{} We also obtain the following ratios: 0.9 0.2, 0.20 0.01, 2-1 -(0.60 0.01). Thanks to a better control of the ratios of form factors, the ratio of the $\bar B$ decays into $\pi$ over the $\rho$ can be predicted, to a good accuracy. It becomes compatible with the prediction obtained by only retaining the contribution of the vector component of the form factors. Our predictions are compatible with the ones in [@DOSCH] except for $\Gamma_+/\Gamma_-$, where the one in [@DOSCH] is about one order of magnitude smaller. The difference of two of these three quantities with ones in [@SN1] (the large branching ratio into $\rho$ over $\pi$ and the positive value of the asymmetry $\alpha$ in [@SN1] and in most other pole dominance models for $A^B_1$) is mainly due to the different $q^2-$behaviour of $A^B_1$ used here.
Conclusions
===========
We have studied, using the QCD $hybrid$ sum rule, the $M_b$- and $q^2$-behaviours of the heavy-to-light transition form factors. We find that these quantities are dominated in a $universal$ way by the light-quark condensate contribution.
The $M_b$-dependence obtained here is in perfect agreement with the HQET and lattice expectations.
The $q^2$-dependence of the $A^B_1$ form factor, which is mainly due to the one from the light-quark condensate contribution, is in clear contradiction with the one expected from a pole parametrization. The other form factors can mimic $numerically$ this pole parametrization. Our QCD-analytic $q^2$-behaviours confirm the previous numerical results given in [@DOSCH].
We have also shown that it can be incorrect to derive the $M_b$-behaviour of the form factors at $q^2=0$ by combining the HQET result at $q^2_{max}$ with the pole parametrization.
We have also shown that the unusual $q^2-$behaviour of the $A^B_1$ form factor affects strongly the branching ratio of $B \rar \rho
$ over $B\rar \pi$ and the $\rho$-polarisation parameter $\alpha$. A measurement of these quantities complemented by the one of the $q^2-$behaviour of the form factor should provide a good test of the sum rules approach.
We want also to stress that the extrapolation of the results obtained in this paper to the case of the $D$-meson would be too audacious: the uses of the HSR in that case cannot be $rigorously$ justified since the value of the $c$-quark mass is smaller, although it may lead to acceptable phenomenological results. We are investigating this point at present.
Acknowledgements {#acknowledgements .unnumbered}
================
I thank Olivier Pène and Chris Sachrajda for discussions of the lattice results.
Figure captions {#figure-captions .unnumbered}
===============
$M_b$-dependence of the perturbative spectral integrals at $q^2$ = 0.
$q^2$-behaviour of the normalized form factors: $R_1 \equiv A_1^B(q^2)/A_1^B(0)$, $R_2 \equiv A_2^B(q^2)/A_2^B(0)$, $R_V \equiv V^B(q^2)/V^B(0)$ and $R_\pi \equiv f_+^B(q^2)/f_+^B(0)$. The squared points in the timelike region are from [@DOSCH].
$\tau'$- and $n$-dependences of the ratios of form factors at $q^2$ = 0: $r_2 \equiv A_2^B(0)/A_1^B(0)$, $r_V \equiv V^B(0)/A_1^B(0)$ and $r_\gamma \equiv A_1^B(0)/F_1^B(0)$.
[999]{}
S. Narison, [*Phys. Lett.*]{} [**B283**]{} (1992) 384; [*Z. Phys.*]{} [**C55**]{} (1992) 55. S. Narison, ([*Phys. Lett.*]{}[**B327**]{} (1994) 354. P. Ball, [*Phys. Rev.*]{} [**D48**]{} (1993) 3190;\
P. Ball, V.M. Braun and H.G. Dosch, [*Phys. Rev.*]{} [**D44**]{} (1991) 3567; [*Phys. Lett.*]{} [**B273**]{} (1991) 316; S. Narison, [*Phys. Lett.*]{} [**B198**]{} (1987) 104; [**B308**]{} (1993) 365 and [*Talk given at the Third $\tau$Cf Workshop, 1-6 June 1993, Marbella, Spain*]{}, CERN preprint TH-7042/93 (1993) and references therein;\
S. Narison and K. Zalewski, [*Phys. Lett.*]{} [**B320**]{} (1994) 369; A. Ovchinnikov and V.A. Slobodenyuk, [*Z. Phys.*]{} [**C44**]{} (1989) 433;\
V.N. Baier and A.G. Grozin, [*Z. Phys.*]{} [**C47**]{} (1990) 669. S. Narison, [*Lecture Notes in Physics, Vol. 26, QCD Spectral Sum Rules*]{} (World Scientific, Singapore, 1989) and references therein; ELC: A. Abada et al., LPTHE Orsay-93/20 (1993). N. Isgur and M.B. Wise, [*Phys. Rev.*]{} [**D42**]{} (1990) 2388;\
G. Burdman and J.F. Donoghue, [*Phys. Lett.*]{} [**B270**]{} (1991) 55. A.A. Ovchinnikov, [*Phys. Lett.*]{} [**B229**]{} (1989) 127. V.M. Belyaev, A. Khodjamirian and R. Rückl, [*Z. Phys.*]{} [**C60**]{} (1993) 349. A. Ali, V.M. Braun and H. Simma, CERN preprint TH-7118/93 (1993).
|
---
abstract: 'The turnover, or peak, magnitude in a galaxy’s globular cluster luminosity function (GCLF) may provide a standard candle for an independent distance estimator. Here we examine the GCLF of the giant elliptical NGC 4365 using photometry of $\sim$ 350 globular clusters from the [*Hubble Space Telescope’s*]{} Wide Field and Planetary Camera 2 (WFPC2). The WFPC2 data have several advantages over equivalent ground–based imaging. The membership of NGC 4365 in the Virgo cluster has been the subject of recent debate. We have fit a Gaussian and $t_5$ profile to the luminosity function and find that it can be well represented by a turnover magnitude of $m_V^0$ = 24.2 $\pm$ 0.3 and a dispersion $\sigma$ = 1.28 $\pm$ 0.15. After applying a small metallicity correction to the ‘universal’ globular cluster turnover magnitude, we derive a distance modulus of (m – M) = 31.6 $\pm$ 0.3 which is in reasonable agreement with that from surface brightness fluctuation measurements. This result places NGC 4365 about 6 Mpc beyond the Virgo cluster core. For a V$_{CMB}$ = 1592$\pm$ 24 km s$^{-1}$ the Hubble constant is H$_{\circ}$ = 72$_{-12}^{+10}$ km s$^{-1}$ Mpc$^{-1}$. We also describe our method for estimating a local specific frequency for the GC system within the central 5 h$^{-1}$ kpc which has fewer uncertain corrections than a total estimate. The resulting value of 6.4 $\pm$ 1.5 indicates that NGC 4365 has a GC richness similar to other early type galaxies.'
author:
- 'Duncan A. Forbes'
title: 'Globular Cluster Luminosity Functions and the Hubble Constant from WFPC2 Imaging: The Giant Elliptical NGC 4365'
---
Introduction
============
The measurement and interpretation of extragalactic distances are problematic and often the subject of dispute. In a review of techniques for measuring distances, Jacoby (1992) describe five methods that can be applied to elliptical galaxies, namely planetary nebula luminosity functions (PNLF), novae, surface brightness fluctuations (SBF), the D$_n$ – $\sigma$ relation and globular cluster luminosity functions (GCLF). In the case of the GCLF method, the ‘standard candle’ is the magnitude of the turnover, or peak, in the luminosity function. Although there is no generally accepted theoretical basis, all well–studied globular cluster (GC) systems reveal a similar GCLF shape (often approximated by a Gaussian) with a turnover magnitude of M$_V^0$ $\sim$ –7.5. Measurement of distances, and hence the Hubble constant H$_{\circ}$, with this method, rely on the assumption that the turnover is the same for all galaxies. Recently there have been claims that the turnover magnitude is not quite constant, but has a slight dependency ($\sim$ 0.2 mag) on GC metallicity (Ashman, Conti & Zepf 1995) or the dispersion in the GCLF, which in turn may reflect a galaxy Hubble type dependence (Secker & Harris 1993).
There have been two recent developments of direct relevance to GCLF–determined distances. The first is due to the much improved Strehl ratio (image concentration) of the second Wide Field and Planetary Camera (WFPC2) on the [*Hubble Space Telescope*]{}. Now even relatively short exposures of ellipticals with WFPC2 can contain hundreds of GCs, several magnitudes fainter than typical ground–based observations. To date, published distance measurements based on WFPC2 studies of the GCLF have been carried out by Baum (1996) on NGC 4881 in Coma and by Whitmore (1995) on M87 in Virgo. These studies quote values of H$_{\circ}$ = 67 and 78 km s$^{-1}$ Mpc$^{-1}$ respectively. The second development is a new calibration of Galactic GC distances based on revised RR Lyrae luminosities by Sandage & Tammann (1995) which gives an absolute turnover magnitude of M$_V^0$ = –7.60 $\pm$ 0.11 for our Galaxy. Combined with the halo GCs in M31, they derive M$_V^0$ = –7.62 with an internal error of $\pm$ 0.08 and external error of $\pm$ 0.2 mags.
Sandage & Tammann (1995) went on to re–analyze ground–based GCLFs for five Virgo ellipticals. Although they derive a distance modulus similar to that of Secker & Harris (1993) for Virgo, they disagree on a couple of issues. In particular, Sandage & Tammann question the dependence of M$_V^0$ on the GCLF dispersion claiming that it is in part an artifact of the fitting procedure. They also disagree on the cluster membership of one galaxy – NGC 4365. Secker & Harris claim that it lies slightly more distant than the Virgo cluster in the W cloud, as supported by the surface brightness fluctuation (SBF) measurements of Tonry, Ajhar & Luppino (1990). Sandage & Tammann (1995), on the other hand, suggest that the high metallicity of NGC 4365 makes the SBF method unreliable. Both of these GCLF studies used data from Harris (1991) and they both derived a $m_V^0$ for NGC 4365 to be about 0.8 mags fainter than typical Virgo ellipticals. However the uncertainty on this value is much larger than for the other Virgo ellipticals in the Harris (1991) data set.
A WFPC2 study of the GCLF for NGC 4365 would have several advantages to help resolve the issue of Virgo membership and further test the hypothesis of a universal GCLF. As well as providing an independent data set, the benefits include very low background contamination, no serious blending effects, accurate photometry and the ability to probe to faint magnitude levels. The galaxy itself is also of interest as it has a relatively high Mg$_2$ index (Davies 1987), contains a kinematically–distinct core which is detectable as a disk–like structure in both the kinematics (Surma 1992) and photometry (Forbes 1994), and has a notably blue nucleus (Carollo 1996).
Forbes (1996) presented WFPC2 data on the GCs in 14 ellipticals with kinematically–distinct cores (KDC). In that paper we discussed the colors, radial and azimuthal distribution of the GCs. Analysis of the GCLFs were not attempted. Here we analyze the GCLF of NGC 4365, the richest GC system in the Forbes study. After determining a completeness function and quantifying the photometric errors, we use the maximum likelihood method of Secker & Harris (1993) to determine the turnover magnitude and dispersion of the GCLF. Within the assumptions of the GCLF–distance method, this leads to an independent estimate of the Hubble constant H$_{\circ}$.
Observations and Data Reduction
===============================
Details of the WFPC2 data for the NGC 4365 GCs are presented, along with 13 other KDC ellipticals, in Forbes (1996). Briefly, two 500s F555W images were combined, as were two 230s F814W images. Using DAOPHOT (Stetson 1987), GCs were detected in the F555W image only, their magnitudes measured and then the corresponding F814W magnitudes were determined. The magnitudes have been converted into the standard Johnson–Cousins V, I system and corrected for Galactic extinction. We chose a fairly conservative detection criteria based on flux threshold, shape, sharpness and size. Additionally, we checked the positions of GCs against a list of known hot pixels. After these criteria have been applied, we are confident that the contamination from cosmic rays, hot pixels, foreground stars and background galaxies is small (less than a few percent) in our object list. Forbes (1996) made one additional cut, namely $\pm$3$\sigma$ about the mean color of V–I = 1.10. Here we have chosen to use only the V band data (which is of higher signal-to-noise than the I band data) and not apply any selection based on color.
The detection flux threshold in the PC image was set lower than the WFC images, to compensate for the different point source sensitivity (i.e. $\sim$0.3 mags; Burrows 1993). In Figure 1 we show the fraction of actual GCs detected as a function of GC magnitude, normalized at V = 25. This figure shows that the resulting detection fractions are similar between all four CCDs.
Modeling
========
Completeness Function
---------------------
Forbes (1996) carried out simulations to quantify the ability of DAOPHOT to detect GCs as a function of magnitude. A typical WFC image was chosen for the simulation. The resulting completeness function showed that all GCs brighter than V $\sim$ 24 were detected, and the completeness dropped off rapidly to V $\sim$ 25. As it is crucial for GCLF studies to have a well–determined completeness function we have decided to redo the simulation using the WF3 image of NGC 4365 to ensure that we have the same photon and read noise characteristics as the data. We note that there is no evidence for a significant variation between CCDs. We have simulated GCs using the [*addstar*]{} task and then used DAOPHOT with the same detection criteria as for the actual GCs. In particular, we have excluded all objects with FWHM sizes greater than three pixels. For these simulations $\le$ 1% of the objects are excluded by this criterion. As with the real data, we have not attempted to reject any objects based on color.
The completeness function resulting from simulations of 900 artifical GCs is shown in Figure 2a. This function is similar to that given in Forbes (1996). We derive a 50% completeness level at V = 24.7. For the subsequent analysis the completeness function is set to zero for magnitudes fainter than this to avoid incompleteness corrections larger than a factor of two.
Photometric Errors
------------------
In addition to the completeness function we need to quantify the photometric, or measurement, error from DAOPHOT. Photometric errors can cause a shift of the GCLF peak to brighter magnitudes, as the fainter GCs, with relatively large errors, move into brighter magnitude bins. This ‘bin jumping’ effect is described in detail by Secker & Harris (1993) and taken into account in their maximum likelihood technique. Here we have fit the DAOPHOT determined errors with an exponential of the form:\
p.e. = exp \[a (V – b)\]\
The photometric error and the fit as a function of V magnitude are shown in Figure 2b. Reassuringly these errors are similar to those found by comparing the input and measured magnitudes of the simulated GCs. A typical photometric error is $\pm$ 0.1 mag at V = 24.
Background Contamination
------------------------
One of the advantages of using WFPC2 data for GCLF studies is the ability to exclude most foreground stars and background galaxies based on angular size. This means that the contamination from such sources is very low. Nevertheless we estimated the background contamination on a similar exposure time WFPC2 image from the Medium Deep Survey (Forbes 1994). Again we used DAOPHOT to detect objects with the same detection parameters as before, including the same size criteria as for the GCs. No color selection was used. We estimate a background contamination of seven objects, brighter than V = 24.7, in the WFPC2 field-of-view.
Maximum Likelihood Technique
----------------------------
In this study we use the maximum likelihood technique of Secker & Harris (1993) to accurately determine the GCLF peak magnitude and dispersion. Their technique is designed to take proper account of detection incompleteness at faint magnitudes, photometric error and background contamination. It calculates the convolution product of the photometric error and the intrinsic GCLF, weighted by the completeness function. This is then compared to the raw data set, and after allowing for the background contamination, gives the most likely parameters of the intrinsic GCLF. As well as the commonly used Gaussian profile, we fit a $t_5$ distribution which is less susceptible to variations at the extremes of the luminosity function. The $t_5$ distribution function has the form:\
N$(m) = A (1 + (m - m^0 )^2 / 5
\sigma_t^2)^{-3}$\
Where A is a scaling constant and $\sigma_t$ is the GCLF dispersion, which is related to the dispersion of a Gaussian by $\sigma_t \sim 0.78 \sigma_G$ (see Secker 1992 for details of the $t_5$ function).
Results and Discussion
======================
After applying the maximum likelihood code to our sample of 346 GCs with V $<$ 24.7, we find the best estimate and uncertainty for a Gaussian profile fit to the GCLF to be $m_V^0$ = 24.17 (+0.3,–0.3), $\sigma$ = 1.36 (+0.14,–0.15). For a $t_5$ profile fit, we find $m_V^0$ = 24.00 (+0.3,–0.2), $\sigma$ = 1.17 (+0.15,–0.13). These errors represent the collapsed one–dimensional confidence limits for one standard deviation. The probability contours output from the maximum likelihood code for the Gaussian fit, over a range of 0.5–3 standard deviations, are shown in Figure 3. The contours are skewed towards a larger dispersion and fainter magnitudes, giving rise to the asymmetric errors quoted above. A similar effect can be seen in the ground–based data of NGC 4365 by Secker & Harris (1993). Although the quoted errors represent the internal error of the fitting procedure, they dominate over any contribution from photometric errors. As a test, we increased the photometric errors by 20% (which represents the extreme range of photometric errors from DAOPHOT) and refit the data. This gave a turnover magnitude and dispersion different by $<$ 3%. In Figure 4 we show a binned GCLF and our best–fit Gaussian superposed. Note that the fitting procedure does not use binned data but rather treats each data point individually.
Secker & Harris (1993) have shown that the GCLF parameters will be systematically biased towards brighter magnitudes and smaller dispersions if the limiting magnitude is close to or brighter than the true turnover magnitude. Our limiting magnitude has been set at the 50% completeness level, i.e. V = 24.7. Using their figure 6, the true turnover magnitude for a $t_5$ distribution is $\sim$ 0.1 mag fainter and the dispersion 0.05 mag larger. The results of the two fitting methods, after applying this bias correction to both results, are listed in Table 1. Averaging the results from the two fitting methods, gives $m_V^0$ = 24.2 $\pm$ 0.3 and $\sigma$ = 1.28 $\pm$ 0.15, which is a reasonable representation of the turnover and dispersion of our V band data for the GCLF of NGC 4365. We also list the results of Secker & Harris for the B band GCLF. If we assume that the NGC 4365 GCs have B–V $\sim$ 0.9, then their turnover is fainter by $\sim$ 0.2 mags. We find a slightly smaller GCLF dispersion. The quoted errors are similar between the two studies.
An additional small correction may be required if we wish to compare the magnitudes of the NGC 4365 GCs with the ‘universal value’ (i.e. the mean of the Milky Way and M31 systems from Sandage & Tammann 1995). As mentioned in the introduction, Secker & Harris (1993) and also Fleming (1995) suggest that the GCLF turnover depends on Hubble type. Compared to nearby spirals, the more luminous ellipticals have a fainter turnover. On a more quantitative basis, Ashman (1995) have suggested that GC metallicity is the second parameter, with more metal rich GCs having a fainter turnover. These ideas can be connected via the GC metallicity–galaxy luminosity relation (e.g. Brodie & Huchra 1991, Forbes 1996) in which more luminous galaxies (e.g. giant ellipticals) have relatively metal rich GC systems. Ashman showed that if these metallicity–based corrections were applied to the turnover GC absolute magnitude, then the systematic offset between GCLF distance estimates and other methods (see Jacoby 1992) could be largely eliminated.
In the absence of spectroscopic measures, the mean metallicity of the GC system in NGC 4365 can be estimated crudely from the V–I colors of the GCs. Using the GC sample described in section 2, we calculate a mean metallicity, assuming \[Fe/H\] = 5.051 (V–I) – 6.096 (Couture 1990), of \[Fe/H\] = –0.6. The mean metallicity of the Milky Way and M31 GCs, using the same relative weighting as Sandage & Tammann (1995) is \[Fe/H\] = –1.4. Applying Table 3 of Ashman gives $\Delta$M$_B^0$ = 0.37 and $\Delta$M$_V^0$ = 0.23 for a metallicity difference of 0.8 dex. (A metallicity difference of 0.7–0.9 dex would correspond to roughly $\Delta$M$_B^0$ = 0.32–0.45 and $\Delta$M$_V^0$ = 0.18–26.) These corrections make the combined Milky Way and M31 peaks of M$_B^0$ = –6.93 $\pm$ 0.08 and M$_V^0$ = –7.62 $\pm$ 0.08 (Sandage & Tammann 1995) fainter by 0.37 and 0.22 respectively. Combining these M$^0$ values with $m^0$ from Table 1 gives the distance modulus for both the Gaussian and $t_5$ fits. The fits and the averages (with errors added in quadrature and divided by $\sqrt{N}$) are listed in Table 2. From our data the $t_5$ and Gaussian fits give $(m - M)$ = 31.6 $\pm$ 0.3.
An independent estimate of the distance modulus comes from surface brightness fluctuation (SBF) measurements (Tonry, Ajhar & Luppino 1990). Ajhar (1994) quote an SBF distance modulus of 31.74 $\pm$ 0.16 for NGC 4365, based on the latest calibration of Tonry (1991), which is given in Table 2. The two GCLF distance determinations are in good agreement with that from the SBF method.
We also list in Table 2, recent determinations for the distance modulus to the Virgo galaxies NGC 4472 (M49), NGC 4486 (M87) and NGC 4649 using the GCLF method. Again the distance modulus is calculated using either M$_B^0$ = –6.93 $\pm$ 0.08 or M$_V^0$ = –7.62 $\pm$ 0.08 and a metallicity correction from Ashman (1995), with GC mean metallicities as compiled by Perelmuter (1995). These three galaxies give a GCLF distance modulus to Virgo of about $(m - M)$ = 31.25 $\pm$ 0.15 (internal error only), which can be compared to the weighted mean from 6 different methods (i.e. novae, SN Ia, Tully–Fisher, PNLF, SBF and D$_n$–$\sigma$) of $(m - M)$ = 30.97 $\pm$ 0.18.
To summarize, the GCLF method indicates a similar distance to the Virgo cluster as other distance methods, and the GCLF and SBF distances to NGC 4365 are in good agreement. However, the distance modulus for NGC 4365 is 0.5–0.75 magnitudes fainter than that for the Virgo core. This would suggest it is 25–40% more distant than the Virgo core. Using a representative GCLF distance modulus of $(m - M)$ = 31.74 $\pm$ 0.3 gives a distance of 22.28 (+3.31, –2.87) Mpc.
It is of course interesting to take the distance calculation one step further and estimate the Hubble constant from this one galaxy. The velocity of NGC 4365 with respect to the cosmic microwave background is V$_{CMB}$ = 1592 km s$^{-1}$ (Faber 1989). We assume an error on this value to be the fractional error from the radial velocity measurement by Huchra (1983), i.e. $\pm$ 24 km s$^{-1}$. Dividing this velocity by 22.28 Mpc gives a Hubble constant of 72 (+10, –12) km s$^{-1}$ Mpc$^{-1}$. The velocity from the D$_n$–$\sigma$ relation is similar, i.e. 1509 $\pm$ 250 km s$^{-1}$ after making a $\sim$3% correction for the Malmquist bias (Faber 1989). This would give a Hubble constant of 68 (+21, –20) km s$^{-1}$ Mpc$^{-1}$.
Another measure of interest is the specific frequency ($S$) of the GC system, which gives an indication of the relative richness of the GC system, and is defined by:\
$S = N 10^{0.4(M_V + 15)}$
Where N is usually the total number of GCs and M$_V$ the total galaxy magnitude. Estimates of $S$ for galaxies beyond the local group require two, sometimes large and uncertain, corrections for the number of faint GCs that weren’t detected and the limited areal coverage. Starting with the first correction, by integrating under our profile fit to the GCLF we can make a fairly accurate estimate of the total number of GCs within the WFPC2 field-of-view. The $t_5$ and Gaussian fits give a total of 522 and 554 GCs respectively, over all magnitudes. Taking the average of these we get 538 GCs. We find that 84.7% of the NGC 4365 GCs lie within a 180$^{\circ}$ hemisphere of radius 100$^{''}$ (5 h$^{-1}$ kpc). The total number of GCs within a 100$^{''}$ radius circle is twice this amount or 911 (with an estimated error of $\pm$12%). Knowing the integrated galaxy absolute magnitude within this radius will give us a ‘local’ $S$ value. From the surface photometry of Goudfrooij (1994), we calculate a magnitude of V = 11.36 $\pm$ 0.1, and using $(m - M)$ = 31.74 $\pm$ 0.3, gives a localized specific frequency of $S$ = 6.4 $\pm$ 2.7. The second correction, calculating the total number in the GC system, is much more uncertain. This can be estimated by integrating the density profile found by Forbes (1996) out to large radii, with the boundary condition that at r = 100$^{''}$, the number of GCs is 911. This gives a total for the GC system of N = 2511 $\pm$ 1000, which can be compared to N = 3500 $\pm$ 1200, from ground–based imaging, estimated by Harris (1991). Using a total V = 9.65 $\pm$ 0.1 (Faber 1989) and the same distance modulus as above, gives $M_V$ = –22.09 $\pm$ 0.32 and $S$ = 3.7 $\pm$ 2.4. Our estimated $S$ values, 6.4 and 3.7, are similar to the average value of 5.1 for 34 E+S0 galaxies (van den Bergh 1995). Harris (1991) quoted 7.7 $\pm$ 2.7 for NGC 4365. Part of the difference is due to our lower number of GCs and also because Harris, assumed that NGC 4365 was in Virgo with a distance modulus of 31.3. For N = 3500 $\pm$ 1200 and our a distance modulus of 31.74 $\pm$ 0.3, $S$ = 5.1 $\pm$ 3.3.
Conclusions
===========
We have used the [*HST*]{} WFPC2 data of Forbes (1996) to examine the luminosity function of $\sim$350 globular clusters in the central regions of the giant (M$_V$ = –22.1) elliptical NGC 4365. In particular, we fit the globular cluster luminosity function (GCLF) by both a Gaussian and $t_5$ distribution, using the maximum likelihood analysis of Secker & Harris (1993). The GCLF is well fit by a turnover magnitude of $m_V^0$ = 24.2 $\pm$ 0.3 and dispersion $\sigma$ = 1.28 $\pm$ 0.15 (the two fitting profiles give similar results). Our results are compared to previous work on the GCLF of NGC 4365 and other Virgo ellipticals. Using the most recent determination of the Milky Way and M31 galaxy’s GCLF turnover magnitude of Sandage & Tammann (1995), and a metallicity correction based on the precepts of Ashman (1995), we derive a distance modulus of 31.6 $\pm$ 0.3. This is in reasonable agreement with $(m - M)$ = 31.74 $\pm$ 0.16 derived from surface brightness fluctuation measurements of NGC 4365 and provides further support to the hypothesis that the absolute turnover magnitude of GCLFs is approximately constant or ‘universal’ for all galaxies. Our distance modulus also supports the previous findings from ground–based data that the GCLF turnover is $\sim$0.7 magnitudes fainter, or $\sim$6 Mpc more distant, than that of ellipticals in the Virgo cluster core. As such NGC 4365 may lie in the W$^{'}$ group of the SW extension of the Virgo cluster or in the background W cloud (Binggeli, Tammann, & Sandage 1987).
Adopting a velocity, with respect to the cosmic microwave background, for NGC 4365 of V$_{CMB}$ = 1592 $\pm$ 24 km s$^{-1}$ gives a Hubble constant of H$_{\circ}$ = 72 (+10, –12) km s$^{-1}$ Mpc$^{-1}$ (internal errors only). This value lies between recent determinations of H$_{\circ}$ from the GCLF of NGC 4881 and M87 using WFPC2. After correcting for undetected objects, we have estimated the total number of globular clusters in a 5 h$^{-1}$ kpc radius circle about the galaxy center to be 911. Using the integrated galaxy light within this region, we derive a ‘local’ specific frequency of $S$ = 6.4 $\pm$ 1.5. This measure has the advantage of requiring fewer uncertain corrections than a total estimate. For a distance modulus of 31.74, the ground–based total specific frequency becomes 5 $\pm$ 2, which is similar to our local $S$ value and in excellent agreement with the average $S$ value for a large sample of early type galaxies.
[**Acknowledgments**]{}\
We are particularly grateful to J. Secker for the use of his maximum likelihood code and useful suggestions. We also thank R. Elson, C. Grillmair and R. Guzmán for helpful discussions. This research was funded by the HST grant AR-05794.01-94A\
[**References**]{}
Ajhar, E. A., Blakeslee, J. P., & Tonry, J. L. 1994, AJ, 108, 2087 (ABT94)\
Ashman, K. M., Conti, A., & Zepf, S. E. 1995, AJ, 110, 1164\
Baum, W. A., 1995, AJ, 110, 2537\
Binggeli, B, Tammann, G. A., & Sandage, A. 1987, AJ, 94, 251\
Brodie, J. P., & Huchra, J. 1991, ApJ, 379, 157\
Burrows, C., 1993, Hubble Space Telescope Wide Field and Planetary Camera 2 Instrument Handbook, STScI\
Carollo, C. M., Franx, M., Illingworth, G. D., & Forbes, D. A. 1996, ApJ, submitted\
Couture, J., Harris, W. E., & Allwright, J. W. B., 1990, ApJS, 73, 671\
Davies, R. L., 1987, ApJS, 64, 581\
Faber, S. M., 1989, ApJS, 69, 763\
Forbes, D. A. 1994, AJ, 107, 2017\
Forbes, D. A., Elson, R. A. W., Phillips, A. C., Illingworth, G. D. & Koo, D. C. 1994, ApJ, 437, L17\
Forbes, D. A., Franx, M., Illingworth, G. D., & Carollo, C. M. 1996, ApJ, in press\
Fleming, D. E. B., Harris, W. E., Pritchet, C. J., & Hanes, D. A. 1995, AJ, 109, 1044\
Goudfrooij, P., Hansen, L., Jorgensen, H. E., & Norgaard Nielson, H. U. 1994, A & AS, 105, 341\
Harris, W. E. 1991, ARAA, 29, 543\
Harris, W. E., Allwright, J. W. B., Pritchet, C. J., & van den Bergh, S. 1991, ApJS, 276, 491\
Huchra, J., Davis, M., Latham, D., & Tonry, J. 1983, ApJS, 52, 89\
Jacoby, G. H., 1992, PASP, 104, 599 (J92)\
Perelmuter, J. L. 1995, ApJ, 454, 762\
Sandage, A., & Tammann, G. A. 1995, ApJ, 446, 1\
Secker, J. 1992, AJ, 104, 1472\
Secker, J., & Harris, W. E. 1993, AJ, 105, 1358 (SH93)\
Stetson, P. B., 1987, PASP, 99, 191\
Surma, P. 1992, Structure, Dynamics and Chemical Evolution of Elliptical Galaxies, ed. I. J. Danziger, W. W. Zeilinger and K. Kjar, ESO: Garching, p. 669\
Tonry, J. L., Ajhar, E. A., & Luppino, G. A. 1990, AJ, 100, 1416\
Tonry, J. L. 1991, ApJ, 373, L1\
van den Bergh, S. 1995, AJ, 110, 2700\
Whitmore, B. C., Sparks, W. B., Lucas, R. A., Macchetto, F. D., & Biretta, J. A. 1995, ApJ, in press (W95)\
|
---
abstract: 'In this paper, a multiple-relay network in considered, in which $K$ single-antenna relays assist a single-antenna transmitter to communicate with a single-antenna receiver in a half-duplex mode. A new Amplify and Forward (AF) scheme is proposed for this network and is shown to achieve the optimum diversity-multiplexing trade-off curve.'
author:
- |
Shahab Oveis Gharan, Alireza Bayesteh, and Amir K. Khandani\
Coding & Signal Transmission Laboratory\
Department of Electrical & Computer Engineering\
University of Waterloo\
Waterloo, ON, N2L 3G1\
@cst.uwaterloo.ca\
title: 'Optimum Diversity-Multiplexing Tradeoff in the Multiple Relays Network [^1]'
---
System Model
============
The system , as in [@laneman], [@azarian], and [@yuksel], consists of $K$ relays assisting the transmitter and the receiver in the half-duplex mode, i.e. in each time, the relays can either transmit or receive. The channels between each two node is assumed to be quasi-static flat Rayleigh-fading, i.e. the channel gains remain constant during a block of transmission and changes independently from one block to another. However, we assume that there is no direct link between the transmitter and the receiver. This assumption is reasonable when the transmitter and the receiver are far from each other or when the receiver is supposed to have connection with just the relay nodes to avoid the complexity of the network. As in [@azarian] and [@yang_belfiore], each node is assumed to know the state of its backward channel and, moreover, the receiver is supposed to know the equivalent channel gain from the transmitter to the receiver. No feedback to the transmitting node is permitted. All nodes have the same power constraint. Also, we assume that a capacity achieving gaussian random codebook can be generated at each node of the network. Hence, the code design problem is not considered in this paper.
Proposed $K$-Slot Switching N-sub-block Markovian Scheme (SM)
=============================================================
In the proposed scheme, the entire block of transmission is divided into $N$ sub-blocks. Each sub-block consists of $K$ slots. Each slot has $T'$ symbols. Hence, the entire block consists of $T=NKT'$ symbols. In order to transmit a message $w$, the transmitter selects the corresponding codeword of a gaussian random codebook consisting of $2^{NKT'r}$ codewords of length $\frac{NK-1}{NK}T$ and transmits the codeword during the first $NK-1$ slots. In each sub-block, each relay receives the signal in one of the slots and transmits the received signal in the next slot. So, each relay is off in $\frac{K-2}{2}$ of time. More precisely, in the $k$’ slot of the $n$’the sub-block ($1 \leq n \leq N, 1 \leq k \leq K, nk \neq NK$), the $k$’th relay receives the signals the transmitter is sending, and amplifies and forwards it to the receiver in the next slot. The receiver starts receiving the signal from the second slot. After receiving the last slot ($NK$’th slot) signal, the receiver decodes the transmitted message by using the signal of $NK-1$ slot received from $K$ relays. It will be shown in the next section that the equivalent point-to-point channel from the transmitter to the receiver would act as a lower-triangular MIMO channel.
Diversity-Multiplexing Tradeoff
===============================
In this section, we show that the proposed method achieves the optimum achievable diversity-multiplexing curve. First, according to the cut-set bound theorem [@cover_book], the point-to-point capacity of the uplink channel (the channel from the transmitter to the relays) is an upper-bound for the capacity of this system. Accordingly, the diversity-multiplexing curve of a $1 \times K$ SIMO system which is a straight line from multiplexing gain $1$ to the diversity gain $K$ is an upper-bound for the diversity-multiplexing curve of our system. In this section, we prove that the tradeoff curve of the proposed method achieves the upper-bound and thus, it is optimum. First, we prove the statement for the case that there is no link between the relays. Next, we prove the statement for the general case.
No Interfering Relays
---------------------
Assume, the link gain between the $k$’th relay and the transmitter and the $k$’th relay and the receiver are $h_k$ and $g_k$, respectively. Furthermore, assume that there is no link between the relays. Accordingly, at the $k$’th relay we have $$\mathbf{r}_k=h_k\mathbf{x}+\mathbf{n}_k,$$ where $\mathbf{r}_k$ is the received signal vector of the $k$’th relay, $\mathbf{x}$ is the transmitter signal vector and $\mathbf{n}_k \sim \mathcal{N}(0, \mathbf{I}_{T'})$ is the noise vector of the channel. At the receiver side, we have $$\mathbf{y}=\sum_{k=1}^{K}{g_k\mathbf{t}_k}+\mathbf{z},$$ where $\mathbf{t}_k$ is the transmitted signal vector of the $k$’th relay, $\mathbf{y}$ is the received signal vector at the receiver side and $\mathbf{z} \sim \mathcal{N}(0,\mathbf{I}_{T'})$ is the noise vector of the downlink channel. The output power constraint $\mathbb{E} \left\{\left\|\mathbf{x}\right\|^2\right\}, \mathbb{E}
\left\{\left\|\mathbf{t}_k\right\|^2\right\} \leq T'P$ holds at the transmitter and relays side. To obtain the DM tradeoff curve of the proposed scheme, we are looking for the end-to-end probability of outage from the rate $r\log\left( P \right)$, as $P$ goes to infinity.
![DM Tradeoff for the proposed Switching Markovian Scheme and various values of (K,N), No interfering relays case[]{data-label="fig:dm_ni"}](dm_ni2.eps)
Assume a half-duplex parallel relay scenario with $K$ no interfering relays. The proposed SM scheme achieves the diversity gain $$d_{SM,NI}(r)=\max \left\{0, K\left(1-r\right)- \frac{1}{N},
K\left(1-r\right)- \frac{K r}{N-1} \right\},$$ which achieves the optimum achievable DM tradeoff curve $d_{opt}(r)=K(1-r)$ as $N \to \infty$.
Let us define $\mathbf{x}_{n,k}, \mathbf{n}_{n,k}, \mathbf{r}_{n,k},
\mathbf{t}_{n,k}, \mathbf{z}_{n,k}, \mathbf{y}_{n,k}$ as the signal/noise transmitted/received by the transmitter/relay/receiver to the $k$’th relay/receiver in the $k$’th slot of the $n$’th sub-block. Also, let us define $(k)\equiv k-2~\mod~K + 1$ and $(n)\equiv n-\lfloor{\frac{(k)}{K}}\rfloor$. Thus, we have $$\begin{aligned}
\mathbf{y}_{n,k}&=&g_k\mathbf{t}_{n,k}+\mathbf{z}_{n,k} \nonumber \\
&=&g_k{\alpha}_{(k)}\left(h_{(k)}\mathbf{x}_{(n),(k)}+\mathbf{n}_{(n),(k)}\right)+\mathbf{z}_{n,k},\end{aligned}$$ where ${\alpha}_k=\frac{P}{|h_k|^2P+1}$ is the amplification coefficient performed in the $k$’th relay. Defining the event $\mathcal{E}_k$ as the event of outage from the rate $r\log(P)$ in the $k$’th sub-channel consisting of the transmitter, the $k$’th relay, and the receiver, we have $$\begin{aligned}
\mathbb{P}\{\mathcal{E}_k\}&=&\mathbb{P}\left\{\log\left[1+P|g_k|^2|{\alpha}_k|^2|h_k|^2\left(1+|g_k|^2|{\alpha}_k|^2\right)^{-1}\right]
\leq r\log(P)\right\} \nonumber \\
&\doteq & \min \left\{ \mbox{sign}(r),
\mathbb{P}\left\{|g_k|^2|{\alpha}_k|^2|h_k|^2\left(1+|g_k|^2|{\alpha}_k|^2\right)^{-1}
\leq P^{r-1} \right\} \right\} \nonumber \\
& \stackrel{(a)}{\doteq} & \min \left\{ \mbox{sign}(r),
\mathbb{P}\left\{|g_k|^2|{\alpha}_k|^2|h_k|^2 \min
\left\{\frac{1}{2}, \frac{1}{2|g_k|^2|{\alpha}_k|^2} \right\} \leq
P^{r-1} \right\} \right\} \nonumber \\
& \stackrel{(b)}{\doteq} & \min \left\{\mbox{sign}(r), \mathbb{P}
\left\{|h_k|^2 \leq 2P^{r-1} \right\} + \mathbb{P} \left\{
|g_k|^2|{\alpha}_k|^2|h_k|^2 \leq 2
P^{r-1} \right\} \right\} \nonumber \\
& \stackrel{(c)}{\doteq} & \min \left\{ \mbox{sign}(r), P^{-(1-r)} +
\mathbb{P} \left\{ |g_k|^2\min \left\{ \frac{1}{2},
\frac{|h_k|^2P}{2} \right\} \leq 2 P^{r-1} \right\} \right\}
\nonumber \\
& \stackrel{(d)}{\doteq} & \min \left\{ \mbox{sign}(r), P^{-(1-r)} +
\mathbb{P} \left\{ |g_k|^2 \leq 4 P^{r-1} \right\} + \mathbb{P}
\left\{ |g_k|^2|h_k|^2 \leq 4 P^{r-2} \right\} \right\} \nonumber\\
& \stackrel{(e)}{\doteq} & \min \left\{ \mbox{sign}(r), P^{-(1-r)}
\right\}, \label{eq:sbch_ni}\end{aligned}$$ where $\mbox{sign}(r)$ is the sign function, i.e. $\mbox{sign}(r)=1,
r \geq 0, \mbox{sign}(r)=0, r<0$. Here, (a) follows from the fact that $\frac{1}{1+|g_k|^2|{\alpha}_k|^2} \doteq \min
\left\{\frac{1}{2}, \frac{1}{2|g_k|^2|{\alpha}_k|^2} \right\}$, (b) and (d) follow from the union bound inequality, (c) follows from the fact that $|{\alpha}_k|^2|h_k|^2 \doteq \min \left\{ \frac{1}{2},
\frac{|h_k|^2P}{2} \right\}$ and the pdf distribution of the rayleigh-fading parameter near zero, and (e) follows from the fact that the product of two independent rayleigh-fading parameters behave as a rayleigh-fading parameter near zero. (\[eq:sbch\_ni\]) shows that each sub-channel’s tradeoff curve performs as a single-antenna point-to-point channel.
Defining $R_k(P)$ as the random variable showing the rate of the $k$’th sub-channel consisting of the transmitter, the $k$’th relay, and the receiver in terms of $P$, the outage event of the entire channel from the $r\log(P)$, the event $\mathcal{E}$, is equal to $$\mathbb{P}\left\{\mathcal{E}\right\} = \mathbb{P}\left\{N
\sum_{k=1}^{K-1}{R_k(P)}+(N-1)R_K(P) \leq NKr\log(P) \right\}$$ Assuming $R_k(P)= r_k\log(P)$, we have $$\mathbb{P}\left\{\mathcal{E}\right\} \doteq
\mathbb{P}\left\{N\sum_{k=1}^{K-1}{r_k}+(N-1)r_K \leq NKr \right\}$$ $\mathbb{P} \left\{R_k(P) \leq r_k \log(P)\right\}$ is known by (\[eq:sbch\_ni\]). Defining the region $\mathcal{R}$ as $$\mathcal{R} = \left\{ \left(r_1, r_2, \cdots, r_K\right) | 0 \leq
r_k \leq 1, N\sum_{k=1}^{K-1}{r_k}+(N-1)r_K \leq NKr \right\}
\label{eq:R_df_ni}$$ it is easy to check that all the vectors $\left(r_1, r_2, \cdots,
r_K\right)$ that result in the outage event almost surely lie in $\mathcal{R}$. In fact, according to (\[eq:sbch\_ni\]), for all $k$ we know $r_k \geq 0$. Also, for $r_k
> 1$, $\mathbb{P} \left\{R_k(P) \geq r_k \log(P)\right\} \leq
e^{-P^{r-1}}$ which is exponential in terms of $P$. Hence, $r_k > 1$ can be disregarded for the outage region. As a result, $\mathbb{P}
\left\{ \mathcal{E} \right\} \doteq \mathbb{P} \left\{ \mathbf{r}
\in \mathcal{R} \right\}$.
On the other hand, by (\[eq:sbch\_ni\]) and the fact that $r_k$’s are independent, we have $$\mathbb{P} \left\{ r_1 \leq r_1^0, r_2 \leq r_2^0, \cdots, r_K \leq
r_K^0 \right\} \doteq P^{-\left(K-\sum_{k=1}^{K}{r_k^0}\right)}
\label{eq:cdf_ni}$$ Now, we show that $\mathbb{P} \left\{ \mathcal{E} \right\} \doteq
P^{-\min_{\mathbf{r} \in
\mathcal{R}}{K-\mathbf{1}\cdot\mathbf{r}}}$. First of all, by taking derivative of (\[eq:cdf\_ni\]) with respect to $r_1,r_2,\cdots,r_K$, it is easy to see that the probability density function of $\mathbf{r}$ behaves the same as the probability function in (\[eq:cdf\_ni\]), i.e. $f_r(\mathbf{r}) \doteq
P^{-\left( {K-\mathbf{1}\cdot\mathbf{r}} \right) }$. Hence, the outage probability is equal to $$\begin{aligned}
\mathbb{P} \left\{ \mathcal{E} \right\} & \doteq & \int_{\mathbf{r}
\in \mathcal{R}}{f_r(\mathbf{\mathbf{r}})d\mathbf{r}} \nonumber \\
& \dot{\leq} & vol(\mathcal R)P^{-\min_{\mathbf{r} \in
\mathcal{R}}{K-\mathbf{1}\cdot\mathbf{r}}} \nonumber \\
& \stackrel{(a)}{\doteq} & P^{-\min_{\mathbf{r} \in
\mathcal{R}}{K-\mathbf{1}\cdot\mathbf{r}}} \label{eq:R_ub_ni}\end{aligned}$$ Here, (a) follows from the fact that $\mathcal{R}$ is a fixed bounded region whose volume is independent of $P$. On the other hand, by continuity of $P^{-\left( {K-\mathbf{1}\cdot\mathbf{r}}
\right) }$ over $\mathbf{r}$, we have $\mathbb{P} \left\{
\mathcal{E} \right\} \dot{\geq} P^{-\min_{\mathbf{r} \in
\mathcal{R}}{K-\mathbf{1}\cdot\mathbf{r}}}$ which combining with (\[eq:R\_ub\_ni\]), results into $\mathbb{P} \left\{ \mathcal{E}
\right\} \doteq P^{-\min_{\mathbf{r} \in
\mathcal{R}}{K-\mathbf{1}\cdot\mathbf{r}}}$. Defining $l(\mathbf{r})=K-\mathbf{1}\cdot\mathbf{r}$, we have to solve the following linear programming optimization problem $\min_{\mathbf{r}
\in \mathcal{R}}{l(\mathbf{r})}$. Notice that the region $\mathcal{R}$ is defined by a set of linear inequality constraints. To solve the problem, we have $$\begin{aligned}
l(\mathbf{r}) & \stackrel{(a)}{\geq} & \max \left\{0, K - \frac{NKr + r_K}{N}, K - \frac{NKr-\sum_{k=1}^{K-1}r_k}{N-1} \right\}\nonumber \\
& \stackrel{(b)}{\geq} & \max \left\{0, K(1-r)- \frac{1}{N}, K(1-r)-
\frac{K r}{N-1} \right\}.\end{aligned}$$ Here, (a) follows from the inequality constraint in (\[eq:R\_df\_ni\]) governing $\mathcal{R}$, and (b) follows from the fact that $r_K \leq 1$ and $\forall k<K: r_k \geq 0$. Now, we partition the range $0 \leq r \leq 1$ into three intervals. First, in the case that $r>1-\frac{1}{NK}$, the feasible point $\mathbf{r}=\mathbf{1}$ achieves the lower bound $0$. Second, in the case that $r<\frac{1}{K}-\frac{1}{NK}$, the feasible point $\mathbf{r}=\left(0,0,\cdots,0,\frac{NKr}{N-1}\right)$, achieves the lower bound $K(1-r)- \frac{K r}{N-1}$. Finally, in the case that $\frac{1}{K}-\frac{1}{NK} \leq r \leq 1-\frac{1}{NK}$, The lower bound $K(1-r)- \frac{1}{N}$ is achievable by the feasible point $\mathbf{r}, \forall k<K:~ r_k=\frac{NKr-N+1}{N(K-1)}, r_K=1$. Hence, we have $\min_{\mathbf{r} \in \mathcal{R}} l(\mathbf{r}) =
\max \left\{ 0, K(1-r)- \frac{1}{N}, K(1-r)- \frac{K r}{N-1}
\right\}$. This completes the proof.
*Remark -* It is worth noting that as long as the graph $G(V,
E)$ whose vertices are the relay nodes and edges are the non interfering relay node pairs includes a hamiltonian cycle [^2], the result of this subsection remains valid.
General Case
------------
In the general case, an interference term due to the neighboring relay adds at the receiver antenna of each relay. $$\mathbf{r}_k = h_k \mathbf{x} + i_{(k)} \mathbf{t}_{(k)} +
\mathbf{n}_k,$$ where $i_{(k)}$ is the interference link gain between the $k$’th and $(k)$’th relays. Hence, the amplification coefficient is bounded as $\alpha _k \leq \frac{P}{P \left( \left| h_k \right|^2 + \left|
i_{(k)} \right|^2 \right) + 1}$. Here, we observe that in the case that $\alpha_k >1$, the noise $n_k$ at the receiving side of the $k$’th relay can be boosted at the receiving side of the next relay. Hence, we bound the amplification coefficient as $\alpha_k = \min
\left\{ 1, \frac{P}{P \left( \left| h_k \right|^2 + \left| i_{(k)}
\right|^2 \right) + 1} \right\}$. In this way, it is guaranteed that the noise of relays are not boosted up through the system. This is at the expense working with the output power less than $P$. On the other hand, we know that almost surely [^3] $\left| h_k \right|^2 , \left| i_{(k)}
\right|^2 \dot{\leq} 1$. Hence, almost surely we have $\alpha _k
\doteq 1$. Another change we make in this part is that we assume that the entire time of transmission consists of $NK+1$ slots, and the transmitter sends the data during the first $NK$ slots while the relays send in the last $NK$ slots (from the second slot up to the $NK+1$’th slot). Hence, we have $T=(NK+1)T'$. This assumption makes our analysis easier and the lower bound on the diversity curve tighter. Now, we prove the main theorem of this section.
Consider a half-duplex multiple relays scenario with $K$ interfering relays whose gains are independent rayleigh fading variables. The proposed SM scheme achieves the diversity gain $$d_{SM,I}(r) \geq \max \left\{ 0, K \left( 1 - r \right) -
\frac{r}{N} \right\},$$ which achieves the optimum achievable DM tradeoff curve $d_{opt}(r)=K(1-r)$ as $N \to \infty$.
First, we show that the entire channel matrix acts as a lower triangular matrix. At the receiver side, we have $$\begin{aligned}
\mathbf{y}_{n,k} & = & g_k \mathbf{t}_{n, k} + \mathbf{z}_{n, k}
\nonumber \\
& = & g_k \alpha _{(k)} \left( \sum_{0 < n_1, k_1, n_1 (K + 1) + k_1
< n (K + 1) + k}{p_{n-n_1, k, k_1}\left( h_{k_1}\mathbf{x}_{n_1,
k_1} + \mathbf{n}_{n_1, k_1}\right) } \right) + \mathbf{z}_{n, k}\end{aligned}$$ Here, $p_{n, k, k_1}$ has the following recursive formula $p_{0, k,
k}=1, p_{n, k, k_1}=i_{(k)}\alpha_{(k)}p_{(n), (k), k_1}$. Defining the square $NK \times NK$ matrices as $\mathbf{G}= \mathbf{I}_N
\otimes \textit{diag}\left\{ g_1, g_2, \cdots ,g_K \right\}$, $\mathbf{H}= \mathbf{I}_N \otimes \textit{diag}\left\{ h_1, h_2,
\cdots ,h_K \right\}$, $\mathbf{\Omega} = \mathbf{I}_N \otimes
\textit{diag}\left\{ \alpha _1, \alpha _2, \cdots ,\alpha _K
\right\}$, and $$\mathbf{F}= \left(
\begin{array}{ccccc}
1 & 0 & 0 & 0 & \ldots \\
p_{0,2,1} & 1 & 0 & 0 & \ldots \\
p_{0,3,1} & p_{0, 3, 2} & 1 & 0 & \ldots \\
\vdots & \vdots & \vdots & \vdots & \ddots \\
p_{N-1, K, 1} & p_{N-1, K, 2} & \ldots & p_{0, K, K-1} & 1
\end{array},
\right)$$ where $\otimes$ is the Kronecker product[@matrix_book] of matrices and $\mathbf{I}_N$ is the $N \times N$ identity matrix, and the $NK \times 1$ vectors $\mathbf{x}\left(s\right)=[x_{1,1}(s),
x_{1, 2}(s), \cdots ,x_{N, K}(s)]^T$, $\mathbf{n}\left(s\right)=\left[n_{1,1}\left(s\right), n_{1, 2}(s),
\cdots ,n_{N, K}(s)\right]^T$, $\mathbf{z}\left(s\right)=[z_{1,2}(s), z_{1, 3}(s), \cdots ,z_{N+1,
1}(s)]^T$, and $\mathbf{y}\left(s\right)=[y_{1,2}(s), y_{1, 3}(s),
\cdots ,y_{N+1, 1}(s)]^T$, we have $$\mathbf{y}\left(s\right) = \mathbf{G} \mathbf{\Omega} \mathbf{F}
\left( \mathbf{H} \mathbf{x}\left(s\right) +
\mathbf{n}\left(s\right) \right) + \mathbf{z}\left(s\right).$$ Here, we observe that the matrix of the entire channel acts as a lower triangular matrix of a $NK \times NK$ MIMO channel whose noise is colored. The probability of outage of such a channel for the multiplexing gain $r$ is defined as $$\mathbb{P} \left\{ \mathcal{E} \right\}=\mathbb{P} \left\{ \log
\left|\mathbf{I}_{KN} + P \mathbf{H}_{T}\mathbf{H}_{T}^{H}\mathbf{P}_n^{-1} \right| \leq
(NK+1)r \log\left( P \right) \right\},$$ where $\mathbf{P}_N=\mathbf{I}_{NK}+\mathbf{G} \mathbf{\Omega}
\mathbf{F} \mathbf{F}^H \mathbf{\Omega}^H \mathbf{G}^H$, and $\mathbf{H}_T=\mathbf{G} \mathbf{\Omega} \mathbf{F} \mathbf{H}$. Assume $|h(k)|^2=P^{-\mu(k)}$, $|g(k)|^2=P^{-\nu(k)}$, $|i(k)|^2=P^{-\omega(k)}$, and $\mathcal{R}$ as the region in $\mathbb{R}^{3K}$ that defines the outage event $\mathcal{E}$ in terms of the vector $[\mathbf \mu, \mathbf \nu, \mathbf \omega]$, where $\mathbf{\mu}=\left[ \mu(1) \mu(2) \cdots \mu(K) \right]^T,
\mathbf{\nu}=\left[ \nu(1) \nu(2) \cdots \nu(K)
\right]^T,\mathbf{\omega}=\left[ \omega(1) \omega(2) \cdots
\omega(K) \right]^T$. The probability distribution function (and also the inverse of cumulative distribution function) decays exponentially as $P^{-P^{-\delta}}$ for positive values of $\delta$. Hence, the outage region $\mathcal R$ is almost surely equal to $\mathcal{R}_{+}=\mathcal{R} \bigcap \mathbb{R}_{+}^{3K}$. Now, we have $$\begin{aligned}
\mathbb{P} \left\{ \mathcal{E} \right\} & \stackrel{(a)}{\leq} &
\mathbb{P} \left\{ \left| \mathbf{H}_T \right|^2 \left| \mathbf{P}_n
\right|^{-1} \leq
P^{-NK \left( 1-r \right) +r}\right\} \nonumber \\
& \stackrel{(b)}{\leq} & \mathbb{P} \left\{ -N
\sum_{k=1}^{K}{\mu(k)+\nu(k)- \min \left\{ 0, \mu(k), \omega((k))
\right\}} + \right. \nonumber \\
&& \frac{NK\log(3) - \log \left| \mathbf{P}_{N} \right|}{\log \left(
P \right)} \leq -NK(1-r)+r
\Bigg\} \nonumber \\
& \stackrel{(c)}{\dot{\leq}} & \mathbb{P} \left\{ NK \frac{\log(3) -
\log (N^2K^2+1)}{\log (P)} + NK\left( 1-r \right) - r \leq N
\sum_{k=1}^{K}{\mu(k) + \nu(k)},\right. \nonumber \\
&& \mu(k),\nu(k),\omega(k) \geq 0 \Bigg\}. \label{eq:R_hat_wi}\end{aligned}$$ Here, (a) follows from the fact that for a positive semidefinite matrix $\mathbf A$ we have $\left| \mathbf{I} + \mathbf{A} \right|
\geq \left| \mathbf{A} \right|$, (b) follows from the fact that $$\alpha (k) = \min \left\{ 1, \frac{P}{P^{1-\mu (k)} + P^{1-\omega
((k))} + 1} \right\} \geq \frac{1}{3} \min \left\{ 1, P, P^{\mu
(k)}, P^{\omega ((k))} \right\} \nonumber$$ and assuming $P$ is large enough such that $P \geq 1$, and (c) follows from the fact that $\alpha (k) \leq 1$ and accordingly, $p_{n,k,k_1} \leq 1$, and knowing that the sum of the entries of each row in $\mathbf{F}\mathbf{F}^H$ is less than $N^2K^2$, we have[^4] $\mathbf{F}\mathbf{F}^H \preccurlyeq N^2K^2 \mathbf{I}_{NK}$, and $\mathbb P \left\{ \mathcal{R} \right\} \doteq \mathbb P \left\{
\mathcal{R}_{+} \right\}$, and conditioned on $\mathcal{R}_{+}$, we have $\min \left\{ 0, \mu(k), \omega ((k)) \right\} = 0$ and $\nu
(k) \geq 0$ and consecutively $\mathbf{P}_N \preccurlyeq (N^2K^2 +
1) \mathbf{I}_{KN}$.
On the other hand, we know for vectors $\mathbf{\mu}^0,
\mathbf{\nu}^0, \mathbf{\omega}^0 \geq \mathbf 0$, we have $\mathbb{P} \left\{\mathbf{\mu} \geq \mathbf{\mu}^0, \mathbf{\nu}
\geq \mathbf{\nu}^0, \mathbf{\omega} \geq \mathbf{\omega}^0 \right\}
\doteq P^{-\mathbf{1} \cdot \left( \mathbf{\mu}^0 + \mathbf{\nu}^0 +
\mathbf{\omega}^0 \right)}$. Similarly to the proof of Theorem 1, by taking derivative with respect to $\mathbf \mu, \mathbf \nu$ we have $f_{\mathbf \mu, \mathbf \nu}(\mathbf \mu, \mathbf \nu) \doteq P^{-
\mathbf 1 \cdot \left( \mathbf \mu + \mathbf \nu \right)}$ .Defining the lower bound $l_0$ as $l_0 = \frac{\log(3) - \log
(N^2K^2+1)}{\log (P)} + \left( 1-r \right) - \frac{r}{NK} $, the new region $\hat{\mathcal{R}}$ as $\hat{\mathcal{R}} = \left\{
\mathbf{\mu},\mathbf{\nu} \geq \mathbf{0}, \frac{1}{K} \mathbf{1}
\cdot \left( \mathbf{\mu} + \mathbf{\nu}\right) \geq l_0 \right\}$, the cube $\mathcal I$ as $\mathcal I = \left[0, Kl_0 \right]^{2K}$, and for $1 \leq i \leq 2K$, $\mathcal{I}_i^c=[0, \infty )^{i-1}
\times [Kl_0, \infty ) \times [0, \infty )^{2K-i}$, we observe $$\begin{aligned}
\mathbb{P} \left\{ \mathcal E \right\} & \stackrel{(a)}{\dot{\leq}}
& \mathbb{P} \{ \hat{\mathcal R} \} \nonumber \\
& \leq & \int_{\mathcal{\hat{R}} \bigcap \mathcal{I}}{f_{\mathbf
\mu, \mathbf \nu}\left(\mathbf \mu, \mathbf \nu \right) d\mathbf \mu
d \mathbf \nu} + \sum_{i=1}^{2K}{\mathbb{P} \left\{ [\mathbf \mu,
\mathbf \nu] \in \mathcal{\hat{R}} \cap \mathcal{I}_i^c \right\}}
\nonumber
\\
&\dot{\leq} & vol (\mathcal{\hat{R}} \cap \mathcal{I})
P^{-\min_{\left[ \mathbf{\mu}^0, \mathbf{\nu}^0 \right] \in
\mathcal{\hat{R}} \bigcap \mathcal{I}} \mathbf{1} \cdot \left(
\mathbf{\mu}^0 + \mathbf{\nu}^0 \right) } + 2K P^{-Kl_0}
\nonumber \\
& \stackrel{(b)}{\doteq} & P^{-Kl_0} \nonumber \\
& \doteq & P^{-\left[K \left( 1 - r \right) - \frac{r}{N} \right]}.
\label{eq:t2_r_wi}\end{aligned}$$ Here, (a) follows from (\[eq:R\_hat\_wi\]) and (b) follows from the fact that $\mathcal{\hat{R}} \bigcap \mathcal{I}$ is a bounded region whose volume is independent of $P$. (\[eq:t2\_r\_wi\]) completes the proof.
![DM Tradeoff for the proposed Switching Markovian Scheme and various values of (K,N), Interfering relays case[]{data-label="fig:dm_wi"}](dm_wi.eps)
*Remark -* The statement in the above theorem holds for the general case in which any arbitrary set of relay pairs are non-interfering. Hence, the proposed scheme achieves the upper-bound of the tradeoff curve in the asymptotic case of $N \to \infty$ for any graph topology on the interfering relay pairs.
Figure (\[fig:dm\_wi\]) shows the D-M tradeoff curve of the scheme for the case of interfering relays and varying number of $K$ and $N$.
[1]{} , “[Cooperative diversity in wireless networks: efficient protocols and outage behavior]{},” *IEEE Trans. Inform. Theory*, vol. 50, no. 12, pp. 3062–3080, Dec. 2004.
, “[On the Achievable Diversity-Multiplexing Tradeoff in Half-Duplex Cooperative Channels]{},” *IEEE Trans. Inform. Theory*, vol. 51, no. 12, pp. 4152–4172, Dec. 2005.
, “[Cooperative Wireless Systems: A Diversity-Multiplexing Tradeoff Perspective]{},” *IEEE Trans. Inform. Theory*, Aug. 2006, under Review.
, “[A Novel Two-Relay Three-Slot Amplify-and-Forward Cooperative Scheme]{},” *IEEE Trans. Inform. Theory*, vol. 51, no. 12, pp. 4152–4172, Dec. 2005.
T. M. Cover and J. A. Thomas, *Elements of Information Theory*.1em plus 0.5em minus 0.4emNew york: Wiley, 1991.
R. A. Horn and C. R. Johnson, *Matrix Analysis*.1em plus 0.5em minus 0.4emCambridge University Press, 1985.
[^1]: Financial supports provided by Nortel, and the corresponding matching funds by the Federal government: Natural Sciences and Engineering Research Council of Canada (NSERC) and Province of Ontario: Ontario Centres of Excellence (OCE) are gratefully acknowledged.
[^2]: By hamiltonian cycle, we mean a simple cycle $v_1v_2\cdots
v_K v_1$ that goes exactly one time through each vertex of the graph.
[^3]: By almost surely, we mean its probability is greater than $1-P^{-\delta}$, for all values of $\delta$.
[^4]: This can be verified by the fact that every symmetric real matrix $\mathbf{A}$ which has the property that for every $i$, $a_{i,i} \geq \sum_{i \neq j}|a_{i,j}|$ is positive semidefinite.
|
---
author:
- |
, Kazutaka Yamaoka, Mizuki Fukuyama, Takehiro G. Miyakawa and Atsumasa Yoshida\
Department of Physics and Mathematics, Aoyama Gakuin University, Sagamihara, Japan\
E-mail: , , , ,
- |
Jeroen Homan\
MIT Kavli Institute for Astrophysics and Space Reseach, Cambridge, USA\
E-mail:
title: 'RXTE spectra of the Galactic microquasar GRO J1655–40 during the 2005 outburst'
---
Introduction and Observations
=============================
The black-hole transient GRO J1655–40 is a well known Galactic superluminal jet source [@1][@2]. After its 1996–1997 outburst it remained in quiescence for more than 7 years, until *RXTE*/ PCA detected a rise in the X-ray flux on 2005 February 17 [@3]. Figure 1 shows [*RXTE*]{}/ASM light curves of the 1996–1997 and 2005 outbursts. The 2005 outburst lasted for 8 months and its maximum luminosity was about a factor of 1.4 higher than in the 1996–1997 outburst (which had a duration of 16 months).
We have analyzed 389 *RXTE*/PCA and HEXTE observations of GRO J1655–40, which were carried out between 2005 March 7 and 2005 September 17, and 80 observations from the period between 1996 May 9 and 1997 September 11. Spectra were extracted using FTOOLS v5.3.1. PCA spectra were produced using ’Standard 2 mode’ data from PCU2. We corrected not only for PCA dead-time but also for pile-up[^1] and a 2% systematic error was added. We found that pile-up effects could be as strong as $\sim$10%, but fitting parameters did not change significantly even for such high values. HEXTE spectra were produced from ’Archive mode’ data of cluster A (0–3) and cluster B (0,1,3). The PCA data were fitted between 3 and 20 keV and the HEXTE data between 17 and 240 keV (or lower energies, in case of low source counts).
![2-10 keV [*RXTE*]{}/ASM one-day averaged light curves for GRO J1655–40. In the 2005 outburst (right panel), the source returned to the hard state in about half the time compared to the 1996–1997 outburst. The colors of the bars near the top of the panels are explained in more detail in Section 2.[]{data-label="1"}](asm1655.eps){width="15.3cm"}
![Time history of the spectral parameters of GRO J1655–40 during the 1996–1997 (left) and 2005 (right) outbursts. From top to bottom: (1) the inner disk temperature $T_{\rm in}$, (2) the inner disk radius $R_{\rm in}$, (3) the photon index of the power-law component, $\Gamma$, and (4) the disk luminosity $L_{\rm disk}$ (filled star) and the luminosity of 3-100 keV power-law component $L_{\rm hard}$ (open square). Epochs were color-coded: *red* (Epoch 1), *green* (Epoch 2), *blue* (Epoch 3), *yellow* (Epoch 4), *aqua* (Epoch 5) in 2005 and *green* (Epoch 1), *blue* (Epoch 2), *aqua* (Epoch 3) in 1996–1997.[]{data-label="2"}](mjd-tin.eps "fig:"){width="15.2cm"} ![Time history of the spectral parameters of GRO J1655–40 during the 1996–1997 (left) and 2005 (right) outbursts. From top to bottom: (1) the inner disk temperature $T_{\rm in}$, (2) the inner disk radius $R_{\rm in}$, (3) the photon index of the power-law component, $\Gamma$, and (4) the disk luminosity $L_{\rm disk}$ (filled star) and the luminosity of 3-100 keV power-law component $L_{\rm hard}$ (open square). Epochs were color-coded: *red* (Epoch 1), *green* (Epoch 2), *blue* (Epoch 3), *yellow* (Epoch 4), *aqua* (Epoch 5) in 2005 and *green* (Epoch 1), *blue* (Epoch 2), *aqua* (Epoch 3) in 1996–1997.[]{data-label="2"}](mjd-rin.eps "fig:"){width="15.2cm"} ![Time history of the spectral parameters of GRO J1655–40 during the 1996–1997 (left) and 2005 (right) outbursts. From top to bottom: (1) the inner disk temperature $T_{\rm in}$, (2) the inner disk radius $R_{\rm in}$, (3) the photon index of the power-law component, $\Gamma$, and (4) the disk luminosity $L_{\rm disk}$ (filled star) and the luminosity of 3-100 keV power-law component $L_{\rm hard}$ (open square). Epochs were color-coded: *red* (Epoch 1), *green* (Epoch 2), *blue* (Epoch 3), *yellow* (Epoch 4), *aqua* (Epoch 5) in 2005 and *green* (Epoch 1), *blue* (Epoch 2), *aqua* (Epoch 3) in 1996–1997.[]{data-label="2"}](mjd-pi.eps "fig:"){width="15.2cm"} ![Time history of the spectral parameters of GRO J1655–40 during the 1996–1997 (left) and 2005 (right) outbursts. From top to bottom: (1) the inner disk temperature $T_{\rm in}$, (2) the inner disk radius $R_{\rm in}$, (3) the photon index of the power-law component, $\Gamma$, and (4) the disk luminosity $L_{\rm disk}$ (filled star) and the luminosity of 3-100 keV power-law component $L_{\rm hard}$ (open square). Epochs were color-coded: *red* (Epoch 1), *green* (Epoch 2), *blue* (Epoch 3), *yellow* (Epoch 4), *aqua* (Epoch 5) in 2005 and *green* (Epoch 1), *blue* (Epoch 2), *aqua* (Epoch 3) in 1996–1997.[]{data-label="2"}](mjd-l.eps "fig:"){width="15.2cm"}
Results
=======
We applied a simple phenomenological spectral model consisting of a multi-color disk black body and a power-law component. Following [@4] we added an absorption line at 6.8 keV and absorption edges at 7.7 keV, 8.8 keV, 9.3 keV and 10.8 keV to our model. The hydrogen column density ($N_{\rm H}$) was fixed to 7.1$\times 10^{21}$ atoms cm$^{-2}$. We were mainly interested in the evolution of the disk parameters and since no disk component could be detected in the hard state of GRO J1655–40, observations from that state were excluded from further consideration. Figure 2 shows the time evolution of the spectral parameters during both outbursts and Figure 3 shows typical PCA and HEXTE energy spectra of GRO J1655–40 for the 2005 outburst.
The values of $R_{\rm in}$ were corrected by $ R_{\rm in} =
\kappa^2\xi r_{\rm in} $ [@5], where $\kappa$ is a spectral hardening factor [@6], $\xi$ is a correction factor for the inner boundary condition [@7], and $r_{\rm in}$ is the apparent inner disk radius. Although the value of $\kappa$ depends on a luminosity, we followed [@5] and corrected the values of $R_{\rm in}$ with $\kappa=1.7$ and $\xi=0.41$ as an approximation. The values of $R_{\rm in}$ and $L_{\rm disk}$ are calculated for D = 3.2 kpc [@2] and an inclination of 70$^\circ$ [@8]. If D $\leq$ 1.7 kpc [@9], $R_{\rm in}$ and $L_{\rm disk}$ decrease by a factor of $\sim$53% and $\sim$30%, respectively. When $\Gamma \geq 3$ and the power-law component could not be well constrained (i.e. when it was weak), $\Gamma$ was fixed to 2.1, which was nominal soft-state value.
Based on the spectral fit results, the observations of the 2005 outburst could be divided into 5 Epochs (this excludes the hard state). Epoch 1: the hard component was strong and the disk component was underestimated ($R_{\rm in}$ was smaller than the value in Epoch 5, which is discussed below). Epoch 2: the spectra were very soft and the source was rarely detected with HEXTE. $R_{\rm in}$ was small ($\sim$ 10–18 km) and $T_{\rm in}$ was high ($\sim$ 1.4–1.5 keV). During the transitions from Epoch 1 to Epoch 2 and from Epoch 2 to Epoch 3 the hardness fluctuated, so sharp boundaries between these Epochs were difficult to define. Epoch 3: the luminosity reached its maximum of the 2005 outburst and power-law component increased dramatically. $R_{\rm in}$ and $T_{\rm in}$ exhibited large fluctuations. Epoch 4: the hard component was strong and similar to Epoch 1. Epoch 5: $R_{\rm in}$ was fairly constant at $\sim$26km and $T_{\rm in}$ changed in accordance with the changes of $L_{\rm disk}$. Although the hardness was not constant and there were days that $\Gamma$ was large (but not well constrained), the parameters of the disk component changed little and fitting was still acceptable when $\Gamma$ was fixed at 2.1. Since the disk component dominated the X-ray spectrum in Epoch 5, we consider the fitted disk parameters to be more reliable than in the other Epochs.
The 1996–1997 outburst could be roughly classified into 3 Epochs. Epoch 1, 2 and 3 in the 1996–1997 outburst were similar to Epoch 2, 3 and 5 in the 2005 outburst, respectively. GRO J1655–40 therefore passed through similar states in the roughly the same sequence in both outbursts. However, there was no period in the 1996–1997 outburst corresponding to Epoch 1 in the 2005 outburst, probably because the initial phase of the 1996–1997 outburst was not well covered by pointed [*RXTE*]{} observations. In addition, it is not clear whether the 1996–1997 outburst has a period similar to Epoch 4 in the 2005 outburst because the object went into the solar exclusion zone after Epoch 2 in the 1996–1997 outburst [@10].
Although we tried to add emission lines and/or absorption edges [@10][@11][@12], our fits for Epoch 2 in the 2005 outburst and Epoch 1 in the 1996–1997 outburst were poor ($\chi^2/d.o.f \sim$ 2-4). As can be seen from Figure 4, when the power-law component became strong, the disk component was underestimated and $R_{\rm in}$ decreases. However, the distribution of Epoch 2 in 2005 and Epoch 1 in 1996–1997 was quite different from other Epochs in the sense that they did not fall on the main branch traced out by the other Epochs. Perhaps the state of the accretion disk was different in these two Epochs and other disk models need to be applied.
Finally, Figure 5 shows the correlation between $T_{\rm in}$ and $L_{\rm disk}/L_{\rm E}$. $L_{\rm E}$ was calculated for the mass of the compact object $M =7M_\odot$ [@8]. By comparing the 2005 outburst with the 1996–1997 outburst, we find that both outbursts traced out very similar tracks and have nearly the same critical luminosity ($\sim0.2L_{\rm E}$) at which the source starts to leave the $L_{\rm disk} \propto T_{\rm in}^4$ relation \[5\]. This result indicates that there is likely a fixed physical process that causes the departure from the $L_{\rm disk} \propto T_{\rm in}^4$ relation in both outbursts.
: We would like to thank J. A. Tomsick for providing his PCA pile-up correction program. This report has made use of data obtained through the High Energy Astrophysics Science Archive Research Center on-line service, provided by NASA/Goddard Space Flight Center.
[99]{}
S. J. Tingay et al., *Relativistic motion in a nearby bright X-ray source*, Nature, 374, 141 (1995).
R. M. Hjellming, & M. P. Rupen, *Episodic ejection of relativistic jets by the X-ray transient GRO J1655–40*, Nature, 375, 464 (1995).
C. B. Markwardt, & J. H. Swank, *New Outburst of GRO J1655–40?*, ATel, 414 (2005).
K. Yamaoka, Y. Ueda, H. Inoue, F. Nagase, K. Ebisawa, T. Kotani, Y. Tanaka, & S. N. Zhang, *ASCA Observation of the Superluminal Jet Source GRO J1655–40 in the 1997 Outburst*, PASJ, 53, 179 (2001).
A. Kubota, K. Makishima, & K. Ebisawa, *Observational Evidence for Strong Disk Comptonization in GRO J1655–40*, ApJ, 560, 147 (2001) \[[astro-ph/0105426]{}\].
T. Shimura, & F. Takahara, *On the spectral hardening factor of the X-ray emission from accretion disks in black hole candidates*, ApJ, 445, 780 (1995).
A. Kubota, Y. Tanaka, K. Makishima, Y. Ueda, T. Dotani, H. Inoue, & K. Yamaoka, *Evidence for a Black Hole in the X-Ray Transient GRS 1009-45*, PASJ, 50, 667 (1998).
J. A. Orosz, & C. D. Bailyn, *Optical Observation of GRO J1655–40 in Quiescence. I. A. Precise Mass for the Black Hole Primary*, ApJ, 477, 876 (1997) \[[astro-ph/9610211]{}\].
C. Foellmi, E. Depagne, T. H. Dall, & I. F. Mirabel, *On the distance of GRO J1655–40*, A&A, 457, 249 (2006) \[[astro-ph/0606269]{}\].
G. J. Sobczak, J. E. McClintock, R. A. Remillard, C. D. Bailyn, & J. A. Orosz, **RXTE* Spectral Observations of the 1996–1997 Outburst of the Microquasar GRO J1655–40*, ApJ, 520, 776 (1999) \[[astro-ph/9809195]{}\].
C. Brocksopp et al., *The 2005 outburst of GRO J1655–40: spectral evolution of the rise, as observed by *Swift**, MNRAS, 365, 1203 (2006) \[[astro-ph/0510775]{}\].
G .Sala, J. Greiner, J. Vink, F. Haberl, E. Kendziorra, & X. L. Zhang, *The highly ionized disk wind of GRO J1655–40*, A&A, in press (2006) \[[astro-ph/0606272]{}\].
![Plot of $L_{\rm hard}$ vs $R_{\rm in}$. The data of the 1996–1997 outburst are plotted by open squares and the 2005 outburst are plotted by filled stars. The colors of the Epoch are same as Figure 2.[]{data-label="fig:two"}](spectra.eps){width="5.2cm"}
![Plot of $L_{\rm hard}$ vs $R_{\rm in}$. The data of the 1996–1997 outburst are plotted by open squares and the 2005 outburst are plotted by filled stars. The colors of the Epoch are same as Figure 2.[]{data-label="fig:two"}](rin-lhard.eps){width="7.5cm"}
![Correlation between $L_{\rm disk}/L_{\rm E}$ and $T_{\rm in}$. The plotted symbols are same as Figure 4 and the colors of the Epoch are same as Figure 2. []{data-label="5"}](tin-ldisk.eps){width="11cm"}
[^1]: From J. A. Tomsick and P. Kaaret (1998): http://astrophysics.gsfc.nasa.gov/xrays/programs/rxte/pca/.
|
---
abstract: |
We study the spectrum of metric fluctuation in $\kappa$-deformed inflationary universe. We write the theory of scalar metric fluctuations in the $\kappa-$deformed Robertson-Walker space, which is represented as a non-local theory in the conventional Robertson-Walker space. One important consequence of the deformation is that the mode generation time is naturally determined by the structure of the $\kappa-$deformation.
We expand the non-local action in $H^2/\kappa^2$, with $H$ being the Hubble parameter and $\kappa$ the deformation parameter, and then compute the power spectra of scalar metric fluctuations both for the cases of exponential and power law inflations up to the first order in $H^2/\kappa^2$. We show that the power spectra of the metric fluctuation have non-trivial corrections on the time dependence and on the momentum dependence compared to the commutative space results. Especially for the power law inflation case, the power spectrum for UV modes is weakly blue shifted early in the inflation and its strength decreases in time. The power spectrum of far-IR modes has cutoff proportional to $k^3$ which may explain the low CMB quadrupole moment.
author:
- 'Hyeong-Chan Kim'
- Jae Hyung Yee
- Chaiho Rim
title: 'Density fluctuations in $\kappa$-deformed inflationary universe'
---
Introduction
============
The history of the studies on the Cosmic Microwave Background (CMB) anisotropies and on the cosmological fluctuations is closely linked to that of the study of the standard cosmological model [@weinberg; @kolb]. We now have high resolution maps of the anisotropies in the temperature of the cosmic microwave background [@bennet], and its accuracy of the resolution is improving further. In relation to these observational data, overviews of the theory of cosmological perturbation applied to inflationary cosmology have been presented in Refs. [@branden; @giovan]. The cosmological observations reveal that the Universe has non-random fluctuations on all scales smaller than the present Hubble radius. In the most currently studied models of the very early universe it is assumed that the perturbations originate from quantum vacuum fluctuations, which was first proposed in a paper by Sakharov [@sakharov]. With this, the inflationary cosmology bears in it the ‘trans-Planckian problem": Since inflation has to last for long enough time to solve several problems of big-bang model and to provide a causal generation mechanism for CMB fluctuations, the corresponding physical wavelength of these fluctuations has to be smaller than the Planck length at the beginning of the inflation [@branden2]. Both of the theories of gravity and of matter break down at the trans-Planckian scale. Many methods have been proposed to cure the problem. The modification of the dispersion relation, which was used to study the thermal spectrum of black hole radiation [@unruh], was applied to cosmology [@martin]. Modifications of the evolution of cosmological fluctuations due to the string-motivated space-time uncertainty relations, $\delta x_{\rm phys}\delta t \geq l_s^2$, have been introduced by Brandenberger and Ho [@ho]. It was shown that the uncertainty relation plays a significant role in the spectrum of the metric fluctuation [@cai]. Greene et al. [@greene] proposed the initial states which give an oscillatory contribution to the primordial power spectrum of inflationary density perturbations. There have also been some attempts to explain the low CMB quadrupole moment contribution [@piao] by using the pre-Big Bang scenario in string theory [@veneziano]. The ambiguity of the action in the presence of a minimal length cutoff in inflation by the boundary terms are studied by Ashoorioon, Kempf, and Mann [@asho]. However, it is not easy to construct a consistent field theoretic model which satisfies both the stringy space-time uncertainty relation and the spatial homogeneity and isotropy of the Robertson-Walker space. A direct non-commutative deformation of the commutation relation, $$\begin{aligned}
\label{NCF}
[x^\mu, x^\nu]= i \theta^{\mu\nu},\end{aligned}$$ introduces a preferred direction in space, which breaks the isotropy of 3-dimensional space. Therefore, it would be interesting to construct a space-time non-commutative theory which keeps the spatial homogeneity and isotropy demanded by the Robertson-Walker space-time.
Much attention has been given on the possibility of explaining the observational data as a quantum gravitational effect. As a theoretical framework to study these quantum gravity effects phenomenologically “Doubly Special Relativity" (DSR, also called Deformed Special Relativity) was proposed by Amelino-Camelia [@giovanni], where there exist two relativistically invariant scales, the speed of light and the Planck scale, and extensive studies have been followed [@girelli; @okon; @freidel; @amelino; @livine; @amelino2]. Recently, it was argued that the coordinate space of the DSR theory defined in curved momentum space is described by the $\kappa-$Minkowski space. Therefore, a good candidate to study the quantum gravity effect to cosmology is to extend the $\kappa-$Minkowski theory to $\kappa-$Robertson-Walker Space ($\kappa-$RWS) and to study the effect of the deformation in the cosmological evolution. In $\kappa-$Minkowski space the space-time coordinates are non-commuting generators of a quantized Minkowski space-time. The $\kappa$-deformed Minkowski space-time introduces a dimensionful quantum deformation parameter, $\kappa$, which can be chosen to have the dimension of mass [@kosinski; @glikman; @daszkie]. A natural choice of this deformation parameter is the Planck mass $\kappa=M_P$. It is therefore important to construct consistent quantum field theoretic framework in $\kappa-$Minkowski space, and explore the physical effects [@amelino] in cosmological evolution. In this respect, Kowalski-Glikman [@kowalski] have studied the effects on the density fluctuations of the quantum $\kappa-$Poincaré algebra.
In Sec. II, we construct the theory of cosmological fluctuations in $\kappa$-Deformed Inflationary Universe ($\kappa$-DIU). Starting from the scalar-gravity theory in a flat Robertson-Walker space we briefly summarize the linearized theory of scalar metric fluctuations. After developing the $\kappa-$RWS, we write the theory of scalar metric fluctuations in the deformed space. We show that the scalar theory in $\kappa-$RWS space is described by a nonlocal field theory in the conventional Robertson-Walker space. The nonlocal action is series expanded in $H^2/\kappa^2$ in Sec. III, where $H$ is the Hubble parameter, and is quantized. In Sec. IV and V, we calculate the power spectra of scalar metric fluctuations for the cases of exponential and power law inflations, respectively. We show that the $\kappa-$deformation alters both the time dependence and the frequency dependence of the power spectrum nontrivially. In Sec. V, we summarize our results.
Density fluctuation in $\kappa$-deformed Robertson-Walker space-time
====================================================================
If the Universe is quantum mechanically created with vacuum energy dominance, it will inflate from the beginning. Even though there may be other choices for the pre-inflationary universes, we assume that the inflation starts from the beginning of the universe.
The calculations in this paper are carried out in 4-dimensional spatially flat Robertson-Walker metric, $$\begin{aligned}
\label{metric}
ds^2=-d t^2+a^2(t)(d r^2+ r^2 d\Omega^2),\end{aligned}$$ where $a(t)$ denotes the scale factor of expanding universe. For later use, we introduce the conformal time $\eta$ defined by $$\begin{aligned}
\label{eta}
\eta=\int \frac{dt}{a(t)}.\end{aligned}$$
The Einstein-Hilbert action for gravity coupled to scalar matter field is $$\begin{aligned}
\label{S:sG}
S=\int d^4 x \sqrt{-g}\left[-\frac{R}{16\pi
G}+\frac{1}{2}\partial_\mu \varphi \partial^\mu
\varphi-V(\varphi)\right],\end{aligned}$$ where $R$ is the Ricci curvature scalar. Starting from the action (\[S:sG\]), it was shown that the scalar and tensor parts of the linear metric fluctuation are described by the action (See Refs. [@ho; @giovan]), $$\begin{aligned}
\label{S:s}
S=\frac{1}{2}\int_{\bf k} d\eta \left[(\partial_\eta{v_{\bf
k}})^2+\left(\frac{\partial_\eta ^2 z}{z}-{\bf k}^2\right)
v_{\bf k}^2 \right],\end{aligned}$$ where $\displaystyle \int_{\bf k}\equiv \int \frac{d^3 {\bf
k}}{(2\pi)^3}$ and $$\begin{aligned}
\label{z}
z\equiv \frac{a(t)\dot \varphi_0}{H},\end{aligned}$$ with $\varphi_0$ being the scalar zero mode and $H=\dot a/a$ the Hubble parameter. It is noted that in cases of power law inflation and of slow roll inflation, $H$ is proportional to $\dot \varphi_0$, hence $z \propto a$. The action (\[S:s\]) can be cast into the covariant form $$\begin{aligned}
\label{S:s2}
S&=&-\frac{1}{2}\int d^4x \sqrt{-g}g^{\mu \nu}
\partial_\mu {\cal R}\partial_\nu{\cal R},\end{aligned}$$ where ${\cal R}$ is the gauge invariant metric fluctuation and plays the role of a massless scalar field in the inflating universe. The Fourier mode of ${\cal R}$, ${\cal R}_{\bf k}$, is related to the field by $v_{\bf k}=z {\cal R}_{\bf k}$. Therefore, its power spectrum is $$\begin{aligned}
\label{df:t}
P_{\bf k}=\frac{|{\bf k}|^3}{2\pi^2 z^2(t)} \langle 0|v_{\bf k}^2 |0
\rangle
.\end{aligned}$$
We now want to write the action (\[S:s2\]) in $\kappa-$RWS. For this purpose, we develop scalar field theory in $\kappa-$RWS in the following subsections. The $4$-dimensional field theory in $\kappa-$Minkowski space has been constructed in Ref. [@amelino; @girelli; @kosinski] and references therein. We briefly summarize the results obtained in these references in the next subsection.
$\kappa$-deformed Minikowski space-time
----------------------------------------
The $\kappa$-deformed Hopf algebra $H_x$ describing the $\kappa-$Minkowski space is generated by the coordinates $\hat
x_\mu$ determined by the following relations: $$\begin{aligned}
\label{algebra}
~[\hat x_0, \hat x_i]&=& \frac{i}{\kappa} \hat x_i,~~ [\hat x_i,
\hat
x_j]=0,~~\Delta(\hat x_\mu)=\hat x_\mu \otimes 1
+1 \otimes \hat x_\mu .\end{aligned}$$ The dual Hopf algebra $H_k$ of functions on $\kappa-$deformed four-momenta is described by the Hopf subalgebra of the $\kappa-$deformed Poincaré algebra as follows: $$\begin{aligned}
\label{algebra:p}
~[k_\mu, k_\nu]&=&0,~~\Delta(k_i)=k_i\otimes
e^{-k_0/\kappa}+1\otimes k_i,~~
\Delta(k_0)=k_0\otimes 1+1\otimes k_0 .\end{aligned}$$ The first Casimir operator of the algebra (\[algebra\]) and (\[algebra:p\]) is $$\begin{aligned}
\label{Casimir}
M^2=\left(2 \kappa \sinh \frac{k_0}{2 \kappa }\right)^2- {\bf k}^2
e^{k_0/\kappa} .\end{aligned}$$ It follows from this that for positive $\kappa$ the on-shell three-momentum is bounded from above by $$\begin{aligned}
\label{klimit}
{\bf k}^2 \leq \kappa^2,\end{aligned}$$ and the maximal value of momentum results in an infinite energy [@kow].
By using $\kappa-$deformed Fourier transform, the fields on $\kappa-$Minkowski space with non-commutative space-time coordinates $\hat x=(\hat x_0,\hat x_i)$ is written as ($k\hat x\equiv k_i \hat
x_i-k_0 \hat x_0$): $$\begin{aligned}
\label{fourier}
\Phi(\hat x) = \int_k \tilde \Phi(k)
:e^{i k \hat x}: ,\end{aligned}$$ where $\displaystyle \int_k \equiv\int \frac{d^4k}{(2\pi)^4}$ and $\tilde\Phi(k)$ is a classical function on commuting four-momentum space $k=(k_0,k_i)$ and the normal ordering is defined by $$\begin{aligned}
\label{NOrdering}
:e^{i k \hat x}:\equiv e^{-i k_0 \hat x_0} e^{i \bf k \hat x}.\end{aligned}$$ Multiplication of two normal ordered $\kappa-$deformed exponentials follows from Eqs. (\[algebra\]) and (\[NOrdering\]): $$\begin{aligned}
\label{normal ordering}
:e^{i k \hat x}::e^{i q \hat x}:= :e^{i ({\bf k}
e^{-q_0/\kappa}+ {\bf q}){\bf \hat x}-i (k_0+q_0) \hat x_0}: ,\end{aligned}$$ which follows from the four momentum addition rule described by the coproduct (\[algebra:p\]). From this we get the conjugate field, $$\begin{aligned}
\label{HC}
\Phi^\dagger(\hat x)&=& \int_k \tilde \Phi^\dagger (k) :e^{ik\hat
x}:,~~\tilde \Phi^\dagger({\bf k},k_0) = e^{3k_0/\kappa}
\tilde \Phi^*(-e^{k_0/\kappa} {\bf k},-k_0 ).\end{aligned}$$ The multiplication of fields can now be expressed as $$\begin{aligned}
\label{phi^2}
\int \Phi^2(\hat x) d^4x= \int_k \tilde\Phi(k)\tilde{\Phi}(-{\bf k}
e^{k_0/\kappa}, -k_0) .\end{aligned}$$
The differential calculus and its covariance properties under the action of $\kappa-$deformed Poincaré group have been constructed in Ref. [@kosinski; @daszkie]. The left or right partial derivatives $\hat\partial_A$ to define $\kappa$-deformed vector field are given by $$\begin{aligned}
\label{derivative}
\hat \partial_A \Phi(\hat x) =
:\chi_A\left(\frac{1}{i}\partial_\mu\right) \Phi(\hat x) : ,\end{aligned}$$ where $\chi_A :e^{ik\hat x}:= :\chi_A(k_\mu) e^{ik\hat x}:$ and $$\begin{aligned}
\label{vector}
\chi_i(k_\mu)= e^{k_0/\kappa} k_i, ~~\chi_0=\kappa \sinh
\frac{k_0}{\kappa} + \frac{{\bf k}^2}{2\kappa} e^{k_0/\kappa} .\end{aligned}$$ The adjoint derivative $\hat
\partial_A^\dagger$ can be defined to satisfy $$\begin{aligned}
\label{cond:unamb}
\int d^4\hat x \Phi_1(\hat x) \hat
\partial_0 \Phi_2(\hat x)= \int d^4\hat x [\hat\partial_0^\dagger
\Phi_1(\hat x)]\Phi_2(\hat x),\end{aligned}$$ which leads to $$\begin{aligned}
\label{dagger}
\hat \partial_\mu^\dagger \Phi(x) &=&
\chi_\mu ^\dagger(\partial_\mu/i)\Phi(\hat x)
,\end{aligned}$$ where $$\begin{aligned}
\label{..}
\chi_\mu^\dagger(k)=\chi_\mu(-e^{k_0/\kappa} {\bf k},-k_0).\end{aligned}$$ Based on these, the $\lambda \phi^4$ field theory was constructed in Ref. [@kosinski]. In the next subsection we generalize the formulation to the case of the Robertson-Walker space-time.
The scalar field theory in $\kappa$-deformed Robertson Walker space-time
------------------------------------------------------------------------
To generalize the field theories in the $\kappa$-Minkowski space to the curved space case, we must be careful in selecting the coordinates which satisfy the commutation relation (\[algebra\]). Since any non-decreasing reparametrization of $t$ is an equally good time coordinate in commutative space-time, it is important to choose the time coordinate for which the commutation relation, $$\begin{aligned}
\label{com:RW}
[\hat x_0, \hat x_i]=\frac{i}{\kappa} \hat x_i ,\end{aligned}$$ is imposed. We note that a natural time coordinate consistent with the commutation relation is the cosmological time $x_0=t$. This choice ensures the same form of commutation relation satisfied by the locally flat coordinates $(\hat t, \hat X_i=a(\hat t) \hat
x_i)$: $[\hat t,\hat X_i]= a(\hat t) [\hat t, \hat x_i]= i \hat
X_i/\kappa$. This simplicity cannot be attained for other choices of time coordinate. For example, consider a time coordinate $x_0$ defined by the $00$-part of the metric $g_{00}=-s^2(x_0)$. In the locally flat coordinates, the commutator $[s(\hat x_0) \hat
x_0,a(\hat x_0) \hat x_i]=a(\hat x_0)s(\hat x_0)[\hat x_0, \hat
x_i]+a(\hat x_0)[s(\hat x_0), \hat x_i] \hat x_0$ is not simply reduced to a well defined form of Eq. (\[algebra\]). In this sense the natural choice for the time coordinate is the cosmological time $t$ where $s(t)=1$. With the cosmological time $t$, the equations (\[algebra\])$\sim$(\[derivative\]) can be used without modification.
The generalization of $\kappa-$deformed vector fields in $\kappa-$RWS can be written as Eq. (\[derivative\]) with the operator $\chi_\mu$ defined by, $$\begin{aligned}
\label{vector}
\chi_i= e^{\partial_0/i\kappa} \frac{\partial_i}{i}, ~~\chi_0=\kappa
\sinh \frac{\partial_0}{i\kappa} - \frac{1}{2\kappa}
\partial_i e^{\partial_0/{2i\kappa}} g^{ij}
\partial_je^{\partial_0/{2i\kappa}} ,\end{aligned}$$ where the ordering of the time-dependent metric, $g^{ij}$, and derivatives are determined by demanding the adjoint derivatives to satisfy Eq. (\[cond:unamb\]), which gives $$\begin{aligned}
\label{chi:RW}
\chi_i^\dagger(k)&=&-k_i ,\\
\hat \partial_0^\dagger \Phi(\hat x)&=&-\kappa \sinh
\frac{\partial_0}{i\kappa} \Phi(\hat x)-\frac{1}{2\kappa}
e^{\partial_0/(2i\kappa)}\partial_i \left[ e^{\partial_0/(2i\kappa)}
\partial_j \Phi(\hat x) \right]g^{ij}(\hat t) .\end{aligned}$$ In addition to Eq. (\[cond:unamb\]) we demand the condition $$\begin{aligned}
\label{consisit}
\hat
\partial_0^\dagger \Phi^\dagger(\hat x)=(\hat
\partial_0 \Phi(\hat x))^\dagger ,\end{aligned}$$ to determine $\hat \partial_0^\dagger$.
Given the covariant derivatives and its adjoint derivatives, we can write the action of a massless scalar field in $\kappa-$RWS as $$\begin{aligned}
\label{S}
S&=&-\frac{1}{2}\int d^4x(\hat
\partial_\mu^\dagger \Phi^\dagger(\hat
x)) \sqrt{-g} g^{\mu \nu}(\hat t)
\hat \partial_\nu\Phi(\hat x) \\
&=&\frac{1}{2}\int d^4x \left[(\hat\partial_0^\dagger \Phi^\dagger(
\hat x)) a^3(\hat t)
\hat\partial_0\Phi(\hat x)-(\hat\partial_i^\dagger \Phi^\dagger(
\hat x)) a(\hat t)
\hat\partial_i\Phi(\hat x)\right], \nonumber\end{aligned}$$ where we choose the symmetric form in the action so that the metric dependent factor is placed in the middle of the operator products. It turns out that this choice gives the simplest form of the interaction between different modes. In addition we do not consider the change of measure [@moller] due to the complication of the non-commutative multiplication since what we are interested in in this paper is to understand the main feature of the $\kappa-$deformation on the metric fluctuation.
Using Eqs. (\[fourier\]), (\[normal ordering\]), (\[HC\]), (\[vector\]), and (\[chi:RW\]) we obtain for the action $$\begin{aligned}
\label{S:nl}
S &=&\frac{\kappa^2}{4}\int_{\bf k}dt\left\{ g({\bf k},t)\tilde
\Phi_{-\bf k}(t-
\frac{i}{\kappa})\tilde \Phi_{\bf k}(t+\frac{i}{\kappa})
-a^3(t)\left[\rho_3(t)+g_3(t) \bar k^2(t)\right]
\tilde \Phi_{-\bf k}(t)
\tilde \Phi_{\bf k}(t)\right\} ,\end{aligned}$$ where $\bar k$ denotes the relative ratio between the physical momentum ($|{\bf k}|/a$) and the non-commutative scale ($\displaystyle \bar k(t)=\frac{|{\bf k}|}{a(t) \kappa}$) and the coefficients $g({\bf k},t)$, $g_i(t)$ and $\rho_n(t)$ are given by $$\begin{aligned}
\label{rhos}
g({\bf k},t) &=& a^3(t)\left[1+(g_{1}(t)-2)\bar k^2
+\frac{g_2(t)}{2}\bar k^4\right]
, \\
g_1(t)&=& \frac{1}{2}\left[\frac{a^2(t)}{a^2(t+i/(2\kappa))}
+\frac{a^2(t)}{a^2(t-i/(2\kappa))}\right]
, \nonumber \\
g_2(t)&=& \frac{a^4(t)}{a^2(t+i/(2\kappa))a^2(t-i/(2\kappa))}
, \nonumber \\
g_3(t) &=& \frac{1}{2}\left[
\frac{a^3(t-i/\kappa)}{a(t)a^{2}(t-i/(2\kappa))}+
\frac{a^3(t+i/\kappa)}{a(t)a^{2}(t+i/(2\kappa))}\right]
, \nonumber \\
\rho_n(t)&=&
\frac{1}{2}\left[\frac{a^n(t-i/\kappa)}{a^n(t)}
+\frac{a^n(t+i/\kappa)}{a^n(t)}\right]
.\nonumber\end{aligned}$$ Note that the left hand sides of Eq. (\[rhos\]) are defined to satisfy $\displaystyle \lim_{\kappa t\rightarrow \infty} g_i= 1=
\lim_{\kappa t\rightarrow \infty} \rho_n$. The action (\[S:nl\]) is highly non-local in that the fields are non-locally multiplied in the action, in addition to the nonlocal coupling between the background metric and the field modes. Each mode of the field, $\tilde \Phi_{\pm \bf k}$, is diagonalized so that it is not coupled to other modes of different $\bf k$.
First order approximation and the Hamiltonian formulation
=========================================================
It is not possible to solve the nonlocal equation of motion derived from the action (\[S:nl\]) exactly. The canonical formalism for Lagrangians with non-locality of finite extent has been proposed by Woodard [@woodard]. However, we do not follow the formalism since our purpose is to obtain information on how the non-commutativity (\[com:RW\]) affects the evolution of the metric fluctuation in inflationary Universe in a simple calculable form. Instead, we use a perturbative expansion in the parameter $\bar
H^2\equiv H^2/\kappa^2$ and construct the Hamiltonian for the action up to the first order in $\bar H^2$.
To have an approximation of the action (\[S:nl\]), we expand the integrand in $\bar H^2$ as $$\begin{aligned}
\label{fPhi:red}
g(t)\Phi(t+\frac{i}{\kappa})\Phi(t-\frac{i}{\kappa})&=&g_S^{(0)}(t)
\Phi(t)^2+\frac{2 g_A^{(-1)}(t)}{\kappa^2} \dot
\Phi^2(t)+\left[6(g_S^{(-4)}-g^{(-4)}) \right. \\
&&\left.+\kappa^{-2}\left(8g_A^{(-3)}
-4 g_S^{(-2)}-g^{(-2)}\right)-\frac{g}{4\kappa^4}
\right] \ddot \Phi^2(t)+O(\bar H^6
) \nonumber,\end{aligned}$$ where $g^{(-n)}(t)$ denotes the $n^{\rm th}$ indefinite integrals of $g(t)$, and $$\begin{aligned}
\label{sa}
g^{(-n)}_S(t)\equiv
\frac{1}{2}[g^{(-n)}(t+i/\kappa)+g^{(-n)}(t-i/\kappa)],
~~g^{(-n)}_A(t)\equiv\frac{\kappa}{2i}\left[g^{(-n)}(t+i/\kappa)
-g^{(-n)}(t-i/\kappa)\right]
.\end{aligned}$$
From Eqs (\[S:nl\]) and (\[fPhi:red\]), we get the action up to the order $O(\bar H^2)$, $$\begin{aligned}
\label{S:2}
S&=&\frac{1}{2}\int_{\bf k}dt~ a^3(t)\left\{ \mu({\bf k},t)
\dot{\tilde \Phi}_{\bf k}(t) \dot{\tilde\Phi}_{-\bf k}(t)
- \frac{{\bf k}^2}{a^2(t)}\left(g_4-
\frac{g_5 \bar k^2}{4}
\right) \tilde\Phi_{\bf k} \tilde\Phi_{-\bf k}
+ \frac{\gamma({\bf k},t)}{3\kappa^2}\ddot{\tilde\Phi}_{\bf k}
\ddot{\tilde\Phi}_{-\bf k} + \cdots
\right\} ,\end{aligned}$$ where the coefficients are given by $$\begin{aligned}
\label{coef}
\mu({\bf k},t) &=&a^{-3}(t) g_A^{(-1)}({\bf k},t),\\
g_4(t) &=& \rho_1+\frac{1}{4}\left\{g_3- \frac{1}{2}
\left[\frac{a^3(t+i/\kappa)}{a(t)a^2(t+3i/(2\kappa))}+
\frac{a^3(t-i/\kappa)}{a(t)a^2(t-3i/(2\kappa))}
\right]\right\},\nonumber \\
g_5(t)
&=&\frac{1}{2}\left[\frac{a(t)a^3(t+i/\kappa)}{a^2(t+i/2\kappa)
a^2(t+3i/(2\kappa))}+
\frac{a(t)a^3(t-i/\kappa)}{a^2(t-i/2\kappa)
a^2(t-3i/(2\kappa))}
\right], \nonumber \\
\gamma({\bf k},t)&=& \frac{3 }{2a^3(t)}
\left[-6\kappa^4(g^{(-4)}-g_S^{(-4)})+\kappa^{2}\left(8g_A^{(-3)}
-4 g_S^{(-2)}-g^{(-2)}\right)-\frac{g}{4}\right] .\nonumber\end{aligned}$$ The asymptotic values of these coefficient functions are $$\begin{aligned}
\label{g,m:asym}
g_4(\infty)=g_5(\infty)=\mu(0,\infty)=\gamma(0,\infty)=1 ,\end{aligned}$$ which make it easier to guess the asymptotic behaviors of the coefficient functions for large $t$.
For notational simplicity, we use the change of variables $\Phi_{{\bf k},+}=\frac{1}{2}(\tilde\Phi_{\bf k}+\tilde\Phi_{-\bf
k}),~~ \Phi_{{\bf k},-}=\frac{i}{2}(\tilde\Phi_{\bf
k}-\tilde\Phi_{-\bf k})$, to write the action in a diagonal form: $$\begin{aligned}
\label{S:+-}
S&=&\frac{1}{2}\int_{\alpha}dt ~a^3(t)\left\{\mu({\bf k},t)
\dot\Phi^2_{\alpha}(t) - \frac{{\bf k}^2}{a^2}
\left(g_4-\frac{g_5 \bar k^2}{4}
\right) \Phi^2_{\alpha}
+
\frac{\gamma({\bf k},t)}{3\kappa^2}\ddot{\Phi}^2_{\alpha}+\cdots\right\}
,\end{aligned}$$ where $\alpha = ({\bf k}, \pm)$. Note that the coefficients $\mu$, $\gamma$, and $g_i$ are exactly calculable once $a(t)$ is given.
Introducing the conformal time $\eta$ and the rescaling of the field, $\phi_{\alpha}=a(t(\eta))\Phi_\alpha$, we reduce the action (\[S:+-\]) into the form, up to the order $\bar H^2$, $$\begin{aligned}
\label{S:+-2}
S&=&\frac{1}{2}\int_{\alpha}d\eta \left\{\bar \mu
(\partial_\eta \phi_{\alpha})^2 - \omega({\bf k},t)\phi^2_{\alpha}
+ \frac{\gamma({\bf k},t)}{3\kappa^2a^2(t)}
(\partial_\eta^2\phi_{\alpha})^2
\right\},\end{aligned}$$ where $$\begin{aligned}
\label{omega}
\bar \mu &=& \mu +\frac{4\gamma H^2}{3\kappa^2}\left(1+\frac{5\dot
H}{4H^2}+\frac{3\dot\gamma}{4H\gamma}\right) ,\\
\omega({\bf k},t)&=&k^2\left(g_4-\frac{g_5\bar k^2}{
4}
\right)-\frac{\partial_\eta(A^2H)}{a}+\frac{\partial_\eta^2(
\gamma \partial_\eta H)}{3 \kappa^2 a}, \\
A^2&=& a^2(\eta)\left[\mu({\bf k},t)+\frac{\partial_\eta(\gamma
\partial_\eta a)}{3 \kappa^2 a^3}\right] . \nonumber\end{aligned}$$ Note that the action (\[S:+-2\]) contains a higher derivative term, which may lead to nonunitary evolution of the system. Since this higher derivative term is a term of order $\bar H^2$ and our purpose in this paper is to obtain the effect of the deformation on the cosmological evolution up to the $1^{\rm st}$ order in $\bar
H^2$, we require that the higher order derivative term is written as a function of the field, its first time derivative, and time: $$\begin{aligned}
\label{ass:H}
\partial_\eta^2\phi= \Psi(\phi, \partial_\eta \phi, \eta) .\end{aligned}$$ Explicitly, we use the linearized approximation, $$\begin{aligned}
\label{ddphi:01}
\partial_\eta^2\phi= \frac{a(\eta)}{\gamma^{1/2}({\bf k},\eta)}
[c(\eta) \phi +d(\eta) \partial_\eta\phi],\end{aligned}$$ where the coefficients $c$ and $d$ are to be determined by consistency.
This requirement is equivalent to the perturbative calculation up to the 1$^{st}$ order in $\bar H^2$. This can be shown as follows: The equation of motion for $\phi=\phi_0+ \frac{1}{\kappa^2}
\phi_1+\cdots $ can be written as $$\begin{aligned}
\label{eom0,1}
\partial_\eta (\bar \mu \partial_\eta \phi_0)+\omega \phi_0 &=&0, \\
\partial_\eta (\bar \mu \partial_\eta \phi)+\omega \phi&=&
\frac{1}{3\kappa^2} \partial_\eta^2\left(\frac{\gamma}{a^2}
\partial_\eta^2 \phi_0\right) , \nonumber\end{aligned}$$ where the first equation is the 0$^{\rm th}$ order equation and the second is the full equation written explicitly up to the 1$^{\rm
st}$ order. The first equation of Eq. (\[eom0,1\]) defines $\partial_\eta^2 \phi_0$ as a linear function of $\partial_\eta
\phi_0$ and $\phi_0$. Then, the second line can be understood as a defining equation of $\partial_\eta^2\phi$ as a linear function of $\partial_\eta \phi$ and $\phi$ up to $O(\bar H^2)$, which is Eq. (\[ddphi:01\]).
With this reasoning and Eq. (\[ddphi:01\]), the action (\[S:+-2\]) is perturbatively equivalent up to $\bar H^2$ to the following unitary action $$\begin{aligned}
\label{S:fi}
S=\frac{1}{2} \int_{\alpha} d\eta \left[m (\partial_\eta
\phi_{\alpha})^{2}-
f(k,\eta) \phi_{\alpha}^2\right],\end{aligned}$$ where $k=|{\bf k}|$ and we have $$\begin{aligned}
\label{bmu0}
m&=& \bar \mu +\frac{d^2}{3\kappa^2},\\
f &=&\omega({\bf k},t)-\frac{c^2-\partial_\eta(c d)}{3\kappa^2}.
\nonumber\end{aligned}$$ Substituting Eq. (\[ddphi:01\]) into (\[S:+-2\])and requiring the resultant action to be the same as the action (\[S:fi\]) with (\[ddphi:01\]) as its equation of motion, we find $$\begin{aligned}
\label{cd}
c=-\frac{f\gamma^{1/2}}{m a},~~d=-\frac{\dot{m} \gamma^{1/2}}{m } .\end{aligned}$$ Eqs. (\[bmu0\]) and (\[cd\]) imply that $m$ and $f$ satisfy the following differential equations: $$\begin{aligned}
\label{bmu}
m&=& \bar\mu +\frac{\gamma}{3\kappa^2} \frac{(\dot{ m})^2}{m^2}
,\\
\frac{f^2}{3\kappa^2 m^2 a^2}+ f &=&\omega({\bf k},t)
+\frac{a(t)}{3\kappa^2}\frac{d}{dt}\frac{f\gamma
\dot{m}}{a m^2} . \nonumber\end{aligned}$$ Note that these conditions make the action to be a functional of $\partial_\eta \phi$ and $\phi$. The time evolution for the theory (\[S:fi\]) is unitary and quantum mechanically well defined.
We introduce mode dependent conformal time $\eta_k$ by $$\begin{aligned}
\label{etak}
d\eta_k= m^{-1}(k,\eta) d\eta=\frac{dt}{a(t)m(k,t)}.\end{aligned}$$ Then the action (\[S:fi\]) can be written in a simplified form: $$\begin{aligned}
\label{S:final}
S=\frac{1}{2} \int_{\alpha} d\eta_k \left[\phi_{\alpha}'^{2}-
\Omega^2_k(\eta_k) \phi_{\alpha}^2\right],\end{aligned}$$ where $'$ denotes the derivative with respect to $\eta_k$ and $$\begin{aligned}
\label{Om:mf}
\Omega_k^2(\eta_k)\equiv m(k,\eta_k) f(k,\eta_k) .\end{aligned}$$ The Hamiltonian for mode $\alpha$ is the same as that of the time-dependent harmonic oscillator with frequency squared $\Omega_k^2$: $$\begin{aligned}
\label{H}
H_\alpha= \frac{\hat\pi_\alpha^2}{2}+ \frac{1}{2} \Omega_k^2(\eta_k)
\hat \phi_{\alpha}^2 .\end{aligned}$$ The time evolution of each mode can be described by introducing invariant creation and annihilation operators [@lvn], $$\begin{aligned}
\label{Adag}
\hat A_\alpha=-\frac{i}{\hbar^{1/2}} (\varphi_\alpha^* \hat
\pi_\alpha-{\varphi_\alpha^*}' \hat \phi_\alpha ),~~
\hat A_\alpha^\dagger=\frac{i}{\hbar^{1/2}} (\varphi_\alpha \hat
\pi_\alpha-{\varphi_\alpha}' \hat \phi_\alpha ) ,\end{aligned}$$ where $\varphi_\alpha$ is the mode solution of the differential equation (\[diff\]) below and $\hat A_\alpha$ and $\hat
A_\alpha^\dagger$ satisfy the Liouville-von Neumann equation, $$\begin{aligned}
\label{Lvn}
i\hbar \partial_{\eta_k} \hat A_\alpha +[\hat A_\alpha, H_\alpha]=0
.\end{aligned}$$ One may invert Eq. (\[Adag\]) to construct the field operator in terms of the creation and annihilation operators as $$\begin{aligned}
\label{inv:fi}
\hat \phi_\alpha &=& \hbar^{1/2} \left[\varphi_\alpha(\eta_k) \hat
A_\alpha
+\varphi_\alpha^*(\eta_k) \hat A_\alpha^\dagger \right],\\
\hat \pi_\alpha &=& \hbar^{1/2} \left[\varphi_\alpha'(\eta_k) \hat
A_\alpha
+{\varphi_\alpha^*}'(\eta_k) \hat A_\alpha^\dagger \right].\nonumber\end{aligned}$$ Note also that the Liouville-von Neumann equation is equivalent to the following differential equation for the coefficients $\varphi_\alpha$, $$\begin{aligned}
\label{diff}
\varphi_\alpha''(\eta_k) +\Omega^2_k(\eta_k)
\varphi_\alpha(\eta_k)=0 .\end{aligned}$$ The commutation relation $[\hat A_\alpha, \hat A_\beta^\dagger
]=\delta_{\alpha\beta}$ restricts the mode solution $\varphi_\alpha$ to satisfy $\varphi_\alpha {\varphi_\alpha^*}'-\varphi_\alpha'
\varphi_\alpha^*=i$.
We present the first order approximation of $m$ and $f$ for later use. To first order in $1/\kappa^2$, Eq. (\[bmu\]) gives $$\begin{aligned}
\label{dm}
m&\simeq & \bar\mu +\frac{\gamma}{3\kappa^2} \frac{(\dot{ \bar
\mu})^2}{\bar \mu^2}
\simeq \bar \mu ,\\
f&=&2\omega({\bf k},t)\left[1+
\left(1+\frac{4\omega}{3\kappa^2m^2a^2}\right)^{1/2}
\right]^{-1}\simeq \omega
\left(1-\frac{\omega}{3\kappa^2 m^2 a^2}\right). \nonumber\end{aligned}$$ Using the explicit form for $\omega$ and $\mu$, we have $$\begin{aligned}
\label{f0}
m &\simeq& \bar \mu
\simeq 1-\frac{\bar H^2}{6}(1-7\epsilon_1)
-\bar k^2, \\
f&\simeq &k^2\left[1+\frac{\bar H^2}{3}(1-\epsilon_1)
-\frac{7}{12}\bar k^2\right]-2 H^2 a^2\left[1
+\frac{\epsilon_1}{2}-\frac{\bar H^2}{6}
\left(1-16\epsilon_1-\epsilon_1^2+\frac{\epsilon_2}{2}
+\epsilon_3\right )\right]
,
\nonumber\end{aligned}$$ where $\displaystyle \epsilon_n = \frac{H^{(n)}}{(H)^{n+1}}$, with $\displaystyle H^{(n)}= \frac{d^n H}{dt^n}$, are constant numbers for power law inflation and vanish for exponential inflation. From these we have $$\begin{aligned}
\label{Om:gen}
m^{-1}(k,\eta) &\simeq & 1+
\alpha_n\frac{H^2}{\kappa^2}+\frac{k^2}{a^2
\kappa^2} +\cdots , \\
\Omega^2_k(\eta_k)& \simeq & k^2(1-w_1 \bar H^2-w_2 \bar k^2)-
\left(2+\epsilon_1\right) H^2 a^2(\eta)(1-w_3\bar H^2)+\cdots .
\nonumber\end{aligned}$$ For the power law inflation, the values of $\alpha_n$ and $w_i$ are given by $$\begin{aligned}
\label{ws}
\alpha_n=\frac{n+7}{6n},~~ w_1=\frac{1}{6}(13-11/n),~~w_2=19/12,~~
w_3=\frac{1}{3(1-1/(2n))}\left(1+\frac{45}{4n}-\frac{7}{4n^2}
-\frac{3}{n^3}\right) ,\end{aligned}$$ and for the exponential inflation their values are given by the limits $n \rightarrow \infty$ of Eq. (\[ws\]).
metric fluctuations in $\kappa-$DIU: The exponential inflation
==============================================================
The simplest inflationary model is the exponential inflation, in which the scale factor $a(t)$ increases as, $$\begin{aligned}
\label{a:t}
a(t)= a_0 e^{H t} , ~~~-\infty < t <\infty .\end{aligned}$$ Here $a_0$ is the scale factor at $t=0$ and $H$ is the Hubble constant. Using the conformal time $\eta$, we get $$\begin{aligned}
\label{a}
Ht=-\ln (-a_0 H \eta),~~ a(\eta)=\frac{1}{-H\eta} ,\end{aligned}$$ where the conformal time $\eta$ varies from $-\infty$ to $0$ as $t$ varies from $-\infty$ to $\infty$. From Eqs. (\[coef\]) and (\[omega\]), we have $$\begin{aligned}
\label{mu}
\bar \mu &=& \frac{\sin 3
\bar H}{3 \bar H}+\frac{4}{3}\bar H^2
\xi(3\bar H)
+ \left(-\frac{\sin \bar H}{\bar H}+\frac{2}{3}\bar H^2
\xi(\bar H)
\right)\left[2-\cos \bar H-\frac{\bar k^2}{2}\right]\bar
k^2,\\
\gamma({\bf k},t)
&=&\xi(3 \bar H)- \xi(\bar H)
\left[2-\cos \bar H-\frac{\bar k^2}{2}\right]\bar
k^2, \nonumber\end{aligned}$$ where $\displaystyle \xi(x)=3\left[-3 \frac{1-\cos
x}{x^4}+4\frac{\sin x}{x^3}-\frac{4\cos
x+1}{2x^2}-\frac{1}{8}\right] \simeq 1-\frac{13}{80} x^2 +\cdots$.
Since the action (\[S:final\]) is obtained in the $\bar H^2$ expansion, the normalized Hubble constant, $\bar H=H/\kappa$ is assumed to be smaller than one. Moreover, the Eq. (\[k:max1\]) below restricts $\bar k=k/(a(t)\kappa)$ to be smaller than one. Then we get $m(k,\eta)$ from the differential equation (\[bmu\]) and (\[mu\]) by series expansion in $\bar H^2$ and $\bar k$, $$\begin{aligned}
\label{m:s}
m \simeq \bar \mu+\frac{\bar H^2}{3}\frac{\dot {\bar \mu}^2}{H^2\bar
\mu^2} \simeq \left(1-\frac{\bar H^2}{6}\right)(1-\bar H^2 k^2
\eta^2) +O(\bar H^4) .\end{aligned}$$ By integrating $m^{-1}$ over $\eta$ using Eq. (\[etak\]), we get the mode-dependent conformal time $\eta_k$, $$\begin{aligned}
\label{eta:eta}
\eta_k \simeq\frac{1}{2\bar H k(1- \bar H^2/6)}
\ln \left(\frac{1+ \bar H k \eta}{1-
\bar H k \eta}\right)
= \frac{\eta}{1-\bar H^2/6}\left(1+\frac{\bar H^2}{3}
k^2 \eta^2+\cdots \right),\end{aligned}$$ which is normalized so that $\eta_k=0$ at $\eta=0$. A crucial point is that there is a global rescaling of the conformal time due to the non-commutative effect.
From Eq. (\[Om:gen\]), the effective frequency squared $\Omega^2_k$ is written as $$\begin{aligned}
\label{Om:22}
\Omega^2_k(\eta_k)&= & k^2(1-w_1 \bar H^2-w_2 \bar k^2)-
2H^2 a^2(\eta)(1-w_3 \bar H^2)+\cdots \\
&\simeq& \tilde k^2-\frac{\nu^2-1/4}{\eta_k^2}
-w_2\bar H^2 k^4 \eta_k^2, \nonumber\end{aligned}$$ where $w_1=13/6$, $w_2= 19/12$, $w_3=1/3$, and $$\begin{aligned}
\label{nu,k}
\tilde k^2 \equiv k^2\left[1-(w_1+2/3)\bar H^2\right],
~~\nu^2-1/4\equiv\frac{2(1-w_3 \bar H^2)}{(1-\bar H^2/6)^2}\simeq 2.\end{aligned}$$ The explicit value of $\nu$ is $\nu=3/2$ to this order, which is the same as the commutative space result. We note that both the frequency and the mode solution have corrections from non-commutativity for large $|\eta_k|$. The last term of the second line in Eq. (\[Om:22\]) becomes negligible for $\eta_k\sim 0$.
Mode generation and the initial condition
-----------------------------------------
In the $\kappa-$RWS, spatial momentum is also restricted similarly as in (\[klimit\]). In terms of comoving momentum $k=|{\bf k}|$, we have $$\begin{aligned}
\label{k:max0}
\frac{k}{a(t)} \leq \kappa .\end{aligned}$$ This gives the upper bound of $k$ $$\begin{aligned}
\label{k:max1}
k \leq k_{max}(t)\equiv \kappa a(t) .\end{aligned}$$ The maximal value of the wave-number (\[k:max1\]) is very similar to that used by Brandenberger and Ho [@ho] except for the fact that the maximal value used in Ref. [@ho] is determined by an effective scale factor modified by the Moyal star product in the action.
Since $m$ is positive definite, the mode-dependent conformal time $\eta_k$, Eq. (\[etak\]), is well defined and is an increasing function of $t$. The relation (\[k:max1\]) implies that for a given $k$, there exists a conformal time $\eta_k^0$, the time saturating the relation (\[k:max1\]): $$\begin{aligned}
\label{eta:0}
a(t(\eta_k^0)) \equiv \frac{k}{\kappa}.\end{aligned}$$ Then, the mode $\phi_k$ cannot exist before $\eta_k^0$. In other words, $\eta_k^0$ is the generating time of the mode $\varphi_k$. This provides a hint to one of the major issues in which state the fluctuations are generated. To satisfy the continuity of the number of quanta of the $k$ mode when the mode becomes physical at $\eta_k^0$, it must be in the adiabatic vacuum state. This vacuum state can be chosen to be the WKB mode solution, $$\begin{aligned}
\label{ini:ex}
\lim_{\eta_k \rightarrow \eta_k^0}\varphi_k(\eta_k)=\frac{1}{
\sqrt{2 \Omega_k(\eta_k)}}\exp i\left(
\int_{\eta_k^0}^{\eta_{k}}\Omega_k d\eta_k + \psi_k\right)
,\end{aligned}$$ where the constant phase $\psi_k$ can be chosen conveniently.
inflationary evolution
----------------------
The inflationary evolution of the mode solution is determined by identifying $\Omega_k^2$ of Eq. (\[Om:22\]). The corresponding commutative space values can be obtained by setting $\bar H=0$ in Eq. (\[Om:22\]). With the $\Omega_k^2$ we have two different time scales $\eta_k^c$ and $\eta_k^i$ defined by $\Omega'_k(\eta_k^c)=0$ and $\Omega^2_k(\eta_k^i)=0$, respectively. These time scales are given by $$\begin{aligned}
\label{eta:bc}
\eta_k^c=-\frac{(\nu^2-1/4)^{1/4}}{k w_2^{1/4} \bar H^{1/2}} ,~~
\eta_k^i \simeq-\frac{(\nu^2-1/4)^{1/2}}{k}\left(1+
\frac{(\nu^2-1/4)w_2\bar
H^2}{2}+\cdots \right),\end{aligned}$$ where $\Omega_k^2(\eta_k)$ increases while $\eta_k< \eta_k^c$ and decreases later as shown in Fig. 1. It is positive definite when $\eta_k< \eta_k^i$ and negative later. For $\eta_k> \eta_k^i$, the modulus of $\phi_k$ increases in time. By Eq. (\[eta:0\]), the mode $\phi_k$ is generated at $\eta_k^0\simeq -(k \bar H)^{-1}$. Note that these three time scales satisfy $$\begin{aligned}
\label{3times}
\eta_k^0 \ll \eta_k^c\ll \eta_k^i ,\end{aligned}$$ if $\bar H \ll 1$. Note also that during $\eta_k^0 < \eta_k
<\eta_k^c$, the condition for the WKB approximation, $$\begin{aligned}
\label{WKB:cond}
\frac{\partial_\eta\Omega_k^2}{\Omega_k^3}\sim 2 \omega_2 k
|\eta_k|\bar H^2 \ll 1,\end{aligned}$$ is valid.
![Schematic plot of $\Omega^2_k$ for the exponential inflation and for the UV modes in the power law inflation. There is no mode $\phi_k$ for $\eta_k<\eta_k^0$. The mode solution for $\eta_k^0< \eta_k<\eta_k^c$ is given by the WKB solution and the mode solution for $\eta_k> \eta_k^c$ is given by the Bessel functions. The two solutions are matched at $\eta_k^c$.[]{data-label="o2:fig1"}](o2.eps){width=".6\linewidth"}
In Fig. \[o2:fig1\], we present schematic plot of $\Omega_k^2(\eta_k)$ for a given mode $\phi_k$. Therefore, during this period, the WKB mode solution, $$\begin{aligned}
\label{sol:ini}
\varphi_k(\eta_k)=\frac{1}{
\sqrt{2 \Omega(k,\eta_k)}}\exp i\left(
\int_{\eta_k^0}^{\eta_{k}}\Omega_k d\eta_k +\psi_k\right),\end{aligned}$$ can be used to describe the time evolution. We use this solution to determine the matching condition at $\eta_k=\eta_k^c$, $$\begin{aligned}
\label{ini.}
\varphi_{k}(\eta_k^c)=\frac{1}{\sqrt{2\Omega_c}},~~~
\varphi_k'(\eta_k^c)=i\sqrt{\frac{\Omega_c}{2}} ,\end{aligned}$$ where $\Omega_c^2 \equiv \Omega^2_k(\eta_k^c)\simeq \tilde
k^2[1-2(\nu^2-1/4)^{1/2}\bar H ]$, $\tilde k$ given in (68), and $\psi_k$ is chosen to give this matching condition (\[ini.\]).
For $\eta_k>\eta_k^c$, the last term in Eq. (\[Om:22\]) becomes much smaller than other terms. Thus we ignore this term and use the Bessel function as the solution for $\eta_k>\eta_k^c$, $$\begin{aligned}
\label{v:0}
\varphi_{k}(\eta_k)=A_k \sqrt{-\eta_k} J_\nu(-\tilde k \eta_k)+ B_k
\sqrt{-\eta_k} Y_\nu(-\tilde k\eta_k) .\end{aligned}$$ Matching the two solutions at $\eta_k=\eta_k^c$, we get $$\begin{aligned}
\label{AB:L}
A_k&=&\frac{\pi}{2}
\left[i \sqrt{-\eta_k^c}Y_\nu(-\tilde k\eta_k^c)
\varphi_k'(\eta_k^c)
+\frac{Y_\nu(-\tilde k\eta_k^c)
-2\tilde k\eta_k^c Y_\nu'(-\tilde k\eta_k^c)}{
2\sqrt{-\eta_k^c}} \varphi_k(\eta_k^c)\right], \\
B_k&=&-\frac{\pi}{2}
\left[i\sqrt{-\eta_k^c}J_\nu(-\tilde k\eta_k^c)\varphi_k'(\eta_k^c)
+\frac{J_\nu(-\tilde k\eta_k^c)
-2\tilde k\eta_k^cJ_\nu'(-\tilde k\eta_k^c)}{
2\sqrt{-\eta_k^c}} \varphi_k(\eta_k^c)\right] .\nonumber\end{aligned}$$ Note that $A_k$ and $B_k$ are independent of $k$ since $\tilde k
\eta_k^c$ is independent of $k$ due to Eq. (\[eta:bc\]) and $\Omega_c \propto k$.
As $\-k\eta_k\rightarrow 0$, the Bessel functions become $$\begin{aligned}
\label{sol:0}
J_\nu \rightarrow \frac{(-\tilde k \eta_k)^{\nu}}{2^{\nu}
\nu!},~~Y_\nu \rightarrow \frac{2^{\nu}(\nu-1)!}{\pi(-\tilde k
\eta_k)^{\nu}} ,\end{aligned}$$ and the second term in Eq. (\[v:0\]) dominates in the later time ($-k\eta_k \sim 0$). Therefore, the power spectrum of the scalar metric perturbation has the form $$\begin{aligned}
\label{pS:EX}
P_{k}(t)= \frac{k^3}{2\pi^2}\frac{|\varphi_k(t)|^2}{z^2(t)}
\simeq \frac{
[2^\nu(\nu-1)!]^2|B|^2H^2}{2\pi^4(z/a)^2}
\frac{(1+ \bar H^2/6)^2}{(1-17\bar H^2/6)^{3/2}}
\left(\tilde k \eta_k\right)^{3-2\nu}
,\end{aligned}$$ where $z/a$, $\bar H$, $\nu$, and $B$ are constant numbers. Note that $3-2\nu \simeq \frac{2}{3}(w_3-2 \alpha) \bar H^2=0$ in the present case since $w_3=1/3$. Therefore, the spectrum of the metric fluctuation for exponential inflation in $\kappa-$RWS is time independent and is scale invariant up to the first order in $\bar
H^2$. The only effect of the non-commutativity to the power spectrum is the global rescaling of the power spectrum, which is of the order $\bar H^2$. It is an interesting fact that $w_3$ and $2\alpha$ are the same. Note that $\alpha$ originates from the scale factor of the mode-dependent conformal time $\eta_k$ with respect to the conformal time $\eta$, and $w_3$ comes from the $\bar H^2$ order correction term of the frequency squared. Since there is no physical reason for the coincidence, it is possible that the next order correction may give a result of $w_3>1/3$. This is an interesting possibility since this positive power of $k \eta$ makes the power spectrum decrease in time. If the present analysis is applied to the tensor mode fluctuation, the time dependence can be used to solve the gravitational hierarchy problem ($H/M_P \sim
10^{-5}$) [@rubakov]. This is what happens in the power law inflation considered in the next section. The spectrum (\[pS:EX\]) is scale invariant in contrast to that of Ref. [@kowalski] with the same initial vacuum state. The difference may be attributed to the different choice of the initial conditions. At the present case, the initial time is dependent on the mode through Eq. (\[eta:0\]), which uniquely fixes the initial state (\[ini:ex\]) for the mode solutions.
metric fluctuations in $\kappa-$DIU: Power law inflation
========================================================
In this section we calculate the metric fluctuation in the power law inflationary model, in which the scale factor increases as, $$\begin{aligned}
\label{a:t}
a(t)= a_0 (\kappa t)^n, ~~ 0<t<t_f,\end{aligned}$$ where $n \neq 1$, $a_0$ is the scale factor at the Planck time $t=1/\kappa$, and $t_f$ is the instance when the inflation ends. In this model, the variable $z(t)$ in Eq. (\[z\]) is given by $\displaystyle z(t)= \sqrt{\frac{2}{n}} M_P a(t)$. For $n \neq 1$, we have $$\begin{aligned}
\label{a}
\kappa t=\left(\frac{\eta}{\eta_0}\right)^{\frac{3}{2}-\mu},~~
a(\eta)
=a_0\left(\frac{\eta}{\eta_0}\right)^{\frac{1}{2}-\mu} ,\end{aligned}$$ where $\displaystyle \mu=\frac{3n-1}{2(n-1)}$, and $\eta_0$ is the conformal time corresponding to $\kappa t=1$, given by $$\begin{aligned}
\label{eta:I}
\eta_0=-\frac{\mu-3/2}{a_0\kappa} .\end{aligned}$$ $\eta_f= \eta_0 (\kappa t_f)^{-1/(2\mu-3)}$ is the time when the inflation ends.
When $\kappa t\gg 1$ is large and $k^2/(a^2\kappa^2) \ll 1$ is small, we have $$\begin{aligned}
\label{m:s}
m^{-1}(k,\eta) \simeq 1+
\alpha_n\frac{H^2(\eta)}{\kappa^2}+\frac{k^2}{a^2(\eta) \kappa^2}
+\cdots.\end{aligned}$$ The mode-dependent conformal time $\eta_k$ can be approximated as $$\begin{aligned}
\label{eta:eta}
\eta_k \simeq \eta\left[1
+\frac{\alpha_n n^2}{2(\mu-1)} \bar \eta^{2\mu-3}
+\frac{k^2}{2\mu\kappa^2 a_0^2}
\bar \eta^{2\mu-1} +\cdots\right] ,\end{aligned}$$ where we use the notation $$\begin{aligned}
\label{bar eta}
\bar \eta \equiv \frac{\eta}{\eta_0}=-\frac{a_0 \kappa \eta}{\mu-3/2},\end{aligned}$$ and we normalize the time so that $\eta_k=0$ when $\eta=0$. The function $\Omega_k^2$ in Eq. (\[Om:gen\]) for $\bar H^2 \ll 1$ is approximated to be $$\begin{aligned}
\label{Om:pw1}
\Omega^2_k(\eta_k)&\simeq & k^2(1-w_1 H_k^2-w_2 \bar k^2)-
\frac{2n-1}{n}H^2 a^2(\eta)(1-w_3h^2) +\delta ,\end{aligned}$$ where $\delta$ represents the smaller terms proportional to the differences of the Hubble parameter from its time-averaged values, $$\begin{aligned}
\label{delta}
\delta&=&-w_1k^2(\bar
H^2-H_k^2)+2w_3H^2 a^2(\bar H^2-h^2).\end{aligned}$$ $w_i$’s are given in Eq. (57) and the time-averaged values of the Hubble parameters, $H_k$ and $h$ are defined by $$\begin{aligned}
\label{Hk}
H_k^2 \equiv \frac{\int d\eta_k H^2(\eta)}{\kappa^2\int
d\eta_k},~~~~~
h^2 \equiv \frac{ \int d\eta_k a^2(\eta)
H^4(\eta) }{\kappa^2\int d\eta_k a^2(\eta) H^2(\eta) },\end{aligned}$$ with the $\eta_k$ integrations performed over the validity range of the differential equation (\[diff\]) for a given mode solution, which will be clarified in the next subsections. We do not put the index $k$ to $h$ since $h$ depends on $k$ very weakly.
Mode generation and the initial condition
-----------------------------------------
As in the case of the $\kappa-$deformed exponentially inflating universe, due to the condition (\[k:max1\]), the mode $\phi_k$ is generated at the conformal time $\eta(k)$, $$\begin{aligned}
\label{eta0:pw}
\eta(k)=\eta_0\left(\frac{k}{a_0 \kappa}\right)^{-\frac{1}{\mu-1/2}}
,\end{aligned}$$ where we assume the Universe is inflating with $n \gg 1$.
A serious obstacle in finding physics of low comoving momentum modes is that the action (\[S:final\]) is not well defined for large $\bar H^2$ since the action is approximated by expansion in $\bar
H^2$. The condition $\bar H(\eta_m) \sim 1$ is attained at $$\begin{aligned}
\label{eta:1}
\eta_m \sim \eta_0 ~n^{-\frac{1}{2\mu-3}} .\end{aligned}$$ Before $\eta_m$, our approximation for the action (\[S:final\]) is not valid. This condition restricts the validity range of the present approximation to the modes $\phi_k$ with comoving momentum $$\begin{aligned}
\label{valid}
k > k_m \equiv n^{\frac{\mu-1/2}{\mu-3/2}} a_0 \kappa .\end{aligned}$$ In this subsection, we restrict ourselves to the ultra-violet modes satisfying the condition (\[valid\]). To know the behavior of smaller frequency modes, one should use better approximation of the action (\[S:nl\]) instead of (\[S:final\]). We present some reasonable arguments for the evolution of those low frequency modes in Sec. V.C.
For modes $k > k_m$, all the arguments for the initial state in Sec. IV.A hold true. Therefore, the initial state is given by the WKB ground state, $$\begin{aligned}
\label{ini:cond}
\varphi_k(\eta_k)=\frac{1}{
\sqrt{2 \Omega_k(\eta_k)}}\exp i\left(
\int_{\eta_k^0}^{\eta_{k}}\Omega_k(\eta_k) d\eta_k + \psi_k\right)
, ~~ k\geq k_m,\end{aligned}$$ where $\eta_k$ is close to the mode generation time $\eta_k^0$ given by Eq. (\[eta:0\]).
![Schematic plot for the mode generation time in power law inflation. Each curve describes the value of the normalized physical wave-number $\bar k(t)=\frac{k}{a \kappa}$ for a given conformal time $\eta$. Since $a(t(\eta))$ increases for expanding universe, the value of $\bar k$ always decreases. Due to Eq. (\[eta0:pw\]) each mode becomes physical when it crosses the horizontal line $\bar
k=1$. We denote the unphysical part of each modes using the dashed curve. Since $\bar H=1$ at $\eta=\eta_m$, the region $\eta> \eta_m$ is perturbatively reliable and the region $\eta< \eta_m$ is perturbatively unreliable, which are divided by a shaded vertical line. The UV modes resides in the perturbatively reliable region. The IR and the Far Infra-Red ( $ \frac{k}{a_0 \kappa}<1$) modes pass through the perturbatively unreliable region. []{data-label="o2:fig2"}](cut1.eps){width=".6\linewidth"}
inflationary evolution
----------------------
The inflationary evolution of the mode solution is governed by $\Omega_k^2$. When $k>k_m$, we approximate $\Omega_k^2$ in Eq. (\[Om:pw1\]) using Eqs. (\[eta:eta\]) and (\[delta\]), and dropping the term $\delta$ in Eq. (\[Om:pw1\]), as $$\begin{aligned}
\label{Om:pw}
\Omega^2_k(\eta_k)&\simeq & \tilde k^2-\frac{\nu^2-1/4}{\eta_k^2}
- \frac{w_2 k^4}{\kappa^2 a_0^2}\bar \eta_k^{2\mu -1}+ \cdots.\end{aligned}$$ where $$\begin{aligned}
\label{k nu:22}
\tilde k^2\equiv k^2\left[1-(w_1+\frac{2n-1}{\mu n})H_k^2
\right],~~~
\nu^2-\frac{1}{4}\equiv\left(\mu^2-\frac{1}{4}\right)
\left[1+\left(\frac{\alpha_n}{\mu-1}-w_3\right)h^2 \right] .\end{aligned}$$ Eq. (\[Om:pw\]) looks similar to Eq. (\[Om:22\]) except for the final term. With the $\Omega^2_k(\eta_k)$ in Eq. (\[Om:pw\]), we have two different time scales defined by $\Omega_k'(\eta_k^c)=0$ and $\Omega_k^2(\eta_k^i)=0$: $$\begin{aligned}
\label{etas:pw}
\eta_k^c
&=&\eta_0 \left[\frac{\nu^2-1/4}{w_2(\mu-1/2)(\mu-3/2)^2\bar
k_0^4}\right]^{1/(2\mu+1)}
,\\
{\eta_k^i}^2&=&\frac{(\nu^2-1/4)}{\tilde k^2}
\left[1+w_2(\mu-3/2)\left(\frac{\nu^2-1/4}{
\mu-3/2}\right)^{2\mu}\bar k_0^{3-2\mu}\right]+\cdots \nonumber\end{aligned}$$ where $\bar k_0= k/(\kappa a_0)$. Since $\bar k_0 \gg 1$ for modes under consideration, we have $$\begin{aligned}
\label{etas}
\eta_k^0 \ll \eta_k^c \ll \eta_k^i .\end{aligned}$$ Since the condition for the WKB approximation, ${\Omega_k^2}'/\Omega_k^3 \ll 1$, holds during $\eta_k <\eta_k^c$, we may use the WKB solution (\[ini:cond\]). Therefore, we get the mode solution at $\eta_k^c$ given by Eq.(\[ini.\]) with $$\begin{aligned}
\label{Omc}
\Omega_c^2\equiv \Omega_k^2(\eta_k^c)\simeq
\tilde k^2\left[1-\frac{2\mu+1}{2\mu-1}
\frac{\nu^2-1/4}{(\tilde k \eta_k^c)^2} \right] ,\end{aligned}$$ where the second term in the parenthesis is smaller than 1 for $\bar
k_0 \gg 1$. For $\eta_k
> \eta_k^c$, we ignore the last term in Eq. (\[Om:pw\]) since $\bar \eta_k$ is very small there. The solution for the differential equation (\[diff\]) is given by the Bessel function in Eq. (\[v:0\]) with parameters given in Eq. (\[k nu:22\]). With the initial condition (\[ini.\]) and $\Omega_c$ in Eq. (\[Omc\]), $A_k$ and $B_k$ are given by Eq. (\[AB:L\]).
Since we are considering modes with $\bar k \gg 1$, we always have $|\tilde k \eta_k^c| \gg 1$. Using the asymptotic expansion of the Bessel functions, $$\begin{aligned}
\label{vp:H}
J_{\nu}(x) \simeq \sqrt{\frac{2}{\pi x}} \cos [x
-(\nu+\frac{1}{2})\frac{\pi}{2}] , ~~
Y_{\nu}(x) \simeq \sqrt{\frac{2}{\pi x}} \sin [x
-(\nu+\frac{1}{2})\frac{\pi}{2}] ,\end{aligned}$$ we get $$\begin{aligned}
\label{AB:UV}
A_k=\frac{\sqrt{\pi}}{2}\sqrt{\frac{\Omega_c}{\tilde k}}e^{-i(\tilde
k \eta_k^c+(\nu+1/2)\pi/2)}=i B_k .\end{aligned}$$ Note that the absolute values of $A_k$ and $B_k$ are very weakly dependent on $k$ since $\Omega_c\sim \tilde k$.
From this, we have the power spectrum, $$\begin{aligned}
\label{pS:UV}
P_{UV,k}(t)= \frac{k^3}{2\pi^2}\frac{|\varphi_k(t)|^2}{z^2(t)}
\simeq \frac{\kappa^2}{M_P^2}
\frac{n[2^{\nu}(\nu-1)!]^2(\mu-3/2)^{1-2\nu}}{
16\pi^3 }
\left(\frac{k}{a_0\kappa}\right)^{3-2\nu}\left(
\frac{\eta_k}{\eta_0}\right)^{2(\mu-\nu)} +\cdots .\end{aligned}$$ Since we are interested in the time evolution for $\eta_k^c <\eta_k
< \eta_f$, we have the time-averaged Hubble parameters, $$\begin{aligned}
\label{Hk:v}
H_k^2 &\simeq& \frac{n^2}{2(2-\mu)}\frac{(\bar
\eta_f)^{4-2\mu}
-(\bar \eta_k^c)^{4-2\mu}}{\bar \eta_f
-\bar \eta_k^c}
\simeq\frac{n^2}{2(2-\mu)}\frac{1}{(\bar \eta_k^c)^{2\mu-3}} ,\\
h^2 &\simeq & \frac{n^2}{2(2-\mu)} \frac{\bar
\eta_f^{2\mu-4} - (\bar \eta_k^i)^{2\mu-4} }{\bar \eta_f^{-1}
-(\bar \eta_k^i)^{-1} }
\simeq \frac{n^2}{2(2-\mu)} (\bar
\eta_f)^{2\mu -3} .\end{aligned}$$ The mode-dependent conformal time $\eta_f$ at the end of the inflation is almost independent of the comoving momentum $k$. Note that from Eq. (\[k nu:22\]), $$\begin{aligned}
\label{mu:nu}
\nu &\simeq & \mu - c h^2, ~~
c\equiv\frac{\mu^2-1/4}{2\mu}\left(w_3-\frac{\alpha_n}{\mu-1}\right)
, \\
3-2\nu &\simeq& 3-2\mu+ 2c h^2=-\frac{2}{n-1}+ 2 ch^2, \nonumber\end{aligned}$$ where $c$ is a positive number for $n> 1.53$. $3-2\nu$ changes sign from positive to negative at $\eta_k=\eta_{flat}$, where $$\begin{aligned}
\label{flat}
\eta_{flat}=
\left[\frac{(2-\mu)(2\mu-3)}{cn^2}\right]^{\frac{1}{2\mu-3}}\eta_0 .\end{aligned}$$ Since $|3-2\nu| \ll 1$ always, the slightly blue-shifted spectrum changes into the slightly red-shifted spectrum at $\eta=
\eta_{flat}$.
The power spectrum decreases in time since $\mu-\nu$ is positive even though it is very small. For example, for $n=14$, we have $a(\eta_{flat})/a_0 \sim e^{41}$ and $\displaystyle P_k \sim 0.075
\frac{\kappa^2}{M_P^2}$. Since the currently discussed minimal duration of inflation is roughly $a/a_0 \sim e^{60}$ [@giovan], $\eta_f$ must be later than $\eta_{flat}$ and the size of the spectrum decreases in time. We will observe the decreasing red spectrum of the scalar density fluctuation for high comoving momentum $k>k_m$.
If $n=19$, on the other hand, we have $a(\eta_{flat})/a_0\sim
e^{60}$ and we will observe fully scale invariant power spectrum for the high comoving momentum modes for minimal inflation.
There are a couple of features in our result (\[pS:UV\]) which can be used to resolve the gravitational hierarchy problem, the extreme weakness of the gravitational wave (the tensor mode fluctuation) relative to the large scale density fluctuations. The first is the fact that the power spectrum decreases as $k$ increases for later time, which is also the case in the commutative space results. The second is the time-dependence of the power spectrum, which is a new feature of the present paper, that may describe the difference of the power spectra between the structure formation time and the observation time. These features may solve the gravitational hierarchy problem without imposing any fine tuning on $H$ or on the non-commutativity scale $\kappa$.
Speculation for IR modes
------------------------
In the previous subsection, we deferred the discussion on the evolution of the modes of smaller frequencies than $k_m$. Due to the definition of $k_m$, (\[valid\]), these modes always satisfy in its physical region, $$\begin{aligned}
\label{fir:1}
\frac{k}{a(t) \kappa} \ll 1.\end{aligned}$$ The Far Infra-Red (FIR) modes are created at $\eta=\eta_k^0<
\eta_0$. For $\eta_k^0<\eta <\eta_m$, the condition (\[fir:1\]) can be interpreted as $$\begin{aligned}
\label{k:H}
k^2 \ll a^2 H^2, ~~ \mbox{ for } \eta < \eta_m ,\end{aligned}$$ since $\eta_m$ is the time for $H/\kappa \sim 1$.
To figure out the dynamics of these modes, we must have reasonable approximation on the action around $H^2/\kappa^2 \sim 1$. We assume that the action (\[S:final\]) is still valid even though the explicit functional forms of $m$ and $f$ are not known. We keep Eq. (\[bmu\]) since the equation comes from the requirements of quantization and unitarity of the time evolution. To show that the mode solution at time $\eta_m$ is almost independent of $k$, we use several steps of reasoning. First, we show that $m(k,\eta_k)$ is very small until $\eta_k <\eta_m$. Next, we argue that $\Omega^2_k$ is also very small. Finally, we argue that $\Omega_k^2$ is almost independent of $k$ for this region of time. With these, we have the result that the matching condition of $\varphi_k(\eta_k)$ at $\eta_k
\sim \eta_m$ is almost independent of $k$.
$m(k,\eta)$ is defined by the first order differential equation (\[bmu\]) with $m=1$ at $\kappa t\rightarrow \infty$. The value of $m$ increases in time asymptotically approaching to, $m\sim
\bar \mu$ at $\eta=0$. Therefore, to get the behavior of $m$, we need to analyze its behavior around $m\sim 0$. In this case, the differential equation (\[bmu\]) becomes $$\begin{aligned}
\label{dm:0}
\frac{\dot m}{m} \simeq \sqrt{-\frac{3\kappa^2\bar \mu}{\gamma}}.\end{aligned}$$ Thus, to have a consistent, real-numbered value of $m$, we must have $ \bar \mu /\gamma<0$ in this region and $$\begin{aligned}
\label{mm}
m(k,\eta_k) \simeq
\left[\Lambda-\int_{\eta_k^0}^{\eta_k}\sqrt{\frac{-3\kappa^2 \bar
\mu a^2}{\gamma}} d\eta_k \right]^{-1} ,\end{aligned}$$ where the integration constant $\Lambda$ is a large number which ensures the value of $m$ to be small. The value of $m$ increases to $O(1)$ around $\bar \mu /\gamma \sim 0$ since for positive $\bar
\mu$ we have $m \sim \bar \mu$ as a 0$^{\rm th}$ order approximation. Let $\eta_k^c$ be the value of $\eta_k$ satisfying $\bar \mu(\eta_k^c)/\gamma(\eta_k^c) =0$. Then, we have $m \sim
1/\Lambda \ll 1$ for $\eta_k \ll \eta_k^c$ and $m \sim \bar \mu$ for $\eta_k > \eta_k^c$. Rough estimation on the value of $\eta_k^c$ is possible by using the asymptotic form of $m(k, \eta_k)\sim \bar
\mu$. Since $$\begin{aligned}
\label{..}
\frac{k^2}{\kappa^2 a^2(t_k^c)}\simeq 1- \alpha_n
\frac{H^2(t_k^c)}{\kappa^2} <1,\end{aligned}$$ where $t_k^c$ is the time corresponding to the mode-dependent conformal time $\eta_k^c$, and the mode $\phi_k$ is generated before $\eta_k^c$. Since $H(\eta_m)\sim \kappa$, we must have $\eta_k^c\sim
\eta_m+\cdots $. Therefore, the zeroth order of $\eta_k^c$ is independent of $k$.
Next, we consider the function $f$ of Eq. (46) for $\eta_k \ll
\eta_k^c$. From Eqs. (\[bmu\]), (\[dm:0\]), and the change of variable $\displaystyle \bar f=\frac{\sqrt{-\gamma \bar
\mu}}{\kappa^2 a m} f$, we get $$\begin{aligned}
\label{f:0}
\frac{3\omega}{\kappa^2}+\partial_\eta{\bar f}\simeq -\frac{\bar
f^2}{\gamma \bar \mu}
\geq 0,\end{aligned}$$ where we have ignored the term proportional to $m$ since $m$ is very small. For this differential equation to be well defined, $\bar f$ must be a function of $O(1)$. Thus, $\displaystyle f=\frac{\kappa^2
a m}{\sqrt{-\gamma \bar \mu}} \bar f$ is of the same order as $m$, and is very small. Therefore, the potential ($\propto m f \propto
1/\Lambda^2$) is very small.
Finally, note that $\Omega_k^2/(a^2\kappa^2)$ is a function of $k^2/(a^2\kappa^2)$ and $H^2/\kappa^2$. Because of Eq. (\[k:H\]) for FIR modes, we guess that $\Omega_k$ may be almost independent of $k$ for $\eta_k <\eta_k^c$. This implies that the initial mode solution created at $\eta_k^0$ of the FIR modes are almost independent of $k$.
Since $m$ is small, small variation $\delta\eta$ corresponds to a large variation in the mode dependent conformal time $\delta
\eta_k\sim \Lambda\delta\eta$ in this region. Since the multiplication of potential and the time-interval, $$\begin{aligned}
\label{..dphi}
\frac{\delta \phi_k'}{\phi_k}\sim \Omega_k^2 \delta \eta_k \sim
\frac{1}{\Lambda},\end{aligned}$$ is small, the mode solution does not change much until $\eta_k^c$ after the mode generation at $\eta_k^0$. Then, the state at $\eta_k=\eta_k^c$ is not much different from its ground state at $\eta_k^0$.
![Schematic plot of $\Omega_k^2$ for IR modes. .[]{data-label="o2:fig3"}](omIR.eps){width=".6\linewidth"}
Collecting all of the above arguments, we propose that the initial mode solutions, which are almost independent of $k$, vary only by a small amount during $\eta_k^0<\eta<\eta_k^c\sim \eta_m$. The evolution for $\eta_k>\eta_k^c$ will be described by the same method as that of the UV modes with the matching condition at $\eta_k^c$. The mode solution is given by Eq. (\[v:0\]) and its coefficients are given by Eq. (\[AB:L\]). For these modes we have $-k \eta_k^c(
\sim -k \eta_m) <n(=- k_m \eta_m )$. In the case of a FIR modes, we have $|k \eta_k^c| \ll 1$. Then, we can use the form (\[sol:0\]) of the Bessel function to determine $B_k$: $$\begin{aligned}
\label{B:IR}
B_k \sim -\frac{\pi(\nu+1/2)}{2^{\nu+3/2}\nu
\nu!}\sqrt{\frac{k}{\Omega_k(\eta_k^c)}} (-k \eta_k^c)^{\nu-1/2} ,\end{aligned}$$ where we have assumed $\eta_k^c\Omega_k(\eta_k^c)\ll 1$.
Thus the power spectrum for the FIR modes is given by $$\begin{aligned}
\label{FIR}
P_{FIR}(k) \simeq \frac{n(\nu+1/2)^2}{32\pi^2\nu^2}
\frac{k^3}{a^2(\eta_m) M_P^2\Omega_k(\eta_m)}
\left(\frac{\eta_k}{\eta_k^c}\right)^{2(\mu-\nu)} .\end{aligned}$$ Since $\eta_k^c \sim \eta_m$, we have strong (proportional to $k^3$) blue spectrum for this FIR modes. Since $\mu-\nu$ is a small positive number, the size of the spectrum slowly decreases in $\eta_k$. It was argued that, in relation to the pre-big-bang scenario, this kinds of cutoff of power spectrum in the low frequency region can explain the low CMB quadrupole moments [@piao].
For the spectrum to be continuous, the red spectrum in the UV region must be continuously deformed to the blue spectrum in the FIR region. Therefore, the spectrum for $a_0 \kappa < k < k_m$ must be continuously deformed from the weak red spectrum to flat and then to weak blue spectrum.
Summary and discussion
======================
We have studied the efffects of the $\kappa-$deformation of Robertson-Walker space on the evolution of metric fluctuations in expanding cosmological background. For a given noncommutative $\kappa-$deformed inflationary universe the cosmological background is still described by the Einstein equation since the background fields only depend on one variable $t$ so that the homogeneity and isotropy of the Robertson-Walker space are kept. The equation for linear fluctuations, however, are modified. We have shown that the modification takes the form of nonlocal interaction of the fluctuating field with itself and with the background. We have analyzed the system by perturbatively expanding the action up to the first order $H^2/\kappa^2$.
An important consequence of the space-time non-commutativity is that for each wave number $k$, there exists an earliest time $\eta_k^0$ at which the fluctuating mode is created. The origin of the mode generation phenomena is a direct consequence of the $\kappa-$deformation which introduces an upper bound of the comoving wave-number by Eqs. (\[klimit\]) and (\[k:max1\]). We assume that the fluctuation starts out with its vacuum amplitude at $\eta_k^0$ since the number of excitations should be conserved during the creation process. Moreover, this condition restricts the physical frequency to be smaller than $\kappa$. This condition determines the initial condition of given modes for a given initial time. The deformation also generates the correction terms proportional to $H^2/\kappa^2$ and $k^2/(\kappa^2a^2)$ to the frequency squared, which determines the time evolution of mode $k$.
There are two main results for the corrections to the power spectrum of the metric fluctuation due to the deformation. The first is that the deformation alters both the time dependence and the momentum dependence of the power spectrum. Especially, in the case of a power law inflation, we have shown that the power spectrum slowly decreases in time as $(\eta_k/\eta_0)^{2(\mu-\nu)}$, where $\mu-\nu\simeq c h^2$ is a very small but positive. This time dependence of the power spectrum is a new feature of the present approach in contrast to the result of the commutative space case and to the result of Ref. [@ho]. The fact that the power spectrum decreases in time in addition to the existence of the red shifted spectrum, can be used to resolve the gravitational hierarchy problem for the tensor mode fluctuation. Another consequence of the deformation is that the momentum dependence of the power spectrum for UV modes is also dependent on time as $k^{3-2\nu}$, where $\nu$ is given by Eq. (92). Note that $3-2\nu$ is a small positive number for $\eta_f< \eta_{flat}$ and is a small negative number for $\eta_f>\eta_{flat}$. Therefore, there is a period ($\eta <
\eta_{flat}$) of blue spectrum in the earlier time of the inflation and the spectrum becomes red after the time $\eta_{flat}$, which is determined by $n$. Maximal red shift occurs to the power spectrum as $\eta_f \rightarrow \infty$ in which case $\mu=\nu$. Another interesting effect appears in the power spectrum of infra-red modes. Although we cannot obtain the dynamics of infra-red modes explicitly, we have suggested a form of the power spectrum from the consistency requirements. The power spectrum of the far infra-red modes ($k< a_0 \kappa$), with $a_0$ the scale factor at $t=1/\kappa$, have a cutoff proportional to $k^3$ even though the explicit procedure needs much refining since the dynamics at early times are not known. One can use the existence of this cutoff to explain the low CMB quadrupole moment. For the spectrum to be continuous, this $k^3$ type power may change as $k$ increases. We know that for ultra-violet mode ($k>k_m$, Eq. (89)) the spectrum becomes slightly red shifted for $\eta > \eta_{flat}$. Therefore, the spectrum may change from weak blue to flat spectrum for $ a_0
\kappa < k< k_m$.
Brandenberger and Ho [@ho] computed the effect of the stringy space-time uncertainty relation to the power spectrum of metric fluctuations for power law inflation in the Robertson-Walker space. It is interesting to compare their results with that of the present paper since we have started from a different commutation relation (\[com:RW\]). We start from the basic commutation relation (\[com:RW\]) and construct the theory from the first principle, although we have to use the perturbation to compute the physical effects of the deformation. Let us consider the power spectra for time $\eta_f>\eta_{flat}$. The spectra of Ref. [@ho] changes from weak red for ultra-violet modes to weak blue for infra-red modes. In our case, the far infra-red modes behaves as $k^3$. This is due to the fact that the far infra-red modes becomes almost independent of $k$ for the very early times $\eta < \eta_m$. Since the generation time for the ultra-violet modes is of similar form as in Ref. [@ho], the behavior of the spectrum in our case is similar to theirs for UV modes. The spectrum for $k>a_0 \kappa$ changes from weak blue to weak red as $k$ increases. A totally new phenomena due to the $\kappa-$deformation, which is absent in the cases of Ref. [@ho] and the metric fluctuations in the commutative Robertson-Walker model, is the time-dependence of the spectra. We have shown that both the spectra of the ultra-violet and infra-red modes decrease slowly in time.
We have used the perturbative approximation to obtain the generic feature of the physical effect of the $\kappa-$deformation on the cosmological evolution. It would be interesting if one could develop some nonperturbative approximation methods to extract better information from the nonlocal theory described by the action (\[S:nl\]).
This work was supported in part by Korea Research Foundation under Project number KRF-2003-005-C00010 (H.-C.K. and J.H.Y.) and by Korea Science and Engineering Foundation Grant number R01-2004-000-10526-0(CR).
[10]{}
S. Weinberg, [*Gravitation and Cosmology*]{}, (John Wiley & Sons, New York, US. 1972).
E. W. Kolb and M. S. Turner, [*The Early Universe*]{}, (Addison-Wesley, US, 1990).
C. L. Bennett [*et al.*]{}, Astrophys. J. Suppl. [**148**]{}, 1 (2003) \[astro-ph/0302207\]; D. N. Spergel [*et al.*]{}, Astrophys. J. Suppl. [**148**]{}, 175 (2003), \[astro-ph/0302209\]; G. Hinshaw [*et al.*]{}, Astrophys. J. Suppl. [**148**]{}, 135 (2003), \[astro-ph/0302217\]; S. Sarkar, [*Measuring the cosmological density perturbation*]{}, \[hep-ph/0503271\] (2005).
R. H. Brandenberger, [*Theory of cosmological perturbations and applications to superstring cosmology*]{}, \[hep-th/0501033\], (2005).
M. Giovannini, [*Theoretical tools for CMB physics*]{}, \[astro-ph/0412601\] (2004).
A. D. Sakharov, Sov. Phys. JETP [**22**]{}, 241 (1965) \[Zh. Eksp. Teor. Fiz. [**49**]{}, 345 (1965)\].
R. H. Brandenberger, Inflationary cosmology: Progress and problems. In [*1st Iranian International School on Cosmology: large Scale Structure Formation, Kish Island, Iran, 22 Jan.-4 Feb. 1999.*]{} Kluwer, Dordrecht, 2000. (Astrophys. Space Sci. Libr. ; 247), ed. by R. Mansouri and R. Brandenberger (Kluwer, Dordrecht, 2000). \[hep-ph/9910410\].
W. G. Unruh, Phys. Rev. [**D 51**]{}, 2827 (1995); S. Corley and T. Jacobson, Phys. Rev. [**D 54**]{}, 1568 (1996) \[hep-th/9601073\].
J. Martin and R. H. Brandenberger, Phys. Rev. [**D 63**]{}, 123501 (2001) \[hep-th/0005209\]; Mod. Phys. Lett. [**A 16**]{}, 999 (2001) \[astro-ph/0005432\].
R. Brandenberger and P.-M. Ho, Phys. Rev. [**D 66**]{}, 023517 (2002) \[hep-th/0203119\]; Rong-Gen Cai, Phys. Lett. [**B 593**]{}, 1 (2004) \[hep-th/0403134\].
Q. G. Huang and Miao Li, JCAP 0311 001 (2003) \[astro-ph/0308458\]; JHEP 0306, 014 (2003) \[hep-th/0304203\].
B. Greene, K. Schalm, J.P. Schaar, and G. Shiu, [*Extracting New Physics from the CMB*]{}, \[astro-ph/0503458\].
Yun-Song Piao, [*A possible explanation to low CMB quadrupole*]{}, \[astro-ph/0502343\].
G. Veneziano, [*String cosmology: The pre-Big Bang scenario*]{}, \[hep-th/0002094\].
A. Ashoorioon, A. Kempf, and R. B. Mann, Phys.Rev. [**D 71**]{}, 023503 (2005), \[astro-ph/0410139\]; A. Ashoorioon, and R. B. Mann, Nucl. Phys. [**B 716**]{}, 261 (2005), \[gr-qc/0411056\].
Giovanni Amelino-Camelia, Phys. Lett. [**B 510**]{} 255, (2001) \[hep-th/0012238\]; Int.J.Mod.Phys. [**D 11**]{} 35, (2002) \[gr-qc/0012051\].
F. Girelli and R. Livine, [*Physics of Deformed Special Relativity; Relativity Principle revisited*]{}, \[gr-qc/0412079\].
C. Chryssomalakos and E. Okon, Int. J. Mod. Phys. [**D 13**]{} 2003, (2004), \[hep-th/0410211\].
L. Freidel, J. Kowalski-Glikman, L. Smolin, Phys. Rev. [**D 69**]{} (2004) 044001, \[hep-th/0307085\].
G Amelino-Camelia, [*The three perspectives on the quantum-gravity problem and their implications for the fate of Lorentz symmetry*]{}, \[gr-qc/0309054\]; J. Kowalski-Glikman, [*Intorduction to Doubly Special Relativity*]{}, \[hep-th/0405273\]; G Amelino-Camelia, L. Smolin, A. Starodubtsev, Class. Quant. Grav. [**21**]{} (2004) 3095, \[hep-th/0306134\]; Giovanni Amelino-Camelia, \[gr-qc/0506117\].
F. Girelli, E.R. Livine, and D. Oriti, [*Deformed Special Relativity as an effective flat limit of quantum gravity*]{}, \[gr-qc/0406100\].
A. Agostini, G Amelino-Camelia, and F. D’Andrea, Int. J. Mod. Phys. [**A 19**]{}, 5187 (2004), \[hep-th/0306013\].
P. Kosiński, J. Lukierski, and P. Maślanka, Phys. Rev. [**D 62**]{}, 025004 (2000).
J. Kowalski-Glikman and S. Nowak, [*Quantum $\kappa-$Poincaré algebra from de Sitter Space of Momenta*]{}, \[hep-th/0411154\].
M. Daszkiewicz, K. Imilkowska, J. Kowalski-Glikman and S. Nowak, [*Scalar field theory on $\kappa-$Minkowski space-time and doubly special relativity*]{}, \[hep-th/0410058\].
J. Kowalski-Glikman, Phys. Lett. [**B 499**]{}, 1 (2001), \[astro-ph/0006250\].
J. Kowalski-Glikman and S. Nowak, Phys. Lett. [**B 539**]{}, 126 (2002), \[hep-th/0203040\].
L. Möller, [*A symmetry invariant integral on $\kappa-$deformed space-time*]{}, \[hep-th/0409128\].
R. P. Woodard, Phys. Rev. [**A 62**]{}, 052105 (2000).
H. R. Lewis Jr., Phys. Rev. Lett. [**27**]{}, 510 (1967); H. R. Lewis Jr. and W. B. Riesenfeld, J. Math. Phys. [**10**]{}, 1458 (1969); S. P. Kim, Class. Quant. Grav. [**13**]{}, 1377 (1996); Phys. Rev. [**D 55**]{}, 7511 (1997); H.-C. Kim, M.-H. Lee, J.-Y. Ji, and J. K. Kim, Phys. Rev. [**A 53**]{}, 3767 (1996).
V.A. Rubakov, M.V. Sazhin, and A.V. Veryaskin, Phys. Lett. [**115 B**]{}, 189 (1982); L.F. Abbot and M.B. Wise, Nucl Phys. [**B 244**]{}, 541 (1984).
|
---
abstract: 'Quasiparticle recombination in a superconductor with an *s*-wave gap is typically dominated by a phonon bottleneck effect. We have studied how a magnetic field changes this recombination process in metallic thin-film superconductors, finding that the quasiparticle recombination process is significantly slowed as the field increases. While we observe this for all field orientations, we focus here on the results for a field applied parallel to the thin film surface, minimizing the influence of vortices. The magnetic field disrupts the time-reversal symmetry of the pairs, giving them a finite lifetime and decreasing the energy gap. The field could also polarize the quasiparticle spins, producing different populations of spin-up and spin-down quasiparticles. Both processes favor slower recombination; in our materials we conclude that strong spin-orbit scattering reduces the spin polarization, leaving the field-induced gap reduction as the dominant effect and accounting quantitatively for the observed recombination rate reduction.'
author:
- Xiaoxiang Xi
- 'J. Hwang'
- 'C. Martin'
- 'D. H. Reitze'
- 'C. J. Stanton'
- 'D. B. Tanner'
- 'G. L. Carr'
title: Effect of a magnetic field on the quasiparticle recombination in superconductors
---
An excitation from the superconducting condensate requires finite energy (the energy gap 2$\Delta$) and produces two quasiparticles. A quasiparticle excited to very high energy (compared to $\Delta$) quickly decays via a number of fast scattering processes to near the gap edge, where it recombines with a partner to form a Cooper pair. The pair’s binding energy is emitted mainly as 2$\Delta$ phonons [@Ginsberg1962; @Schrieffer1962; @Tewordt1962]. The recombination process is delayed by a phonon bottleneck: each recombination-generated phonon can break another Cooper pair, causing energy to be trapped in a coupled system of 2$\Delta$ phonons and excess gap-edge quasiparticles [@Rothwarf1967; @Schuller1975; @Chi1981]. Quasiparticle recombination has been widely studied in both metallic superconductors, to investigate the non-equilibrium processes in the many-body BCS system [@Federici1992; @Carr2000; @Demsar2003; @Beck2011], and high-temperature superconductors, to gain new insight into the pairing mechanism [@Gedik2004; @Kusar2008; @Mertelj2009; @Mihailovic1999]. Theories of the recombination process considered the reaction kinetics and interactions of quasiparticles and phonons [@Kaplan1976; @Chang1977]; while experiments obtained the dependence of the quasiparticle lifetime on temperature, film thickness, and excitation strength [@Levine1968; @Perrin1982]. A magnetic field is known to couple to the electron orbital motion and to align the spin; both effects weaken superconductivity [@Fulde2010]. The consequence of magnetic field on the quasiparticle recombination [@Holdik1985] has not been examined in detail by optical pump-probe methods.
![Experimental set-up. Electrons circulate in bunches in the synchrotron storage ring, generating pulses of far-infrared radiation with a repetition frequency of 52.9 MHz. The Ti:sapphire laser produces pulses with a repetition frequency of 105.8 MHz and a pulse picker selects every other pulse to match the synchrotron pulse pattern. The selected laser pulses are delivered over a fiber optic cable to the sample and the synchrotron pulse probes the photoinduced transmission at a fixed time delay afterward. To synchronize the synchrotron and laser pulses, the 52.9 MHz bunch timing signal from a pair of electrodes inside the synchrotron ring chamber is used by the Synchro-Lock laser control system as a reference for the laser pulse emission. The pulse generator introduces an adjustable delay between the laser and synchrotron pulses. The transmitted far-infrared light is detected by a bolometer detector and recorded on a computer.[]{data-label="Fig1"}](Fig1.eps){width="49.00000%"}
We use a novel time-resolved laser-pump synchrotron-probe spectroscopic technique to study the quasiparticle recombination dynamics in superconducting thin films, under applied magnetic field. Samples studied include a 10 nm thick Nb$_{0.5}$Ti$_{0.5}$N film on a crystal quartz substrate and a 70 nm thick NbN film on a MgO substrate. These substrates are essentially transparent in the far-infrared spectral range. The films were grown by reactive magnetron sputtering, using NbTi cathode in Ar/N$_2$ gas for Nb$_{0.5}$Ti$_{0.5}$N and Nb cathode in N$_2$ gas for NbN. The two films have critical temperatures of 10.2 K and 12.8 K, and a zero-temperature, zero-field, energy gap 2$\Delta$ of 2.7 meV and 4.5 meV, respectively. Four-probe resistivity measurements with magnetic field parallel to the films determined their upper critical field to be greater than 20 T at $T\le3$ K.
The samples were mounted in a $^4$He Oxford cryostat equipped with a 10 T superconducting magnet, and probed by far-infrared radiation produced in a bending magnet at beamline U4IR of the National Synchrotron Light Source, Brookhaven National Laboratory. The experiment, illustrated in FIG. \[Fig1\], exploits the fact that the synchrotron radiation is emitted in $\sim$300 ps long pulses (governed by the electron bunch structure in the storage ring). We applied mode-locked near-infrared Ti:sapphire laser pulses ($\sim$2 ps in duration and $\sim$1.5 eV in photon energy) as the source for photoexcitation. The synchrotron probe beam measures the photoinduced optical properties due to the excess quasiparticles as a function of time delay relative to the arrival of the pump beam. The synchrotron pulse has a Gaussian profile with a FWHM of $\sim$300 ps, determining the time resolution of the experiment. At selected delay times $t$, we measure the spectrally integrated photoinduced transmission $S(t)\equiv -\Delta\mathcal{T}(t)$ over the spectral range spanning the superconductor’s energy gap ($\sim$3 meV). The spectral shape is determined primarily by the optical components carrying the beam, and the detector.
If the laser were turned on and off to measure the photoinduced response, there would be a temperature modulation as well as the photoexcited quasiparticle modulation. To reduce these thermal effects we dither the laser pulse back and forth by a few tens of picoseconds at each delay setting, keeping the incident laser power constant. The dither is achieved by phase modulating the laser pulse using the internal oscillator of a lock-in amplifier. The directly obtained quantity is therefore a differential signal, $dS/dt$. This signal was detected using a B-doped Si bolometer in combination with the lock-in amplifier. Numerical integration yields the photoinduced transmission $S(t)$, which directly follows the excess quasiparticle density [@Lobo2005].
![Photoinduced transmission $S(t)$ vs. time $t$ for Nb$_{0.5}$Ti$_{0.5}$N (a and b) and for NbN (c and d), all measured in parallel fields at $T\le2$ K. Low-fluence and high-fluence data are compared. Note the semilog scale; simple exponential decay produces a straight line.[]{data-label="Fig2"}](Fig2.eps){width="49.00000%"}
To study the effect of magnetic fields and excess carrier density on the recombination dynamics, we measured the magnetic-field and laser-fluence dependent photoinduced transmission for Nb$_{0.5}$Ti$_{0.5}$N and NbN thin films. The samples were fully immersed in superfluid $^4$He ($T\le2$ K) to minimize heating. At this low temperature, the thermal quasiparticle population is small (compared to the number of broken pairs at high fluence) but not zero. The field was applied parallel to the film surface to avoid the complexity of vortex effects. (See Ref. [@Xi2010]). Typical results are shown in FIG. \[Fig2\], where the photoinduced signal $S(t)$ (excess quasiparticle density) is plotted against delay time. At both low (FIG. \[Fig2\]a and FIG. \[Fig2\]c) and high laser fluences (FIG. \[Fig2\]b and FIG. \[Fig2\]d), a longer time is required for recombination as the magnetic field is increased. The pulse width of the synchrotron probe beam gives rise to the initial upturn in the data, which is skipped in the following data analysis.
We have discovered a revealing perspective to display our results, shown in FIG. \[Fig3\]. We define an effective instantaneous recombination rate $1/\tau_{\mathrm{eff}}(t) \equiv -[dS(t)/dt]/S(t)$ and plot $1/\tau_{\mathrm{eff}}(t)$ vs $S(t)$ at various fields and fluences. Here short times are at the right (large $S(t)$) and long times at the left. In this presentation, data at the same field but for different pump fluences scale to the same straight line. As will be shown below, this behavior is expected for bimolecular recombination where the lifetime for a given particle is proportional to the availability of other particles with which to combine.
![Effective instantaneous recombination rate vs. photoinduced transmission. (a) For Nb$_{0.5}$Ti$_{0.5}$N, data at each field include fluences ranging from 0.4 to 10.7 nJ/cm$^2$. (b) For NbN, data at each field include fluences ranging from 2.4 to 18.1 nJ/cm$^2$ except for 8 T and 10 T, where data were collected at 18.1 nJ/cm$^2$. A 4-point moving average was performed on the data to reduce noise. The lines are linear fits to the data.[]{data-label="Fig3"}](Fig3.eps){width="30.00000%"}
{width="62.00000%"}
The scaling can be understood as follows. The phonon bottleneck was first discussed by Rothwarf and Taylor [@Rothwarf1967] using two rate equations, one for the quasiparticles and the other for the 2$\Delta$ phonons. The quasiparticles, which directly correspond to our signal $S(t)$, follow a simple model that captures the feature of bimolecular recombination, meanwhile taking into account the phonon bottleneck. The decay rate of the total quasiparticle density $N(t)$ toward the equilibrium density is proportional to $N^2$, because recombination requires the presence of two quasiparticles. Motivated by the Rothwarf-Taylor [@Rothwarf1967] equations, we write $$\frac{dN}{dt}=-2R(N^2-N_{\mathrm{th}}^2). \label{eq1}$$ A thermal term $N_{\mathrm{th}}^2$ is subtracted from $N^2$, because at equilibrium $N = N_{\mathrm{th}}$ and the quasiparticle density must remain constant. The phonon bottleneck is introduced into the model through the recombination rate coefficient $R$. (See Section 1 of the Supplemental Material.) A factor of 2 is included because each recombination event depletes two quasiparticles. Now, $N(t) = N_{\mathrm{th}} + N_{\mathrm{ex}}(t)$, with $N_{\mathrm{th}}$ the thermal density and $N_{\mathrm{ex}}(t)$ the photoinduced excess density. At a given temperature and magnetic field, $N_{\mathrm{th}}$ is time-independent, making Eq. become $-(dN_{\mathrm{ex}}/dt)/N_{\mathrm{ex}} = 2R(N_{\mathrm{ex}}(t) + 2N_{\mathrm{th}})$. We identify $-(dN_{\mathrm{ex}}/dt)/N_{\mathrm{ex}}$ as the effective instantaneous relaxation rate $1/\tau_{\mathrm{eff}}(t)$ defined earlier, because the photoinduced transmission $S(t)$ is proportional to the excess quasiparticle density [@Lobo2005], $S = CN_{\mathrm{ex}}$, where $C$ is just a constant to convert from signal to quasiparticle density. Hence, $$-\frac{1}{S(t)}\frac{dS(t)}{dt} = \frac{2R}{C}(S(t)+2S_{\mathrm{th}}), \label{eq2}$$ with $S_{\mathrm{th}} = CN_{\mathrm{th}}$. Eq. is consistent with the linear behavior demonstrated in FIG. \[Fig3\]. The field dependence requires the prefactor $R$ to decrease with field.
To interpret the field dependence shown in FIG. \[Fig3\], it is a prerequisite to understand how the field changes the electronic states of the superconductor. If spin-orbit scattering is small, the magnetic field could make the majority of quasiparticles have one spin direction. (This is the same polarization that gives Pauli paramagnetism to metals.) Spin polarization will slow the recombination because only quasiparticles with opposite spins can recombine. A recombination model including this spin polarization effect is discussed in Section 2 of the Supplemental Material. In this case, the recombination equation remains in the same form as Eq. , but with the coefficient $2R/C$ replaced by $(8R/C)(N^{\uparrow}N^{\downarrow}/N^2)$, where $N^{\uparrow}$ and $N^{\downarrow}$ are respectively the densities of spin-up and spin-down quasiparticles. The quasiparticle spin-polarization factor $N^{\uparrow}N^{\downarrow}$ would depend on the magnetic field in the limit of weak spin-orbit coupling, just as in the Pauli paramagnetism of metals. According to the BCS theory, electrons form spin-singlet pairs condensed in the ground state; the spin susceptibility vanishes as the temperature approaches 0. The studies of superconductor spin susceptibility were done on thin films with thickness so small that the effect of a magnetic field on the electron orbit could be neglected. Paramagnetic splitting of the quasiparticle density of states was observed in 5 nm aluminum films in a parallel magnetic field [@Meservey1970]. In a study of magnetic field effects on far-infrared absorption of thin superconducting aluminum films, van Bentum and Wyder [@Bentum1986] concluded that paramagnetic splitting was important in their thinnest films, but did not allow for quasiparticle spin polarization. If a high degree of spin polarization existed, the recombination rate would be slowed much more than observed. However, spin-orbit scattering must be considered. Tedrow and Meservey observed the spin-state mixing in thin aluminum films due to spin-orbit scattering [@Tedrow1971]. They defined a spin-orbit scattering parameter $b\equiv\hbar/3\Delta\tau_{\mathrm{so}}$ to describe the degree of spin-orbit scattering, where $\tau_{\mathrm{so}}$ is the spin-orbit scattering time. They calculated that, as $b$ is increased to 0.5, the spin-up and spin-down quasiparticle density of states completely mix, leaving no clear signature of the two-peak feature in the density of states due to Zeeman splitting. Considering the short spin-orbit scattering time measured [@Hake1967] in NbTi, $\tau_{\mathrm{so}} = 3.0\times 10^{-14}$ s, and using the $\Delta$ of Nb$_{0.5}$Ti$_{0.5}$N and NbN, we estimate that $b = 4.2$ and 3.3 for Nb$_{0.5}$Ti$_{0.5}$N and NbN, respectively. We believe that spin is not a good quantum number in our samples, requiring us to look beyond spin polarization to understand the recombination.
In a study [@Xi2010] of the optical conductivity of Nb$_{0.5}$Ti$_{0.5}$N, we found that a parallel magnetic field breaks the time-reversal symmetry of the Cooper pairs and decreases the superconducting energy gap. The physics is similar to magnetic-impurity-induced pair-breaking effects, as originally formulated by Abrikosov and Gor’kov [@Abrikosov1961]. In a magnetic field, one must distinguish between the spectroscopic energy gap 2$\Omega_G$ and the pair-correlation gap $\Delta$. These gaps [@Skalski1964] are plotted in FIG. \[Fig4\]a as squares and triangles respectively. The real part of the optical conductivity, corresponding to the electromagnetic absorption, shows a clear suppression of the energy gap 2$\Omega_G$ with field (squares in FIG. \[Fig4\]a). The imaginary conductivity is a measure of the superconducting condensate density $N_{\mathrm{sc}}$, which goes as $\Delta^2$. The field dependences of $\Delta$ and of $\sqrt{N_{\mathrm{sc}}}$ (shown as circles) agree well, providing clear evidence for a weakening of superconductivity by the magnetic field. The quantities $\Omega_G$, $\Delta$, and $N_{\mathrm{sc}}$ for NbN, obtained using the same technique (in Section 3 of the Supplemental Material), are plotted in FIG. \[Fig4\]b. The NbN field dependence is qualitatively different from that of Nb$_{0.5}$Ti$_{0.5}$N because in this thicker film the applied field induces a spatial variation in the order parameter, making the weakening of superconductivity be proportional to the field, rather than being quadratic in field as in the much thinner Nb$_{0.5}$Ti$_{0.5}$N [@Parks1969]. The energy gaps will be used in the following analysis.
The field dependence, shown in FIG. \[Fig3\], is dominated by the recombination rate coefficient $R$. On the one hand, by explicitly solving Eq. one can identify a low-fluence recombination rate $1/\tau_{\mathrm{eff}} = 4RN_{\mathrm{th}}$. (See Section 4 of the Supplemental Material.) The field dependence of the thermal quasiparticle density $N_{\mathrm{th}}$ results from the field-dependent energy gap and the quasiparticle density of states [@Bayrle1989]. On the other hand, the effective lifetime of the excess quasiparticles is modified from the intrinsic value $\tau_R$, and is tied to the rates at which the phonons, produced in recombination events, re-break pairs (1/$\tau_B$) or escape from the film (1/$\tau_{\gamma}$) [@Gray1971]. The quasi-equilibrium values of $\tau_R$ and $\tau_B$ were derived by Kaplan *et al.* [@Kaplan1976]. Magnetic-field-induced pair breaking decreases the spectroscopic energy gap (FIG. \[Fig4\]) and modifies the quasiparticle density of states, resulting in a decrease in $\tau_R$ and an increase in $\tau_B$. The field independent phonon escape time is determined by the film thickness and the acoustic mismatch between the film and the environment [@Chang1978]. The recombination rate coefficient $R$ (and, hence, the slope of $1/\tau_{\mathrm{eff}}$ in FIG. \[Fig3\]) is therefore field-dependent through $N_{\mathrm{th}}$, $\tau_R$ and $\tau_B$. (See FIG. S7 in the Supplemental Material.) The equation is involved but, when we compute the slope vs $\Omega_G$ for Nb$_{0.5}$Ti$_{0.5}$N and NbN, shown in FIG. \[Fig4\]c and FIG. \[Fig4\]d, we obtain a basically linear relation. This calculation implies a connection between the field-dependent quasiparticle recombination and the field-induced pair breaking. The linear relation can be explained by considering only the field-induced gap reduction. (See FIG. S7 in the Supplemental Material.) The finite y-intercepts in FIG. \[Fig4\]c and FIG. \[Fig4\]d are intriguing, bringing out the question of how the photoexcited quasiparticles relax in a gapless superconductor, motivating challenging experiments to probe the gapless regime.
In conclusion, our time-resolved pump-probe measurements on metallic *s*-wave superconductors reveal a slowing of quasiparticle recombination in an external magnetic field. The field was aligned parallel to the thin-film sample surface, to minimize effects due to vortices. There are two possible causes of the observed slowing: field-induced spin imbalance and field-induced gap reduction. The spin imbalance is unlikely to be important in Nb$_{0.5}$Ti$_{0.5}$N and NbN due to strong spin-orbit scattering. This scenario can be tested by investigating materials with small spin-orbit scattering. The field-induced gap reduction alone can explain quantitatively the slowing of recombination, and we conclude it to be the dominant effect observed in our experiment.
We thank P. Bosland and E. Jacques for providing the samples, J. J. Tu for access to the magnet, R. P. S. M. Lobo for data acquisition software development, G. Nintzel and R. Smith for technical assistance, and P. J. Hirschfeld for discussions. This work was supported by the U.S. Department of Energy through contracts DE-ACO2-98CH10886 (Brookhaven National Laboratory) and DEFG02-02ER45984 (University of Florida), and by the National Research Foundation of Korea through Grant No. 20100008552.
[99]{}
D. M. Ginsberg, Phys. Rev. Lett. **8**, 204 (1962).
J. R. Schrieffer and D. M. Ginsberg, Phys. Rev. Lett. **8**, 207 (1962).
L. Tewordt, Phys. Rev. **127**, 371 (1962).
A. Rothwarf and N. Taylor, Phys. Rev. Lett. **19**, 27 (1967).
I. Schuller and K. E. Gray, Phys. Rev. B **12**, 2629 (1975).
C. C. Chi, M. M. T. Loy, and D. C. Cronemeyer, Phys. Rev. B **23**, 124 (1981).
J. F. Federici, B. I. Greene, P. N. Saeta, D. R. Dykaar, F. Sharifi, and R. C. Dynes, Phys. Rev. B **46**, 11153 (1992).
G. L. Carr, R. P. S. M. Lobo, J. LaVeigne, D. H. Reitze, and D. B. Tanner, Phys. Rev. Lett. **85**, 3001 (2000).
J. Demsar, R. D. Averitt, A. J. Taylor, V. V. Kabanov, W. N. Kang, H. J. Kim, E. M. Choi, and S. I. Lee, Phys. Rev. Lett. **91**, 267002 (2003).
M. Beck, M. Klammer, S. Lang, P. Leiderer, V. V. Kabanov, G. N. Gol’tsman, and J. Demsar, Phys. Rev. Lett. **107**, 177007 (2011).
N. Gedik, P. Blake, R. C. Spitzer, J. Orenstein, Ruixing Liang, D. A. Bonn, and W. N. Hardy, Phys. Rev. B **70**, 014504 (2004).
P. Kusar, V. V. Kabanov, J. Demsar, T. Mertelj, S. Sugai, and D. Mihailovic, Phys. Rev. Lett. **101**, 227001 (2008).
T. Mertelj, V. V. Kabanov, C. Gadermaier, N. D. Zhigadlo, S. Katrych, J. Karpinski, and D. Mihailovic, Phys. Rev. Lett. **102**, 117002 (2009).
D. Mihailovic and J. Demsar, *Spectroscopy of Superconducting Materials* (American Chemical Society, 1999), pp. 230–244.
S. B. Kaplan, C. C. Chi, D. N. Langenberg, J. J. Chang, S. Jafarey, and D. J. Scalapino, Phys. Rev. B **14**, 4854 (1976).
J. J. Chang and D. J. Scalapino, Phys. Rev. B **15**, 2651 (1977).
J. L. Levine amd S. Y. Hsieh, Phys. Rev. Lett. **20**, 994 (1968).
N. Perrin, Phys. Lett. A **90**, 67 (1982).
P. Fulde, Mod. Phys. Lett. B **24**, 2601 (2010).
K. Holdik, M. Welte, and W. Einsenmenger, J. Low Temp. Phys. **58**, 379 (1985).
R. P. S. M. Lobo, J. D. LaVeigne, D. H. Reitze, D. B. Tanner, Z. H. Barber, E. Jacques, P. Bosland, M. J. Burns, and G. L. Carr, Phys. Rev. B **72**, 024510 (2005).
X. Xi, J. Hwang, C. Martin, D. B. Tanner, and G. L. Carr, Phys. Rev. Lett. **105**, 257006 (2010).
R. Meservey, P. M. Tedrow, and P. Fulde, Phys. Rev. Lett. **25**, 1270 (1970).
P. J. M. van Bentum and P. Wyder, Phys. Rev. B **34**, 1582 (1986).
P. M. Tedrow and R. Meservey, Phys. Rev. Lett. **27**, 919 (1971).
R. R. Hake, Appl. Phys. Lett. **10**, 189 (1967).
A. A. Abrikosov and L. P. Gor’kov, Sov. Phys. JETP **12**, 1243 (1961).
S. Skalski, O. Betbeder-Matibet, and P. R. Weiss, Phys. Rev. **136**, A1500 (1964).
R. D. Parks, *Superconductivity* (Marcel Dekker, New York, 1969).
R. Bayrle and O. Weis, J. Low Temp. Phys. **76**, 143 (1978).
K. E. Gray, J. Phys. F: Met. Phys. **1**, 290 (1971).
J. J. Chang and D. J. Scalapino, J. Low Temp. Phys. **31**, 1 (1978).
**Supplemental Material**
1. P
====
In this section we show that the phonon bottleneck effect can be included in the effective recombination rate $R$ as a proportionality coefficient. Consider the Rothwarf-Taylor equations [@Rothwarf1967] for the coupled populations of quasiparticles and phonons in the absence of external quasiparticle injection, $$\frac{dN}{dt} = -\tau_R^{-1}N^2+\tau_B^{-1}N_{\omega}, \label{eqS1}\tag{S1}$$ $$\frac{dN_{\omega}}{dt} = \frac{1}{2}\tau_R^{-1}N^2-\frac{1}{2}\tau_B^{-1}N_{\omega}-\tau_{\gamma}^{-1}(N_{\omega}-N_{\omega,\mathrm{th}}),\label{eqS2}\tag{S2}$$ where $N(t)$ and $N_{\omega}(t)$ are respectively the densities of quasiparticles and high-energy (defined as $\hbar\omega\ge 2\Delta$) phonons, $\tau_R^{-1}$ is the intrinsic quasiparticle recombination rate, $\tau_B^{-1}$ is the phonon pair-breaking rate, and $\tau_{\gamma}^{-1}$ is the phonon escape rate. (The phonons may enter the substrate, enter the helium bath, or decay anharmonically to energies $\hbar\omega<2\Delta$.) In thermal equilibrium, the quasiparticles $N(t)$ and high-energy phonons $N_{\omega}(t)$ reach time-independent equilibrium values, linked by $N_{\mathrm{th}}=\sqrt{N_{\omega,\mathrm{th}}\tau_B^{-1}/\tau_R^{-1}}$, where $N_{\mathrm{th}}$ and $N_{\omega,\mathrm{th}}$ are for the quasiparticles and high-energy phonons, respectively. The coupled non-linear equations and can be solved numerically, e.g., using the Runge-Kutta integration method. To illustrate the solutions, we set $\tau_R^{-1}=1$ ns$^{-1}$ and $\tau_B^{-1}=10$ ns$^{-1}$, typical values for these quantities at low temperatures [@Lobo2005]. Without loss of generality we set $N_{\omega,\mathrm{th}}=1$, which determines $N_{\mathrm{th}}=\sqrt{10}$. The initial value of $N$ is determined by the pump laser fluence. We consider three cases, $N(0) = 2N_{\mathrm{th}}$ for low fluence, $N(0) = 5N_{\mathrm{th}}$ for intermediate fluence, and $N(0) = 10N_{\mathrm{th}}$ for high fluence. $N_{\omega}(0)$ is set to $N_{\omega,\mathrm{th}}$. For each case, we consider a range of phonon escape rates ($\tau_{\gamma}^{-1}$) from 1 ns$^{-1}$ to 10 ns$^{-1}$, spanning the strong to the weak phonon-bottleneck regime.
The numerical solutions of the Rothwarf-Taylor equations are shown in the first row of FIG. \[FigS1\]. After a short period of approximately 0.05 ns, the phonon bottleneck effect becomes clear. As expected, when the phonon escape rate increases, the phonon bottleneck effect becomes weaker and recombination becomes faster. In the second row of FIG. \[FigS1\], the rate of change of the excess quasiparticle density $N_{\mathrm{ex}}(t)=N(t)-N_{\mathrm{th}}$ is plotted vs. $N_{\mathrm{ex}}$. The early stage ($t<0.05$ ns) shown in the first row of FIG. \[FigS1\] has been skipped, because the quasiparticle and phonon populations are not yet equilibrated and it does not give information about the phonon bottleneck. Moreover, this stage is not temporally resolved in our experiment.
The quasi-linear relation shown in the second row of FIG. \[FigS1\] suggests we can define an effective recombination rate $R$ as $-(dN_{\mathrm{ex}}/dt)/N_{\mathrm{ex}}$. This rate, defined as the slope of $-dN_{\mathrm{ex}}/dt$ vs. $N_{\mathrm{ex}}$, is plotted vs. $\tau_{\gamma}^{-1}$ in FIG. \[FigS2\]. As $N(0)$ increases from 2$N_{\mathrm{th}}$ to 10$N_{\mathrm{th}}$, the relation between $R$ and $\tau_{\gamma}^{-1}$ becomes almost linear. We estimated that, for our experimental conditions, $N(0)$ in our samples was three orders of magnitude greater than $N_{\mathrm{th}}$ at the lowest laser fluence. Therefore it is safe to conclude that the effective recombination rate $R$ scales linearly with the phonon escape rate $\tau_{\gamma}^{-1}$.
2. P
====
Our recombination model can be extended to include the magnetic-field dependence due to quasiparticle spin polarization. In a quasiparticle recombination event, both a spin-up and a spin-down quasiparticle are needed to form a Cooper pair. We use an equation similar to the band-to-band recombination equation in a semiconductor to describe the quasiparticle recombination, $$\frac{dN}{dt}=\frac{dN^{\uparrow}}{dt}+\frac{dN^{\downarrow}}{dt} = -8R(N^{\uparrow}N^{\downarrow}-N_{\mathrm{th}}^{\uparrow}N_{\mathrm{th}}^{\downarrow}),\label{eqS3}\tag{S3}$$ where $\uparrow$ and $\downarrow$ denote the spin-up and spin-down populations, respectively. This reduces to Eq. (1) in the main text when the spin-up and spin-down populations are equal. At a given condition, Eq. can be rewritten as $-(dN_{\mathrm{ex}}/dt)/N_{\mathrm{ex}}=8R(N^{\uparrow}N^{\downarrow})(N_{\mathrm{ex}}+2N_{\mathrm{th}})/N^2$, where $N_{\mathrm{ex}} = N-N_{\mathrm{th}}$. Relating the measured photoinduced transmission $S$ to the excess quasiparticle density, $S = CN_{\mathrm{ex}}$, where $C$ is a constant, the recombination equation can be further reduced to $$-\frac{1}{S}\frac{dS}{dt}=8\frac{R}{CN^2}(N^{\uparrow}N^{\downarrow})(S+2S_{\mathrm{th}}),\label{eqS4}\tag{S4}$$ where $S_{\mathrm{th}}=CN_{\mathrm{th}}$.
{width="100.00000%"}
![The effective recombination rate as a function of the phonon escape rate at low, intermediate and high fluence.[]{data-label="FigS2"}](FigS2.eps){width="40.00000%"}
![Majority spin fraction. The theory is based on Eqs. and . The data points are extracted from the slopes in FIG. 3 in the main text, all scaled so that the majority spin fraction is 1/2 at 0 T.[]{data-label="FigS3"}](FigS3.eps){width="40.00000%"}
The majority spin fraction $N^{\downarrow}/N$ can be calculated from the paramagnetic model, assuming Fermi-Dirac distribution of quasiparticles $f(E)$ and the quasiparticle density of states $D(E)$ from the BCS theory, $$\begin{aligned}
N^{\downarrow} &= 2\int_0^{\infty}f(E)D(E+\mu H)dE,\label{eqS5}\tag{S5}\\
N^{\uparrow} &= 2\int_0^{\infty}f(E)D(E-\mu H)dE.\label{eqS6}\tag{S6}\end{aligned}$$ If we assume that the field dependence we see in FIG. 3 in the main text is only through the product $N^{\uparrow}N^{\downarrow}$, we can extract the majority spin fraction at different fields. The results are compared with the calculation in FIG. \[FigS3\]. Pure Pauli paramagnetism predicts a stronger magnetic field dependence than observed in our data.
The theory, however, must consider the strong spin-orbit scattering in NbN and Nb$_{0.5}$Ti$_{0.5}$N. In the main text, we estimate the strong spin-state mixing in the Nb$_{0.5}$Ti$_{0.5}$N and NbN samples. Based on that argument and the analysis shown in this section, we expect that the spin polarization factor $N^{\uparrow}N^{\downarrow}$ is weakly dependent on the field for our samples.
3. A NN
=======
![The transmission and reflection of NbN in parallel fields and 2 K, normalized to the corresponding normal-state values. The angle of incidence $\theta_i$ for both transmission and reflection was 30$^{\circ}$.[]{data-label="FigS4"}](FigS4.eps){width="49.00000%"}
We studied the magnetic-field-induced effects in the Nb$_{0.5}$Ti$_{0.5}$N and NbN thin films using Fourier transform far-infrared spectroscopy. The experimental technique and the analysis for Nb$_{0.5}$Ti$_{0.5}$N can be found in Ref. [@Xi2010]. Here we analyze the superconducting-state to normal-state transmission ratio $\mathcal{T}_s/\mathcal{T}_n$ and reflection ratio $\mathcal{R}_s/\mathcal{R}_n$ for NbN shown in FIG. \[FigS4\], measured at 2 K with the magnetic field parallel to the film. The data were taken with 4 cm$^{-1}$ (0.5 meV) resolution, so that the fringes due to the multiple internal reflections in the substrate were not resolved. The angle of incidence for both transmission and reflection was 30$^{\circ}$. The NbN thin film has a normal-state conductivity $\sigma_n = 2.0\times 10^3~\mathrm{Ohm}^{-1}\mathrm{cm}^{-1}$, determined from its normal-state transmittance and thickness. The MgO substrate has a refractive index $n \approx 3.0$ and negligible absorption in the far-infrared. The zero-field gap $\Delta_0 = 17.9$ cm$^{-1}$ is obtained from fitting the zero-field optical conductivity with the Mattis-Bardeen theory.
![The real (circles) and imaginary (triangles) parts of the optical conductivity of NbN at various parallel fields and $T=2$ K, normalized to the normal-state conductivity $\sigma_n$. The solid lines are fits to $\sigma_1/\sigma_n$ using the pair-breaking theory [@Skalski1964]. The dashed lines show the corresponding $\sigma_2/\sigma_n$ as determined by a Kramers-Kronig transform of the fit to the real part.[]{data-label="FigS5"}](FigS5.eps){width="49.00000%"}
![Field dependence of the pair-breaking parameter $\Gamma$ determined from the optical conductivity of NbN, normalized to the zero-field gap. The dashed line is a linear fit.[]{data-label="FigS6"}](FigS6.eps){width="40.00000%"}
From the transmission and reflection ratios, we extracted the real ($\sigma_1$) and imaginary ($\sigma_2$) parts of the optical conductivity using the method discussed in Ref. [@Xi2010]. The results are shown in FIG. \[FigS5\]. We found that the field dependence can be explained well by the pair-breaking theory. (See details of the theory in Ref. [@Skalski1964].)
The pair-breaking parameter $\Gamma$ is the only fitting parameter, describing the strength of pair breaking due to the magnetic field. Its value at different fields is shown in FIG. \[FigS6\]. From $\Gamma$ we calculated the pair-correlation gap $\Delta$ and the effective spectroscopic gap $\Omega_G$ using $\mathrm{ln}(\Delta/\Delta_0) = -\pi\Gamma/4\Delta$ and $\Omega_G = \Delta[1-(\Gamma/\Delta)^{2/3}]^{3/2}$. These quantities are shown in FIG. 4b in the main text. The superconducting condensate density $N_{\mathrm{sc}}$ is estimated from the below-gap part of $\sigma_2$ at $T\ll T_c$, which has the form, $$\sigma_{2}(\omega) = \frac{N_{\mathrm{sc}}e^2}{m\omega},\label{eqS7}\tag{S7}$$ where $e$ and $m$ are the electron charge and mass, respectively.
4. E
====
The recombination equation, Eq. (2) in the main text, links the measured $S(t)$ to the model, $$-\frac{1}{S(t)}\frac{dS(t)}{dt} = \frac{2R}{C}(S(t)+2S_{\mathrm{th}}), \nonumber$$ where $S_{\mathrm{th}} = CN_{\mathrm{th}}$. This equation has the following exact solution: $$S(t) = S(0)\frac{2e^{-t/\tau}}{2+\frac{\displaystyle S(0)}{\displaystyle S_{\mathrm{th}}}(1-e^{-t/\tau})}.\label{eqS8}\tag{S8}$$ Here $\tau$ is the effective lifetime: $\tau =1/4RN_{\mathrm{th}}$. In the low-fluence regime, especially when $S(0)\ll S_{\mathrm{th}}$, the solution is close to an exponential decay. As the fluence increases, the deviation from a simple exponential decay becomes significant.
5. E
====
On the one hand, the quasiparticles, interacting with phonons in the system, decay with an effective lifetime $\tau_{\mathrm{eff}}= \tau_{\gamma}+(1/2)\tau_R(1+\tau_{\gamma}/\tau_B)$, where $\tau_{\gamma}$, $\tau_R$, and $\tau_B$ are the same quantities as defined above and in the main text [@Gray1971]. At low temperatures, $\tau_R\gg\tau_{\gamma}$ and $\tau_{\gamma}\gg\tau_B$ [@Lobo2005]. The effective lifetime can be approximated as $\tau_{\mathrm{eff}}=\tau_R\tau_{\gamma}/2\tau_B$. On the other hand, by solving the recombination equation proposed in the main text, one can identify an effective quasiparticle lifetime $\tau_{\mathrm{eff}}=1/4RN_{\mathrm{th}}$, as shown in the previous section. As a result, $$R=\frac{\tau_B}{2\tau_{\gamma}\tau_RN_{\mathrm{th}}}=\frac{\tau_{\gamma}^{-1}\tau_R^{-1}}{2\tau_B^{-1}N_{\mathrm{th}}}. \label{eqS9}\tag{S9}$$ The phonon escape rate $\tau_{\gamma}^{-1}$ is expected to be independent of the gap, as discussed in the main text. A theory for $\tau_R^{-1}$ and $\tau_B^{-1}$ has been given by Kaplan *et al*. [@Kaplan1976], $$\begin{aligned}
\tau_R^{-1}&(\omega)=\frac{\tau_0^{-1}}{(k_BT_c)^3[1-f(\omega)]}\int_{\omega+\Delta}^{\infty}\frac{\Omega^2(\Omega-\omega)}{\sqrt{(\Omega-\omega)^2-\Delta^2}}\nonumber\\
&\cdot\left[1+\frac{\Delta^2}{\omega(\Omega-\omega)}\right][n(\Omega)+1]f(\Omega-\omega)d\Omega,\label{eqS10}\tag{S10}\end{aligned}$$ $$\begin{aligned}
\tau_B^{-1}&(\Omega) = \frac{\tau_{0,\mathrm{ph}}^{-1}}{\pi\Delta(0)}\int_{\Delta}^{\Omega-\Delta}\frac{\omega}{\sqrt{\omega^2-\Delta^2}}\frac{\Omega-\omega}{\sqrt{(\Omega-\omega)^2-\Delta^2}}\nonumber\\
&\cdot\left[1+\frac{\Delta^2}{\omega(\Omega-\omega)}\right][1-f(\omega)-f(\Omega-\omega)]d\omega,\label{eqS11}\tag{S11}\end{aligned}$$ where $\tau_0^{-1}$ and $\tau_{0,\mathrm{ph}}^{-1}$ are respectively the characteristic lifetimes of the quasiparticles and phonons, determined by the electron-phonon coupling and phonon density of states. The quantities $f$ and $n$ are the Fermi-Dirac and Bose-Einstein distribution functions, respectively. $\tau_R^{-1}$ should be evaluated at the quasiparticle gap energy $\omega=\Delta$ and $\tau_B^{-1}$ should be evaluated at the phonon energy $\Omega = 2\Delta$ for the estimation of their near-equilibrium values. For $\tau_R^{-1}$ $$\begin{aligned}
\tau_R^{-1}(\Delta)&\approx \frac{\tau_0^{-1}}{(k_BT_c)^3}\int_{2\Delta}^{\infty}d\Omega\frac{\Omega^3 e^{-(\Omega-\Delta)/k_BT}}{\sqrt{(\Omega-\Delta)^2-\Delta^2}}\nonumber\\
&= \frac{\tau_0^{-1}e^{-\Delta/k_BT}}{(k_BT_c)^3}\int_0^{\infty}dx\frac{(x+2\Delta)^{5/2}}{x^{1/2}}e^{-x/k_BT},\label{eqS12}\tag{S12}\end{aligned}$$ in which we have replaced the Bose factor $n(\Omega)$ and the Fermi factor $f(\Delta)$ by their low-temperature values. Because only small $x$ in the integrand contributes significantly to the integral, an expansion of the numerator yields $$\begin{aligned}
\tau_R^{-1}(\Delta)&\approx \frac{\tau_0^{-1}e^{-\Delta/k_BT}}{(k_BT_c)^3}(2\Delta)^{5/2}\Big[\int_0^{\infty}dx x^{-1/2}e^{-x/k_BT}\nonumber\\
&\qquad\qquad\qquad\qquad+\frac{5}{4\Delta}\int_0^{\infty}dx x^{1/2}e^{-x/k_BT}\Big]\nonumber\\
& = \frac{\tau_0^{-1}e^{-\Delta/k_BT}}{(k_BT_c)^3}(2\Delta)^{5/2}\sqrt{2\pi k_BT}\left(1+\frac{5k_BT}{8\Delta}\right).\label{eqS13}\tag{S13}\end{aligned}$$
![The dependence of $\tau_R$, $\tau_B$, $N_{\mathrm{th}}$, and $R$, on the spectroscopic gap, calculated directly from Eqs. , , , and without approximations.[]{data-label="FigS7"}](FigS7.eps){width="40.00000%"}
The phonon pair-breaking rate $\tau_B^{-1}$ has a simple form for its near-equilibrium state, given in Ref. [@Kaplan1976] as $$\tau_B^{-1}(2\Delta)=\tau_{0,\mathrm{ph}}^{-1}\frac{\Delta}{\Delta_0}[1-2f(\Delta)]\approx \tau_{0,\mathrm{ph}}^{-1}\frac{\Delta}{\Delta_0}.\label{eqS14}\tag{S14}$$ The quasiparticle density is $$N_{\mathrm{th}}\approx N(0)\sqrt{2\pi\Delta k_BT}e^{-\Delta/k_BT}.\label{eqS15}\tag{S15}$$ Substituting Eqs. – into Eq. yields $$R\approx\frac{2\sqrt{2}\Delta_0}{N(0)}\frac{\tau_{\gamma}^{-1}\tau_0^{-1}}{\tau_{0,\mathrm{ph}}^{-1}}\frac{1}{(k_BT_c)^3}\left(1+\frac{5k_BT}{8\Delta}\right)\Delta.\label{eqS16}\tag{S16}$$ In the context of pair breaking, $\Delta$ should everywhere be replaced by the field-dependent spectroscopic gap $\Omega_G$.
The gap dependence of $\tau_R$, $\tau_B$, $N_{\mathrm{th}}$, and $R$ can also be numerically evaluated directly using Eqs. , , , and without approximations. The results are shown in FIG. \[FigS7\], confirming the linear relation between $R$ and $\Omega_G$, discussed in the main text and shown in FIG. 4c and 4d.
[99]{}
A. Rothwarf and N. Taylor, Phys. Rev. Lett. **19**, 27 (1967).
R. P. S. M. Lobo, J. D. LaVeigne, D. H. Reitze, D. B. Tanner, Z. H. Barber, E. Jacques, P. Bosland, M. J. Burns, and G. L. Carr, Phys. Rev. B **72**, 024510 (2005).
X. Xi, J. Hwang, C. Martin, D. B. Tanner, and G. L. Carr, Phys. Rev. Lett. **105**, 257006 (2010).
S. Skalski, O. Betbeder-Matibet, and P. R. Weiss, Phys. Rev. **136**, A1500 (1964).
K. E. Gray, J. Phys. F: Met. Phys. **1**, 290 (1971).
S. B. Kaplan, C. C. Chi, D. N. Langenberg, J. J. Chang, S. Jafarey, and D. J. Scalapino, Phys. Rev. B **14**, 4854 (1976).
|
---
abstract: 'At the Large Hadron Collider (LHC), the most abundant processes which take place in proton-proton collisions are the generation of multijet events. These final states rely heavily on phenomenological models and perturbative corrections which are not fully understood, and yet for many physics searches at the LHC, multijet processes are an important background to deal with. It is therefore imperative that the modelling of multijet processes is better understood and improved. For this reason, a study has been done with several state-of-the-art Monte Carlo event generators, and their predictions are tested against ATLAS data using the [<span style="font-variant:small-caps;">Rivet</span>]{}framework. The results display a mix of agreement and disagreement between the predictions and data, depending on which variables are studied. Several points for improvement on the modelling of multijet processes are stated and discussed.'
address: 'School of Physics, University of the Witwatersrand, Johannesburg 2050, South Africa.'
author:
- Stefan von Buddenbrock
bibliography:
- 'ref.bib'
title: Performance of various event generators in describing multijet final states at the LHC
---
Introduction {#introduction .unnumbered}
============
The biggest challenges to deal with in proton-proton ($pp$) collisions arise from multijet processes, as far as Standard Model (SM) backgrounds are considered. Due to the nature of quantum chromodynamics (QCD), multijet production processes have the largest cross sections at the Large Hadron Collider (LHC). In addition to this, their partial reliance on non-perturbative QCD makes them difficult to deal with from a theoretical perspective. This is because simulation of fragmentation and hadronisation depend on a non-perturbative calculations, these often being done using phenomenological models. It is therefore of importance to study the performance of event generators in describing multijet final states, since certain combinations of matrix element (ME) calculations, parton shower (PS) and hadronisation models do not always provide an accurate description of the data.
In ATLAS, a number of generators are used to model multijet processes. These are discussed in detail in the next section. The predictions of these generators can be compared both to each other and to data corrected for detector effects (unfolded datasets). The simplest way of doing this is by using the [<span style="font-variant:small-caps;">Rivet</span>]{}analysis system [@Buckley:2010ar], which has a large set of built in analyses and distributions of unfolded data from various experiments. This short paper will present a subset of distributions relating to multijet processes, and compare the current set of ATLAS Monte Carlo (MC) multijet samples to unfolded data. From these results, information can be extracted about how to improve the modelling of the generators for future generation of samples in ATLAS.
Multijet event generators in ATLAS {#multijet-event-generators-in-atlas .unnumbered}
==================================
A variety of MC event generators are used for studying multijet topologies in ATLAS. These involve different combinations of ME and PS programs. For a general review of event generators currently used in LHC physics, the reader is encouraged to look at Ref. [@Buckley:2011ms]. Below is a list of the event generators considered in this study, as well as a few notes about their set up:
- [<span style="font-variant:small-caps;">Pythia</span>]{}8 [@Sjostrand:2014zea]: The prediction by [<span style="font-variant:small-caps;">Pythia</span>]{}8 is sliced up by jet [$p_{\text{T}}$]{}using filters. The lowest [$p_{\text{T}}$]{}filtered samples use the [<span style="font-variant:small-caps;">Pythia</span>]{}8 built in diffractive scattering processes (`SoftQCD`) to generate events. The rest of the slices use the elastic scattering processes (`HardQCD`). The chosen tune for the [<span style="font-variant:small-caps;">Pythia</span>]{}8 samples is the A14 tune [@ATL-PHYS-PUB-2014-021], which assumes the `NNPDF23LO` parton density function (PDF).
- [<span style="font-variant:small-caps;">Sherpa</span>]{} [@Hoeche:2012yf]: The official [<span style="font-variant:small-caps;">Sherpa</span>]{}samples make use of a [$2\to3$]{}ME calculation,[^1] matched with a CKKW scheme to a default [<span style="font-variant:small-caps;">Sherpa</span>]{}PS that use the CT10 tune. This sample has known issues with forward jets. The [<span style="font-variant:small-caps;">Sherpa</span>]{}prediction is also sliced in jet [$p_{\text{T}}$]{}.
- [[<span style="font-variant:small-caps;">Powheg</span>]{}+[<span style="font-variant:small-caps;">Pythia</span>]{}]{}8 [@Alioli:2010xd]: The [<span style="font-variant:small-caps;">Powheg</span>]{}ME is generated using the `Dijet` code that is provided with version 2 of the [<span style="font-variant:small-caps;">Powheg-Box</span>]{}. It is passed to the [<span style="font-variant:small-caps;">Pythia</span>]{}8 PS, with the A14 tune. The sample is also sliced in jet [$p_{\text{T}}$]{}.
- [<span style="font-variant:small-caps;">Herwig</span>++]{} [@Bahr:2008pv]: Like [<span style="font-variant:small-caps;">Pythia</span>]{}8, the [<span style="font-variant:small-caps;">Herwig</span>++]{}sample makes use of the built-in `MEMinBias` process to simulate diffractive scattering for the lowest two slices in jet [$p_{\text{T}}$]{}, and the `MEQCD2to2` for the remaining slices. These samples make use of the UE-EE5 tune, and therefore the `CTEQ6L1` PDF.
- [MG5\_aMC@NLO+[<span style="font-variant:small-caps;">Pythia</span>]{}8]{} [@Alwall:2014hca]: The [MG5\_aMC@NLO+[<span style="font-variant:small-caps;">Pythia</span>]{}8]{}samples use a [$2\to4$]{}ME matched with a [<span style="font-variant:small-caps;">Pythia</span>]{}8 PS using the CKKW-L scheme. The ME makes use of the `NNPDF30NLO` PDF, while the PS uses the A14 tune as described above. These samples are sliced at the ME level in parton [$H_{\text{T}}$]{}.
Key comparisons to data {#key-comparisons-to-data .unnumbered}
=======================
As mentioned above, [<span style="font-variant:small-caps;">Rivet</span>]{}is used to compare the predictions of these variables against each other and unfolded data. In this short paper, the predictions are compared in the context of three different aspects of jet physics, namely azimuthal decorrelations, jet fragmentation and jet shapes. Note that all jets considered in the following analyses are constructed using the anti-$k_\text{T}$ algorithm [@Cacciari:2008gp] with a radius parameter of $R=0.6$.
Azimuthal decorrelations {#azimuthal-decorrelations .unnumbered}
------------------------
Purely elastic scattering of QCD partons most often results in a dijet event – that is, exactly two well separated jets in the final state. In such a case, the azimuthal separation between the two jets should be $\pi$ radians. However, in theory, one expects to see more QCD interactions in the elastic scattering of quarks and gluons. This extra activity can produce more jet activity in multijet events. Depending on how much more activity is found in the event, the azimuthal angle between the two leading jets will deviate from $\pi$. This is known as an azimuthal decorrelation.
In order to study azimuthal decorrelations, one typically looks at the azimuthal angle between the two leading jets in multijet events. ATLAS performed a differential cross section measurement of azimuthal decorrelation variables with the Run 1 7 TeV dataset [@STDM-2012-17]. The corresponding [<span style="font-variant:small-caps;">Rivet</span>]{}routine for this analysis is `ATLAS_2014_I1307243`. In , some comparison plots are shown from this analysis. The different generators mostly perform well against the data, although discrepancies arise in different regions of the distributions, particularly for [<span style="font-variant:small-caps;">Herwig</span>++]{}.
![Differential cross section measurements as a function of the azimuthal angle between the two leading jets in multijet events [@STDM-2012-17]. The plots are made for inclusive multijet events in bins of rapidity separation, with $2<\Delta y<3$ on the left and $4<\Delta y<5$ on the right.[]{data-label="fig:AD"}](images/AD_y2to3.pdf "fig:"){width="49.00000%"} ![Differential cross section measurements as a function of the azimuthal angle between the two leading jets in multijet events [@STDM-2012-17]. The plots are made for inclusive multijet events in bins of rapidity separation, with $2<\Delta y<3$ on the left and $4<\Delta y<5$ on the right.[]{data-label="fig:AD"}](images/AD_y4to5.pdf "fig:"){width="49.00000%"}
Jet fragmentation {#jet-fragmentation .unnumbered}
-----------------
The behaviour of the fragmentation function used in different PS models is most commonly studied by looking at the densities of jet constituents in selected jets. The simplest [<span style="font-variant:small-caps;">Rivet</span>]{}routine to use when studying jet fragmentation is `ATLAS_2011_I929691`, which is a 7 TeV measurement of charged jet constituent densities as a function of three different variables [@STDM-2011-14]. Firstly, the variable $z$ is scanned, which is the fraction of longitudinal momentum carried by a jet constituent: $$z=\frac{\vec{p}_\text{jet}\cdot\vec{p}_\text{ch}}{\left|\vec{p}_\text{jet}\right|^2}.$$ Here, $\vec{p}_\text{ch}$ denotes the 3-momentum of the charged jet constituent, and $\vec{p}_\text{jet}$ is the 3-momentum of the jet. Secondly, the distance between the jet axis and the jet constituent in units of $\phi$ and $y$ is scanned over (denoted by $r$). And thirdly, the jet constituent’s momentum transverse to the jet axis, $p_\text{T}^\text{rel}$, is scanned over: $$p_\text{T}^\text{rel}=\frac{\left|\vec{p}_\text{ch}\times\vec{p}_\text{jet}\right|}{\left|\vec{p}_\text{jet}\right|}.$$
![Ratio plots of the different event generators compared to the data in measurements of jet constituent densities as a function of $z$ (left) and $p_\text{T}^\text{rel}$ (right) [@STDM-2011-14]. These plots are both shown in the same bin of jet [$p_{\text{T}}$]{}.[]{data-label="fig:JF"}](images/JF_z.pdf "fig:"){width="49.00000%"} ![Ratio plots of the different event generators compared to the data in measurements of jet constituent densities as a function of $z$ (left) and $p_\text{T}^\text{rel}$ (right) [@STDM-2011-14]. These plots are both shown in the same bin of jet [$p_{\text{T}}$]{}.[]{data-label="fig:JF"}](images/JF_pt.pdf "fig:"){width="49.00000%"}
In , plots are shown for jet constituent densities as a function of two of these variables. In this case, [<span style="font-variant:small-caps;">Herwig</span>++]{}arguably performs the best compared with the data, and [<span style="font-variant:small-caps;">Sherpa</span>]{}tends to perform the poorest. This is most probably due to the old version of [<span style="font-variant:small-caps;">Sherpa</span>]{}used by ATLAS in the official samples. Recent studies on newer [<span style="font-variant:small-caps;">Sherpa</span>]{}samples in ATLAS have seen the problems with jet fragmentation fixed, although these results could not be shown in this short paper.
Jet shapes {#jet-shapes .unnumbered}
----------
Jet algorithms can tell us about the geometry of the constituents of a jet. But to understand how energy is distributed in the average jet, it is more instructive to look at jet shapes. Similarly to jet fragmentation measurements, jet shapes are studied through looking at jet constituent densities. These are distributed as a function of the distance away from the axis of a jet, $r$. Typically, we look at the jet [$p_{\text{T}}$]{}weighted density in bins of annulus areas in the jet cone, $$\rho(r)=\frac{1}{\Delta rN_\text{jet}}\sum_\text{jets}\frac{{\ensuremath{p_{\text{T}}}\xspace}(r-\Delta r/2, r+\Delta r/2)}{{\ensuremath{p_{\text{T}}}\xspace}(0,R)},$$ where $p_\text{T}(r_1,r_2)$ is the sum of the jet constituent [$p_{\text{T}}$]{}between $r_1$ and $r_2$ away from the jet cone axis. In addition to this, we measure the integrated [$p_{\text{T}}$]{}weighted density, $$\Psi(r)=\frac{1}{N_\text{jet}}\sum_\text{jets}\frac{{\ensuremath{p_{\text{T}}}\xspace}(0,r)}{{\ensuremath{p_{\text{T}}}\xspace}(0,R)}.$$
The [<span style="font-variant:small-caps;">Rivet</span>]{}routine `ATLAS_2011_S8924791` contains a large set of doubly differential jet shapes corresponding to an ATLAS 7 TeV measurement [@STDM-2010-10]. In , some plots are shown in a single bin of the ATLAS analysis. Here, most of the generators considered agree relatively well with the data. It should be noted that [[<span style="font-variant:small-caps;">Powheg</span>]{}+[<span style="font-variant:small-caps;">Pythia</span>]{}]{}seems to predict a different jet shape than what is seen in the data.
![Jet shape measurements in terms of differential [$p_{\text{T}}$]{}density (left) and integrated [$p_{\text{T}}$]{}density (right) [@STDM-2010-10].[]{data-label="fig:JS"}](images/JS_rho.pdf "fig:"){width="49.00000%"} ![Jet shape measurements in terms of differential [$p_{\text{T}}$]{}density (left) and integrated [$p_{\text{T}}$]{}density (right) [@STDM-2010-10].[]{data-label="fig:JS"}](images/JS_psi.pdf "fig:"){width="49.00000%"}
Summary {#summary .unnumbered}
=======
Using the ATLAS multijet samples, comparisons have been made to unfolded data using [<span style="font-variant:small-caps;">Rivet</span>]{}. Three different measurements have been considered in this short paper. In each, it can be seen that the different generators tend to perform better in some regions of the phase space than others, while there is no clear choice for one generator performing systematically better than any of the others.
However, the information from these comparisons is still useful for the ATLAS collaboration to improve the modelling of multijet processes by knowing where the current predictions fail. There are yet many more measurements that can be considered in this study, and in future these studies will be extended to a more comprehensive study.
[^1]: That is, up to three partons can be generated in the final state.
|
---
abstract: 'The volume of data and the velocity with which it is being generated by computational experiments on high performance computing (HPC) systems is quickly outpacing our ability to effectively store this information in its full fidelity. Therefore, it is critically important to identify and study compression methodologies that retain as much information as possible, particularly in the most salient regions of the simulation space. In this paper, we cast this in terms of a general decision-theoretic problem and discuss a wavelet-based compression strategy for its solution. We provide a heuristic argument as justification and illustrate our methodology on several examples. Finally, we will discuss how our proposed methodology may be utilized in an HPC environment on large-scale computational experiments.'
author:
- 'Henry Scharf[^1]'
- Ryan Elmore
- Kenny Gruchalla
bibliography:
- 'PWC.bib'
title: '**Prioritized Data Compression using Wavelets**'
---
\#1
[*Keywords:*]{} Wavelets, Data Compression, High Performance Computing
Introduction {#sec:intro}
============
The US Department of Energy recently published a document highlighting several issues related to data-intensive computing on future high-performance computing (HPC) systems [@chen2013synergistic] with particular emphasis given to data analysis and visualization in Chapter 4. The authors highlight the growing disparity between I/O and storage capabilities, and computational capabilities. They warn that “our ability to produce data is rapidly outstripping our ability to use it”, particularly in a meaningful manner. This statement echoes the sentiments expressed in previous DOE publications, [*e.g.*]{} @ahern2011scientific, @ashby2010opportunities, among others. The problem has recently manifested itself at the National Renewable Energy Laboratory in the form of large-scale wind-turbine array simulations. That is, data analysis tools and the computational machinery that supports them have not been able to scale with the HPC systems that are generating the wind-turbine array simulations.
These considerations motivate the following research on what we term a prioritized wavelet-based data compression methodology. We propose storing data in varying fidelities within the simulation space based on regions of saliency. That is, salient regions will be stored in a high fidelity whereas the less important regions will be more compressed. As an example, one might imagine the area directly in the wake of a wind turbine as being more important in future analysis/visualization experiments and, thus, we would like to retain the simulated data in its fullest fidelity in these regions. On the other hand, data on the periphery of the wind farm (less turbulent) might be less interesting from a subsequent analytic perspective, and may be stored in a lower fidelity in order to save space.
The wavelet representation of a signal has a history of use in both data compression and denoising [@nayson_g._p._wavelets_2008]. The two applications use the same basic algorithm of (1) performing a discrete wavelet transform, (2) setting all coefficients in the representation whose magnitude is below a threshold to zero, and then (3) reconstructing the signal based on the sparse wavelet representation. Wavelet compression has most commonly been used for images [@skodras_jpeg_2001] and time-series such as electrocardiogram signals [@hilton_wavelet_1997].
Optimal data compression sensitive to secondary analysis has not been generally investigated, though there are some specific applications in image processing. While not framed as an explicit secondary analysis, there have been algorithms created to find optimal compression of an image sensitive to human visual perception (see for example [@chandler_dynamic_2005] or the JPEG-2000 standards [@skodras_jpeg_2001]). We presume in our methodology that the form of secondary analysis can be expressed explicitly as a mathematical function on the data, however we expect the approach taken here may be extended to include more loosely defined secondary analyses.
The use of wavelet-based compression schemes is becoming increasingly popular in the data visualization domain, see for example @gruchalla-2009, @gruchalla2009visualization, and @gruchalla2011segmentation. We fully expect this trend to continue with their inclusion as the default compression tool in the VAPOR software package [@clyne2010vapor]. One of our aims with this research is to develop a compression strategy that allows for heterogeneous levels of compression throughout the simulation domain while remaining consistent with VAPOR’s use of the discrete wavelet transform.
While our current work does not address the motivating problem [*per se*]{} ( compression strategies for exascale-type problems), our intentionally narrow focus provides the foundation upon which future research may be built. We lay out the mathematical background, our current problem of interest, and provide a heuristic justification for our proposed solution in Section \[sec:notation\]. We illustrate the novel approach on several examples in Section \[sec:examples\]. Finally, in Section \[sec:conc\], we summarize our results and discuss future research directions as they relate to problems in HPC environments.
Notation/Formulation {#sec:notation}
====================
Brief Introduction to Wavelets
------------------------------
The notion of a wavelet is suggested by the name. They are ‘little waves’, in the sense that they possess some quality of oscillation, but have small, localized support. A single wavelet is one member of a complete set of basis functions with which we can represent a time series or, in general, a function. Though there are many such sets of basis functions which qualify as wavelets, it is useful to consider a particular set widely considered to be the simplest. The wavelets which make up this set are called Haar wavelets, and they are defined by translations and dilations of the so called *mother* Haar wavelet defined as $$\begin{aligned}
\psi(t) &=
\begin{cases}
1 \; &t \in \left[0, \frac{1}{2}\right)\\
-1 &t \in \left[\frac{1}{2}, 1\right)\\
0 & \text{otherwise}.
\end{cases}\end{aligned}$$ By choosing $j$ and $k$ appropriately, we can build a complete basis which spans $L^2(\mathbb{R})$, and thus represent all square integrable functions as linear combinations of these wavelets $$\begin{aligned}
\psi_{j, k}(t) &= 2^{j/2}\psi(2^j t - k)\end{aligned}$$ with time series representation $$\begin{aligned}
f(t) &= \sum_j\sum_k a_{j, k}\psi_{j, k}(t).\end{aligned}$$ These equations describe the continuous wavelet transform, but there are analogous forms for discrete representations as well. For a more thorough introduction to wavelets, see [@nayson_g._p._wavelets_2008].
These basis functions are useful for many reasons, including the fact that the wavelet representation is relatively efficient to compute, and the representation is robust in the presence of discontinuities compared to related methods. Additionally, the Haar wavelet is a member of a class of wavelets which form an orthonormal basis and provides the following useful form $$\begin{aligned}
\sum_{t=1}^n f^2(t) &=
\sum_k \sum_j a^2_{j, k}.
\label{eqn:parseval}\end{aligned}$$ This identity, sometimes referred to as Parseval’s relation, serves as the foundation for a class of estimators $\hat{\f_B}$ that we propose in Section \[sec:f\_hat\_B\].
Decision-Theoretic Prioritization {#sec:decision_theoretic}
---------------------------------
This formulation considers the situation where the secondary analysis to be performed upon $\f=\left(f(1), \dots, f(n)\right)^\prime$, some finite vector of data indexed by $t=1, \dots, n$, may be explicitly expressed as a transformation $\g\!:\! \mathbb{R}^n \rightarrow \mathbb{R}^m$. The flexible nature of $\g$ gives this procedure a wide range of applicability. Take for example the following hypothetical situation.
\[ex:moments\] Suppose we have data $\f$ for which we want to estimate the first four moments. Our secondary function then is given by the following map from $\mathbb{R}^n$ to $\mathbb{R}^4$: $$\begin{aligned}
\g(\f) =
\g(f(1), \dots, f(n)) &=
\left(
\sum_{t=1}^nf(t),\; \sum_{t=1}^nf^2(t),\;
\sum_{t=1}^nf^3(t),\; \sum_{t=1}^nf^4(t)
\right)^\prime.\end{aligned}$$ We will use the notation $g_i(\f)$ to refer to a single component of the vector $\g(\f)$, for example $$\begin{aligned}
g_i(\f) = \sum_{t=1}^nf^i(t), \quad i=1, \dots, 4.\end{aligned}$$ We will revisit this particular example in Example \[ex:mom\] in Section \[sec:examples\].
We suppose now that storing the full-fidelity data $\f$ is impractical, and we will instead be forced to make do with $\hat{\f}$, a compressed version of $\f$, which for the moment need not be wavelet-based. In order to give some simple but meaningful measure to the amount of error in $\hat{\f}$, and $\g(\hat{\f})$, we propose modeling the errors $\hat{\f} - \f$ as a random vector generated from some distribution with mean $\mathbf{0}$ and covariance matrix $\bmSigma$. In fact, for the case of wavelet-based compression, each $\hat{f}(t) - f(t)$ will not be random, but entirely deterministic. For even moderately large $n$ though, these values may behave similarly to random variables. By modeling the errors in this way, we are able to develop this problem from a decision-theoretic perspective [@casella2002] and minimize the expected squared distance between $\bmtheta = \g(\f)$ and a candidate estimator $\hat{\bmtheta} =
\g(\hat{\f})$. Specifically, we use the squared error loss function $$\begin{aligned}
\label{eqn:loss}
L(\bmtheta, \bm{a}) &=
\left(
\bmtheta - \bm{a}\right)^\prime
\left(\bmtheta - \bm{a}\right) =
||\bmtheta - \bm{a}||_2^2\end{aligned}$$ with corresponding risk for a candidate estimator $\bmdelta$ defined by $$\begin{aligned}
R(\bmtheta, \bmdelta) & =
\mathbb{E}\left[L(\bmtheta, \bmdelta)\right]. \nonumber\end{aligned}$$ Note that we are operating under the constraint that all candidate approximations, $\hat{\f}$, must be of the same fixed size, where size will be a measure of the total cost of storing the approximation $\hat{\f}$. In the case of wavelet-based compression, this is defined to be the number of non-zero wavelet coefficients. For complicated $\g$, $\f$, and approximations $\hat{\f}$, it may be an extremely intensive or even impossible computation to find the optimal such $\hat{\f}$. We therefore impose two limitations. First, we will require that we be able to linearize $\g$ and use the first-order Taylor approximation: $$\begin{aligned}
\g(\hat{\f}) - \g(\f) &\approx J_{\g} \cdot (\hat{\f} - \f). \nonumber\end{aligned}$$ where $J_{\g}$ is the $m \times n$ Jacobian matrix whose $i, t$ element is $\partial g_i(\f)/\partial f(t)$, and $i=1, \dots, m$, $t=1, \dots,
n$. Second, we limit ourselves to a subset of the class of all fixed-size $\hat{\f}$, defined in Section \[sec:f\_hat\_B\]. This class is defined such that approximations are relatively easy to compute, but still flexible enough in their structure to take into account the demands of the specific secondary analysis $\g$.
A class of ‘magnifying glass’ approximations $\hat{\f}_B$ {#sec:f_hat_B}
---------------------------------------------------------
From now on, $\hat{\f}$ will refer to an estimator made through (1) generating a wavelet transform of $\f$, (2) setting a fixed number of coefficients to zero, and then (3) reconstructing $\hat{\f}$ with this sparse wavelet representation. The class we propose yields estimators which are practically straightforward to produce, are mathematically tractable, and yield estimators with significant improvements compared to a natural baseline estimator $\tilde{\f}$. The baseline estimator is what would be produced if we completely ignored the secondary function $\g$, and instead followed the wavelet compression procedure which minimized the squared error norm $||\tilde{\f} -
\f||^2_2$ (where $\tilde{\f}$ is taken to be the same size as $\hat{\f}_B$). See the top plot in Figure \[fig:f\_hat\_B\_sample\] for one example.
First, we write the general $\hat{\f}$ with errors partitioned into two disjoint sets $B$ and $B^c$ whose union is the indexing set $T=\left\{1, \dots,
n\right\}$. The $t^{th}$ element may be written as $$\begin{aligned}
\hat{f}_{B}(t) &= f(t) + \varepsilon_B(t)\mathbf{1}_B(t) +
\varepsilon_{B^c}(t)\mathbf{1}_{B^c}(t) \nonumber\end{aligned}$$ where $\mathbf{1}_B(t)$ is the usual indicator function equal to one if $t \in
B$ and zero otherwise. Our class of $\hat{\f}_B$ will be those for which the mean squared error in $B$ is proportional to the mean squared error in $B^c$. That is, we define$$\begin{aligned}
\sigma^2_B &:=
|B|^{-1} \sum_{t \in B}\varepsilon_B^2(t), \ \text{and} \nonumber \\
\sigma^2_{B^c} &=
\frac{1}{\kappa} \sigma^2_{B} \label{eqn:kappa}\end{aligned}$$ where $|B| = $ the number of elements in $B$.
We can easily generate $\hat{\f}_B$ such that (\[eqn:kappa\]) holds (to a high level of precision) when we have orthonormal wavelets, and the set $B$ is made up of a small number of connected intervals in $T$. The reason we need $B$ to be this sort of set is so that we can make use of the compact support of our wavelets. In our toy examples we only use the Haar wavelet, but the procedure by which we generate $\hat{\f}_B$ may be extended to any orthonormal wavelet basis with minor modifications. We use the localized nature of the Haar wavelet basis functions along with (\[eqn:parseval\]) in the following way.
First, we define a set $\calB$, which is a set indexing all wavelet basis functions whose support overlaps the region $B \in T$. We then order from smallest to largest the coefficients of the wavelets in $\calB$ and ${\calB}^c$ by their squared value $$\begin{aligned}
(a^2_{\calB (1)}, \dots, a^2_{\calB (n_J)}), \text{ and }
(a^2_{\calB^c (1)}, \dots, a^2_{\calB^c (n_L)})\end{aligned}$$ and choose the unique pair of threshold values $\alpha^2_{\calB}$ and $\alpha^2_{\calB^c}$ so that $$\begin{aligned}
n^{-1}_J \sum_{a^2_{\calB (j)}<\alpha^2_{\calB}}a^2_{\calB (j)} &=
\kappa n^{-1}_L \sum_{a^2_{\calB^c (l)}<\alpha^2_{\calB^c}}a^2_{\calB^b (l)}.
\label{eqn:energy}\end{aligned}$$ The values $\alpha^2_{\calB}$ and $\alpha^2_{\calB^c}$ are unique because we have fixed the number of non-zero coefficients.
Because of the localized support of the wavelets, when $B$ is made up of a small number of connected subsets of $T$, the number of wavelets with support in both $B$ and $B^c$ will be small compared to the number of wavelets with support entirely in $B$ or $B^c$. Moreover, the coefficients which typically have small magnitudes are those which correspond to basis functions at finer scales, which means the wavelet coefficients which are most likely to be effected by thresholding are the ones who tend to have support entirely in $B$ or $B^c$. Therefore we will have $|B| \approx n_J$ and also $$\begin{aligned}
\frac1{|B|}\sum_{t \in B}(\hat{f}_{B}(t) - f(t))^2 &\approx
\frac1{n_J}\sum_{a^2_{\calB(j)}<\alpha^2_{\calB}}a^2_{\calB (j)}\end{aligned}$$ as well as the analogous result for $B^c$ and $\calB^c$. Therefore, ensuring (\[eqn:energy\]) in turn ensures that (\[eqn:kappa\]) approximately holds. In practice, when we implemented this procedure, we were able to generate $\hat{\f}_B$ for which $\hat{\kappa}$, the realized value of the ratio of mean squared errors, came very close to $\kappa$. When we specified $\kappa = 0.1$, for example, our realized ratio was generally in $(0.09, 0.11)$.
These $\hat{\f}_B$ are in some sense the simplest possible way to take into account the demands of the secondary analysis. We are partitioning the indexing set $T$ into two subsets, where values $f(t)$, $t \in B$ are *more important* than values $f(t^*)$, $t^* \notin B$ in accurately estimating $\g(\f)$ by a factor of $\kappa$. Therefore, a datum $f(t)$ is either important or *not* important, there is no spectrum of importance. Familiarly, we call these approximations $\hat{\f}_B$ ‘magnifying glass’ estimators with magnification factor $\kappa^{-1}$ since they effectively give us a closer look at region $B$ compared to $B^c$ (see bottom plot of Figure \[fig:f\_hat\_B\_sample\]). More subtle schemes are certainly worth investigating, but this first-order approach already yields promising results.
![The top plot shows the full fidelity time series in gray (‘Doppler’) with the baseline wavelet-based approximation in black. This approximation ignores the secondary analysis. The bottom plot shows the same time series in gray with an approximation $\hat{\f}_B$ in black. In this example, $\kappa$ was set at 0.02, with realized value $\hat{\kappa}=0.0209$, and $B=\{401, \dots, 600\}$ (inducated in both plots by the light shading). We set the size of this approximation so that only 10% of the wavelet coefficients were non-zero. The ‘magnifying glass’ approximation is noticeably closer to the truth in $B$ than in $B^c$.[]{data-label="fig:f_hat_B_sample"}](f_hat_B_sample-eps-converted-to.pdf){width="\linewidth"}
Figure \[fig:f\_hat\_B\_sample\] shows an implementation of this method. In the top plot, the gray curve shows the full-fidelity time series (‘Doppler’ [@nayson_g._p._wavelets_2008]) and the black step function is the baseline approximation $\tilde{\f}$. In the bottom plot, the gray curve is the same, but the approximation shown is a magnifying glass estimator with $B=\{401, \dots,
600\}$ and $\kappa=0.02$ ($\hat{\kappa}=0.0209$). The size of the approximation is such that 10% of the wavelet coefficients were allowed to be non-zero. The higher fidelity region appears to extend slightly beyond $B$ on both ends because of the presence of wavelet basis functions with overlapping support.
The values $\sigma^2_B$ and $\sigma^2_{B^c}$ will depend both on $\kappa$ and the size of the approximation. For instance, the smallest possible $\sigma^2_B$ and $\sigma^2_{B^c}$ for a given approximation size will occur when $\kappa=1$. As $\kappa$ decreases toward 0, we are sacrificing some overall increase in the error in $\hat{\f}_B$ in exchange for better estimating the parts of $\f$ that are most important for the secondary analysis $\g$, resulting in an overall improvement in the precision of $\g(\hat{\f}_B)$.
Importance Function {#sec:importance_function}
-------------------
In order to add some degree of saliency to regions within the simulation space, we introduce the ‘importance function’ concept. Recall the loss function defined in \[eqn:loss\] and note that it can be re-written (approximately) as $$\begin{aligned}
L(\g, \hat{\f}) &\approx
\left[J_{\g} \cdot (\hat{\f} - \f)\right]^\prime
\left[J_{\g} \cdot (\hat{\f} - \f)\right]
\\ \nonumber
&= (\hat{\f} - \f)^\prime J_{\g}^\prime J_{\g} (\hat{\f} - \f)\\ \nonumber
&= (\hat{\f} - \f)^\prime \M (\hat{\f} - \f).\end{aligned}$$ We will refer to $\M:=J_{\g}^\prime J_{\g}$ as the ‘importance matrix’ and its diagonal elements as the ‘importance function’. This name will become clear later when we show that the diagonal of this matrix largely determines the form of the optimal approximation. These $n$ diagonal elements can be thought of as defining a level of importance or weight for each datum $f(t)$.
We will now model the deterministic errors $\hat{\f} - \f$ as a random vector with mean vector $\mathbf{0}$ and covariance matrix $\bmSigma$. For approximations of type $\hat{\f}_B$, a natural covariance matrix to ascribe to this random variable is a diagonal matrix with elements equal to either $\kappa\sigma^2_{B^c}$ or $\sigma^2_{B^c}$. Using this model, our loss function takes a quadratic form, and the risk is approximated by $$\begin{aligned}
R(\g, \hat{\f}_B) &\approx
\E\left[(\hat{\f} - \f)^T\M(\hat{\f} - \f)\right] \nonumber\\
&=\tr(\M\Sigma) \nonumber\\
&=\sigma^2_{B^c}\left[\kappa\sum_{t \in
B}\sum_{i=1}^m \left(\frac{\partial g_i}{\partial f(t)}\right)^2
+ \sum_{t \notin B}\sum_{i=1}^m \left(\frac{\partial g_i}{\partial
f(t)}\right)^2\right]. \label{eqn:risk}\end{aligned}$$ In order to minimize $R(\g, \hat{\f}_B)$, we will need to take into account two effects determined by $B$. First, as the size of $B$ increases, $\sigma^2_{B^c}$ will increase, since more and more error in $B^c$ will be sacrificed to maintain the higher fidelity in $B$. Second, the choice of $B$ will have an impact on the two sums. If we choose $B$ such that the largest values of $\sum_{i=1}^m
\left(\partial g_i/\partial f(t)\right)^2$ are in the first term of Equation (\[eqn:risk\]), then we will reduce our risk function $R(\g, \hat{\f}_B)$ because $\kappa<1$. Taking these together, we can intuitively expect that the optimal set $B$ will be the smallest possible set such that the large elements on the diagonal of $\M$ have index inside $B$.
Idealized Secondary Analysis {#sec:contrived}
----------------------------
We now turn our attention to one idealized setting where the secondary analysis takes a contrived form as a basic check on casting the problem in this decision-theoretic manner. Consider the following secondary calculation $\g$ defined by $$\begin{aligned}
\g(\f)_i & := f(i)(\lambda\1_A(i) + \1_{A^c}(i)), \quad i = 1, \dots, n\end{aligned}$$ or written more explicitly $$\begin{aligned}
g_i(\f) &=
\begin{cases}
\lambda f(i), \quad & i\in A\\
f(i), & i \notin A
\end{cases} \\
i &\in \{1, \dots, 1024\}.\end{aligned}$$ This represents the case where our secondary analysis consists merely of multiplying some elements of $\f$ by a scalar $\lambda$, and retaining the rest unaltered. This is akin to placing some region of the data under a magnifying glass with a magnification factor of $\lambda$, and we therefore expect an estimator of type $\hat{\f}_B$ to be a good choice. For $\lambda$ greater than one, we expect that the optimal approximation $\hat{\f}_B$ ought to be one where $B \approx A$, since this is clearly the ‘important’ part of $\f$. The importance matrix is straightforward to calculate, since it is the product of two diagonal matrices with elements equal to either $\lambda$ or 1 depending on the index’s membership in $A$.
In the argument that follows, we let $\lambda$ go to infinity, and $\kappa$ go to zero, which corresponds to the case where $\g$ and $\f$ show infinite preference for region $A$ and $B$ respectively. In this extreme we will see that it is possible to verify $B=A$.
Starting from (\[eqn:risk\]) we have $$\begin{aligned}
R(\g, \hat{\f}_B) &=
\sigma^2_{B^c}\left[
\kappa\left(
\sum_{t \in A \cap B}\lambda^2 +
\sum_{t \in A^c \cap B}1\right) +
\left(
\sum_{t \in A \cap B^c}\lambda^2 +
\sum_{t \in A^c \cap B^c}1\right)
\right]\\
&= \sigma^2_{B^c}\left[
\kappa\left(
\lambda^2|A \cap B| + |A^c \cap B|\right) +
\lambda^2|A \cap B^c| + |A^c \cap B^C|
\right].\end{aligned}$$ As $\lambda$ approaches infinity, the risk will also become infinite, so we next normalize by the constant $\lambda^2$ to get $$\begin{aligned}
\frac{R(\g, \hat{\f}_B)}{\lambda^2} &=
\sigma^2_{B^c}\left[
\kappa\left(
|A \cap B| + \frac{|A^c \cap B|}{\lambda^2}\right) +
|A \cap B^c| + \frac{|A^c \cap B^c|}{\lambda^2}
\right]. \label{eqn:normrisk}\end{aligned}$$ Minimizing (\[eqn:normrisk\]) is equivalent to minimizing the risk, but now as we let $\lambda^2$ grow large, the quantity of interest simplifies to $$\begin{aligned}
\lim_{\lambda^2 \rightarrow \infty}
\left(
\frac{R(\g, \hat{\f}_B)}{\lambda^2}
\right) &=
\sigma^2_{B^c}
\left(
\kappa |A \cap B| + |A \cap B^c|
\right). \label{eqn:limnormrisk}\end{aligned}$$ To get any further, we need to know more about how $\sigma^2_{B^c}$ depends on our choice of $B$. In practice, this relationship will be complicated, but roughly speaking we expect $\sigma^2_{B^c}$ to be monotone increasing as $|B|$ increases as discussed in Section \[sec:importance\_function\] (see Figure \[fig:sigmaB\_vs\_sizeB\]). From (\[eqn:limnormrisk\]), we can see that the second factor in the limiting risk is independent of the size of $A^c \cap B$, and so we might as well choose $B^\prime = A \cap B$ instead of $B$. That is, $$\begin{aligned}
\lim_{\lambda^2 \rightarrow \infty}
\left(
\frac{R(\g, \hat{\f}_B)}{\lambda^2}
\right) &\geq
\lim_{\lambda^2 \rightarrow \infty}
\left(
\frac{R(\g, \hat{\f}_{A \cap B})}{\lambda^2}
\right),\end{aligned}$$ since $\sigma^2_{B^c}$ is monotone increasing in $|B|$. It is clear the optimal $B$ will be contained within $A$.
Letting $\kappa$ go to zero now yields $$\begin{aligned}
\lim_{\lambda^2 \rightarrow \infty, \; \kappa \rightarrow 0}
\left(
\frac{R(\g, \hat{\f}_B)}{\lambda^2}
\right) &=
\sigma^2_{B^c}|A \cap B^c|.\end{aligned}$$ Now as $B$ increases in size to approach $B=A$, $\sigma^2_{B^c}$ will increase, but when $B=A$, the second term is zero and the normalized risk vanishes. We will see in Examples \[ex:ind\_dop\] and \[ex:ind\_bumps\] that even when $\lambda$ and $\kappa$ are far from infinity and zero, this result still holds to large degree.
![This plot shows the roughly monotone behavior of $\sigma^2_{B^c}$ as we increase the size of $B$ for fixed $\kappa=0.1$. In this case, we used the ‘Doppler’ time series and let $B$ increase from $\{511, 512, 513\}$ to $\{91, \dots, 931\}$, increasing the width by 4 in each direction at each step. The units on the $y$-axis are specific to this example, but the monotone behavior is generally applicable.[]{data-label="fig:sigmaB_vs_sizeB"}](sigmaB_vs_sizeB-eps-converted-to.pdf){width="\linewidth"}
Examples {#sec:examples}
========
In considering possible sets $B \subset T$, it is computationally intractable to consider all $2^{|T|}$ possible subsets. Often though, we may have reason to believe *a priori* that the most important parts of $\f$ lie predominantly in a few connected subsets of the space $T$. For instance, if the data are spatially or temporally arranged, we may expect strong positive correlation in importance between neighboring data. In our motivating example related to wind turbine arrays, we expect that the most important regions are in and around the wakes. In the following examples we restrict the space of possible $\hat{\f}_B$ to include only those for which $B$ is a single connected interval in $T$.
We performed a nearly-exhaustive (see Appendix \[app:searches\]) search for the optimal $B$ for several toy examples. In each example our data $\f$ is one of two different one-dimensional time series, both widely used in simulation studies involving denoising and density estimation ([@donoho1994b] and [@donoho1995adapting]), and also used for illustration in [@nayson_g._p._wavelets_2008]. In each case we also specify a secondary analysis $\g$, and a value for $\kappa$. In all examples, the size of the approximation is such that 10% of the wavelet coefficients were allowed to be non-zero. The nearly-exhaustive search considers almost all possible uninterrupted intervals within $T=\{1, \dots, 1024\}$ for a fixed value $\kappa$. For each interval, the relative squared error is defined as $$\begin{aligned}
SE_B&=L(\g, \hat{\f}_B), \\
SE_0&=L(\g, \tilde{\f}), \& \\
\mbox{relSE}&:=SE_B/SE_0.\end{aligned}$$ Therefore, values of $\mbox{relSE}$ less than 1 represent improvements over the baseline wavelet compression, $\tilde{\f}$, which ignores the secondary function. The interval corresponding to the smallest $\mbox{relSE}$ is (nearly) optimal in the space of single uninterrupted intervals.
\[ex:ind\_dop\]
We offer here the results of implementation on a particular example of the idealized case mentioned in Section \[sec:contrived\]. For demonstration, the secondary function is $$\begin{aligned}
\g_1(\f)_i &= f(i)\left[
5\times \mathbf{1}_{\{401, \dots, 600\}}(i) +
\mathbf{1}_{\{401, \dots, 600\}^c}(i)
\right], \quad i=1, \dots, n
\end{aligned}$$ so that $\lambda = 5$ and $A = \{401, \dots, 600\} \subset T=\{1, \dots,
1024\}$. We therefore expect that the optimal $\hat{\f}_B$ will occur when $B
\approx \{401, \dots, 600\}$. Figure \[fig:ind\_dop\] shows the results of a nearly-exhaustive search over all single intervals $B$ in $T$ with fixed $\kappa=1/10$ for our first time series (‘Doppler’). The top plot is the full fidelity time series $\f$, with results overlaid. The bottom plot shows the importance function ($\diag(\M)$) with results overlaid. Each overlaid segment represents a proposed interval $B$, plotted at a height proportional to $\mbox{relSE}$, the ratio of squared error in $\hat{\f}_B$ to $\tilde{\f}$, the approximation which ignores $\g$. The axis to the right shows these $\mbox{relSE}$. Since this search procedure is naive, many of the proposed $\hat{\f}_B$ are in fact much worse than $\tilde{\f}$. We have included only the top performing $B$, in this case the top 2%.
The interval with the smallest $\mbox{relSE}$ is $\{413, \dots, 597\}$, which is almost exactly equal to the interval where the diagonal of $\M$ is large. As we saw in Figure \[fig:f\_hat\_B\_sample\], our method for generating $\hat{\f}_B$ tends to produce an approximation to $\f$ in which the high fidelity region bleeds slightly beyond the specified $B$, which may explain why the optimal interval is slightly narrower than we expected. Additionally, we can see in the bottom plot that the top 2% of intervals are all stably located near $\{401, \dots, 600\}$, and this can be verified for a range of $\kappa$ values (we verified $\kappa=(1/5, 1/10, 1/20)$. In fact, this stability is visible well beyond the top 2% of $\mbox{relSE}$, but plots including a larger proportion of proposed segments are cluttered and difficult to interpret.
![Top 2% of $B$ for the ‘Doppler’ time series and secondary function $\g_1$ in Example \[ex:ind\_dop\]. Each horizontal line segment represents an interval $B$, and the height of each segment is proportional to the improvement over the approximation which ignores secondary analysis, $\tilde{\f}$. The ratio of squared error loss for $\hat{\f}_B$ to that of $\tilde{\f}$ is shown in the right axis. The histogram to the right of each plot shows the distribution of the top 2% of proposed intervals. Some segments $B$ have the same $\mbox{relSE}$, so the histogram helps to illuminate where segments have overlapped.[]{data-label="fig:ind_dop"}](indicator_dop_FO_color.pdf){width="\linewidth"}
\[ex:ind\_bumps\]
In this example, we keep the same secondary analysis $\g_1$, but we examine a different time series called ‘Bumps’. Figure \[fig:ind\_bumps\] shows the same pair of plots as in Figure \[fig:ind\_dop\] for this new time series. It is interesting to note the influence here of not just the importance function, but also the nature of $\f$, in particular the places where the time series is equal to zero. In these regions, the approximation that the errors $\hat{\f} -
\f$ are randomly distributed with variance $\sigma^2_B$ or $\sigma^2_{B^c}$ is unreasonable. Since it is trivial to represent data that are identically zero, the errors here will be zero for any reasonably sized approximation, [ *i.e.*]{}, for any approximation with more than a very small number of non-zero wavelet coefficients), regardless of our specification of $\kappa$. The optimal interval is $\{405, \dots, 461\}$. As is visible in the top plot, this region corresponds roughly to the portion of $A$ where $\f$ is non-zero.
![Top 5% of $B$ for the ‘Bumps’ time series and secondary function $\g_1$ in Example \[ex:ind\_bumps\]. Each horizontal line segment represents an interval $B$, and the height of each segment is proportional to the improvement over the approximation which ignores secondary analysis, $\tilde{\f}$. The ratio of squared error loss for $\hat{\f}_B$ to that of $\tilde{\f}$ is shown in the right axis. The histogram to the right of each plot shows the distribution of the top 2% of proposed intervals. Some segments $B$ have the same $\mbox{relSE}$, so the histogram helps to illuminate where segments have overlapped. In this example, it is interesting to note the influence of not just the importance function $\diag(\M)$, but also the nature of the data $\f$ themselves. In particular, there are many indices for which the data are close to zero. The optimal interval (most easily seen on the middle figure) is $\{405, \dots, 461\}$, which corresponds roughly to the portion of $A$ where $\f$ is non-zero.[]{data-label="fig:ind_bumps"}](indicator_bumps_FO_color.pdf){width="\linewidth"}
\[ex:expsin\]
We next consider a secondary analysis for which we will have little intuition. The function $\g$ is still chosen to be a map to $\mathbb{R}^n$, but the function is more complicated than the magnifying glass situation. We define the secondary analysis to be $$\begin{aligned}
\g_3(\f)_i &= e^{f(i)/6}\sin\left(f(i)\right), \quad i = 1, \dots, n.
\end{aligned}$$ The shape of the importance function is now something more complicated than the function in the previous two examples. Without looking at the importance function, it is difficult to guess *a priori* where the most ‘important’ data will be, and where the lens of our magnifying glass approximation should lie. In this example we reuse the ‘Doppler’ data.
Figure \[fig:expsin\] shows results overlaying the data (top) and the importance function (bottom). The best proposed intervals $B$ cover the region of the importance function where there are the most large values. Though the importance function attains large values in the interval $\{600, \dots, 900\}$ as well as in the interval $\{350, \dots, 425\}$, the former has more such values, and since we are only considering $B$ that are single intervals, it makes sense that the best $B$ are clustered more or less in this range.
![Top 5% of $B$ for the ‘Doppler’ time series and secondary function $\g_3$ in Example \[ex:expsin\]. Each horizontal line segment represents an interval $B$, and the height of each segment is proportional to the improvement over the approximation which ignores secondary analysis, $\tilde{\f}$. The ratio of squared error loss for $\hat{\f}_B$ to that of $\tilde{\f}$ is shown in the right axis. The histogram to the right of each plot shows the distribution of the top 2% of proposed intervals. Some segments $B$ have the same $\mbox{relSE}$, so the histogram helps to illuminate where segments have overlapped.[]{data-label="fig:expsin"}](expsin_dop_FO_color.pdf){width="\linewidth"}
\[ex:mom\] We consider one final secondary analysis on the ‘Doppler’ data, first mentioned in Example \[ex:moments\]. In this example, we consider a secondary analysis that summarizes the raw data using the first four moments $$\begin{aligned}
\g_4(\f) &=
\left(
\sum_{t}f(t), \sum_{t}f^2(t),
\sum_{t}f^3(t), \sum_{t}f^4(t)
\right)^\prime.
\end{aligned}$$ The corresponding Jacobian and importance functions, respectively, are $$\begin{aligned}
J_{\g_4} &=
\begin{bmatrix}
1 & 1 & \dots & 1\\
2f(1) & 2f(2) & \dots & 2f(n)\\
3f^2(1) & 3f^2(2) & \dots & 3f^2(n)\\
4f^3(1) & 4f^3(2) & \dots & 4f^3(n)
\end{bmatrix}
\end{aligned}$$ and $$\begin{aligned}
\diag(\M)(t) &= 1 + (2f(t))^2 + (3f^2(t))^2 + (4f^3(t))^2, \quad t=1, \dots, n.
\end{aligned}$$ This is our first example for which $m \neq n$, and is intended to be the most realistic. However, these results may be interpreted in much the same manner as the other examples.
Figure \[fig:mom\_dop\] shows the results overlaying the ‘Doppler’ time series (top) and the importance function (bottom). As in the previous examples, we can see that the intervals $B$ corresponding to the best magnifying glass type approximations include the portions of the importance function with the largest values.
![Top 2% of $B$ for the ‘Doppler’ time series and secondary function $\g_4$ in Example \[ex:mom\]. Each horizontal line segment represents an interval $B$, and the height of each segment is proportional to the improvement over the approximation which ignores secondary analysis, $\tilde{\f}$. The ratio of squared error loss for $\hat{\f}_B$ to that of $\tilde{\f}$ is shown in the right axis. The histogram to the right of each plot shows the distribution of the top 2% of proposed intervals. Some segments $B$ have the same $\mbox{relSE}$, so the histogram helps to illuminate where segments have overlapped.[]{data-label="fig:mom_dop"}](mom_dop_FO_color.pdf){width="\linewidth"}
Discussion and Future Work {#sec:conc}
==========================
There are many hurdles to clear in DOE’s race towards exascale high performance computing environments including future system architectures, power/energy compliance, and many others. In this paper, we discuss one such problem related to the overwhelming demands being placed on I/O and storage systems due to the increased volume of data being generated by computational experiments. We remind the reader that our treatment of this larger-scale problem is in no way complete. However, we do believe that our codification of a simpler class of heterogeneous wavelet-based compression strategies will provide practitioners and theoreticians a systematic framework for future work. Our initial experiments given in Section \[sec:examples\] certainly supports this assertion.
In closing, we will briefly discuss the obvious extension for this current work. First, it may be unrealistic to assume that we know an importance function [*a priori*]{}. Consider our motivating example involving wind turbine arrays. We might know the approximate locations within the simulation domain that we would like to store in a high fidelity, however, the precise locations will be unknown at run-time. In other words, we will need to estimate the optimal prioritized region without using $\g(\f)$ and an exhaustive (or near-exhaustive) search. A smart [*in situ*]{} sampling strategy of the data should enable the development of a reasonably good estimator, $\hat{\M}$.
[**Acknowledgements**]{}
The authors would like to thank Professor Jay Breidt and David Biagioni for their suggestions on improving early drafts of this manuscript.
This work was supported by the Laboratory Directed Research and Development (LDRD) Program at the National Renewable Energy Laboratory. NREL is a national laboratory of the U.S. Department of Energy Office of Energy Efficiency and Renewable Energy operated by the Alliance for Sustainable Energy, LLC.
Nearly Exhaustive Searches {#app:searches}
==========================
All data used in the examples in this paper were of size 1024. When searching for the optimal $B$, we were unable to consider every interval within $T$, however we we able to consider nearly every interval, and we do not expect substantive changes would occur in our results if we did check all possibilities.
By nearly exhaustive, we mean the following. Instead of all intervals $$\begin{aligned}
\{a, \dots, b\}, \quad 1 \leq a < b \leq n,\end{aligned}$$ we considered only intervals of the form where $$\begin{aligned}
a &= 4x + 1, \quad &x=1, \dots, 255\\
b &= 4y + a, &\text{for all } y = 1, \dots, 81 \text{ such that } b<1024.\end{aligned}$$ We therefore considered all intervals of length equal to a multiple of 4 up to 324, and left boundary equal to all integers with a value of 1 modulo 4 up to 1021, [*e.g.*]{}, we checked $513
\dots 513 + 32$ but [*not*]{} $514 \dots 514 + 32$. For the cases where this system led to ranges that would extend beyond $\{1, \dots, 1024\}$, we simply took the intersection of the proposed $B$ with $T$. The convenience of this systematic approach outweighed the slight preference shown for boundaries that included points near the right edge of $\{1, \dots, 1024\}$
This nearly-exhaustive method was used for convenience. The computational demands for implementation were reasonable for a personal laptop computer (on the order of 10 minutes per search).
[^1]: Correspondence author. Email: henry.scharf@colostate.edu
|
---
abstract: 'In a recent eprint [@Povh:2016kvg] it is argued that the experimental determinations of the spin-dependent structure function $g_1$ have been done incorrectly and that a reanalysis of those data suggests that the original motivation to argue for a “spin crisis", namely the small contribution of quark spins to the nucleon spin, is invalid. In a subsequent note [@Leader:2016sli] the theoretical understanding, as it has evolved from almost 30 years of theoretical and experimental scrutiny, has been shortly summarised. In this short note, arguments are presented that the line of reasoning in Ref. [@Povh:2016kvg] does not apply, at least not for the [[Compass]{}]{}data.'
author:
- 'F. Bradamante[^1] and G.K. Mallot[^2]'
date: 16 April 2016
title: 'Why there is no crisis of the “spin crisis"'
---
A restitution of the strongly violated Ellis–Jaffe sum rule due to the consideration of diffractive events has been put forward in a recent eprint [@Povh:2016kvg]. The main argument is the claim that in the Deeply Inelastic Scattering (DIS) cross-section a fraction $f$ of the events is non-perturbative, i.e. diffractive, which shows no spin asymmetry, and that the fraction $f$ is large, of order 0.3–0.4. Consequently the analysing power is reduced, and one has to rescale the results for the polarised cross-section asymmetries by multiplying them by a factor $1/(1-f)$. Rescaling by the same factor is then required also for the first moments of the spin-dependent structure functions $g_1$ of the proton and the neutron, defined as $$\Gamma_1^{p,n} = \int_0^1 g_1^{p,n}(x,Q^2) \, dx \, .$$ As shown in Fig. 2 of Ref. [@Povh:2016kvg], when $f$ lies between 0.3–0.4, both the [[Hermes]{}]{}and the [[Compass]{}]{}data should be rescaled by factors of 1.4–1.6, which would bring the singlet axial coupling $a_0$ (and consequently $\Delta\Sigma$, the contribution of the quark spins to the spin of the nucleon) to about $0.6$, the value originally expected from the Ellis–Jaffe sum rule. This suggestion is motivated in the eprint [@Povh:2016kvg] by referring to results from the H1 [@Ahmed:1992qc] and the [[Zeus]{}]{}[@Derrick:1994dt] experiments at [[Hera]{}]{}, which measured the ratio between the photoproduction cross-section for diffractive events and the total photoproduction cross-section. This ratio is measured to be 0.30–0.40. Their assumption is that the fraction $f$ has to be the same in DIS, where analyses typically require the photon virtuality $Q^2$ to be larger than 1 GeV$^2$.
We have two comments to these considerations:
The first comment regards the amount of diffractive events in the [[Compass]{}]{} DIS data. Here only interactions of the resolved photon have to be considered, since hard diffraction events arise from point-like virtual photon interactions with partons from the intrinsic proton structure. The contribution of diffractive events to the [[Compass]{}]{}inclusive and semi-inclusive DIS (SIDIS) event samples has been studied e.g. in the context of the analysis of hadron multiplicities [@Adolph:2016bga]. For the SIDIS events at least one hadron is detected in addition to the scattered lepton. The main motivations of this analysis are the extraction of the hadron multiplicities, $p_T$ distributions and azimuthal modulations of the unpolarized cross-section normalised to the inclusive cross-section. Rather than relying on particular assumptions and on measurements done at $Q^2 = 0$, the diffractive production of vector mesons ($\rho^0$, $\omega$, $\phi$, …), was estimated for the actual [[Compass]{}]{}kinematics. The evaluation is based on two MC simulations, one using the LEPTO [@Ingelman:1996mq] event generator simulating SIDIS free of diffractive contributions, and the other one using the HEPGEN [@Sandacz:2012at] generator simulating diffractive $\rho^0$ and $\phi$ production, normalised to the GPD model of Ref. [@Goloskokov:2007nt]. Further channels, which are characterised by smaller cross-sections and more particles in the final state, are not taken into account. Besides events with the nucleon staying intact, also events with diffractive dissociation of the target nucleon are simulated. The simulation of these events includes nuclear effects, i.e. coherent production and nuclear absorption as described in Refs. [@Alexakhin:2007mw; @Adolph:2012ht]. Taking into account all these effects, the $f$ values [[Compass]{}]{}obtains for the inclusive event sample range from 0.04 at low $x$ and $Q^2$ to 0.003 at high $x$ and $Q^2$ [@Adolph:2016bga], a result which is in line with what is known in the literature on the amount of diffraction in DIS [@BaPr2002]. Consequently the effective dilution of the virtual photon polarisation in the [[Compass]{}]{}measurements is 10–100 times smaller than what has been assumed in Ref. [@Povh:2016kvg] and well inside the systematic uncertainties.
The second comment regards the Bjorken sum rule. If the arguments in Ref. [@Povh:2016kvg] were correct and indeed the first moments $\Gamma_1^p$ and $\Gamma_1^n$ had to be rescaled by a factor $1/(1-f) \simeq 1.5$, the Bjorken sum rule as measured from the [[Compass]{}]{}data alone would be violated by almost 4 standard deviations. The Bjorken sum rule was formulated already in 1956 using current algebra, and reformulated in QCD. No discrepancy with the available data on $g_1^p$, $g_1^n$ and $g_1^d$ has ever been reported (apart from an early measurement corrected later). If the rescaling suggested in Ref. [@Povh:2016kvg] would be applied, a major problem would open up for QCD.
This might be the reason, why the authors of Ref. [@Povh:2016kvg] claim an under-exhaustion of the fundamental Bjorken sum rule despite its verification to the 9% level by [[Compass]{}]{}[@Adolph:2015saz]. In their eprint, they assert that in Ref. [@Adolph:2015saz] the Bjorken sum is derived from a fit to the scaled world data. This is incorrect, the [[Compass]{}]{}result is obtained directly from the measured [[Compass]{}]{}$g_1^\mathrm{NS}$ non-singlet data points in the region $0.0025<x< 0.7$ corresponding to 93.8% of the full first moment. The extrapolations to $x=0$ and $x=1$ amount to 3.6% and 2.6%, respectively. For these small extrapolations a fit to the [[Compass]{}]{}non-singlet data is used. Therefore, there is no way to turn the experimental confirmation of the Bjorken sum rule into an under-exhaustion.
In the same way the Ellis–Jaffe sums are determined [@Adolph:2015saz] at $Q^2=3~\mathrm{GeV}^2$. We obtain for the proton $\Gamma_1^p = 0.139 \pm 0.010$ and for the neutron $\Gamma_1^n = -0.041\pm0.013$. The theoretical values are $0.172 \pm 0.003$ and $-0.017\pm 0.003$, respectively. It remains unclear why in Ref. [@Povh:2016kvg] it is stated that “... however with the realization of the idea presented in this paper the Bjorken sum as well as the Ellis-Jaffe-sum rule are in accord with the data naturally", whereas “this realisation" leads to a clear violation of the Bjorken sum rule.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank our [[Compass]{}]{}colleagues for fruitful discussions and clarifications, in particular Y. Bedfer, M. Faessler, F. Kunne, E.M. Kabuß, A. Kotzinian, A. Sandacz, E. Seder, M. Stolarski, and M. Wilfert. Useful discussions with V. Barone are also acknowledged.
[99]{} B. Povh and T. Walcher, arXiv:1603.05884 \[hep-ph\]. E. Leader, arXiv:1604.00305 \[hep-ph\]. T. Ahmed [*et al.*]{} \[H1 Collaboration\], Phys. Lett. B [**299**]{} (1993) 374.
M. Derrick [*et al.*]{} \[[[Zeus]{}]{}Collaboration\], Z. Phys. C [**63**]{} (1994) 391.
G. Ingelman, A. Edin and J. Rathsman, Comput. Phys. Commun. [**101**]{} (1997) 108.
A. Sandacz and P. Sznajder, \[arXiv:1207.0333 \[hep-ph\]\].
S. V. Goloskokov and P. Kroll, Eur. Phys. J. C [**53**]{} (2008) 367.
V. Y. Alexakhin [*et al.*]{} \[[[Compass]{}]{}Collaboration\], Eur. Phys. J. C [**52**]{} (2007) 255 \[arXiv:0704.1863 \[hep-ex\]\]. C. Adolph [*et al.*]{} \[[[Compass]{}]{}Collaboration\], Nucl. Phys. B [**865**]{} (2012) 1 \[arXiv:1207.4301 \[hep-ex\]\]. C. Adolph [*et al.*]{} \[[[Compass]{}]{}Collaboration\], arXiv:1604.02695 \[hep-ex\]. V. Barone and E. Predazzi, “High-Energy Particle Diffraction", ISSN 0172-5998 (Springer, 2002) p. 283ff.
C. Adolph [*et al.*]{} \[[[Compass]{}]{}Collaboration\], Phys. Lett. B [**753**]{} (2016) 18 \[arXiv:1503.08935 \[hep-ex\]\].
[^1]: INFN and University of Trieste, Department of Physics, 34127 Trieste, Italy
[^2]: CERN, 1211 Geneva 23, Switzerland
|
---
abstract: 'Using the maximum entropy method, spectral functions of the pseudo-scalar and vector mesons are extracted from lattice Monte Carlo data of the imaginary time Green’s functions. The resonance and continuum structures as well as the ground state peaks are successfully obtained. Error analysis of the resultant spectral functions is also given on the basis of the Bayes probability theory.'
address:
- 'Department of Physics,Nagoya University, Nagoya 464 - 8602, Japan'
- 'Physics Department, Kyoto University, Kyoto 606-8502, Japan'
author:
- 'Y. Nakahara$^a$, M. Asakawa, and T. Hatsuda'
title: 'Spectral Functions of Hadrons in Lattice QCD[^1]'
---
Introduction
============
The spectral functions (SPFs) of hadrons play a special role in physical observables in QCD (See the examples in [@shuryak; @negele]). However, the lattice QCD simulations so far have difficulties in accessing the dynamical quantities in the Minkowski space, because measurements on the lattice can only be carried out for discrete points in imaginary time. The analytic continuation from the imaginary time to the real time using the noisy lattice data is highly non-trivial and is even classified as an ill-posed problem.
Recently we made a first serious attempt to extract SPFs of hadrons from lattice QCD data without making a priori assumptions on the spectral shape [@nah]. We use the maximum entropy method (MEM), which has been successfully applied for similar problems in quantum Monte Carlo simulations in condensed matter physics, image reconstruction in crystallography and astrophysics, and so forth [@physrep; @linden]. In this report, we present the results for the pseudo-scalar (PS) and vector (V) channels at $T=0$ using the continuum kernel and the lattice kernel of the integral transform. The latter analysis has not been reported in [@nah].
Basic idea of MEM
=================
The Euclidean correlation function $D(\tau)$ of an operator ${\cal O}(\tau,\vec{x})$ and its spectral decomposition at zero three-momentum read $$\begin{aligned}
D(\tau ) &=& \int
\langle {\cal O}^{\dagger}(\tau,\vec{x}){\cal O}(0,\vec{0})\rangle d^3 x\label{KA}\\\nonumber
&=& \int_{0}^{\infty} \!\! K(\tau, \omega) A(\omega ) d\omega,\end{aligned}$$ where $\tau > 0$, $\omega$ is a real frequency, and $A(\omega)$ is SPF (or sometimes called the [*image*]{}), which is positive semi-definite. The kernel $K(\tau, \omega)$ is proportional to the Fourier transform of a free boson propagator with mass $\omega$: At $T=0$ in the continuum limit, $K = K_{cont}(\tau, \omega)
=\exp(-\tau\omega)$.
Monte Carlo simulation provides $D(\tau_i)$ on the discrete set of temporal points $0 \le \tau_i /a \le N_\tau$. From this data with statistical noise, we need to reconstruct the spectral function $A(\omega)$ with continuous variable $\omega$. This is a typical ill-posed problem, where the number of data is much smaller than the number of degrees of freedom to be reconstructed. This makes the standard likelihood analysis and its variants inapplicable [@others] unless strong assumptions on the spectral shape are made. MEM is a method to circumvent this difficulty through Bayesian statistical inference of the most probable [*image*]{} together with its reliability [@physrep].
MEM is based on the Bayes’ theorem in probability theory: $P[X|Y] = P[Y|X]P[X]/P[Y]$, where $P[X|Y]$ is the conditional probability of $X$ given $Y$. The most probable image $A(\omega )$ for given lattice data $D$ is obtained by maximizing the conditional probability $P[A|DH]$, where $H$ summarizes all the definitions and prior knowledge such as $A(\omega) \ge 0$. By the Bayes’ theorem, $$\label{bayes_latt}
P[A|DH] \propto P[D|AH]P[A|H] ,$$ where $P[D|AH]$ ($P[A|H]$) is called the likelihood function (the prior probability).
For the likelihood function, the standard $\chi^2$ is adopted, namely $P[D|AH]= Z_L^{-1} \exp (-L)$ with $$\begin{aligned}
&&\mbox{\hspace{-0.5cm}}L = {1 \over 2} \sum_{i,j}
(D(\tau_i)-D^A(\tau_i)) \\[-0.3cm]
&&\mbox{\hspace{2.5cm}} \times \, C^{-1}_{ij}
(D(\tau_j)-D^A(\tau_j)).\nonumber \label{chi2}\end{aligned}$$ $Z_L$ is a normalization factor given by $Z_L = (2\pi)^{N/2} \sqrt{\det C}$ with $N={\tau}_{max}/a - {\tau}_{min}/a+1$. $D(\tau_i )$ is the lattice data averaged over gauge configurations and $D^A(\tau_i )$ is the correlation function defined by the right hand side of (\[KA\]). $C$ is an $N \times N$ covariance matrix of the data with $N$ being the number of temporal points to be used in the MEM analysis. The lattice data have generally strong correlations among different $\tau$’s, and it is essential to take into account the off-diagonal components of $C$.
Axiomatic construction as well as intuitive “monkey argument” [@skilling] show that, for positive distributions such as SPF, the prior probability can be written with parameters $\alpha$ and $m$ as $P[A|H\alpha m]= Z_S^{-1} \exp (\alpha S)$. Here $S$ is the Shannon-Jaynes entropy, $$\begin{aligned}
&&\mbox{\hspace{-.5cm}}S =\\
&&\mbox{\hspace{-.5cm}} \ \ \int_0^{\infty} \left [ A(\omega ) - m(\omega )
- A(\omega)\log \left ( \frac{A(\omega)}{m(\omega )} \right ) \right ]
d\omega .\nonumber\end{aligned}$$ $Z_S$ is a normalization factor: $Z_S \equiv \int e^{\alpha S} {\cal D}A$. $\alpha$ is a real and positive parameter and $m(\omega )$ is a real function called the default model.
In the state-of-art MEM [@physrep], the output image $A_{out}$ is given by a weighted average over $A$ and $\alpha$: $$\begin{aligned}
&&\mbox{\hspace{-.5cm}}A_{out}(\omega) \nonumber \\
&&= \int A(\omega) \
P[A|DH\alpha m]P[\alpha|DHm] \ {\cal D} A \ d\alpha \nonumber \\
&& \simeq \int A_{\alpha}(\omega) \ P[\alpha|DHm] \ d\alpha .
\label{final}\end{aligned}$$ Here $A_{\alpha}(\omega)$ is obtained by maximizing the “free-energy” $$\begin{aligned}
Q \equiv \alpha S - L,\end{aligned}$$ for a given $\alpha$. Here we assumed that $P[A|DH\alpha m] $ is sharply peaked around $A_{\alpha}(\omega)$. $\alpha$ dictates the relative weight of the entropy $S$ (which tends to fit $A$ to the default model $m$) and the likelihood function $L$ (which tends to fit $A$ to the lattice data). Note, however, that $\alpha$ appears only in the intermediate step and is integrated out in the final result. Our lattice data show that the weight factor $P[\alpha|DHm]$, which is calculable using $Q$ [@physrep], is highly peaked around its maximum $\alpha = \hat{\alpha}$. We have also studied the stability of the $A_{out}(\omega)$ against a reasonable variation of $m(\omega)$.
The non-trivial part of the MEM analysis is to find a global maximum of $Q$ in the functional space of $A(\omega)$, which has typically 750 degrees of freedom in our case. We have utilized the singular value decomposition (SVD) of the kernel to define the search direction in this functional space. The method works successfully to find the global maximum within reasonable iteration steps.
MEM with mock data
==================
To check our MEM code and to see the dependence of the MEM image on the quality of the data, we made the following test using mock data. (i) We start with an input image $A_{in}(\omega) \equiv \omega^2 \rho_{in}(\omega)$ in the $\rho$-meson channel which simulates the experimental $e^+e^-$ cross section. Then we calculate $D_{in}(\tau)$ from $A_{in}(\omega)$ using eq.(\[KA\]). (ii) By taking $D_{in}(\tau_i)$ at $N$ discrete points and adding a Gaussian noise, we create a mock data $D_{mock}(\tau_i)$. The variance of the noise $\sigma (\tau_i)$ is given by $\sigma (\tau_i)= b \times D_{in}(\tau_i) \times \tau_i /a$ with a parameter $b$, which controls the noise level [@noise]. (iii) We construct the output image $A_{out}(\omega) \equiv \omega^2 \rho_{out}(\omega)$ using MEM with $D_{mock}(\tau_{min} \le \tau_i \le \tau_{max})$ and compare the result with $A_{in}(\omega)$. In this test, we have assumed that $C$ is diagonal for simplicity.
In Fig.1, we show $\rho_{in}(\omega)$, and $\rho_{out} (\omega)$ for two sets of parameters, (I) and (II). As for $m$, we choose a form $m(\omega) = m_0 \omega^2$ with $m_0 = 0.027$, which is motivated by the asymptotic behavior of $A$ in perturbative QCD, $A(\omega \gg 1 {\rm GeV})
= (1/4 \pi^2) (1+\alpha_s / \pi) \omega^2$. The final result is, however, insensitive to the variation of $m_0$ even by factor 5 or 1/5. The calculation of $A_{out}(\omega)$ has been done by discretizing the $\omega$-space with an equal separation of 10 MeV between adjacent points. This number is chosen for the reason we shall discuss below. The comparison of the dashed line (set (I)) and the dash-dotted line (set (II)) shows that increasing $\tau_{max}$ and reducing the noise level $b$ lead to better SPFs closer to the input SPF.
We have also checked that MEM can nicely reproduce other forms of the mock SPFs. In particular, MEM works very well to reproduce not only the broad structure but also the sharp peaks close to the delta-function as far as the noise level is sufficiently small.
=8.2cm
MEM with lattice data
=====================
To apply MEM to actual lattice data, quenched lattice QCD simulations have been done with the plaquette gluon action and the Wilson quark action by the open MILC code with minor modifications [@milc]. The lattice size is $20^3 \times 24$ with $\beta =6.0$, which corresponds to $ a = 0.0847$ fm ($a^{-1} = 2.33$ GeV), $\kappa_c = 0.1571$ [@kc], and the spatial size of the lattice $L_s a = 1.69 $ fm. Gauge configurations are generated by the heat-bath and over-relaxation algorithms with a ratio $1:4$. Each configuration is separated by 1000 sweeps. Hopping parameters are chosen to be $\kappa =$ 0.153, 0.1545, and 0.1557 with $N_{conf}=161$ for each $\kappa$. For the quark propagator, the Dirichlet (periodic) boundary condition is employed for the temporal (spatial) direction. We have also done the simulation with periodic boundary condition in the temporal direction and obtained qualitatively the same results. To calculate the two-point correlation functions, we adopt a point-source at $\vec{x}=0$ and a point-sink averaged over the spatial lattice-points.
We use data at $1 \le \tau_i/a \le 12 (24) $ for the Dirichlet (periodic) boundary condition in the temporal direction. To avoid the known pathological behavior of the eigenvalues of $C$ [@physrep], we take $N_{conf} \gg N$.
We define SPFs for the PS and V channels as $$A(\omega) = \omega^2 \rho_{_{PS},_{V}}(\omega) ,$$ so that $\rho_{_{PS,V}}(\omega \rightarrow {\rm large}) $ approaches a finite constant as predicted by perturbative QCD. For the MEM analysis, we need to discretize the $\omega$-integration in (\[KA\]). Since $\Delta \omega$ (the mesh size) $\ll 1/\tau_{max}$ should be satisfied to suppress the discretization error, we take $\Delta \omega$ = 10 MeV. $\omega_{max}$ (the upper limit for the $\omega$ integration) should be comparable to the maximum available momentum on the lattice: $\omega_{max} \sim \pi /a \sim 7.3$ GeV. We have checked that larger values of $\omega_{max} $ do not change the result of $A(\omega)$ substantially, while smaller values of $\omega_{max} $ distort the high energy end of the spectrum. The dimension of the image to be reconstructed is $N_{\omega} \equiv
\omega_{max}/\Delta \omega \sim 750$, which is in fact much larger than the maximum number of Monte Carlo data $N = 25$.
In Fig.2 (a) and (b), we show the reconstructed images for each $\kappa$ in the case of the Dirichlet boundary condition. Here we use the continuum kernel $K_{cont} = \exp(-\tau \omega)$ in the Laplace transform. In these figures, we have used $m = m_0 \omega^2$ with $m_0 = 2.0 (0.86)$ for PS (V) channel motivated by the perturbative estimate of $m_0$ (see eq.(\[cont-V\]) and the text below). We have checked that the result is not sensitive, within the statistical significance of the image, to the variation of $m_0$ by factor 5 or 1/5. The obtained images have a common structure: the low-energy peaks corresponding to $\pi$ and $\rho$, and the broad structure in the high-energy region. From the position of the pion peaks in Fig.2(a), we extract $\kappa_c = 0.1570(3)$, which is consistent with $ 0.1571 $ [@kc] determined from the asymptotic behavior of $D(\tau)$. The mass of the $\rho$-meson in the chiral limit extracted from the peaks in Fig.2(b) reads $m_{\rho}a = 0.348(15)$. This is also consistent with $m_{\rho}a = 0.331(22) $ [@kc] determined by the asymptotic behavior. Although our maximum value of the fitting range $\tau_{max}/a =12$ marginally covers the asymptotic limit in $\tau$, we can extract reasonable masses for $\pi$ and $\rho$. The width of $\pi$ and $\rho$ in Fig.2 is an artifact due to the statistical errors of the lattice data. In fact, in the quenched approximation, there is no room for the $\rho$-meson to decay into two pions.
As for the second peaks in the PS and V channels, the error analysis discussed in Fig.4 shows that their spectral “shape" does not have much statistical significance, although the existence of the non-vanishing spectral strength is significant. Under this reservation, we fit the position of the second peaks and made linear extrapolation to the chiral limit with the results, $m^{2nd}/m_{\rho} = 1.88(8)
(2.44(11))$ for the PS (V) channel. These numbers should be compared with the experimental values: $m_{\pi(1300)}/m_{\rho} = 1.68$, and $m_{\rho(1450)}/m_{\rho} = 1.90$ or $m_{\rho(1700)}/m_{\rho} = 2.20$.
One should remark here that, in the standard two-mass fit of $D(\tau)$, the mass of the second resonance is highly sensitive to the lower limit of the fitting range, e.g., $m^{2nd}/m_{\rho} = 2.21(27) (1.58(26))$ for $\tau_{min}/a = 8 (9)$ in the $V$ channel with $\beta=6.0$ [@kc]. This is because the contamination from the short distance contributions from $\tau < \tau_{min}$ is not under control in such an approach. On the other hand, MEM does not suffer from this difficulty and can utilize the full information down to $\tau_{min}/a=1$. Therefore, MEM opens a possibility of systematic study of higher resonances with lattice QCD data.
=8.2cm
As for the third bumps in Fig.2, the spectral “shape" is statistically not significant as is discussed in Fig.4, and they should rather be considered a part of the perturbative continuum instead of a single resonance. Fig.2 also shows that SPF decreases substantially above 6 GeV; MEM automatically detects the existence of the momentum cutoff on the lattice $\sim \pi/a$. It is expected that MEM with the data on finer lattices leads to larger ultraviolet cut-offs in the spectra. The height of the asymptotic form of the spectrum at high energy is estimated as $$\begin{aligned}
&&\mbox{\hspace{-.5cm}} \rho_{_V}(\omega \simeq 6 {\rm GeV})\\
&&\mbox{\hspace{.5cm}} = {1 \over 4 \pi^2} \left ( 1 + {\alpha_s \over \pi} \right )
\left ( {1 \over 2\kappa Z_{_V}} \right )^2 \nonumber
\simeq 0.86 . \nonumber
\label{cont-V}\end{aligned}$$ The first two factors are the $q \bar{q}$ continuum expected from perturbative QCD. The third factor contains the non-perturbative renormalization constant for the lattice composite operator. We adopt $Z_{_V} = 0.57$ determined from the two-point functions at $\beta$ = 6.0 [@mm86] together with $\alpha_s = 0.21$ and $\kappa = 0.1557$. Our estimate in eq.(\[cont-V\]) is consistent with the high energy part of the spectrum in Fig.2(b) after averaging over $\omega$. We made a similar estimate for the PS channel using $Z_{_{PS}} = 0.49 $ [@shi] and obtained $\rho_{_{PS}}(\omega \simeq 6 {\rm GeV}) \simeq 2.0$. This is also consistent with Fig. 2(a). We note here that an independent analysis of the imaginary time correlation functions [@negele] also shows that the lattice data at short distance is dominated by the perturbative continuum.
In Fig.3(a) and (b), the results using the lattice kernel $K_{lat}$ are shown. $K_{lat}$ is obtained from the free boson propagator on the lattice. It reduces to $K_{cont}$ when $a \rightarrow 0$. The other parameters and boundary conditions are the same with Fig.2(a,b). The difference of Fig.2 and Fig.3 can be interpreted as a systematic error due to the finiteness of the lattice spacing $a$.
=8.2cm
Error analsis
=============
The statistical significance of the reconstructed image can be studied by the following procedure [@physrep]. Assuming that $P[A|DH\alpha m]$ has a Gaussian distribution around the most probable image $\hat{A}$, we estimate the error by the covariance of the image, $- \langle (\delta_A \delta_A Q)^{-1} \rangle_{A=\hat{A}}$, where $\delta_A$ is a functional derivative and $\langle \cdot \rangle $ is an average over a given energy interval. The final error for $A_{out}$ is obtained by averaging the covariance over $\alpha$ with a weight factor $P[\alpha |DHm]$. Shown in Fig.4 is the MEM image in the V channel for $\kappa= 0.1557$ with errors obtained in the above procedure. The height of each horizontal bar is $\langle\rho_{out}(\omega)\rangle$ in each $\omega$ interval. The vertical bar indicates the error of $\langle\rho_{out}(\omega)\rangle$. The small error for the lowest peak in Fig.4 supports our identification of the peak with $\rho$. Although the existence of the non-vanishing spectral strength of the 2nd peak and 3rd bump is statistically significant, their spectral “shape” is either marginal or insignificant. Lattice data with better quality are called for to obtain better SPFs.
=8.2cm
Summary
=======
We have made a first serious attempt to reconstruct SPFs of hadrons from lattice QCD data. We have used MEM, which allows us to study SPFs without making a priori assumption on the spectral shape. The method works well for the mock data and actual lattice data. MEM produces resonance and continuum-like structures in addition to the ground state peaks. The statistical significance of the image can be also analyzed. Better data with finer and larger lattice will produce better images with smaller errors, and our study is a first attempt towards this goal.
There are many problems which can be explored by MEM combined with lattice QCD data. Some of the applications in the baryon excited states, hadrons at finite temperature, and heavy quark systems will be reported in future publications [@nextpaper].
We appreciate MILC collaboration for their open codes for lattice QCD simulations, which has enabled this research. Our simulation was carried out on a Hitachi SR2201 parallel computer at Japan Atomic Energy Research Institute. M. A. (T. H.) was partly supported by Grant-in-Aid for Scientific Research No. 10740112 (No. 10874042) of the Japanese Ministry of Education, Science, and Culture.
[99]{}
E. V. Shuryak, Rev. Mod. Phys. [**65**]{},1 (1993). M. -C. Chu, J. M. Grandy, S. Huang, and J. W. Negele, Phys. Rev. D [**48**]{}, 3340 (1993). Y. Nakahara, M. Asakawa and T. Hatsuda, hep-lat/9905034 (Phys. Rev. D in press). See the review, M. Jarrell and J. E. Gubernatis, Phys. Rep. [**269**]{}, 133 (1996). R. N. Silver et al., Phys. Rev. Lett. [**65**]{}, 496 (1990); W. von der Linden, R. Preuss, and W. Hanke, J. Phys. [**8**]{}, 3881 (1996); N. Wu, [*The Maximum Entropy Method*]{}, (Springer-Verlag, Berlin, 1997).
D. B. Leinweber, Phys. Rev. D [**51**]{}, 6369 (1995); D. Makovoz and G. A. Miller, Nucl. Phys. [**B468**]{}, 293 (1996); C. Allton and S. Capitani, Nucl. Phys. [**B526**]{}, 463 (1998); Ph. de Forcrand et al., Nucl. Phys. B (Proc. Suppl.) [**63A-C**]{}, 460 (1998). See e.g., J. Skilling, in [*Maximum Entropy and Bayesian Methods*]{}, ed. J. Skilling (Kluwer, London, 1989), pp.45-52; S. F. Gull, ibid. pp.53-71. This formula is motivated by our lattice QCD data. The MILC code ver. 5,\
http://cliodhna.cop.uop.edu/\~hetrick/milc . Y. Iwasaki et al., Phys. Rev. D [**53**]{}, 6443 (1996); T. Bhattacharya et al., ibid. 6486. L. Maiani and G. Martinelli, Phys. Lett. [**B178**]{}, 265 (1986). M. Göckeler et al., Nucl. Phys. [**B544**]{}, 699 (1999). M. Asakawa, T. Hatsuda and Y. Nakahara, in preparation.
[^1]: Talk given by Y. Nakahara and M. Asakawa at LATTICE99.
|
---
abstract: 'The operator product expansion is used to compute the matrix elements of composite renormalized operators on the lattice. We study the product of two fundamental fields in the two-dimensional $\sigma$-model and discuss the possible sources of systematic errors. The key problem turns out to be the violation of asymptotic scaling.'
address:
- 'Scuola Normale Superiore and INFN, Sezione di Pisa, I-56100 Pisa, ITALIA'
- 'Dipartimento di Fisica and INFN, Sezione di Roma I, Università degli Studi di Roma “La Sapienza”, I-00185 Roma, ITALIA'
author:
- 'Sergio Caracciolo${}^{\rm a}$, Andrea Montanari, and Andrea Pelissetto'
title: 'Composite operators from the operator product expansion: what can go wrong?'
---
The Operator Product Expansion $$\begin{aligned}
A(x;\mu)B(-x;\mu) \sim \sum_C W_{AB}^C(x;\mu)C(0;\mu)
\label{ShortDistance}\end{aligned}$$ is widely thought to hold beyond perturbation theory.
The use of Eq. (\[ShortDistance\]) in lattice simulations [@DeltaI; @NostroLat98; @Schierolz] is still in its infancy. In Ref. [@DeltaI] it was suggested to use Eq. (\[ShortDistance\]) in order to compute renormalized matrix elements. The OPE approach consists in the following steps: one computes the (matrix element of the) l.h.s. of Eq. (\[ShortDistance\]), then renormalizes $A$ and $B$ in some scheme, and finally obtains (the matrix element of) $C$ through a fit, using some perturbative approximation of the Wilson coefficients.
The main sources of systematic errors in this approach are the following:
1. finite-size effects and corrections to scaling, i.e. lattice artifacts; \[ScalingProblem\]
2. “power-correction effects” which are due to the fact that we truncate the expansion (\[ShortDistance\]) to some finite order in $x^2$; \[HTProblem\]
3. corrections to asymptotic scaling which must be taken in account since the Wilson coefficients in Eq. (\[ShortDistance\]) have to be substituted by the first few terms of their perturbative expansion. \[AsymptoticProblem\]
Errors of type \[ScalingProblem\] are widely studied and do not need more explanations. Here we shall focus on errors of type \[HTProblem\] and \[AsymptoticProblem\].
The use of Eq. (\[ShortDistance\]) in a continuum scheme, for which only a perturbative computation of the Wilson coefficients is available, poses a restriction on the operators which can be obtained in this approach. Only the operators of lowest dimension for each spin sector, i.e. those of lowest twist, can be computed using Eq. (\[ShortDistance\]). Higher-twist operators give rise to systematic errors of order $O(x^2)$. Moreover, because of statistical errors, only the operators of low dimension can be reliably extracted: the remaining ones are strongly subleading in the region of validity of the OPE.
Problem \[AsymptoticProblem\] can be stated in a cleaner way if we get rid of the scale dependence which is introduced in this approach somehow artificially through the Wilson coefficients. One can rewrite Eq. (\[ShortDistance\]) by making use of renormalization-group invariant operators defined as follows: $$\begin{aligned}
\lefteqn{ Q^{RGI}(x)\equiv Q(x;\mu)/F_Q(g(\mu))\; ,} \\
\lefteqn{ F_Q(g) \equiv g^{\frac{\gamma^Q_0}{\beta_0}}
\exp\left\{\int_0^{g} \left[\frac{\gamma^Q(x)}{\beta(x)}-
\frac{\gamma^Q_0}{\beta_0 x}\right]dx\right\} \; . }\end{aligned}$$ The Wilson coefficients obviously become $\mu$-independent and their general perturbative form is $$\begin{aligned}
W^{RGI}(\Lambda x) = g(\Lambda x)^{\frac{\gamma_0^W}{\beta_0}}
\sum_{k=0}^{\infty} c_k g(\Lambda x)^k \; ,
\label{RGIPT}\end{aligned}$$ where $\Lambda$ is the “lambda parameter” of the theory. The use of a truncation of Eq. (\[RGIPT\]) introduces systematic errors of order $O(\log^{-k}(\Lambda x))$. Notice, however, that this approach allows to compute directly “infinite-energy” quantities (i.e. the renormalization-group invariant matrix elements) which are of interest in phenomenological applications. Errors of order $\log^{-k} (\Lambda x)$ arise also in the widely used “non-perturbative renormalization method” [@NonPerturbativeMethod] in which perturbation theory is used to “evolve” the renormalization constants computed at some energy scale achievable on the lattice up to high energies.
We have considered several products of operators for the $O(N)$ nonlinear $\sigma$-model in two dimensions, with lattice action $$\begin{aligned}
S(\sg) \equiv \frac{1}{2g_L}\sum_{x,\mu}(\partial_{\mu}\sg)_x^2 \; ,\end{aligned}$$ where $\sg\in S^{N-1}$, $(\partial_{\mu}f)_x \equiv f_{x+\mu}-f_x$, and $N=3$. Here we shall refer to the following (respectively scalar and symmetric) products of fundamental fields: $$\begin{aligned}
\lefteqn{\sg(x)\cdot\sg(-x)\sim W_0(x) +O(x^2)\; ,}
\label{ProdottoScalare}\\
\lefteqn{\sigma^a(x)\sigma^b(-x)+(a\leftrightarrow b)-
\mbox{trace}\sim}\nonumber\\
&&\sim W_2(x)\left[\sigma^a\sigma^b-\mbox{trace}\right](0)+O(x^2) \; .
\label{ProdottoSimmetrico}\end{aligned}$$ The Monte Carlo data presented refer to two lattices: the first one of size $L\times T=128\times 256$ and correlation length $(am)^{-1} = 13.632(6)$ ($m$ is the mass gap); the second with $L\times T=256\times 512$ and correlation length $(am)^{-1} = 27.094(43)$.
The expectation value of the product (\[ProdottoSimmetrico\]) between states of momentum $p$ is shown in Fig. \[FunzioneDiCorrelazione\] for the two different lattices: corrections to scaling, i.e. errors of type \[ScalingProblem\], are completely under control in our simulations.\
Our general procedure consists in choosing a truncation of the expansion (\[ShortDistance\]) and in using it to fit the data in the region $\rho< |x| < R$. The results are independent of $\rho$ for $1.5\ a\le\rho\le 3\ a$. We use the stability of the fit with respect to the truncation and to the choice of $R$ as a criterion to distinguish whether errors of type \[HTProblem\] and \[AsymptoticProblem\] are relevant or not.
The quality of the fits obtained is well represented by the curves shown in Fig. \[FunzioneDiCorrelazione\], where the two-loop expression was used for the first term in the expansion (\[ProdottoSimmetrico\]) and the tree-level form for the terms of order $O(x^2)$.
An additional conclusion can be drawn from Fig. \[FunzioneDiCorrelazione\]: in order to describe the symmetric product (\[ProdottoSimmetrico\]) up to distances $2x\sim m^{-1}, p^{-1}$, it is necessary (and almost sufficient) to include terms of order $O(x^2)$. However this does not mean that the terms of order $x^2$ with the same symmetry of the leading one — the higher-twist terms — can be obtained from the fit. Indeed, their matrix elements extracted from the fits are very unstable with respect to changes of $R$.
In order to test the stability of the fit, we considered the vacuum expectation value of the product (\[ProdottoScalare\]), that is the two-point function. Here the renormalization constant of the field is a fit parameter. The results are reported in Fig. \[Z\] and refer to the lattice with correlation length $(ma)^{-1}\simeq 27.094$; we used $\overline{\mu}a \simeq 10$. As a manifestation of asymptotic freedom, the various curves shrink when $R\to 0$. Their $R$-dependence becomes weaker as more terms of the perturbative expansion are included. Nevertheless, even if we use the three-loop Wilson coefficient, we do not obtain a value of $Z$ independent of $R$ within the statistical errors, as it should be in the asymptotic-scaling regime. Two remarks are in order here: first, it is notoriously difficult to reach asymptotic scaling in the $O(3)$ nonlinear $\sigma$-model [@Asymptotic]; second, we are studing a small effect which could be negligible in QCD applications with respect to other errors.
Note that the addition of $O(x^2)$ terms, which, in this case, have spin $0$ and are therefore higher twists, seems to improve the situation, making the curve flatter. However, this must be regarded as a spurious effect. Higher twists are simply mimicking the contribution of higher orders in perturbation theory: otherwise, the curves obtained without including them should show a $(mR)^2$ behavior. From Fig. \[Z\] we can estimate the systematic error due to the use of two- or three-loop Wilson coefficients to fit the data in this range of $mR$: the error is approximately $5\%$.\
We report in Fig. \[ElementoDiMatrice\] the results of the fit presented in Fig. \[FunzioneDiCorrelazione\]. Each type of truncation, including or not $O(x^2)$ corrections, gives a reasonable, that is $p$ independent and (almost) $R$ independent, answer. The difference between them should be interpreted as a violation of asymptotic scaling and, indeed, it is of the same magnitude of the systematic error estimated for the renormalization constant of the field.
We thank R. Petronzio for useful discussions.
[99]{}
C. Dawson, G. Martinelli, G. C. Rossi, C. T. Sachrajda, S. Sharpe, M. Talevi, and M. Testa, Nucl. Phys. B514 (1998) 313
S. Caracciolo, A. Montanari, and A. Pelissetto, Nucl. Phys. B (Proc. Suppl.) 73 (1999) 273
Talk given by G. Schierholz at this conference.
G. Martinelli, C. Pittori, C. T. Sachrajda, M. Testa, and A. Vladikas, Nucl. Phys. B445 (1995) 81
S. Caracciolo, R. G. Edwards, A. Pelissetto, and A. D. Sokal, Phys. Rev. Lett. 75 (1995) 1891
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.